The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

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

remove(x', Cons(x, xs)) → remove[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
remove(x, Nil) → Nil
minsort(Cons(x, xs)) → appmin(x, xs, Cons(x, xs))
minsort(Nil) → Nil
appmin(min, Cons(x, xs), xs') → appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
appmin(min, Nil, xs) → Cons(min, minsort(remove(min, xs)))
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → minsort(xs)

The (relative) TRS S consists of the following rules:

!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0, S(y)) → False
!EQ(S(x), 0) → False
!EQ(0, 0) → True
<(S(x), S(y)) → <(x, y)
<(0, S(y)) → True
<(x, 0) → False
remove[Ite][True][Ite](False, x', Cons(x, xs)) → Cons(x, remove(x', xs))
appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') → appmin(x, xs, xs')
remove[Ite][True][Ite](True, x', Cons(x, xs)) → xs
appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') → appmin(min, xs, xs')

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:

remove(x', Cons(x, xs)) → remove[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
remove(x, Nil) → Nil
minsort(Cons(x, xs)) → appmin(x, xs, Cons(x, xs))
minsort(Nil) → Nil
appmin(min, Cons(x, xs), xs') → appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
appmin(min, Nil, xs) → Cons(min, minsort(remove(min, xs)))
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → minsort(xs)

The (relative) TRS S consists of the following rules:

!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
remove[Ite][True][Ite](False, x', Cons(x, xs)) → Cons(x, remove(x', xs))
appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') → appmin(x, xs, xs')
remove[Ite][True][Ite](True, x', Cons(x, xs)) → xs
appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') → appmin(min, xs, xs')

Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
remove(x', Cons(x, xs)) → remove[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
remove(x, Nil) → Nil
minsort(Cons(x, xs)) → appmin(x, xs, Cons(x, xs))
minsort(Nil) → Nil
appmin(min, Cons(x, xs), xs') → appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
appmin(min, Nil, xs) → Cons(min, minsort(remove(min, xs)))
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → minsort(xs)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
remove[Ite][True][Ite](False, x', Cons(x, xs)) → Cons(x, remove(x', xs))
appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') → appmin(x, xs, xs')
remove[Ite][True][Ite](True, x', Cons(x, xs)) → xs
appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') → appmin(min, xs, xs')

Types:
remove :: S:0' → Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
remove[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
!EQ :: S:0' → S:0' → True:False
Nil :: Cons:Nil
minsort :: Cons:Nil → Cons:Nil
appmin :: S:0' → Cons:Nil → Cons:Nil → Cons:Nil
appmin[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

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

Heuristically decided to analyse the following defined symbols:
remove, !EQ, minsort, appmin, <

They will be analysed ascendingly in the following order:
!EQ < remove
remove < appmin
minsort = appmin
< < appmin

(6) Obligation:

Innermost TRS:
Rules:
remove(x', Cons(x, xs)) → remove[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
remove(x, Nil) → Nil
minsort(Cons(x, xs)) → appmin(x, xs, Cons(x, xs))
minsort(Nil) → Nil
appmin(min, Cons(x, xs), xs') → appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
appmin(min, Nil, xs) → Cons(min, minsort(remove(min, xs)))
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → minsort(xs)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
remove[Ite][True][Ite](False, x', Cons(x, xs)) → Cons(x, remove(x', xs))
appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') → appmin(x, xs, xs')
remove[Ite][True][Ite](True, x', Cons(x, xs)) → xs
appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') → appmin(min, xs, xs')

Types:
remove :: S:0' → Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
remove[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
!EQ :: S:0' → S:0' → True:False
Nil :: Cons:Nil
minsort :: Cons:Nil → Cons:Nil
appmin :: S:0' → Cons:Nil → Cons:Nil → Cons:Nil
appmin[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

The following defined symbols remain to be analysed:
!EQ, remove, minsort, appmin, <

They will be analysed ascendingly in the following order:
!EQ < remove
remove < appmin
minsort = appmin
< < appmin

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

Proved the following rewrite lemma:
!EQ(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → False, rt ∈ Ω(0)

Induction Base:
!EQ(gen_S:0'5_0(0), gen_S:0'5_0(+(1, 0))) →_R^Ω(0)
False

Induction Step:
!EQ(gen_S:0'5_0(+(n7_0, 1)), gen_S:0'5_0(+(1, +(n7_0, 1)))) →_R^Ω(0)
!EQ(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) →_IH
False

We have rt ∈ Ω(1) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n⁰).

