| import Mathlib |
| import OAI.Analysis.HardSphere.Paths.AcceptedIntegrand |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal BigOperators |
| namespace HardSphere.DyadicSkeleton |
| open HardSphere.Lattice |
| attribute [local instance] Classical.propDecidable |
|
|
| def currentAdmissiblePair {K : ℕ} [NeZero K] (x y : DiscreteTorus K) : Prop := |
| ∃ n : ℕ, K≤10*n ∧ 8*n≤K ∧ ∃ d : Fin 2 → ℤ, (∀ j, |d j|≤n) ∧ |
| y=x+![(n:ZMod K),(d 0:ZMod K),(d 1:ZMod K)] |
|
|
| def currentAdmissiblePairs (K : ℕ) [NeZero K] : Finset (DiscreteTorus K × DiscreteTorus K) := |
| Finset.univ.filter (fun z ↦ currentAdmissiblePair z.1 z.2) |
|
|
| lemma current_admissible_lift {K : ℕ} [NeZero K] {x y : DiscreteTorus K} |
| (h : currentAdmissiblePair x y) : |
| ∃ n : ℕ, 0<n ∧ 8*n≤K ∧ K≤10*n ∧ ∃ v : Vertex, ∃ d : Fin 2 → ℤ, |
| (∀j, |d j|≤n) ∧ torusProjection K v=x ∧ |
| torusProjection K (v+![(n:ℤ),d 0,d 1])=y := by |
| obtain ⟨n,hl,hu,d,hd,hy⟩ := h |
| let v : Vertex := fun j ↦ (x j).val |
| have hK : 0<K := NeZero.pos K |
| refine ⟨n,by omega,hu,hl,v,d,hd,?_,?_⟩ |
| · ext j; simp [torusProjection,v] |
| · rw [hy] |
| ext j; fin_cases j <;> simp [torusProjection,v] |
|
|
| lemma current_admissible_card {K : ℕ} [NeZero K] (hK : 400≤K) : |
| K^6≤2000*(currentAdmissiblePairs K).card := by |
| let m := K/10 |
| let I := Finset.Icc (K/10+1) (K/8) |
| let J := Finset.range (2*m+1) |
| let S : Finset (DiscreteTorus K × ℕ × ℕ × ℕ) := |
| Finset.univ ×ˢ (I ×ˢ (J ×ˢ J)) |
| let f : DiscreteTorus K × ℕ × ℕ × ℕ → DiscreteTorus K × DiscreteTorus K := |
| fun z ↦ (z.1,z.1+![(z.2.1:ZMod K),((z.2.2.1:ℤ)-m:ZMod K),((z.2.2.2:ℤ)-m:ZMod K)]) |
| have hI (n : ℕ) (hn : n∈I) : K≤10*n ∧ 8*n≤K ∧ n<K ∧ m<n := by |
| have hn := Finset.mem_Icc.mp hn |
| dsimp [m] |
| omega |
| have hJ (b : ℕ) (hb : b∈J) : b<K ∧ |(b:ℤ)-m|≤(m:ℤ) := by |
| have hb := Finset.mem_range.mp hb |
| dsimp [m] at * |
| constructor |
| · omega |
| · rw [abs_le]; constructor <;> omega |
| have hsub : S.image f⊆currentAdmissiblePairs K := by |
| intro z hz |
| obtain ⟨⟨x,n,b,c⟩,hu,rfl⟩ := Finset.mem_image.mp hz |
| simp only [S,Finset.mem_product] at hu |
| obtain ⟨_,hn,hb,hc⟩ := hu |
| have hin := hI n hn |
| apply Finset.mem_filter.mpr |
| refine ⟨Finset.mem_univ _,n,hin.1,hin.2.1,![(b:ℤ)-m,(c:ℤ)-m],?_,?_⟩ |
| · intro j |
| fin_cases j |
| · exact (hJ b hb).2.trans (by exact_mod_cast hin.2.2.2.le) |
| · exact (hJ c hc).2.trans (by exact_mod_cast hin.2.2.2.le) |
| · simp [f] |
| have hinj : Set.InjOn f (↑S : Set _) := by |
| rintro ⟨x,n,b,c⟩ hx ⟨x',n',b',c'⟩ hx' he |
| have he₁ : x=x' := congrArg Prod.fst he |
| subst x' |
| have he₂ : ![(n:ZMod K),((b:ℤ)-m:ZMod K),((c:ℤ)-m:ZMod K)]= |
| ![(n':ZMod K),((b':ℤ)-m:ZMod K),((c':ℤ)-m:ZMod K)] := |
| add_left_cancel (congrArg Prod.snd he) |
| change (x,n,b,c)∈S at hx |
| change (x,n',b',c')∈S at hx' |
| simp only [S,Finset.mem_product] at hx hx' |
| obtain ⟨_,hn,hb,hc⟩ := hx |
| obtain ⟨_,hn',hb',hc'⟩ := hx' |
| have hnval := congrArg (fun z : Fin 3 → ZMod K ↦ (z 0).val) he₂ |
