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67aee1d | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 | import Mathlib
import OAI.Analysis.HardSphere.Collisions.LocalInsertion
namespace OAI
noncomputable section
open MeasureTheory Filter Set
open scoped ENNReal Topology BigOperators
open scoped BigOperators ComplexConjugate
open MeasureTheory ProbabilityTheory Filter Set
open scoped ENNReal NNReal Topology BigOperators
open MeasureTheory ProbabilityTheory Real Complex
open scoped ENNReal ComplexConjugate
open ProbabilityTheory
namespace HardSphere
open MeasureTheory Set Filter PeriodicCutoff TightCutoff
open scoped ComplexConjugate
open scoped Classical
lemma closeLabels_cons {n : ℕ} {L : Length} (r : ℝ) (w : SmoothMultiplier 1 L)
(x : Torus L) (Y : Configuration n L) :
(closeLabels r w (Fin.cons x Y)).card=(closeLabels r w Y).card+
(if ∃ y∈tsupport w.orbital, torusDistance L y x ≤ 2*r then 1 else 0) := by
classical
simp only [closeLabels,Finset.card_filter]
rw [Fin.sum_univ_succ]
simp only [Fin.cons_zero,Fin.cons_succ]
exact Nat.add_comm _ _
lemma closeLabels_insert_le {S : Type*} [DecidableEq S] {n : ℕ} {L : Length} (r : ℝ)
(w : S → SmoothMultiplier 1 L)
(hsep : ∀ z v : S, z≠v → ∀ x : Torus L, (w v).orbital x≠0 →
¬ ∃ y∈tsupport (w z).orbital, torusDistance L y x ≤ 2*r)
(v z : S) (X : Configuration (n+1) L) (hx : (w v).orbital (X 0)≠0) :
(closeLabels r (w z) X).card ≤ (closeLabels r (w z) (Fin.tail X)).card+(if v=z then 1 else 0) := by
classical
have he : Fin.cons (X 0) (Fin.tail X)=X := by funext i; cases i using Fin.cases <;> rfl
rw [←he,closeLabels_cons]
simp only [Fin.tail_cons]
by_cases hvz : v=z
· subst v
simp only [ite_true]
split_ifs <;> omega
· have hzv : z≠v := Ne.symm hvz
rw [ite_eq_right hvz,ite_eq_right (hsep z v hzv (X 0) hx)]
noncomputable def insertionCount {S : Type*} [DecidableEq S] (site : ℕ → S) (t : ℕ) (z : S) : ℕ :=
((Finset.range t).filter (fun i ↦ site i=z)).card
lemma insertionCount_zero {S : Type*} [DecidableEq S] (site : ℕ → S) (z : S) :
insertionCount site 0 z=0 := by simp [insertionCount]
lemma insertionCount_succ {S : Type*} [DecidableEq S] (site : ℕ → S) (t : ℕ) (z : S) :
insertionCount site (t+1) z=insertionCount site t z+(if site t=z then 1 else 0) := by
simp only [insertionCount,Finset.card_filter,Finset.sum_range_succ]
lemma insertionCount_mono {S : Type*} [DecidableEq S] (site : ℕ → S) {t T : ℕ} (ht : t ≤ T) (z : S) :
insertionCount site t z ≤ insertionCount site T z := by
apply Finset.card_le_card
intro i hi
obtain ⟨hi,hiz⟩ := Finset.mem_filter.mp hi
exact Finset.mem_filter.mpr ⟨Finset.mem_range.mpr ((Finset.mem_range.mp hi).trans_le ht),hiz⟩
lemma local_insertion_iterated : ∃ C : ℝ, 0<C ∧
∀ (S : Type) [DecidableEq S] (n : ℕ) (L : Length) (r : ℝ), 0<r →
