### (0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
max(x, 0) → x
max(0, y) → y
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
gcd(s(x), s(y), z) → gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
gcd(x, s(y), s(z)) → gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
gcd(s(x), y, s(z)) → gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
gcd(x, 0, 0) → x
gcd(0, y, 0) → y
gcd(0, 0, z) → z

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
min(s(x), s(y)) →+ s(min(x, y))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
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:

min(x, 0') → 0'
min(0', y) → 0'
min(s(x), s(y)) → s(min(x, y))
max(x, 0') → x
max(0', y) → y
max(s(x), s(y)) → s(max(x, y))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
gcd(s(x), s(y), z) → gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
gcd(x, s(y), s(z)) → gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
gcd(s(x), y, s(z)) → gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
gcd(x, 0', 0') → x
gcd(0', y, 0') → y
gcd(0', 0', z) → z

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
min(x, 0') → 0'
min(0', y) → 0'
min(s(x), s(y)) → s(min(x, y))
max(x, 0') → x
max(0', y) → y
max(s(x), s(y)) → s(max(x, y))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
gcd(s(x), s(y), z) → gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
gcd(x, s(y), s(z)) → gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
gcd(s(x), y, s(z)) → gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
gcd(x, 0', 0') → x
gcd(0', y, 0') → y
gcd(0', 0', z) → z

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
- :: 0':s → 0':s → 0':s
gcd :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

### (7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
min, max, -, gcd

They will be analysed ascendingly in the following order:
min < gcd
max < gcd
- < gcd

### (8) Obligation:

TRS:
Rules:
min(x, 0') → 0'
min(0', y) → 0'
min(s(x), s(y)) → s(min(x, y))
max(x, 0') → x
max(0', y) → y
max(s(x), s(y)) → s(max(x, y))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
gcd(s(x), s(y), z) → gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
gcd(x, s(y), s(z)) → gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
gcd(s(x), y, s(z)) → gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
gcd(x, 0', 0') → x
gcd(0', y, 0') → y
gcd(0', 0', z) → z

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
- :: 0':s → 0':s → 0':s
gcd :: 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:
min, max, -, gcd

They will be analysed ascendingly in the following order:
min < gcd
max < gcd
- < gcd

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

Proved the following rewrite lemma:
min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n40)

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

Induction Step:
min(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(+(n4_0, 1))) →RΩ(1)
s(min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0))) →IH
s(gen_0':s2_0(c5_0))

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

### (11) Obligation:

TRS:
Rules:
min(x, 0') → 0'
min(0', y) → 0'
min(s(x), s(y)) → s(min(x, y))
max(x, 0') → x
max(0', y) → y
max(s(x), s(y)) → s(max(x, y))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
gcd(s(x), s(y), z) → gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
gcd(x, s(y), s(z)) → gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
gcd(s(x), y, s(z)) → gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
gcd(x, 0', 0') → x
gcd(0', y, 0') → y
gcd(0', 0', z) → z

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
- :: 0':s → 0':s → 0':s
gcd :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(n4_0), 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:
max, -, gcd

They will be analysed ascendingly in the following order:
max < gcd
- < gcd

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

Proved the following rewrite lemma:
max(gen_0':s2_0(n333_0), gen_0':s2_0(n333_0)) → gen_0':s2_0(n333_0), rt ∈ Ω(1 + n3330)

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

Induction Step:
max(gen_0':s2_0(+(n333_0, 1)), gen_0':s2_0(+(n333_0, 1))) →RΩ(1)
s(max(gen_0':s2_0(n333_0), gen_0':s2_0(n333_0))) →IH
s(gen_0':s2_0(c334_0))

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

### (14) Obligation:

TRS:
Rules:
min(x, 0') → 0'
min(0', y) → 0'
min(s(x), s(y)) → s(min(x, y))
max(x, 0') → x
max(0', y) → y
max(s(x), s(y)) → s(max(x, y))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
gcd(s(x), s(y), z) → gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
gcd(x, s(y), s(z)) → gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
gcd(s(x), y, s(z)) → gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
gcd(x, 0', 0') → x
gcd(0', y, 0') → y
gcd(0', 0', z) → z

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
- :: 0':s → 0':s → 0':s
gcd :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n40)
max(gen_0':s2_0(n333_0), gen_0':s2_0(n333_0)) → gen_0':s2_0(n333_0), rt ∈ Ω(1 + n3330)

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:
-, gcd

They will be analysed ascendingly in the following order:
- < gcd

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

Proved the following rewrite lemma:
-(gen_0':s2_0(n748_0), gen_0':s2_0(n748_0)) → gen_0':s2_0(0), rt ∈ Ω(1 + n7480)

