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

le(0, y, z) → greater(y, z)
le(s(x), 0, z) → false
le(s(x), s(y), 0) → false
le(s(x), s(y), s(z)) → le(x, y, z)
greater(x, 0) → first
greater(0, s(y)) → second
greater(s(x), s(y)) → greater(x, y)
double(0) → 0
double(s(x)) → s(s(double(x)))
triple(x) → if(le(x, x, double(x)), x, 0, 0)
if(false, x, y, z) → true
if(first, x, y, z) → if(le(s(x), y, s(z)), s(x), y, s(z))
if(second, x, y, z) → if(le(s(x), s(y), z), s(x), s(y), z)

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

le'(0', y, z) → greater'(y, z)
le'(s'(x), 0', z) → false'
le'(s'(x), s'(y), 0') → false'
le'(s'(x), s'(y), s'(z)) → le'(x, y, z)
greater'(x, 0') → first'
greater'(0', s'(y)) → second'
greater'(s'(x), s'(y)) → greater'(x, y)
double'(0') → 0'
double'(s'(x)) → s'(s'(double'(x)))
triple'(x) → if'(le'(x, x, double'(x)), x, 0', 0')
if'(false', x, y, z) → true'
if'(first', x, y, z) → if'(le'(s'(x), y, s'(z)), s'(x), y, s'(z))
if'(second', x, y, z) → if'(le'(s'(x), s'(y), z), s'(x), s'(y), z)

Rewrite Strategy: INNERMOST

Infered types.

Rules:
le'(0', y, z) → greater'(y, z)
le'(s'(x), 0', z) → false'
le'(s'(x), s'(y), 0') → false'
le'(s'(x), s'(y), s'(z)) → le'(x, y, z)
greater'(x, 0') → first'
greater'(0', s'(y)) → second'
greater'(s'(x), s'(y)) → greater'(x, y)
double'(0') → 0'
double'(s'(x)) → s'(s'(double'(x)))
triple'(x) → if'(le'(x, x, double'(x)), x, 0', 0')
if'(false', x, y, z) → true'
if'(first', x, y, z) → if'(le'(s'(x), y, s'(z)), s'(x), y, s'(z))
if'(second', x, y, z) → if'(le'(s'(x), s'(y), z), s'(x), s'(y), z)

Types:
le' :: 0':s' → 0':s' → 0':s' → false':first':second'
0' :: 0':s'
greater' :: 0':s' → 0':s' → false':first':second'
s' :: 0':s' → 0':s'
false' :: false':first':second'
first' :: false':first':second'
second' :: false':first':second'
double' :: 0':s' → 0':s'
triple' :: 0':s' → true'
if' :: false':first':second' → 0':s' → 0':s' → 0':s' → true'
true' :: true'
_hole_false':first':second'1 :: false':first':second'
_hole_0':s'2 :: 0':s'
_hole_true'3 :: true'
_gen_0':s'4 :: Nat → 0':s'

Heuristically decided to analyse the following defined symbols:
le', greater', double', if'

They will be analysed ascendingly in the following order:
greater' < le'
le' < if'

Rules:
le'(0', y, z) → greater'(y, z)
le'(s'(x), 0', z) → false'
le'(s'(x), s'(y), 0') → false'
le'(s'(x), s'(y), s'(z)) → le'(x, y, z)
greater'(x, 0') → first'
greater'(0', s'(y)) → second'
greater'(s'(x), s'(y)) → greater'(x, y)
double'(0') → 0'
double'(s'(x)) → s'(s'(double'(x)))
triple'(x) → if'(le'(x, x, double'(x)), x, 0', 0')
if'(false', x, y, z) → true'
if'(first', x, y, z) → if'(le'(s'(x), y, s'(z)), s'(x), y, s'(z))
if'(second', x, y, z) → if'(le'(s'(x), s'(y), z), s'(x), s'(y), z)

Types:
le' :: 0':s' → 0':s' → 0':s' → false':first':second'
0' :: 0':s'
greater' :: 0':s' → 0':s' → false':first':second'
s' :: 0':s' → 0':s'
false' :: false':first':second'
first' :: false':first':second'
second' :: false':first':second'
double' :: 0':s' → 0':s'
triple' :: 0':s' → true'
if' :: false':first':second' → 0':s' → 0':s' → 0':s' → true'
true' :: true'
_hole_false':first':second'1 :: false':first':second'
_hole_0':s'2 :: 0':s'
_hole_true'3 :: true'
_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:
greater', le', double', if'

