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:

sort(nil) → nil
sort(cons(x, y)) → insert(x, sort(y))
insert(x, nil) → cons(x, nil)
insert(x, cons(v, w)) → choose(x, cons(v, w), x, v)
choose(x, cons(v, w), y, 0) → cons(x, cons(v, w))
choose(x, cons(v, w), 0, s(z)) → cons(v, insert(x, w))
choose(x, cons(v, w), s(y), s(z)) → choose(x, cons(v, w), y, z)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
sort(cons(x, y)) →⁺ insert(x, sort(y))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [y / cons(x, 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:

sort(nil) → nil
sort(cons(x, y)) → insert(x, sort(y))
insert(x, nil) → cons(x, nil)
insert(x, cons(v, w)) → choose(x, cons(v, w), x, v)
choose(x, cons(v, w), y, 0') → cons(x, cons(v, w))
choose(x, cons(v, w), 0', s(z)) → cons(v, insert(x, w))
choose(x, cons(v, w), s(y), s(z)) → choose(x, cons(v, w), y, z)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
sort(nil) → nil
sort(cons(x, y)) → insert(x, sort(y))
insert(x, nil) → cons(x, nil)
insert(x, cons(v, w)) → choose(x, cons(v, w), x, v)
choose(x, cons(v, w), y, 0') → cons(x, cons(v, w))
choose(x, cons(v, w), 0', s(z)) → cons(v, insert(x, w))
choose(x, cons(v, w), s(y), s(z)) → choose(x, cons(v, w), y, z)

Types:
sort :: nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
insert :: 0':s → nil:cons → nil:cons
choose :: 0':s → nil:cons → 0':s → 0':s → nil:cons
0' :: 0':s
s :: 0':s → 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
gen_nil:cons3_0 :: Nat → nil:cons
gen_0':s4_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
sort, insert, choose

They will be analysed ascendingly in the following order:
insert < sort
insert = choose

(8) Obligation:

TRS:
Rules:
sort(nil) → nil
sort(cons(x, y)) → insert(x, sort(y))
insert(x, nil) → cons(x, nil)
insert(x, cons(v, w)) → choose(x, cons(v, w), x, v)
choose(x, cons(v, w), y, 0') → cons(x, cons(v, w))
choose(x, cons(v, w), 0', s(z)) → cons(v, insert(x, w))
choose(x, cons(v, w), s(y), s(z)) → choose(x, cons(v, w), y, z)

Types:
sort :: nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
insert :: 0':s → nil:cons → nil:cons
choose :: 0':s → nil:cons → 0':s → 0':s → nil:cons
0' :: 0':s
s :: 0':s → 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
gen_nil:cons3_0 :: Nat → nil:cons
gen_0':s4_0 :: Nat → 0':s

Generator Equations:
gen_nil:cons3_0(0) ⇔ nil
gen_nil:cons3_0(+(x, 1)) ⇔ cons(0', gen_nil:cons3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
choose, sort, insert

They will be analysed ascendingly in the following order:
insert < sort
insert = choose

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

Proved the following rewrite lemma:
choose(gen_0':s4_0(a), gen_nil:cons3_0(1), gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → cons(gen_0':s4_0(a), gen_nil:cons3_0(1)), rt ∈ Ω(1 + n6₀)

Induction Base:
choose(gen_0':s4_0(a), gen_nil:cons3_0(1), gen_0':s4_0(0), gen_0':s4_0(0)) →_R^Ω(1)
cons(gen_0':s4_0(a), cons(0', gen_nil:cons3_0(0)))

Induction Step:
choose(gen_0':s4_0(a), gen_nil:cons3_0(1), gen_0':s4_0(+(n6_0, 1)), gen_0':s4_0(+(n6_0, 1))) →_R^Ω(1)
choose(gen_0':s4_0(a), cons(0', gen_nil:cons3_0(0)), gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) →_IH
cons(gen_0':s4_0(a), gen_nil:cons3_0(1))

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

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
sort(nil) → nil
sort(cons(x, y)) → insert(x, sort(y))
insert(x, nil) → cons(x, nil)
insert(x, cons(v, w)) → choose(x, cons(v, w), x, v)
choose(x, cons(v, w), y, 0') → cons(x, cons(v, w))
choose(x, cons(v, w), 0', s(z)) → cons(v, insert(x, w))
choose(x, cons(v, w), s(y), s(z)) → choose(x, cons(v, w), y, z)

