### (0) Obligation:

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

lt(0, s(x)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
logarithm(x) → ifa(lt(0, x), x)
ifa(true, x) → help(x, 1)
ifa(false, x) → logZeroError
help(x, y) → ifb(lt(y, x), x, y)
ifb(true, x, y) → help(half(x), s(y))
ifb(false, x, y) → y
half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

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

lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
logarithm(x) → ifa(lt(0', x), x)
ifa(true, x) → help(x, 1')
ifa(false, x) → logZeroError
help(x, y) → ifb(lt(y, x), x, y)
ifb(true, x, y) → help(half(x), s(y))
ifb(false, x, y) → y
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
logarithm(x) → ifa(lt(0', x), x)
ifa(true, x) → help(x, 1')
ifa(false, x) → logZeroError
help(x, y) → ifb(lt(y, x), x, y)
ifb(true, x, y) → help(half(x), s(y))
ifb(false, x, y) → y
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))

Types:
lt :: 0':s:1':logZeroError → 0':s:1':logZeroError → true:false
0' :: 0':s:1':logZeroError
s :: 0':s:1':logZeroError → 0':s:1':logZeroError
true :: true:false
false :: true:false
logarithm :: 0':s:1':logZeroError → 0':s:1':logZeroError
ifa :: true:false → 0':s:1':logZeroError → 0':s:1':logZeroError
help :: 0':s:1':logZeroError → 0':s:1':logZeroError → 0':s:1':logZeroError
1' :: 0':s:1':logZeroError
logZeroError :: 0':s:1':logZeroError
ifb :: true:false → 0':s:1':logZeroError → 0':s:1':logZeroError → 0':s:1':logZeroError
half :: 0':s:1':logZeroError → 0':s:1':logZeroError
hole_true:false1_0 :: true:false
hole_0':s:1':logZeroError2_0 :: 0':s:1':logZeroError
gen_0':s:1':logZeroError3_0 :: Nat → 0':s:1':logZeroError

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

Heuristically decided to analyse the following defined symbols:
lt, help, half

They will be analysed ascendingly in the following order:
lt < help
half < help

### (8) Obligation:

TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
logarithm(x) → ifa(lt(0', x), x)
ifa(true, x) → help(x, 1')
ifa(false, x) → logZeroError
help(x, y) → ifb(lt(y, x), x, y)
ifb(true, x, y) → help(half(x), s(y))
ifb(false, x, y) → y
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))

Types:
lt :: 0':s:1':logZeroError → 0':s:1':logZeroError → true:false
0' :: 0':s:1':logZeroError
s :: 0':s:1':logZeroError → 0':s:1':logZeroError
true :: true:false
false :: true:false
logarithm :: 0':s:1':logZeroError → 0':s:1':logZeroError
ifa :: true:false → 0':s:1':logZeroError → 0':s:1':logZeroError
help :: 0':s:1':logZeroError → 0':s:1':logZeroError → 0':s:1':logZeroError
1' :: 0':s:1':logZeroError
logZeroError :: 0':s:1':logZeroError
ifb :: true:false → 0':s:1':logZeroError → 0':s:1':logZeroError → 0':s:1':logZeroError
half :: 0':s:1':logZeroError → 0':s:1':logZeroError
hole_true:false1_0 :: true:false
hole_0':s:1':logZeroError2_0 :: 0':s:1':logZeroError
gen_0':s:1':logZeroError3_0 :: Nat → 0':s:1':logZeroError

Generator Equations:
gen_0':s:1':logZeroError3_0(0) ⇔ 0'
gen_0':s:1':logZeroError3_0(+(x, 1)) ⇔ s(gen_0':s:1':logZeroError3_0(x))

The following defined symbols remain to be analysed:
lt, help, half

They will be analysed ascendingly in the following order:
lt < help
half < help

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

Proved the following rewrite lemma:
lt(gen_0':s:1':logZeroError3_0(n5_0), gen_0':s:1':logZeroError3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n50)

Induction Base:
lt(gen_0':s:1':logZeroError3_0(0), gen_0':s:1':logZeroError3_0(+(1, 0))) →RΩ(1)
true

Induction Step:
lt(gen_0':s:1':logZeroError3_0(+(n5_0, 1)), gen_0':s:1':logZeroError3_0(+(1, +(n5_0, 1)))) →RΩ(1)
lt(gen_0':s:1':logZeroError3_0(n5_0), gen_0':s:1':logZeroError3_0(+(1, n5_0))) →IH
true

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

### (11) Obligation:

TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
logarithm(x) → ifa(lt(0', x), x)
ifa(true, x) → help(x, 1')
ifa(false, x) → logZeroError
help(x, y) → ifb(lt(y, x), x, y)
ifb(true, x, y) → help(half(x), s(y))
ifb(false, x, y) → y
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))

