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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 90 91 92 93 | import OAI.Probability.ContinuumTransition.Stability
namespace OAI
section
open scoped Topology BigOperators ENNReal
open Filter MeasureTheory Set
namespace ContinuumTemperature
noncomputable section
open scoped ContDiff
open scoped InnerProductSpace
noncomputable def cubeCells (L : ℝ) : Finset Lattice :=
Fintype.piFinset (fun _ : Fin 3 => Finset.Icc ⌊-2 * L⌋ ⌊2 * L⌋)
theorem cellIndex_mem_cubeCells {L : ℝ} {x : Space} (hx : x ∈ cube L) :
cellIndex x ∈ cubeCells L := by
classical
simp only [cubeCells, Fintype.mem_piFinset, Finset.mem_Icc, cellIndex]
intro k
constructor
· apply Int.floor_mono
have h := (hx k).1
linarith
· apply Int.floor_mono
have h := (hx k).2
linarith
theorem card_cubeCells_le {L : ℝ} (hL : 0 ≤ L) :
((cubeCells L).card : ℝ) ≤ (4 * L + 2) ^ 3 := by
classical
let m := (Finset.Icc ⌊-2 * L⌋ ⌊2 * L⌋).card
have horder : ⌊-2 * L⌋ ≤ ⌊2 * L⌋ + 1 := by
have hh := Int.floor_mono (show -2 * L ≤ 2 * L by linarith)
omega
have hcInt := Int.card_Icc_of_le ⌊-2 * L⌋ ⌊2 * L⌋ horder
have hcReal : (m : ℝ) = (⌊2 * L⌋ : ℝ) + 1 - (⌊-2 * L⌋ : ℝ) := by
dsimp only [m]
exact_mod_cast hcInt
have hm : (m : ℝ) ≤ 4 * L + 2 := by
rw [hcReal]
have hu := Int.floor_le (2 * L)
have hl := Int.lt_floor_add_one (-2 * L)
linarith
have hc : ((cubeCells L).card : ℝ) = (m : ℝ) ^ 3 := by
simp [cubeCells, Fintype.card_piFinset, m]
rw [hc]
exact pow_le_pow_left₀ (Nat.cast_nonneg m) hm 3
theorem physicalProfile_energy_coercive {t L : ℝ} (ht : 0 ≤ t) (hL : 0 ≤ L)
{σ R : ℕ → ℝ} (hσ : ∀ n, 0 ≤ σ n) (hR : ∀ n, 1 ≤ R n) (hs : Summable σ)
(hsmall : 2197 * t * kernelHeight * (∑' n, σ n) / 2 ≤ t ^ 2 / 8)
(N : ℕ) (x : Fin N → Space) (hx : ∀ i, x i ∈ cube L) :
(((t ^ 2 / 8 - 2197 * t * kernelHeight * (∑' n, σ n) / 2) *
((N : ℝ) ^ 2 / (4 * L + 2) ^ 3) - t ^ 2 / 2 * N : ℝ) : EReal) ≤
energy (physicalProfile t σ R) x := by
classical
let κ := (cubeCells L : Type)
let c (i : Fin N) : κ := ⟨cellIndex (x i), cellIndex_mem_cubeCells (hx i)⟩
let n (k : κ) : ℝ := Fintype.card {i // c i = k}
have hsN : (∑ k : κ, n k) = N := by
have hh := sum_comp_eq_sum_fibres c (fun _ => (1 : ℝ))
simpa only [n, mul_one, Finset.sum_const, Finset.card_univ, Fintype.card_fin,
nsmul_eq_mul, mul_one] using hh.symm
have hquad : (N : ℝ) ^ 2 ≤ (Fintype.card κ : ℝ) * ∑ k, n k ^ 2 := by
have hh := Finset.sum_mul_sq_le_sq_mul_sq (Finset.univ : Finset κ) (fun _ => (1 : ℝ)) n
simpa only [one_mul, one_pow, Finset.sum_const, Finset.card_univ, nsmul_eq_mul,
mul_one, hsN] using hh
have hquad' : (N : ℝ) ^ 2 ≤ (4 * L + 2) ^ 3 * ∑ k, n k ^ 2 := by
apply hquad.trans
apply mul_le_mul_of_nonneg_right _ (Finset.sum_nonneg (fun _ _ => sq_nonneg _))
simpa only [κ, Fintype.card_coe] using card_cubeCells_le hL
have hden : 0 < (4 * L + 2) ^ 3 := by positivity
have hquot : (N : ℝ) ^ 2 / (4 * L + 2) ^ 3 ≤ ∑ k, n k ^ 2 := by
apply (div_le_iff₀ hden).mpr
simpa only [mul_comm] using hquad'
have hsuper := physicalProfile_superstable ht hσ hR hs x c
(fun k : κ => (k : Lattice)) Subtype.val_injective (fun _ => rfl)
have he := coe_pairSum_le_energy (physicalProfile t σ R) x
apply le_trans _ he
apply EReal.coe_le_coe_iff.mpr
have hm := mul_le_mul_of_nonneg_left hquot (sub_nonneg.mpr hsmall)
simp only [Fintype.card_fin] at hsuper
exact (sub_le_sub_right hm _).trans hsuper
end
end ContinuumTemperature
end
end OAI
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