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:

even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
half(0) → 0
half(s(s(x))) → s(half(x))
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → if_times(even(s(x)), s(x), y)
if_times(true, s(x), y) → plus(times(half(s(x)), y), times(half(s(x)), y))
if_times(false, s(x), y) → plus(y, times(x, y))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

even(0') → true
even(s(0')) → false
even(s(s(x))) → even(x)
half(0') → 0'
half(s(s(x))) → s(half(x))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → if_times(even(s(x)), s(x), y)
if_times(true, s(x), y) → plus(times(half(s(x)), y), times(half(s(x)), y))
if_times(false, s(x), y) → plus(y, times(x, y))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
even(0') → true
even(s(0')) → false
even(s(s(x))) → even(x)
half(0') → 0'
half(s(s(x))) → s(half(x))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → if_times(even(s(x)), s(x), y)
if_times(true, s(x), y) → plus(times(half(s(x)), y), times(half(s(x)), y))
if_times(false, s(x), y) → plus(y, times(x, y))

Types:
even :: 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
half :: 0':s → 0':s
plus :: 0':s → 0':s → 0':s
times :: 0':s → 0':s → 0':s
if_times :: true:false → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
even, half, plus, times

They will be analysed ascendingly in the following order:
even < times
half < times
plus < times

(8) Obligation:

TRS:
Rules:
even(0') → true
even(s(0')) → false
even(s(s(x))) → even(x)
half(0') → 0'
half(s(s(x))) → s(half(x))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → if_times(even(s(x)), s(x), y)
if_times(true, s(x), y) → plus(times(half(s(x)), y), times(half(s(x)), y))
if_times(false, s(x), y) → plus(y, times(x, y))

Types:
even :: 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
half :: 0':s → 0':s
plus :: 0':s → 0':s → 0':s
times :: 0':s → 0':s → 0':s
if_times :: true:false → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
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:
even, half, plus, times

They will be analysed ascendingly in the following order:
even < times
half < times
plus < times

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

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

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

Induction Step:
even(gen_0':s3_0(*(2, +(n5_0, 1)))) →_R^Ω(1)
even(gen_0':s3_0(*(2, n5_0))) →_IH
true

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

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
even(0') → true
even(s(0')) → false
even(s(s(x))) → even(x)
half(0') → 0'
half(s(s(x))) → s(half(x))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → if_times(even(s(x)), s(x), y)
if_times(true, s(x), y) → plus(times(half(s(x)), y), times(half(s(x)), y))
if_times(false, s(x), y) → plus(y, times(x, y))

Types:
even :: 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
half :: 0':s → 0':s
plus :: 0':s → 0':s → 0':s
times :: 0':s → 0':s → 0':s
if_times :: true:false → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
even(gen_0':s3_0(*(2, 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:
half, plus, times

They will be analysed ascendingly in the following order:
half < times
plus < times

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

Proved the following rewrite lemma:
half(gen_0':s3_0(*(2, n149_0))) → gen_0':s3_0(n149_0), rt ∈ Ω(1 + n149₀)

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

Induction Step:
half(gen_0':s3_0(*(2, +(n149_0, 1)))) →_R^Ω(1)
s(half(gen_0':s3_0(*(2, n149_0)))) →_IH
s(gen_0':s3_0(c150_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:
even(0') → true
even(s(0')) → false
even(s(s(x))) → even(x)
half(0') → 0'
half(s(s(x))) → s(half(x))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → if_times(even(s(x)), s(x), y)
if_times(true, s(x), y) → plus(times(half(s(x)), y), times(half(s(x)), y))
if_times(false, s(x), y) → plus(y, times(x, y))

Types:
even :: 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
half :: 0':s → 0':s
plus :: 0':s → 0':s → 0':s
times :: 0':s → 0':s → 0':s
if_times :: true:false → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
even(gen_0':s3_0(*(2, n5_0))) → true, rt ∈ Ω(1 + n5₀)
half(gen_0':s3_0(*(2, n149_0))) → gen_0':s3_0(n149_0), rt ∈ Ω(1 + n149₀)

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:
plus, times

They will be analysed ascendingly in the following order:
plus < times

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

Proved the following rewrite lemma:
plus(gen_0':s3_0(n355_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n355_0, b)), rt ∈ Ω(1 + n355₀)

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

Induction Step:
plus(gen_0':s3_0(+(n355_0, 1)), gen_0':s3_0(b)) →_R^Ω(1)
s(plus(gen_0':s3_0(n355_0), gen_0':s3_0(b))) →_IH
s(gen_0':s3_0(+(b, c356_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:
even(0') → true
even(s(0')) → false
even(s(s(x))) → even(x)
half(0') → 0'
half(s(s(x))) → s(half(x))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → if_times(even(s(x)), s(x), y)
if_times(true, s(x), y) → plus(times(half(s(x)), y), times(half(s(x)), y))
if_times(false, s(x), y) → plus(y, times(x, y))

