### (0) Obligation:

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

check(0) → zero
check(s(0)) → odd
check(s(s(0))) → even
check(s(s(s(x)))) → check(s(x))
half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → timesIter(x, y, 0)
timesIter(x, y, z) → if(check(x), x, y, z, plus(z, y))
p(s(x)) → x
p(0) → 0
if(zero, x, y, z, u) → z
if(odd, x, y, z, u) → timesIter(p(x), y, u)
if(even, x, y, z, u) → plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z))))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
check(s(s(s(x)))) →+ check(s(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:

check(0') → zero
check(s(0')) → odd
check(s(s(0'))) → even
check(s(s(s(x)))) → check(s(x))
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → timesIter(x, y, 0')
timesIter(x, y, z) → if(check(x), x, y, z, plus(z, y))
p(s(x)) → x
p(0') → 0'
if(zero, x, y, z, u) → z
if(odd, x, y, z, u) → timesIter(p(x), y, u)
if(even, x, y, z, u) → plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z))))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
check(0') → zero
check(s(0')) → odd
check(s(s(0'))) → even
check(s(s(s(x)))) → check(s(x))
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → timesIter(x, y, 0')
timesIter(x, y, z) → if(check(x), x, y, z, plus(z, y))
p(s(x)) → x
p(0') → 0'
if(zero, x, y, z, u) → z
if(odd, x, y, z, u) → timesIter(p(x), y, u)
if(even, x, y, z, u) → plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z))))

Types:
check :: 0':s → zero:odd:even
0' :: 0':s
zero :: zero:odd:even
s :: 0':s → 0':s
odd :: zero:odd:even
even :: zero:odd:even
half :: 0':s → 0':s
plus :: 0':s → 0':s → 0':s
times :: 0':s → 0':s → 0':s
timesIter :: 0':s → 0':s → 0':s → 0':s
if :: zero:odd:even → 0':s → 0':s → 0':s → 0':s → 0':s
p :: 0':s → 0':s
hole_zero:odd:even1_0 :: zero:odd:even
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:
check, half, plus, timesIter

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

### (8) Obligation:

TRS:
Rules:
check(0') → zero
check(s(0')) → odd
check(s(s(0'))) → even
check(s(s(s(x)))) → check(s(x))
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → timesIter(x, y, 0')
timesIter(x, y, z) → if(check(x), x, y, z, plus(z, y))
p(s(x)) → x
p(0') → 0'
if(zero, x, y, z, u) → z
if(odd, x, y, z, u) → timesIter(p(x), y, u)
if(even, x, y, z, u) → plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z))))

Types:
check :: 0':s → zero:odd:even
0' :: 0':s
zero :: zero:odd:even
s :: 0':s → 0':s
odd :: zero:odd:even
even :: zero:odd:even
half :: 0':s → 0':s
plus :: 0':s → 0':s → 0':s
times :: 0':s → 0':s → 0':s
timesIter :: 0':s → 0':s → 0':s → 0':s
if :: zero:odd:even → 0':s → 0':s → 0':s → 0':s → 0':s
p :: 0':s → 0':s
hole_zero:odd:even1_0 :: zero:odd:even
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:
check, half, plus, timesIter

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

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

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

Induction Base:
check(gen_0':s3_0(+(1, *(2, 0)))) →RΩ(1)
odd

Induction Step:
check(gen_0':s3_0(+(1, *(2, +(n5_0, 1))))) →RΩ(1)
check(s(gen_0':s3_0(*(2, n5_0)))) →IH
odd

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

### (11) Obligation:

TRS:
Rules:
check(0') → zero
check(s(0')) → odd
check(s(s(0'))) → even
check(s(s(s(x)))) → check(s(x))
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → timesIter(x, y, 0')
timesIter(x, y, z) → if(check(x), x, y, z, plus(z, y))
p(s(x)) → x
p(0') → 0'
if(zero, x, y, z, u) → z
if(odd, x, y, z, u) → timesIter(p(x), y, u)
if(even, x, y, z, u) → plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z))))

Types:
check :: 0':s → zero:odd:even
0' :: 0':s
zero :: zero:odd:even
s :: 0':s → 0':s
odd :: zero:odd:even
even :: zero:odd:even
half :: 0':s → 0':s
plus :: 0':s → 0':s → 0':s
times :: 0':s → 0':s → 0':s
timesIter :: 0':s → 0':s → 0':s → 0':s
if :: zero:odd:even → 0':s → 0':s → 0':s → 0':s → 0':s
p :: 0':s → 0':s
hole_zero:odd:even1_0 :: zero:odd:even
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
check(gen_0':s3_0(+(1, *(2, n5_0)))) → odd, 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, timesIter

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

### (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))) →RΩ(1)
0'

Induction Step:
half(gen_0':s3_0(*(2, +(n249_0, 1)))) →RΩ(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 ircR ∈ Ω(n).

