The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

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

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
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)
log(x) → logIter(x, 0)
logIter(x, y) → if(le(s(0), x), le(s(s(0)), x), quot(x, s(s(0))), inc(y))
if(false, b, x, y) → logZeroError
if(true, false, x, s(y)) → y
if(true, true, x, y) → logIter(x, y)

Rewrite Strategy: INNERMOST

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

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

minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
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')
log(x) → logIter(x, 0')
logIter(x, y) → if(le(s(0'), x), le(s(s(0')), x), quot(x, s(s(0'))), inc(y))
if(false, b, x, y) → logZeroError
if(true, false, x, s(y)) → y
if(true, true, x, y) → logIter(x, y)

S is empty.
Rewrite Strategy: INNERMOST

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
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')
log(x) → logIter(x, 0')
logIter(x, y) → if(le(s(0'), x), le(s(s(0')), x), quot(x, s(s(0'))), inc(y))
if(false, b, x, y) → logZeroError
if(true, false, x, s(y)) → y
if(true, true, x, y) → logIter(x, y)

Types:
minus :: 0':s:logZeroError → 0':s:logZeroError → 0':s:logZeroError
0' :: 0':s:logZeroError
s :: 0':s:logZeroError → 0':s:logZeroError
quot :: 0':s:logZeroError → 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
log :: 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
hole_0':s:logZeroError1_0 :: 0':s:logZeroError
hole_true:false2_0 :: true:false
gen_0':s:logZeroError3_0 :: Nat → 0':s:logZeroError

(5) OrderProof (LOWER BOUND(ID) transformation)

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

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

(6) Obligation:

Innermost TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
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')
log(x) → logIter(x, 0')
logIter(x, y) → if(le(s(0'), x), le(s(s(0')), x), quot(x, s(s(0'))), inc(y))
if(false, b, x, y) → logZeroError
if(true, false, x, s(y)) → y
if(true, true, x, y) → logIter(x, y)

Types:
minus :: 0':s:logZeroError → 0':s:logZeroError → 0':s:logZeroError
0' :: 0':s:logZeroError
s :: 0':s:logZeroError → 0':s:logZeroError
quot :: 0':s:logZeroError → 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
log :: 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
hole_0':s:logZeroError1_0 :: 0':s:logZeroError
hole_true:false2_0 :: true:false
gen_0':s:logZeroError3_0 :: Nat → 0':s:logZeroError

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

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

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

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
minus(gen_0':s:logZeroError3_0(n5_0), gen_0':s:logZeroError3_0(n5_0)) → gen_0':s:logZeroError3_0(0), rt ∈ Ω(1 + n5₀)

Induction Base:
minus(gen_0':s:logZeroError3_0(0), gen_0':s:logZeroError3_0(0)) →_R^Ω(1)
gen_0':s:logZeroError3_0(0)

Induction Step:
minus(gen_0':s:logZeroError3_0(+(n5_0, 1)), gen_0':s:logZeroError3_0(+(n5_0, 1))) →_R^Ω(1)
minus(gen_0':s:logZeroError3_0(n5_0), gen_0':s:logZeroError3_0(n5_0)) →_IH
gen_0':s:logZeroError3_0(0)

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

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
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')
log(x) → logIter(x, 0')
logIter(x, y) → if(le(s(0'), x), le(s(s(0')), x), quot(x, s(s(0'))), inc(y))
if(false, b, x, y) → logZeroError
if(true, false, x, s(y)) → y
if(true, true, x, y) → logIter(x, y)

Types:
minus :: 0':s:logZeroError → 0':s:logZeroError → 0':s:logZeroError
0' :: 0':s:logZeroError
s :: 0':s:logZeroError → 0':s:logZeroError
quot :: 0':s:logZeroError → 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
log :: 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
hole_0':s:logZeroError1_0 :: 0':s:logZeroError
hole_true:false2_0 :: true:false
gen_0':s:logZeroError3_0 :: Nat → 0':s:logZeroError

