The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).

(0) Obligation:

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

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
eq(0, 0) → true
eq(0, s(y)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
if1(true, x, y, xs) → min(x, xs)
if1(false, x, y, xs) → min(y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
minsort(nil) → nil
minsort(cons(x, y)) → cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) → x
min(x, cons(y, z)) → if1(le(x, y), x, y, z)
del(x, nil) → nil
del(x, cons(y, z)) → if2(eq(x, y), x, y, z)

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:

le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
if1(true, x, y, xs) → min(x, xs)
if1(false, x, y, xs) → min(y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
minsort(nil) → nil
minsort(cons(x, y)) → cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) → x
min(x, cons(y, z)) → if1(le(x, y), x, y, z)
del(x, nil) → nil
del(x, cons(y, z)) → if2(eq(x, y), x, y, z)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
if1(true, x, y, xs) → min(x, xs)
if1(false, x, y, xs) → min(y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
minsort(nil) → nil
minsort(cons(x, y)) → cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) → x
min(x, cons(y, z)) → if1(le(x, y), x, y, z)
del(x, nil) → nil
del(x, cons(y, z)) → if2(eq(x, y), x, y, z)

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
eq :: 0':s → 0':s → true:false
if1 :: true:false → 0':s → 0':s → cons:nil → 0':s
min :: 0':s → cons:nil → 0':s
if2 :: true:false → 0':s → 0':s → cons:nil → cons:nil
cons :: 0':s → cons:nil → cons:nil
del :: 0':s → cons:nil → cons:nil
minsort :: cons:nil → cons:nil
nil :: cons:nil
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

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

Heuristically decided to analyse the following defined symbols:
le, eq, min, del, minsort

They will be analysed ascendingly in the following order:
le < min
eq < del
min < minsort
del < minsort

(6) Obligation:

Innermost TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
if1(true, x, y, xs) → min(x, xs)
if1(false, x, y, xs) → min(y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
minsort(nil) → nil
minsort(cons(x, y)) → cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) → x
min(x, cons(y, z)) → if1(le(x, y), x, y, z)
del(x, nil) → nil
del(x, cons(y, z)) → if2(eq(x, y), x, y, z)

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
eq :: 0':s → 0':s → true:false
if1 :: true:false → 0':s → 0':s → cons:nil → 0':s
min :: 0':s → cons:nil → 0':s
if2 :: true:false → 0':s → 0':s → cons:nil → cons:nil
cons :: 0':s → cons:nil → cons:nil
del :: 0':s → cons:nil → cons:nil
minsort :: cons:nil → cons:nil
nil :: cons:nil
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

The following defined symbols remain to be analysed:
le, eq, min, del, minsort

They will be analysed ascendingly in the following order:
le < min
eq < del
min < minsort
del < minsort

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

Induction Base:
le(gen_0':s4_0(0), gen_0':s4_0(0)) →_R^Ω(1)
true

Induction Step:
le(gen_0':s4_0(+(n7_0, 1)), gen_0':s4_0(+(n7_0, 1))) →_R^Ω(1)
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) →_IH
true

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:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
if1(true, x, y, xs) → min(x, xs)
if1(false, x, y, xs) → min(y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
minsort(nil) → nil
minsort(cons(x, y)) → cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) → x
min(x, cons(y, z)) → if1(le(x, y), x, y, z)
del(x, nil) → nil
del(x, cons(y, z)) → if2(eq(x, y), x, y, z)

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
eq :: 0':s → 0':s → true:false
if1 :: true:false → 0':s → 0':s → cons:nil → 0':s
min :: 0':s → cons:nil → 0':s
if2 :: true:false → 0':s → 0':s → cons:nil → cons:nil
cons :: 0':s → cons:nil → cons:nil
del :: 0':s → cons:nil → cons:nil
minsort :: cons:nil → cons:nil
nil :: cons:nil
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

The following defined symbols remain to be analysed:
eq, min, del, minsort

They will be analysed ascendingly in the following order:
eq < del
min < minsort
del < minsort

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

Proved the following rewrite lemma:
eq(gen_0':s4_0(n348_0), gen_0':s4_0(n348_0)) → true, rt ∈ Ω(1 + n348₀)

