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

minus(0, x) → 0
minus(s(x), 0) → s(x)
minus(s(x), s(y)) → minus(x, y)
mod(x, 0) → 0
mod(x, s(y)) → if(lt(x, s(y)), x, s(y))
if(true, x, y) → x
if(false, x, y) → mod(minus(x, y), y)
gcd(x, 0) → x
gcd(0, s(y)) → s(y)
gcd(s(x), s(y)) → gcd(mod(s(x), s(y)), mod(s(y), s(x)))
lt(x, 0) → false
lt(0, s(x)) → true
lt(s(x), s(y)) → lt(x, y)

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

minus'(0', x) → 0'
minus'(s'(x), 0') → s'(x)
minus'(s'(x), s'(y)) → minus'(x, y)
mod'(x, 0') → 0'
mod'(x, s'(y)) → if'(lt'(x, s'(y)), x, s'(y))
if'(true', x, y) → x
if'(false', x, y) → mod'(minus'(x, y), y)
gcd'(x, 0') → x
gcd'(0', s'(y)) → s'(y)
gcd'(s'(x), s'(y)) → gcd'(mod'(s'(x), s'(y)), mod'(s'(y), s'(x)))
lt'(x, 0') → false'
lt'(0', s'(x)) → true'
lt'(s'(x), s'(y)) → lt'(x, y)

Rewrite Strategy: INNERMOST

Infered types.

Rules:
minus'(0', x) → 0'
minus'(s'(x), 0') → s'(x)
minus'(s'(x), s'(y)) → minus'(x, y)
mod'(x, 0') → 0'
mod'(x, s'(y)) → if'(lt'(x, s'(y)), x, s'(y))
if'(true', x, y) → x
if'(false', x, y) → mod'(minus'(x, y), y)
gcd'(x, 0') → x
gcd'(0', s'(y)) → s'(y)
gcd'(s'(x), s'(y)) → gcd'(mod'(s'(x), s'(y)), mod'(s'(y), s'(x)))
lt'(x, 0') → false'
lt'(0', s'(x)) → true'
lt'(s'(x), s'(y)) → lt'(x, y)

Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
mod' :: 0':s' → 0':s' → 0':s'
if' :: true':false' → 0':s' → 0':s' → 0':s'
lt' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
gcd' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_gen_0':s'3 :: Nat → 0':s'

Heuristically decided to analyse the following defined symbols:
minus', mod', lt', gcd'

They will be analysed ascendingly in the following order:
minus' < mod'
lt' < mod'
mod' < gcd'

Rules:
minus'(0', x) → 0'
minus'(s'(x), 0') → s'(x)
minus'(s'(x), s'(y)) → minus'(x, y)
mod'(x, 0') → 0'
mod'(x, s'(y)) → if'(lt'(x, s'(y)), x, s'(y))
if'(true', x, y) → x
if'(false', x, y) → mod'(minus'(x, y), y)
gcd'(x, 0') → x
gcd'(0', s'(y)) → s'(y)
gcd'(s'(x), s'(y)) → gcd'(mod'(s'(x), s'(y)), mod'(s'(y), s'(x)))
lt'(x, 0') → false'
lt'(0', s'(x)) → true'
lt'(s'(x), s'(y)) → lt'(x, y)

Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
mod' :: 0':s' → 0':s' → 0':s'
if' :: true':false' → 0':s' → 0':s' → 0':s'
lt' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
gcd' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_gen_0':s'3 :: Nat → 0':s'

Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))

The following defined symbols remain to be analysed:
minus', mod', lt', gcd'

They will be analysed ascendingly in the following order:
minus' < mod'
lt' < mod'
mod' < gcd'

Proved the following rewrite lemma:
minus'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → _gen_0':s'3(0), rt ∈ Ω(1 + n5)

Induction Base:
minus'(_gen_0':s'3(0), _gen_0':s'3(0)) →RΩ(1)
0'

Induction Step:
minus'(_gen_0':s'3(+(_\$n6, 1)), _gen_0':s'3(+(_\$n6, 1))) →RΩ(1)
minus'(_gen_0':s'3(_\$n6), _gen_0':s'3(_\$n6)) →IH
_gen_0':s'3(0)

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

Rules:
minus'(0', x) → 0'
minus'(s'(x), 0') → s'(x)
minus'(s'(x), s'(y)) → minus'(x, y)
mod'(x, 0') → 0'
mod'(x, s'(y)) → if'(lt'(x, s'(y)), x, s'(y))
if'(true', x, y) → x
if'(false', x, y) → mod'(minus'(x, y), y)
gcd'(x, 0') → x
gcd'(0', s'(y)) → s'(y)
gcd'(s'(x), s'(y)) → gcd'(mod'(s'(x), s'(y)), mod'(s'(y), s'(x)))
lt'(x, 0') → false'
lt'(0', s'(x)) → true'
lt'(s'(x), s'(y)) → lt'(x, y)

Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
mod' :: 0':s' → 0':s' → 0':s'
if' :: true':false' → 0':s' → 0':s' → 0':s'
lt' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
gcd' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_gen_0':s'3 :: Nat → 0':s'

Lemmas:
minus'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → _gen_0':s'3(0), rt ∈ Ω(1 + n5)

Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))

The following defined symbols remain to be analysed:
lt', mod', gcd'

They will be analysed ascendingly in the following order:
lt' < mod'
mod' < gcd'

Proved the following rewrite lemma:
lt'(_gen_0':s'3(_n871), _gen_0':s'3(_n871)) → false', rt ∈ Ω(1 + n871)

Induction Base:
lt'(_gen_0':s'3(0), _gen_0':s'3(0)) →RΩ(1)
false'

Induction Step:
lt'(_gen_0':s'3(+(_\$n872, 1)), _gen_0':s'3(+(_\$n872, 1))) →RΩ(1)
lt'(_gen_0':s'3(_\$n872), _gen_0':s'3(_\$n872)) →IH
false'

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

Rules:
minus'(0', x) → 0'
minus'(s'(x), 0') → s'(x)
minus'(s'(x), s'(y)) → minus'(x, y)
mod'(x, 0') → 0'
mod'(x, s'(y)) → if'(lt'(x, s'(y)), x, s'(y))
if'(true', x, y) → x
if'(false', x, y) → mod'(minus'(x, y), y)
gcd'(x, 0') → x
gcd'(0', s'(y)) → s'(y)
gcd'(s'(x), s'(y)) → gcd'(mod'(s'(x), s'(y)), mod'(s'(y), s'(x)))
lt'(x, 0') → false'
lt'(0', s'(x)) → true'
lt'(s'(x), s'(y)) → lt'(x, y)

Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
mod' :: 0':s' → 0':s' → 0':s'
if' :: true':false' → 0':s' → 0':s' → 0':s'
lt' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
gcd' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_gen_0':s'3 :: Nat → 0':s'

Lemmas:
minus'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → _gen_0':s'3(0), rt ∈ Ω(1 + n5)
lt'(_gen_0':s'3(_n871), _gen_0':s'3(_n871)) → false', rt ∈ Ω(1 + n871)

Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))

The following defined symbols remain to be analysed:
mod', gcd'

They will be analysed ascendingly in the following order:
mod' < gcd'

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

Rules:
minus'(0', x) → 0'
minus'(s'(x), 0') → s'(x)
minus'(s'(x), s'(y)) → minus'(x, y)
mod'(x, 0') → 0'
mod'(x, s'(y)) → if'(lt'(x, s'(y)), x, s'(y))
if'(true', x, y) → x
if'(false', x, y) → mod'(minus'(x, y), y)
gcd'(x, 0') → x
gcd'(0', s'(y)) → s'(y)
gcd'(s'(x), s'(y)) → gcd'(mod'(s'(x), s'(y)), mod'(s'(y), s'(x)))
lt'(x, 0') → false'
lt'(0', s'(x)) → true'
lt'(s'(x), s'(y)) → lt'(x, y)

Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
mod' :: 0':s' → 0':s' → 0':s'
if' :: true':false' → 0':s' → 0':s' → 0':s'
lt' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
gcd' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_gen_0':s'3 :: Nat → 0':s'

Lemmas:
minus'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → _gen_0':s'3(0), rt ∈ Ω(1 + n5)
lt'(_gen_0':s'3(_n871), _gen_0':s'3(_n871)) → false', rt ∈ Ω(1 + n871)

Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))

The following defined symbols remain to be analysed:
gcd'

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

Rules:
minus'(0', x) → 0'
minus'(s'(x), 0') → s'(x)
minus'(s'(x), s'(y)) → minus'(x, y)
mod'(x, 0') → 0'
mod'(x, s'(y)) → if'(lt'(x, s'(y)), x, s'(y))
if'(true', x, y) → x
if'(false', x, y) → mod'(minus'(x, y), y)
gcd'(x, 0') → x
gcd'(0', s'(y)) → s'(y)
gcd'(s'(x), s'(y)) → gcd'(mod'(s'(x), s'(y)), mod'(s'(y), s'(x)))
lt'(x, 0') → false'
lt'(0', s'(x)) → true'
lt'(s'(x), s'(y)) → lt'(x, y)

Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
mod' :: 0':s' → 0':s' → 0':s'
if' :: true':false' → 0':s' → 0':s' → 0':s'
lt' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
gcd' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_gen_0':s'3 :: Nat → 0':s'

Lemmas:
minus'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → _gen_0':s'3(0), rt ∈ Ω(1 + n5)
lt'(_gen_0':s'3(_n871), _gen_0':s'3(_n871)) → false', rt ∈ Ω(1 + n871)

Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
minus'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → _gen_0':s'3(0), rt ∈ Ω(1 + n5)