Lemmas:
minus(gen_0':s:logZeroError3_0(n5_0), gen_0':s:logZeroError3_0(n5_0)) → gen_0':s:logZeroError3_0(0), rt ∈ Ω(1 + n5₀)

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

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

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

(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(11) Obligation:

Innermost TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
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')
log(x) → logIter(x, 0')
logIter(x, y) → if(le(s(0'), x), le(s(s(0')), x), quot(x, s(s(0'))), inc(y))
if(false, b, x, y) → logZeroError
if(true, false, x, s(y)) → y
if(true, true, x, y) → logIter(x, y)

Types:
minus :: 0':s:logZeroError → 0':s:logZeroError → 0':s:logZeroError
0' :: 0':s:logZeroError
s :: 0':s:logZeroError → 0':s:logZeroError
quot :: 0':s:logZeroError → 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
log :: 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
hole_0':s:logZeroError1_0 :: 0':s:logZeroError
hole_true:false2_0 :: true:false
gen_0':s:logZeroError3_0 :: Nat → 0':s:logZeroError

Lemmas:
minus(gen_0':s:logZeroError3_0(n5_0), gen_0':s:logZeroError3_0(n5_0)) → gen_0':s:logZeroError3_0(0), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s:logZeroError3_0(0) ⇔ 0'
gen_0':s:logZeroError3_0(+(x, 1)) ⇔ s(gen_0':s:logZeroError3_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:logZeroError3_0(n455_0), gen_0':s:logZeroError3_0(n455_0)) → true, rt ∈ Ω(1 + n455₀)

Induction Base:
le(gen_0':s:logZeroError3_0(0), gen_0':s:logZeroError3_0(0)) →_R^Ω(1)
true

Induction Step:
le(gen_0':s:logZeroError3_0(+(n455_0, 1)), gen_0':s:logZeroError3_0(+(n455_0, 1))) →_R^Ω(1)
le(gen_0':s:logZeroError3_0(n455_0), gen_0':s:logZeroError3_0(n455_0)) →_IH
true

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

(13) Complex Obligation (BEST)

(14) Obligation:

Innermost TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
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')
log(x) → logIter(x, 0')
logIter(x, y) → if(le(s(0'), x), le(s(s(0')), x), quot(x, s(s(0'))), inc(y))
if(false, b, x, y) → logZeroError
if(true, false, x, s(y)) → y
if(true, true, x, y) → logIter(x, y)

Types:
minus :: 0':s:logZeroError → 0':s:logZeroError → 0':s:logZeroError
0' :: 0':s:logZeroError
s :: 0':s:logZeroError → 0':s:logZeroError
quot :: 0':s:logZeroError → 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
log :: 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
hole_0':s:logZeroError1_0 :: 0':s:logZeroError
hole_true:false2_0 :: true:false
gen_0':s:logZeroError3_0 :: Nat → 0':s:logZeroError

Lemmas:
minus(gen_0':s:logZeroError3_0(n5_0), gen_0':s:logZeroError3_0(n5_0)) → gen_0':s:logZeroError3_0(0), rt ∈ Ω(1 + n5₀)
le(gen_0':s:logZeroError3_0(n455_0), gen_0':s:logZeroError3_0(n455_0)) → true, rt ∈ Ω(1 + n455₀)

Generator Equations:
gen_0':s:logZeroError3_0(0) ⇔ 0'
gen_0':s:logZeroError3_0(+(x, 1)) ⇔ s(gen_0':s:logZeroError3_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:logZeroError3_0(n822_0)) → gen_0':s:logZeroError3_0(+(1, n822_0)), rt ∈ Ω(1 + n822₀)

Induction Base:
inc(gen_0':s:logZeroError3_0(0)) →_R^Ω(1)
s(0')

Induction Step:
inc(gen_0':s:logZeroError3_0(+(n822_0, 1))) →_R^Ω(1)
s(inc(gen_0':s:logZeroError3_0(n822_0))) →_IH
s(gen_0':s:logZeroError3_0(+(1, c823_0)))

