The Runtime Complexity (full) 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:

double(0) → 0
double(s(x)) → s(s(double(x)))
half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
if(0, y, z) → y
if(s(x), y, z) → z
half(double(x)) → x

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
double(s(x)) →⁺ s(s(double(x)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [x / s(x)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

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

double(0') → 0'
double(s(x)) → s(s(double(x)))
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
if(0', y, z) → y
if(s(x), y, z) → z
half(double(x)) → x

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
double(0') → 0'
double(s(x)) → s(s(double(x)))
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
if(0', y, z) → y
if(s(x), y, z) → z
half(double(x)) → x

Types:
double :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
half :: 0':s → 0':s
- :: 0':s → 0':s → 0':s
if :: 0':s → if → if → if
hole_0':s1_0 :: 0':s
hole_if2_0 :: if
gen_0':s3_0 :: Nat → 0':s

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
double, half, -

(8) Obligation:

TRS:
Rules:
double(0') → 0'
double(s(x)) → s(s(double(x)))
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
if(0', y, z) → y
if(s(x), y, z) → z
half(double(x)) → x

Types:
double :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
half :: 0':s → 0':s
- :: 0':s → 0':s → 0':s
if :: 0':s → if → if → if
hole_0':s1_0 :: 0':s
hole_if2_0 :: if
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:
double, half, -

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
double(gen_0':s3_0(n5_0)) → gen_0':s3_0(*(2, n5_0)), rt ∈ Ω(1 + n5₀)

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

Induction Step:
double(gen_0':s3_0(+(n5_0, 1))) →_R^Ω(1)
s(s(double(gen_0':s3_0(n5_0)))) →_IH
s(s(gen_0':s3_0(*(2, c6_0))))

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

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
double(0') → 0'
double(s(x)) → s(s(double(x)))
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
if(0', y, z) → y
if(s(x), y, z) → z
half(double(x)) → x

Types:
double :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
half :: 0':s → 0':s
- :: 0':s → 0':s → 0':s
if :: 0':s → if → if → if
hole_0':s1_0 :: 0':s
hole_if2_0 :: if
gen_0':s3_0 :: Nat → 0':s

Lemmas:
double(gen_0':s3_0(n5_0)) → gen_0':s3_0(*(2, n5_0)), 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:
half, -

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
half(gen_0':s3_0(*(2, n249_0))) → gen_0':s3_0(n249_0), rt ∈ Ω(1 + n249₀)

Induction Base:
half(gen_0':s3_0(*(2, 0))) →_R^Ω(1)
0'

Induction Step:
half(gen_0':s3_0(*(2, +(n249_0, 1)))) →_R^Ω(1)
s(half(gen_0':s3_0(*(2, n249_0)))) →_IH
s(gen_0':s3_0(c250_0))

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

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
Rules:
double(0') → 0'
double(s(x)) → s(s(double(x)))
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
if(0', y, z) → y
if(s(x), y, z) → z
half(double(x)) → x

Types:
double :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
half :: 0':s → 0':s
- :: 0':s → 0':s → 0':s
if :: 0':s → if → if → if
hole_0':s1_0 :: 0':s
hole_if2_0 :: if
gen_0':s3_0 :: Nat → 0':s

Lemmas:
double(gen_0':s3_0(n5_0)) → gen_0':s3_0(*(2, n5_0)), rt ∈ Ω(1 + n5₀)
half(gen_0':s3_0(*(2, n249_0))) → gen_0':s3_0(n249_0), rt ∈ Ω(1 + n249₀)

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:
-

(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
-(gen_0':s3_0(n676_0), gen_0':s3_0(n676_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n676₀)

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

Induction Step:
-(gen_0':s3_0(+(n676_0, 1)), gen_0':s3_0(+(n676_0, 1))) →_R^Ω(1)
-(gen_0':s3_0(n676_0), gen_0':s3_0(n676_0)) →_IH
gen_0':s3_0(0)

