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

0 CpxTRS
↳1 RenamingProof (⇔, 0 ms)
↳2 CpxRelTRS
↳3 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 typed CpxTrs
↳5 OrderProof (LOWER BOUND(ID), 0 ms)
↳6 typed CpxTrs
↳7 RewriteLemmaProof (LOWER BOUND(ID), 863 ms)
↳8 BEST
    ↳9 typed CpxTrs
        ↳10 RewriteLemmaProof (LOWER BOUND(ID), 21 ms)
        ↳11 BEST
            ↳12 typed CpxTrs
                ↳13 LowerBoundsProof (⇔, 0 ms)
                ↳14 BOUNDS(n^1, INF)
            ↳15 typed CpxTrs
                ↳16 LowerBoundsProof (⇔, 0 ms)
                ↳17 BOUNDS(n^1, INF)
    ↳18 typed CpxTrs
        ↳19 LowerBoundsProof (⇔, 0 ms)
        ↳20 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: INNERMOST

(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: INNERMOST

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

Infered types.

(4) Obligation:

Innermost 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:

Innermost 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)) →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).

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost 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(n471_0), gen_0':s2_0(n471_0)) → gen_0':s2_0(0), rt ∈ Ω(1 + n4710)

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

Induction Step:
-(gen_0':s2_0(+(n471_0, 1)), gen_0':s2_0(+(n471_0, 1))) →RΩ(1)
-(gen_0':s2_0(n471_0), gen_0':s2_0(n471_0)) →IH
gen_0':s2_0(0)

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

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost 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(n471_0), gen_0':s2_0(n471_0)) → gen_0':s2_0(0), rt ∈ Ω(1 + n4710)

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:

Innermost 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(n471_0), gen_0':s2_0(n471_0)) → gen_0':s2_0(0), rt ∈ Ω(1 + n4710)

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:

Innermost 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)