### (14) Obligation:

TRS:
Rules:
check(0') → zero
check(s(0')) → odd
check(s(s(0'))) → even
check(s(s(s(x)))) → check(s(x))
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → timesIter(x, y, 0')
timesIter(x, y, z) → if(check(x), x, y, z, plus(z, y))
p(s(x)) → x
p(0') → 0'
if(zero, x, y, z, u) → z
if(odd, x, y, z, u) → timesIter(p(x), y, u)
if(even, x, y, z, u) → plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z))))

Types:
check :: 0':s → zero:odd:even
0' :: 0':s
zero :: zero:odd:even
s :: 0':s → 0':s
odd :: zero:odd:even
even :: zero:odd:even
half :: 0':s → 0':s
plus :: 0':s → 0':s → 0':s
times :: 0':s → 0':s → 0':s
timesIter :: 0':s → 0':s → 0':s → 0':s
if :: zero:odd:even → 0':s → 0':s → 0':s → 0':s → 0':s
p :: 0':s → 0':s
hole_zero:odd:even1_0 :: zero:odd:even
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
check(gen_0':s3_0(+(1, *(2, n5_0)))) → odd, 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:
plus, timesIter

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

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

Proved the following rewrite lemma:
plus(gen_0':s3_0(n571_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n571_0, b)), rt ∈ Ω(1 + n5710)

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(+(n571_0, 1)), gen_0':s3_0(b)) →RΩ(1)
s(plus(gen_0':s3_0(n571_0), gen_0':s3_0(b))) →IH
s(gen_0':s3_0(+(b, c572_0)))

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

### (17) Obligation:

TRS:
Rules:
check(0') → zero
check(s(0')) → odd
check(s(s(0'))) → even
check(s(s(s(x)))) → check(s(x))
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → timesIter(x, y, 0')
timesIter(x, y, z) → if(check(x), x, y, z, plus(z, y))
p(s(x)) → x
p(0') → 0'
if(zero, x, y, z, u) → z
if(odd, x, y, z, u) → timesIter(p(x), y, u)
if(even, x, y, z, u) → plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z))))

Types:
check :: 0':s → zero:odd:even
0' :: 0':s
zero :: zero:odd:even
s :: 0':s → 0':s
odd :: zero:odd:even
even :: zero:odd:even
half :: 0':s → 0':s
plus :: 0':s → 0':s → 0':s
times :: 0':s → 0':s → 0':s
timesIter :: 0':s → 0':s → 0':s → 0':s
if :: zero:odd:even → 0':s → 0':s → 0':s → 0':s → 0':s
p :: 0':s → 0':s
hole_zero:odd:even1_0 :: zero:odd:even
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
check(gen_0':s3_0(+(1, *(2, n5_0)))) → odd, rt ∈ Ω(1 + n50)
half(gen_0':s3_0(*(2, n249_0))) → gen_0':s3_0(n249_0), rt ∈ Ω(1 + n2490)
plus(gen_0':s3_0(n571_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n571_0, b)), rt ∈ Ω(1 + n5710)

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:
timesIter

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

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

### (19) Obligation:

TRS:
Rules:
check(0') → zero
check(s(0')) → odd
check(s(s(0'))) → even
check(s(s(s(x)))) → check(s(x))
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → timesIter(x, y, 0')
timesIter(x, y, z) → if(check(x), x, y, z, plus(z, y))
p(s(x)) → x
p(0') → 0'
if(zero, x, y, z, u) → z
if(odd, x, y, z, u) → timesIter(p(x), y, u)
if(even, x, y, z, u) → plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z))))

