### (0) Obligation:

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

times(x, 0) → 0
times(x, s(y)) → plus(times(x, y), x)
plus(x, 0) → x
plus(0, x) → x
plus(x, s(y)) → s(plus(x, y))
plus(s(x), y) → s(plus(x, y))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

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

times(x, 0') → 0'
times(x, s(y)) → plus(times(x, y), x)
plus(x, 0') → x
plus(0', x) → x
plus(x, s(y)) → s(plus(x, y))
plus(s(x), y) → s(plus(x, y))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
times(x, 0') → 0'
times(x, s(y)) → plus(times(x, y), x)
plus(x, 0') → x
plus(0', x) → x
plus(x, s(y)) → s(plus(x, y))
plus(s(x), y) → s(plus(x, y))

Types:
times :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
plus :: 0':s → 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:
times, plus

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

### (8) Obligation:

TRS:
Rules:
times(x, 0') → 0'
times(x, s(y)) → plus(times(x, y), x)
plus(x, 0') → x
plus(0', x) → x
plus(x, s(y)) → s(plus(x, y))
plus(s(x), y) → s(plus(x, y))

Types:
times :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
plus :: 0':s → 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:
plus, times

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

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

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

Induction Base:
plus(gen_0':s2_0(a), gen_0':s2_0(0)) →RΩ(1)
gen_0':s2_0(a)

Induction Step:
plus(gen_0':s2_0(a), gen_0':s2_0(+(n4_0, 1))) →RΩ(1)
s(plus(gen_0':s2_0(a), gen_0':s2_0(n4_0))) →IH
s(gen_0':s2_0(+(a, c5_0)))

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

### (11) Obligation:

TRS:
Rules:
times(x, 0') → 0'
times(x, s(y)) → plus(times(x, y), x)
plus(x, 0') → x
plus(0', x) → x
plus(x, s(y)) → s(plus(x, y))
plus(s(x), y) → s(plus(x, y))

Types:
times :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
plus :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), 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:
times

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

Proved the following rewrite lemma:
times(gen_0':s2_0(a), gen_0':s2_0(n544_0)) → gen_0':s2_0(*(n544_0, a)), rt ∈ Ω(1 + a·n5440 + n5440)

Induction Base:
times(gen_0':s2_0(a), gen_0':s2_0(0)) →RΩ(1)
0'

Induction Step:
times(gen_0':s2_0(a), gen_0':s2_0(+(n544_0, 1))) →RΩ(1)
plus(times(gen_0':s2_0(a), gen_0':s2_0(n544_0)), gen_0':s2_0(a)) →IH
plus(gen_0':s2_0(*(c545_0, a)), gen_0':s2_0(a)) →LΩ(1 + a)
gen_0':s2_0(+(a, *(n544_0, a)))

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

### (14) Obligation:

TRS:
Rules:
times(x, 0') → 0'
times(x, s(y)) → plus(times(x, y), x)
plus(x, 0') → x
plus(0', x) → x
plus(x, s(y)) → s(plus(x, y))
plus(s(x), y) → s(plus(x, y))

Types:
times :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
plus :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n40)
times(gen_0':s2_0(a), gen_0':s2_0(n544_0)) → gen_0':s2_0(*(n544_0, a)), rt ∈ Ω(1 + a·n5440 + n5440)

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.

### (15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
times(gen_0':s2_0(a), gen_0':s2_0(n544_0)) → gen_0':s2_0(*(n544_0, a)), rt ∈ Ω(1 + a·n5440 + n5440)

### (17) Obligation:

TRS:
Rules:
times(x, 0') → 0'
times(x, s(y)) → plus(times(x, y), x)
plus(x, 0') → x
plus(0', x) → x
plus(x, s(y)) → s(plus(x, y))
plus(s(x), y) → s(plus(x, y))

Types:
times :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
plus :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n40)
times(gen_0':s2_0(a), gen_0':s2_0(n544_0)) → gen_0':s2_0(*(n544_0, a)), rt ∈ Ω(1 + a·n5440 + n5440)

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.

### (18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
times(gen_0':s2_0(a), gen_0':s2_0(n544_0)) → gen_0':s2_0(*(n544_0, a)), rt ∈ Ω(1 + a·n5440 + n5440)

### (20) Obligation:

TRS:
Rules:
times(x, 0') → 0'
times(x, s(y)) → plus(times(x, y), x)
plus(x, 0') → x
plus(0', x) → x
plus(x, s(y)) → s(plus(x, y))
plus(s(x), y) → s(plus(x, y))

Types:
times :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
plus :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), 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.

### (21) LowerBoundsProof (EQUIVALENT transformation)

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