The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

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

le(s(x), 0) → false
le(0, y) → true
le(s(x), s(y)) → le(x, y)
double(0) → 0
double(s(x)) → s(s(double(x)))
log(0) → logError
log(s(x)) → loop(s(x), s(0), 0)
loop(x, s(y), z) → if(le(x, s(y)), x, s(y), z)
if(true, x, y, z) → z
if(false, x, y, z) → loop(x, double(y), s(z))

Rewrite Strategy: INNERMOST

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

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

le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
log(0') → logError
log(s(x)) → loop(s(x), s(0'), 0')
loop(x, s(y), z) → if(le(x, s(y)), x, s(y), z)
if(true, x, y, z) → z
if(false, x, y, z) → loop(x, double(y), s(z))

S is empty.
Rewrite Strategy: INNERMOST

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
log(0') → logError
log(s(x)) → loop(s(x), s(0'), 0')
loop(x, s(y), z) → if(le(x, s(y)), x, s(y), z)
if(true, x, y, z) → z
if(false, x, y, z) → loop(x, double(y), s(z))

Types:
le :: s:0':logError → s:0':logError → false:true
s :: s:0':logError → s:0':logError
0' :: s:0':logError
false :: false:true
true :: false:true
double :: s:0':logError → s:0':logError
log :: s:0':logError → s:0':logError
logError :: s:0':logError
loop :: s:0':logError → s:0':logError → s:0':logError → s:0':logError
if :: false:true → s:0':logError → s:0':logError → s:0':logError → s:0':logError
hole_false:true1_0 :: false:true
hole_s:0':logError2_0 :: s:0':logError
gen_s:0':logError3_0 :: Nat → s:0':logError

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
le, double, loop

They will be analysed ascendingly in the following order:
le < loop
double < loop

(6) Obligation:

Innermost TRS:
Rules:
le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
log(0') → logError
log(s(x)) → loop(s(x), s(0'), 0')
loop(x, s(y), z) → if(le(x, s(y)), x, s(y), z)
if(true, x, y, z) → z
if(false, x, y, z) → loop(x, double(y), s(z))

Types:
le :: s:0':logError → s:0':logError → false:true
s :: s:0':logError → s:0':logError
0' :: s:0':logError
false :: false:true
true :: false:true
double :: s:0':logError → s:0':logError
log :: s:0':logError → s:0':logError
logError :: s:0':logError
loop :: s:0':logError → s:0':logError → s:0':logError → s:0':logError
if :: false:true → s:0':logError → s:0':logError → s:0':logError → s:0':logError
hole_false:true1_0 :: false:true
hole_s:0':logError2_0 :: s:0':logError
gen_s:0':logError3_0 :: Nat → s:0':logError

Generator Equations:
gen_s:0':logError3_0(0) ⇔ 0'
gen_s:0':logError3_0(+(x, 1)) ⇔ s(gen_s:0':logError3_0(x))

The following defined symbols remain to be analysed:
le, double, loop

They will be analysed ascendingly in the following order:
le < loop
double < loop

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
le(gen_s:0':logError3_0(+(1, n5_0)), gen_s:0':logError3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)

Induction Base:
le(gen_s:0':logError3_0(+(1, 0)), gen_s:0':logError3_0(0)) →_R^Ω(1)
false

Induction Step:
le(gen_s:0':logError3_0(+(1, +(n5_0, 1))), gen_s:0':logError3_0(+(n5_0, 1))) →_R^Ω(1)
le(gen_s:0':logError3_0(+(1, n5_0)), gen_s:0':logError3_0(n5_0)) →_IH
false

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
log(0') → logError
log(s(x)) → loop(s(x), s(0'), 0')
loop(x, s(y), z) → if(le(x, s(y)), x, s(y), z)
if(true, x, y, z) → z
if(false, x, y, z) → loop(x, double(y), s(z))

Types:
le :: s:0':logError → s:0':logError → false:true
s :: s:0':logError → s:0':logError
0' :: s:0':logError
false :: false:true
true :: false:true
double :: s:0':logError → s:0':logError
log :: s:0':logError → s:0':logError
logError :: s:0':logError
loop :: s:0':logError → s:0':logError → s:0':logError → s:0':logError
if :: false:true → s:0':logError → s:0':logError → s:0':logError → s:0':logError
hole_false:true1_0 :: false:true
hole_s:0':logError2_0 :: s:0':logError
gen_s:0':logError3_0 :: Nat → s:0':logError

Lemmas:
le(gen_s:0':logError3_0(+(1, n5_0)), gen_s:0':logError3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_s:0':logError3_0(0) ⇔ 0'
gen_s:0':logError3_0(+(x, 1)) ⇔ s(gen_s:0':logError3_0(x))

The following defined symbols remain to be analysed:
double, loop

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

(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
double(gen_s:0':logError3_0(n318_0)) → gen_s:0':logError3_0(*(2, n318_0)), rt ∈ Ω(1 + n318₀)

Induction Base:
double(gen_s:0':logError3_0(0)) →_R^Ω(1)
0'

Induction Step:
double(gen_s:0':logError3_0(+(n318_0, 1))) →_R^Ω(1)
s(s(double(gen_s:0':logError3_0(n318_0)))) →_IH
s(s(gen_s:0':logError3_0(*(2, c319_0))))

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost TRS:
Rules:
le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
log(0') → logError
log(s(x)) → loop(s(x), s(0'), 0')
loop(x, s(y), z) → if(le(x, s(y)), x, s(y), z)
if(true, x, y, z) → z
if(false, x, y, z) → loop(x, double(y), s(z))