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

(16) Complex Obligation (BEST)

(17) Obligation:

Innermost TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
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')
log(x) → logIter(x, 0')
logIter(x, y) → if(le(s(0'), x), le(s(s(0')), x), quot(x, s(s(0'))), inc(y))
if(false, b, x, y) → logZeroError
if(true, false, x, s(y)) → y
if(true, true, x, y) → logIter(x, y)

Types:
minus :: 0':s:logZeroError → 0':s:logZeroError → 0':s:logZeroError
0' :: 0':s:logZeroError
s :: 0':s:logZeroError → 0':s:logZeroError
quot :: 0':s:logZeroError → 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
log :: 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
hole_0':s:logZeroError1_0 :: 0':s:logZeroError
hole_true:false2_0 :: true:false
gen_0':s:logZeroError3_0 :: Nat → 0':s:logZeroError

Lemmas:
minus(gen_0':s:logZeroError3_0(n5_0), gen_0':s:logZeroError3_0(n5_0)) → gen_0':s:logZeroError3_0(0), rt ∈ Ω(1 + n5₀)
le(gen_0':s:logZeroError3_0(n455_0), gen_0':s:logZeroError3_0(n455_0)) → true, rt ∈ Ω(1 + n455₀)
inc(gen_0':s:logZeroError3_0(n822_0)) → gen_0':s:logZeroError3_0(+(1, n822_0)), rt ∈ Ω(1 + n822₀)

Generator Equations:
gen_0':s:logZeroError3_0(0) ⇔ 0'
gen_0':s:logZeroError3_0(+(x, 1)) ⇔ s(gen_0':s:logZeroError3_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:

Innermost TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
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')
log(x) → logIter(x, 0')
logIter(x, y) → if(le(s(0'), x), le(s(s(0')), x), quot(x, s(s(0'))), inc(y))
if(false, b, x, y) → logZeroError
if(true, false, x, s(y)) → y
if(true, true, x, y) → logIter(x, y)

Types:
minus :: 0':s:logZeroError → 0':s:logZeroError → 0':s:logZeroError
0' :: 0':s:logZeroError
s :: 0':s:logZeroError → 0':s:logZeroError
quot :: 0':s:logZeroError → 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
log :: 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
hole_0':s:logZeroError1_0 :: 0':s:logZeroError
hole_true:false2_0 :: true:false
gen_0':s:logZeroError3_0 :: Nat → 0':s:logZeroError

Lemmas:
minus(gen_0':s:logZeroError3_0(n5_0), gen_0':s:logZeroError3_0(n5_0)) → gen_0':s:logZeroError3_0(0), rt ∈ Ω(1 + n5₀)
le(gen_0':s:logZeroError3_0(n455_0), gen_0':s:logZeroError3_0(n455_0)) → true, rt ∈ Ω(1 + n455₀)
inc(gen_0':s:logZeroError3_0(n822_0)) → gen_0':s:logZeroError3_0(+(1, n822_0)), rt ∈ Ω(1 + n822₀)

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

No more defined symbols left to analyse.

(20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
minus(gen_0':s:logZeroError3_0(n5_0), gen_0':s:logZeroError3_0(n5_0)) → gen_0':s:logZeroError3_0(0), rt ∈ Ω(1 + n5₀)

(21) BOUNDS(n^1, INF)

(22) Obligation:

Innermost TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
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')
log(x) → logIter(x, 0')
logIter(x, y) → if(le(s(0'), x), le(s(s(0')), x), quot(x, s(s(0'))), inc(y))
if(false, b, x, y) → logZeroError
if(true, false, x, s(y)) → y
if(true, true, x, y) → logIter(x, y)

Types:
minus :: 0':s:logZeroError → 0':s:logZeroError → 0':s:logZeroError
0' :: 0':s:logZeroError
s :: 0':s:logZeroError → 0':s:logZeroError
quot :: 0':s:logZeroError → 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
log :: 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
hole_0':s:logZeroError1_0 :: 0':s:logZeroError
hole_true:false2_0 :: true:false
gen_0':s:logZeroError3_0 :: Nat → 0':s:logZeroError

