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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 | import OAI.Probability.ContinuumTransition.FreeGeometry
namespace OAI
section
open scoped BigOperators
open Set
namespace ContinuumTemperature
open Classical
theorem freeConfig_roots {ι : Type} [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) :
∃ s : Finset ι, s.card ≤ packingNumber L ∧ Fintype.card ι ≤ 4*s.card ∧
∀ j, ∃ i ∈ s, ‖x j-x i‖ ≤ (1+shellDelta)*Real.exp (-t) := by
let Q := Quotient (freeSetoid ht hx)
let s : Finset ι := Finset.univ.image (fun c : Q => c.out)
have ho : Function.Injective (fun c : Q => c.out) := by
intro a b hab
have he := congrArg (Quotient.mk (freeSetoid ht hx)) hab
simpa only [Quotient.out_eq] using he
have hs : s.card = Fintype.card Q := by
exact (Finset.card_image_of_injective _ ho).trans (Finset.card_univ)
have hc (j : ι) : ∃ i ∈ s, j ∈ freeCluster x i := by
refine ⟨(Quotient.mk (freeSetoid ht hx) j).out, ?_, ?_⟩
· exact Finset.mem_image.mpr ⟨Quotient.mk (freeSetoid ht hx) j, Finset.mem_univ _, rfl⟩
· apply mem_freeCluster.mpr
exact Quotient.exact (Quotient.out_eq (Quotient.mk (freeSetoid ht hx) j))
refine ⟨s, ?_, ?_, ?_⟩
· rw [hs]
exact free_component_card_le_packing ht hx hL
· have hu : Finset.univ ⊆ s.biUnion (freeCluster x) := by
intro j _
obtain ⟨i,hi,hj⟩ := hc j
exact Finset.mem_biUnion.mpr ⟨i,hi,hj⟩
calc
Fintype.card ι = (Finset.univ : Finset ι).card := Finset.card_univ.symm
_ ≤ (s.biUnion (freeCluster x)).card := Finset.card_le_card hu
_ ≤ ∑ i ∈ s, (freeCluster x i).card := Finset.card_biUnion_le
_ ≤ ∑ _i ∈ s, 4 := Finset.sum_le_sum (fun i _ => freeCluster_card_le_four ht hx i)
_ = 4*s.card := by simp only [Finset.sum_const, smul_eq_mul, Nat.mul_comm]
· intro j
obtain ⟨i,hi,hj⟩ := hc j
exact ⟨i,hi,by simpa only [norm_sub_rev] using freeCluster_close_root hx hj⟩
end ContinuumTemperature
end
end OAI
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