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a2ee19a | 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 | import OAI.Probability.RadialTransition.ClusterEmbedding
namespace OAI
universe uIndex
noncomputable section
open scoped BigOperators ENNReal NNReal
open MeasureTheory Set Filter
namespace RadialTransition
attribute [local instance] Classical.propDecidable
def CoreFree {ι : Type uIndex} (t : ℝ) (x : ι → Position) : Prop :=
∀ i j, i ≠ j → dist (x i) (x j) < 1 →
microLength t ≤ dist (x i) (x j) ∧
dist (x i) (x j) ≤ (1+(1:ℝ)/1000)*microLength t
lemma same_cell_dist_lt {x y : Position} (h : cellOf x = cellOf y) : dist x y < 1 := by
have hc (k : Fin 3) : |x k-y k| ≤ 1/4 := by
have hh : ⌊4*x k⌋ = ⌊4*y k⌋ := congrFun h k
have hxl := Int.floor_le (4*x k)
have hxu := Int.lt_floor_add_one (4*x k)
have hyl := Int.floor_le (4*y k)
have hyu := Int.lt_floor_add_one (4*y k)
rw [← hh] at hyl hyu
exact abs_le.mpr ⟨by linarith, by linarith⟩
have hs : dist x y ^ 2 ≤ 3*(1/4:ℝ)^2 := by
rw [EuclideanSpace.dist_sq_eq]
calc
_ ≤ ∑ k : Fin 3, (1/4:ℝ)^2 := Finset.sum_le_sum (fun k _ => by
simpa only [Real.dist_eq] using (sq_le_sq₀ (abs_nonneg _) (by norm_num)).mpr (hc k))
_ = _ := by simp
nlinarith [dist_nonneg (x := x) (y := y)]
lemma CoreFree.dist_bound {ι : Type uIndex} {t : ℝ} {x : ι → Position} (hx : CoreFree t x)
(i j : ι) (hij : dist (x i) (x j) < 1) :
dist (x i) (x j) ≤ (1+(1:ℝ)/1000)*microLength t := by
by_cases he : i = j
· subst j; simp only [dist_self]; exact mul_nonneg (by norm_num) (microLength_pos t).le
· exact (hx i j he hij).2
def coreFreeSetoid {ι : Type uIndex} {t : ℝ} {x : ι → Position} (hx : CoreFree t x)
(ht : 2*((1+(1:ℝ)/1000)*microLength t) < 1) : Setoid ι where
r i j := dist (x i) (x j) < 1
iseqv := ⟨fun _ => by simp, fun h => by simpa only [dist_comm] using h, by
intro i j k hij hjk
have h1 := hx.dist_bound i j hij
have h2 := hx.dist_bound j k hjk
linarith [dist_triangle (x i) (x j) (x k)]⟩
lemma coreFree_fiber_card {ι : Type uIndex} [Fintype ι] [DecidableEq ι]
{t : ℝ} {x : ι → Position} (hx : CoreFree t x)
(ht : 2*((1+(1:ℝ)/1000)*microLength t) < 1)
(q : Quotient (coreFreeSetoid hx ht)) :
(Finset.univ.filter (fun i => Quotient.mk (coreFreeSetoid hx ht) i = q)).card ≤ 4 := by
classical
let S := Finset.univ.filter (fun i => Quotient.mk (coreFreeSetoid hx ht) i = q)
have hclique : (shellGraph x (microLength t)).IsClique S := by
intro i hi j hj hij
have hiq : Quotient.mk (coreFreeSetoid hx ht) i = q := (Finset.mem_filter.mp hi).2
have hjq : Quotient.mk (coreFreeSetoid hx ht) j = q := (Finset.mem_filter.mp hj).2
exact ⟨hij, hx i j hij (Quotient.exact (hiq.trans hjq.symm))⟩
by_contra hc
change ¬ S.card ≤ 4 at hc
have h5 : 5 ≤ S.card := by omega
exact (SimpleGraph.CliqueFree.mono h5 (shellGraph_cliqueFree x (microLength_pos t))) S
⟨hclique, rfl⟩
lemma coreFree_roots {ι : Type uIndex} [Fintype ι] {t L : ℝ} {x : ι → Position}
(hx : CoreFree t x) (ht : 2*((1+(1:ℝ)/1000)*microLength t) < 1)
(hbox : ∀ i, x i ∈ cube L) :
Fintype.card (Quotient (coreFreeSetoid hx ht)) ≤ packingNumber L ∧
Function.Injective (fun q : Quotient (coreFreeSetoid hx ht) => cellOf (x q.out)) := by
classical
let Q := Quotient (coreFreeSetoid hx ht)
have hsep (q r : Q) (hqr : q ≠ r) : 1 ≤ dist (x q.out) (x r.out) := by
by_contra! hh
apply hqr
have he : Quotient.mk (coreFreeSetoid hx ht) q.out = Quotient.mk (coreFreeSetoid hx ht) r.out :=
Quotient.sound hh
simpa only [Quotient.out_eq] using he
have hinj : Function.Injective (fun q : Q => x q.out) := by
intro q r he
by_contra hne
have hh := hsep q r hne
change x q.out = x r.out at he
rw [he, dist_self] at hh
norm_num at hh
constructor
· let S := Finset.univ.image (fun q : Q => x q.out)
have hp : IsPacking L S := by
constructor
· intro y hy
obtain ⟨q, _, rfl⟩ := Finset.mem_image.mp hy
exact hbox _
· intro y hy z hz hyz
obtain ⟨q, _, rfl⟩ := Finset.mem_image.mp hy
obtain ⟨r, _, rfl⟩ := Finset.mem_image.mp hz
exact hsep q r (fun he => hyz (congrArg (fun q : Q => x q.out) he))
have hh := packing_card_le hp
simpa only [S, Finset.card_image_of_injective _ hinj, Finset.card_univ] using hh
· intro q r he
by_contra hne
exact (not_lt_of_ge (hsep q r hne)) (same_cell_dist_lt he)
end RadialTransition
end
end OAI
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