The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

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

a__nats → cons(0, incr(nats))
a__pairs → cons(0, incr(odds))
a__odds → a__incr(a__pairs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__head(cons(X, XS)) → mark(X)
a__tail(cons(X, XS)) → mark(XS)
mark(nats) → a__nats
mark(incr(X)) → a__incr(mark(X))
mark(pairs) → a__pairs
mark(odds) → a__odds
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__nats → nats
a__incr(X) → incr(X)
a__pairs → pairs
a__odds → odds
a__head(X) → head(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(X18065_3, X28066_3)))) →⁺ a__tail(cons(mark(X18065_3), X28066_3))
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [X28066_3 / tail(cons(X18065_3, X28066_3))].
The result substitution is [X / mark(X18065_3)].

(2) BOUNDS(n^1, INF)

(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__nats → cons(0', incr(nats))
a__pairs → cons(0', incr(odds))
a__odds → a__incr(a__pairs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__head(cons(X, XS)) → mark(X)
a__tail(cons(X, XS)) → mark(XS)
mark(nats) → a__nats
mark(incr(X)) → a__incr(mark(X))
mark(pairs) → a__pairs
mark(odds) → a__odds
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(s(X)) → s(mark(X))
a__nats → nats
a__incr(X) → incr(X)
a__pairs → pairs
a__odds → odds
a__head(X) → head(X)
a__tail(X) → tail(X)

S is empty.
Rewrite Strategy: FULL

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

TRS:
Rules:
a__nats → cons(0', incr(nats))
a__pairs → cons(0', incr(odds))
a__odds → a__incr(a__pairs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__head(cons(X, XS)) → mark(X)
a__tail(cons(X, XS)) → mark(XS)
mark(nats) → a__nats
mark(incr(X)) → a__incr(mark(X))
mark(pairs) → a__pairs
mark(odds) → a__odds
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(s(X)) → s(mark(X))
a__nats → nats
a__incr(X) → incr(X)
a__pairs → pairs
a__odds → odds
a__head(X) → head(X)
a__tail(X) → tail(X)

Types:
a__nats :: 0':nats:incr:cons:odds:s:pairs:head:tail
cons :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
0' :: 0':nats:incr:cons:odds:s:pairs:head:tail
incr :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
nats :: 0':nats:incr:cons:odds:s:pairs:head:tail
a__pairs :: 0':nats:incr:cons:odds:s:pairs:head:tail
odds :: 0':nats:incr:cons:odds:s:pairs:head:tail
a__odds :: 0':nats:incr:cons:odds:s:pairs:head:tail
a__incr :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
s :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
mark :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
a__head :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
a__tail :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
pairs :: 0':nats:incr:cons:odds:s:pairs:head:tail
head :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
tail :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
hole_0':nats:incr:cons:odds:s:pairs:head:tail1_0 :: 0':nats:incr:cons:odds:s:pairs:head:tail
gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0 :: Nat → 0':nats:incr:cons:odds:s:pairs:head:tail

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

Heuristically decided to analyse the following defined symbols:
a__odds, a__incr, mark, a__head, a__tail

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

(8) Obligation:

TRS:
Rules:
a__nats → cons(0', incr(nats))
a__pairs → cons(0', incr(odds))
a__odds → a__incr(a__pairs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__head(cons(X, XS)) → mark(X)
a__tail(cons(X, XS)) → mark(XS)
mark(nats) → a__nats
mark(incr(X)) → a__incr(mark(X))
mark(pairs) → a__pairs
mark(odds) → a__odds
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(s(X)) → s(mark(X))
a__nats → nats
a__incr(X) → incr(X)
a__pairs → pairs
a__odds → odds
a__head(X) → head(X)
a__tail(X) → tail(X)

Types:
a__nats :: 0':nats:incr:cons:odds:s:pairs:head:tail
cons :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
0' :: 0':nats:incr:cons:odds:s:pairs:head:tail
incr :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
nats :: 0':nats:incr:cons:odds:s:pairs:head:tail
a__pairs :: 0':nats:incr:cons:odds:s:pairs:head:tail
odds :: 0':nats:incr:cons:odds:s:pairs:head:tail
a__odds :: 0':nats:incr:cons:odds:s:pairs:head:tail
a__incr :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
s :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
mark :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
a__head :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
a__tail :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
pairs :: 0':nats:incr:cons:odds:s:pairs:head:tail
head :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
tail :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
hole_0':nats:incr:cons:odds:s:pairs:head:tail1_0 :: 0':nats:incr:cons:odds:s:pairs:head:tail
gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0 :: Nat → 0':nats:incr:cons:odds:s:pairs:head:tail

Generator Equations:
gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(0) ⇔ 0'
gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(+(x, 1)) ⇔ cons(gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(x), 0')

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

They will be analysed ascendingly in the following order:
a__odds = a__incr
a__odds = mark
a__odds = a__head
a__odds = a__tail
a__incr = mark
a__incr = a__head
a__incr = a__tail
mark = a__head
mark = a__tail
a__head = 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__nats → cons(0', incr(nats))
a__pairs → cons(0', incr(odds))
a__odds → a__incr(a__pairs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__head(cons(X, XS)) → mark(X)
a__tail(cons(X, XS)) → mark(XS)
mark(nats) → a__nats
mark(incr(X)) → a__incr(mark(X))
mark(pairs) → a__pairs
mark(odds) → a__odds
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(s(X)) → s(mark(X))
a__nats → nats
a__incr(X) → incr(X)
a__pairs → pairs
a__odds → odds
a__head(X) → head(X)
a__tail(X) → tail(X)

Types:
a__nats :: 0':nats:incr:cons:odds:s:pairs:head:tail
cons :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
0' :: 0':nats:incr:cons:odds:s:pairs:head:tail
incr :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
nats :: 0':nats:incr:cons:odds:s:pairs:head:tail
a__pairs :: 0':nats:incr:cons:odds:s:pairs:head:tail
odds :: 0':nats:incr:cons:odds:s:pairs:head:tail
a__odds :: 0':nats:incr:cons:odds:s:pairs:head:tail
a__incr :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
s :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
mark :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
a__head :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
a__tail :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
pairs :: 0':nats:incr:cons:odds:s:pairs:head:tail
head :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
tail :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
hole_0':nats:incr:cons:odds:s:pairs:head:tail1_0 :: 0':nats:incr:cons:odds:s:pairs:head:tail
gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0 :: Nat → 0':nats:incr:cons:odds:s:pairs:head:tail

Generator Equations:
gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(0) ⇔ 0'
gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(+(x, 1)) ⇔ cons(gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(x), 0')

The following defined symbols remain to be analysed:
mark, a__odds, a__head, a__tail

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

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

Proved the following rewrite lemma:
mark(gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(n76935_0)) → gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(n76935_0), rt ∈ Ω(1 + n76935₀)

Induction Base:
mark(gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(0)) →_R^Ω(1)
0'

Induction Step:
mark(gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(+(n76935_0, 1))) →_R^Ω(1)
cons(mark(gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(n76935_0)), 0') →_IH
cons(gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(c76936_0), 0')

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

(12) Complex Obligation (BEST)

(13) Obligation:

TRS:
Rules:
a__nats → cons(0', incr(nats))
a__pairs → cons(0', incr(odds))
a__odds → a__incr(a__pairs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__head(cons(X, XS)) → mark(X)
a__tail(cons(X, XS)) → mark(XS)
mark(nats) → a__nats
mark(incr(X)) → a__incr(mark(X))
mark(pairs) → a__pairs
mark(odds) → a__odds
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(s(X)) → s(mark(X))
a__nats → nats
a__incr(X) → incr(X)
a__pairs → pairs
a__odds → odds
a__head(X) → head(X)
a__tail(X) → tail(X)

Types:
a__nats :: 0':nats:incr:cons:odds:s:pairs:head:tail
cons :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
0' :: 0':nats:incr:cons:odds:s:pairs:head:tail
incr :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
nats :: 0':nats:incr:cons:odds:s:pairs:head:tail
a__pairs :: 0':nats:incr:cons:odds:s:pairs:head:tail
odds :: 0':nats:incr:cons:odds:s:pairs:head:tail
a__odds :: 0':nats:incr:cons:odds:s:pairs:head:tail
a__incr :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
s :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
mark :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
a__head :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
a__tail :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
pairs :: 0':nats:incr:cons:odds:s:pairs:head:tail
head :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
tail :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
hole_0':nats:incr:cons:odds:s:pairs:head:tail1_0 :: 0':nats:incr:cons:odds:s:pairs:head:tail
gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0 :: Nat → 0':nats:incr:cons:odds:s:pairs:head:tail

Lemmas:
mark(gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(n76935_0)) → gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(n76935_0), rt ∈ Ω(1 + n76935₀)

Generator Equations:
gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(0) ⇔ 0'
gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(+(x, 1)) ⇔ cons(gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(x), 0')

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

They will be analysed ascendingly in the following order:
a__odds = a__incr
a__odds = mark
a__odds = a__head
a__odds = a__tail
a__incr = mark
a__incr = a__head
a__incr = a__tail
mark = a__head
mark = a__tail
a__head = 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__nats → cons(0', incr(nats))
a__pairs → cons(0', incr(odds))
a__odds → a__incr(a__pairs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__head(cons(X, XS)) → mark(X)
a__tail(cons(X, XS)) → mark(XS)
mark(nats) → a__nats
mark(incr(X)) → a__incr(mark(X))
mark(pairs) → a__pairs
mark(odds) → a__odds
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(s(X)) → s(mark(X))
a__nats → nats
a__incr(X) → incr(X)
a__pairs → pairs
a__odds → odds
a__head(X) → head(X)
a__tail(X) → tail(X)

