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

(0) Obligation:

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

qsort(nil) → nil
qsort(.(x, y)) → ++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y))))
lowers(x, nil) → nil
lowers(x, .(y, z)) → if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
greaters(x, nil) → nil
greaters(x, .(y, z)) → if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2ⁿ):
The rewrite sequence
lowers(x, .(y, z)) →⁺ if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1].
The pumping substitution is [z / .(y, z)].
The result substitution is [ ].

The rewrite sequence
lowers(x, .(y, z)) →⁺ if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
gives rise to a decreasing loop by considering the right hand sides subterm at position [2].
The pumping substitution is [z / .(y, z)].
The result substitution is [ ].

(2) BOUNDS(2^n, 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:

qsort(nil) → nil
qsort(.(x, y)) → ++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y))))
lowers(x, nil) → nil
lowers(x, .(y, z)) → if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
greaters(x, nil) → nil
greaters(x, .(y, z)) → if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
qsort(nil) → nil
qsort(.(x, y)) → ++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y))))
lowers(x, nil) → nil
lowers(x, .(y, z)) → if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
greaters(x, nil) → nil
greaters(x, .(y, z)) → if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))

Types:
qsort :: nil:.:++:if → nil:.:++:if
nil :: nil:.:++:if
. :: a → nil:.:++:if → nil:.:++:if
++ :: nil:.:++:if → nil:.:++:if → nil:.:++:if
lowers :: a → nil:.:++:if → nil:.:++:if
greaters :: a → nil:.:++:if → nil:.:++:if
if :: <= → nil:.:++:if → nil:.:++:if → nil:.:++:if
<= :: a → a → <=
hole_nil:.:++:if1_0 :: nil:.:++:if
hole_a2_0 :: a
hole_<=3_0 :: <=
gen_nil:.:++:if4_0 :: Nat → nil:.:++:if

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

Heuristically decided to analyse the following defined symbols:
qsort, lowers, greaters

They will be analysed ascendingly in the following order:
lowers < qsort
greaters < qsort

(8) Obligation:

TRS:
Rules:
qsort(nil) → nil
qsort(.(x, y)) → ++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y))))
lowers(x, nil) → nil
lowers(x, .(y, z)) → if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
greaters(x, nil) → nil
greaters(x, .(y, z)) → if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))

Types:
qsort :: nil:.:++:if → nil:.:++:if
nil :: nil:.:++:if
. :: a → nil:.:++:if → nil:.:++:if
++ :: nil:.:++:if → nil:.:++:if → nil:.:++:if
lowers :: a → nil:.:++:if → nil:.:++:if
greaters :: a → nil:.:++:if → nil:.:++:if
if :: <= → nil:.:++:if → nil:.:++:if → nil:.:++:if
<= :: a → a → <=
hole_nil:.:++:if1_0 :: nil:.:++:if
hole_a2_0 :: a
hole_<=3_0 :: <=
gen_nil:.:++:if4_0 :: Nat → nil:.:++:if

Generator Equations:
gen_nil:.:++:if4_0(0) ⇔ nil
gen_nil:.:++:if4_0(+(x, 1)) ⇔ .(hole_a2_0, gen_nil:.:++:if4_0(x))

The following defined symbols remain to be analysed:
lowers, qsort, greaters

They will be analysed ascendingly in the following order:
lowers < qsort
greaters < qsort

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) Obligation:

TRS:
Rules:
qsort(nil) → nil
qsort(.(x, y)) → ++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y))))
lowers(x, nil) → nil
lowers(x, .(y, z)) → if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
greaters(x, nil) → nil
greaters(x, .(y, z)) → if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))

Types:
qsort :: nil:.:++:if → nil:.:++:if
nil :: nil:.:++:if
. :: a → nil:.:++:if → nil:.:++:if
++ :: nil:.:++:if → nil:.:++:if → nil:.:++:if
lowers :: a → nil:.:++:if → nil:.:++:if
greaters :: a → nil:.:++:if → nil:.:++:if
if :: <= → nil:.:++:if → nil:.:++:if → nil:.:++:if
<= :: a → a → <=
hole_nil:.:++:if1_0 :: nil:.:++:if
hole_a2_0 :: a
hole_<=3_0 :: <=
gen_nil:.:++:if4_0 :: Nat → nil:.:++:if

Generator Equations:
gen_nil:.:++:if4_0(0) ⇔ nil
gen_nil:.:++:if4_0(+(x, 1)) ⇔ .(hole_a2_0, gen_nil:.:++:if4_0(x))

The following defined symbols remain to be analysed:
greaters, qsort

They will be analysed ascendingly in the following order:
greaters < qsort

(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(12) Obligation:

TRS:
Rules:
qsort(nil) → nil
qsort(.(x, y)) → ++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y))))
lowers(x, nil) → nil
lowers(x, .(y, z)) → if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
greaters(x, nil) → nil
greaters(x, .(y, z)) → if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))

Types:
qsort :: nil:.:++:if → nil:.:++:if
nil :: nil:.:++:if
. :: a → nil:.:++:if → nil:.:++:if
++ :: nil:.:++:if → nil:.:++:if → nil:.:++:if
lowers :: a → nil:.:++:if → nil:.:++:if
greaters :: a → nil:.:++:if → nil:.:++:if
if :: <= → nil:.:++:if → nil:.:++:if → nil:.:++:if
<= :: a → a → <=
hole_nil:.:++:if1_0 :: nil:.:++:if
hole_a2_0 :: a
hole_<=3_0 :: <=
gen_nil:.:++:if4_0 :: Nat → nil:.:++:if

Generator Equations:
gen_nil:.:++:if4_0(0) ⇔ nil
gen_nil:.:++:if4_0(+(x, 1)) ⇔ .(hole_a2_0, gen_nil:.:++:if4_0(x))

The following defined symbols remain to be analysed:
qsort

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

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

(14) Obligation:

TRS:
Rules:
qsort(nil) → nil
qsort(.(x, y)) → ++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y))))
lowers(x, nil) → nil
lowers(x, .(y, z)) → if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
greaters(x, nil) → nil
greaters(x, .(y, z)) → if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))

Types:
qsort :: nil:.:++:if → nil:.:++:if
nil :: nil:.:++:if
. :: a → nil:.:++:if → nil:.:++:if
++ :: nil:.:++:if → nil:.:++:if → nil:.:++:if
lowers :: a → nil:.:++:if → nil:.:++:if
greaters :: a → nil:.:++:if → nil:.:++:if
if :: <= → nil:.:++:if → nil:.:++:if → nil:.:++:if
<= :: a → a → <=
hole_nil:.:++:if1_0 :: nil:.:++:if
hole_a2_0 :: a
hole_<=3_0 :: <=
gen_nil:.:++:if4_0 :: Nat → nil:.:++:if

Generator Equations:
gen_nil:.:++:if4_0(0) ⇔ nil
gen_nil:.:++:if4_0(+(x, 1)) ⇔ .(hole_a2_0, gen_nil:.:++:if4_0(x))

No more defined symbols left to analyse.