The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(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) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) 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

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(4) 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

(5) OrderProof (LOWER BOUND(ID) transformation)

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

(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

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:
+', -

(7) 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)) →_R^Ω(1)
gen_0':s2_0(b)

Induction Step:
+'(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(b)) →_R^Ω(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 irc_R ∈ Ω(n).

(8) Complex Obligation (BEST)

(9) 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:
-

(10) 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)) →_R^Ω(1)
0'

Induction Step:
-(gen_0':s2_0(+(n421_0, 1)), gen_0':s2_0(+(n421_0, 1))) →_R^Ω(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 irc_R ∈ Ω(n).

(11) Complex Obligation (BEST)

(12) 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.

(13) 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)

(14) BOUNDS(n^1, INF)

(15) 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.

(16) 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) BOUNDS(n^1, INF)

(18) 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.

(19) 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) BOUNDS(n^1, INF)