Types:
a__nats :: 0':nats:incr:cons:odds:s:pairs:head:tail
cons :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
0' :: 0':nats:incr:cons:odds:s:pairs:head:tail
incr :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
nats :: 0':nats:incr:cons:odds:s:pairs:head:tail
a__pairs :: 0':nats:incr:cons:odds:s:pairs:head:tail
odds :: 0':nats:incr:cons:odds:s:pairs:head:tail
a__odds :: 0':nats:incr:cons:odds:s:pairs:head:tail
a__incr :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
s :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
mark :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
a__head :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
a__tail :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
pairs :: 0':nats:incr:cons:odds:s:pairs:head:tail
head :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
tail :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
hole_0':nats:incr:cons:odds:s:pairs:head:tail1_0 :: 0':nats:incr:cons:odds:s:pairs:head:tail
gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0 :: Nat → 0':nats:incr:cons:odds:s:pairs:head:tail

Lemmas:
mark(gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(n76935_0)) → gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(n76935_0), rt ∈ Ω(1 + n76935₀)

Generator Equations:
gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(0) ⇔ 0'
gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(+(x, 1)) ⇔ cons(gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(x), 0')

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

They will be analysed ascendingly in the following order:
a__odds = a__incr
a__odds = mark
a__odds = a__head
a__odds = a__tail
a__incr = mark
a__incr = a__head
a__incr = a__tail
mark = a__head
mark = a__tail
a__head = 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__nats → cons(0', incr(nats))
a__pairs → cons(0', incr(odds))
a__odds → a__incr(a__pairs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__head(cons(X, XS)) → mark(X)
a__tail(cons(X, XS)) → mark(XS)
mark(nats) → a__nats
mark(incr(X)) → a__incr(mark(X))
mark(pairs) → a__pairs
mark(odds) → a__odds
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(s(X)) → s(mark(X))
a__nats → nats
a__incr(X) → incr(X)
a__pairs → pairs
a__odds → odds
a__head(X) → head(X)
a__tail(X) → tail(X)

Types:
a__nats :: 0':nats:incr:cons:odds:s:pairs:head:tail
cons :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
0' :: 0':nats:incr:cons:odds:s:pairs:head:tail
incr :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
nats :: 0':nats:incr:cons:odds:s:pairs:head:tail
a__pairs :: 0':nats:incr:cons:odds:s:pairs:head:tail
odds :: 0':nats:incr:cons:odds:s:pairs:head:tail
a__odds :: 0':nats:incr:cons:odds:s:pairs:head:tail
a__incr :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
s :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
mark :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
a__head :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
a__tail :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
pairs :: 0':nats:incr:cons:odds:s:pairs:head:tail
head :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
tail :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
hole_0':nats:incr:cons:odds:s:pairs:head:tail1_0 :: 0':nats:incr:cons:odds:s:pairs:head:tail
gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0 :: Nat → 0':nats:incr:cons:odds:s:pairs:head:tail

Lemmas:
mark(gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(n76935_0)) → gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(n76935_0), rt ∈ Ω(1 + n76935₀)

Generator Equations:
gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(0) ⇔ 0'
gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(+(x, 1)) ⇔ cons(gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(x), 0')

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__head
a__odds = a__tail
a__incr = mark
a__incr = a__head
a__incr = a__tail
mark = a__head
mark = a__tail
a__head = 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__nats → cons(0', incr(nats))
a__pairs → cons(0', incr(odds))
a__odds → a__incr(a__pairs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__head(cons(X, XS)) → mark(X)
a__tail(cons(X, XS)) → mark(XS)
mark(nats) → a__nats
mark(incr(X)) → a__incr(mark(X))
mark(pairs) → a__pairs
mark(odds) → a__odds
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(s(X)) → s(mark(X))
a__nats → nats
a__incr(X) → incr(X)
a__pairs → pairs
a__odds → odds
a__head(X) → head(X)
a__tail(X) → tail(X)

Types:
a__nats :: 0':nats:incr:cons:odds:s:pairs:head:tail
cons :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
0' :: 0':nats:incr:cons:odds:s:pairs:head:tail
incr :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
nats :: 0':nats:incr:cons:odds:s:pairs:head:tail
a__pairs :: 0':nats:incr:cons:odds:s:pairs:head:tail
odds :: 0':nats:incr:cons:odds:s:pairs:head:tail
a__odds :: 0':nats:incr:cons:odds:s:pairs:head:tail
a__incr :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
s :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
mark :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
a__head :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
a__tail :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
pairs :: 0':nats:incr:cons:odds:s:pairs:head:tail
head :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
tail :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
hole_0':nats:incr:cons:odds:s:pairs:head:tail1_0 :: 0':nats:incr:cons:odds:s:pairs:head:tail
gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0 :: Nat → 0':nats:incr:cons:odds:s:pairs:head:tail

