### (0) Obligation:

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

half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
inc(s(x)) → s(inc(x))
inc(0) → s(0)
logarithm(x) → logIter(x, 0)
logIter(x, y) → if(le(s(0), x), le(s(s(0)), x), half(x), inc(y))
if(false, b, x, y) → logZeroError
if(true, false, x, s(y)) → y
if(true, true, x, y) → logIter(x, y)
fg
fh

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

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

half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
inc(s(x)) → s(inc(x))
inc(0') → s(0')
logarithm(x) → logIter(x, 0')
logIter(x, y) → if(le(s(0'), x), le(s(s(0')), x), half(x), inc(y))
if(false, b, x, y) → logZeroError
if(true, false, x, s(y)) → y
if(true, true, x, y) → logIter(x, y)
fg
fh

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
inc(s(x)) → s(inc(x))
inc(0') → s(0')
logarithm(x) → logIter(x, 0')
logIter(x, y) → if(le(s(0'), x), le(s(s(0')), x), half(x), inc(y))
if(false, b, x, y) → logZeroError
if(true, false, x, s(y)) → y
if(true, true, x, y) → logIter(x, y)
fg
fh

Types:
half :: 0':s:logZeroError → 0':s:logZeroError
0' :: 0':s:logZeroError
s :: 0':s:logZeroError → 0':s:logZeroError
le :: 0':s:logZeroError → 0':s:logZeroError → true:false
true :: true:false
false :: true:false
inc :: 0':s:logZeroError → 0':s:logZeroError
logarithm :: 0':s:logZeroError → 0':s:logZeroError
logIter :: 0':s:logZeroError → 0':s:logZeroError → 0':s:logZeroError
if :: true:false → true:false → 0':s:logZeroError → 0':s:logZeroError → 0':s:logZeroError
logZeroError :: 0':s:logZeroError
f :: g:h
g :: g:h
h :: g:h
hole_0':s:logZeroError1_0 :: 0':s:logZeroError
hole_true:false2_0 :: true:false
hole_g:h3_0 :: g:h
gen_0':s:logZeroError4_0 :: Nat → 0':s:logZeroError

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

Heuristically decided to analyse the following defined symbols:
half, le, inc, logIter

They will be analysed ascendingly in the following order:
half < logIter
le < logIter
inc < logIter

### (8) Obligation:

TRS:
Rules:
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
inc(s(x)) → s(inc(x))
inc(0') → s(0')
logarithm(x) → logIter(x, 0')
logIter(x, y) → if(le(s(0'), x), le(s(s(0')), x), half(x), inc(y))
if(false, b, x, y) → logZeroError
if(true, false, x, s(y)) → y
if(true, true, x, y) → logIter(x, y)
fg
fh

Types:
half :: 0':s:logZeroError → 0':s:logZeroError
0' :: 0':s:logZeroError
s :: 0':s:logZeroError → 0':s:logZeroError
le :: 0':s:logZeroError → 0':s:logZeroError → true:false
true :: true:false
false :: true:false
inc :: 0':s:logZeroError → 0':s:logZeroError
logarithm :: 0':s:logZeroError → 0':s:logZeroError
logIter :: 0':s:logZeroError → 0':s:logZeroError → 0':s:logZeroError
if :: true:false → true:false → 0':s:logZeroError → 0':s:logZeroError → 0':s:logZeroError
logZeroError :: 0':s:logZeroError
f :: g:h
g :: g:h
h :: g:h
hole_0':s:logZeroError1_0 :: 0':s:logZeroError
hole_true:false2_0 :: true:false
hole_g:h3_0 :: g:h
gen_0':s:logZeroError4_0 :: Nat → 0':s:logZeroError

Generator Equations:
gen_0':s:logZeroError4_0(0) ⇔ 0'
gen_0':s:logZeroError4_0(+(x, 1)) ⇔ s(gen_0':s:logZeroError4_0(x))

The following defined symbols remain to be analysed:
half, le, inc, logIter

They will be analysed ascendingly in the following order:
half < logIter
le < logIter
inc < logIter

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

Proved the following rewrite lemma:
half(gen_0':s:logZeroError4_0(*(2, n6_0))) → gen_0':s:logZeroError4_0(n6_0), rt ∈ Ω(1 + n60)

Induction Base:
half(gen_0':s:logZeroError4_0(*(2, 0))) →RΩ(1)
0'

Induction Step:
half(gen_0':s:logZeroError4_0(*(2, +(n6_0, 1)))) →RΩ(1)
s(half(gen_0':s:logZeroError4_0(*(2, n6_0)))) →IH
s(gen_0':s:logZeroError4_0(c7_0))

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

### (11) Obligation:

TRS:
Rules:
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
inc(s(x)) → s(inc(x))
inc(0') → s(0')
logarithm(x) → logIter(x, 0')
logIter(x, y) → if(le(s(0'), x), le(s(s(0')), x), half(x), inc(y))
if(false, b, x, y) → logZeroError
if(true, false, x, s(y)) → y
if(true, true, x, y) → logIter(x, y)
fg
fh

Types:
half :: 0':s:logZeroError → 0':s:logZeroError
0' :: 0':s:logZeroError
s :: 0':s:logZeroError → 0':s:logZeroError
le :: 0':s:logZeroError → 0':s:logZeroError → true:false
true :: true:false
false :: true:false
inc :: 0':s:logZeroError → 0':s:logZeroError
logarithm :: 0':s:logZeroError → 0':s:logZeroError
logIter :: 0':s:logZeroError → 0':s:logZeroError → 0':s:logZeroError
if :: true:false → true:false → 0':s:logZeroError → 0':s:logZeroError → 0':s:logZeroError
logZeroError :: 0':s:logZeroError
f :: g:h
g :: g:h
h :: g:h
hole_0':s:logZeroError1_0 :: 0':s:logZeroError
hole_true:false2_0 :: true:false
hole_g:h3_0 :: g:h
gen_0':s:logZeroError4_0 :: Nat → 0':s:logZeroError

Lemmas:
half(gen_0':s:logZeroError4_0(*(2, n6_0))) → gen_0':s:logZeroError4_0(n6_0), rt ∈ Ω(1 + n60)

Generator Equations:
gen_0':s:logZeroError4_0(0) ⇔ 0'
gen_0':s:logZeroError4_0(+(x, 1)) ⇔ s(gen_0':s:logZeroError4_0(x))

The following defined symbols remain to be analysed:
le, inc, logIter

They will be analysed ascendingly in the following order:
le < logIter
inc < logIter

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

Proved the following rewrite lemma:
le(gen_0':s:logZeroError4_0(n320_0), gen_0':s:logZeroError4_0(n320_0)) → true, rt ∈ Ω(1 + n3200)

Induction Base:
le(gen_0':s:logZeroError4_0(0), gen_0':s:logZeroError4_0(0)) →RΩ(1)
true

Induction Step:
le(gen_0':s:logZeroError4_0(+(n320_0, 1)), gen_0':s:logZeroError4_0(+(n320_0, 1))) →RΩ(1)
le(gen_0':s:logZeroError4_0(n320_0), gen_0':s:logZeroError4_0(n320_0)) →IH
true

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

### (14) Obligation:

TRS:
Rules:
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
inc(s(x)) → s(inc(x))
inc(0') → s(0')
logarithm(x) → logIter(x, 0')
logIter(x, y) → if(le(s(0'), x), le(s(s(0')), x), half(x), inc(y))
if(false, b, x, y) → logZeroError
if(true, false, x, s(y)) → y
if(true, true, x, y) → logIter(x, y)
fg
fh

Types:
half :: 0':s:logZeroError → 0':s:logZeroError
0' :: 0':s:logZeroError
s :: 0':s:logZeroError → 0':s:logZeroError
le :: 0':s:logZeroError → 0':s:logZeroError → true:false
true :: true:false
false :: true:false
inc :: 0':s:logZeroError → 0':s:logZeroError
logarithm :: 0':s:logZeroError → 0':s:logZeroError
logIter :: 0':s:logZeroError → 0':s:logZeroError → 0':s:logZeroError
if :: true:false → true:false → 0':s:logZeroError → 0':s:logZeroError → 0':s:logZeroError
logZeroError :: 0':s:logZeroError
f :: g:h
g :: g:h
h :: g:h
hole_0':s:logZeroError1_0 :: 0':s:logZeroError
hole_true:false2_0 :: true:false
hole_g:h3_0 :: g:h
gen_0':s:logZeroError4_0 :: Nat → 0':s:logZeroError

Lemmas:
half(gen_0':s:logZeroError4_0(*(2, n6_0))) → gen_0':s:logZeroError4_0(n6_0), rt ∈ Ω(1 + n60)
le(gen_0':s:logZeroError4_0(n320_0), gen_0':s:logZeroError4_0(n320_0)) → true, rt ∈ Ω(1 + n3200)

Generator Equations:
gen_0':s:logZeroError4_0(0) ⇔ 0'
gen_0':s:logZeroError4_0(+(x, 1)) ⇔ s(gen_0':s:logZeroError4_0(x))

The following defined symbols remain to be analysed:
inc, logIter

They will be analysed ascendingly in the following order:
inc < logIter

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

Proved the following rewrite lemma:
inc(gen_0':s:logZeroError4_0(n627_0)) → gen_0':s:logZeroError4_0(+(1, n627_0)), rt ∈ Ω(1 + n6270)

Induction Base:
inc(gen_0':s:logZeroError4_0(0)) →RΩ(1)
s(0')

Induction Step:
inc(gen_0':s:logZeroError4_0(+(n627_0, 1))) →RΩ(1)
s(inc(gen_0':s:logZeroError4_0(n627_0))) →IH
s(gen_0':s:logZeroError4_0(+(1, c628_0)))

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

### (17) Obligation:

TRS:
Rules:
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
inc(s(x)) → s(inc(x))
inc(0') → s(0')
logarithm(x) → logIter(x, 0')
logIter(x, y) → if(le(s(0'), x), le(s(s(0')), x), half(x), inc(y))
if(false, b, x, y) → logZeroError
if(true, false, x, s(y)) → y
if(true, true, x, y) → logIter(x, y)
fg
fh

Types:
half :: 0':s:logZeroError → 0':s:logZeroError
0' :: 0':s:logZeroError
s :: 0':s:logZeroError → 0':s:logZeroError
le :: 0':s:logZeroError → 0':s:logZeroError → true:false
true :: true:false
false :: true:false
inc :: 0':s:logZeroError → 0':s:logZeroError
logarithm :: 0':s:logZeroError → 0':s:logZeroError
logIter :: 0':s:logZeroError → 0':s:logZeroError → 0':s:logZeroError
if :: true:false → true:false → 0':s:logZeroError → 0':s:logZeroError → 0':s:logZeroError
logZeroError :: 0':s:logZeroError
f :: g:h
g :: g:h
h :: g:h
hole_0':s:logZeroError1_0 :: 0':s:logZeroError
hole_true:false2_0 :: true:false
hole_g:h3_0 :: g:h
gen_0':s:logZeroError4_0 :: Nat → 0':s:logZeroError

Lemmas:
half(gen_0':s:logZeroError4_0(*(2, n6_0))) → gen_0':s:logZeroError4_0(n6_0), rt ∈ Ω(1 + n60)
le(gen_0':s:logZeroError4_0(n320_0), gen_0':s:logZeroError4_0(n320_0)) → true, rt ∈ Ω(1 + n3200)
inc(gen_0':s:logZeroError4_0(n627_0)) → gen_0':s:logZeroError4_0(+(1, n627_0)), rt ∈ Ω(1 + n6270)

Generator Equations:
gen_0':s:logZeroError4_0(0) ⇔ 0'
gen_0':s:logZeroError4_0(+(x, 1)) ⇔ s(gen_0':s:logZeroError4_0(x))

The following defined symbols remain to be analysed:
logIter

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

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

### (19) Obligation:

TRS:
Rules:
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
inc(s(x)) → s(inc(x))
inc(0') → s(0')
logarithm(x) → logIter(x, 0')
logIter(x, y) → if(le(s(0'), x), le(s(s(0')), x), half(x), inc(y))
if(false, b, x, y) → logZeroError
if(true, false, x, s(y)) → y
if(true, true, x, y) → logIter(x, y)
fg
fh

Types:
half :: 0':s:logZeroError → 0':s:logZeroError
0' :: 0':s:logZeroError
s :: 0':s:logZeroError → 0':s:logZeroError
le :: 0':s:logZeroError → 0':s:logZeroError → true:false
true :: true:false
false :: true:false
inc :: 0':s:logZeroError → 0':s:logZeroError
logarithm :: 0':s:logZeroError → 0':s:logZeroError
logIter :: 0':s:logZeroError → 0':s:logZeroError → 0':s:logZeroError
if :: true:false → true:false → 0':s:logZeroError → 0':s:logZeroError → 0':s:logZeroError
logZeroError :: 0':s:logZeroError
f :: g:h
g :: g:h
h :: g:h
hole_0':s:logZeroError1_0 :: 0':s:logZeroError
hole_true:false2_0 :: true:false
hole_g:h3_0 :: g:h
gen_0':s:logZeroError4_0 :: Nat → 0':s:logZeroError

Lemmas:
half(gen_0':s:logZeroError4_0(*(2, n6_0))) → gen_0':s:logZeroError4_0(n6_0), rt ∈ Ω(1 + n60)
le(gen_0':s:logZeroError4_0(n320_0), gen_0':s:logZeroError4_0(n320_0)) → true, rt ∈ Ω(1 + n3200)
inc(gen_0':s:logZeroError4_0(n627_0)) → gen_0':s:logZeroError4_0(+(1, n627_0)), rt ∈ Ω(1 + n6270)

Generator Equations:
gen_0':s:logZeroError4_0(0) ⇔ 0'
gen_0':s:logZeroError4_0(+(x, 1)) ⇔ s(gen_0':s:logZeroError4_0(x))

No more defined symbols left to analyse.

### (20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
half(gen_0':s:logZeroError4_0(*(2, n6_0))) → gen_0':s:logZeroError4_0(n6_0), rt ∈ Ω(1 + n60)

### (22) Obligation:

TRS:
Rules:
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
inc(s(x)) → s(inc(x))
inc(0') → s(0')
logarithm(x) → logIter(x, 0')
logIter(x, y) → if(le(s(0'), x), le(s(s(0')), x), half(x), inc(y))
if(false, b, x, y) → logZeroError
if(true, false, x, s(y)) → y
if(true, true, x, y) → logIter(x, y)
fg
fh

Types:
half :: 0':s:logZeroError → 0':s:logZeroError
0' :: 0':s:logZeroError
s :: 0':s:logZeroError → 0':s:logZeroError
le :: 0':s:logZeroError → 0':s:logZeroError → true:false
true :: true:false
false :: true:false
inc :: 0':s:logZeroError → 0':s:logZeroError
logarithm :: 0':s:logZeroError → 0':s:logZeroError
logIter :: 0':s:logZeroError → 0':s:logZeroError → 0':s:logZeroError
if :: true:false → true:false → 0':s:logZeroError → 0':s:logZeroError → 0':s:logZeroError
logZeroError :: 0':s:logZeroError
f :: g:h
g :: g:h
h :: g:h
hole_0':s:logZeroError1_0 :: 0':s:logZeroError
hole_true:false2_0 :: true:false
hole_g:h3_0 :: g:h
gen_0':s:logZeroError4_0 :: Nat → 0':s:logZeroError

Lemmas:
half(gen_0':s:logZeroError4_0(*(2, n6_0))) → gen_0':s:logZeroError4_0(n6_0), rt ∈ Ω(1 + n60)
le(gen_0':s:logZeroError4_0(n320_0), gen_0':s:logZeroError4_0(n320_0)) → true, rt ∈ Ω(1 + n3200)
inc(gen_0':s:logZeroError4_0(n627_0)) → gen_0':s:logZeroError4_0(+(1, n627_0)), rt ∈ Ω(1 + n6270)

Generator Equations:
gen_0':s:logZeroError4_0(0) ⇔ 0'
gen_0':s:logZeroError4_0(+(x, 1)) ⇔ s(gen_0':s:logZeroError4_0(x))

No more defined symbols left to analyse.

### (23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
half(gen_0':s:logZeroError4_0(*(2, n6_0))) → gen_0':s:logZeroError4_0(n6_0), rt ∈ Ω(1 + n60)

### (25) Obligation:

TRS:
Rules:
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
inc(s(x)) → s(inc(x))
inc(0') → s(0')
logarithm(x) → logIter(x, 0')
logIter(x, y) → if(le(s(0'), x), le(s(s(0')), x), half(x), inc(y))
if(false, b, x, y) → logZeroError
if(true, false, x, s(y)) → y
if(true, true, x, y) → logIter(x, y)
fg
fh

Types:
half :: 0':s:logZeroError → 0':s:logZeroError
0' :: 0':s:logZeroError
s :: 0':s:logZeroError → 0':s:logZeroError
le :: 0':s:logZeroError → 0':s:logZeroError → true:false
true :: true:false
false :: true:false
inc :: 0':s:logZeroError → 0':s:logZeroError
logarithm :: 0':s:logZeroError → 0':s:logZeroError
logIter :: 0':s:logZeroError → 0':s:logZeroError → 0':s:logZeroError
if :: true:false → true:false → 0':s:logZeroError → 0':s:logZeroError → 0':s:logZeroError
logZeroError :: 0':s:logZeroError
f :: g:h
g :: g:h
h :: g:h
hole_0':s:logZeroError1_0 :: 0':s:logZeroError
hole_true:false2_0 :: true:false
hole_g:h3_0 :: g:h
gen_0':s:logZeroError4_0 :: Nat → 0':s:logZeroError

Lemmas:
half(gen_0':s:logZeroError4_0(*(2, n6_0))) → gen_0':s:logZeroError4_0(n6_0), rt ∈ Ω(1 + n60)
le(gen_0':s:logZeroError4_0(n320_0), gen_0':s:logZeroError4_0(n320_0)) → true, rt ∈ Ω(1 + n3200)

Generator Equations:
gen_0':s:logZeroError4_0(0) ⇔ 0'
gen_0':s:logZeroError4_0(+(x, 1)) ⇔ s(gen_0':s:logZeroError4_0(x))

No more defined symbols left to analyse.

### (26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
half(gen_0':s:logZeroError4_0(*(2, n6_0))) → gen_0':s:logZeroError4_0(n6_0), rt ∈ Ω(1 + n60)

### (28) Obligation:

TRS:
Rules:
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
inc(s(x)) → s(inc(x))
inc(0') → s(0')
logarithm(x) → logIter(x, 0')
logIter(x, y) → if(le(s(0'), x), le(s(s(0')), x), half(x), inc(y))
if(false, b, x, y) → logZeroError
if(true, false, x, s(y)) → y
if(true, true, x, y) → logIter(x, y)
fg
fh

Types:
half :: 0':s:logZeroError → 0':s:logZeroError
0' :: 0':s:logZeroError
s :: 0':s:logZeroError → 0':s:logZeroError
le :: 0':s:logZeroError → 0':s:logZeroError → true:false
true :: true:false
false :: true:false
inc :: 0':s:logZeroError → 0':s:logZeroError
logarithm :: 0':s:logZeroError → 0':s:logZeroError
logIter :: 0':s:logZeroError → 0':s:logZeroError → 0':s:logZeroError
if :: true:false → true:false → 0':s:logZeroError → 0':s:logZeroError → 0':s:logZeroError
logZeroError :: 0':s:logZeroError
f :: g:h
g :: g:h
h :: g:h
hole_0':s:logZeroError1_0 :: 0':s:logZeroError
hole_true:false2_0 :: true:false
hole_g:h3_0 :: g:h
gen_0':s:logZeroError4_0 :: Nat → 0':s:logZeroError

Lemmas:
half(gen_0':s:logZeroError4_0(*(2, n6_0))) → gen_0':s:logZeroError4_0(n6_0), rt ∈ Ω(1 + n60)

Generator Equations:
gen_0':s:logZeroError4_0(0) ⇔ 0'
gen_0':s:logZeroError4_0(+(x, 1)) ⇔ s(gen_0':s:logZeroError4_0(x))

No more defined symbols left to analyse.

### (29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
half(gen_0':s:logZeroError4_0(*(2, n6_0))) → gen_0':s:logZeroError4_0(n6_0), rt ∈ Ω(1 + n60)