Types:
even :: 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
half :: 0':s → 0':s
plus :: 0':s → 0':s → 0':s
times :: 0':s → 0':s → 0':s
if_times :: true:false → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
even(gen_0':s3_0(*(2, n5_0))) → true, rt ∈ Ω(1 + n5₀)
half(gen_0':s3_0(*(2, n149_0))) → gen_0':s3_0(n149_0), rt ∈ Ω(1 + n149₀)
plus(gen_0':s3_0(n355_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n355_0, b)), rt ∈ Ω(1 + n355₀)

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:
times

(18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(19) Obligation:

TRS:
Rules:
even(0') → true
even(s(0')) → false
even(s(s(x))) → even(x)
half(0') → 0'
half(s(s(x))) → s(half(x))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → if_times(even(s(x)), s(x), y)
if_times(true, s(x), y) → plus(times(half(s(x)), y), times(half(s(x)), y))
if_times(false, s(x), y) → plus(y, times(x, y))

Types:
even :: 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
half :: 0':s → 0':s
plus :: 0':s → 0':s → 0':s
times :: 0':s → 0':s → 0':s
if_times :: true:false → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
even(gen_0':s3_0(*(2, n5_0))) → true, rt ∈ Ω(1 + n5₀)
half(gen_0':s3_0(*(2, n149_0))) → gen_0':s3_0(n149_0), rt ∈ Ω(1 + n149₀)
plus(gen_0':s3_0(n355_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n355_0, b)), rt ∈ Ω(1 + n355₀)

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.

(20) LowerBoundsProof (EQUIVALENT transformation)

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

(21) BOUNDS(n^1, INF)

(22) Obligation:

TRS:
Rules:
even(0') → true
even(s(0')) → false
even(s(s(x))) → even(x)
half(0') → 0'
half(s(s(x))) → s(half(x))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → if_times(even(s(x)), s(x), y)
if_times(true, s(x), y) → plus(times(half(s(x)), y), times(half(s(x)), y))
if_times(false, s(x), y) → plus(y, times(x, y))

Types:
even :: 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
half :: 0':s → 0':s
plus :: 0':s → 0':s → 0':s
times :: 0':s → 0':s → 0':s
if_times :: true:false → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
even(gen_0':s3_0(*(2, n5_0))) → true, rt ∈ Ω(1 + n5₀)
half(gen_0':s3_0(*(2, n149_0))) → gen_0':s3_0(n149_0), rt ∈ Ω(1 + n149₀)
plus(gen_0':s3_0(n355_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n355_0, b)), rt ∈ Ω(1 + n355₀)

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.

(23) LowerBoundsProof (EQUIVALENT transformation)

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

(24) BOUNDS(n^1, INF)

(25) Obligation:

TRS:
Rules:
even(0') → true
even(s(0')) → false
even(s(s(x))) → even(x)
half(0') → 0'
half(s(s(x))) → s(half(x))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → if_times(even(s(x)), s(x), y)
if_times(true, s(x), y) → plus(times(half(s(x)), y), times(half(s(x)), y))
if_times(false, s(x), y) → plus(y, times(x, y))

Types:
even :: 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
half :: 0':s → 0':s
plus :: 0':s → 0':s → 0':s
times :: 0':s → 0':s → 0':s
if_times :: true:false → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
even(gen_0':s3_0(*(2, n5_0))) → true, rt ∈ Ω(1 + n5₀)
half(gen_0':s3_0(*(2, n149_0))) → gen_0':s3_0(n149_0), rt ∈ Ω(1 + n149₀)

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.

(26) LowerBoundsProof (EQUIVALENT transformation)

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

(27) BOUNDS(n^1, INF)

(28) Obligation:

TRS:
Rules:
even(0') → true
even(s(0')) → false
even(s(s(x))) → even(x)
half(0') → 0'
half(s(s(x))) → s(half(x))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → if_times(even(s(x)), s(x), y)
if_times(true, s(x), y) → plus(times(half(s(x)), y), times(half(s(x)), y))
if_times(false, s(x), y) → plus(y, times(x, y))

Types:
even :: 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
half :: 0':s → 0':s
plus :: 0':s → 0':s → 0':s
times :: 0':s → 0':s → 0':s
if_times :: true:false → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
even(gen_0':s3_0(*(2, 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.

(29) LowerBoundsProof (EQUIVALENT transformation)

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

(30) BOUNDS(n^1, INF)