They will be analysed ascendingly in the following order:
greater' < le'
le' < if'

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

Induction Base:
greater'(_gen_0':s'4(0), _gen_0':s'4(0)) →RΩ(1)
first'

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

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

Rules:
le'(0', y, z) → greater'(y, z)
le'(s'(x), 0', z) → false'
le'(s'(x), s'(y), 0') → false'
le'(s'(x), s'(y), s'(z)) → le'(x, y, z)
greater'(x, 0') → first'
greater'(0', s'(y)) → second'
greater'(s'(x), s'(y)) → greater'(x, y)
double'(0') → 0'
double'(s'(x)) → s'(s'(double'(x)))
triple'(x) → if'(le'(x, x, double'(x)), x, 0', 0')
if'(false', x, y, z) → true'
if'(first', x, y, z) → if'(le'(s'(x), y, s'(z)), s'(x), y, s'(z))
if'(second', x, y, z) → if'(le'(s'(x), s'(y), z), s'(x), s'(y), z)

Types:
le' :: 0':s' → 0':s' → 0':s' → false':first':second'
0' :: 0':s'
greater' :: 0':s' → 0':s' → false':first':second'
s' :: 0':s' → 0':s'
false' :: false':first':second'
first' :: false':first':second'
second' :: false':first':second'
double' :: 0':s' → 0':s'
triple' :: 0':s' → true'
if' :: false':first':second' → 0':s' → 0':s' → 0':s' → true'
true' :: true'
_hole_false':first':second'1 :: false':first':second'
_hole_0':s'2 :: 0':s'
_hole_true'3 :: true'
_gen_0':s'4 :: Nat → 0':s'

Lemmas:
greater'(_gen_0':s'4(_n6), _gen_0':s'4(_n6)) → first', 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:
le', double', if'

They will be analysed ascendingly in the following order:
le' < if'

Proved the following rewrite lemma:
le'(_gen_0':s'4(+(1, _n711)), _gen_0':s'4(_n711), _gen_0':s'4(_n711)) → false', rt ∈ Ω(1 + n711)

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

Induction Step:
le'(_gen_0':s'4(+(1, +(_\$n712, 1))), _gen_0':s'4(+(_\$n712, 1)), _gen_0':s'4(+(_\$n712, 1))) →RΩ(1)
le'(_gen_0':s'4(+(1, _\$n712)), _gen_0':s'4(_\$n712), _gen_0':s'4(_\$n712)) →IH
false'

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

Rules:
le'(0', y, z) → greater'(y, z)
le'(s'(x), 0', z) → false'
le'(s'(x), s'(y), 0') → false'
le'(s'(x), s'(y), s'(z)) → le'(x, y, z)
greater'(x, 0') → first'
greater'(0', s'(y)) → second'
greater'(s'(x), s'(y)) → greater'(x, y)
double'(0') → 0'
double'(s'(x)) → s'(s'(double'(x)))
triple'(x) → if'(le'(x, x, double'(x)), x, 0', 0')
if'(false', x, y, z) → true'
if'(first', x, y, z) → if'(le'(s'(x), y, s'(z)), s'(x), y, s'(z))
if'(second', x, y, z) → if'(le'(s'(x), s'(y), z), s'(x), s'(y), z)

Types:
le' :: 0':s' → 0':s' → 0':s' → false':first':second'
0' :: 0':s'
greater' :: 0':s' → 0':s' → false':first':second'
s' :: 0':s' → 0':s'
false' :: false':first':second'
first' :: false':first':second'
second' :: false':first':second'
double' :: 0':s' → 0':s'
triple' :: 0':s' → true'
if' :: false':first':second' → 0':s' → 0':s' → 0':s' → true'
true' :: true'
_hole_false':first':second'1 :: false':first':second'
_hole_0':s'2 :: 0':s'
_hole_true'3 :: true'
_gen_0':s'4 :: Nat → 0':s'

Lemmas:
greater'(_gen_0':s'4(_n6), _gen_0':s'4(_n6)) → first', rt ∈ Ω(1 + n6)
le'(_gen_0':s'4(+(1, _n711)), _gen_0':s'4(_n711), _gen_0':s'4(_n711)) → false', rt ∈ Ω(1 + n711)

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', if'