Types:
sort :: nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
insert :: 0':s → nil:cons → nil:cons
choose :: 0':s → nil:cons → 0':s → 0':s → nil:cons
0' :: 0':s
s :: 0':s → 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
gen_nil:cons3_0 :: Nat → nil:cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
choose(gen_0':s4_0(a), gen_nil:cons3_0(1), gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → cons(gen_0':s4_0(a), gen_nil:cons3_0(1)), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_nil:cons3_0(0) ⇔ nil
gen_nil:cons3_0(+(x, 1)) ⇔ cons(0', gen_nil:cons3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
insert, sort

They will be analysed ascendingly in the following order:
insert < sort
insert = choose

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(13) Obligation:

TRS:
Rules:
sort(nil) → nil
sort(cons(x, y)) → insert(x, sort(y))
insert(x, nil) → cons(x, nil)
insert(x, cons(v, w)) → choose(x, cons(v, w), x, v)
choose(x, cons(v, w), y, 0') → cons(x, cons(v, w))
choose(x, cons(v, w), 0', s(z)) → cons(v, insert(x, w))
choose(x, cons(v, w), s(y), s(z)) → choose(x, cons(v, w), y, z)

Types:
sort :: nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
insert :: 0':s → nil:cons → nil:cons
choose :: 0':s → nil:cons → 0':s → 0':s → nil:cons
0' :: 0':s
s :: 0':s → 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
gen_nil:cons3_0 :: Nat → nil:cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
choose(gen_0':s4_0(a), gen_nil:cons3_0(1), gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → cons(gen_0':s4_0(a), gen_nil:cons3_0(1)), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_nil:cons3_0(0) ⇔ nil
gen_nil:cons3_0(+(x, 1)) ⇔ cons(0', gen_nil:cons3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
sort

(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(15) Obligation:

TRS:
Rules:
sort(nil) → nil
sort(cons(x, y)) → insert(x, sort(y))
insert(x, nil) → cons(x, nil)
insert(x, cons(v, w)) → choose(x, cons(v, w), x, v)
choose(x, cons(v, w), y, 0') → cons(x, cons(v, w))
choose(x, cons(v, w), 0', s(z)) → cons(v, insert(x, w))
choose(x, cons(v, w), s(y), s(z)) → choose(x, cons(v, w), y, z)

Types:
sort :: nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
insert :: 0':s → nil:cons → nil:cons
choose :: 0':s → nil:cons → 0':s → 0':s → nil:cons
0' :: 0':s
s :: 0':s → 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
gen_nil:cons3_0 :: Nat → nil:cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
choose(gen_0':s4_0(a), gen_nil:cons3_0(1), gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → cons(gen_0':s4_0(a), gen_nil:cons3_0(1)), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_nil:cons3_0(0) ⇔ nil
gen_nil:cons3_0(+(x, 1)) ⇔ cons(0', gen_nil:cons3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
choose(gen_0':s4_0(a), gen_nil:cons3_0(1), gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → cons(gen_0':s4_0(a), gen_nil:cons3_0(1)), rt ∈ Ω(1 + n6₀)

(17) BOUNDS(n^1, INF)

(18) Obligation:

TRS:
Rules:
sort(nil) → nil
sort(cons(x, y)) → insert(x, sort(y))
insert(x, nil) → cons(x, nil)
insert(x, cons(v, w)) → choose(x, cons(v, w), x, v)
choose(x, cons(v, w), y, 0') → cons(x, cons(v, w))
choose(x, cons(v, w), 0', s(z)) → cons(v, insert(x, w))
choose(x, cons(v, w), s(y), s(z)) → choose(x, cons(v, w), y, z)

Types:
sort :: nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
insert :: 0':s → nil:cons → nil:cons
choose :: 0':s → nil:cons → 0':s → 0':s → nil:cons
0' :: 0':s
s :: 0':s → 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
gen_nil:cons3_0 :: Nat → nil:cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
choose(gen_0':s4_0(a), gen_nil:cons3_0(1), gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → cons(gen_0':s4_0(a), gen_nil:cons3_0(1)), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_nil:cons3_0(0) ⇔ nil
gen_nil:cons3_0(+(x, 1)) ⇔ cons(0', gen_nil:cons3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
choose(gen_0':s4_0(a), gen_nil:cons3_0(1), gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → cons(gen_0':s4_0(a), gen_nil:cons3_0(1)), rt ∈ Ω(1 + n6₀)

(20) BOUNDS(n^1, INF)