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
remove(x', Cons(x, xs)) → remove[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
remove(x, Nil) → Nil
minsort(Cons(x, xs)) → appmin(x, xs, Cons(x, xs))
minsort(Nil) → Nil
appmin(min, Cons(x, xs), xs') → appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
appmin(min, Nil, xs) → Cons(min, minsort(remove(min, xs)))
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → minsort(xs)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
remove[Ite][True][Ite](False, x', Cons(x, xs)) → Cons(x, remove(x', xs))
appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') → appmin(x, xs, xs')
remove[Ite][True][Ite](True, x', Cons(x, xs)) → xs
appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') → appmin(min, xs, xs')

Types:
remove :: S:0' → Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
remove[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
!EQ :: S:0' → S:0' → True:False
Nil :: Cons:Nil
minsort :: Cons:Nil → Cons:Nil
appmin :: S:0' → Cons:Nil → Cons:Nil → Cons:Nil
appmin[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
!EQ(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → False, rt ∈ Ω(0)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

The following defined symbols remain to be analysed:
remove, minsort, appmin, <

They will be analysed ascendingly in the following order:
remove < appmin
minsort = appmin
< < appmin

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

Proved the following rewrite lemma:
remove(gen_S:0'5_0(1), gen_Cons:Nil4_0(n366_0)) → gen_Cons:Nil4_0(n366_0), rt ∈ Ω(1 + n366₀)

Induction Base:
remove(gen_S:0'5_0(1), gen_Cons:Nil4_0(0)) →_R^Ω(1)
Nil

Induction Step:
remove(gen_S:0'5_0(1), gen_Cons:Nil4_0(+(n366_0, 1))) →_R^Ω(1)
remove[Ite][True][Ite](!EQ(gen_S:0'5_0(1), 0'), gen_S:0'5_0(1), Cons(0', gen_Cons:Nil4_0(n366_0))) →_R^Ω(0)
remove[Ite][True][Ite](False, gen_S:0'5_0(1), Cons(0', gen_Cons:Nil4_0(n366_0))) →_R^Ω(0)
Cons(0', remove(gen_S:0'5_0(1), gen_Cons:Nil4_0(n366_0))) →_IH
Cons(0', gen_Cons:Nil4_0(c367_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:
remove(x', Cons(x, xs)) → remove[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
remove(x, Nil) → Nil
minsort(Cons(x, xs)) → appmin(x, xs, Cons(x, xs))
minsort(Nil) → Nil
appmin(min, Cons(x, xs), xs') → appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
appmin(min, Nil, xs) → Cons(min, minsort(remove(min, xs)))
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → minsort(xs)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
remove[Ite][True][Ite](False, x', Cons(x, xs)) → Cons(x, remove(x', xs))
appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') → appmin(x, xs, xs')
remove[Ite][True][Ite](True, x', Cons(x, xs)) → xs
appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') → appmin(min, xs, xs')

Types:
remove :: S:0' → Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
remove[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
!EQ :: S:0' → S:0' → True:False
Nil :: Cons:Nil
minsort :: Cons:Nil → Cons:Nil
appmin :: S:0' → Cons:Nil → Cons:Nil → Cons:Nil
appmin[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
!EQ(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → False, rt ∈ Ω(0)
remove(gen_S:0'5_0(1), gen_Cons:Nil4_0(n366_0)) → gen_Cons:Nil4_0(n366_0), rt ∈ Ω(1 + n366₀)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

The following defined symbols remain to be analysed:
<, minsort, appmin

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

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

Proved the following rewrite lemma:
<(gen_S:0'5_0(n882_0), gen_S:0'5_0(+(1, n882_0))) → True, rt ∈ Ω(0)

Induction Base:
<(gen_S:0'5_0(0), gen_S:0'5_0(+(1, 0))) →_R^Ω(0)
True

Induction Step:
<(gen_S:0'5_0(+(n882_0, 1)), gen_S:0'5_0(+(1, +(n882_0, 1)))) →_R^Ω(0)
<(gen_S:0'5_0(n882_0), gen_S:0'5_0(+(1, n882_0))) →_IH
True

We have rt ∈ Ω(1) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n⁰).