Induction Base:
eq(gen_0':s4_0(0), gen_0':s4_0(0)) →_R^Ω(1)
true

Induction Step:
eq(gen_0':s4_0(+(n348_0, 1)), gen_0':s4_0(+(n348_0, 1))) →_R^Ω(1)
eq(gen_0':s4_0(n348_0), gen_0':s4_0(n348_0)) →_IH
true

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:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
if1(true, x, y, xs) → min(x, xs)
if1(false, x, y, xs) → min(y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
minsort(nil) → nil
minsort(cons(x, y)) → cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) → x
min(x, cons(y, z)) → if1(le(x, y), x, y, z)
del(x, nil) → nil
del(x, cons(y, z)) → if2(eq(x, y), x, y, z)

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
eq :: 0':s → 0':s → true:false
if1 :: true:false → 0':s → 0':s → cons:nil → 0':s
min :: 0':s → cons:nil → 0':s
if2 :: true:false → 0':s → 0':s → cons:nil → cons:nil
cons :: 0':s → cons:nil → cons:nil
del :: 0':s → cons:nil → cons:nil
minsort :: cons:nil → cons:nil
nil :: cons:nil
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
eq(gen_0':s4_0(n348_0), gen_0':s4_0(n348_0)) → true, rt ∈ Ω(1 + n348₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

The following defined symbols remain to be analysed:
min, del, minsort

They will be analysed ascendingly in the following order:
min < minsort
del < minsort

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

Proved the following rewrite lemma:
min(gen_0':s4_0(0), gen_cons:nil5_0(n929_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n929₀)

Induction Base:
min(gen_0':s4_0(0), gen_cons:nil5_0(0)) →_R^Ω(1)
gen_0':s4_0(0)

Induction Step:
min(gen_0':s4_0(0), gen_cons:nil5_0(+(n929_0, 1))) →_R^Ω(1)
if1(le(gen_0':s4_0(0), 0'), gen_0':s4_0(0), 0', gen_cons:nil5_0(n929_0)) →_L^Ω(1)
if1(true, gen_0':s4_0(0), 0', gen_cons:nil5_0(n929_0)) →_R^Ω(1)
min(gen_0':s4_0(0), gen_cons:nil5_0(n929_0)) →_IH
gen_0':s4_0(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:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
if1(true, x, y, xs) → min(x, xs)
if1(false, x, y, xs) → min(y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
minsort(nil) → nil
minsort(cons(x, y)) → cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) → x
min(x, cons(y, z)) → if1(le(x, y), x, y, z)
del(x, nil) → nil
del(x, cons(y, z)) → if2(eq(x, y), x, y, z)

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
eq :: 0':s → 0':s → true:false
if1 :: true:false → 0':s → 0':s → cons:nil → 0':s
min :: 0':s → cons:nil → 0':s
if2 :: true:false → 0':s → 0':s → cons:nil → cons:nil
cons :: 0':s → cons:nil → cons:nil
del :: 0':s → cons:nil → cons:nil
minsort :: cons:nil → cons:nil
nil :: cons:nil
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
eq(gen_0':s4_0(n348_0), gen_0':s4_0(n348_0)) → true, rt ∈ Ω(1 + n348₀)
min(gen_0':s4_0(0), gen_cons:nil5_0(n929_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n929₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

The following defined symbols remain to be analysed:
del, minsort

They will be analysed ascendingly in the following order:
del < minsort

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

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

(17) Obligation:

Innermost TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
if1(true, x, y, xs) → min(x, xs)
if1(false, x, y, xs) → min(y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
minsort(nil) → nil
minsort(cons(x, y)) → cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) → x
min(x, cons(y, z)) → if1(le(x, y), x, y, z)
del(x, nil) → nil
del(x, cons(y, z)) → if2(eq(x, y), x, y, z)

