The Runtime Complexity (innermost) 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__2nd(cons(X, cons(Y, Z))) → mark(Y)
a__from(X) → cons(mark(X), from(s(X)))
mark(2nd(X)) → a__2nd(mark(X))
mark(from(X)) → a__from(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__2nd(X) → 2nd(X)
a__from(X) → from(X)

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
a__2nd(cons(X, cons(2nd(cons(X118_0, X219_0)), Z))) →⁺ a__2nd(cons(mark(X118_0), X219_0))
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [X219_0 / cons(2nd(cons(X118_0, X219_0)), Z)].
The result substitution is [X / mark(X118_0)].

(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__2nd(cons(X, cons(Y, Z))) → mark(Y)
a__from(X) → cons(mark(X), from(s(X)))
mark(2nd(X)) → a__2nd(mark(X))
mark(from(X)) → a__from(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__2nd(X) → 2nd(X)
a__from(X) → from(X)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
a__2nd(cons(X, cons(Y, Z))) → mark(Y)
a__from(X) → cons(mark(X), from(s(X)))
mark(2nd(X)) → a__2nd(mark(X))
mark(from(X)) → a__from(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__2nd(X) → 2nd(X)
a__from(X) → from(X)

Types:
a__2nd :: cons:s:from:2nd → cons:s:from:2nd
cons :: cons:s:from:2nd → cons:s:from:2nd → cons:s:from:2nd
mark :: cons:s:from:2nd → cons:s:from:2nd
a__from :: cons:s:from:2nd → cons:s:from:2nd
from :: cons:s:from:2nd → cons:s:from:2nd
s :: cons:s:from:2nd → cons:s:from:2nd
2nd :: cons:s:from:2nd → cons:s:from:2nd
hole_cons:s:from:2nd1_0 :: cons:s:from:2nd
gen_cons:s:from:2nd2_0 :: Nat → cons:s:from:2nd

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

Heuristically decided to analyse the following defined symbols:
a__2nd, mark, a__from

They will be analysed ascendingly in the following order:
a__2nd = mark
a__2nd = a__from
mark = a__from

(8) Obligation:

Innermost TRS:
Rules:
a__2nd(cons(X, cons(Y, Z))) → mark(Y)
a__from(X) → cons(mark(X), from(s(X)))
mark(2nd(X)) → a__2nd(mark(X))
mark(from(X)) → a__from(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__2nd(X) → 2nd(X)
a__from(X) → from(X)

Types:
a__2nd :: cons:s:from:2nd → cons:s:from:2nd
cons :: cons:s:from:2nd → cons:s:from:2nd → cons:s:from:2nd
mark :: cons:s:from:2nd → cons:s:from:2nd
a__from :: cons:s:from:2nd → cons:s:from:2nd
from :: cons:s:from:2nd → cons:s:from:2nd
s :: cons:s:from:2nd → cons:s:from:2nd
2nd :: cons:s:from:2nd → cons:s:from:2nd
hole_cons:s:from:2nd1_0 :: cons:s:from:2nd
gen_cons:s:from:2nd2_0 :: Nat → cons:s:from:2nd

Generator Equations:
gen_cons:s:from:2nd2_0(0) ⇔ hole_cons:s:from:2nd1_0
gen_cons:s:from:2nd2_0(+(x, 1)) ⇔ cons(gen_cons:s:from:2nd2_0(x), hole_cons:s:from:2nd1_0)

The following defined symbols remain to be analysed:
mark, a__2nd, a__from

They will be analysed ascendingly in the following order:
a__2nd = mark
a__2nd = a__from
mark = a__from

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

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

(10) Obligation:

Innermost TRS:
Rules:
a__2nd(cons(X, cons(Y, Z))) → mark(Y)
a__from(X) → cons(mark(X), from(s(X)))
mark(2nd(X)) → a__2nd(mark(X))
mark(from(X)) → a__from(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__2nd(X) → 2nd(X)
a__from(X) → from(X)