Proved the following rewrite lemma:
double'(_gen_0':s'4(_n2094)) → _gen_0':s'4(*(2, _n2094)), rt ∈ Ω(1 + n2094)

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

Induction Step:
double'(_gen_0':s'4(+(_\$n2095, 1))) →RΩ(1)
s'(s'(double'(_gen_0':s'4(_\$n2095)))) →IH
s'(s'(_gen_0':s'4(*(2, _\$n2095))))

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

Rules:
le'(0', y, z) → greater'(y, z)
le'(s'(x), 0', z) → false'
le'(s'(x), s'(y), 0') → false'
le'(s'(x), s'(y), s'(z)) → le'(x, y, z)
greater'(x, 0') → first'
greater'(0', s'(y)) → second'
greater'(s'(x), s'(y)) → greater'(x, y)
double'(0') → 0'
double'(s'(x)) → s'(s'(double'(x)))
triple'(x) → if'(le'(x, x, double'(x)), x, 0', 0')
if'(false', x, y, z) → true'
if'(first', x, y, z) → if'(le'(s'(x), y, s'(z)), s'(x), y, s'(z))
if'(second', x, y, z) → if'(le'(s'(x), s'(y), z), s'(x), s'(y), z)

Types:
le' :: 0':s' → 0':s' → 0':s' → false':first':second'
0' :: 0':s'
greater' :: 0':s' → 0':s' → false':first':second'
s' :: 0':s' → 0':s'
false' :: false':first':second'
first' :: false':first':second'
second' :: false':first':second'
double' :: 0':s' → 0':s'
triple' :: 0':s' → true'
if' :: false':first':second' → 0':s' → 0':s' → 0':s' → true'
true' :: true'
_hole_false':first':second'1 :: false':first':second'
_hole_0':s'2 :: 0':s'
_hole_true'3 :: true'
_gen_0':s'4 :: Nat → 0':s'

Lemmas:
greater'(_gen_0':s'4(_n6), _gen_0':s'4(_n6)) → first', rt ∈ Ω(1 + n6)
le'(_gen_0':s'4(+(1, _n711)), _gen_0':s'4(_n711), _gen_0':s'4(_n711)) → false', rt ∈ Ω(1 + n711)
double'(_gen_0':s'4(_n2094)) → _gen_0':s'4(*(2, _n2094)), rt ∈ Ω(1 + n2094)

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:
if'

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

Rules:
le'(0', y, z) → greater'(y, z)
le'(s'(x), 0', z) → false'
le'(s'(x), s'(y), 0') → false'
le'(s'(x), s'(y), s'(z)) → le'(x, y, z)
greater'(x, 0') → first'
greater'(0', s'(y)) → second'
greater'(s'(x), s'(y)) → greater'(x, y)
double'(0') → 0'
double'(s'(x)) → s'(s'(double'(x)))
triple'(x) → if'(le'(x, x, double'(x)), x, 0', 0')
if'(false', x, y, z) → true'
if'(first', x, y, z) → if'(le'(s'(x), y, s'(z)), s'(x), y, s'(z))
if'(second', x, y, z) → if'(le'(s'(x), s'(y), z), s'(x), s'(y), z)

Types:
le' :: 0':s' → 0':s' → 0':s' → false':first':second'
0' :: 0':s'
greater' :: 0':s' → 0':s' → false':first':second'
s' :: 0':s' → 0':s'
false' :: false':first':second'
first' :: false':first':second'
second' :: false':first':second'
double' :: 0':s' → 0':s'
triple' :: 0':s' → true'
if' :: false':first':second' → 0':s' → 0':s' → 0':s' → true'
true' :: true'
_hole_false':first':second'1 :: false':first':second'
_hole_0':s'2 :: 0':s'
_hole_true'3 :: true'
_gen_0':s'4 :: Nat → 0':s'

Lemmas:
greater'(_gen_0':s'4(_n6), _gen_0':s'4(_n6)) → first', rt ∈ Ω(1 + n6)
le'(_gen_0':s'4(+(1, _n711)), _gen_0':s'4(_n711), _gen_0':s'4(_n711)) → false', rt ∈ Ω(1 + n711)
double'(_gen_0':s'4(_n2094)) → _gen_0':s'4(*(2, _n2094)), rt ∈ Ω(1 + n2094)

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:
greater'(_gen_0':s'4(_n6), _gen_0':s'4(_n6)) → first', rt ∈ Ω(1 + n6)