| simp only [Matrix.cons_val_zero,ZMod.val_natCast_of_lt (hI n hn).2.2.1, |
| ZMod.val_natCast_of_lt (hI n' hn').2.2.1] at hnval |
| have hbeq : (b:ZMod K)=(b':ZMod K) := by |
| have h := congrFun he₂ 1 |
| simpa using h |
| have hceq : (c:ZMod K)=(c':ZMod K) := by |
| have h := congrFun he₂ 2 |
| simpa using h |
| have hbval := congrArg ZMod.val hbeq |
| have hcval := congrArg ZMod.val hceq |
| simp only [ZMod.val_natCast_of_lt (hJ b hb).1,ZMod.val_natCast_of_lt (hJ b' hb').1] at hbval |
| simp only [ZMod.val_natCast_of_lt (hJ c hc).1,ZMod.val_natCast_of_lt (hJ c' hc').1] at hcval |
| simp only [Prod.mk.injEq] |
| exact ⟨trivial,hnval,hbval,hcval⟩ |
| have hcard : S.card≤(currentAdmissiblePairs K).card := |
| (Finset.card_image_iff.mpr hinj).symm.trans_le (Finset.card_le_card hsub) |
| have hcardS : S.card=K^3*I.card*J.card^2 := by |
| dsimp only [S] |
| rw [Finset.card_product,Finset.card_product,Finset.card_product] |
| simp [DiscreteTorus,mul_assoc,pow_two] |
| have hcardI : K≤50*I.card := by |
| dsimp [I] |
| rw [Nat.card_Icc] |
| omega |
| have hcardJ : K≤6*J.card := by |
| simp only [J,Finset.card_range,m] |
| omega |
| rw [hcardS] at hcard |
| calc |
| K^6=K^3*K*K^2 := by ring |
| _ ≤ K^3*(50*I.card)*(6*J.card)^2 := Nat.mul_le_mul |
| (Nat.mul_le_mul_left _ hcardI) (Nat.pow_le_pow_left hcardJ 2) |
| _ = 1800*(K^3*I.card*J.card^2) := by ring |
| _ ≤ 1800*(currentAdmissiblePairs K).card := Nat.mul_le_mul_left _ hcard |
| _ ≤ _ := Nat.mul_le_mul_right _ (by decide) |
|
|
| end HardSphere.DyadicSkeleton |
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal BigOperators |
| namespace HardSphere.DyadicSkeleton |
| open HardSphere.Lattice |
|
|
| theorem current_good_paths (R : ℕ) : |
| ∃ p₀ κ : ℝ, 0<p₀ ∧ 0<κ ∧ ∃ K₀ : ℕ, ∀ K : ℕ, K₀≤K → ∀ _ : NeZero K, |
| K^6≤2000*(currentAdmissiblePairs K).card ∧ |
| ∀ {Ω : Type*} [MeasurableSpace Ω], ∀ P : Measure Ω, ∀ _ : IsProbabilityMeasure P, |
| ∀ b : Ω → Finset (DiscreteTorus K), Measurable b → |
| ∀ p : ℝ, 0≤p → p≤p₀ → |
| (∀ A : Finset (DiscreteTorus K), P {d | A⊆b d}≤(ENNReal.ofReal p)^A.card) → |
| ∀ x y : DiscreteTorus K, currentAdmissiblePair x y → |
| ∃ E : Set Ω, MeasurableSet E ∧ (1/2:ℝ≥0∞)≤P E ∧ |
| ∃ ν : Kernel Ω (SimplePath x y), IsMarkovKernel ν ∧ |
| (∀ e∈E, ν e {q | simplePathGood (b e) q}=1) ∧ |
| (∫⁻ e, E.indicator (fun e ↦ ∫⁻ q, ∫⁻ q', |
| ENNReal.ofReal (Real.exp (κ*(simpleEncounter R q q':ℝ))) ∂ν e ∂ν e) e ∂P)≤2 := by |
| obtain ⟨p₀,κ,hp₀,hκ,K₀,hall⟩ := dyadic_good_paths_lift R |
| refine ⟨p₀,κ,hp₀,hκ,max K₀ 400,?_⟩ |
| intro K hK inst |
| refine ⟨current_admissible_card ((le_max_right K₀ 400).trans hK),?_⟩ |
| intro Ω mΩ P hP b hb p hp hpp hbad x y hxy |
| obtain ⟨n,hn,hnK,_,v,d,hd,hv,hw⟩ := current_admissible_lift hxy |
| subst x y |
| exact hall K ((le_max_left K₀ 400).trans hK) inst P hP b hb p hp hpp hbad n hn hnK d hd v |
|
|
| end HardSphere.DyadicSkeleton |
| end |
| end |
|
|
| end OAI |
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|
