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

a__and(true, X) → mark(X)
a__and(false, Y) → false
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
a__from(X) → cons(X, from(s(X)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(X)
mark(true) → true
mark(false) → false
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

a__and'(true', X) → mark'(X)
a__and'(false', Y) → false'
a__if'(true', X, Y) → mark'(X)
a__if'(false', X, Y) → mark'(Y)
a__first'(0', X) → nil'
a__first'(s'(X), cons'(Y, Z)) → cons'(Y, first'(X, Z))
a__from'(X) → cons'(X, from'(s'(X)))
mark'(and'(X1, X2)) → a__and'(mark'(X1), X2)
mark'(if'(X1, X2, X3)) → a__if'(mark'(X1), X2, X3)
mark'(first'(X1, X2)) → a__first'(mark'(X1), mark'(X2))
mark'(from'(X)) → a__from'(X)
mark'(true') → true'
mark'(false') → false'
mark'(0') → 0'
mark'(s'(X)) → s'(X)
mark'(nil') → nil'
mark'(cons'(X1, X2)) → cons'(X1, X2)
a__and'(X1, X2) → and'(X1, X2)
a__if'(X1, X2, X3) → if'(X1, X2, X3)
a__first'(X1, X2) → first'(X1, X2)
a__from'(X) → from'(X)

Rewrite Strategy: INNERMOST

Sliced the following arguments:
s'/0
cons'/0
cons'/1
a__from'/0
from'/0

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

a__and'(true', X) → mark'(X)
a__and'(false', Y) → false'
a__if'(true', X, Y) → mark'(X)
a__if'(false', X, Y) → mark'(Y)
a__first'(0', X) → nil'
a__first'(s', cons') → cons'
a__from'cons'
mark'(and'(X1, X2)) → a__and'(mark'(X1), X2)
mark'(if'(X1, X2, X3)) → a__if'(mark'(X1), X2, X3)
mark'(first'(X1, X2)) → a__first'(mark'(X1), mark'(X2))
mark'(from') → a__from'
mark'(true') → true'
mark'(false') → false'
mark'(0') → 0'
mark'(s') → s'
mark'(nil') → nil'
mark'(cons') → cons'
a__and'(X1, X2) → and'(X1, X2)
a__if'(X1, X2, X3) → if'(X1, X2, X3)
a__first'(X1, X2) → first'(X1, X2)
a__from'from'

Rewrite Strategy: INNERMOST

Infered types.

Rules:
a__and'(true', X) → mark'(X)
a__and'(false', Y) → false'
a__if'(true', X, Y) → mark'(X)
a__if'(false', X, Y) → mark'(Y)
a__first'(0', X) → nil'
a__first'(s', cons') → cons'
a__from'cons'
mark'(and'(X1, X2)) → a__and'(mark'(X1), X2)
mark'(if'(X1, X2, X3)) → a__if'(mark'(X1), X2, X3)
mark'(first'(X1, X2)) → a__first'(mark'(X1), mark'(X2))
mark'(from') → a__from'
mark'(true') → true'
mark'(false') → false'
mark'(0') → 0'
mark'(s') → s'
mark'(nil') → nil'
mark'(cons') → cons'
a__and'(X1, X2) → and'(X1, X2)
a__if'(X1, X2, X3) → if'(X1, X2, X3)
a__first'(X1, X2) → first'(X1, X2)
a__from'from'

Types:

Heuristically decided to analyse the following defined symbols:
mark'

Rules:
a__and'(true', X) → mark'(X)
a__and'(false', Y) → false'
a__if'(true', X, Y) → mark'(X)
a__if'(false', X, Y) → mark'(Y)
a__first'(0', X) → nil'
a__first'(s', cons') → cons'
a__from'cons'
mark'(and'(X1, X2)) → a__and'(mark'(X1), X2)
mark'(if'(X1, X2, X3)) → a__if'(mark'(X1), X2, X3)
mark'(first'(X1, X2)) → a__first'(mark'(X1), mark'(X2))
mark'(from') → a__from'
mark'(true') → true'
mark'(false') → false'
mark'(0') → 0'
mark'(s') → s'
mark'(nil') → nil'
mark'(cons') → cons'
a__and'(X1, X2) → and'(X1, X2)
a__if'(X1, X2, X3) → if'(X1, X2, X3)
a__first'(X1, X2) → first'(X1, X2)
a__from'from'

Types:

Generator Equations:

The following defined symbols remain to be analysed:
mark'

Proved the following rewrite lemma:

Induction Base:
true'

Induction Step:
mark'(true') →RΩ(1)
true'

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

Rules:
a__and'(true', X) → mark'(X)
a__and'(false', Y) → false'
a__if'(true', X, Y) → mark'(X)
a__if'(false', X, Y) → mark'(Y)
a__first'(0', X) → nil'
a__first'(s', cons') → cons'
a__from'cons'
mark'(and'(X1, X2)) → a__and'(mark'(X1), X2)
mark'(if'(X1, X2, X3)) → a__if'(mark'(X1), X2, X3)
mark'(first'(X1, X2)) → a__first'(mark'(X1), mark'(X2))
mark'(from') → a__from'
mark'(true') → true'
mark'(false') → false'
mark'(0') → 0'
mark'(s') → s'
mark'(nil') → nil'
mark'(cons') → cons'
a__and'(X1, X2) → and'(X1, X2)
a__if'(X1, X2, X3) → if'(X1, X2, X3)
a__first'(X1, X2) → first'(X1, X2)
a__from'from'

Types: