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5d5048e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 | import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationIntegerParameters
namespace OAI
/-! # Three comparisons for the stable mixer order
Only strict count inversions are exchanged. Equal counts retain the original
index order, exactly as in the lexicographic mathematical sorting convention.
-/
namespace LowSpaceSubsetSum.MixerOrderExecution
def sort (counts : Fin 3 → ℕ) : Equiv.Perm (Fin 3) :=
if counts 0 < counts 1 then
if counts 0 < counts 2 then
if counts 1 < counts 2 then Equiv.swap 0 2
else (Equiv.swap 1 2).trans (Equiv.swap 0 1)
else Equiv.swap 0 1
else
if counts 1 < counts 2 then
if counts 0 < counts 2 then (Equiv.swap 0 1).trans (Equiv.swap 1 2)
else Equiv.swap 1 2
else Equiv.refl _
theorem sort_value (counts : Fin 3 → ℕ) : sort counts = mixerOrderFromCounts counts := by
unfold sort
split_ifs <;> apply Tuple.eq_sort_iff.mpr <;> constructor
all_goals
intro i j hij
fin_cases i <;> fin_cases j <;>
simp_all [Equiv.swap_apply_def] <;> omega
def run (counts : Vector ℕ 3) : Vector (Fin 3) 3 × ℕ :=
(Vector.ofFn (sort (fun i => counts[i.val])), 40)
theorem run_value (counts : Vector ℕ 3) (i : Fin 3) :
(run counts).1[i.val] = mixerOrderFromCounts (fun j => counts[j.val]) i := by
simp only [run, Vector.getElem_ofFn, sort_value]
theorem run_work (counts : Vector ℕ 3) : (run counts).2 ≤ 40 := le_rfl
end LowSpaceSubsetSum.MixerOrderExecution
end OAI
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