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e320e82 | 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 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 | import OAI.Combinatorics.ExpectationThreshold.SelectorCountingEncoding
import OAI.Combinatorics.ExpectationThreshold.SelectorCountingEncodingRows
import OAI.Combinatorics.ExpectationThreshold.SelectorProfile
namespace OAI
/-!
# The fixed-profile encoding bound
Each actual coloring maps to the tuple of its changed row subsets and the
original colors on them. Reconstruction proves this map injective. The row
record permits arbitrary subsets of the fixed ambient sets, so overlap between
those ambient sets can only enlarge the number of records.
-/
noncomputable section
open scoped BigOperators
namespace TalagrandThreshold.Selector
variable {α : Type*} [Fintype α] [DecidableEq α] {s : ℕ}
omit [Fintype α] [DecidableEq α] in
/-- Restricting a labeling to a finite set counts exactly its fibers there. -/
theorem wordProfile_restrict_finset {H : Type*} [DecidableEq H]
(U : Finset α) (f : α → H) (h : H) :
wordProfile (fun x : ↥U => f x.val) h =
(U.filter fun x => f x = h).card := by
unfold wordProfile
apply Finset.card_bij (fun x _ => x.val)
· intro x hx
exact Finset.mem_filter.mpr ⟨x.property, (Finset.mem_filter.mp hx).2⟩
· intro x _ y _ hxy
exact Subtype.ext hxy
· intro x hx
refine ⟨⟨x, (Finset.mem_filter.mp hx).1⟩, ?_, rfl⟩
exact Finset.mem_filter.mpr ⟨Finset.mem_univ _, (Finset.mem_filter.mp hx).2⟩
omit [DecidableEq α] in
/-- The recorded original labels have precisely the actual profile row. -/
theorem movedRow_wordProfile (a z : Coloring α s) (i : Fin s) :
wordProfile (fun x : ↥(movedRow a z i) => a x.val) = profile a z i := by
funext h
rw [wordProfile_restrict_finset, profile_eq_card_filter_movedRow]
/-- The literal row record with its proved size and histogram constraints. -/
def profileRowEncoding (a z : Coloring α s) (n : Profile s)
(R : Fin s → Finset α) (hn : profile a z = n)
(hR : ∀ i, movedRow a z i ⊆ R i) (i : Fin s) :
RowEncoding (R i) (rowCount n i) (n i) :=
⟨⟨movedRow a z i, hR i, by rw [movedRow_card_eq_rowCount, hn]⟩,
⟨fun x => a x.val, (mem_wordFiber _ _ _).mpr
((movedRow_wordProfile a z i).trans (congrFun hn i))⟩⟩
/-- Forgetting the constraints retains the actual changed subset and labels. -/
def rowEncodingForget {R : Finset α} {t : ℕ} {n : Color s → ℕ}
(E : RowEncoding R t n) : RawRowEncoding α s :=
⟨E.1.val, E.2.val⟩
@[simp] theorem rowEncodingForget_profileRowEncoding
(a z : Coloring α s) (n : Profile s) (R : Fin s → Finset α)
(hn : profile a z = n) (hR : ∀ i, movedRow a z i ⊆ R i) (i : Fin s) :
rowEncodingForget (profileRowEncoding a z n R hn hR i) =
rawRowEncoding a z i := rfl
/-- Count actual originals of a fixed final coloring and profile. Every
injectivity and histogram condition is proved from the actual row encoding. -/
theorem card_le_profile_encoding
(G : Finset (Coloring α s)) (z : Coloring α s) (n : Profile s)
(R : Fin s → Finset α)
(hEarlier : ∀ a ∈ G, Earlier a z)
(hprofile : ∀ a ∈ G, profile a z = n)
(hrows : ∀ a ∈ G, ∀ i, movedRow a z i ⊆ R i) :
G.card ≤ ∏ i : Fin s,
Nat.choose (R i).card (rowCount n i) * wordCount (rowCount n i) (n i) := by
classical
let encode : ↥G → ∀ i : Fin s, RowEncoding (R i) (rowCount n i) (n i) :=
fun a => profileRowEncoding a.val z n R
(hprofile a.val a.property) (hrows a.val a.property)
have hinj : Function.Injective encode := by
intro a b hab
apply Subtype.ext
apply rawRowEncoding_injective
(hEarlier a.val a.property) (hEarlier b.val b.property)
intro i
have hi := congrArg (fun f => rowEncodingForget (f i)) hab
simpa only [encode, rowEncodingForget_profileRowEncoding] using hi
have hcard := Fintype.card_le_of_injective encode hinj
simpa only [Fintype.card_coe, Fintype.card_pi, card_rowEncoding] using hcard
/-- Specialization to the actual selected bad originals in one profile group. -/
theorem groupOriginals_card_le_profile_encoding
{F : Family α} (mu : MassFamily F) (z : Coloring α s) (n : Profile s)
(R : Fin s → Finset α)
(hrows : ∀ a ∈ groupOriginals mu z n, ∀ i, movedRow a z i ⊆ R i) :
(groupOriginals mu z n).card ≤ ∏ i : Fin s,
Nat.choose (R i).card (rowCount n i) * wordCount (rowCount n i) (n i) := by
apply card_le_profile_encoding (groupOriginals mu z n) z n R
· intro a ha
obtain ⟨_, ⟨H₀, hmin⟩, _⟩ := (mem_groupOriginals mu a z n).mp ha
exact hmin.1.1.1
· intro a ha
exact ((mem_groupOriginals mu a z n).mp ha).2.2
· exact hrows
end TalagrandThreshold.Selector
end
end OAI
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