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:

1024 → 1024_1(0)
1024_1(x) → if(lt(x, 10), x)
if(true, x) → double(1024_1(s(x)))
if(false, x) → s(0)
lt(0, s(y)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
double(0) → 0
double(s(x)) → s(s(double(x)))
10 → double(s(double(s(s(0)))))

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:

1024' → 1024_1(0')
1024_1(x) → if(lt(x, 10'), x)
if(true, x) → double(1024_1(s(x)))
if(false, x) → s(0')
lt(0', s(y)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
10' → double(s(double(s(s(0')))))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
1024' → 1024_1(0')
1024_1(x) → if(lt(x, 10'), x)
if(true, x) → double(1024_1(s(x)))
if(false, x) → s(0')
lt(0', s(y)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
10' → double(s(double(s(s(0')))))

Types:
1024' :: 0':s
1024_1 :: 0':s → 0':s
0' :: 0':s
if :: true:false → 0':s → 0':s
lt :: 0':s → 0':s → true:false
10' :: 0':s
true :: true:false
double :: 0':s → 0':s
s :: 0':s → 0':s
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
1024_1, lt, double

They will be analysed ascendingly in the following order:
lt < 1024_1
double < 1024_1

(6) Obligation:

Innermost TRS:
Rules:
1024' → 1024_1(0')
1024_1(x) → if(lt(x, 10'), x)
if(true, x) → double(1024_1(s(x)))
if(false, x) → s(0')
lt(0', s(y)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
10' → double(s(double(s(s(0')))))

Types:
1024' :: 0':s
1024_1 :: 0':s → 0':s
0' :: 0':s
if :: true:false → 0':s → 0':s
lt :: 0':s → 0':s → true:false
10' :: 0':s
true :: true:false
double :: 0':s → 0':s
s :: 0':s → 0':s
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
lt, 1024_1, double

They will be analysed ascendingly in the following order:
lt < 1024_1
double < 1024_1

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

Proved the following rewrite lemma:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)

Induction Base:
lt(gen_0':s3_0(0), gen_0':s3_0(+(1, 0))) →_R^Ω(1)
true

Induction Step:
lt(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(1, +(n5_0, 1)))) →_R^Ω(1)
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) →_IH
true

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:
1024' → 1024_1(0')
1024_1(x) → if(lt(x, 10'), x)
if(true, x) → double(1024_1(s(x)))
if(false, x) → s(0')
lt(0', s(y)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
10' → double(s(double(s(s(0')))))

Types:
1024' :: 0':s
1024_1 :: 0':s → 0':s
0' :: 0':s
if :: true:false → 0':s → 0':s
lt :: 0':s → 0':s → true:false
10' :: 0':s
true :: true:false
double :: 0':s → 0':s
s :: 0':s → 0':s
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
double, 1024_1

They will be analysed ascendingly in the following order:
double < 1024_1

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

Proved the following rewrite lemma:
double(gen_0':s3_0(n318_0)) → gen_0':s3_0(*(2, n318_0)), rt ∈ Ω(1 + n318₀)

Induction Base:
double(gen_0':s3_0(0)) →_R^Ω(1)
0'

Induction Step:
double(gen_0':s3_0(+(n318_0, 1))) →_R^Ω(1)
s(s(double(gen_0':s3_0(n318_0)))) →_IH
s(s(gen_0':s3_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:
1024' → 1024_1(0')
1024_1(x) → if(lt(x, 10'), x)
if(true, x) → double(1024_1(s(x)))
if(false, x) → s(0')
lt(0', s(y)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
10' → double(s(double(s(s(0')))))

Types:
1024' :: 0':s
1024_1 :: 0':s → 0':s
0' :: 0':s
if :: true:false → 0':s → 0':s
lt :: 0':s → 0':s → true:false
10' :: 0':s
true :: true:false
double :: 0':s → 0':s
s :: 0':s → 0':s
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)
double(gen_0':s3_0(n318_0)) → gen_0':s3_0(*(2, n318_0)), rt ∈ Ω(1 + n318₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
1024_1

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

Could not prove a rewrite lemma for the defined symbol 1024_1.

(14) Obligation:

Innermost TRS:
Rules:
1024' → 1024_1(0')
1024_1(x) → if(lt(x, 10'), x)
if(true, x) → double(1024_1(s(x)))
if(false, x) → s(0')
lt(0', s(y)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
10' → double(s(double(s(s(0')))))

Types:
1024' :: 0':s
1024_1 :: 0':s → 0':s
0' :: 0':s
if :: true:false → 0':s → 0':s
lt :: 0':s → 0':s → true:false
10' :: 0':s
true :: true:false
double :: 0':s → 0':s
s :: 0':s → 0':s
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)
double(gen_0':s3_0(n318_0)) → gen_0':s3_0(*(2, n318_0)), rt ∈ Ω(1 + n318₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)

(16) BOUNDS(n^1, INF)

(17) Obligation:

Innermost TRS:
Rules:
1024' → 1024_1(0')
1024_1(x) → if(lt(x, 10'), x)
if(true, x) → double(1024_1(s(x)))
if(false, x) → s(0')
lt(0', s(y)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
10' → double(s(double(s(s(0')))))

Types:
1024' :: 0':s
1024_1 :: 0':s → 0':s
0' :: 0':s
if :: true:false → 0':s → 0':s
lt :: 0':s → 0':s → true:false
10' :: 0':s
true :: true:false
double :: 0':s → 0':s
s :: 0':s → 0':s
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)
double(gen_0':s3_0(n318_0)) → gen_0':s3_0(*(2, n318_0)), rt ∈ Ω(1 + n318₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

Innermost TRS:
Rules:
1024' → 1024_1(0')
1024_1(x) → if(lt(x, 10'), x)
if(true, x) → double(1024_1(s(x)))
if(false, x) → s(0')
lt(0', s(y)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
10' → double(s(double(s(s(0')))))

Types:
1024' :: 0':s
1024_1 :: 0':s → 0':s
0' :: 0':s
if :: true:false → 0':s → 0':s
lt :: 0':s → 0':s → true:false
10' :: 0':s
true :: true:false
double :: 0':s → 0':s
s :: 0':s → 0':s
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)

(22) BOUNDS(n^1, INF)