### (0) Obligation:

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

a__natscons(0, incr(nats))
a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__tail(cons(X, XS)) → mark(XS)
mark(nats) → a__nats
mark(pairs) → a__pairs
mark(odds) → a__odds
mark(incr(X)) → a__incr(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__natsnats
a__pairspairs
a__oddsodds
a__incr(X) → incr(X)
a__tail(X) → tail(X)

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
a__tail(cons(X, tail(cons(X18666_3, X28667_3)))) →+ a__tail(cons(mark(X18666_3), X28667_3))
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [X28667_3 / tail(cons(X18666_3, X28667_3))].
The result substitution is [X / mark(X18666_3)].

### (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__natscons(0', incr(nats))
a__pairscons(0', incr(odds))
a__oddsa__incr(a__pairs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__tail(cons(X, XS)) → mark(XS)
mark(nats) → a__nats
mark(pairs) → a__pairs
mark(odds) → a__odds
mark(incr(X)) → a__incr(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__natsnats
a__pairspairs
a__oddsodds
a__incr(X) → incr(X)
a__tail(X) → tail(X)

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
a__natscons(0', incr(nats))
a__pairscons(0', incr(odds))
a__oddsa__incr(a__pairs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__tail(cons(X, XS)) → mark(XS)
mark(nats) → a__nats
mark(pairs) → a__pairs
mark(odds) → a__odds
mark(incr(X)) → a__incr(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__natsnats
a__pairspairs
a__oddsodds
a__incr(X) → incr(X)
a__tail(X) → tail(X)

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__odds = a__incr
a__odds = mark
a__odds = a__tail
a__incr = mark
a__incr = a__tail
mark = a__tail

### (8) Obligation:

TRS:
Rules:
a__natscons(0', incr(nats))
a__pairscons(0', incr(odds))
a__oddsa__incr(a__pairs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__tail(cons(X, XS)) → mark(XS)
mark(nats) → a__nats
mark(pairs) → a__pairs
mark(odds) → a__odds
mark(incr(X)) → a__incr(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__natsnats
a__pairspairs
a__oddsodds
a__incr(X) → incr(X)
a__tail(X) → tail(X)

Types:

Generator Equations:

The following defined symbols remain to be analysed:

They will be analysed ascendingly in the following order:
a__odds = a__incr
a__odds = mark
a__odds = a__tail
a__incr = mark
a__incr = a__tail
mark = a__tail

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

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

### (10) Obligation:

TRS:
Rules:
a__natscons(0', incr(nats))
a__pairscons(0', incr(odds))
a__oddsa__incr(a__pairs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__tail(cons(X, XS)) → mark(XS)
mark(nats) → a__nats
mark(pairs) → a__pairs
mark(odds) → a__odds
mark(incr(X)) → a__incr(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__natsnats
a__pairspairs
a__oddsodds
a__incr(X) → incr(X)
a__tail(X) → tail(X)

Types:

Generator Equations:

The following defined symbols remain to be analysed:

They will be analysed ascendingly in the following order:
a__odds = a__incr
a__odds = mark
a__odds = a__tail
a__incr = mark
a__incr = a__tail
mark = a__tail

### (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__natscons(0', incr(nats))
a__pairscons(0', incr(odds))
a__oddsa__incr(a__pairs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__tail(cons(X, XS)) → mark(XS)
mark(nats) → a__nats
mark(pairs) → a__pairs
mark(odds) → a__odds
mark(incr(X)) → a__incr(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__natsnats
a__pairspairs
a__oddsodds
a__incr(X) → incr(X)
a__tail(X) → tail(X)

Types:

Lemmas:

Generator Equations:

The following defined symbols remain to be analysed:

They will be analysed ascendingly in the following order:
a__odds = a__incr
a__odds = mark
a__odds = a__tail
a__incr = mark
a__incr = a__tail
mark = a__tail

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

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

### (15) Obligation:

TRS:
Rules:
a__natscons(0', incr(nats))
a__pairscons(0', incr(odds))
a__oddsa__incr(a__pairs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__tail(cons(X, XS)) → mark(XS)
mark(nats) → a__nats
mark(pairs) → a__pairs
mark(odds) → a__odds
mark(incr(X)) → a__incr(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__natsnats
a__pairspairs
a__oddsodds
a__incr(X) → incr(X)
a__tail(X) → tail(X)

Types:

Lemmas:

Generator Equations:

The following defined symbols remain to be analysed:

They will be analysed ascendingly in the following order:
a__odds = a__incr
a__odds = mark
a__odds = a__tail
a__incr = mark
a__incr = a__tail
mark = a__tail

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

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

### (17) Obligation:

TRS:
Rules:
a__natscons(0', incr(nats))
a__pairscons(0', incr(odds))
a__oddsa__incr(a__pairs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__tail(cons(X, XS)) → mark(XS)
mark(nats) → a__nats
mark(pairs) → a__pairs
mark(odds) → a__odds
mark(incr(X)) → a__incr(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__natsnats
a__pairspairs
a__oddsodds
a__incr(X) → incr(X)
a__tail(X) → tail(X)

Types:

Lemmas:

Generator Equations:

The following defined symbols remain to be analysed:
a__tail, a__incr

They will be analysed ascendingly in the following order:
a__odds = a__incr
a__odds = mark
a__odds = a__tail
a__incr = mark
a__incr = a__tail
mark = a__tail

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

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

### (19) Obligation:

TRS:
Rules:
a__natscons(0', incr(nats))
a__pairscons(0', incr(odds))
a__oddsa__incr(a__pairs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__tail(cons(X, XS)) → mark(XS)
mark(nats) → a__nats
mark(pairs) → a__pairs
mark(odds) → a__odds
mark(incr(X)) → a__incr(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__natsnats
a__pairspairs
a__oddsodds
a__incr(X) → incr(X)
a__tail(X) → tail(X)

Types:

Lemmas:

Generator Equations:

The following defined symbols remain to be analysed:
a__incr

They will be analysed ascendingly in the following order:
a__odds = a__incr
a__odds = mark
a__odds = a__tail
a__incr = mark
a__incr = a__tail
mark = a__tail

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

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

### (21) Obligation:

TRS:
Rules:
a__natscons(0', incr(nats))
a__pairscons(0', incr(odds))
a__oddsa__incr(a__pairs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__tail(cons(X, XS)) → mark(XS)
mark(nats) → a__nats
mark(pairs) → a__pairs
mark(odds) → a__odds
mark(incr(X)) → a__incr(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__natsnats
a__pairspairs
a__oddsodds
a__incr(X) → incr(X)
a__tail(X) → tail(X)

Types:

Lemmas:

Generator Equations:

No more defined symbols left to analyse.

### (22) LowerBoundsProof (EQUIVALENT transformation)

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

### (24) Obligation:

TRS:
Rules:
a__natscons(0', incr(nats))
a__pairscons(0', incr(odds))
a__oddsa__incr(a__pairs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__tail(cons(X, XS)) → mark(XS)
mark(nats) → a__nats
mark(pairs) → a__pairs
mark(odds) → a__odds
mark(incr(X)) → a__incr(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__natsnats
a__pairspairs
a__oddsodds
a__incr(X) → incr(X)
a__tail(X) → tail(X)

Types:

Lemmas: