### (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(0) → 0
inc(s(x)) → s(inc(x))
log(x) → log2(x, 0)
log2(x, y) → if(le(x, s(0)), x, inc(y))
if(true, x, s(y)) → y
if(false, x, y) → log2(half(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
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(0') → 0'
inc(s(x)) → s(inc(x))
log(x) → log2(x, 0')
log2(x, y) → if(le(x, s(0')), x, inc(y))
if(true, x, s(y)) → y
if(false, x, y) → log2(half(x), y)

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(0') → 0'
inc(s(x)) → s(inc(x))
log(x) → log2(x, 0')
log2(x, y) → if(le(x, s(0')), x, inc(y))
if(true, x, s(y)) → y
if(false, x, y) → log2(half(x), y)

Types:
half :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
inc :: 0':s → 0':s
log :: 0':s → 0':s
log2 :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
half, le, inc, log2

They will be analysed ascendingly in the following order:
half < log2
le < log2
inc < log2

### (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(0') → 0'
inc(s(x)) → s(inc(x))
log(x) → log2(x, 0')
log2(x, y) → if(le(x, s(0')), x, inc(y))
if(true, x, s(y)) → y
if(false, x, y) → log2(half(x), y)

Types:
half :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
inc :: 0':s → 0':s
log :: 0':s → 0':s
log2 :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
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:
half, le, inc, log2

They will be analysed ascendingly in the following order:
half < log2
le < log2
inc < log2

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

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

Induction Base:
half(gen_0':s3_0(*(2, 0))) →RΩ(1)
0'

Induction Step:
half(gen_0':s3_0(*(2, +(n5_0, 1)))) →RΩ(1)
s(half(gen_0':s3_0(*(2, n5_0)))) →IH
s(gen_0':s3_0(c6_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(0') → 0'
inc(s(x)) → s(inc(x))
log(x) → log2(x, 0')
log2(x, y) → if(le(x, s(0')), x, inc(y))
if(true, x, s(y)) → y
if(false, x, y) → log2(half(x), y)

Types:
half :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
inc :: 0':s → 0':s
log :: 0':s → 0':s
log2 :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
half(gen_0':s3_0(*(2, n5_0))) → gen_0':s3_0(n5_0), 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:
le, inc, log2

They will be analysed ascendingly in the following order:
le < log2
inc < log2

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

Proved the following rewrite lemma:
le(gen_0':s3_0(n307_0), gen_0':s3_0(n307_0)) → true, rt ∈ Ω(1 + n3070)

Induction Base:
le(gen_0':s3_0(0), gen_0':s3_0(0)) →RΩ(1)
true

Induction Step:
le(gen_0':s3_0(+(n307_0, 1)), gen_0':s3_0(+(n307_0, 1))) →RΩ(1)
le(gen_0':s3_0(n307_0), gen_0':s3_0(n307_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(0') → 0'
inc(s(x)) → s(inc(x))
log(x) → log2(x, 0')
log2(x, y) → if(le(x, s(0')), x, inc(y))
if(true, x, s(y)) → y
if(false, x, y) → log2(half(x), y)

Types:
half :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
inc :: 0':s → 0':s
log :: 0':s → 0':s
log2 :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
half(gen_0':s3_0(*(2, n5_0))) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n50)
le(gen_0':s3_0(n307_0), gen_0':s3_0(n307_0)) → true, rt ∈ Ω(1 + n3070)

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:
inc, log2

They will be analysed ascendingly in the following order:
inc < log2

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

Proved the following rewrite lemma:
inc(gen_0':s3_0(n596_0)) → gen_0':s3_0(n596_0), rt ∈ Ω(1 + n5960)

Induction Base:
inc(gen_0':s3_0(0)) →RΩ(1)
0'

Induction Step:
inc(gen_0':s3_0(+(n596_0, 1))) →RΩ(1)
s(inc(gen_0':s3_0(n596_0))) →IH
s(gen_0':s3_0(c597_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(0') → 0'
inc(s(x)) → s(inc(x))
log(x) → log2(x, 0')
log2(x, y) → if(le(x, s(0')), x, inc(y))
if(true, x, s(y)) → y
if(false, x, y) → log2(half(x), y)

Types:
half :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
inc :: 0':s → 0':s
log :: 0':s → 0':s
log2 :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
half(gen_0':s3_0(*(2, n5_0))) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n50)
le(gen_0':s3_0(n307_0), gen_0':s3_0(n307_0)) → true, rt ∈ Ω(1 + n3070)
inc(gen_0':s3_0(n596_0)) → gen_0':s3_0(n596_0), rt ∈ Ω(1 + n5960)

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:
log2

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

Could not prove a rewrite lemma for the defined symbol log2.

### (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(0') → 0'
inc(s(x)) → s(inc(x))
log(x) → log2(x, 0')
log2(x, y) → if(le(x, s(0')), x, inc(y))
if(true, x, s(y)) → y
if(false, x, y) → log2(half(x), y)

Types:
half :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
inc :: 0':s → 0':s
log :: 0':s → 0':s
log2 :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
half(gen_0':s3_0(*(2, n5_0))) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n50)
le(gen_0':s3_0(n307_0), gen_0':s3_0(n307_0)) → true, rt ∈ Ω(1 + n3070)
inc(gen_0':s3_0(n596_0)) → gen_0':s3_0(n596_0), rt ∈ Ω(1 + n5960)

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

### (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(0') → 0'
inc(s(x)) → s(inc(x))
log(x) → log2(x, 0')
log2(x, y) → if(le(x, s(0')), x, inc(y))
if(true, x, s(y)) → y
if(false, x, y) → log2(half(x), y)

Types:
half :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
inc :: 0':s → 0':s
log :: 0':s → 0':s
log2 :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
half(gen_0':s3_0(*(2, n5_0))) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n50)
le(gen_0':s3_0(n307_0), gen_0':s3_0(n307_0)) → true, rt ∈ Ω(1 + n3070)
inc(gen_0':s3_0(n596_0)) → gen_0':s3_0(n596_0), rt ∈ Ω(1 + n5960)

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

### (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(0') → 0'
inc(s(x)) → s(inc(x))
log(x) → log2(x, 0')
log2(x, y) → if(le(x, s(0')), x, inc(y))
if(true, x, s(y)) → y
if(false, x, y) → log2(half(x), y)

Types:
half :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
inc :: 0':s → 0':s
log :: 0':s → 0':s
log2 :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
half(gen_0':s3_0(*(2, n5_0))) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n50)
le(gen_0':s3_0(n307_0), gen_0':s3_0(n307_0)) → true, rt ∈ Ω(1 + n3070)

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

### (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(0') → 0'
inc(s(x)) → s(inc(x))
log(x) → log2(x, 0')
log2(x, y) → if(le(x, s(0')), x, inc(y))
if(true, x, s(y)) → y
if(false, x, y) → log2(half(x), y)

Types:
half :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
inc :: 0':s → 0':s
log :: 0':s → 0':s
log2 :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

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