### (0) Obligation:

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

+(0, y) → y
+(s(x), y) → s(+(x, y))
+(p(x), y) → p(+(x, y))
minus(0) → 0
minus(s(x)) → p(minus(x))
minus(p(x)) → s(minus(x))
*(0, y) → 0
*(s(x), y) → +(*(x, y), y)
*(p(x), y) → +(*(x, y), minus(y))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

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

+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
+'(p(x), y) → p(+'(x, y))
minus(0') → 0'
minus(s(x)) → p(minus(x))
minus(p(x)) → s(minus(x))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
*'(p(x), y) → +'(*'(x, y), minus(y))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
+'(p(x), y) → p(+'(x, y))
minus(0') → 0'
minus(s(x)) → p(minus(x))
minus(p(x)) → s(minus(x))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
*'(p(x), y) → +'(*'(x, y), minus(y))

Types:
+' :: 0':s:p → 0':s:p → 0':s:p
0' :: 0':s:p
s :: 0':s:p → 0':s:p
p :: 0':s:p → 0':s:p
minus :: 0':s:p → 0':s:p
*' :: 0':s:p → 0':s:p → 0':s:p
hole_0':s:p1_0 :: 0':s:p
gen_0':s:p2_0 :: Nat → 0':s:p

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

Heuristically decided to analyse the following defined symbols:
+', minus, *'

They will be analysed ascendingly in the following order:
+' < *'
minus < *'

### (8) Obligation:

TRS:
Rules:
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
+'(p(x), y) → p(+'(x, y))
minus(0') → 0'
minus(s(x)) → p(minus(x))
minus(p(x)) → s(minus(x))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
*'(p(x), y) → +'(*'(x, y), minus(y))

Types:
+' :: 0':s:p → 0':s:p → 0':s:p
0' :: 0':s:p
s :: 0':s:p → 0':s:p
p :: 0':s:p → 0':s:p
minus :: 0':s:p → 0':s:p
*' :: 0':s:p → 0':s:p → 0':s:p
hole_0':s:p1_0 :: 0':s:p
gen_0':s:p2_0 :: Nat → 0':s:p

Generator Equations:
gen_0':s:p2_0(0) ⇔ 0'
gen_0':s:p2_0(+(x, 1)) ⇔ s(gen_0':s:p2_0(x))

The following defined symbols remain to be analysed:
+', minus, *'

They will be analysed ascendingly in the following order:
+' < *'
minus < *'

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

Proved the following rewrite lemma:
+'(gen_0':s:p2_0(n4_0), gen_0':s:p2_0(b)) → gen_0':s:p2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)

Induction Base:
+'(gen_0':s:p2_0(0), gen_0':s:p2_0(b)) →RΩ(1)
gen_0':s:p2_0(b)

Induction Step:
+'(gen_0':s:p2_0(+(n4_0, 1)), gen_0':s:p2_0(b)) →RΩ(1)
s(+'(gen_0':s:p2_0(n4_0), gen_0':s:p2_0(b))) →IH
s(gen_0':s:p2_0(+(b, c5_0)))

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

### (11) Obligation:

TRS:
Rules:
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
+'(p(x), y) → p(+'(x, y))
minus(0') → 0'
minus(s(x)) → p(minus(x))
minus(p(x)) → s(minus(x))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
*'(p(x), y) → +'(*'(x, y), minus(y))

Types:
+' :: 0':s:p → 0':s:p → 0':s:p
0' :: 0':s:p
s :: 0':s:p → 0':s:p
p :: 0':s:p → 0':s:p
minus :: 0':s:p → 0':s:p
*' :: 0':s:p → 0':s:p → 0':s:p
hole_0':s:p1_0 :: 0':s:p
gen_0':s:p2_0 :: Nat → 0':s:p

Lemmas:
+'(gen_0':s:p2_0(n4_0), gen_0':s:p2_0(b)) → gen_0':s:p2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)

Generator Equations:
gen_0':s:p2_0(0) ⇔ 0'
gen_0':s:p2_0(+(x, 1)) ⇔ s(gen_0':s:p2_0(x))

The following defined symbols remain to be analysed:
minus, *'

They will be analysed ascendingly in the following order:
minus < *'

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

Proved the following rewrite lemma:
minus(gen_0':s:p2_0(+(1, n595_0))) → *3_0, rt ∈ Ω(n5950)

Induction Base:
minus(gen_0':s:p2_0(+(1, 0)))

Induction Step:
minus(gen_0':s:p2_0(+(1, +(n595_0, 1)))) →RΩ(1)
p(minus(gen_0':s:p2_0(+(1, n595_0)))) →IH
p(*3_0)

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

### (14) Obligation:

TRS:
Rules:
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
+'(p(x), y) → p(+'(x, y))
minus(0') → 0'
minus(s(x)) → p(minus(x))
minus(p(x)) → s(minus(x))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
*'(p(x), y) → +'(*'(x, y), minus(y))

Types:
+' :: 0':s:p → 0':s:p → 0':s:p
0' :: 0':s:p
s :: 0':s:p → 0':s:p
p :: 0':s:p → 0':s:p
minus :: 0':s:p → 0':s:p
*' :: 0':s:p → 0':s:p → 0':s:p
hole_0':s:p1_0 :: 0':s:p
gen_0':s:p2_0 :: Nat → 0':s:p

Lemmas:
+'(gen_0':s:p2_0(n4_0), gen_0':s:p2_0(b)) → gen_0':s:p2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)
minus(gen_0':s:p2_0(+(1, n595_0))) → *3_0, rt ∈ Ω(n5950)

Generator Equations:
gen_0':s:p2_0(0) ⇔ 0'
gen_0':s:p2_0(+(x, 1)) ⇔ s(gen_0':s:p2_0(x))

The following defined symbols remain to be analysed:
*'

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

Proved the following rewrite lemma:
*'(gen_0':s:p2_0(n1725_0), gen_0':s:p2_0(b)) → gen_0':s:p2_0(*(n1725_0, b)), rt ∈ Ω(1 + b·n172502 + n17250)

Induction Base:
*'(gen_0':s:p2_0(0), gen_0':s:p2_0(b)) →RΩ(1)
0'

Induction Step:
*'(gen_0':s:p2_0(+(n1725_0, 1)), gen_0':s:p2_0(b)) →RΩ(1)
+'(*'(gen_0':s:p2_0(n1725_0), gen_0':s:p2_0(b)), gen_0':s:p2_0(b)) →IH
+'(gen_0':s:p2_0(*(c1726_0, b)), gen_0':s:p2_0(b)) →LΩ(1 + b·n17250)
gen_0':s:p2_0(+(*(n1725_0, b), b))

We have rt ∈ Ω(n3) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n3).

### (17) Obligation:

TRS:
Rules:
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
+'(p(x), y) → p(+'(x, y))
minus(0') → 0'
minus(s(x)) → p(minus(x))
minus(p(x)) → s(minus(x))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
*'(p(x), y) → +'(*'(x, y), minus(y))

Types:
+' :: 0':s:p → 0':s:p → 0':s:p
0' :: 0':s:p
s :: 0':s:p → 0':s:p
p :: 0':s:p → 0':s:p
minus :: 0':s:p → 0':s:p
*' :: 0':s:p → 0':s:p → 0':s:p
hole_0':s:p1_0 :: 0':s:p
gen_0':s:p2_0 :: Nat → 0':s:p

Lemmas:
+'(gen_0':s:p2_0(n4_0), gen_0':s:p2_0(b)) → gen_0':s:p2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)
minus(gen_0':s:p2_0(+(1, n595_0))) → *3_0, rt ∈ Ω(n5950)
*'(gen_0':s:p2_0(n1725_0), gen_0':s:p2_0(b)) → gen_0':s:p2_0(*(n1725_0, b)), rt ∈ Ω(1 + b·n172502 + n17250)

Generator Equations:
gen_0':s:p2_0(0) ⇔ 0'
gen_0':s:p2_0(+(x, 1)) ⇔ s(gen_0':s:p2_0(x))

No more defined symbols left to analyse.

### (18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n3) was proven with the following lemma:
*'(gen_0':s:p2_0(n1725_0), gen_0':s:p2_0(b)) → gen_0':s:p2_0(*(n1725_0, b)), rt ∈ Ω(1 + b·n172502 + n17250)

### (20) Obligation:

TRS:
Rules:
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
+'(p(x), y) → p(+'(x, y))
minus(0') → 0'
minus(s(x)) → p(minus(x))
minus(p(x)) → s(minus(x))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
*'(p(x), y) → +'(*'(x, y), minus(y))

Types:
+' :: 0':s:p → 0':s:p → 0':s:p
0' :: 0':s:p
s :: 0':s:p → 0':s:p
p :: 0':s:p → 0':s:p
minus :: 0':s:p → 0':s:p
*' :: 0':s:p → 0':s:p → 0':s:p
hole_0':s:p1_0 :: 0':s:p
gen_0':s:p2_0 :: Nat → 0':s:p

Lemmas:
+'(gen_0':s:p2_0(n4_0), gen_0':s:p2_0(b)) → gen_0':s:p2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)
minus(gen_0':s:p2_0(+(1, n595_0))) → *3_0, rt ∈ Ω(n5950)
*'(gen_0':s:p2_0(n1725_0), gen_0':s:p2_0(b)) → gen_0':s:p2_0(*(n1725_0, b)), rt ∈ Ω(1 + b·n172502 + n17250)

Generator Equations:
gen_0':s:p2_0(0) ⇔ 0'
gen_0':s:p2_0(+(x, 1)) ⇔ s(gen_0':s:p2_0(x))

No more defined symbols left to analyse.

### (21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n3) was proven with the following lemma:
*'(gen_0':s:p2_0(n1725_0), gen_0':s:p2_0(b)) → gen_0':s:p2_0(*(n1725_0, b)), rt ∈ Ω(1 + b·n172502 + n17250)

### (23) Obligation:

TRS:
Rules:
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
+'(p(x), y) → p(+'(x, y))
minus(0') → 0'
minus(s(x)) → p(minus(x))
minus(p(x)) → s(minus(x))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
*'(p(x), y) → +'(*'(x, y), minus(y))

Types:
+' :: 0':s:p → 0':s:p → 0':s:p
0' :: 0':s:p
s :: 0':s:p → 0':s:p
p :: 0':s:p → 0':s:p
minus :: 0':s:p → 0':s:p
*' :: 0':s:p → 0':s:p → 0':s:p
hole_0':s:p1_0 :: 0':s:p
gen_0':s:p2_0 :: Nat → 0':s:p

Lemmas:
+'(gen_0':s:p2_0(n4_0), gen_0':s:p2_0(b)) → gen_0':s:p2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)
minus(gen_0':s:p2_0(+(1, n595_0))) → *3_0, rt ∈ Ω(n5950)

Generator Equations:
gen_0':s:p2_0(0) ⇔ 0'
gen_0':s:p2_0(+(x, 1)) ⇔ s(gen_0':s:p2_0(x))

No more defined symbols left to analyse.

### (24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
+'(gen_0':s:p2_0(n4_0), gen_0':s:p2_0(b)) → gen_0':s:p2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)

### (26) Obligation:

TRS:
Rules:
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
+'(p(x), y) → p(+'(x, y))
minus(0') → 0'
minus(s(x)) → p(minus(x))
minus(p(x)) → s(minus(x))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
*'(p(x), y) → +'(*'(x, y), minus(y))

Types:
+' :: 0':s:p → 0':s:p → 0':s:p
0' :: 0':s:p
s :: 0':s:p → 0':s:p
p :: 0':s:p → 0':s:p
minus :: 0':s:p → 0':s:p
*' :: 0':s:p → 0':s:p → 0':s:p
hole_0':s:p1_0 :: 0':s:p
gen_0':s:p2_0 :: Nat → 0':s:p

Lemmas:
+'(gen_0':s:p2_0(n4_0), gen_0':s:p2_0(b)) → gen_0':s:p2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)

Generator Equations:
gen_0':s:p2_0(0) ⇔ 0'
gen_0':s:p2_0(+(x, 1)) ⇔ s(gen_0':s:p2_0(x))

No more defined symbols left to analyse.

### (27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
+'(gen_0':s:p2_0(n4_0), gen_0':s:p2_0(b)) → gen_0':s:p2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)