Types:
a__2nd :: cons:s:from:2nd → cons:s:from:2nd
cons :: cons:s:from:2nd → cons:s:from:2nd → cons:s:from:2nd
mark :: cons:s:from:2nd → cons:s:from:2nd
a__from :: cons:s:from:2nd → cons:s:from:2nd
from :: cons:s:from:2nd → cons:s:from:2nd
s :: cons:s:from:2nd → cons:s:from:2nd
2nd :: cons:s:from:2nd → cons:s:from:2nd
hole_cons:s:from:2nd1_0 :: cons:s:from:2nd
gen_cons:s:from:2nd2_0 :: Nat → cons:s:from:2nd

Generator Equations:
gen_cons:s:from:2nd2_0(0) ⇔ hole_cons:s:from:2nd1_0
gen_cons:s:from:2nd2_0(+(x, 1)) ⇔ cons(gen_cons:s:from:2nd2_0(x), hole_cons:s:from:2nd1_0)

The following defined symbols remain to be analysed:
a__2nd, a__from

They will be analysed ascendingly in the following order:
a__2nd = mark
a__2nd = a__from
mark = a__from

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

Could not prove a rewrite lemma for the defined symbol a__2nd.

(12) Obligation:

Innermost TRS:
Rules:
a__2nd(cons(X, cons(Y, Z))) → mark(Y)
a__from(X) → cons(mark(X), from(s(X)))
mark(2nd(X)) → a__2nd(mark(X))
mark(from(X)) → a__from(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__2nd(X) → 2nd(X)
a__from(X) → from(X)

Types:
a__2nd :: cons:s:from:2nd → cons:s:from:2nd
cons :: cons:s:from:2nd → cons:s:from:2nd → cons:s:from:2nd
mark :: cons:s:from:2nd → cons:s:from:2nd
a__from :: cons:s:from:2nd → cons:s:from:2nd
from :: cons:s:from:2nd → cons:s:from:2nd
s :: cons:s:from:2nd → cons:s:from:2nd
2nd :: cons:s:from:2nd → cons:s:from:2nd
hole_cons:s:from:2nd1_0 :: cons:s:from:2nd
gen_cons:s:from:2nd2_0 :: Nat → cons:s:from:2nd

Generator Equations:
gen_cons:s:from:2nd2_0(0) ⇔ hole_cons:s:from:2nd1_0
gen_cons:s:from:2nd2_0(+(x, 1)) ⇔ cons(gen_cons:s:from:2nd2_0(x), hole_cons:s:from:2nd1_0)

The following defined symbols remain to be analysed:
a__from

They will be analysed ascendingly in the following order:
a__2nd = mark
a__2nd = a__from
mark = a__from

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

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

(14) Obligation:

Innermost TRS:
Rules:
a__2nd(cons(X, cons(Y, Z))) → mark(Y)
a__from(X) → cons(mark(X), from(s(X)))
mark(2nd(X)) → a__2nd(mark(X))
mark(from(X)) → a__from(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__2nd(X) → 2nd(X)
a__from(X) → from(X)

Types:
a__2nd :: cons:s:from:2nd → cons:s:from:2nd
cons :: cons:s:from:2nd → cons:s:from:2nd → cons:s:from:2nd
mark :: cons:s:from:2nd → cons:s:from:2nd
a__from :: cons:s:from:2nd → cons:s:from:2nd
from :: cons:s:from:2nd → cons:s:from:2nd
s :: cons:s:from:2nd → cons:s:from:2nd
2nd :: cons:s:from:2nd → cons:s:from:2nd
hole_cons:s:from:2nd1_0 :: cons:s:from:2nd
gen_cons:s:from:2nd2_0 :: Nat → cons:s:from:2nd

Generator Equations:
gen_cons:s:from:2nd2_0(0) ⇔ hole_cons:s:from:2nd1_0
gen_cons:s:from:2nd2_0(+(x, 1)) ⇔ cons(gen_cons:s:from:2nd2_0(x), hole_cons:s:from:2nd1_0)

No more defined symbols left to analyse.