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9b6e014 | 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 | import OAI.Probability.ContinuumTransition.PackingGeometry
namespace OAI
universe u_1
section
open scoped BigOperators
open Set
namespace ContinuumTemperature
open Classical
def FreeConfig {ι : Type u_1} (t : ℝ) (x : ι → Space) : Prop :=
∀ i j, i ≠ j → ‖x i-x j‖ < 1 →
Real.exp (-t) ≤ ‖x i-x j‖ ∧ ‖x i-x j‖ ≤ (1+shellDelta)*Real.exp (-t)
theorem freeConfig_injective {ι : Type u_1} {t : ℝ} {x : ι → Space}
(hx : FreeConfig t x) : Function.Injective x := by
intro i j hij
by_contra hne
have hh := (hx i j hne (by rw [hij, sub_self, norm_zero]; norm_num)).1
rw [hij,sub_self,norm_zero] at hh
exact (not_le_of_gt (Real.exp_pos _)) hh
theorem freeConfig_close_trans {ι : Type u_1} {t : ℝ} {x : ι → Space}
(ht : 2*(1+shellDelta)*Real.exp (-t) < 1) (hx : FreeConfig t x)
{i j k : ι} (hij : ‖x i-x j‖ < 1) (hjk : ‖x j-x k‖ < 1) :
‖x i-x k‖ < 1 := by
by_cases he : i = j
· simpa only [he] using hjk
by_cases hf : j = k
· simpa only [hf] using hij
have hi := (hx i j he hij).2
have hj := (hx j k hf hjk).2
have hh := norm_sub_le_norm_sub_add_norm_sub (x i) (x j) (x k)
linarith only [hi,hj,hh,ht]
noncomputable def freeSetoid {ι : Type u_1} {t : ℝ} {x : ι → Space}
(ht : 2*(1+shellDelta)*Real.exp (-t) < 1) (hx : FreeConfig t x) : Setoid ι where
r i j := ‖x i-x j‖ < 1
iseqv := ⟨by intro i; simp,
by intro i j h; rwa [norm_sub_rev],
by intro i j k; exact freeConfig_close_trans ht hx⟩
noncomputable def freeCluster {ι : Type u_1} [Fintype ι] (x : ι → Space) (i : ι) : Finset ι :=
Finset.univ.filter (fun j => ‖x i-x j‖ < 1)
theorem mem_freeCluster {ι : Type u_1} [Fintype ι] {x : ι → Space} {i j : ι} :
j ∈ freeCluster x i ↔ ‖x i-x j‖ < 1 := by simp [freeCluster]
theorem freeCluster_card_le_four {ι : Type u_1} [Fintype ι] {t : ℝ} {x : ι → Space}
(ht : 2*(1+shellDelta)*Real.exp (-t) < 1) (hx : FreeConfig t x) (i : ι) :
(freeCluster x i).card ≤ 4 := by
have hc : (shellGraph x (Real.exp (-t))).IsClique (freeCluster x i : Set ι) := by
intro j hj k hk hjk
have hi : ‖x j-x i‖ < 1 := by simpa only [norm_sub_rev] using mem_freeCluster.mp hj
have hh := freeConfig_close_trans ht hx hi (mem_freeCluster.mp hk)
exact ⟨hjk,hx j k hjk hh⟩
have hcf := shellGraph_cliqueFree x (Real.exp_pos (-t))
by_contra hn
obtain ⟨s,hs,hcard⟩ := Finset.exists_subset_card_eq (by omega : 5 ≤ (freeCluster x i).card)
exact hcf s ⟨hc.subset (by exact_mod_cast hs), hcard⟩
theorem freeCluster_close_root {ι : Type u_1} [Fintype ι] {t : ℝ} {x : ι → Space}
(hx : FreeConfig t x) {i j : ι} (hj : j ∈ freeCluster x i) :
‖x i-x j‖ ≤ (1+shellDelta)*Real.exp (-t) := by
by_cases he : i=j
· rw [he,sub_self,norm_zero]
unfold shellDelta
positivity
exact (hx i j he (mem_freeCluster.mp hj)).2
theorem free_component_card_le_packing {ι : Type u_1} [Fintype ι] {t L : ℝ} {x : ι → Space}
(ht : 2*(1+shellDelta)*Real.exp (-t) < 1) (hx : FreeConfig t x)
(hL : ∀ i, x i ∈ cube L) :
Fintype.card (Quotient (freeSetoid ht hx)) ≤ packingNumber L := by
let f : Quotient (freeSetoid ht hx) → Space := fun c => x c.out
apply card_le_packingNumber f (fun c => hL c.out)
intro a b hab
by_contra hlt
have hh : (freeSetoid ht hx).r a.out b.out := lt_of_not_ge hlt
have he := Quotient.sound hh
exact hab (by simpa only [Quotient.out_eq] using he)
end ContinuumTemperature
end
end OAI
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