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:

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, y))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y)))
transform(x) → s(s(x))
transform(cons(x, y)) → cons(cons(x, x), x)
transform(cons(x, y)) → y
transform(s(x)) → s(s(transform(x)))
cons(x, y) → y
cons(x, cons(y, s(z))) → cons(y, x)
cons(cons(x, z), s(y)) → transform(x)

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:

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, y))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y)))
transform(x) → s(s(x))
transform(cons(x, y)) → cons(cons(x, x), x)
transform(cons(x, y)) → y
transform(s(x)) → s(s(transform(x)))
cons(x, y) → y
cons(x, cons(y, s(z))) → cons(y, x)
cons(cons(x, z), s(y)) → transform(x)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost 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, y))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y)))
transform(x) → s(s(x))
transform(cons(x, y)) → cons(cons(x, x), x)
transform(cons(x, y)) → y
transform(s(x)) → s(s(transform(x)))
cons(x, y) → y
cons(x, cons(y, s(z))) → cons(y, x)
cons(cons(x, z), s(y)) → transform(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
gcd :: 0':s → 0':s → gcd
transform :: 0':s → 0':s
cons :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_gcd2_0 :: gcd
gen_0':s3_0 :: Nat → 0':s

(5) OrderProof (LOWER BOUND(ID) transformation)

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

They will be analysed ascendingly in the following order:
min < gcd
max < gcd
minus < gcd
transform < gcd
transform = cons

(6) Obligation:

Innermost 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, y))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y)))
transform(x) → s(s(x))
transform(cons(x, y)) → cons(cons(x, x), x)
transform(cons(x, y)) → y
transform(s(x)) → s(s(transform(x)))
cons(x, y) → y
cons(x, cons(y, s(z))) → cons(y, x)
cons(cons(x, z), s(y)) → transform(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
gcd :: 0':s → 0':s → gcd
transform :: 0':s → 0':s
cons :: 0':s → 0':s → 0':s
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, gcd, transform, cons

They will be analysed ascendingly in the following order:
min < gcd
max < gcd
minus < gcd
transform < gcd
transform = cons

(7) 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).

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost 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, y))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y)))
transform(x) → s(s(x))
transform(cons(x, y)) → cons(cons(x, x), x)
transform(cons(x, y)) → y
transform(s(x)) → s(s(transform(x)))
cons(x, y) → y
cons(x, cons(y, s(z))) → cons(y, x)
cons(cons(x, z), s(y)) → transform(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
gcd :: 0':s → 0':s → gcd
transform :: 0':s → 0':s
cons :: 0':s → 0':s → 0':s
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, gcd, transform, cons

They will be analysed ascendingly in the following order:
max < gcd
minus < gcd
transform < gcd
transform = cons

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

Proved the following rewrite lemma:
max(gen_0':s3_0(n453_0), gen_0':s3_0(n453_0)) → gen_0':s3_0(n453_0), rt ∈ Ω(1 + n453₀)

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(+(n453_0, 1)), gen_0':s3_0(+(n453_0, 1))) →_R^Ω(1)
s(max(gen_0':s3_0(n453_0), gen_0':s3_0(n453_0))) →_IH
s(gen_0':s3_0(c454_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:
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, y))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y)))
transform(x) → s(s(x))
transform(cons(x, y)) → cons(cons(x, x), x)
transform(cons(x, y)) → y
transform(s(x)) → s(s(transform(x)))
cons(x, y) → y
cons(x, cons(y, s(z))) → cons(y, x)
cons(cons(x, z), s(y)) → transform(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
gcd :: 0':s → 0':s → gcd
transform :: 0':s → 0':s
cons :: 0':s → 0':s → 0':s
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(n453_0), gen_0':s3_0(n453_0)) → gen_0':s3_0(n453_0), rt ∈ Ω(1 + n453₀)

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, transform, cons

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

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

Proved the following rewrite lemma:
minus(gen_0':s3_0(n1021_0), gen_0':s3_0(n1021_0)) → gen_0':s3_0(n1021_0), rt ∈ Ω(1 + n1021₀)

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

Induction Step:
minus(gen_0':s3_0(+(n1021_0, 1)), gen_0':s3_0(+(n1021_0, 1))) →_R^Ω(1)
s(minus(gen_0':s3_0(n1021_0), gen_0':s3_0(n1021_0))) →_IH
s(gen_0':s3_0(c1022_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:
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, y))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y)))
transform(x) → s(s(x))
transform(cons(x, y)) → cons(cons(x, x), x)
transform(cons(x, y)) → y
transform(s(x)) → s(s(transform(x)))
cons(x, y) → y
cons(x, cons(y, s(z))) → cons(y, x)
cons(cons(x, z), s(y)) → transform(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
gcd :: 0':s → 0':s → gcd
transform :: 0':s → 0':s
cons :: 0':s → 0':s → 0':s
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(n453_0), gen_0':s3_0(n453_0)) → gen_0':s3_0(n453_0), rt ∈ Ω(1 + n453₀)
minus(gen_0':s3_0(n1021_0), gen_0':s3_0(n1021_0)) → gen_0':s3_0(n1021_0), rt ∈ Ω(1 + n1021₀)

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:
cons, gcd, transform

They will be analysed ascendingly in the following order:
transform < gcd
transform = cons

(16) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(17) Obligation:

Innermost 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, y))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y)))
transform(x) → s(s(x))
transform(cons(x, y)) → cons(cons(x, x), x)
transform(cons(x, y)) → y
transform(s(x)) → s(s(transform(x)))
cons(x, y) → y
cons(x, cons(y, s(z))) → cons(y, x)
cons(cons(x, z), s(y)) → transform(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
gcd :: 0':s → 0':s → gcd
transform :: 0':s → 0':s
cons :: 0':s → 0':s → 0':s
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(n453_0), gen_0':s3_0(n453_0)) → gen_0':s3_0(n453_0), rt ∈ Ω(1 + n453₀)
minus(gen_0':s3_0(n1021_0), gen_0':s3_0(n1021_0)) → gen_0':s3_0(n1021_0), rt ∈ Ω(1 + n1021₀)

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:
transform, gcd

They will be analysed ascendingly in the following order:
transform < gcd
transform = cons

(18) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
transform(gen_0':s3_0(+(1, n1558_0))) → *4_0, rt ∈ Ω(n1558₀)

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

Induction Step:
transform(gen_0':s3_0(+(1, +(n1558_0, 1)))) →_R^Ω(1)
s(s(transform(gen_0':s3_0(+(1, n1558_0))))) →_IH
s(s(*4_0))

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

(19) Complex Obligation (BEST)

(20) Obligation:

Innermost 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, y))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y)))
transform(x) → s(s(x))
transform(cons(x, y)) → cons(cons(x, x), x)
transform(cons(x, y)) → y
transform(s(x)) → s(s(transform(x)))
cons(x, y) → y
cons(x, cons(y, s(z))) → cons(y, x)
cons(cons(x, z), s(y)) → transform(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
gcd :: 0':s → 0':s → gcd
transform :: 0':s → 0':s
cons :: 0':s → 0':s → 0':s
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(n453_0), gen_0':s3_0(n453_0)) → gen_0':s3_0(n453_0), rt ∈ Ω(1 + n453₀)
minus(gen_0':s3_0(n1021_0), gen_0':s3_0(n1021_0)) → gen_0':s3_0(n1021_0), rt ∈ Ω(1 + n1021₀)
transform(gen_0':s3_0(+(1, n1558_0))) → *4_0, rt ∈ Ω(n1558₀)

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:
cons, gcd

They will be analysed ascendingly in the following order:
transform < gcd
transform = cons

(21) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(22) Obligation:

Innermost 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, y))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y)))
transform(x) → s(s(x))
transform(cons(x, y)) → cons(cons(x, x), x)
transform(cons(x, y)) → y
transform(s(x)) → s(s(transform(x)))
cons(x, y) → y
cons(x, cons(y, s(z))) → cons(y, x)
cons(cons(x, z), s(y)) → transform(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
gcd :: 0':s → 0':s → gcd
transform :: 0':s → 0':s
cons :: 0':s → 0':s → 0':s
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(n453_0), gen_0':s3_0(n453_0)) → gen_0':s3_0(n453_0), rt ∈ Ω(1 + n453₀)
minus(gen_0':s3_0(n1021_0), gen_0':s3_0(n1021_0)) → gen_0':s3_0(n1021_0), rt ∈ Ω(1 + n1021₀)
transform(gen_0':s3_0(+(1, n1558_0))) → *4_0, rt ∈ Ω(n1558₀)