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
eq :: 0':s → 0':s → true:false
if1 :: true:false → 0':s → 0':s → cons:nil → 0':s
min :: 0':s → cons:nil → 0':s
if2 :: true:false → 0':s → 0':s → cons:nil → cons:nil
cons :: 0':s → cons:nil → cons:nil
del :: 0':s → cons:nil → cons:nil
minsort :: cons:nil → cons:nil
nil :: cons:nil
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
eq(gen_0':s4_0(n348_0), gen_0':s4_0(n348_0)) → true, rt ∈ Ω(1 + n348₀)
min(gen_0':s4_0(0), gen_cons:nil5_0(n929_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n929₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

The following defined symbols remain to be analysed:
minsort

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

Proved the following rewrite lemma:
minsort(gen_cons:nil5_0(n1626_0)) → gen_cons:nil5_0(n1626_0), rt ∈ Ω(1 + n1626₀ + n1626₀²)

Induction Base:
minsort(gen_cons:nil5_0(0)) →_R^Ω(1)
nil

Induction Step:
minsort(gen_cons:nil5_0(+(n1626_0, 1))) →_R^Ω(1)
cons(min(0', gen_cons:nil5_0(n1626_0)), minsort(del(min(0', gen_cons:nil5_0(n1626_0)), cons(0', gen_cons:nil5_0(n1626_0))))) →_L^{Ω(1 + n1626₀)}
cons(gen_0':s4_0(0), minsort(del(min(0', gen_cons:nil5_0(n1626_0)), cons(0', gen_cons:nil5_0(n1626_0))))) →_L^{Ω(1 + n1626₀)}
cons(gen_0':s4_0(0), minsort(del(gen_0':s4_0(0), cons(0', gen_cons:nil5_0(n1626_0))))) →_R^Ω(1)
cons(gen_0':s4_0(0), minsort(if2(eq(gen_0':s4_0(0), 0'), gen_0':s4_0(0), 0', gen_cons:nil5_0(n1626_0)))) →_L^Ω(1)
cons(gen_0':s4_0(0), minsort(if2(true, gen_0':s4_0(0), 0', gen_cons:nil5_0(n1626_0)))) →_R^Ω(1)
cons(gen_0':s4_0(0), minsort(gen_cons:nil5_0(n1626_0))) →_IH
cons(gen_0':s4_0(0), gen_cons:nil5_0(c1627_0))

We have rt ∈ Ω(n²) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n²).

(19) Complex Obligation (BEST)

(20) Obligation:

Innermost TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
if1(true, x, y, xs) → min(x, xs)
if1(false, x, y, xs) → min(y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
minsort(nil) → nil
minsort(cons(x, y)) → cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) → x
min(x, cons(y, z)) → if1(le(x, y), x, y, z)
del(x, nil) → nil
del(x, cons(y, z)) → if2(eq(x, y), x, y, z)

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
eq :: 0':s → 0':s → true:false
if1 :: true:false → 0':s → 0':s → cons:nil → 0':s
min :: 0':s → cons:nil → 0':s
if2 :: true:false → 0':s → 0':s → cons:nil → cons:nil
cons :: 0':s → cons:nil → cons:nil
del :: 0':s → cons:nil → cons:nil
minsort :: cons:nil → cons:nil
nil :: cons:nil
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
eq(gen_0':s4_0(n348_0), gen_0':s4_0(n348_0)) → true, rt ∈ Ω(1 + n348₀)
min(gen_0':s4_0(0), gen_cons:nil5_0(n929_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n929₀)
minsort(gen_cons:nil5_0(n1626_0)) → gen_cons:nil5_0(n1626_0), rt ∈ Ω(1 + n1626₀ + n1626₀²)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
minsort(gen_cons:nil5_0(n1626_0)) → gen_cons:nil5_0(n1626_0), rt ∈ Ω(1 + n1626₀ + n1626₀²)

(22) BOUNDS(n^2, INF)

(23) Obligation:

Innermost TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
if1(true, x, y, xs) → min(x, xs)
if1(false, x, y, xs) → min(y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
minsort(nil) → nil
minsort(cons(x, y)) → cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) → x
min(x, cons(y, z)) → if1(le(x, y), x, y, z)
del(x, nil) → nil
del(x, cons(y, z)) → if2(eq(x, y), x, y, z)

