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

f(true, x, y) → f(and(gt(x, y), gt(y, s(s(0)))), plus(s(0), x), double(y))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(u, v)
and(x, true) → x
and(x, false) → false
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
double(0) → 0
double(s(x)) → s(s(double(x)))

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

f'(true', x, y) → f'(and'(gt'(x, y), gt'(y, s'(s'(0')))), plus'(s'(0'), x), double'(y))
gt'(0', v) → false'
gt'(s'(u), 0') → true'
gt'(s'(u), s'(v)) → gt'(u, v)
and'(x, true') → x
and'(x, false') → false'
plus'(n, 0') → n
plus'(n, s'(m)) → s'(plus'(n, m))
double'(0') → 0'
double'(s'(x)) → s'(s'(double'(x)))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
f'(true', x, y) → f'(and'(gt'(x, y), gt'(y, s'(s'(0')))), plus'(s'(0'), x), double'(y))
gt'(0', v) → false'
gt'(s'(u), 0') → true'
gt'(s'(u), s'(v)) → gt'(u, v)
and'(x, true') → x
and'(x, false') → false'
plus'(n, 0') → n
plus'(n, s'(m)) → s'(plus'(n, m))
double'(0') → 0'
double'(s'(x)) → s'(s'(double'(x)))

Types:
f' :: true':false' → 0':s' → 0':s' → f'
true' :: true':false'
and' :: true':false' → true':false' → true':false'
gt' :: 0':s' → 0':s' → true':false'
s' :: 0':s' → 0':s'
0' :: 0':s'
plus' :: 0':s' → 0':s' → 0':s'
double' :: 0':s' → 0':s'
false' :: true':false'
_hole_f'1 :: f'
_hole_true':false'2 :: true':false'
_hole_0':s'3 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'

Heuristically decided to analyse the following defined symbols:
f', gt', plus', double'

They will be analysed ascendingly in the following order:
gt' < f'
plus' < f'
double' < f'

Rules:
f'(true', x, y) → f'(and'(gt'(x, y), gt'(y, s'(s'(0')))), plus'(s'(0'), x), double'(y))
gt'(0', v) → false'
gt'(s'(u), 0') → true'
gt'(s'(u), s'(v)) → gt'(u, v)
and'(x, true') → x
and'(x, false') → false'
plus'(n, 0') → n
plus'(n, s'(m)) → s'(plus'(n, m))
double'(0') → 0'
double'(s'(x)) → s'(s'(double'(x)))

Types:
f' :: true':false' → 0':s' → 0':s' → f'
true' :: true':false'
and' :: true':false' → true':false' → true':false'
gt' :: 0':s' → 0':s' → true':false'
s' :: 0':s' → 0':s'
0' :: 0':s'
plus' :: 0':s' → 0':s' → 0':s'
double' :: 0':s' → 0':s'
false' :: true':false'
_hole_f'1 :: f'
_hole_true':false'2 :: true':false'
_hole_0':s'3 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'

Generator Equations:
_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:
gt', f', plus', double'

They will be analysed ascendingly in the following order:
gt' < f'
plus' < f'
double' < f'

Proved the following rewrite lemma:
gt'(_gen_0':s'4(_n6), _gen_0':s'4(_n6)) → false', rt ∈ Ω(1 + n6)

Induction Base:
gt'(_gen_0':s'4(0), _gen_0':s'4(0)) →RΩ(1)
false'

Induction Step:
gt'(_gen_0':s'4(+(_\$n7, 1)), _gen_0':s'4(+(_\$n7, 1))) →RΩ(1)
gt'(_gen_0':s'4(_\$n7), _gen_0':s'4(_\$n7)) →IH
false'

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

Rules:
f'(true', x, y) → f'(and'(gt'(x, y), gt'(y, s'(s'(0')))), plus'(s'(0'), x), double'(y))
gt'(0', v) → false'
gt'(s'(u), 0') → true'
gt'(s'(u), s'(v)) → gt'(u, v)
and'(x, true') → x
and'(x, false') → false'
plus'(n, 0') → n
plus'(n, s'(m)) → s'(plus'(n, m))
double'(0') → 0'
double'(s'(x)) → s'(s'(double'(x)))

Types:
f' :: true':false' → 0':s' → 0':s' → f'
true' :: true':false'
and' :: true':false' → true':false' → true':false'
gt' :: 0':s' → 0':s' → true':false'
s' :: 0':s' → 0':s'
0' :: 0':s'
plus' :: 0':s' → 0':s' → 0':s'
double' :: 0':s' → 0':s'
false' :: true':false'
_hole_f'1 :: f'
_hole_true':false'2 :: true':false'
_hole_0':s'3 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'

Lemmas:
gt'(_gen_0':s'4(_n6), _gen_0':s'4(_n6)) → false', rt ∈ Ω(1 + n6)

Generator Equations:
_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:
plus', f', double'

They will be analysed ascendingly in the following order:
plus' < f'
double' < f'

Proved the following rewrite lemma:
plus'(_gen_0':s'4(a), _gen_0':s'4(_n453)) → _gen_0':s'4(+(_n453, a)), rt ∈ Ω(1 + n453)

Induction Base:
plus'(_gen_0':s'4(a), _gen_0':s'4(0)) →RΩ(1)
_gen_0':s'4(a)

