The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).

(0) Obligation:

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

le(s(x), 0) → false
le(0, y) → true
le(s(x), s(y)) → le(x, y)
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
log(x, 0) → baseError
log(x, s(0)) → baseError
log(0, s(s(b))) → logZeroError
log(s(x), s(s(b))) → loop(s(x), s(s(b)), s(0), 0)
loop(x, s(s(b)), s(y), z) → if(le(x, s(y)), x, s(s(b)), s(y), z)
if(true, x, b, y, z) → z
if(false, x, b, y, z) → loop(x, b, times(b, y), s(z))

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 [ ].

(2) BOUNDS(n^1, INF)

(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(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
log(x, 0') → baseError
log(x, s(0')) → baseError
log(0', s(s(b))) → logZeroError
log(s(x), s(s(b))) → loop(s(x), s(s(b)), s(0'), 0')
loop(x, s(s(b)), s(y), z) → if(le(x, s(y)), x, s(s(b)), s(y), z)
if(true, x, b, y, z) → z
if(false, x, b, y, z) → loop(x, b, times(b, y), s(z))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
log(x, 0') → baseError
log(x, s(0')) → baseError
log(0', s(s(b))) → logZeroError
log(s(x), s(s(b))) → loop(s(x), s(s(b)), s(0'), 0')
loop(x, s(s(b)), s(y), z) → if(le(x, s(y)), x, s(s(b)), s(y), z)
if(true, x, b, y, z) → z
if(false, x, b, y, z) → loop(x, b, times(b, y), s(z))

Types:
le :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → false:true
s :: s:0':baseError:logZeroError → s:0':baseError:logZeroError
0' :: s:0':baseError:logZeroError
false :: false:true
true :: false:true
plus :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
times :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
log :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
baseError :: s:0':baseError:logZeroError
logZeroError :: s:0':baseError:logZeroError
loop :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
if :: false:true → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
hole_false:true1_0 :: false:true
hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError
gen_s:0':baseError:logZeroError3_0 :: Nat → s:0':baseError:logZeroError

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

Heuristically decided to analyse the following defined symbols:
le, plus, times, loop

They will be analysed ascendingly in the following order:
le < loop
plus < times
times < loop

(8) Obligation:

TRS:
Rules:
le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
log(x, 0') → baseError
log(x, s(0')) → baseError
log(0', s(s(b))) → logZeroError
log(s(x), s(s(b))) → loop(s(x), s(s(b)), s(0'), 0')
loop(x, s(s(b)), s(y), z) → if(le(x, s(y)), x, s(s(b)), s(y), z)
if(true, x, b, y, z) → z
if(false, x, b, y, z) → loop(x, b, times(b, y), s(z))

Types:
le :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → false:true
s :: s:0':baseError:logZeroError → s:0':baseError:logZeroError
0' :: s:0':baseError:logZeroError
false :: false:true
true :: false:true
plus :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
times :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
log :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
baseError :: s:0':baseError:logZeroError
logZeroError :: s:0':baseError:logZeroError
loop :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
if :: false:true → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
hole_false:true1_0 :: false:true
hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError
gen_s:0':baseError:logZeroError3_0 :: Nat → s:0':baseError:logZeroError

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

The following defined symbols remain to be analysed:
le, plus, times, loop

They will be analysed ascendingly in the following order:
le < loop
plus < times
times < loop

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

Proved the following rewrite lemma:
le(gen_s:0':baseError:logZeroError3_0(+(1, n5_0)), gen_s:0':baseError:logZeroError3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)

Induction Base:
le(gen_s:0':baseError:logZeroError3_0(+(1, 0)), gen_s:0':baseError:logZeroError3_0(0)) →_R^Ω(1)
false

Induction Step:
le(gen_s:0':baseError:logZeroError3_0(+(1, +(n5_0, 1))), gen_s:0':baseError:logZeroError3_0(+(n5_0, 1))) →_R^Ω(1)
le(gen_s:0':baseError:logZeroError3_0(+(1, n5_0)), gen_s:0':baseError:logZeroError3_0(n5_0)) →_IH
false

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

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
log(x, 0') → baseError
log(x, s(0')) → baseError
log(0', s(s(b))) → logZeroError
log(s(x), s(s(b))) → loop(s(x), s(s(b)), s(0'), 0')
loop(x, s(s(b)), s(y), z) → if(le(x, s(y)), x, s(s(b)), s(y), z)
if(true, x, b, y, z) → z
if(false, x, b, y, z) → loop(x, b, times(b, y), s(z))

