File size: 2,892 Bytes
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 | import OAI.Combinatorics.ExpectationThreshold.SelectorDefinitions
namespace OAI
/-!
# Reconstruction from the actual changed rows
For an earlier coloring, every point outside the changed rows retains its final
color. Inside a changed row its original color is recorded. Thus the changed
subsets and their original labels determine the entire original coloring.
-/
noncomputable section
namespace TalagrandThreshold.Selector
variable {α : Type*} [Fintype α] [DecidableEq α] {s : ℕ}
omit [DecidableEq α] in
/-- An original coloring is reconstructed from its changed rows and the
original labels on those rows. No disjointness of ambient row supersets is used. -/
theorem coloring_eq_of_movedRows
{a b z : Coloring α s} (ha : Earlier a z) (hb : Earlier b z)
(hrows : ∀ i, movedRow a z i = movedRow b z i)
(hlabels : ∀ i x, x ∈ movedRow a z i → a x = b x) : a = b := by
funext x
by_cases hxa : z x < a x
· let i : Fin s := ⟨(z x).val,
Nat.lt_of_lt_of_le hxa (Nat.le_of_lt_succ (a x).isLt)⟩
apply hlabels i x
simp only [movedRow, Finset.mem_filter, Finset.mem_univ, true_and]
exact ⟨rfl, hxa⟩
· have haeq : a x = z x := le_antisymm (le_of_not_gt hxa) (ha x)
have hxb : ¬ z x < b x := by
intro hxb
let i : Fin s := ⟨(z x).val,
Nat.lt_of_lt_of_le hxb (Nat.le_of_lt_succ (b x).isLt)⟩
have hx : x ∈ movedRow b z i := by
simp only [movedRow, Finset.mem_filter, Finset.mem_univ, true_and]
exact ⟨rfl, hxb⟩
rw [← hrows i] at hx
exact hxa (Finset.mem_filter.mp hx).2.2
exact haeq.trans (le_antisymm (le_of_not_gt hxb) (hb x)).symm
/-- The unrestricted row record consists of its actual support and its labels. -/
abbrev RawRowEncoding (α : Type*) (s : ℕ) :=
Σ U : Finset α, (↥U → Color s)
def rawRowEncoding (a z : Coloring α s) (i : Fin s) : RawRowEncoding α s :=
⟨movedRow a z i, fun x => a x.1⟩
/-- Reading one point avoids any dependence on the chosen subtype proof. -/
def rawRowLookup (E : RawRowEncoding α s) (x : α) : Option (Color s) :=
if hx : x ∈ E.1 then some (E.2 ⟨x, hx⟩) else none
/-- The literal row encoding is injective on originals moved only earlier. -/
theorem rawRowEncoding_injective
{a b z : Coloring α s} (ha : Earlier a z) (hb : Earlier b z)
(he : ∀ i, rawRowEncoding a z i = rawRowEncoding b z i) : a = b := by
have hrows : ∀ i, movedRow a z i = movedRow b z i :=
fun i => congrArg Sigma.fst (he i)
apply coloring_eq_of_movedRows ha hb hrows
intro i x hx
have hx' : x ∈ movedRow b z i := by rw [← hrows i]; exact hx
have hlookup := congrArg (fun E => rawRowLookup E x) (he i)
dsimp only [rawRowLookup, rawRowEncoding] at hlookup
simp only [dite_eq_left hx, dite_eq_left hx'] at hlookup
exact Option.some.inj hlookup
end TalagrandThreshold.Selector
end
end OAI
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