### (0) Obligation:

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

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__half(0) → 0
a__half(s(0)) → 0
a__half(s(s(X))) → s(a__half(mark(X)))
a__half(dbl(X)) → mark(X)
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)
a__half(X) → half(X)

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

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

The rewrite sequence
mark(terms(X)) →+ cons(recip(a__sqr(mark(mark(X)))), terms(s(mark(X))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
The pumping substitution is [X / terms(X)].
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__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0') → 0'
a__dbl(0') → 0'
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__first(0', X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__half(0') → 0'
a__half(s(0')) → 0'
a__half(s(s(X))) → s(a__half(mark(X)))
a__half(dbl(X)) → mark(X)
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)
a__half(X) → half(X)

S is empty.
Rewrite Strategy: FULL

### (5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
cons/1

### (6) Obligation:

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

a__terms(N) → cons(recip(a__sqr(mark(N))))
a__sqr(0') → 0'
a__dbl(0') → 0'
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__first(0', X) → nil
a__first(s(X), cons(Y)) → cons(mark(Y))
a__half(0') → 0'
a__half(s(0')) → 0'
a__half(s(s(X))) → s(a__half(mark(X)))
a__half(dbl(X)) → mark(X)
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
mark(cons(X1)) → cons(mark(X1))
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)
a__half(X) → half(X)

S is empty.
Rewrite Strategy: FULL

Infered types.

### (8) Obligation:

TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(N))))
a__sqr(0') → 0'
a__dbl(0') → 0'
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__first(0', X) → nil
a__first(s(X), cons(Y)) → cons(mark(Y))
a__half(0') → 0'
a__half(s(0')) → 0'
a__half(s(s(X))) → s(a__half(mark(X)))
a__half(dbl(X)) → mark(X)
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
mark(cons(X1)) → cons(mark(X1))
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)
a__half(X) → half(X)

Types:

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

Heuristically decided to analyse the following defined symbols:
a__terms, a__sqr, mark, a__add, a__dbl, a__half

They will be analysed ascendingly in the following order:
a__terms = a__sqr
a__terms = mark
a__terms = a__dbl
a__terms = a__half
a__sqr = mark
a__sqr = a__dbl
a__sqr = a__half
mark = a__dbl
mark = a__half
a__dbl = a__half

### (10) Obligation:

TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(N))))
a__sqr(0') → 0'
a__dbl(0') → 0'
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__first(0', X) → nil
a__first(s(X), cons(Y)) → cons(mark(Y))
a__half(0') → 0'
a__half(s(0')) → 0'
a__half(s(s(X))) → s(a__half(mark(X)))
a__half(dbl(X)) → mark(X)
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
mark(cons(X1)) → cons(mark(X1))
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)
a__half(X) → half(X)

Types:

Generator Equations:

The following defined symbols remain to be analysed:
a__sqr, a__terms, mark, a__add, a__dbl, a__half

They will be analysed ascendingly in the following order:
a__terms = a__sqr
a__terms = mark
a__terms = a__dbl
a__terms = a__half
a__sqr = mark
a__sqr = a__dbl
a__sqr = a__half
mark = a__dbl
mark = a__half
a__dbl = a__half

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

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

### (12) Obligation:

TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(N))))
a__sqr(0') → 0'
a__dbl(0') → 0'
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__first(0', X) → nil
a__first(s(X), cons(Y)) → cons(mark(Y))
a__half(0') → 0'
a__half(s(0')) → 0'
a__half(s(s(X))) → s(a__half(mark(X)))
a__half(dbl(X)) → mark(X)
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
mark(cons(X1)) → cons(mark(X1))
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)
a__half(X) → half(X)

Types:

Generator Equations:

The following defined symbols remain to be analysed:

They will be analysed ascendingly in the following order:
a__terms = a__sqr
a__terms = mark
a__terms = a__dbl
a__terms = a__half
a__sqr = mark
a__sqr = a__dbl
a__sqr = a__half
mark = a__dbl
mark = a__half
a__dbl = a__half

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

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

### (14) Obligation:

TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(N))))
a__sqr(0') → 0'
a__dbl(0') → 0'
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__first(0', X) → nil
a__first(s(X), cons(Y)) → cons(mark(Y))
a__half(0') → 0'
a__half(s(0')) → 0'
a__half(s(s(X))) → s(a__half(mark(X)))
a__half(dbl(X)) → mark(X)
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
mark(cons(X1)) → cons(mark(X1))
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)
a__half(X) → half(X)

Types:

Generator Equations:

The following defined symbols remain to be analysed:
mark, a__terms, a__dbl, a__half

They will be analysed ascendingly in the following order:
a__terms = a__sqr
a__terms = mark
a__terms = a__dbl
a__terms = a__half
a__sqr = mark
a__sqr = a__dbl
a__sqr = a__half
mark = a__dbl
mark = a__half
a__dbl = a__half

### (15) 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).

### (17) Obligation:

TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(N))))
a__sqr(0') → 0'
a__dbl(0') → 0'
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__first(0', X) → nil
a__first(s(X), cons(Y)) → cons(mark(Y))
a__half(0') → 0'
a__half(s(0')) → 0'
a__half(s(s(X))) → s(a__half(mark(X)))
a__half(dbl(X)) → mark(X)
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
mark(cons(X1)) → cons(mark(X1))
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)
a__half(X) → half(X)

