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

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__half(0) → 0
a__half(s(0)) → 0
a__half(s(s(X))) → s(a__half(mark(X)))
a__half(dbl(X)) → mark(X)
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)
a__half(X) → half(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__terms'(N) → cons'(recip'(a__sqr'(mark'(N))), terms'(s'(N)))
a__sqr'(0') → 0'
a__sqr'(s'(X)) → s'(a__add'(a__sqr'(mark'(X)), a__dbl'(mark'(X))))
a__dbl'(0') → 0'
a__dbl'(s'(X)) → s'(s'(a__dbl'(mark'(X))))
a__add'(0', X) → mark'(X)
a__add'(s'(X), Y) → s'(a__add'(mark'(X), mark'(Y)))
a__first'(0', X) → nil'
a__first'(s'(X), cons'(Y, Z)) → cons'(mark'(Y), first'(X, Z))
a__half'(0') → 0'
a__half'(s'(0')) → 0'
a__half'(s'(s'(X))) → s'(a__half'(mark'(X)))
a__half'(dbl'(X)) → mark'(X)
mark'(terms'(X)) → a__terms'(mark'(X))
mark'(sqr'(X)) → a__sqr'(mark'(X))
mark'(add'(X1, X2)) → a__add'(mark'(X1), mark'(X2))
mark'(dbl'(X)) → a__dbl'(mark'(X))
mark'(first'(X1, X2)) → a__first'(mark'(X1), mark'(X2))
mark'(half'(X)) → a__half'(mark'(X))
mark'(cons'(X1, X2)) → cons'(mark'(X1), X2)
mark'(recip'(X)) → recip'(mark'(X))
mark'(s'(X)) → s'(mark'(X))
mark'(0') → 0'
mark'(nil') → nil'
a__terms'(X) → terms'(X)
a__sqr'(X) → sqr'(X)
a__add'(X1, X2) → add'(X1, X2)
a__dbl'(X) → dbl'(X)
a__first'(X1, X2) → first'(X1, X2)
a__half'(X) → half'(X)

Rewrite Strategy: INNERMOST

Sliced the following arguments:
cons'/1

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

a__terms'(N) → cons'(recip'(a__sqr'(mark'(N))))
a__sqr'(0') → 0'
a__sqr'(s'(X)) → s'(a__add'(a__sqr'(mark'(X)), a__dbl'(mark'(X))))
a__dbl'(0') → 0'
a__dbl'(s'(X)) → s'(s'(a__dbl'(mark'(X))))
a__add'(0', X) → mark'(X)
a__add'(s'(X), Y) → s'(a__add'(mark'(X), mark'(Y)))
a__first'(0', X) → nil'
a__first'(s'(X), cons'(Y)) → cons'(mark'(Y))
a__half'(0') → 0'
a__half'(s'(0')) → 0'
a__half'(s'(s'(X))) → s'(a__half'(mark'(X)))
a__half'(dbl'(X)) → mark'(X)
mark'(terms'(X)) → a__terms'(mark'(X))
mark'(sqr'(X)) → a__sqr'(mark'(X))
mark'(add'(X1, X2)) → a__add'(mark'(X1), mark'(X2))
mark'(dbl'(X)) → a__dbl'(mark'(X))
mark'(first'(X1, X2)) → a__first'(mark'(X1), mark'(X2))
mark'(half'(X)) → a__half'(mark'(X))
mark'(cons'(X1)) → cons'(mark'(X1))
mark'(recip'(X)) → recip'(mark'(X))
mark'(s'(X)) → s'(mark'(X))
mark'(0') → 0'
mark'(nil') → nil'
a__terms'(X) → terms'(X)
a__sqr'(X) → sqr'(X)
a__add'(X1, X2) → add'(X1, X2)
a__dbl'(X) → dbl'(X)
a__first'(X1, X2) → first'(X1, X2)
a__half'(X) → half'(X)

Rewrite Strategy: INNERMOST

Infered types.

