### (0) Obligation:

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

half(0) → 0
half(s(s(x))) → s(half(x))
log(s(0)) → 0
log(s(s(x))) → s(log(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
half(s(s(x))) →+ s(half(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:

half(0') → 0'
half(s(s(x))) → s(half(x))
log(s(0')) → 0'
log(s(s(x))) → s(log(s(half(x))))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
half(0') → 0'
half(s(s(x))) → s(half(x))
log(s(0')) → 0'
log(s(s(x))) → s(log(s(half(x))))

Types:
half :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
log :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
half, log

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

### (8) Obligation:

TRS:
Rules:
half(0') → 0'
half(s(s(x))) → s(half(x))
log(s(0')) → 0'
log(s(s(x))) → s(log(s(half(x))))

Types:
half :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
log :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

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

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

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

Proved the following rewrite lemma:
half(gen_0':s2_0(*(2, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n40)

Induction Base:
half(gen_0':s2_0(*(2, 0))) →RΩ(1)
0'

Induction Step:
half(gen_0':s2_0(*(2, +(n4_0, 1)))) →RΩ(1)
s(half(gen_0':s2_0(*(2, n4_0)))) →IH
s(gen_0':s2_0(c5_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(s(x))) → s(half(x))
log(s(0')) → 0'
log(s(s(x))) → s(log(s(half(x))))

Types:
half :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
log :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
half(gen_0':s2_0(*(2, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n40)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
log

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

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

### (13) Obligation:

TRS:
Rules:
half(0') → 0'
half(s(s(x))) → s(half(x))
log(s(0')) → 0'
log(s(s(x))) → s(log(s(half(x))))

Types:
half :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
log :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
half(gen_0':s2_0(*(2, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n40)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

### (14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
half(gen_0':s2_0(*(2, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n40)

### (16) Obligation:

TRS:
Rules:
half(0') → 0'
half(s(s(x))) → s(half(x))
log(s(0')) → 0'
log(s(s(x))) → s(log(s(half(x))))

Types:
half :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
log :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
half(gen_0':s2_0(*(2, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n40)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

### (17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
half(gen_0':s2_0(*(2, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n40)