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) CpxTrsToCpxRelTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed TRS to relative TRS where S is empty.

(2) 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

(3) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
./0
lowers/0
greaters/0
if/0
<=/0
<=/1

(4) Obligation:

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

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

S is empty.
Rewrite Strategy: FULL

(5) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2ⁿ):
The rewrite sequence
lowers(.(z)) →⁺ if(.(lowers(z)), lowers(z))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [z / .(z)].
The result substitution is [ ].

The rewrite sequence
lowers(.(z)) →⁺ if(.(lowers(z)), lowers(z))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [z / .(z)].
The result substitution is [ ].

(6) BOUNDS(2^n, INF)