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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 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 82 | import OAI.Combinatorics.LowSpaceSubsetSum.Counting.Model
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Powerset
namespace OAI
/-!
# Equal-weight records and exact dictionary capacities
An additional prime can distinguish unequal weights, but cannot distinguish
different masks of equal weight. These deterministic bounds handle that case,
including the full-mask endpoint of the right-prefix refinement.
-/
namespace LowSpaceSubsetSum
variable {ι : Type*} [DecidableEq ι]
instance : DecidableEq (Entry ι) := fun a b =>
if h : a.value = b.value ∧ a.mask = b.mask then
isTrue (by cases a; cases b; simp_all)
else isFalse (by intro hab; exact h (hab ▸ ⟨rfl, rfl⟩))
/-- Equal-weight records are uniquely identified by their top masks. -/
theorem same_weight_card_le_masks (U : Finset (Entry ι)) (x : ℤ) :
(U.filter fun e => e.value = x).card ≤ (U.image Entry.mask).card := by
classical
have hinj : Set.InjOn Entry.mask (↑(U.filter fun e => e.value = x) : Set (Entry ι)) := by
intro e he f hf hmask
have heq := (Finset.mem_filter.mp he).2
have hfq := (Finset.mem_filter.mp hf).2
cases e
cases f
simp_all
calc
_ = ((U.filter fun e => e.value = x).image Entry.mask).card :=
(Finset.card_image_iff.mpr hinj).symm
_ ≤ (U.image Entry.mask).card :=
Finset.card_le_card (Finset.image_subset_image (Finset.filter_subset _ _))
/-- Prescribed mask size bounds all equal-weight records by a binomial count. -/
theorem same_weight_card_le_choose (U : Finset (Entry ι)) (top : Finset ι)
(k : ℕ) (hmask : ∀ e ∈ U, e.mask ⊆ top ∧ e.mask.card = k) (x : ℤ) :
(U.filter fun e => e.value = x).card ≤ top.card.choose k := by
apply (same_weight_card_le_masks U x).trans
calc
_ ≤ (top.powersetCard k).card := by
apply Finset.card_le_card
intro D hD
obtain ⟨e, he, rfl⟩ := Finset.mem_image.mp hD
exact Finset.mem_powersetCard.mpr (hmask e he)
_ = _ := Finset.card_powersetCard _ _
/-- Fixing both the exact weight and the entire mask leaves at most one key. -/
theorem full_mask_same_weight_card_le_one (U : Finset (Entry ι))
(x : ℤ) (D : Finset ι) :
(U.filter fun e => e.value = x ∧ e.mask = D).card ≤ 1 := by
apply Finset.card_le_one.mpr
intro e he f hf
obtain ⟨he0, he1⟩ := (Finset.mem_filter.mp he).2
obtain ⟨hf0, hf1⟩ := (Finset.mem_filter.mp hf).2
cases e
cases f
simp_all
omit [DecidableEq ι] in
/-- The complete bucket fits when both its equal- and unequal-weight parts fit.
This counts keys, rather than raw tuples producing those keys. -/
theorem bucket_card_le_of_parts (U : Finset (Entry ι)) (x : ℤ) (C A B : ℕ)
(hneq : (U.filter fun e => e.value ≠ x).card ≤ A)
(heq : (U.filter fun e => e.value = x).card ≤ B) (hcap : A + B ≤ C) :
U.card ≤ C := by
have h := Finset.card_filter_add_card_filter_not (s := U) (p := fun e => e.value = x)
have hcard : (U.filter fun e => e.value = x).card +
(U.filter fun e => e.value ≠ x).card = U.card := by
simpa only [ne_eq] using h
omega
end LowSpaceSubsetSum
end OAI
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