### (0) Obligation:

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

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(0, y) → 0
minus(s(x), y) → if_minus(le(s(x), y), s(x), y)
if_minus(true, s(x), y) → 0
if_minus(false, s(x), y) → s(minus(x, y))
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
log(s(0)) → 0
log(s(s(x))) → s(log(s(quot(x, s(s(0))))))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

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

le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
minus(0', y) → 0'
minus(s(x), y) → if_minus(le(s(x), y), s(x), y)
if_minus(true, s(x), y) → 0'
if_minus(false, s(x), y) → s(minus(x, y))
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
log(s(0')) → 0'
log(s(s(x))) → s(log(s(quot(x, s(s(0'))))))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
minus(0', y) → 0'
minus(s(x), y) → if_minus(le(s(x), y), s(x), y)
if_minus(true, s(x), y) → 0'
if_minus(false, s(x), y) → s(minus(x, y))
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
log(s(0')) → 0'
log(s(s(x))) → s(log(s(quot(x, s(s(0'))))))

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
minus :: 0':s → 0':s → 0':s
if_minus :: true:false → 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s
log :: 0':s → 0':s
hole_true:false1_0 :: true:false
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:
le, minus, quot, log

They will be analysed ascendingly in the following order:
le < minus
minus < quot
quot < log

### (8) Obligation:

TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
minus(0', y) → 0'
minus(s(x), y) → if_minus(le(s(x), y), s(x), y)
if_minus(true, s(x), y) → 0'
if_minus(false, s(x), y) → s(minus(x, y))
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
log(s(0')) → 0'
log(s(s(x))) → s(log(s(quot(x, s(s(0'))))))

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
minus :: 0':s → 0':s → 0':s
if_minus :: true:false → 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s
log :: 0':s → 0':s
hole_true:false1_0 :: true:false
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:
le, minus, quot, log

They will be analysed ascendingly in the following order:
le < minus
minus < quot
quot < log

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

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

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

Induction Step:
le(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) →RΩ(1)
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) →IH
true

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

### (11) Obligation:

TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
minus(0', y) → 0'
minus(s(x), y) → if_minus(le(s(x), y), s(x), y)
if_minus(true, s(x), y) → 0'
if_minus(false, s(x), y) → s(minus(x, y))
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
log(s(0')) → 0'
log(s(s(x))) → s(log(s(quot(x, s(s(0'))))))

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
minus :: 0':s → 0':s → 0':s
if_minus :: true:false → 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s
log :: 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, 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:
minus, quot, log

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

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

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

### (13) Obligation:

TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
minus(0', y) → 0'
minus(s(x), y) → if_minus(le(s(x), y), s(x), y)
if_minus(true, s(x), y) → 0'
if_minus(false, s(x), y) → s(minus(x, y))
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
log(s(0')) → 0'
log(s(s(x))) → s(log(s(quot(x, s(s(0'))))))

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
minus :: 0':s → 0':s → 0':s
if_minus :: true:false → 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s
log :: 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, 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:
quot, log

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

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

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

### (15) Obligation:

TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
minus(0', y) → 0'
minus(s(x), y) → if_minus(le(s(x), y), s(x), y)
if_minus(true, s(x), y) → 0'
if_minus(false, s(x), y) → s(minus(x, y))
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
log(s(0')) → 0'
log(s(s(x))) → s(log(s(quot(x, s(s(0'))))))

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
minus :: 0':s → 0':s → 0':s
if_minus :: true:false → 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s
log :: 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, 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:
log

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

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

### (17) Obligation:

TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
minus(0', y) → 0'
minus(s(x), y) → if_minus(le(s(x), y), s(x), y)
if_minus(true, s(x), y) → 0'
if_minus(false, s(x), y) → s(minus(x, y))
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
log(s(0')) → 0'
log(s(s(x))) → s(log(s(quot(x, s(s(0'))))))

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
minus :: 0':s → 0':s → 0':s
if_minus :: true:false → 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s
log :: 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, 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.

### (18) LowerBoundsProof (EQUIVALENT transformation)

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

### (20) Obligation:

TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
minus(0', y) → 0'
minus(s(x), y) → if_minus(le(s(x), y), s(x), y)
if_minus(true, s(x), y) → 0'
if_minus(false, s(x), y) → s(minus(x, y))
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
log(s(0')) → 0'
log(s(s(x))) → s(log(s(quot(x, s(s(0'))))))

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
minus :: 0':s → 0':s → 0':s
if_minus :: true:false → 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s
log :: 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, 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.

### (21) LowerBoundsProof (EQUIVALENT transformation)

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