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

(23) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(24) Obligation:

Innermost 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, y))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y)))
transform(x) → s(s(x))
transform(cons(x, y)) → cons(cons(x, x), x)
transform(cons(x, y)) → y
transform(s(x)) → s(s(transform(x)))
cons(x, y) → y
cons(x, cons(y, s(z))) → cons(y, x)
cons(cons(x, z), s(y)) → transform(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
gcd :: 0':s → 0':s → gcd
transform :: 0':s → 0':s
cons :: 0':s → 0':s → 0':s
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(n453_0), gen_0':s3_0(n453_0)) → gen_0':s3_0(n453_0), rt ∈ Ω(1 + n453₀)
minus(gen_0':s3_0(n1021_0), gen_0':s3_0(n1021_0)) → gen_0':s3_0(n1021_0), rt ∈ Ω(1 + n1021₀)
transform(gen_0':s3_0(+(1, n1558_0))) → *4_0, rt ∈ Ω(n1558₀)

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:

Innermost 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, y))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y)))
transform(x) → s(s(x))
transform(cons(x, y)) → cons(cons(x, x), x)
transform(cons(x, y)) → y
transform(s(x)) → s(s(transform(x)))
cons(x, y) → y
cons(x, cons(y, s(z))) → cons(y, x)
cons(cons(x, z), s(y)) → transform(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
gcd :: 0':s → 0':s → gcd
transform :: 0':s → 0':s
cons :: 0':s → 0':s → 0':s
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(n453_0), gen_0':s3_0(n453_0)) → gen_0':s3_0(n453_0), rt ∈ Ω(1 + n453₀)
minus(gen_0':s3_0(n1021_0), gen_0':s3_0(n1021_0)) → gen_0':s3_0(n1021_0), rt ∈ Ω(1 + n1021₀)
transform(gen_0':s3_0(+(1, n1558_0))) → *4_0, rt ∈ Ω(n1558₀)

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:

Innermost 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, y))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y)))
transform(x) → s(s(x))
transform(cons(x, y)) → cons(cons(x, x), x)
transform(cons(x, y)) → y
transform(s(x)) → s(s(transform(x)))
cons(x, y) → y
cons(x, cons(y, s(z))) → cons(y, x)
cons(cons(x, z), s(y)) → transform(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
gcd :: 0':s → 0':s → gcd
transform :: 0':s → 0':s
cons :: 0':s → 0':s → 0':s
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(n453_0), gen_0':s3_0(n453_0)) → gen_0':s3_0(n453_0), rt ∈ Ω(1 + n453₀)
minus(gen_0':s3_0(n1021_0), gen_0':s3_0(n1021_0)) → gen_0':s3_0(n1021_0), rt ∈ Ω(1 + n1021₀)

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)

(33) Obligation:

Innermost 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, y))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y)))
transform(x) → s(s(x))
transform(cons(x, y)) → cons(cons(x, x), x)
transform(cons(x, y)) → y
transform(s(x)) → s(s(transform(x)))
cons(x, y) → y
cons(x, cons(y, s(z))) → cons(y, x)
cons(cons(x, z), s(y)) → transform(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
gcd :: 0':s → 0':s → gcd
transform :: 0':s → 0':s
cons :: 0':s → 0':s → 0':s
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(n453_0), gen_0':s3_0(n453_0)) → gen_0':s3_0(n453_0), rt ∈ Ω(1 + n453₀)

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.

(34) 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)

(35) BOUNDS(n^1, INF)

(36) Obligation:

Innermost 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, y))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y)))
transform(x) → s(s(x))
transform(cons(x, y)) → cons(cons(x, x), x)
transform(cons(x, y)) → y
transform(s(x)) → s(s(transform(x)))
cons(x, y) → y
cons(x, cons(y, s(z))) → cons(y, x)
cons(cons(x, z), s(y)) → transform(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
gcd :: 0':s → 0':s → gcd
transform :: 0':s → 0':s
cons :: 0':s → 0':s → 0':s
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.

(37) 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)

(38) BOUNDS(n^1, INF)