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:

isort(Cons(x, xs), r) → isort(xs, insert(x, r))
isort(Nil, r) → Nil
insert(S(x), r) → insert[Ite](<(S(x), x), S(x), r)
inssort(xs) → isort(xs, Nil)

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

<(S(x), S(y)) → <(x, y)
<(0, S(y)) → True
<(x, 0) → False
insert[Ite](False, x', Cons(x, xs)) → Cons(x, insert(x', xs))
insert[Ite](True, x, r) → Cons(x, r)

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:

isort(Cons(x, xs), r) → isort(xs, insert(x, r))
isort(Nil, r) → Nil
insert(S(x), r) → insert[Ite](<(S(x), x), S(x), r)
inssort(xs) → isort(xs, Nil)

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

<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
insert[Ite](False, x', Cons(x, xs)) → Cons(x, insert(x', xs))
insert[Ite](True, x, r) → Cons(x, r)

Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
isort(Cons(x, xs), r) → isort(xs, insert(x, r))
isort(Nil, r) → Nil
insert(S(x), r) → insert[Ite](<(S(x), x), S(x), r)
inssort(xs) → isort(xs, Nil)
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
insert[Ite](False, x', Cons(x, xs)) → Cons(x, insert(x', xs))
insert[Ite](True, x, r) → Cons(x, r)

Types:
isort :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
insert :: S:0' → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
S :: S:0' → S:0'
insert[Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
inssort :: Cons:Nil → Cons:Nil
0' :: S:0'
True :: True:False
False :: True:False
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:
isort, insert, <

They will be analysed ascendingly in the following order:
insert < isort
< < insert

(6) Obligation:

Innermost TRS:
Rules:
isort(Cons(x, xs), r) → isort(xs, insert(x, r))
isort(Nil, r) → Nil
insert(S(x), r) → insert[Ite](<(S(x), x), S(x), r)
inssort(xs) → isort(xs, Nil)
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
insert[Ite](False, x', Cons(x, xs)) → Cons(x, insert(x', xs))
insert[Ite](True, x, r) → Cons(x, r)

Types:
isort :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
insert :: S:0' → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
S :: S:0' → S:0'
insert[Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
inssort :: Cons:Nil → Cons:Nil
0' :: S:0'
True :: True:False
False :: True:False
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:
<, isort, insert

They will be analysed ascendingly in the following order:
insert < isort
< < insert

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

Proved the following rewrite lemma:
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_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(+(n7_0, 1)), gen_S:0'5_0(+(1, +(n7_0, 1)))) →_R^Ω(0)
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) →_IH
True

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:
isort(Cons(x, xs), r) → isort(xs, insert(x, r))
isort(Nil, r) → Nil
insert(S(x), r) → insert[Ite](<(S(x), x), S(x), r)
inssort(xs) → isort(xs, Nil)
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
insert[Ite](False, x', Cons(x, xs)) → Cons(x, insert(x', xs))
insert[Ite](True, x, r) → Cons(x, r)

Types:
isort :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
insert :: S:0' → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
S :: S:0' → S:0'
insert[Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
inssort :: Cons:Nil → Cons:Nil
0' :: S:0'
True :: True:False
False :: True:False
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:
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_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:
insert, isort

They will be analysed ascendingly in the following order:
insert < isort

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

Proved the following rewrite lemma:
insert(gen_S:0'5_0(1), gen_Cons:Nil4_0(n240_0)) → *6_0, rt ∈ Ω(n240₀)

Induction Base:
insert(gen_S:0'5_0(1), gen_Cons:Nil4_0(0))