Induction Step:
plus'(_gen_0':s'4(_a586), _gen_0':s'4(+(_\$n454, 1))) →RΩ(1)
s'(plus'(_gen_0':s'4(_a586), _gen_0':s'4(_\$n454))) →IH
s'(_gen_0':s'4(+(_\$n454, _a586)))

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

Rules:
f'(true', x, y) → f'(and'(gt'(x, y), gt'(y, s'(s'(0')))), plus'(s'(0'), x), double'(y))
gt'(0', v) → false'
gt'(s'(u), 0') → true'
gt'(s'(u), s'(v)) → gt'(u, v)
and'(x, true') → x
and'(x, false') → false'
plus'(n, 0') → n
plus'(n, s'(m)) → s'(plus'(n, m))
double'(0') → 0'
double'(s'(x)) → s'(s'(double'(x)))

Types:
f' :: true':false' → 0':s' → 0':s' → f'
true' :: true':false'
and' :: true':false' → true':false' → true':false'
gt' :: 0':s' → 0':s' → true':false'
s' :: 0':s' → 0':s'
0' :: 0':s'
plus' :: 0':s' → 0':s' → 0':s'
double' :: 0':s' → 0':s'
false' :: true':false'
_hole_f'1 :: f'
_hole_true':false'2 :: true':false'
_hole_0':s'3 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'

Lemmas:
gt'(_gen_0':s'4(_n6), _gen_0':s'4(_n6)) → false', rt ∈ Ω(1 + n6)
plus'(_gen_0':s'4(a), _gen_0':s'4(_n453)) → _gen_0':s'4(+(_n453, a)), rt ∈ Ω(1 + n453)

Generator Equations:
_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:
double', f'

They will be analysed ascendingly in the following order:
double' < f'

Proved the following rewrite lemma:
double'(_gen_0':s'4(_n1059)) → _gen_0':s'4(*(2, _n1059)), rt ∈ Ω(1 + n1059)

Induction Base:
double'(_gen_0':s'4(0)) →RΩ(1)
0'

Induction Step:
double'(_gen_0':s'4(+(_\$n1060, 1))) →RΩ(1)
s'(s'(double'(_gen_0':s'4(_\$n1060)))) →IH
s'(s'(_gen_0':s'4(*(2, _\$n1060))))

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

Rules:
f'(true', x, y) → f'(and'(gt'(x, y), gt'(y, s'(s'(0')))), plus'(s'(0'), x), double'(y))
gt'(0', v) → false'
gt'(s'(u), 0') → true'
gt'(s'(u), s'(v)) → gt'(u, v)
and'(x, true') → x
and'(x, false') → false'
plus'(n, 0') → n
plus'(n, s'(m)) → s'(plus'(n, m))
double'(0') → 0'
double'(s'(x)) → s'(s'(double'(x)))

Types:
f' :: true':false' → 0':s' → 0':s' → f'
true' :: true':false'
and' :: true':false' → true':false' → true':false'
gt' :: 0':s' → 0':s' → true':false'
s' :: 0':s' → 0':s'
0' :: 0':s'
plus' :: 0':s' → 0':s' → 0':s'
double' :: 0':s' → 0':s'
false' :: true':false'
_hole_f'1 :: f'
_hole_true':false'2 :: true':false'
_hole_0':s'3 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'

Lemmas:
gt'(_gen_0':s'4(_n6), _gen_0':s'4(_n6)) → false', rt ∈ Ω(1 + n6)
plus'(_gen_0':s'4(a), _gen_0':s'4(_n453)) → _gen_0':s'4(+(_n453, a)), rt ∈ Ω(1 + n453)
double'(_gen_0':s'4(_n1059)) → _gen_0':s'4(*(2, _n1059)), rt ∈ Ω(1 + n1059)

Generator Equations:
_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:
f'

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

Rules:
f'(true', x, y) → f'(and'(gt'(x, y), gt'(y, s'(s'(0')))), plus'(s'(0'), x), double'(y))
gt'(0', v) → false'
gt'(s'(u), 0') → true'
gt'(s'(u), s'(v)) → gt'(u, v)
and'(x, true') → x
and'(x, false') → false'
plus'(n, 0') → n
plus'(n, s'(m)) → s'(plus'(n, m))
double'(0') → 0'
double'(s'(x)) → s'(s'(double'(x)))

Types:
f' :: true':false' → 0':s' → 0':s' → f'
true' :: true':false'
and' :: true':false' → true':false' → true':false'
gt' :: 0':s' → 0':s' → true':false'
s' :: 0':s' → 0':s'
0' :: 0':s'
plus' :: 0':s' → 0':s' → 0':s'
double' :: 0':s' → 0':s'
false' :: true':false'
_hole_f'1 :: f'
_hole_true':false'2 :: true':false'
_hole_0':s'3 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'

Lemmas:
gt'(_gen_0':s'4(_n6), _gen_0':s'4(_n6)) → false', rt ∈ Ω(1 + n6)
plus'(_gen_0':s'4(a), _gen_0':s'4(_n453)) → _gen_0':s'4(+(_n453, a)), rt ∈ Ω(1 + n453)
double'(_gen_0':s'4(_n1059)) → _gen_0':s'4(*(2, _n1059)), rt ∈ Ω(1 + n1059)

Generator Equations:
_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:
gt'(_gen_0':s'4(_n6), _gen_0':s'4(_n6)) → false', rt ∈ Ω(1 + n6)