### (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: 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 [ ].

### (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)
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: FULL

Infered types.

### (6) Obligation:

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

### (7) 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

### (8) Obligation:

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

### (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)) →RΩ(1)
gen_0':s3_0(0)

Induction Step:
-(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) →RΩ(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 ircR ∈ Ω(n).

### (11) Obligation:

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

### (12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
min(gen_0':s3_0(n253_0), gen_0':s3_0(n253_0)) → gen_0':s3_0(n253_0), rt ∈ Ω(1 + n2530)

Induction Base:
min(gen_0':s3_0(0), gen_0':s3_0(0)) →RΩ(1)
0'

Induction Step:
min(gen_0':s3_0(+(n253_0, 1)), gen_0':s3_0(+(n253_0, 1))) →RΩ(1)
s(min(gen_0':s3_0(n253_0), gen_0':s3_0(n253_0))) →IH
s(gen_0':s3_0(c254_0))

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

### (14) Obligation:

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(n253_0), gen_0':s3_0(n253_0)) → gen_0':s3_0(n253_0), rt ∈ Ω(1 + n2530)

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

### (15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
twice(gen_0':s3_0(n554_0)) → gen_0':s3_0(*(2, n554_0)), rt ∈ Ω(1 + n5540)

Induction Base:
twice(gen_0':s3_0(0)) →RΩ(1)
0'

Induction Step:
twice(gen_0':s3_0(+(n554_0, 1))) →RΩ(1)
s(s(twice(gen_0':s3_0(n554_0)))) →IH
s(s(gen_0':s3_0(*(2, c555_0))))

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

### (17) Obligation:

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(n253_0), gen_0':s3_0(n253_0)) → gen_0':s3_0(n253_0), rt ∈ Ω(1 + n2530)
twice(gen_0':s3_0(n554_0)) → gen_0':s3_0(*(2, n554_0)), rt ∈ Ω(1 + n5540)

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)
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(n253_0), gen_0':s3_0(n253_0)) → gen_0':s3_0(n253_0), rt ∈ Ω(1 + n2530)
twice(gen_0':s3_0(n554_0)) → gen_0':s3_0(*(2, n554_0)), rt ∈ Ω(1 + n5540)

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(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) Obligation:

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(n253_0), gen_0':s3_0(n253_0)) → gen_0':s3_0(n253_0), rt ∈ Ω(1 + n2530)
twice(gen_0':s3_0(n554_0)) → gen_0':s3_0(*(2, n554_0)), rt ∈ Ω(1 + n5540)

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(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) Obligation:

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(n253_0), gen_0':s3_0(n253_0)) → gen_0':s3_0(n253_0), rt ∈ Ω(1 + n2530)

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)

### (28) Obligation:

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.

### (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)