Types:
le :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → false:true
s :: s:0':baseError:logZeroError → s:0':baseError:logZeroError
0' :: s:0':baseError:logZeroError
false :: false:true
true :: false:true
plus :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
times :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
log :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
baseError :: s:0':baseError:logZeroError
logZeroError :: s:0':baseError:logZeroError
loop :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
if :: false:true → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
hole_false:true1_0 :: false:true
hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError
gen_s:0':baseError:logZeroError3_0 :: Nat → s:0':baseError:logZeroError

Lemmas:
le(gen_s:0':baseError:logZeroError3_0(+(1, n5_0)), gen_s:0':baseError:logZeroError3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)

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

The following defined symbols remain to be analysed:
plus, times, loop

They will be analysed ascendingly in the following order:
plus < times
times < loop

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

Proved the following rewrite lemma:
plus(gen_s:0':baseError:logZeroError3_0(n300_0), gen_s:0':baseError:logZeroError3_0(b)) → gen_s:0':baseError:logZeroError3_0(+(n300_0, b)), rt ∈ Ω(1 + n300₀)

Induction Base:
plus(gen_s:0':baseError:logZeroError3_0(0), gen_s:0':baseError:logZeroError3_0(b)) →_R^Ω(1)
gen_s:0':baseError:logZeroError3_0(b)

Induction Step:
plus(gen_s:0':baseError:logZeroError3_0(+(n300_0, 1)), gen_s:0':baseError:logZeroError3_0(b)) →_R^Ω(1)
s(plus(gen_s:0':baseError:logZeroError3_0(n300_0), gen_s:0':baseError:logZeroError3_0(b))) →_IH
s(gen_s:0':baseError:logZeroError3_0(+(b, c301_0)))

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

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
Rules:
le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
log(x, 0') → baseError
log(x, s(0')) → baseError
log(0', s(s(b))) → logZeroError
log(s(x), s(s(b))) → loop(s(x), s(s(b)), s(0'), 0')
loop(x, s(s(b)), s(y), z) → if(le(x, s(y)), x, s(s(b)), s(y), z)
if(true, x, b, y, z) → z
if(false, x, b, y, z) → loop(x, b, times(b, y), s(z))

Types:
le :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → false:true
s :: s:0':baseError:logZeroError → s:0':baseError:logZeroError
0' :: s:0':baseError:logZeroError
false :: false:true
true :: false:true
plus :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
times :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
log :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
baseError :: s:0':baseError:logZeroError
logZeroError :: s:0':baseError:logZeroError
loop :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
if :: false:true → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
hole_false:true1_0 :: false:true
hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError
gen_s:0':baseError:logZeroError3_0 :: Nat → s:0':baseError:logZeroError

Lemmas:
le(gen_s:0':baseError:logZeroError3_0(+(1, n5_0)), gen_s:0':baseError:logZeroError3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)
plus(gen_s:0':baseError:logZeroError3_0(n300_0), gen_s:0':baseError:logZeroError3_0(b)) → gen_s:0':baseError:logZeroError3_0(+(n300_0, b)), rt ∈ Ω(1 + n300₀)

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

The following defined symbols remain to be analysed:
times, loop

They will be analysed ascendingly in the following order:
times < loop

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

Proved the following rewrite lemma:
times(gen_s:0':baseError:logZeroError3_0(n897_0), gen_s:0':baseError:logZeroError3_0(b)) → gen_s:0':baseError:logZeroError3_0(*(n897_0, b)), rt ∈ Ω(1 + b·n897₀ + n897₀)

Induction Base:
times(gen_s:0':baseError:logZeroError3_0(0), gen_s:0':baseError:logZeroError3_0(b)) →_R^Ω(1)
0'

Induction Step:
times(gen_s:0':baseError:logZeroError3_0(+(n897_0, 1)), gen_s:0':baseError:logZeroError3_0(b)) →_R^Ω(1)
plus(gen_s:0':baseError:logZeroError3_0(b), times(gen_s:0':baseError:logZeroError3_0(n897_0), gen_s:0':baseError:logZeroError3_0(b))) →_IH
plus(gen_s:0':baseError:logZeroError3_0(b), gen_s:0':baseError:logZeroError3_0(*(c898_0, b))) →_L^{Ω(1 + b)}
gen_s:0':baseError:logZeroError3_0(+(b, *(n897_0, b)))

We have rt ∈ Ω(n²) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n²).

(16) Complex Obligation (BEST)

(17) Obligation:

TRS:
Rules:
le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
log(x, 0') → baseError
log(x, s(0')) → baseError
log(0', s(s(b))) → logZeroError
log(s(x), s(s(b))) → loop(s(x), s(s(b)), s(0'), 0')
loop(x, s(s(b)), s(y), z) → if(le(x, s(y)), x, s(s(b)), s(y), z)
if(true, x, b, y, z) → z
if(false, x, b, y, z) → loop(x, b, times(b, y), s(z))

Types:
le :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → false:true
s :: s:0':baseError:logZeroError → s:0':baseError:logZeroError
0' :: s:0':baseError:logZeroError
false :: false:true
true :: false:true
plus :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
times :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
log :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
baseError :: s:0':baseError:logZeroError
logZeroError :: s:0':baseError:logZeroError
loop :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
if :: false:true → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
hole_false:true1_0 :: false:true
hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError
gen_s:0':baseError:logZeroError3_0 :: Nat → s:0':baseError:logZeroError

Lemmas:
le(gen_s:0':baseError:logZeroError3_0(+(1, n5_0)), gen_s:0':baseError:logZeroError3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)
plus(gen_s:0':baseError:logZeroError3_0(n300_0), gen_s:0':baseError:logZeroError3_0(b)) → gen_s:0':baseError:logZeroError3_0(+(n300_0, b)), rt ∈ Ω(1 + n300₀)
times(gen_s:0':baseError:logZeroError3_0(n897_0), gen_s:0':baseError:logZeroError3_0(b)) → gen_s:0':baseError:logZeroError3_0(*(n897_0, b)), rt ∈ Ω(1 + b·n897₀ + n897₀)

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

The following defined symbols remain to be analysed:
loop

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

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

(19) Obligation:

TRS:
Rules:
le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
log(x, 0') → baseError
log(x, s(0')) → baseError
log(0', s(s(b))) → logZeroError
log(s(x), s(s(b))) → loop(s(x), s(s(b)), s(0'), 0')
loop(x, s(s(b)), s(y), z) → if(le(x, s(y)), x, s(s(b)), s(y), z)
if(true, x, b, y, z) → z
if(false, x, b, y, z) → loop(x, b, times(b, y), s(z))

Types:
le :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → false:true
s :: s:0':baseError:logZeroError → s:0':baseError:logZeroError
0' :: s:0':baseError:logZeroError
false :: false:true
true :: false:true
plus :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
times :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
log :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
baseError :: s:0':baseError:logZeroError
logZeroError :: s:0':baseError:logZeroError
loop :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
if :: false:true → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
hole_false:true1_0 :: false:true
hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError
gen_s:0':baseError:logZeroError3_0 :: Nat → s:0':baseError:logZeroError

Lemmas:
le(gen_s:0':baseError:logZeroError3_0(+(1, n5_0)), gen_s:0':baseError:logZeroError3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)
plus(gen_s:0':baseError:logZeroError3_0(n300_0), gen_s:0':baseError:logZeroError3_0(b)) → gen_s:0':baseError:logZeroError3_0(+(n300_0, b)), rt ∈ Ω(1 + n300₀)
times(gen_s:0':baseError:logZeroError3_0(n897_0), gen_s:0':baseError:logZeroError3_0(b)) → gen_s:0':baseError:logZeroError3_0(*(n897_0, b)), rt ∈ Ω(1 + b·n897₀ + n897₀)

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

No more defined symbols left to analyse.

(20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
times(gen_s:0':baseError:logZeroError3_0(n897_0), gen_s:0':baseError:logZeroError3_0(b)) → gen_s:0':baseError:logZeroError3_0(*(n897_0, b)), rt ∈ Ω(1 + b·n897₀ + n897₀)

(21) BOUNDS(n^2, INF)

(22) Obligation:

TRS:
Rules:
le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
log(x, 0') → baseError
log(x, s(0')) → baseError
log(0', s(s(b))) → logZeroError
log(s(x), s(s(b))) → loop(s(x), s(s(b)), s(0'), 0')
loop(x, s(s(b)), s(y), z) → if(le(x, s(y)), x, s(s(b)), s(y), z)
if(true, x, b, y, z) → z
if(false, x, b, y, z) → loop(x, b, times(b, y), s(z))