Types:
le :: s:0':logError → s:0':logError → false:true
s :: s:0':logError → s:0':logError
0' :: s:0':logError
false :: false:true
true :: false:true
double :: s:0':logError → s:0':logError
log :: s:0':logError → s:0':logError
logError :: s:0':logError
loop :: s:0':logError → s:0':logError → s:0':logError → s:0':logError
if :: false:true → s:0':logError → s:0':logError → s:0':logError → s:0':logError
hole_false:true1_0 :: false:true
hole_s:0':logError2_0 :: s:0':logError
gen_s:0':logError3_0 :: Nat → s:0':logError

Lemmas:
le(gen_s:0':logError3_0(+(1, n5_0)), gen_s:0':logError3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)
double(gen_s:0':logError3_0(n318_0)) → gen_s:0':logError3_0(*(2, n318_0)), rt ∈ Ω(1 + n318₀)

Generator Equations:
gen_s:0':logError3_0(0) ⇔ 0'
gen_s:0':logError3_0(+(x, 1)) ⇔ s(gen_s:0':logError3_0(x))

The following defined symbols remain to be analysed:
loop

(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol loop.

(14) Obligation:

Innermost TRS:
Rules:
le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
log(0') → logError
log(s(x)) → loop(s(x), s(0'), 0')
loop(x, s(y), z) → if(le(x, s(y)), x, s(y), z)
if(true, x, y, z) → z
if(false, x, y, z) → loop(x, double(y), s(z))

Types:
le :: s:0':logError → s:0':logError → false:true
s :: s:0':logError → s:0':logError
0' :: s:0':logError
false :: false:true
true :: false:true
double :: s:0':logError → s:0':logError
log :: s:0':logError → s:0':logError
logError :: s:0':logError
loop :: s:0':logError → s:0':logError → s:0':logError → s:0':logError
if :: false:true → s:0':logError → s:0':logError → s:0':logError → s:0':logError
hole_false:true1_0 :: false:true
hole_s:0':logError2_0 :: s:0':logError
gen_s:0':logError3_0 :: Nat → s:0':logError

Lemmas:
le(gen_s:0':logError3_0(+(1, n5_0)), gen_s:0':logError3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)
double(gen_s:0':logError3_0(n318_0)) → gen_s:0':logError3_0(*(2, n318_0)), rt ∈ Ω(1 + n318₀)

Generator Equations:
gen_s:0':logError3_0(0) ⇔ 0'
gen_s:0':logError3_0(+(x, 1)) ⇔ s(gen_s:0':logError3_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
le(gen_s:0':logError3_0(+(1, n5_0)), gen_s:0':logError3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)

(16) BOUNDS(n^1, INF)

(17) Obligation:

Innermost TRS:
Rules:
le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
log(0') → logError
log(s(x)) → loop(s(x), s(0'), 0')
loop(x, s(y), z) → if(le(x, s(y)), x, s(y), z)
if(true, x, y, z) → z
if(false, x, y, z) → loop(x, double(y), s(z))

Types:
le :: s:0':logError → s:0':logError → false:true
s :: s:0':logError → s:0':logError
0' :: s:0':logError
false :: false:true
true :: false:true
double :: s:0':logError → s:0':logError
log :: s:0':logError → s:0':logError
logError :: s:0':logError
loop :: s:0':logError → s:0':logError → s:0':logError → s:0':logError
if :: false:true → s:0':logError → s:0':logError → s:0':logError → s:0':logError
hole_false:true1_0 :: false:true
hole_s:0':logError2_0 :: s:0':logError
gen_s:0':logError3_0 :: Nat → s:0':logError

Lemmas:
le(gen_s:0':logError3_0(+(1, n5_0)), gen_s:0':logError3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)
double(gen_s:0':logError3_0(n318_0)) → gen_s:0':logError3_0(*(2, n318_0)), rt ∈ Ω(1 + n318₀)

Generator Equations:
gen_s:0':logError3_0(0) ⇔ 0'
gen_s:0':logError3_0(+(x, 1)) ⇔ s(gen_s:0':logError3_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
le(gen_s:0':logError3_0(+(1, n5_0)), gen_s:0':logError3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

Innermost TRS:
Rules:
le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
log(0') → logError
log(s(x)) → loop(s(x), s(0'), 0')
loop(x, s(y), z) → if(le(x, s(y)), x, s(y), z)
if(true, x, y, z) → z
if(false, x, y, z) → loop(x, double(y), s(z))

Types:
le :: s:0':logError → s:0':logError → false:true
s :: s:0':logError → s:0':logError
0' :: s:0':logError
false :: false:true
true :: false:true
double :: s:0':logError → s:0':logError
log :: s:0':logError → s:0':logError
logError :: s:0':logError
loop :: s:0':logError → s:0':logError → s:0':logError → s:0':logError
if :: false:true → s:0':logError → s:0':logError → s:0':logError → s:0':logError
hole_false:true1_0 :: false:true
hole_s:0':logError2_0 :: s:0':logError
gen_s:0':logError3_0 :: Nat → s:0':logError

Lemmas:
le(gen_s:0':logError3_0(+(1, n5_0)), gen_s:0':logError3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_s:0':logError3_0(0) ⇔ 0'
gen_s:0':logError3_0(+(x, 1)) ⇔ s(gen_s:0':logError3_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
le(gen_s:0':logError3_0(+(1, n5_0)), gen_s:0':logError3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)

(22) BOUNDS(n^1, INF)