Types:
lt :: 0':s:1':logZeroError → 0':s:1':logZeroError → true:false
0' :: 0':s:1':logZeroError
s :: 0':s:1':logZeroError → 0':s:1':logZeroError
true :: true:false
false :: true:false
logarithm :: 0':s:1':logZeroError → 0':s:1':logZeroError
ifa :: true:false → 0':s:1':logZeroError → 0':s:1':logZeroError
help :: 0':s:1':logZeroError → 0':s:1':logZeroError → 0':s:1':logZeroError
1' :: 0':s:1':logZeroError
logZeroError :: 0':s:1':logZeroError
ifb :: true:false → 0':s:1':logZeroError → 0':s:1':logZeroError → 0':s:1':logZeroError
half :: 0':s:1':logZeroError → 0':s:1':logZeroError
hole_true:false1_0 :: true:false
hole_0':s:1':logZeroError2_0 :: 0':s:1':logZeroError
gen_0':s:1':logZeroError3_0 :: Nat → 0':s:1':logZeroError

Lemmas:
lt(gen_0':s:1':logZeroError3_0(n5_0), gen_0':s:1':logZeroError3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n50)

Generator Equations:
gen_0':s:1':logZeroError3_0(0) ⇔ 0'
gen_0':s:1':logZeroError3_0(+(x, 1)) ⇔ s(gen_0':s:1':logZeroError3_0(x))

The following defined symbols remain to be analysed:
half, help

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

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

Proved the following rewrite lemma:
half(gen_0':s:1':logZeroError3_0(*(2, n288_0))) → gen_0':s:1':logZeroError3_0(n288_0), rt ∈ Ω(1 + n2880)

Induction Base:
half(gen_0':s:1':logZeroError3_0(*(2, 0))) →RΩ(1)
0'

Induction Step:
half(gen_0':s:1':logZeroError3_0(*(2, +(n288_0, 1)))) →RΩ(1)
s(half(gen_0':s:1':logZeroError3_0(*(2, n288_0)))) →IH
s(gen_0':s:1':logZeroError3_0(c289_0))

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

### (14) Obligation:

TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
logarithm(x) → ifa(lt(0', x), x)
ifa(true, x) → help(x, 1')
ifa(false, x) → logZeroError
help(x, y) → ifb(lt(y, x), x, y)
ifb(true, x, y) → help(half(x), s(y))
ifb(false, x, y) → y
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))

Types:
lt :: 0':s:1':logZeroError → 0':s:1':logZeroError → true:false
0' :: 0':s:1':logZeroError
s :: 0':s:1':logZeroError → 0':s:1':logZeroError
true :: true:false
false :: true:false
logarithm :: 0':s:1':logZeroError → 0':s:1':logZeroError
ifa :: true:false → 0':s:1':logZeroError → 0':s:1':logZeroError
help :: 0':s:1':logZeroError → 0':s:1':logZeroError → 0':s:1':logZeroError
1' :: 0':s:1':logZeroError
logZeroError :: 0':s:1':logZeroError
ifb :: true:false → 0':s:1':logZeroError → 0':s:1':logZeroError → 0':s:1':logZeroError
half :: 0':s:1':logZeroError → 0':s:1':logZeroError
hole_true:false1_0 :: true:false
hole_0':s:1':logZeroError2_0 :: 0':s:1':logZeroError
gen_0':s:1':logZeroError3_0 :: Nat → 0':s:1':logZeroError

Lemmas:
lt(gen_0':s:1':logZeroError3_0(n5_0), gen_0':s:1':logZeroError3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n50)
half(gen_0':s:1':logZeroError3_0(*(2, n288_0))) → gen_0':s:1':logZeroError3_0(n288_0), rt ∈ Ω(1 + n2880)

Generator Equations:
gen_0':s:1':logZeroError3_0(0) ⇔ 0'
gen_0':s:1':logZeroError3_0(+(x, 1)) ⇔ s(gen_0':s:1':logZeroError3_0(x))

The following defined symbols remain to be analysed:
help

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

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

### (16) Obligation:

TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
logarithm(x) → ifa(lt(0', x), x)
ifa(true, x) → help(x, 1')
ifa(false, x) → logZeroError
help(x, y) → ifb(lt(y, x), x, y)
ifb(true, x, y) → help(half(x), s(y))
ifb(false, x, y) → y
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))

Types:
lt :: 0':s:1':logZeroError → 0':s:1':logZeroError → true:false
0' :: 0':s:1':logZeroError
s :: 0':s:1':logZeroError → 0':s:1':logZeroError
true :: true:false
false :: true:false
logarithm :: 0':s:1':logZeroError → 0':s:1':logZeroError
ifa :: true:false → 0':s:1':logZeroError → 0':s:1':logZeroError
help :: 0':s:1':logZeroError → 0':s:1':logZeroError → 0':s:1':logZeroError
1' :: 0':s:1':logZeroError
logZeroError :: 0':s:1':logZeroError
ifb :: true:false → 0':s:1':logZeroError → 0':s:1':logZeroError → 0':s:1':logZeroError
half :: 0':s:1':logZeroError → 0':s:1':logZeroError
hole_true:false1_0 :: true:false
hole_0':s:1':logZeroError2_0 :: 0':s:1':logZeroError
gen_0':s:1':logZeroError3_0 :: Nat → 0':s:1':logZeroError

Lemmas:
lt(gen_0':s:1':logZeroError3_0(n5_0), gen_0':s:1':logZeroError3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n50)
half(gen_0':s:1':logZeroError3_0(*(2, n288_0))) → gen_0':s:1':logZeroError3_0(n288_0), rt ∈ Ω(1 + n2880)

Generator Equations:
gen_0':s:1':logZeroError3_0(0) ⇔ 0'
gen_0':s:1':logZeroError3_0(+(x, 1)) ⇔ s(gen_0':s:1':logZeroError3_0(x))

No more defined symbols left to analyse.

### (17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
lt(gen_0':s:1':logZeroError3_0(n5_0), gen_0':s:1':logZeroError3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n50)

### (19) Obligation:

TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
logarithm(x) → ifa(lt(0', x), x)
ifa(true, x) → help(x, 1')
ifa(false, x) → logZeroError
help(x, y) → ifb(lt(y, x), x, y)
ifb(true, x, y) → help(half(x), s(y))
ifb(false, x, y) → y
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))

Types:
lt :: 0':s:1':logZeroError → 0':s:1':logZeroError → true:false
0' :: 0':s:1':logZeroError
s :: 0':s:1':logZeroError → 0':s:1':logZeroError
true :: true:false
false :: true:false
logarithm :: 0':s:1':logZeroError → 0':s:1':logZeroError
ifa :: true:false → 0':s:1':logZeroError → 0':s:1':logZeroError
help :: 0':s:1':logZeroError → 0':s:1':logZeroError → 0':s:1':logZeroError
1' :: 0':s:1':logZeroError
logZeroError :: 0':s:1':logZeroError
ifb :: true:false → 0':s:1':logZeroError → 0':s:1':logZeroError → 0':s:1':logZeroError
half :: 0':s:1':logZeroError → 0':s:1':logZeroError
hole_true:false1_0 :: true:false
hole_0':s:1':logZeroError2_0 :: 0':s:1':logZeroError
gen_0':s:1':logZeroError3_0 :: Nat → 0':s:1':logZeroError

Lemmas:
lt(gen_0':s:1':logZeroError3_0(n5_0), gen_0':s:1':logZeroError3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n50)
half(gen_0':s:1':logZeroError3_0(*(2, n288_0))) → gen_0':s:1':logZeroError3_0(n288_0), rt ∈ Ω(1 + n2880)

Generator Equations:
gen_0':s:1':logZeroError3_0(0) ⇔ 0'
gen_0':s:1':logZeroError3_0(+(x, 1)) ⇔ s(gen_0':s:1':logZeroError3_0(x))

No more defined symbols left to analyse.

### (20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
lt(gen_0':s:1':logZeroError3_0(n5_0), gen_0':s:1':logZeroError3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n50)

### (22) Obligation:

TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
logarithm(x) → ifa(lt(0', x), x)
ifa(true, x) → help(x, 1')
ifa(false, x) → logZeroError
help(x, y) → ifb(lt(y, x), x, y)
ifb(true, x, y) → help(half(x), s(y))
ifb(false, x, y) → y
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))

Types:
lt :: 0':s:1':logZeroError → 0':s:1':logZeroError → true:false
0' :: 0':s:1':logZeroError
s :: 0':s:1':logZeroError → 0':s:1':logZeroError
true :: true:false
false :: true:false
logarithm :: 0':s:1':logZeroError → 0':s:1':logZeroError
ifa :: true:false → 0':s:1':logZeroError → 0':s:1':logZeroError
help :: 0':s:1':logZeroError → 0':s:1':logZeroError → 0':s:1':logZeroError
1' :: 0':s:1':logZeroError
logZeroError :: 0':s:1':logZeroError
ifb :: true:false → 0':s:1':logZeroError → 0':s:1':logZeroError → 0':s:1':logZeroError
half :: 0':s:1':logZeroError → 0':s:1':logZeroError
hole_true:false1_0 :: true:false
hole_0':s:1':logZeroError2_0 :: 0':s:1':logZeroError
gen_0':s:1':logZeroError3_0 :: Nat → 0':s:1':logZeroError

Lemmas:
lt(gen_0':s:1':logZeroError3_0(n5_0), gen_0':s:1':logZeroError3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n50)

Generator Equations:
gen_0':s:1':logZeroError3_0(0) ⇔ 0'
gen_0':s:1':logZeroError3_0(+(x, 1)) ⇔ s(gen_0':s:1':logZeroError3_0(x))

No more defined symbols left to analyse.

### (23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
lt(gen_0':s:1':logZeroError3_0(n5_0), gen_0':s:1':logZeroError3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n50)