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

(16) Complex Obligation (BEST)

(17) Obligation:

TRS:
Rules:
double(0') → 0'
double(s(x)) → s(s(double(x)))
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
if(0', y, z) → y
if(s(x), y, z) → z
half(double(x)) → x

Types:
double :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
half :: 0':s → 0':s
- :: 0':s → 0':s → 0':s
if :: 0':s → if → if → if
hole_0':s1_0 :: 0':s
hole_if2_0 :: if
gen_0':s3_0 :: Nat → 0':s

Lemmas:
double(gen_0':s3_0(n5_0)) → gen_0':s3_0(*(2, n5_0)), rt ∈ Ω(1 + n5₀)
half(gen_0':s3_0(*(2, n249_0))) → gen_0':s3_0(n249_0), rt ∈ Ω(1 + n249₀)
-(gen_0':s3_0(n676_0), gen_0':s3_0(n676_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n676₀)

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:
double(gen_0':s3_0(n5_0)) → gen_0':s3_0(*(2, n5_0)), rt ∈ Ω(1 + n5₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

TRS:
Rules:
double(0') → 0'
double(s(x)) → s(s(double(x)))
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
if(0', y, z) → y
if(s(x), y, z) → z
half(double(x)) → x

Types:
double :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
half :: 0':s → 0':s
- :: 0':s → 0':s → 0':s
if :: 0':s → if → if → if
hole_0':s1_0 :: 0':s
hole_if2_0 :: if
gen_0':s3_0 :: Nat → 0':s

Lemmas:
double(gen_0':s3_0(n5_0)) → gen_0':s3_0(*(2, n5_0)), rt ∈ Ω(1 + n5₀)
half(gen_0':s3_0(*(2, n249_0))) → gen_0':s3_0(n249_0), rt ∈ Ω(1 + n249₀)
-(gen_0':s3_0(n676_0), gen_0':s3_0(n676_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n676₀)

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:
double(gen_0':s3_0(n5_0)) → gen_0':s3_0(*(2, n5_0)), rt ∈ Ω(1 + n5₀)

(22) BOUNDS(n^1, INF)

(23) Obligation:

TRS:
Rules:
double(0') → 0'
double(s(x)) → s(s(double(x)))
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
if(0', y, z) → y
if(s(x), y, z) → z
half(double(x)) → x

Types:
double :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
half :: 0':s → 0':s
- :: 0':s → 0':s → 0':s
if :: 0':s → if → if → if
hole_0':s1_0 :: 0':s
hole_if2_0 :: if
gen_0':s3_0 :: Nat → 0':s

Lemmas:
double(gen_0':s3_0(n5_0)) → gen_0':s3_0(*(2, n5_0)), rt ∈ Ω(1 + n5₀)
half(gen_0':s3_0(*(2, n249_0))) → gen_0':s3_0(n249_0), rt ∈ Ω(1 + n249₀)

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.

(24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
double(gen_0':s3_0(n5_0)) → gen_0':s3_0(*(2, n5_0)), rt ∈ Ω(1 + n5₀)

(25) BOUNDS(n^1, INF)

(26) Obligation:

TRS:
Rules:
double(0') → 0'
double(s(x)) → s(s(double(x)))
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
if(0', y, z) → y
if(s(x), y, z) → z
half(double(x)) → x

Types:
double :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
half :: 0':s → 0':s
- :: 0':s → 0':s → 0':s
if :: 0':s → if → if → if
hole_0':s1_0 :: 0':s
hole_if2_0 :: if
gen_0':s3_0 :: Nat → 0':s

Lemmas:
double(gen_0':s3_0(n5_0)) → gen_0':s3_0(*(2, n5_0)), 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.

(27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
double(gen_0':s3_0(n5_0)) → gen_0':s3_0(*(2, n5_0)), rt ∈ Ω(1 + n5₀)

(28) BOUNDS(n^1, INF)