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

Induction Step:
-(gen_0':s2_0(+(n748_0, 1)), gen_0':s2_0(+(n748_0, 1))) →RΩ(1)
-(gen_0':s2_0(n748_0), gen_0':s2_0(n748_0)) →IH
gen_0':s2_0(0)

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

### (17) Obligation:

TRS:
Rules:
min(x, 0') → 0'
min(0', y) → 0'
min(s(x), s(y)) → s(min(x, y))
max(x, 0') → x
max(0', y) → y
max(s(x), s(y)) → s(max(x, y))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
gcd(s(x), s(y), z) → gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
gcd(x, s(y), s(z)) → gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
gcd(s(x), y, s(z)) → gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
gcd(x, 0', 0') → x
gcd(0', y, 0') → y
gcd(0', 0', z) → z

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
- :: 0':s → 0':s → 0':s
gcd :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n40)
max(gen_0':s2_0(n333_0), gen_0':s2_0(n333_0)) → gen_0':s2_0(n333_0), rt ∈ Ω(1 + n3330)
-(gen_0':s2_0(n748_0), gen_0':s2_0(n748_0)) → gen_0':s2_0(0), rt ∈ Ω(1 + n7480)

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:
gcd

### (18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

### (19) Obligation:

TRS:
Rules:
min(x, 0') → 0'
min(0', y) → 0'
min(s(x), s(y)) → s(min(x, y))
max(x, 0') → x
max(0', y) → y
max(s(x), s(y)) → s(max(x, y))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
gcd(s(x), s(y), z) → gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
gcd(x, s(y), s(z)) → gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
gcd(s(x), y, s(z)) → gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
gcd(x, 0', 0') → x
gcd(0', y, 0') → y
gcd(0', 0', z) → z

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
- :: 0':s → 0':s → 0':s
gcd :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n40)
max(gen_0':s2_0(n333_0), gen_0':s2_0(n333_0)) → gen_0':s2_0(n333_0), rt ∈ Ω(1 + n3330)
-(gen_0':s2_0(n748_0), gen_0':s2_0(n748_0)) → gen_0':s2_0(0), rt ∈ Ω(1 + n7480)

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.

### (20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n40)

### (22) Obligation:

TRS:
Rules:
min(x, 0') → 0'
min(0', y) → 0'
min(s(x), s(y)) → s(min(x, y))
max(x, 0') → x
max(0', y) → y
max(s(x), s(y)) → s(max(x, y))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
gcd(s(x), s(y), z) → gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
gcd(x, s(y), s(z)) → gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
gcd(s(x), y, s(z)) → gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
gcd(x, 0', 0') → x
gcd(0', y, 0') → y
gcd(0', 0', z) → z

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
- :: 0':s → 0':s → 0':s
gcd :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n40)
max(gen_0':s2_0(n333_0), gen_0':s2_0(n333_0)) → gen_0':s2_0(n333_0), rt ∈ Ω(1 + n3330)
-(gen_0':s2_0(n748_0), gen_0':s2_0(n748_0)) → gen_0':s2_0(0), rt ∈ Ω(1 + n7480)

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.

### (23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n40)

### (25) Obligation:

TRS:
Rules:
min(x, 0') → 0'
min(0', y) → 0'
min(s(x), s(y)) → s(min(x, y))
max(x, 0') → x
max(0', y) → y
max(s(x), s(y)) → s(max(x, y))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
gcd(s(x), s(y), z) → gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
gcd(x, s(y), s(z)) → gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
gcd(s(x), y, s(z)) → gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
gcd(x, 0', 0') → x
gcd(0', y, 0') → y
gcd(0', 0', z) → z

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
- :: 0':s → 0':s → 0':s
gcd :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n40)
max(gen_0':s2_0(n333_0), gen_0':s2_0(n333_0)) → gen_0':s2_0(n333_0), rt ∈ Ω(1 + n3330)

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.

### (26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n40)

### (28) Obligation:

TRS:
Rules:
min(x, 0') → 0'
min(0', y) → 0'
min(s(x), s(y)) → s(min(x, y))
max(x, 0') → x
max(0', y) → y
max(s(x), s(y)) → s(max(x, y))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
gcd(s(x), s(y), z) → gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
gcd(x, s(y), s(z)) → gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
gcd(s(x), y, s(z)) → gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
gcd(x, 0', 0') → x
gcd(0', y, 0') → y
gcd(0', 0', z) → z

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
- :: 0':s → 0':s → 0':s
gcd :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(n4_0), 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.

### (29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n40)