∀ (w : S → SmoothMultiplier 1 L),
(∀ z X, conj ((w z).toFun X)=(w z).toFun X) →
(∀ z, (∫ y : Torus L, ‖(w z).orbital y‖^2)=1) →
(∀ z v : S, z≠v → ∀ x : Torus L, (w v).orbital x≠0 →
¬ ∃ y∈tsupport (w z).orbital, torusDistance L y x ≤ 2*r) →
∀ (T k m : ℕ) (site : ℕ → S) (B E : ℝ),
(∀ z, (w z).orbitalBound ≤ B) → (∀ z, (w z).orbitalEnergy ≤ E) →
(∀ z, insertionCount site T z ≤ m) → C*B*(k+m)*r^3 ≤ 1/4 →
∀ u : Jet n L, u∈dirichletDomain n L r →
(∀ᵐ Y ∂volume, u.1 Y≠0 → ∀ z, (closeLabels r (w z) Y).card ≤ k) →
∃ v : Jet (n+T) L, v∈dirichletDomain (n+T) L r ∧ ‖v.1‖=‖u.1‖ ∧
energy v ≤ energy u+T*(C*(E+B*(k+m)*r))*‖u.1‖^2 := by
obtain ⟨C,hC,hins⟩ := local_insertion_supported
refine ⟨C,hC,?_⟩
intro S _ n L r hr w hw hn hsep T k m site B E hB hE hct hs u hu hk
have hrec (t : ℕ) (ht : t ≤ T) : ∃ v : Jet (n+t) L,
v∈dirichletDomain (n+t) L r ∧ ‖v.1‖=‖u.1‖ ∧
energy v ≤ energy u+t*(C*(E+B*(k+m)*r))*‖u.1‖^2 ∧
(∀ᵐ X ∂volume, v.1 X≠0 → ∀ z,
(closeLabels r (w z) X).card ≤ k+insertionCount site t z) := by
induction t with
| zero =>
refine ⟨u,hu,rfl,?_,?_⟩
· simp only [Nat.cast_zero,zero_mul,add_zero,le_refl]
· simpa only [insertionCount_zero,Nat.add_zero] using hk
| succ t ih =>
obtain ⟨v,hv,hvn,hve,hvk⟩ := ih (Nat.le_trans (Nat.le_succ t) ht)
have hmass : C*(w (site t)).orbitalBound*(k+m:ℕ)*r^3 ≤ 1/4 := by
apply le_trans _ hs
simp only [Nat.cast_add]
apply mul_le_mul_of_nonneg_right _ (by positivity)
apply mul_le_mul_of_nonneg_right _ (by positivity)
exact mul_le_mul_of_nonneg_left (hB _) hC.le
have hbudget : ∀ᵐ X ∂volume, v.1 X≠0 → (closeLabels r (w (site t)) X).card ≤ k+m := by
filter_upwards [hvk] with X hX
intro hne
exact (hX hne (site t)).trans (Nat.add_le_add_left
((insertionCount_mono site (Nat.le_trans (Nat.le_succ t) ht) _).trans (hct _)) k)
obtain ⟨v',hv',hvn',hve',hvs⟩ := hins (n+t) L r hr (w (site t)) (hw _) (hn _) (k+m) hmass v hv hbudget
refine ⟨v',hv',hvn'.trans hvn,?_,?_⟩
· have hbound : C*((w (site t)).orbitalEnergy+(w (site t)).orbitalBound*(k+m:ℕ)*r) ≤
C*(E+B*(k+m)*r) := by
apply mul_le_mul_of_nonneg_left _ hC.le
apply add_le_add (hE _)
rw [Nat.cast_add]
exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right (hB _) (by positivity)) hr.le
rw [hvn] at hve'
have he := hve'.trans (add_le_add le_rfl (mul_le_mul_of_nonneg_right hbound (sq_nonneg _)))
rw [Nat.cast_add,Nat.cast_one]
nlinarith
· filter_upwards [hvs,(configurationTail_quasiPreserving (n+t) L).ae hvk] with X hX hOld
intro hne z
obtain ⟨hOldne,hOrb⟩ := hX hne
have hh := closeLabels_insert_le r w hsep (site t) z X hOrb
have hh' := hOld hOldne z
rw [insertionCount_succ]
omega
obtain ⟨v,hv,hvn,hve,_⟩ := hrec T le_rfl
exact ⟨v,hv,hvn,hve⟩
end HardSphere
end
end OAI
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