The Runtime Complexity (full) 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:

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))
minus(x, 0) → x
minus(s(x), s(y)) → s(minus(x, any(y)))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, y)), s(min(x, y)))
any(s(x)) → s(s(any(x)))
any(x) → x

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 [ ].

(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:

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))
minus(x, 0') → x
minus(s(x), s(y)) → s(minus(x, any(y)))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, y)), s(min(x, y)))
any(s(x)) → s(s(any(x)))
any(x) → x

S is empty.
Rewrite Strategy: FULL

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

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))
minus(x, 0') → x
minus(s(x), s(y)) → s(minus(x, any(y)))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, y)), s(min(x, y)))
any(s(x)) → s(s(any(x)))
any(x) → x

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
minus :: 0':s → 0':s → 0':s
any :: 0':s → 0':s
gcd :: 0':s → 0':s → gcd
hole_0':s1_0 :: 0':s
hole_gcd2_0 :: gcd
gen_0':s3_0 :: Nat → 0':s

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

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

They will be analysed ascendingly in the following order:
min < gcd
max < gcd
any < minus
minus < 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))
minus(x, 0') → x
minus(s(x), s(y)) → s(minus(x, any(y)))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, y)), s(min(x, y)))
any(s(x)) → s(s(any(x)))
any(x) → x

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
minus :: 0':s → 0':s → 0':s
any :: 0':s → 0':s
gcd :: 0':s → 0':s → gcd
hole_0':s1_0 :: 0':s
hole_gcd2_0 :: gcd
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, max, minus, any, gcd

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

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

Proved the following rewrite lemma:
min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n5₀)

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

Induction Step:
min(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) →_R^Ω(1)
s(min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0))) →_IH
s(gen_0':s3_0(c6_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:
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))
minus(x, 0') → x
minus(s(x), s(y)) → s(minus(x, any(y)))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, y)), s(min(x, y)))
any(s(x)) → s(s(any(x)))
any(x) → x

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
minus :: 0':s → 0':s → 0':s
any :: 0':s → 0':s
gcd :: 0':s → 0':s → gcd
hole_0':s1_0 :: 0':s
hole_gcd2_0 :: gcd
gen_0':s3_0 :: Nat → 0':s

Lemmas:
min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(n5_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:
max, minus, any, gcd

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

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

Proved the following rewrite lemma:
max(gen_0':s3_0(n313_0), gen_0':s3_0(n313_0)) → gen_0':s3_0(n313_0), rt ∈ Ω(1 + n313₀)

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

Induction Step:
max(gen_0':s3_0(+(n313_0, 1)), gen_0':s3_0(+(n313_0, 1))) →_R^Ω(1)
s(max(gen_0':s3_0(n313_0), gen_0':s3_0(n313_0))) →_IH
s(gen_0':s3_0(c314_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:
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))
minus(x, 0') → x
minus(s(x), s(y)) → s(minus(x, any(y)))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, y)), s(min(x, y)))
any(s(x)) → s(s(any(x)))
any(x) → x

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
minus :: 0':s → 0':s → 0':s
any :: 0':s → 0':s
gcd :: 0':s → 0':s → gcd
hole_0':s1_0 :: 0':s
hole_gcd2_0 :: gcd
gen_0':s3_0 :: Nat → 0':s

Lemmas:
min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n5₀)
max(gen_0':s3_0(n313_0), gen_0':s3_0(n313_0)) → gen_0':s3_0(n313_0), rt ∈ Ω(1 + n313₀)

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:
any, minus, gcd

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

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

Proved the following rewrite lemma:
any(gen_0':s3_0(+(1, n701_0))) → *4_0, rt ∈ Ω(n701₀)

Induction Base:
any(gen_0':s3_0(+(1, 0)))

Induction Step:
any(gen_0':s3_0(+(1, +(n701_0, 1)))) →_R^Ω(1)
s(s(any(gen_0':s3_0(+(1, n701_0))))) →_IH
s(s(*4_0))

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

(16) Complex Obligation (BEST)

(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))
minus(x, 0') → x
minus(s(x), s(y)) → s(minus(x, any(y)))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, y)), s(min(x, y)))
any(s(x)) → s(s(any(x)))
any(x) → x

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
minus :: 0':s → 0':s → 0':s
any :: 0':s → 0':s
gcd :: 0':s → 0':s → gcd
hole_0':s1_0 :: 0':s
hole_gcd2_0 :: gcd
gen_0':s3_0 :: Nat → 0':s

