The Runtime Complexity (innermost) 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:

-(x, 0) → x
-(s(x), s(y)) → -(x, y)
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
twice(0) → 0
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(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:

-(x, 0') → x
-(s(x), s(y)) → -(x, y)
min(x, 0') → 0'
min(0', y) → 0'
min(s(x), s(y)) → s(min(x, y))
twice(0') → 0'
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, 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)
min(x, 0') → 0'
min(0', y) → 0'
min(s(x), s(y)) → s(min(x, y))
twice(0') → 0'
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

Types:
- :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
min :: 0':s → 0':s → 0':s
twice :: 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:
-, min, twice, f

They will be analysed ascendingly in the following order:
- < f
min < f
twice < f

(6) Obligation:

Innermost TRS:
Rules:
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
min(x, 0') → 0'
min(0', y) → 0'
min(s(x), s(y)) → s(min(x, y))
twice(0') → 0'
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

Types:
- :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
min :: 0':s → 0':s → 0':s
twice :: 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:
-, min, twice, f

They will be analysed ascendingly in the following order:
- < f
min < f
twice < 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)
min(x, 0') → 0'
min(0', y) → 0'
min(s(x), s(y)) → s(min(x, y))
twice(0') → 0'
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

Types:
- :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
min :: 0':s → 0':s → 0':s
twice :: 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:
min, twice, f

They will be analysed ascendingly in the following order:
min < f
twice < f

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

Proved the following rewrite lemma:
min(gen_0':s3_0(n309_0), gen_0':s3_0(n309_0)) → gen_0':s3_0(n309_0), rt ∈ Ω(1 + n309₀)

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

Induction Step:
min(gen_0':s3_0(+(n309_0, 1)), gen_0':s3_0(+(n309_0, 1))) →_R^Ω(1)
s(min(gen_0':s3_0(n309_0), gen_0':s3_0(n309_0))) →_IH
s(gen_0':s3_0(c310_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)
min(x, 0') → 0'
min(0', y) → 0'
min(s(x), s(y)) → s(min(x, y))
twice(0') → 0'
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

Types:
- :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
min :: 0':s → 0':s → 0':s
twice :: 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)
min(gen_0':s3_0(n309_0), gen_0':s3_0(n309_0)) → gen_0':s3_0(n309_0), rt ∈ Ω(1 + n309₀)

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:
twice, f

They will be analysed ascendingly in the following order:
twice < f

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

Proved the following rewrite lemma:
twice(gen_0':s3_0(n659_0)) → gen_0':s3_0(*(2, n659_0)), rt ∈ Ω(1 + n659₀)

Induction Base:
twice(gen_0':s3_0(0)) →_R^Ω(1)
0'

Induction Step:
twice(gen_0':s3_0(+(n659_0, 1))) →_R^Ω(1)
s(s(twice(gen_0':s3_0(n659_0)))) →_IH
s(s(gen_0':s3_0(*(2, c660_0))))

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

(14) Complex Obligation (BEST)

(15) Obligation:

Innermost TRS:
Rules:
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
min(x, 0') → 0'
min(0', y) → 0'
min(s(x), s(y)) → s(min(x, y))
twice(0') → 0'
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

Types:
- :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
min :: 0':s → 0':s → 0':s
twice :: 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)
min(gen_0':s3_0(n309_0), gen_0':s3_0(n309_0)) → gen_0':s3_0(n309_0), rt ∈ Ω(1 + n309₀)
twice(gen_0':s3_0(n659_0)) → gen_0':s3_0(*(2, n659_0)), rt ∈ Ω(1 + n659₀)

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)
min(x, 0') → 0'
min(0', y) → 0'
min(s(x), s(y)) → s(min(x, y))
twice(0') → 0'
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

Types:
- :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
min :: 0':s → 0':s → 0':s
twice :: 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)
min(gen_0':s3_0(n309_0), gen_0':s3_0(n309_0)) → gen_0':s3_0(n309_0), rt ∈ Ω(1 + n309₀)
twice(gen_0':s3_0(n659_0)) → gen_0':s3_0(*(2, n659_0)), rt ∈ Ω(1 + n659₀)

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(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)

(19) BOUNDS(n^1, INF)

(20) Obligation:

Innermost TRS:
Rules:
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
min(x, 0') → 0'
min(0', y) → 0'
min(s(x), s(y)) → s(min(x, y))
twice(0') → 0'
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

Types:
- :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
min :: 0':s → 0':s → 0':s
twice :: 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)
min(gen_0':s3_0(n309_0), gen_0':s3_0(n309_0)) → gen_0':s3_0(n309_0), rt ∈ Ω(1 + n309₀)
twice(gen_0':s3_0(n659_0)) → gen_0':s3_0(*(2, n659_0)), rt ∈ Ω(1 + n659₀)

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(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)

(22) BOUNDS(n^1, INF)

(23) Obligation:

Innermost TRS:
Rules:
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
min(x, 0') → 0'
min(0', y) → 0'
min(s(x), s(y)) → s(min(x, y))
twice(0') → 0'
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

Types:
- :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
min :: 0':s → 0':s → 0':s
twice :: 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)
min(gen_0':s3_0(n309_0), gen_0':s3_0(n309_0)) → gen_0':s3_0(n309_0), rt ∈ Ω(1 + n309₀)

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)
min(x, 0') → 0'
min(0', y) → 0'
min(s(x), s(y)) → s(min(x, y))
twice(0') → 0'
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

Types:
- :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
min :: 0':s → 0':s → 0':s
twice :: 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)