Lemmas:
minus(gen_0':s:logZeroError3_0(n5_0), gen_0':s:logZeroError3_0(n5_0)) → gen_0':s:logZeroError3_0(0), rt ∈ Ω(1 + n5₀)
le(gen_0':s:logZeroError3_0(n455_0), gen_0':s:logZeroError3_0(n455_0)) → true, rt ∈ Ω(1 + n455₀)
inc(gen_0':s:logZeroError3_0(n822_0)) → gen_0':s:logZeroError3_0(+(1, n822_0)), rt ∈ Ω(1 + n822₀)

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

No more defined symbols left to analyse.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
minus(gen_0':s:logZeroError3_0(n5_0), gen_0':s:logZeroError3_0(n5_0)) → gen_0':s:logZeroError3_0(0), rt ∈ Ω(1 + n5₀)

(24) BOUNDS(n^1, INF)

(25) Obligation:

Innermost TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
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')
log(x) → logIter(x, 0')
logIter(x, y) → if(le(s(0'), x), le(s(s(0')), x), quot(x, s(s(0'))), inc(y))
if(false, b, x, y) → logZeroError
if(true, false, x, s(y)) → y
if(true, true, x, y) → logIter(x, y)

Types:
minus :: 0':s:logZeroError → 0':s:logZeroError → 0':s:logZeroError
0' :: 0':s:logZeroError
s :: 0':s:logZeroError → 0':s:logZeroError
quot :: 0':s:logZeroError → 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
log :: 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
hole_0':s:logZeroError1_0 :: 0':s:logZeroError
hole_true:false2_0 :: true:false
gen_0':s:logZeroError3_0 :: Nat → 0':s:logZeroError

Lemmas:
minus(gen_0':s:logZeroError3_0(n5_0), gen_0':s:logZeroError3_0(n5_0)) → gen_0':s:logZeroError3_0(0), rt ∈ Ω(1 + n5₀)
le(gen_0':s:logZeroError3_0(n455_0), gen_0':s:logZeroError3_0(n455_0)) → true, rt ∈ Ω(1 + n455₀)

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

No more defined symbols left to analyse.

(26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
minus(gen_0':s:logZeroError3_0(n5_0), gen_0':s:logZeroError3_0(n5_0)) → gen_0':s:logZeroError3_0(0), rt ∈ Ω(1 + n5₀)

(27) BOUNDS(n^1, INF)

(28) Obligation:

Innermost TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
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')
log(x) → logIter(x, 0')
logIter(x, y) → if(le(s(0'), x), le(s(s(0')), x), quot(x, s(s(0'))), inc(y))
if(false, b, x, y) → logZeroError
if(true, false, x, s(y)) → y
if(true, true, x, y) → logIter(x, y)

Types:
minus :: 0':s:logZeroError → 0':s:logZeroError → 0':s:logZeroError
0' :: 0':s:logZeroError
s :: 0':s:logZeroError → 0':s:logZeroError
quot :: 0':s:logZeroError → 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
log :: 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
hole_0':s:logZeroError1_0 :: 0':s:logZeroError
hole_true:false2_0 :: true:false
gen_0':s:logZeroError3_0 :: Nat → 0':s:logZeroError

Lemmas:
minus(gen_0':s:logZeroError3_0(n5_0), gen_0':s:logZeroError3_0(n5_0)) → gen_0':s:logZeroError3_0(0), rt ∈ Ω(1 + n5₀)

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

No more defined symbols left to analyse.

(29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
minus(gen_0':s:logZeroError3_0(n5_0), gen_0':s:logZeroError3_0(n5_0)) → gen_0':s:logZeroError3_0(0), rt ∈ Ω(1 + n5₀)

(30) BOUNDS(n^1, INF)