(14) Complex Obligation (BEST)

(15) Obligation:

Innermost TRS:
Rules:
remove(x', Cons(x, xs)) → remove[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
remove(x, Nil) → Nil
minsort(Cons(x, xs)) → appmin(x, xs, Cons(x, xs))
minsort(Nil) → Nil
appmin(min, Cons(x, xs), xs') → appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
appmin(min, Nil, xs) → Cons(min, minsort(remove(min, xs)))
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → minsort(xs)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
remove[Ite][True][Ite](False, x', Cons(x, xs)) → Cons(x, remove(x', xs))
appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') → appmin(x, xs, xs')
remove[Ite][True][Ite](True, x', Cons(x, xs)) → xs
appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') → appmin(min, xs, xs')

Types:
remove :: S:0' → Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
remove[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
!EQ :: S:0' → S:0' → True:False
Nil :: Cons:Nil
minsort :: Cons:Nil → Cons:Nil
appmin :: S:0' → Cons:Nil → Cons:Nil → Cons:Nil
appmin[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
!EQ(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → False, rt ∈ Ω(0)
remove(gen_S:0'5_0(1), gen_Cons:Nil4_0(n366_0)) → gen_Cons:Nil4_0(n366_0), rt ∈ Ω(1 + n366₀)
<(gen_S:0'5_0(n882_0), gen_S:0'5_0(+(1, n882_0))) → True, rt ∈ Ω(0)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

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

They will be analysed ascendingly in the following order:
minsort = appmin

(16) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
appmin(gen_S:0'5_0(0), gen_Cons:Nil4_0(n1193_0), gen_Cons:Nil4_0(0)) → gen_Cons:Nil4_0(1), rt ∈ Ω(1 + n1193₀)

Induction Base:
appmin(gen_S:0'5_0(0), gen_Cons:Nil4_0(0), gen_Cons:Nil4_0(0)) →_R^Ω(1)
Cons(gen_S:0'5_0(0), minsort(remove(gen_S:0'5_0(0), gen_Cons:Nil4_0(0)))) →_R^Ω(1)
Cons(gen_S:0'5_0(0), minsort(Nil)) →_R^Ω(1)
Cons(gen_S:0'5_0(0), Nil)

Induction Step:
appmin(gen_S:0'5_0(0), gen_Cons:Nil4_0(+(n1193_0, 1)), gen_Cons:Nil4_0(0)) →_R^Ω(1)
appmin[Ite][True][Ite](<(0', gen_S:0'5_0(0)), gen_S:0'5_0(0), Cons(0', gen_Cons:Nil4_0(n1193_0)), gen_Cons:Nil4_0(0)) →_R^Ω(0)
appmin[Ite][True][Ite](False, gen_S:0'5_0(0), Cons(0', gen_Cons:Nil4_0(n1193_0)), gen_Cons:Nil4_0(0)) →_R^Ω(0)
appmin(gen_S:0'5_0(0), gen_Cons:Nil4_0(n1193_0), gen_Cons:Nil4_0(0)) →_IH
gen_Cons:Nil4_0(1)

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

(17) Complex Obligation (BEST)

(18) Obligation:

Innermost TRS:
Rules:
remove(x', Cons(x, xs)) → remove[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
remove(x, Nil) → Nil
minsort(Cons(x, xs)) → appmin(x, xs, Cons(x, xs))
minsort(Nil) → Nil
appmin(min, Cons(x, xs), xs') → appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
appmin(min, Nil, xs) → Cons(min, minsort(remove(min, xs)))
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → minsort(xs)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
remove[Ite][True][Ite](False, x', Cons(x, xs)) → Cons(x, remove(x', xs))
appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') → appmin(x, xs, xs')
remove[Ite][True][Ite](True, x', Cons(x, xs)) → xs
appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') → appmin(min, xs, xs')

Types:
remove :: S:0' → Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
remove[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
!EQ :: S:0' → S:0' → True:False
Nil :: Cons:Nil
minsort :: Cons:Nil → Cons:Nil
appmin :: S:0' → Cons:Nil → Cons:Nil → Cons:Nil
appmin[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
!EQ(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → False, rt ∈ Ω(0)
remove(gen_S:0'5_0(1), gen_Cons:Nil4_0(n366_0)) → gen_Cons:Nil4_0(n366_0), rt ∈ Ω(1 + n366₀)
<(gen_S:0'5_0(n882_0), gen_S:0'5_0(+(1, n882_0))) → True, rt ∈ Ω(0)
appmin(gen_S:0'5_0(0), gen_Cons:Nil4_0(n1193_0), gen_Cons:Nil4_0(0)) → gen_Cons:Nil4_0(1), rt ∈ Ω(1 + n1193₀)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

The following defined symbols remain to be analysed:
minsort

They will be analysed ascendingly in the following order:
minsort = appmin

(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(20) Obligation:

Innermost TRS:
Rules:
remove(x', Cons(x, xs)) → remove[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
remove(x, Nil) → Nil
minsort(Cons(x, xs)) → appmin(x, xs, Cons(x, xs))
minsort(Nil) → Nil
appmin(min, Cons(x, xs), xs') → appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
appmin(min, Nil, xs) → Cons(min, minsort(remove(min, xs)))
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → minsort(xs)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
remove[Ite][True][Ite](False, x', Cons(x, xs)) → Cons(x, remove(x', xs))
appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') → appmin(x, xs, xs')
remove[Ite][True][Ite](True, x', Cons(x, xs)) → xs
appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') → appmin(min, xs, xs')

Types:
remove :: S:0' → Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
remove[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
!EQ :: S:0' → S:0' → True:False
Nil :: Cons:Nil
minsort :: Cons:Nil → Cons:Nil
appmin :: S:0' → Cons:Nil → Cons:Nil → Cons:Nil
appmin[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
!EQ(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → False, rt ∈ Ω(0)
remove(gen_S:0'5_0(1), gen_Cons:Nil4_0(n366_0)) → gen_Cons:Nil4_0(n366_0), rt ∈ Ω(1 + n366₀)
<(gen_S:0'5_0(n882_0), gen_S:0'5_0(+(1, n882_0))) → True, rt ∈ Ω(0)
appmin(gen_S:0'5_0(0), gen_Cons:Nil4_0(n1193_0), gen_Cons:Nil4_0(0)) → gen_Cons:Nil4_0(1), rt ∈ Ω(1 + n1193₀)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
remove(gen_S:0'5_0(1), gen_Cons:Nil4_0(n366_0)) → gen_Cons:Nil4_0(n366_0), rt ∈ Ω(1 + n366₀)

(22) BOUNDS(n^1, INF)

(23) Obligation:

Innermost TRS:
Rules:
remove(x', Cons(x, xs)) → remove[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
remove(x, Nil) → Nil
minsort(Cons(x, xs)) → appmin(x, xs, Cons(x, xs))
minsort(Nil) → Nil
appmin(min, Cons(x, xs), xs') → appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
appmin(min, Nil, xs) → Cons(min, minsort(remove(min, xs)))
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → minsort(xs)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
remove[Ite][True][Ite](False, x', Cons(x, xs)) → Cons(x, remove(x', xs))
appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') → appmin(x, xs, xs')
remove[Ite][True][Ite](True, x', Cons(x, xs)) → xs
appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') → appmin(min, xs, xs')

Types:
remove :: S:0' → Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
remove[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
!EQ :: S:0' → S:0' → True:False
Nil :: Cons:Nil
minsort :: Cons:Nil → Cons:Nil
appmin :: S:0' → Cons:Nil → Cons:Nil → Cons:Nil
appmin[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
!EQ(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → False, rt ∈ Ω(0)
remove(gen_S:0'5_0(1), gen_Cons:Nil4_0(n366_0)) → gen_Cons:Nil4_0(n366_0), rt ∈ Ω(1 + n366₀)
<(gen_S:0'5_0(n882_0), gen_S:0'5_0(+(1, n882_0))) → True, rt ∈ Ω(0)
appmin(gen_S:0'5_0(0), gen_Cons:Nil4_0(n1193_0), gen_Cons:Nil4_0(0)) → gen_Cons:Nil4_0(1), rt ∈ Ω(1 + n1193₀)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

No more defined symbols left to analyse.