Lemmas:
mark(gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(n76935_0)) → gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(n76935_0), rt ∈ Ω(1 + n76935₀)

Generator Equations:
gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(0) ⇔ 0'
gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(+(x, 1)) ⇔ cons(gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(x), 0')

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__head
a__odds = a__tail
a__incr = mark
a__incr = a__head
a__incr = a__tail
mark = a__head
mark = a__tail
a__head = 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__nats → cons(0', incr(nats))
a__pairs → cons(0', incr(odds))
a__odds → a__incr(a__pairs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__head(cons(X, XS)) → mark(X)
a__tail(cons(X, XS)) → mark(XS)
mark(nats) → a__nats
mark(incr(X)) → a__incr(mark(X))
mark(pairs) → a__pairs
mark(odds) → a__odds
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(s(X)) → s(mark(X))
a__nats → nats
a__incr(X) → incr(X)
a__pairs → pairs
a__odds → odds
a__head(X) → head(X)
a__tail(X) → tail(X)

Types:
a__nats :: 0':nats:incr:cons:odds:s:pairs:head:tail
cons :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
0' :: 0':nats:incr:cons:odds:s:pairs:head:tail
incr :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
nats :: 0':nats:incr:cons:odds:s:pairs:head:tail
a__pairs :: 0':nats:incr:cons:odds:s:pairs:head:tail
odds :: 0':nats:incr:cons:odds:s:pairs:head:tail
a__odds :: 0':nats:incr:cons:odds:s:pairs:head:tail
a__incr :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
s :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
mark :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
a__head :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
a__tail :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
pairs :: 0':nats:incr:cons:odds:s:pairs:head:tail
head :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
tail :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
hole_0':nats:incr:cons:odds:s:pairs:head:tail1_0 :: 0':nats:incr:cons:odds:s:pairs:head:tail
gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0 :: Nat → 0':nats:incr:cons:odds:s:pairs:head:tail

Lemmas:
mark(gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(n76935_0)) → gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(n76935_0), rt ∈ Ω(1 + n76935₀)

Generator Equations:
gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(0) ⇔ 0'
gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(+(x, 1)) ⇔ cons(gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(x), 0')

No more defined symbols left to analyse.

(22) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(n76935_0)) → gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(n76935_0), rt ∈ Ω(1 + n76935₀)

(23) BOUNDS(n^1, INF)

(24) Obligation:

TRS:
Rules:
a__nats → cons(0', incr(nats))
a__pairs → cons(0', incr(odds))
a__odds → a__incr(a__pairs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__head(cons(X, XS)) → mark(X)
a__tail(cons(X, XS)) → mark(XS)
mark(nats) → a__nats
mark(incr(X)) → a__incr(mark(X))
mark(pairs) → a__pairs
mark(odds) → a__odds
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(s(X)) → s(mark(X))
a__nats → nats
a__incr(X) → incr(X)
a__pairs → pairs
a__odds → odds
a__head(X) → head(X)
a__tail(X) → tail(X)

Types:
a__nats :: 0':nats:incr:cons:odds:s:pairs:head:tail
cons :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
0' :: 0':nats:incr:cons:odds:s:pairs:head:tail
incr :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
nats :: 0':nats:incr:cons:odds:s:pairs:head:tail
a__pairs :: 0':nats:incr:cons:odds:s:pairs:head:tail
odds :: 0':nats:incr:cons:odds:s:pairs:head:tail
a__odds :: 0':nats:incr:cons:odds:s:pairs:head:tail
a__incr :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
s :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
mark :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
a__head :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
a__tail :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
pairs :: 0':nats:incr:cons:odds:s:pairs:head:tail
head :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
tail :: 0':nats:incr:cons:odds:s:pairs:head:tail → 0':nats:incr:cons:odds:s:pairs:head:tail
hole_0':nats:incr:cons:odds:s:pairs:head:tail1_0 :: 0':nats:incr:cons:odds:s:pairs:head:tail
gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0 :: Nat → 0':nats:incr:cons:odds:s:pairs:head:tail

Lemmas:
mark(gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(n76935_0)) → gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(n76935_0), rt ∈ Ω(1 + n76935₀)

Generator Equations:
gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(0) ⇔ 0'
gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(+(x, 1)) ⇔ cons(gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(x), 0')

No more defined symbols left to analyse.

(25) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(n76935_0)) → gen_0':nats:incr:cons:odds:s:pairs:head:tail2_0(n76935_0), rt ∈ Ω(1 + n76935₀)

(26) BOUNDS(n^1, INF)