Types:
le :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → false:true
s :: s:0':baseError:logZeroError → s:0':baseError:logZeroError
0' :: s:0':baseError:logZeroError
false :: false:true
true :: false:true
plus :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
times :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
log :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
baseError :: s:0':baseError:logZeroError
logZeroError :: s:0':baseError:logZeroError
loop :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
if :: false:true → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
hole_false:true1_0 :: false:true
hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError
gen_s:0':baseError:logZeroError3_0 :: Nat → s:0':baseError:logZeroError

Lemmas:
le(gen_s:0':baseError:logZeroError3_0(+(1, n5_0)), gen_s:0':baseError:logZeroError3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)
plus(gen_s:0':baseError:logZeroError3_0(n300_0), gen_s:0':baseError:logZeroError3_0(b)) → gen_s:0':baseError:logZeroError3_0(+(n300_0, b)), rt ∈ Ω(1 + n300₀)
times(gen_s:0':baseError:logZeroError3_0(n897_0), gen_s:0':baseError:logZeroError3_0(b)) → gen_s:0':baseError:logZeroError3_0(*(n897_0, b)), rt ∈ Ω(1 + b·n897₀ + n897₀)

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

No more defined symbols left to analyse.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
times(gen_s:0':baseError:logZeroError3_0(n897_0), gen_s:0':baseError:logZeroError3_0(b)) → gen_s:0':baseError:logZeroError3_0(*(n897_0, b)), rt ∈ Ω(1 + b·n897₀ + n897₀)

(24) BOUNDS(n^2, INF)

(25) Obligation:

TRS:
Rules:
le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
log(x, 0') → baseError
log(x, s(0')) → baseError
log(0', s(s(b))) → logZeroError
log(s(x), s(s(b))) → loop(s(x), s(s(b)), s(0'), 0')
loop(x, s(s(b)), s(y), z) → if(le(x, s(y)), x, s(s(b)), s(y), z)
if(true, x, b, y, z) → z
if(false, x, b, y, z) → loop(x, b, times(b, y), s(z))

Types:
le :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → false:true
s :: s:0':baseError:logZeroError → s:0':baseError:logZeroError
0' :: s:0':baseError:logZeroError
false :: false:true
true :: false:true
plus :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
times :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
log :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
baseError :: s:0':baseError:logZeroError
logZeroError :: s:0':baseError:logZeroError
loop :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
if :: false:true → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
hole_false:true1_0 :: false:true
hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError
gen_s:0':baseError:logZeroError3_0 :: Nat → s:0':baseError:logZeroError

Lemmas:
le(gen_s:0':baseError:logZeroError3_0(+(1, n5_0)), gen_s:0':baseError:logZeroError3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)
plus(gen_s:0':baseError:logZeroError3_0(n300_0), gen_s:0':baseError:logZeroError3_0(b)) → gen_s:0':baseError:logZeroError3_0(+(n300_0, b)), rt ∈ Ω(1 + n300₀)

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

No more defined symbols left to analyse.

(26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
le(gen_s:0':baseError:logZeroError3_0(+(1, n5_0)), gen_s:0':baseError:logZeroError3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)

(27) BOUNDS(n^1, INF)

(28) Obligation:

TRS:
Rules:
le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
log(x, 0') → baseError
log(x, s(0')) → baseError
log(0', s(s(b))) → logZeroError
log(s(x), s(s(b))) → loop(s(x), s(s(b)), s(0'), 0')
loop(x, s(s(b)), s(y), z) → if(le(x, s(y)), x, s(s(b)), s(y), z)
if(true, x, b, y, z) → z
if(false, x, b, y, z) → loop(x, b, times(b, y), s(z))

Types:
le :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → false:true
s :: s:0':baseError:logZeroError → s:0':baseError:logZeroError
0' :: s:0':baseError:logZeroError
false :: false:true
true :: false:true
plus :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
times :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
log :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
baseError :: s:0':baseError:logZeroError
logZeroError :: s:0':baseError:logZeroError
loop :: s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
if :: false:true → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError → s:0':baseError:logZeroError
hole_false:true1_0 :: false:true
hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError
gen_s:0':baseError:logZeroError3_0 :: Nat → s:0':baseError:logZeroError

Lemmas:
le(gen_s:0':baseError:logZeroError3_0(+(1, n5_0)), gen_s:0':baseError:logZeroError3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)

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

No more defined symbols left to analyse.

(29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
le(gen_s:0':baseError:logZeroError3_0(+(1, n5_0)), gen_s:0':baseError:logZeroError3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)

(30) BOUNDS(n^1, INF)