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

int(0, 0) → .(0, nil)
int(0, s(y)) → .(0, int(s(0), s(y)))
int(s(x), 0) → nil
int(s(x), s(y)) → int_list(int(x, y))
int_list(nil) → nil
int_list(.(x, y)) → .(s(x), int_list(y))

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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


int'(0', 0') → .'(0', nil')
int'(0', s'(y)) → .'(0', int'(s'(0'), s'(y)))
int'(s'(x), 0') → nil'
int'(s'(x), s'(y)) → int_list'(int'(x, y))
int_list'(nil') → nil'
int_list'(.'(x, y)) → .'(s'(x), int_list'(y))

Rewrite Strategy: INNERMOST

Sliced the following arguments:
.'/0

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


int'(0', 0') → .'(nil')
int'(0', s'(y)) → .'(int'(s'(0'), s'(y)))
int'(s'(x), 0') → nil'
int'(s'(x), s'(y)) → int_list'(int'(x, y))
int_list'(nil') → nil'
int_list'(.'(y)) → .'(int_list'(y))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
int'(0', 0') → .'(nil')
int'(0', s'(y)) → .'(int'(s'(0'), s'(y)))
int'(s'(x), 0') → nil'
int'(s'(x), s'(y)) → int_list'(int'(x, y))
int_list'(nil') → nil'
int_list'(.'(y)) → .'(int_list'(y))

Types:
int' :: 0':s' → 0':s' → nil':.'
0' :: 0':s'
.' :: nil':.' → nil':.'
nil' :: nil':.'
s' :: 0':s' → 0':s'
int_list' :: nil':.' → nil':.'
_hole_nil':.'1 :: nil':.'
_hole_0':s'2 :: 0':s'
_gen_nil':.'3 :: Nat → nil':.'
_gen_0':s'4 :: Nat → 0':s'

Heuristically decided to analyse the following defined symbols:
int', int_list'

They will be analysed ascendingly in the following order:
int_list' < int'

Rules:
int'(0', 0') → .'(nil')
int'(0', s'(y)) → .'(int'(s'(0'), s'(y)))
int'(s'(x), 0') → nil'
int'(s'(x), s'(y)) → int_list'(int'(x, y))
int_list'(nil') → nil'
int_list'(.'(y)) → .'(int_list'(y))

Types:
int' :: 0':s' → 0':s' → nil':.'
0' :: 0':s'
.' :: nil':.' → nil':.'
nil' :: nil':.'
s' :: 0':s' → 0':s'
int_list' :: nil':.' → nil':.'
_hole_nil':.'1 :: nil':.'
_hole_0':s'2 :: 0':s'
_gen_nil':.'3 :: Nat → nil':.'
_gen_0':s'4 :: Nat → 0':s'

Generator Equations:
_gen_nil':.'3(0) ⇔ nil'
_gen_nil':.'3(+(x, 1)) ⇔ .'(_gen_nil':.'3(x))
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))

The following defined symbols remain to be analysed:
int_list', int'

They will be analysed ascendingly in the following order:
int_list' < int'

Proved the following rewrite lemma:
int_list'(_gen_nil':.'3(_n6)) → _gen_nil':.'3(_n6), rt ∈ Ω(1 + n6)

Induction Base:
int_list'(_gen_nil':.'3(0)) →RΩ(1)
nil'

Induction Step:
int_list'(_gen_nil':.'3(+(_$n7, 1))) →RΩ(1)
.'(int_list'(_gen_nil':.'3(_$n7))) →IH
.'(_gen_nil':.'3(_$n7))

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

Rules:
int'(0', 0') → .'(nil')
int'(0', s'(y)) → .'(int'(s'(0'), s'(y)))
int'(s'(x), 0') → nil'
int'(s'(x), s'(y)) → int_list'(int'(x, y))
int_list'(nil') → nil'
int_list'(.'(y)) → .'(int_list'(y))

Types:
int' :: 0':s' → 0':s' → nil':.'
0' :: 0':s'
.' :: nil':.' → nil':.'
nil' :: nil':.'
s' :: 0':s' → 0':s'
int_list' :: nil':.' → nil':.'
_hole_nil':.'1 :: nil':.'
_hole_0':s'2 :: 0':s'
_gen_nil':.'3 :: Nat → nil':.'
_gen_0':s'4 :: Nat → 0':s'

Lemmas:
int_list'(_gen_nil':.'3(_n6)) → _gen_nil':.'3(_n6), rt ∈ Ω(1 + n6)

Generator Equations:
_gen_nil':.'3(0) ⇔ nil'
_gen_nil':.'3(+(x, 1)) ⇔ .'(_gen_nil':.'3(x))
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))

The following defined symbols remain to be analysed:
int'

Proved the following rewrite lemma:
int'(_gen_0':s'4(_n216), _gen_0':s'4(_n216)) → _gen_nil':.'3(1), rt ∈ Ω(1 + n216)

Induction Base:
int'(_gen_0':s'4(0), _gen_0':s'4(0)) →RΩ(1)
.'(nil')

Induction Step:
int'(_gen_0':s'4(+(_$n217, 1)), _gen_0':s'4(+(_$n217, 1))) →RΩ(1)
int_list'(int'(_gen_0':s'4(_$n217), _gen_0':s'4(_$n217))) →IH
int_list'(_gen_nil':.'3(1)) →LΩ(2)
_gen_nil':.'3(1)

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

Rules:
int'(0', 0') → .'(nil')
int'(0', s'(y)) → .'(int'(s'(0'), s'(y)))
int'(s'(x), 0') → nil'
int'(s'(x), s'(y)) → int_list'(int'(x, y))
int_list'(nil') → nil'
int_list'(.'(y)) → .'(int_list'(y))

Types:
int' :: 0':s' → 0':s' → nil':.'
0' :: 0':s'
.' :: nil':.' → nil':.'
nil' :: nil':.'
s' :: 0':s' → 0':s'
int_list' :: nil':.' → nil':.'
_hole_nil':.'1 :: nil':.'
_hole_0':s'2 :: 0':s'
_gen_nil':.'3 :: Nat → nil':.'
_gen_0':s'4 :: Nat → 0':s'

Lemmas:
int_list'(_gen_nil':.'3(_n6)) → _gen_nil':.'3(_n6), rt ∈ Ω(1 + n6)
int'(_gen_0':s'4(_n216), _gen_0':s'4(_n216)) → _gen_nil':.'3(1), rt ∈ Ω(1 + n216)

Generator Equations:
_gen_nil':.'3(0) ⇔ nil'
_gen_nil':.'3(+(x, 1)) ⇔ .'(_gen_nil':.'3(x))
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
int_list'(_gen_nil':.'3(_n6)) → _gen_nil':.'3(_n6), rt ∈ Ω(1 + n6)