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e63eb7f | 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 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 | import OAI.Combinatorics.EditApproximation.Probability.Amplification
import OAI.Combinatorics.EditApproximation.Definitions
namespace OAI
/-!
# The executable upper middle order statistic
The seed pass sorts its integer copy outputs and takes index `length/2`.
Strictly more than half the outputs below an upper bound suffice; ties
and repeated values are retained by the list representation.
-/
namespace EditApproximation
theorem upperMedian_lower (values : List ℕ) (hvalues : values ≠ []) (c : ℕ)
(hlower : ∀ value ∈ values, c ≤ value) : c ≤ upperMedian values := by
have hlength : 0 < values.length := List.length_pos_iff.mpr hvalues
have hindex : values.length / 2 < (values.mergeSort (· ≤ ·)).length := by
rw [List.length_mergeSort]
omega
have hmem : (values.mergeSort (· ≤ ·))[values.length / 2] ∈ values :=
(List.mergeSort_perm values (· ≤ ·)).mem_iff.mp (List.getElem_mem hindex)
have hmedian : upperMedian values = (values.mergeSort (· ≤ ·))[values.length / 2] := by
exact (List.getElem_eq_getD 0).symm
rw [hmedian]
exact hlower _ hmem
theorem upperMedian_upper (values : List ℕ) (hvalues : values ≠ []) (upper : ℕ)
(hmajority : values.length / 2 < values.countP (fun value => value ≤ upper)) :
upperMedian values ≤ upper := by
let sorted := values.mergeSort (· ≤ ·)
have hlength : sorted.length = values.length := List.length_mergeSort values
have hpositive : 0 < values.length := List.length_pos_iff.mpr hvalues
have hindex : values.length / 2 < sorted.length := by omega
have hmedian : upperMedian values = sorted[values.length / 2] :=
(List.getElem_eq_getD 0).symm
by_contra hnot
have habove : upper < sorted[values.length / 2] := by omega
have hdrop : (sorted.drop (values.length / 2)).filter (fun value => value ≤ upper) = [] := by
apply List.filter_eq_nil_iff.mpr
intro value hvalue
obtain ⟨j, hj, heq⟩ := List.mem_iff_getElem.mp hvalue
have hj' : values.length / 2 + j < sorted.length := by
simp only [List.length_drop] at hj
omega
have hge : sorted[values.length / 2] ≤ sorted[values.length / 2 + j] :=
(List.sortedLE_mergeSort (l := values)).getElem_le_getElem_of_le
(hi := hindex) (hj := hj')
(show values.length / 2 ≤ values.length / 2 + j by omega)
have hget : (sorted.drop (values.length / 2))[j] = sorted[values.length / 2 + j] := by
simp only [List.getElem_drop]
rw [hget] at heq
have haboveValue : upper < value := habove.trans_le (hge.trans_eq heq)
simpa only [decide_eq_true_eq] using not_le_of_gt haboveValue
have hdropCount : (sorted.drop (values.length / 2)).countP (fun value => value ≤ upper) = 0 := by
rw [List.countP_eq_length_filter, hdrop]
rfl
have hcount : sorted.countP (fun value => value ≤ upper) ≤ values.length / 2 := by
have hsplit := List.take_append_drop (values.length / 2) sorted
rw [← hsplit, List.countP_append, hdropCount, Nat.add_zero]
exact List.countP_le_length.trans (List.length_take_le _ _)
have hperm := (List.mergeSort_perm values (· ≤ ·)).countP_eq (fun value => value ≤ upper)
change sorted.countP (fun value => value ≤ upper) = values.countP (fun value => value ≤ upper) at hperm
omega
theorem clamped_upperMedian_spec (values : List ℕ) (hvalues : values ≠ [])
(c upper N : ℕ) (hcN : c ≤ N)
(hlower : ∀ value ∈ values, c ≤ value)
(hmajority : values.length / 2 < values.countP (fun value => value ≤ upper)) :
c ≤ min N (upperMedian values) ∧ min N (upperMedian values) ≤ upper :=
⟨le_min hcN (upperMedian_lower values hvalues c hlower),
(min_le_right _ _).trans (upperMedian_upper values hvalues upper hmajority)⟩
end EditApproximation
end OAI
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