Types:

Lemmas:

Generator Equations:

The following defined symbols remain to be analysed:

They will be analysed ascendingly in the following order:
a__terms = a__sqr
a__terms = mark
a__terms = a__dbl
a__terms = a__half
a__sqr = mark
a__sqr = a__dbl
a__sqr = a__half
mark = a__dbl
mark = a__half
a__dbl = a__half

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

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

### (19) Obligation:

TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(N))))
a__sqr(0') → 0'
a__dbl(0') → 0'
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__first(0', X) → nil
a__first(s(X), cons(Y)) → cons(mark(Y))
a__half(0') → 0'
a__half(s(0')) → 0'
a__half(s(s(X))) → s(a__half(mark(X)))
a__half(dbl(X)) → mark(X)
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
mark(cons(X1)) → cons(mark(X1))
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)
a__half(X) → half(X)

Types:

Lemmas:

Generator Equations:

The following defined symbols remain to be analysed:

They will be analysed ascendingly in the following order:
a__terms = a__sqr
a__terms = mark
a__terms = a__dbl
a__terms = a__half
a__sqr = mark
a__sqr = a__dbl
a__sqr = a__half
mark = a__dbl
mark = a__half
a__dbl = a__half

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

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

### (21) Obligation:

TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(N))))
a__sqr(0') → 0'
a__dbl(0') → 0'
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__first(0', X) → nil
a__first(s(X), cons(Y)) → cons(mark(Y))
a__half(0') → 0'
a__half(s(0')) → 0'
a__half(s(s(X))) → s(a__half(mark(X)))
a__half(dbl(X)) → mark(X)
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
mark(cons(X1)) → cons(mark(X1))
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)
a__half(X) → half(X)

Types:

Lemmas:

Generator Equations:

The following defined symbols remain to be analysed:

They will be analysed ascendingly in the following order:
a__terms = a__sqr
a__terms = mark
a__terms = a__dbl
a__terms = a__half
a__sqr = mark
a__sqr = a__dbl
a__sqr = a__half
mark = a__dbl
mark = a__half
a__dbl = a__half

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

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

### (23) Obligation:

TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(N))))
a__sqr(0') → 0'
a__dbl(0') → 0'
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__first(0', X) → nil
a__first(s(X), cons(Y)) → cons(mark(Y))
a__half(0') → 0'
a__half(s(0')) → 0'
a__half(s(s(X))) → s(a__half(mark(X)))
a__half(dbl(X)) → mark(X)
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
mark(cons(X1)) → cons(mark(X1))
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)
a__half(X) → half(X)

Types:

Lemmas:

Generator Equations:

The following defined symbols remain to be analysed:

They will be analysed ascendingly in the following order:
a__terms = a__sqr
a__terms = mark
a__terms = a__dbl
a__terms = a__half
a__sqr = mark
a__sqr = a__dbl
a__sqr = a__half
mark = a__dbl
mark = a__half
a__dbl = a__half

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

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

### (25) Obligation:

TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(N))))
a__sqr(0') → 0'
a__dbl(0') → 0'
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__first(0', X) → nil
a__first(s(X), cons(Y)) → cons(mark(Y))
a__half(0') → 0'
a__half(s(0')) → 0'
a__half(s(s(X))) → s(a__half(mark(X)))
a__half(dbl(X)) → mark(X)
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
mark(cons(X1)) → cons(mark(X1))
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)
a__half(X) → half(X)

Types:

Lemmas:

Generator Equations:

The following defined symbols remain to be analysed:

They will be analysed ascendingly in the following order:
a__terms = a__sqr
a__terms = mark
a__terms = a__dbl
a__terms = a__half
a__sqr = mark
a__sqr = a__dbl
a__sqr = a__half
mark = a__dbl
mark = a__half
a__dbl = a__half

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

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

### (27) Obligation:

TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(N))))
a__sqr(0') → 0'
a__dbl(0') → 0'
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__first(0', X) → nil
a__first(s(X), cons(Y)) → cons(mark(Y))
a__half(0') → 0'
a__half(s(0')) → 0'
a__half(s(s(X))) → s(a__half(mark(X)))
a__half(dbl(X)) → mark(X)
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
mark(cons(X1)) → cons(mark(X1))
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)
a__half(X) → half(X)

Types:

Lemmas:

Generator Equations:

No more defined symbols left to analyse.

### (28) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:

### (30) Obligation:

TRS:
Rules:
a__terms(N) → cons(recip(a__sqr(mark(N))))
a__sqr(0') → 0'
a__dbl(0') → 0'
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__first(0', X) → nil
a__first(s(X), cons(Y)) → cons(mark(Y))
a__half(0') → 0'
a__half(s(0')) → 0'
a__half(s(s(X))) → s(a__half(mark(X)))
a__half(dbl(X)) → mark(X)
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
mark(cons(X1)) → cons(mark(X1))
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)
a__half(X) → half(X)

Types:

Lemmas: