### (0) Obligation:

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

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

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

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

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
- :: 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:
+', -

### (8) Obligation:

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

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
- :: 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:
+', -

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

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

Induction Base:
+'(gen_0':s2_0(0), gen_0':s2_0(b)) →RΩ(1)
gen_0':s2_0(b)

Induction Step:
+'(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(b)) →RΩ(1)
s(+'(gen_0':s2_0(n4_0), gen_0':s2_0(b))) →IH
s(gen_0':s2_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))
-(0', y) → 0'
-(x, 0') → x
-(s(x), s(y)) → -(x, y)

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
- :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s2_0(n4_0), gen_0':s2_0(b)) → gen_0':s2_0(+(n4_0, b)), 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:
-

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

Proved the following rewrite lemma:
-(gen_0':s2_0(n421_0), gen_0':s2_0(n421_0)) → gen_0':s2_0(0), rt ∈ Ω(1 + n4210)

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

Induction Step:
-(gen_0':s2_0(+(n421_0, 1)), gen_0':s2_0(+(n421_0, 1))) →RΩ(1)
-(gen_0':s2_0(n421_0), gen_0':s2_0(n421_0)) →IH
gen_0':s2_0(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))
-(0', y) → 0'
-(x, 0') → x
-(s(x), s(y)) → -(x, y)

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
- :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s2_0(n4_0), gen_0':s2_0(b)) → gen_0':s2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)
-(gen_0':s2_0(n421_0), gen_0':s2_0(n421_0)) → gen_0':s2_0(0), rt ∈ Ω(1 + n4210)

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

### (17) Obligation:

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

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
- :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s2_0(n4_0), gen_0':s2_0(b)) → gen_0':s2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)
-(gen_0':s2_0(n421_0), gen_0':s2_0(n421_0)) → gen_0':s2_0(0), rt ∈ Ω(1 + n4210)

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

### (20) Obligation:

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

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
- :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s2_0(n4_0), gen_0':s2_0(b)) → gen_0':s2_0(+(n4_0, b)), 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:
+'(gen_0':s2_0(n4_0), gen_0':s2_0(b)) → gen_0':s2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)