(0) Obligation:

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

a__incr(nil) → nil
a__incr(cons(X, L)) → cons(s(mark(X)), incr(L))
a__zeroscons(0, zeros)
a__tail(cons(X, L)) → mark(L)
mark(incr(X)) → a__incr(mark(X))
mark(nats) → a__nats
mark(zeros) → a__zeros
mark(tail(X)) → a__tail(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__incr(X) → incr(X)
a__natsnats
a__zeroszeros
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(X114187_3, X214188_3)))) →+ a__tail(cons(mark(X114187_3), X214188_3))
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [X214188_3 / tail(cons(X114187_3, X214188_3))].
The result substitution is [X / mark(X114187_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__incr(nil) → nil
a__incr(cons(X, L)) → cons(s(mark(X)), incr(L))
a__zeroscons(0', zeros)
a__tail(cons(X, L)) → mark(L)
mark(incr(X)) → a__incr(mark(X))
mark(nats) → a__nats
mark(zeros) → a__zeros
mark(tail(X)) → a__tail(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0') → 0'
a__incr(X) → incr(X)
a__natsnats
a__zeroszeros
a__tail(X) → tail(X)

S is empty.
Rewrite Strategy: FULL

Infered types.

(6) Obligation:

TRS:
Rules:
a__incr(nil) → nil
a__incr(cons(X, L)) → cons(s(mark(X)), incr(L))
a__zeroscons(0', zeros)
a__tail(cons(X, L)) → mark(L)
mark(incr(X)) → a__incr(mark(X))
mark(nats) → a__nats
mark(zeros) → a__zeros
mark(tail(X)) → a__tail(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0') → 0'
a__incr(X) → incr(X)
a__natsnats
a__zeroszeros
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__incr = mark
a__incr = a__nats
a__incr = a__tail
mark = a__nats
mark = a__tail
a__nats = a__tail

(8) Obligation:

TRS:
Rules:
a__incr(nil) → nil
a__incr(cons(X, L)) → cons(s(mark(X)), incr(L))
a__zeroscons(0', zeros)
a__tail(cons(X, L)) → mark(L)
mark(incr(X)) → a__incr(mark(X))
mark(nats) → a__nats
mark(zeros) → a__zeros
mark(tail(X)) → a__tail(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0') → 0'
a__incr(X) → incr(X)
a__natsnats
a__zeroszeros
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__incr = mark
a__incr = a__nats
a__incr = a__tail
mark = a__nats
mark = a__tail
a__nats = a__tail

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:

Induction Base:
nil

Induction Step:

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

(11) Obligation:

TRS:
Rules:
a__incr(nil) → nil
a__incr(cons(X, L)) → cons(s(mark(X)), incr(L))
a__zeroscons(0', zeros)
a__tail(cons(X, L)) → mark(L)
mark(incr(X)) → a__incr(mark(X))
mark(nats) → a__nats
mark(zeros) → a__zeros
mark(tail(X)) → a__tail(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0') → 0'
a__incr(X) → incr(X)
a__natsnats
a__zeroszeros
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__incr = mark
a__incr = a__nats
a__incr = a__tail
mark = a__nats
mark = a__tail
a__nats = a__tail

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(13) Obligation:

TRS:
Rules:
a__incr(nil) → nil
a__incr(cons(X, L)) → cons(s(mark(X)), incr(L))
a__zeroscons(0', zeros)
a__tail(cons(X, L)) → mark(L)
mark(incr(X)) → a__incr(mark(X))
mark(nats) → a__nats
mark(zeros) → a__zeros
mark(tail(X)) → a__tail(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0') → 0'
a__incr(X) → incr(X)
a__natsnats
a__zeroszeros
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__incr = mark
a__incr = a__nats
a__incr = a__tail
mark = a__nats
mark = a__tail
a__nats = a__tail

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

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

(15) Obligation:

TRS:
Rules:
a__incr(nil) → nil
a__incr(cons(X, L)) → cons(s(mark(X)), incr(L))
a__zeroscons(0', zeros)
a__tail(cons(X, L)) → mark(L)
mark(incr(X)) → a__incr(mark(X))
mark(nats) → a__nats
mark(zeros) → a__zeros
mark(tail(X)) → a__tail(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0') → 0'
a__incr(X) → incr(X)
a__natsnats
a__zeroszeros
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__incr = mark
a__incr = a__nats
a__incr = a__tail
mark = a__nats
mark = a__tail
a__nats = a__tail

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

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

(17) Obligation:

TRS:
Rules:
a__incr(nil) → nil
a__incr(cons(X, L)) → cons(s(mark(X)), incr(L))
a__zeroscons(0', zeros)
a__tail(cons(X, L)) → mark(L)
mark(incr(X)) → a__incr(mark(X))
mark(nats) → a__nats
mark(zeros) → a__zeros
mark(tail(X)) → a__tail(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0') → 0'
a__incr(X) → incr(X)
a__natsnats
a__zeroszeros
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__incr = mark
a__incr = a__nats
a__incr = a__tail
mark = a__nats
mark = a__tail
a__nats = a__tail

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

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

(19) Obligation:

TRS:
Rules:
a__incr(nil) → nil
a__incr(cons(X, L)) → cons(s(mark(X)), incr(L))
a__zeroscons(0', zeros)
a__tail(cons(X, L)) → mark(L)
mark(incr(X)) → a__incr(mark(X))
mark(nats) → a__nats
mark(zeros) → a__zeros
mark(tail(X)) → a__tail(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0') → 0'
a__incr(X) → incr(X)
a__natsnats
a__zeroszeros
a__tail(X) → tail(X)

Types:

Lemmas:

Generator Equations:

The following defined symbols remain to be analysed:
a__tail

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

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

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

(21) Obligation:

TRS:
Rules:
a__incr(nil) → nil
a__incr(cons(X, L)) → cons(s(mark(X)), incr(L))
a__zeroscons(0', zeros)
a__tail(cons(X, L)) → mark(L)
mark(incr(X)) → a__incr(mark(X))
mark(nats) → a__nats
mark(zeros) → a__zeros
mark(tail(X)) → a__tail(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0') → 0'
a__incr(X) → incr(X)
a__natsnats
a__zeroszeros
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__incr(nil) → nil
a__incr(cons(X, L)) → cons(s(mark(X)), incr(L))
a__zeroscons(0', zeros)
a__tail(cons(X, L)) → mark(L)
mark(incr(X)) → a__incr(mark(X))
mark(nats) → a__nats
mark(zeros) → a__zeros
mark(tail(X)) → a__tail(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0') → 0'
a__incr(X) → incr(X)
a__natsnats
a__zeroszeros
a__tail(X) → tail(X)

Types:

Lemmas: