The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).

(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), y) → f(-(*(s(x), s(y)), s(*(s(x), y))), *(y, 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:

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

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost 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), y) → f(-(*'(s(x), s(y)), s(*'(s(x), y))), *'(y, y))

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 → 0':s → f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat → 0':s

(5) 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

(6) Obligation:

Innermost 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), y) → f(-(*'(s(x), s(y)), s(*'(s(x), y))), *'(y, y))

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 → 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

(7) 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)) →_R^Ω(1)
gen_0':s3_0(0)

Induction Step:
-(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) →_R^Ω(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 irc_R ∈ Ω(n).

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost 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), y) → f(-(*'(s(x), s(y)), s(*'(s(x), y))), *'(y, y))

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 → 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

(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
+'(gen_0':s3_0(n285_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n285_0, b)), rt ∈ Ω(1 + n285₀)

Induction Base:
+'(gen_0':s3_0(0), gen_0':s3_0(b)) →_R^Ω(1)
gen_0':s3_0(b)

Induction Step:
+'(gen_0':s3_0(+(n285_0, 1)), gen_0':s3_0(b)) →_R^Ω(1)
s(+'(gen_0':s3_0(n285_0), gen_0':s3_0(b))) →_IH
s(gen_0':s3_0(+(b, c286_0)))

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

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost 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), y) → f(-(*'(s(x), s(y)), s(*'(s(x), y))), *'(y, y))

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 → 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(n285_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n285_0, b)), rt ∈ Ω(1 + n285₀)

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

(13) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
*'(gen_0':s3_0(a), gen_0':s3_0(n816_0)) → gen_0':s3_0(*(n816_0, a)), rt ∈ Ω(1 + a·n816₀ + n816₀)

Induction Base:
*'(gen_0':s3_0(a), gen_0':s3_0(0)) →_R^Ω(1)
0'

Induction Step:
*'(gen_0':s3_0(a), gen_0':s3_0(+(n816_0, 1))) →_R^Ω(1)
+'(gen_0':s3_0(a), *'(gen_0':s3_0(a), gen_0':s3_0(n816_0))) →_IH
+'(gen_0':s3_0(a), gen_0':s3_0(*(c817_0, a))) →_L^{Ω(1 + a)}
gen_0':s3_0(+(a, *(n816_0, a)))

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

(14) Complex Obligation (BEST)

(15) Obligation:

Innermost 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), y) → f(-(*'(s(x), s(y)), s(*'(s(x), y))), *'(y, y))

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 → 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(n285_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n285_0, b)), rt ∈ Ω(1 + n285₀)
*'(gen_0':s3_0(a), gen_0':s3_0(n816_0)) → gen_0':s3_0(*(n816_0, a)), rt ∈ Ω(1 + a·n816₀ + n816₀)

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

(16) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(17) Obligation:

Innermost 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), y) → f(-(*'(s(x), s(y)), s(*'(s(x), y))), *'(y, y))

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 → 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(n285_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n285_0, b)), rt ∈ Ω(1 + n285₀)
*'(gen_0':s3_0(a), gen_0':s3_0(n816_0)) → gen_0':s3_0(*(n816_0, a)), rt ∈ Ω(1 + a·n816₀ + n816₀)

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.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
*'(gen_0':s3_0(a), gen_0':s3_0(n816_0)) → gen_0':s3_0(*(n816_0, a)), rt ∈ Ω(1 + a·n816₀ + n816₀)

(19) BOUNDS(n^2, INF)

(20) Obligation:

Innermost 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), y) → f(-(*'(s(x), s(y)), s(*'(s(x), y))), *'(y, y))

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 → 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(n285_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n285_0, b)), rt ∈ Ω(1 + n285₀)
*'(gen_0':s3_0(a), gen_0':s3_0(n816_0)) → gen_0':s3_0(*(n816_0, a)), rt ∈ Ω(1 + a·n816₀ + n816₀)

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.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
*'(gen_0':s3_0(a), gen_0':s3_0(n816_0)) → gen_0':s3_0(*(n816_0, a)), rt ∈ Ω(1 + a·n816₀ + n816₀)

(22) BOUNDS(n^2, INF)

(23) Obligation:

Innermost 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), y) → f(-(*'(s(x), s(y)), s(*'(s(x), y))), *'(y, y))

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 → 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(n285_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n285_0, b)), rt ∈ Ω(1 + n285₀)

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.

(24) 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)

(25) BOUNDS(n^1, INF)

(26) Obligation:

Innermost 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), y) → f(-(*'(s(x), s(y)), s(*'(s(x), y))), *'(y, y))

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 → 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.

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