(24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
remove(gen_S:0'5_0(1), gen_Cons:Nil4_0(n366_0)) → gen_Cons:Nil4_0(n366_0), rt ∈ Ω(1 + n366₀)

(25) BOUNDS(n^1, INF)

(26) Obligation:

Innermost TRS:
Rules:
remove(x', Cons(x, xs)) → remove[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
remove(x, Nil) → Nil
minsort(Cons(x, xs)) → appmin(x, xs, Cons(x, xs))
minsort(Nil) → Nil
appmin(min, Cons(x, xs), xs') → appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
appmin(min, Nil, xs) → Cons(min, minsort(remove(min, xs)))
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → minsort(xs)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
remove[Ite][True][Ite](False, x', Cons(x, xs)) → Cons(x, remove(x', xs))
appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') → appmin(x, xs, xs')
remove[Ite][True][Ite](True, x', Cons(x, xs)) → xs
appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') → appmin(min, xs, xs')

Types:
remove :: S:0' → Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
remove[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
!EQ :: S:0' → S:0' → True:False
Nil :: Cons:Nil
minsort :: Cons:Nil → Cons:Nil
appmin :: S:0' → Cons:Nil → Cons:Nil → Cons:Nil
appmin[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
!EQ(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → False, rt ∈ Ω(0)
remove(gen_S:0'5_0(1), gen_Cons:Nil4_0(n366_0)) → gen_Cons:Nil4_0(n366_0), rt ∈ Ω(1 + n366₀)
<(gen_S:0'5_0(n882_0), gen_S:0'5_0(+(1, n882_0))) → True, rt ∈ Ω(0)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

No more defined symbols left to analyse.

(27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
remove(gen_S:0'5_0(1), gen_Cons:Nil4_0(n366_0)) → gen_Cons:Nil4_0(n366_0), rt ∈ Ω(1 + n366₀)

(28) BOUNDS(n^1, INF)

(29) Obligation:

Innermost TRS:
Rules:
remove(x', Cons(x, xs)) → remove[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
remove(x, Nil) → Nil
minsort(Cons(x, xs)) → appmin(x, xs, Cons(x, xs))
minsort(Nil) → Nil
appmin(min, Cons(x, xs), xs') → appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
appmin(min, Nil, xs) → Cons(min, minsort(remove(min, xs)))
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → minsort(xs)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
remove[Ite][True][Ite](False, x', Cons(x, xs)) → Cons(x, remove(x', xs))
appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') → appmin(x, xs, xs')
remove[Ite][True][Ite](True, x', Cons(x, xs)) → xs
appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') → appmin(min, xs, xs')

Types:
remove :: S:0' → Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
remove[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
!EQ :: S:0' → S:0' → True:False
Nil :: Cons:Nil
minsort :: Cons:Nil → Cons:Nil
appmin :: S:0' → Cons:Nil → Cons:Nil → Cons:Nil
appmin[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
!EQ(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → False, rt ∈ Ω(0)
remove(gen_S:0'5_0(1), gen_Cons:Nil4_0(n366_0)) → gen_Cons:Nil4_0(n366_0), rt ∈ Ω(1 + n366₀)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

No more defined symbols left to analyse.

(30) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
remove(gen_S:0'5_0(1), gen_Cons:Nil4_0(n366_0)) → gen_Cons:Nil4_0(n366_0), rt ∈ Ω(1 + n366₀)

(31) BOUNDS(n^1, INF)

(32) Obligation:

Innermost TRS:
Rules:
remove(x', Cons(x, xs)) → remove[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
remove(x, Nil) → Nil
minsort(Cons(x, xs)) → appmin(x, xs, Cons(x, xs))
minsort(Nil) → Nil
appmin(min, Cons(x, xs), xs') → appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
appmin(min, Nil, xs) → Cons(min, minsort(remove(min, xs)))
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → minsort(xs)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
remove[Ite][True][Ite](False, x', Cons(x, xs)) → Cons(x, remove(x', xs))
appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') → appmin(x, xs, xs')
remove[Ite][True][Ite](True, x', Cons(x, xs)) → xs
appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') → appmin(min, xs, xs')

Types:
remove :: S:0' → Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
remove[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
!EQ :: S:0' → S:0' → True:False
Nil :: Cons:Nil
minsort :: Cons:Nil → Cons:Nil
appmin :: S:0' → Cons:Nil → Cons:Nil → Cons:Nil
appmin[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
!EQ(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → False, rt ∈ Ω(0)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

No more defined symbols left to analyse.

(33) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(1) was proven with the following lemma:
!EQ(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → False, rt ∈ Ω(0)

(34) BOUNDS(1, INF)