### (0) Obligation:

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

inc(s(x)) → s(inc(x))
inc(0) → s(0)
plus(x, y) → ifPlus(eq(x, 0), minus(x, s(0)), x, inc(x))
ifPlus(false, x, y, z) → plus(x, z)
ifPlus(true, x, y, z) → y
minus(s(x), s(y)) → minus(x, y)
minus(0, x) → 0
minus(x, 0) → x
minus(x, x) → 0
eq(s(x), s(y)) → eq(x, y)
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(0, 0) → true
eq(x, x) → true
times(x, y) → timesIter(x, y, 0)
timesIter(x, y, z) → ifTimes(eq(x, 0), minus(x, s(0)), y, z, plus(y, z))
ifTimes(true, x, y, z, u) → z
ifTimes(false, x, y, z, u) → timesIter(x, y, u)
fg
fh

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
inc(s(x)) →+ s(inc(x))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x / 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:

inc(s(x)) → s(inc(x))
inc(0') → s(0')
plus(x, y) → ifPlus(eq(x, 0'), minus(x, s(0')), x, inc(x))
ifPlus(false, x, y, z) → plus(x, z)
ifPlus(true, x, y, z) → y
minus(s(x), s(y)) → minus(x, y)
minus(0', x) → 0'
minus(x, 0') → x
minus(x, x) → 0'
eq(s(x), s(y)) → eq(x, y)
eq(0', s(x)) → false
eq(s(x), 0') → false
eq(0', 0') → true
eq(x, x) → true
times(x, y) → timesIter(x, y, 0')
timesIter(x, y, z) → ifTimes(eq(x, 0'), minus(x, s(0')), y, z, plus(y, z))
ifTimes(true, x, y, z, u) → z
ifTimes(false, x, y, z, u) → timesIter(x, y, u)
fg
fh

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
inc(s(x)) → s(inc(x))
inc(0') → s(0')
plus(x, y) → ifPlus(eq(x, 0'), minus(x, s(0')), x, inc(x))
ifPlus(false, x, y, z) → plus(x, z)
ifPlus(true, x, y, z) → y
minus(s(x), s(y)) → minus(x, y)
minus(0', x) → 0'
minus(x, 0') → x
minus(x, x) → 0'
eq(s(x), s(y)) → eq(x, y)
eq(0', s(x)) → false
eq(s(x), 0') → false
eq(0', 0') → true
eq(x, x) → true
times(x, y) → timesIter(x, y, 0')
timesIter(x, y, z) → ifTimes(eq(x, 0'), minus(x, s(0')), y, z, plus(y, z))
ifTimes(true, x, y, z, u) → z
ifTimes(false, x, y, z, u) → timesIter(x, y, u)
fg
fh

Types:
inc :: s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
plus :: s:0' → s:0' → s:0'
ifPlus :: false:true → s:0' → s:0' → s:0' → s:0'
eq :: s:0' → s:0' → false:true
minus :: s:0' → s:0' → s:0'
false :: false:true
true :: false:true
times :: s:0' → s:0' → s:0'
timesIter :: s:0' → s:0' → s:0' → s:0'
ifTimes :: false:true → s:0' → s:0' → s:0' → s:0' → s:0'
f :: g:h
g :: g:h
h :: g:h
hole_s:0'1_0 :: s:0'
hole_false:true2_0 :: false:true
hole_g:h3_0 :: g:h
gen_s:0'4_0 :: Nat → s:0'

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

Heuristically decided to analyse the following defined symbols:
inc, plus, eq, minus, timesIter

They will be analysed ascendingly in the following order:
inc < plus
eq < plus
minus < plus
plus < timesIter
eq < timesIter
minus < timesIter

### (8) Obligation:

TRS:
Rules:
inc(s(x)) → s(inc(x))
inc(0') → s(0')
plus(x, y) → ifPlus(eq(x, 0'), minus(x, s(0')), x, inc(x))
ifPlus(false, x, y, z) → plus(x, z)
ifPlus(true, x, y, z) → y
minus(s(x), s(y)) → minus(x, y)
minus(0', x) → 0'
minus(x, 0') → x
minus(x, x) → 0'
eq(s(x), s(y)) → eq(x, y)
eq(0', s(x)) → false
eq(s(x), 0') → false
eq(0', 0') → true
eq(x, x) → true
times(x, y) → timesIter(x, y, 0')
timesIter(x, y, z) → ifTimes(eq(x, 0'), minus(x, s(0')), y, z, plus(y, z))
ifTimes(true, x, y, z, u) → z
ifTimes(false, x, y, z, u) → timesIter(x, y, u)
fg
fh

Types:
inc :: s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
plus :: s:0' → s:0' → s:0'
ifPlus :: false:true → s:0' → s:0' → s:0' → s:0'
eq :: s:0' → s:0' → false:true
minus :: s:0' → s:0' → s:0'
false :: false:true
true :: false:true
times :: s:0' → s:0' → s:0'
timesIter :: s:0' → s:0' → s:0' → s:0'
ifTimes :: false:true → s:0' → s:0' → s:0' → s:0' → s:0'
f :: g:h
g :: g:h
h :: g:h
hole_s:0'1_0 :: s:0'
hole_false:true2_0 :: false:true
hole_g:h3_0 :: g:h
gen_s:0'4_0 :: Nat → s:0'

Generator Equations:
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

The following defined symbols remain to be analysed:
inc, plus, eq, minus, timesIter

They will be analysed ascendingly in the following order:
inc < plus
eq < plus
minus < plus
plus < timesIter
eq < timesIter
minus < timesIter

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

Proved the following rewrite lemma:
inc(gen_s:0'4_0(n6_0)) → gen_s:0'4_0(+(1, n6_0)), rt ∈ Ω(1 + n60)

Induction Base:
inc(gen_s:0'4_0(0)) →RΩ(1)
s(0')

Induction Step:
inc(gen_s:0'4_0(+(n6_0, 1))) →RΩ(1)
s(inc(gen_s:0'4_0(n6_0))) →IH
s(gen_s:0'4_0(+(1, c7_0)))

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

### (11) Obligation:

TRS:
Rules:
inc(s(x)) → s(inc(x))
inc(0') → s(0')
plus(x, y) → ifPlus(eq(x, 0'), minus(x, s(0')), x, inc(x))
ifPlus(false, x, y, z) → plus(x, z)
ifPlus(true, x, y, z) → y
minus(s(x), s(y)) → minus(x, y)
minus(0', x) → 0'
minus(x, 0') → x
minus(x, x) → 0'
eq(s(x), s(y)) → eq(x, y)
eq(0', s(x)) → false
eq(s(x), 0') → false
eq(0', 0') → true
eq(x, x) → true
times(x, y) → timesIter(x, y, 0')
timesIter(x, y, z) → ifTimes(eq(x, 0'), minus(x, s(0')), y, z, plus(y, z))
ifTimes(true, x, y, z, u) → z
ifTimes(false, x, y, z, u) → timesIter(x, y, u)
fg
fh

Types:
inc :: s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
plus :: s:0' → s:0' → s:0'
ifPlus :: false:true → s:0' → s:0' → s:0' → s:0'
eq :: s:0' → s:0' → false:true
minus :: s:0' → s:0' → s:0'
false :: false:true
true :: false:true
times :: s:0' → s:0' → s:0'
timesIter :: s:0' → s:0' → s:0' → s:0'
ifTimes :: false:true → s:0' → s:0' → s:0' → s:0' → s:0'
f :: g:h
g :: g:h
h :: g:h
hole_s:0'1_0 :: s:0'
hole_false:true2_0 :: false:true
hole_g:h3_0 :: g:h
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
inc(gen_s:0'4_0(n6_0)) → gen_s:0'4_0(+(1, n6_0)), rt ∈ Ω(1 + n60)

Generator Equations:
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

The following defined symbols remain to be analysed:
eq, plus, minus, timesIter

They will be analysed ascendingly in the following order:
eq < plus
minus < plus
plus < timesIter
eq < timesIter
minus < timesIter

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

Proved the following rewrite lemma:
eq(gen_s:0'4_0(n239_0), gen_s:0'4_0(+(1, n239_0))) → false, rt ∈ Ω(1 + n2390)

Induction Base:
eq(gen_s:0'4_0(0), gen_s:0'4_0(+(1, 0))) →RΩ(1)
false

Induction Step:
eq(gen_s:0'4_0(+(n239_0, 1)), gen_s:0'4_0(+(1, +(n239_0, 1)))) →RΩ(1)
eq(gen_s:0'4_0(n239_0), gen_s:0'4_0(+(1, n239_0))) →IH
false

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

### (14) Obligation:

TRS:
Rules:
inc(s(x)) → s(inc(x))
inc(0') → s(0')
plus(x, y) → ifPlus(eq(x, 0'), minus(x, s(0')), x, inc(x))
ifPlus(false, x, y, z) → plus(x, z)
ifPlus(true, x, y, z) → y
minus(s(x), s(y)) → minus(x, y)
minus(0', x) → 0'
minus(x, 0') → x
minus(x, x) → 0'
eq(s(x), s(y)) → eq(x, y)
eq(0', s(x)) → false
eq(s(x), 0') → false
eq(0', 0') → true
eq(x, x) → true
times(x, y) → timesIter(x, y, 0')
timesIter(x, y, z) → ifTimes(eq(x, 0'), minus(x, s(0')), y, z, plus(y, z))
ifTimes(true, x, y, z, u) → z
ifTimes(false, x, y, z, u) → timesIter(x, y, u)
fg
fh

Types:
inc :: s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
plus :: s:0' → s:0' → s:0'
ifPlus :: false:true → s:0' → s:0' → s:0' → s:0'
eq :: s:0' → s:0' → false:true
minus :: s:0' → s:0' → s:0'
false :: false:true
true :: false:true
times :: s:0' → s:0' → s:0'
timesIter :: s:0' → s:0' → s:0' → s:0'
ifTimes :: false:true → s:0' → s:0' → s:0' → s:0' → s:0'
f :: g:h
g :: g:h
h :: g:h
hole_s:0'1_0 :: s:0'
hole_false:true2_0 :: false:true
hole_g:h3_0 :: g:h
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
inc(gen_s:0'4_0(n6_0)) → gen_s:0'4_0(+(1, n6_0)), rt ∈ Ω(1 + n60)
eq(gen_s:0'4_0(n239_0), gen_s:0'4_0(+(1, n239_0))) → false, rt ∈ Ω(1 + n2390)

Generator Equations:
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

The following defined symbols remain to be analysed:
minus, plus, timesIter

They will be analysed ascendingly in the following order:
minus < plus
plus < timesIter
minus < timesIter

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

Proved the following rewrite lemma:
minus(gen_s:0'4_0(n780_0), gen_s:0'4_0(n780_0)) → gen_s:0'4_0(0), rt ∈ Ω(1 + n7800)

Induction Base:
minus(gen_s:0'4_0(0), gen_s:0'4_0(0)) →RΩ(1)
0'

Induction Step:
minus(gen_s:0'4_0(+(n780_0, 1)), gen_s:0'4_0(+(n780_0, 1))) →RΩ(1)
minus(gen_s:0'4_0(n780_0), gen_s:0'4_0(n780_0)) →IH
gen_s:0'4_0(0)

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

### (17) Obligation:

TRS:
Rules:
inc(s(x)) → s(inc(x))
inc(0') → s(0')
plus(x, y) → ifPlus(eq(x, 0'), minus(x, s(0')), x, inc(x))
ifPlus(false, x, y, z) → plus(x, z)
ifPlus(true, x, y, z) → y
minus(s(x), s(y)) → minus(x, y)
minus(0', x) → 0'
minus(x, 0') → x
minus(x, x) → 0'
eq(s(x), s(y)) → eq(x, y)
eq(0', s(x)) → false
eq(s(x), 0') → false
eq(0', 0') → true
eq(x, x) → true
times(x, y) → timesIter(x, y, 0')
timesIter(x, y, z) → ifTimes(eq(x, 0'), minus(x, s(0')), y, z, plus(y, z))
ifTimes(true, x, y, z, u) → z
ifTimes(false, x, y, z, u) → timesIter(x, y, u)
fg
fh

Types:
inc :: s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
plus :: s:0' → s:0' → s:0'
ifPlus :: false:true → s:0' → s:0' → s:0' → s:0'
eq :: s:0' → s:0' → false:true
minus :: s:0' → s:0' → s:0'
false :: false:true
true :: false:true
times :: s:0' → s:0' → s:0'
timesIter :: s:0' → s:0' → s:0' → s:0'
ifTimes :: false:true → s:0' → s:0' → s:0' → s:0' → s:0'
f :: g:h
g :: g:h
h :: g:h
hole_s:0'1_0 :: s:0'
hole_false:true2_0 :: false:true
hole_g:h3_0 :: g:h
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
inc(gen_s:0'4_0(n6_0)) → gen_s:0'4_0(+(1, n6_0)), rt ∈ Ω(1 + n60)
eq(gen_s:0'4_0(n239_0), gen_s:0'4_0(+(1, n239_0))) → false, rt ∈ Ω(1 + n2390)
minus(gen_s:0'4_0(n780_0), gen_s:0'4_0(n780_0)) → gen_s:0'4_0(0), rt ∈ Ω(1 + n7800)

Generator Equations:
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

The following defined symbols remain to be analysed:
plus, timesIter

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

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

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

### (19) Obligation:

TRS:
Rules:
inc(s(x)) → s(inc(x))
inc(0') → s(0')
plus(x, y) → ifPlus(eq(x, 0'), minus(x, s(0')), x, inc(x))
ifPlus(false, x, y, z) → plus(x, z)
ifPlus(true, x, y, z) → y
minus(s(x), s(y)) → minus(x, y)
minus(0', x) → 0'
minus(x, 0') → x
minus(x, x) → 0'
eq(s(x), s(y)) → eq(x, y)
eq(0', s(x)) → false
eq(s(x), 0') → false
eq(0', 0') → true
eq(x, x) → true
times(x, y) → timesIter(x, y, 0')
timesIter(x, y, z) → ifTimes(eq(x, 0'), minus(x, s(0')), y, z, plus(y, z))
ifTimes(true, x, y, z, u) → z
ifTimes(false, x, y, z, u) → timesIter(x, y, u)
fg
fh

Types:
inc :: s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
plus :: s:0' → s:0' → s:0'
ifPlus :: false:true → s:0' → s:0' → s:0' → s:0'
eq :: s:0' → s:0' → false:true
minus :: s:0' → s:0' → s:0'
false :: false:true
true :: false:true
times :: s:0' → s:0' → s:0'
timesIter :: s:0' → s:0' → s:0' → s:0'
ifTimes :: false:true → s:0' → s:0' → s:0' → s:0' → s:0'
f :: g:h
g :: g:h
h :: g:h
hole_s:0'1_0 :: s:0'
hole_false:true2_0 :: false:true
hole_g:h3_0 :: g:h
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
inc(gen_s:0'4_0(n6_0)) → gen_s:0'4_0(+(1, n6_0)), rt ∈ Ω(1 + n60)
eq(gen_s:0'4_0(n239_0), gen_s:0'4_0(+(1, n239_0))) → false, rt ∈ Ω(1 + n2390)
minus(gen_s:0'4_0(n780_0), gen_s:0'4_0(n780_0)) → gen_s:0'4_0(0), rt ∈ Ω(1 + n7800)

Generator Equations:
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

The following defined symbols remain to be analysed:
timesIter

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

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

### (21) Obligation:

TRS:
Rules:
inc(s(x)) → s(inc(x))
inc(0') → s(0')
plus(x, y) → ifPlus(eq(x, 0'), minus(x, s(0')), x, inc(x))
ifPlus(false, x, y, z) → plus(x, z)
ifPlus(true, x, y, z) → y
minus(s(x), s(y)) → minus(x, y)
minus(0', x) → 0'
minus(x, 0') → x
minus(x, x) → 0'
eq(s(x), s(y)) → eq(x, y)
eq(0', s(x)) → false
eq(s(x), 0') → false
eq(0', 0') → true
eq(x, x) → true
times(x, y) → timesIter(x, y, 0')
timesIter(x, y, z) → ifTimes(eq(x, 0'), minus(x, s(0')), y, z, plus(y, z))
ifTimes(true, x, y, z, u) → z
ifTimes(false, x, y, z, u) → timesIter(x, y, u)
fg
fh

Types:
inc :: s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
plus :: s:0' → s:0' → s:0'
ifPlus :: false:true → s:0' → s:0' → s:0' → s:0'
eq :: s:0' → s:0' → false:true
minus :: s:0' → s:0' → s:0'
false :: false:true
true :: false:true
times :: s:0' → s:0' → s:0'
timesIter :: s:0' → s:0' → s:0' → s:0'
ifTimes :: false:true → s:0' → s:0' → s:0' → s:0' → s:0'
f :: g:h
g :: g:h
h :: g:h
hole_s:0'1_0 :: s:0'
hole_false:true2_0 :: false:true
hole_g:h3_0 :: g:h
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
inc(gen_s:0'4_0(n6_0)) → gen_s:0'4_0(+(1, n6_0)), rt ∈ Ω(1 + n60)
eq(gen_s:0'4_0(n239_0), gen_s:0'4_0(+(1, n239_0))) → false, rt ∈ Ω(1 + n2390)
minus(gen_s:0'4_0(n780_0), gen_s:0'4_0(n780_0)) → gen_s:0'4_0(0), rt ∈ Ω(1 + n7800)

Generator Equations:
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

No more defined symbols left to analyse.

### (22) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
inc(gen_s:0'4_0(n6_0)) → gen_s:0'4_0(+(1, n6_0)), rt ∈ Ω(1 + n60)

### (24) Obligation:

TRS:
Rules:
inc(s(x)) → s(inc(x))
inc(0') → s(0')
plus(x, y) → ifPlus(eq(x, 0'), minus(x, s(0')), x, inc(x))
ifPlus(false, x, y, z) → plus(x, z)
ifPlus(true, x, y, z) → y
minus(s(x), s(y)) → minus(x, y)
minus(0', x) → 0'
minus(x, 0') → x
minus(x, x) → 0'
eq(s(x), s(y)) → eq(x, y)
eq(0', s(x)) → false
eq(s(x), 0') → false
eq(0', 0') → true
eq(x, x) → true
times(x, y) → timesIter(x, y, 0')
timesIter(x, y, z) → ifTimes(eq(x, 0'), minus(x, s(0')), y, z, plus(y, z))
ifTimes(true, x, y, z, u) → z
ifTimes(false, x, y, z, u) → timesIter(x, y, u)
fg
fh

Types:
inc :: s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
plus :: s:0' → s:0' → s:0'
ifPlus :: false:true → s:0' → s:0' → s:0' → s:0'
eq :: s:0' → s:0' → false:true
minus :: s:0' → s:0' → s:0'
false :: false:true
true :: false:true
times :: s:0' → s:0' → s:0'
timesIter :: s:0' → s:0' → s:0' → s:0'
ifTimes :: false:true → s:0' → s:0' → s:0' → s:0' → s:0'
f :: g:h
g :: g:h
h :: g:h
hole_s:0'1_0 :: s:0'
hole_false:true2_0 :: false:true
hole_g:h3_0 :: g:h
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
inc(gen_s:0'4_0(n6_0)) → gen_s:0'4_0(+(1, n6_0)), rt ∈ Ω(1 + n60)
eq(gen_s:0'4_0(n239_0), gen_s:0'4_0(+(1, n239_0))) → false, rt ∈ Ω(1 + n2390)
minus(gen_s:0'4_0(n780_0), gen_s:0'4_0(n780_0)) → gen_s:0'4_0(0), rt ∈ Ω(1 + n7800)

Generator Equations:
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

No more defined symbols left to analyse.