Lemmas:
min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n5₀)
max(gen_0':s3_0(n313_0), gen_0':s3_0(n313_0)) → gen_0':s3_0(n313_0), rt ∈ Ω(1 + n313₀)
any(gen_0':s3_0(+(1, n701_0))) → *4_0, rt ∈ Ω(n701₀)

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:
minus, gcd

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

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

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

(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))
minus(x, 0') → x
minus(s(x), s(y)) → s(minus(x, any(y)))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, y)), s(min(x, y)))
any(s(x)) → s(s(any(x)))
any(x) → x

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
minus :: 0':s → 0':s → 0':s
any :: 0':s → 0':s
gcd :: 0':s → 0':s → gcd
hole_0':s1_0 :: 0':s
hole_gcd2_0 :: gcd
gen_0':s3_0 :: Nat → 0':s

Lemmas:
min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n5₀)
max(gen_0':s3_0(n313_0), gen_0':s3_0(n313_0)) → gen_0':s3_0(n313_0), rt ∈ Ω(1 + n313₀)
any(gen_0':s3_0(+(1, n701_0))) → *4_0, rt ∈ Ω(n701₀)

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:
gcd

(20) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(21) 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))
minus(x, 0') → x
minus(s(x), s(y)) → s(minus(x, any(y)))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, y)), s(min(x, y)))
any(s(x)) → s(s(any(x)))
any(x) → x

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
minus :: 0':s → 0':s → 0':s
any :: 0':s → 0':s
gcd :: 0':s → 0':s → gcd
hole_0':s1_0 :: 0':s
hole_gcd2_0 :: gcd
gen_0':s3_0 :: Nat → 0':s

Lemmas:
min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n5₀)
max(gen_0':s3_0(n313_0), gen_0':s3_0(n313_0)) → gen_0':s3_0(n313_0), rt ∈ Ω(1 + n313₀)
any(gen_0':s3_0(+(1, n701_0))) → *4_0, rt ∈ Ω(n701₀)

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.

(22) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n5₀)

(23) BOUNDS(n^1, INF)

(24) 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))
minus(x, 0') → x
minus(s(x), s(y)) → s(minus(x, any(y)))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, y)), s(min(x, y)))
any(s(x)) → s(s(any(x)))
any(x) → x

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
minus :: 0':s → 0':s → 0':s
any :: 0':s → 0':s
gcd :: 0':s → 0':s → gcd
hole_0':s1_0 :: 0':s
hole_gcd2_0 :: gcd
gen_0':s3_0 :: Nat → 0':s

Lemmas:
min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n5₀)
max(gen_0':s3_0(n313_0), gen_0':s3_0(n313_0)) → gen_0':s3_0(n313_0), rt ∈ Ω(1 + n313₀)
any(gen_0':s3_0(+(1, n701_0))) → *4_0, rt ∈ Ω(n701₀)

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.

(25) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n5₀)

(26) BOUNDS(n^1, INF)

(27) 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))
minus(x, 0') → x
minus(s(x), s(y)) → s(minus(x, any(y)))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, y)), s(min(x, y)))
any(s(x)) → s(s(any(x)))
any(x) → x

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
minus :: 0':s → 0':s → 0':s
any :: 0':s → 0':s
gcd :: 0':s → 0':s → gcd
hole_0':s1_0 :: 0':s
hole_gcd2_0 :: gcd
gen_0':s3_0 :: Nat → 0':s

Lemmas:
min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n5₀)
max(gen_0':s3_0(n313_0), gen_0':s3_0(n313_0)) → gen_0':s3_0(n313_0), rt ∈ Ω(1 + n313₀)

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.

(28) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n5₀)

(29) BOUNDS(n^1, INF)

(30) 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))
minus(x, 0') → x
minus(s(x), s(y)) → s(minus(x, any(y)))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, y)), s(min(x, y)))
any(s(x)) → s(s(any(x)))
any(x) → x

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
minus :: 0':s → 0':s → 0':s
any :: 0':s → 0':s
gcd :: 0':s → 0':s → gcd
hole_0':s1_0 :: 0':s
hole_gcd2_0 :: gcd
gen_0':s3_0 :: Nat → 0':s

Lemmas:
min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(n5_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.

(31) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n5₀)

(32) BOUNDS(n^1, INF)