Types:
check :: 0':s → zero:odd:even
0' :: 0':s
zero :: zero:odd:even
s :: 0':s → 0':s
odd :: zero:odd:even
even :: zero:odd:even
half :: 0':s → 0':s
plus :: 0':s → 0':s → 0':s
times :: 0':s → 0':s → 0':s
timesIter :: 0':s → 0':s → 0':s → 0':s
if :: zero:odd:even → 0':s → 0':s → 0':s → 0':s → 0':s
p :: 0':s → 0':s
hole_zero:odd:even1_0 :: zero:odd:even
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
check(gen_0':s3_0(+(1, *(2, n5_0)))) → odd, rt ∈ Ω(1 + n50)
half(gen_0':s3_0(*(2, n249_0))) → gen_0':s3_0(n249_0), rt ∈ Ω(1 + n2490)
plus(gen_0':s3_0(n571_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n571_0, b)), rt ∈ Ω(1 + n5710)

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

### (22) Obligation:

TRS:
Rules:
check(0') → zero
check(s(0')) → odd
check(s(s(0'))) → even
check(s(s(s(x)))) → check(s(x))
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → timesIter(x, y, 0')
timesIter(x, y, z) → if(check(x), x, y, z, plus(z, y))
p(s(x)) → x
p(0') → 0'
if(zero, x, y, z, u) → z
if(odd, x, y, z, u) → timesIter(p(x), y, u)
if(even, x, y, z, u) → plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z))))

Types:
check :: 0':s → zero:odd:even
0' :: 0':s
zero :: zero:odd:even
s :: 0':s → 0':s
odd :: zero:odd:even
even :: zero:odd:even
half :: 0':s → 0':s
plus :: 0':s → 0':s → 0':s
times :: 0':s → 0':s → 0':s
timesIter :: 0':s → 0':s → 0':s → 0':s
if :: zero:odd:even → 0':s → 0':s → 0':s → 0':s → 0':s
p :: 0':s → 0':s
hole_zero:odd:even1_0 :: zero:odd:even
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
check(gen_0':s3_0(+(1, *(2, n5_0)))) → odd, rt ∈ Ω(1 + n50)
half(gen_0':s3_0(*(2, n249_0))) → gen_0':s3_0(n249_0), rt ∈ Ω(1 + n2490)
plus(gen_0':s3_0(n571_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n571_0, b)), rt ∈ Ω(1 + n5710)

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

### (25) Obligation:

TRS:
Rules:
check(0') → zero
check(s(0')) → odd
check(s(s(0'))) → even
check(s(s(s(x)))) → check(s(x))
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → timesIter(x, y, 0')
timesIter(x, y, z) → if(check(x), x, y, z, plus(z, y))
p(s(x)) → x
p(0') → 0'
if(zero, x, y, z, u) → z
if(odd, x, y, z, u) → timesIter(p(x), y, u)
if(even, x, y, z, u) → plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z))))

Types:
check :: 0':s → zero:odd:even
0' :: 0':s
zero :: zero:odd:even
s :: 0':s → 0':s
odd :: zero:odd:even
even :: zero:odd:even
half :: 0':s → 0':s
plus :: 0':s → 0':s → 0':s
times :: 0':s → 0':s → 0':s
timesIter :: 0':s → 0':s → 0':s → 0':s
if :: zero:odd:even → 0':s → 0':s → 0':s → 0':s → 0':s
p :: 0':s → 0':s
hole_zero:odd:even1_0 :: zero:odd:even
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
check(gen_0':s3_0(+(1, *(2, n5_0)))) → odd, 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.

### (26) LowerBoundsProof (EQUIVALENT transformation)

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

### (28) Obligation:

TRS:
Rules:
check(0') → zero
check(s(0')) → odd
check(s(s(0'))) → even
check(s(s(s(x)))) → check(s(x))
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → timesIter(x, y, 0')
timesIter(x, y, z) → if(check(x), x, y, z, plus(z, y))
p(s(x)) → x
p(0') → 0'
if(zero, x, y, z, u) → z
if(odd, x, y, z, u) → timesIter(p(x), y, u)
if(even, x, y, z, u) → plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z))))

Types:
check :: 0':s → zero:odd:even
0' :: 0':s
zero :: zero:odd:even
s :: 0':s → 0':s
odd :: zero:odd:even
even :: zero:odd:even
half :: 0':s → 0':s
plus :: 0':s → 0':s → 0':s
times :: 0':s → 0':s → 0':s
timesIter :: 0':s → 0':s → 0':s → 0':s
if :: zero:odd:even → 0':s → 0':s → 0':s → 0':s → 0':s
p :: 0':s → 0':s
hole_zero:odd:even1_0 :: zero:odd:even
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

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