### (25) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
inc(gen_s:0'4_0(n6_0)) → gen_s:0'4_0(+(1, n6_0)), rt ∈ Ω(1 + n60)

### (27) Obligation:

TRS:
Rules:
inc(s(x)) → s(inc(x))
inc(0') → s(0')
plus(x, y) → ifPlus(eq(x, 0'), minus(x, s(0')), x, inc(x))
ifPlus(false, x, y, z) → plus(x, z)
ifPlus(true, x, y, z) → y
minus(s(x), s(y)) → minus(x, y)
minus(0', x) → 0'
minus(x, 0') → x
minus(x, x) → 0'
eq(s(x), s(y)) → eq(x, y)
eq(0', s(x)) → false
eq(s(x), 0') → false
eq(0', 0') → true
eq(x, x) → true
times(x, y) → timesIter(x, y, 0')
timesIter(x, y, z) → ifTimes(eq(x, 0'), minus(x, s(0')), y, z, plus(y, z))
ifTimes(true, x, y, z, u) → z
ifTimes(false, x, y, z, u) → timesIter(x, y, u)
fg
fh

Types:
inc :: s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
plus :: s:0' → s:0' → s:0'
ifPlus :: false:true → s:0' → s:0' → s:0' → s:0'
eq :: s:0' → s:0' → false:true
minus :: s:0' → s:0' → s:0'
false :: false:true
true :: false:true
times :: s:0' → s:0' → s:0'
timesIter :: s:0' → s:0' → s:0' → s:0'
ifTimes :: false:true → s:0' → s:0' → s:0' → s:0' → s:0'
f :: g:h
g :: g:h
h :: g:h
hole_s:0'1_0 :: s:0'
hole_false:true2_0 :: false:true
hole_g:h3_0 :: g:h
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
inc(gen_s:0'4_0(n6_0)) → gen_s:0'4_0(+(1, n6_0)), rt ∈ Ω(1 + n60)
eq(gen_s:0'4_0(n239_0), gen_s:0'4_0(+(1, n239_0))) → false, rt ∈ Ω(1 + n2390)

Generator Equations:
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

No more defined symbols left to analyse.

### (28) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
inc(gen_s:0'4_0(n6_0)) → gen_s:0'4_0(+(1, n6_0)), rt ∈ Ω(1 + n60)

### (30) Obligation:

TRS:
Rules:
inc(s(x)) → s(inc(x))
inc(0') → s(0')
plus(x, y) → ifPlus(eq(x, 0'), minus(x, s(0')), x, inc(x))
ifPlus(false, x, y, z) → plus(x, z)
ifPlus(true, x, y, z) → y
minus(s(x), s(y)) → minus(x, y)
minus(0', x) → 0'
minus(x, 0') → x
minus(x, x) → 0'
eq(s(x), s(y)) → eq(x, y)
eq(0', s(x)) → false
eq(s(x), 0') → false
eq(0', 0') → true
eq(x, x) → true
times(x, y) → timesIter(x, y, 0')
timesIter(x, y, z) → ifTimes(eq(x, 0'), minus(x, s(0')), y, z, plus(y, z))
ifTimes(true, x, y, z, u) → z
ifTimes(false, x, y, z, u) → timesIter(x, y, u)
fg
fh

Types:
inc :: s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
plus :: s:0' → s:0' → s:0'
ifPlus :: false:true → s:0' → s:0' → s:0' → s:0'
eq :: s:0' → s:0' → false:true
minus :: s:0' → s:0' → s:0'
false :: false:true
true :: false:true
times :: s:0' → s:0' → s:0'
timesIter :: s:0' → s:0' → s:0' → s:0'
ifTimes :: false:true → s:0' → s:0' → s:0' → s:0' → s:0'
f :: g:h
g :: g:h
h :: g:h
hole_s:0'1_0 :: s:0'
hole_false:true2_0 :: false:true
hole_g:h3_0 :: g:h
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
inc(gen_s:0'4_0(n6_0)) → gen_s:0'4_0(+(1, n6_0)), rt ∈ Ω(1 + n60)

Generator Equations:
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

No more defined symbols left to analyse.

### (31) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
inc(gen_s:0'4_0(n6_0)) → gen_s:0'4_0(+(1, n6_0)), rt ∈ Ω(1 + n60)