### (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 [ ].

### (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

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))) →RΩ(1)
true

Induction Step:
even(gen_0':s3_0(*(2, +(n5_0, 1)))) →RΩ(1)
even(gen_0':s3_0(*(2, n5_0))) →IH
true

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

### (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))) →RΩ(1)
0'

Induction Step:
half(gen_0':s3_0(*(2, +(n149_0, 1)))) →RΩ(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 ircR ∈ Ω(n).

### (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)) →RΩ(1)
gen_0':s3_0(b)

Induction Step:
plus(gen_0':s3_0(+(n355_0, 1)), gen_0':s3_0(b)) →RΩ(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 ircR ∈ Ω(n).

### (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)

### (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)

### (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)

### (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)