### (0) Obligation:

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

-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
+(s(x), y) → s(+(x, y))
*(x, 0) → 0
*(x, s(y)) → +(x, *(x, y))
f(s(x)) → f(-(+(*(s(x), s(x)), *(s(x), s(s(s(0))))), *(s(s(x)), s(s(x)))))

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), s(y)) →+ -(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 [ ].

### (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:

-(x, 0') → x
-(s(x), s(y)) → -(x, y)
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(x, *'(x, y))
f(s(x)) → f(-(+'(*'(s(x), s(x)), *'(s(x), s(s(s(0'))))), *'(s(s(x)), s(s(x)))))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(x, *'(x, y))
f(s(x)) → f(-(+'(*'(s(x), s(x)), *'(s(x), s(s(s(0'))))), *'(s(s(x)), s(s(x)))))

Types:
- :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':s → 0':s → 0':s
*' :: 0':s → 0':s → 0':s
f :: 0':s → f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat → 0':s

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

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

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

### (8) Obligation:

TRS:
Rules:
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(x, *'(x, y))
f(s(x)) → f(-(+'(*'(s(x), s(x)), *'(s(x), s(s(s(0'))))), *'(s(s(x)), s(s(x)))))

Types:
- :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':s → 0':s → 0':s
*' :: 0':s → 0':s → 0':s
f :: 0':s → f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat → 0':s

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

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

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

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

Proved the following rewrite lemma:
-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n50)

Induction Base:
-(gen_0':s3_0(0), gen_0':s3_0(0)) →RΩ(1)
gen_0':s3_0(0)

Induction Step:
-(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) →RΩ(1)
-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) →IH
gen_0':s3_0(0)

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

### (11) Obligation:

TRS:
Rules:
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(x, *'(x, y))
f(s(x)) → f(-(+'(*'(s(x), s(x)), *'(s(x), s(s(s(0'))))), *'(s(s(x)), s(s(x)))))

Types:
- :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':s → 0':s → 0':s
*' :: 0':s → 0':s → 0':s
f :: 0':s → f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat → 0':s

Lemmas:
-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n50)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

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

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

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

Proved the following rewrite lemma:
+'(gen_0':s3_0(n237_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n237_0, b)), rt ∈ Ω(1 + n2370)

Induction Base:
+'(gen_0':s3_0(0), gen_0':s3_0(b)) →RΩ(1)
gen_0':s3_0(b)

Induction Step:
+'(gen_0':s3_0(+(n237_0, 1)), gen_0':s3_0(b)) →RΩ(1)
s(+'(gen_0':s3_0(n237_0), gen_0':s3_0(b))) →IH
s(gen_0':s3_0(+(b, c238_0)))

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

### (14) Obligation:

TRS:
Rules:
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(x, *'(x, y))
f(s(x)) → f(-(+'(*'(s(x), s(x)), *'(s(x), s(s(s(0'))))), *'(s(s(x)), s(s(x)))))

Types:
- :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':s → 0':s → 0':s
*' :: 0':s → 0':s → 0':s
f :: 0':s → f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat → 0':s

Lemmas:
-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n50)
+'(gen_0':s3_0(n237_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n237_0, b)), rt ∈ Ω(1 + n2370)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

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

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

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

Proved the following rewrite lemma:
*'(gen_0':s3_0(a), gen_0':s3_0(n708_0)) → gen_0':s3_0(*(n708_0, a)), rt ∈ Ω(1 + a·n7080 + n7080)

Induction Base:
*'(gen_0':s3_0(a), gen_0':s3_0(0)) →RΩ(1)
0'

Induction Step:
*'(gen_0':s3_0(a), gen_0':s3_0(+(n708_0, 1))) →RΩ(1)
+'(gen_0':s3_0(a), *'(gen_0':s3_0(a), gen_0':s3_0(n708_0))) →IH
+'(gen_0':s3_0(a), gen_0':s3_0(*(c709_0, a))) →LΩ(1 + a)
gen_0':s3_0(+(a, *(n708_0, a)))

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

### (17) Obligation:

TRS:
Rules:
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(x, *'(x, y))
f(s(x)) → f(-(+'(*'(s(x), s(x)), *'(s(x), s(s(s(0'))))), *'(s(s(x)), s(s(x)))))

Types:
- :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':s → 0':s → 0':s
*' :: 0':s → 0':s → 0':s
f :: 0':s → f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat → 0':s

Lemmas:
-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n50)
+'(gen_0':s3_0(n237_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n237_0, b)), rt ∈ Ω(1 + n2370)
*'(gen_0':s3_0(a), gen_0':s3_0(n708_0)) → gen_0':s3_0(*(n708_0, a)), rt ∈ Ω(1 + a·n7080 + n7080)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
f

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

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

### (19) Obligation:

TRS:
Rules:
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(x, *'(x, y))
f(s(x)) → f(-(+'(*'(s(x), s(x)), *'(s(x), s(s(s(0'))))), *'(s(s(x)), s(s(x)))))

Types:
- :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':s → 0':s → 0':s
*' :: 0':s → 0':s → 0':s
f :: 0':s → f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat → 0':s

Lemmas:
-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n50)
+'(gen_0':s3_0(n237_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n237_0, b)), rt ∈ Ω(1 + n2370)
*'(gen_0':s3_0(a), gen_0':s3_0(n708_0)) → gen_0':s3_0(*(n708_0, a)), rt ∈ Ω(1 + a·n7080 + n7080)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

### (20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
*'(gen_0':s3_0(a), gen_0':s3_0(n708_0)) → gen_0':s3_0(*(n708_0, a)), rt ∈ Ω(1 + a·n7080 + n7080)

### (22) Obligation:

TRS:
Rules:
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(x, *'(x, y))
f(s(x)) → f(-(+'(*'(s(x), s(x)), *'(s(x), s(s(s(0'))))), *'(s(s(x)), s(s(x)))))

Types:
- :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':s → 0':s → 0':s
*' :: 0':s → 0':s → 0':s
f :: 0':s → f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat → 0':s

Lemmas:
-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n50)
+'(gen_0':s3_0(n237_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n237_0, b)), rt ∈ Ω(1 + n2370)
*'(gen_0':s3_0(a), gen_0':s3_0(n708_0)) → gen_0':s3_0(*(n708_0, a)), rt ∈ Ω(1 + a·n7080 + n7080)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

### (23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
*'(gen_0':s3_0(a), gen_0':s3_0(n708_0)) → gen_0':s3_0(*(n708_0, a)), rt ∈ Ω(1 + a·n7080 + n7080)

### (25) Obligation:

TRS:
Rules:
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(x, *'(x, y))
f(s(x)) → f(-(+'(*'(s(x), s(x)), *'(s(x), s(s(s(0'))))), *'(s(s(x)), s(s(x)))))

Types:
- :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':s → 0':s → 0':s
*' :: 0':s → 0':s → 0':s
f :: 0':s → f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat → 0':s

Lemmas:
-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n50)
+'(gen_0':s3_0(n237_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n237_0, b)), rt ∈ Ω(1 + n2370)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

### (26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n50)

### (28) Obligation:

TRS:
Rules:
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(x, *'(x, y))
f(s(x)) → f(-(+'(*'(s(x), s(x)), *'(s(x), s(s(s(0'))))), *'(s(s(x)), s(s(x)))))

Types:
- :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':s → 0':s → 0':s
*' :: 0':s → 0':s → 0':s
f :: 0':s → f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat → 0':s

Lemmas:
-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n50)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

### (29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n50)