Induction Step:
insert(gen_S:0'5_0(1), gen_Cons:Nil4_0(+(n240_0, 1))) →_R^Ω(1)
insert[Ite](<(S(gen_S:0'5_0(0)), gen_S:0'5_0(0)), S(gen_S:0'5_0(0)), gen_Cons:Nil4_0(+(n240_0, 1))) →_R^Ω(0)
insert[Ite](False, S(gen_S:0'5_0(0)), gen_Cons:Nil4_0(+(1, n240_0))) →_R^Ω(0)
Cons(0', insert(S(gen_S:0'5_0(0)), gen_Cons:Nil4_0(n240_0))) →_IH
Cons(0', *6_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:
isort(Cons(x, xs), r) → isort(xs, insert(x, r))
isort(Nil, r) → Nil
insert(S(x), r) → insert[Ite](<(S(x), x), S(x), r)
inssort(xs) → isort(xs, Nil)
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
insert[Ite](False, x', Cons(x, xs)) → Cons(x, insert(x', xs))
insert[Ite](True, x, r) → Cons(x, r)

Types:
isort :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
insert :: S:0' → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
S :: S:0' → S:0'
insert[Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
inssort :: Cons:Nil → Cons:Nil
0' :: S:0'
True :: True:False
False :: True:False
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:
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → True, rt ∈ Ω(0)
insert(gen_S:0'5_0(1), gen_Cons:Nil4_0(n240_0)) → *6_0, rt ∈ Ω(n240₀)

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:
isort

(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(14) Obligation:

Innermost TRS:
Rules:
isort(Cons(x, xs), r) → isort(xs, insert(x, r))
isort(Nil, r) → Nil
insert(S(x), r) → insert[Ite](<(S(x), x), S(x), r)
inssort(xs) → isort(xs, Nil)
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
insert[Ite](False, x', Cons(x, xs)) → Cons(x, insert(x', xs))
insert[Ite](True, x, r) → Cons(x, r)

Types:
isort :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
insert :: S:0' → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
S :: S:0' → S:0'
insert[Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
inssort :: Cons:Nil → Cons:Nil
0' :: S:0'
True :: True:False
False :: True:False
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:
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → True, rt ∈ Ω(0)
insert(gen_S:0'5_0(1), gen_Cons:Nil4_0(n240_0)) → *6_0, rt ∈ Ω(n240₀)

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.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
insert(gen_S:0'5_0(1), gen_Cons:Nil4_0(n240_0)) → *6_0, rt ∈ Ω(n240₀)

(16) BOUNDS(n^1, INF)

(17) Obligation:

Innermost TRS:
Rules:
isort(Cons(x, xs), r) → isort(xs, insert(x, r))
isort(Nil, r) → Nil
insert(S(x), r) → insert[Ite](<(S(x), x), S(x), r)
inssort(xs) → isort(xs, Nil)
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
insert[Ite](False, x', Cons(x, xs)) → Cons(x, insert(x', xs))
insert[Ite](True, x, r) → Cons(x, r)

Types:
isort :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
insert :: S:0' → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
S :: S:0' → S:0'
insert[Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
inssort :: Cons:Nil → Cons:Nil
0' :: S:0'
True :: True:False
False :: True:False
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:
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → True, rt ∈ Ω(0)
insert(gen_S:0'5_0(1), gen_Cons:Nil4_0(n240_0)) → *6_0, rt ∈ Ω(n240₀)

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.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
insert(gen_S:0'5_0(1), gen_Cons:Nil4_0(n240_0)) → *6_0, rt ∈ Ω(n240₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

Innermost TRS:
Rules:
isort(Cons(x, xs), r) → isort(xs, insert(x, r))
isort(Nil, r) → Nil
insert(S(x), r) → insert[Ite](<(S(x), x), S(x), r)
inssort(xs) → isort(xs, Nil)
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
insert[Ite](False, x', Cons(x, xs)) → Cons(x, insert(x', xs))
insert[Ite](True, x, r) → Cons(x, r)

Types:
isort :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
insert :: S:0' → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
S :: S:0' → S:0'
insert[Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
inssort :: Cons:Nil → Cons:Nil
0' :: S:0'
True :: True:False
False :: True:False
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:
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_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.

(21) LowerBoundsProof (EQUIVALENT transformation)

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

(22) BOUNDS(1, INF)