Rules:
a__terms'(N) → cons'(recip'(a__sqr'(mark'(N))))
a__sqr'(0') → 0'
a__sqr'(s'(X)) → s'(a__add'(a__sqr'(mark'(X)), a__dbl'(mark'(X))))
a__dbl'(0') → 0'
a__dbl'(s'(X)) → s'(s'(a__dbl'(mark'(X))))
a__add'(0', X) → mark'(X)
a__add'(s'(X), Y) → s'(a__add'(mark'(X), mark'(Y)))
a__first'(0', X) → nil'
a__first'(s'(X), cons'(Y)) → cons'(mark'(Y))
a__half'(0') → 0'
a__half'(s'(0')) → 0'
a__half'(s'(s'(X))) → s'(a__half'(mark'(X)))
a__half'(dbl'(X)) → mark'(X)
mark'(terms'(X)) → a__terms'(mark'(X))
mark'(sqr'(X)) → a__sqr'(mark'(X))
mark'(add'(X1, X2)) → a__add'(mark'(X1), mark'(X2))
mark'(dbl'(X)) → a__dbl'(mark'(X))
mark'(first'(X1, X2)) → a__first'(mark'(X1), mark'(X2))
mark'(half'(X)) → a__half'(mark'(X))
mark'(cons'(X1)) → cons'(mark'(X1))
mark'(recip'(X)) → recip'(mark'(X))
mark'(s'(X)) → s'(mark'(X))
mark'(0') → 0'
mark'(nil') → nil'
a__terms'(X) → terms'(X)
a__sqr'(X) → sqr'(X)
a__add'(X1, X2) → add'(X1, X2)
a__dbl'(X) → dbl'(X)
a__first'(X1, X2) → first'(X1, X2)
a__half'(X) → half'(X)

Types:
a__terms' :: recip':cons':0':s':nil':dbl':terms':sqr':add':first':half' → recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'
cons' :: recip':cons':0':s':nil':dbl':terms':sqr':add':first':half' → recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'
recip' :: recip':cons':0':s':nil':dbl':terms':sqr':add':first':half' → recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'
a__sqr' :: recip':cons':0':s':nil':dbl':terms':sqr':add':first':half' → recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'
mark' :: recip':cons':0':s':nil':dbl':terms':sqr':add':first':half' → recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'
0' :: recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'
s' :: recip':cons':0':s':nil':dbl':terms':sqr':add':first':half' → recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'
a__add' :: recip':cons':0':s':nil':dbl':terms':sqr':add':first':half' → recip':cons':0':s':nil':dbl':terms':sqr':add':first':half' → recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'
a__dbl' :: recip':cons':0':s':nil':dbl':terms':sqr':add':first':half' → recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'
a__first' :: recip':cons':0':s':nil':dbl':terms':sqr':add':first':half' → recip':cons':0':s':nil':dbl':terms':sqr':add':first':half' → recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'
nil' :: recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'
a__half' :: recip':cons':0':s':nil':dbl':terms':sqr':add':first':half' → recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'
dbl' :: recip':cons':0':s':nil':dbl':terms':sqr':add':first':half' → recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'
terms' :: recip':cons':0':s':nil':dbl':terms':sqr':add':first':half' → recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'
sqr' :: recip':cons':0':s':nil':dbl':terms':sqr':add':first':half' → recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'
add' :: recip':cons':0':s':nil':dbl':terms':sqr':add':first':half' → recip':cons':0':s':nil':dbl':terms':sqr':add':first':half' → recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'
first' :: recip':cons':0':s':nil':dbl':terms':sqr':add':first':half' → recip':cons':0':s':nil':dbl':terms':sqr':add':first':half' → recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'
half' :: recip':cons':0':s':nil':dbl':terms':sqr':add':first':half' → recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'
_hole_recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'1 :: recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'
_gen_recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'2 :: Nat → recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'

Heuristically decided to analyse the following defined symbols:
a__terms', a__sqr', mark', a__add', a__dbl', a__half'

They will be analysed ascendingly in the following order:
a__terms' = a__sqr'
a__terms' = mark'
a__terms' = a__add'
a__terms' = a__dbl'
a__terms' = a__half'
a__sqr' = mark'
a__sqr' = a__add'
a__sqr' = a__dbl'
a__sqr' = a__half'
mark' = a__add'
mark' = a__dbl'
mark' = a__half'
a__add' = a__dbl'
a__add' = a__half'
a__dbl' = a__half'

Generator Equations:
_gen_recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'2(0) ⇔ 0'
_gen_recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'2(+(x, 1)) ⇔ cons'(_gen_recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'2(x))

The following defined symbols remain to be analysed:
a__sqr', a__terms', mark', a__add', a__dbl', a__half'

Could not prove a rewrite lemma for the defined symbol a__sqr'.

The following defined symbols remain to be analysed:
a__add', a__terms', mark', a__dbl', a__half'

Could not prove a rewrite lemma for the defined symbol a__add'.

The following defined symbols remain to be analysed:
mark', a__terms', a__dbl', a__half'

Proved the following rewrite lemma:
mark'(_gen_recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'2(_n41408)) → _gen_recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'2(_n41408), rt ∈ Ω(1 + _n41408)

Induction Base:
mark'(_gen_recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'2(0)) →_R^Ω(1)
0'

Induction Step:
mark'(_gen_recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'2(+(_$n41409, 1))) →_R^Ω(1)
cons'(mark'(_gen_recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'2(_$n41409))) →_IH
cons'(_gen_recip':cons':0':s':nil':dbl':terms':sqr':add':first':half'2(_$n41409))

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