The Runtime Complexity (full) 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: 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 [ ].

(2) BOUNDS(n^1, INF)

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

S is empty.
Rewrite Strategy: FULL

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

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), 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

(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

(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), 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

(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)) →_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).

(10) Complex Obligation (BEST)

(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), 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

(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)) →_R^Ω(1)
gen_0':s3_0(b)

Induction Step:
+'(gen_0':s3_0(+(n237_0, 1)), gen_0':s3_0(b)) →_R^Ω(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 irc_R ∈ Ω(n).

(13) Complex Obligation (BEST)

(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), 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(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)) →_R^Ω(1)
0'

Induction Step:
*'(gen_0':s3_0(a), gen_0':s3_0(+(n708_0, 1))) →_R^Ω(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))) →_L^{Ω(1 + a)}
gen_0':s3_0(+(a, *(n708_0, a)))

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

(16) Complex Obligation (BEST)

(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), 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(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), 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(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)

(21) BOUNDS(n^2, INF)

(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), 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(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)

(24) BOUNDS(n^2, INF)

(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), 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(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)

(27) BOUNDS(n^1, INF)

(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), 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.

(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)

(30) BOUNDS(n^1, INF)