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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 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 | import OAI.Combinatorics.DiscreteConvexity.Basic
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Powerset
import Mathlib.Data.Finset.Union
namespace OAI
/-!
# Finite objects in the elementary covering argument
Sampling is indexed by ordered pairs of edges, including diagonal pairs.
Different pairs remain different indices even if their unions coincide.
Residual generators are indexed by injective ordered tuples.
-/
namespace TalagrandDiscreteConvexity.Covering
variable {V : Type*} [DecidableEq V]
/-- An ordered tuple of distinct members of a finite edge family. -/
def EdgeTuple (H : Family V) (t : ℕ) :=
{f : Fin t → ↥H // Function.Injective f}
noncomputable instance edgeTupleFintype (H : Family V) (t : ℕ) :
Fintype (EdgeTuple H t) := by
classical
exact Subtype.fintype _
/-- All independent ordered-pair sampling indices. -/
abbrev PairIndex (H : Family V) := ↥H × ↥H
/-- A sampling outcome specifies precisely the pairs selected. -/
abbrev Sample (H : Family V) := Finset (PairIndex H)
def tupleUnion {H : Family V} {t : ℕ} (f : EdgeTuple H t) : Finset V :=
Finset.univ.biUnion (fun i => (f.val i).val)
noncomputable def tuplePairs {H : Family V} {t : ℕ}
(f : EdgeTuple H t) : Finset (PairIndex H) := by
classical
exact Finset.univ.image (fun ij : Fin t × Fin t => (f.val ij.1, f.val ij.2))
@[simp] theorem mem_tupleUnion {H : Family V} {t : ℕ}
{f : EdgeTuple H t} {v : V} :
v ∈ tupleUnion f ↔ ∃ i, v ∈ (f.val i).val := by
simp [tupleUnion]
@[simp] theorem mem_tuplePairs {H : Family V} {t : ℕ}
{f : EdgeTuple H t} {p : PairIndex H} :
p ∈ tuplePairs f ↔ ∃ i j, (f.val i, f.val j) = p := by
classical
simp [tuplePairs, Prod.exists]
/-- The weighted number of edges containing a prescribed vertex set. -/
noncomputable def weightedDegree (H : Family V) (rho : ℝ) (r : ℕ)
(I : Finset V) : ℝ := by
classical
exact ((H.filter (fun e => I ⊆ e)).card : ℝ) * rho ^ (r - I.card)
/-- Edges avoiding all prescribed initial generators. -/
noncomputable def regularFamily (H C0 : Family V) : Family V := by
classical
exact H.filter (fun e => ∀ I ∈ C0, ¬ I ⊆ e)
@[simp] theorem mem_regularFamily {H C0 : Family V} {e : Finset V} :
e ∈ regularFamily H C0 ↔ e ∈ H ∧ ∀ I ∈ C0, ¬ I ⊆ e := by
classical
simp [regularFamily]
theorem regularFamily_subset (H C0 : Family V) : regularFamily H C0 ⊆ H := by
intro e he
exact (mem_regularFamily.mp he).1
section FiniteGround
variable [Fintype V]
/-- The high weighted-degree sets used as the initial generators. -/
noncomputable def initialFamily (H : Family V) (rho : ℝ) (r : ℕ)
(T : ℝ) : Family V := by
classical
exact Finset.univ.filter (fun I => I.card ≤ r ∧ T < weightedDegree H rho r I)
/-- The regular edges for the actual high-degree initial family. -/
noncomputable def regularEdges (H : Family V) (rho : ℝ) (r : ℕ)
(T : ℝ) : Family V := regularFamily H (initialFamily H rho r T)
/-- Sets containing at least `m` distinct edges of the original family. -/
noncomputable def denseFamily (H : Family V) (m : ℕ) : Family V := by
classical
exact Finset.univ.filter (fun S => m ≤ (H.filter (fun e => e ⊆ S)).card)
@[simp] theorem mem_initialFamily {H : Family V} {rho T : ℝ}
{r : ℕ} {I : Finset V} :
I ∈ initialFamily H rho r T ↔ I.card ≤ r ∧ T < weightedDegree H rho r I := by
classical
simp [initialFamily]
@[simp] theorem mem_regularEdges {H : Family V} {rho T : ℝ}
{r : ℕ} {e : Finset V} :
e ∈ regularEdges H rho r T ↔
e ∈ H ∧ ∀ I ∈ initialFamily H rho r T, ¬ I ⊆ e := by
exact mem_regularFamily
@[simp] theorem mem_denseFamily {H : Family V} {m : ℕ} {S : Finset V} :
S ∈ denseFamily H m ↔ m ≤ (H.filter (fun e => e ⊆ S)).card := by
classical
simp [denseFamily]
theorem regularEdges_subset (H : Family V) (rho : ℝ) (r : ℕ) (T : ℝ) :
regularEdges H rho r T ⊆ H := regularFamily_subset _ _
end FiniteGround
end TalagrandDiscreteConvexity.Covering
end OAI
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