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
eq :: 0':s → 0':s → true:false
if1 :: true:false → 0':s → 0':s → cons:nil → 0':s
min :: 0':s → cons:nil → 0':s
if2 :: true:false → 0':s → 0':s → cons:nil → cons:nil
cons :: 0':s → cons:nil → cons:nil
del :: 0':s → cons:nil → cons:nil
minsort :: cons:nil → cons:nil
nil :: cons:nil
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
eq(gen_0':s4_0(n348_0), gen_0':s4_0(n348_0)) → true, rt ∈ Ω(1 + n348₀)
min(gen_0':s4_0(0), gen_cons:nil5_0(n929_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n929₀)
minsort(gen_cons:nil5_0(n1626_0)) → gen_cons:nil5_0(n1626_0), rt ∈ Ω(1 + n1626₀ + n1626₀²)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

No more defined symbols left to analyse.

(24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
minsort(gen_cons:nil5_0(n1626_0)) → gen_cons:nil5_0(n1626_0), rt ∈ Ω(1 + n1626₀ + n1626₀²)

(25) BOUNDS(n^2, INF)

(26) Obligation:

Innermost TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
if1(true, x, y, xs) → min(x, xs)
if1(false, x, y, xs) → min(y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
minsort(nil) → nil
minsort(cons(x, y)) → cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) → x
min(x, cons(y, z)) → if1(le(x, y), x, y, z)
del(x, nil) → nil
del(x, cons(y, z)) → if2(eq(x, y), x, y, z)

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
eq :: 0':s → 0':s → true:false
if1 :: true:false → 0':s → 0':s → cons:nil → 0':s
min :: 0':s → cons:nil → 0':s
if2 :: true:false → 0':s → 0':s → cons:nil → cons:nil
cons :: 0':s → cons:nil → cons:nil
del :: 0':s → cons:nil → cons:nil
minsort :: cons:nil → cons:nil
nil :: cons:nil
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
eq(gen_0':s4_0(n348_0), gen_0':s4_0(n348_0)) → true, rt ∈ Ω(1 + n348₀)
min(gen_0':s4_0(0), gen_cons:nil5_0(n929_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n929₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

No more defined symbols left to analyse.

(27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

(28) BOUNDS(n^1, INF)

(29) Obligation:

Innermost TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
if1(true, x, y, xs) → min(x, xs)
if1(false, x, y, xs) → min(y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
minsort(nil) → nil
minsort(cons(x, y)) → cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) → x
min(x, cons(y, z)) → if1(le(x, y), x, y, z)
del(x, nil) → nil
del(x, cons(y, z)) → if2(eq(x, y), x, y, z)

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
eq :: 0':s → 0':s → true:false
if1 :: true:false → 0':s → 0':s → cons:nil → 0':s
min :: 0':s → cons:nil → 0':s
if2 :: true:false → 0':s → 0':s → cons:nil → cons:nil
cons :: 0':s → cons:nil → cons:nil
del :: 0':s → cons:nil → cons:nil
minsort :: cons:nil → cons:nil
nil :: cons:nil
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
eq(gen_0':s4_0(n348_0), gen_0':s4_0(n348_0)) → true, rt ∈ Ω(1 + n348₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

No more defined symbols left to analyse.

(30) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

(31) BOUNDS(n^1, INF)

(32) Obligation:

Innermost TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
if1(true, x, y, xs) → min(x, xs)
if1(false, x, y, xs) → min(y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
minsort(nil) → nil
minsort(cons(x, y)) → cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) → x
min(x, cons(y, z)) → if1(le(x, y), x, y, z)
del(x, nil) → nil
del(x, cons(y, z)) → if2(eq(x, y), x, y, z)

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
eq :: 0':s → 0':s → true:false
if1 :: true:false → 0':s → 0':s → cons:nil → 0':s
min :: 0':s → cons:nil → 0':s
if2 :: true:false → 0':s → 0':s → cons:nil → cons:nil
cons :: 0':s → cons:nil → cons:nil
del :: 0':s → cons:nil → cons:nil
minsort :: cons:nil → cons:nil
nil :: cons:nil
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

No more defined symbols left to analyse.

(33) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

(34) BOUNDS(n^1, INF)