| import Mathlib |
| import OAI.Analysis.HardSphere.Comparison.PartialLabelMoment |
| import OAI.Analysis.HardSphere.Geometry.PathSlots |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
| noncomputable section |
| open scoped Classical |
| namespace HardSphere.Comparison |
| open _root_.OAI.HardSphere.Lattice MeasureTheory ProbabilityTheory |
| variable {K : ℕ} {x y : DiscreteTorus K} {J : Type*} [DecidableEq J] |
|
|
| def departureSites (p : SimplePath x y) : Finset (DiscreteTorus K) := |
| Finset.univ.image (fun i : Fin p.val.length ↦ pathVertexEmbedding p i.castSucc) |
|
|
| def matchingSites (p q : SimplePath x y) (f g : DiscreteTorus K→J) : Finset (DiscreteTorus K) := |
| (departureSites p∩departureSites q).filter (fun v ↦ f v=g v) |
|
|
| lemma routeSlots_inter_card {d e : ℕ} (f : Fin d↪J) (g : Fin e↪J) : |
| (routeSlots f∩routeSlots g).card=1+((Finset.univ.image f)∩(Finset.univ.image g)).card := by |
| have he : routeSlots f∩routeSlots g=insert none |
| ((((Finset.univ.image f)∩(Finset.univ.image g))).image some) := by |
| ext j |
| cases j <;> simp [routeSlots_exact] |
| rw [he,Finset.card_insert_of_notMem (by simp)] |
| rw [Finset.card_image_of_injective _ (Option.some_injective _),Nat.add_comm] |
|
|
| lemma matched_labels_bound (p q : SimplePath x y) (anchor : J→DiscreteTorus K) |
| (f g : DiscreteTorus K→J) |
| (hf : ∀v∈departureSites p,anchor (f v)=v) |
| (hg : ∀v∈departureSites q,anchor (g v)=v) : |
| ((Finset.univ.image (fun i : Fin p.val.length ↦ f (pathVertexEmbedding p i.castSucc)))∩ |
| (Finset.univ.image (fun i : Fin q.val.length ↦ g (pathVertexEmbedding q i.castSucc)))).card≤ |
| (matchingSites p q f g).card := by |
| let s := (Finset.univ.image (fun i : Fin p.val.length ↦ f (pathVertexEmbedding p i.castSucc)))∩ |
| (Finset.univ.image (fun i : Fin q.val.length ↦ g (pathVertexEmbedding q i.castSucc))) |
| have hip (i : Fin p.val.length) : pathVertexEmbedding p i.castSucc∈departureSites p := |
| Finset.mem_image_of_mem _ (Finset.mem_univ i) |
| have hiq (i : Fin q.val.length) : pathVertexEmbedding q i.castSucc∈departureSites q := |
| Finset.mem_image_of_mem _ (Finset.mem_univ i) |
| have hin : Set.InjOn anchor (s : Set J) := by |
| intro a ha b hb hab |
| obtain ⟨i,_,rfl⟩ := Finset.mem_image.mp (Finset.mem_inter.mp ha).1 |
| obtain ⟨j,_,rfl⟩ := Finset.mem_image.mp (Finset.mem_inter.mp hb).1 |
| rw [hf _ (hip i),hf _ (hip j)] at hab |
| exact congrArg f hab |
| rw [←Finset.card_image_of_injOn hin] |
| apply Finset.card_le_card |
| intro v hv |
| obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hv |
| obtain ⟨i,_,hi⟩ := Finset.mem_image.mp (Finset.mem_inter.mp ha).1 |
| obtain ⟨j,_,hj⟩ := Finset.mem_image.mp (Finset.mem_inter.mp ha).2 |
| have hia : anchor a=pathVertexEmbedding p i.castSucc := by rw [←hi,hf _ (hip i)] |
| have hja : anchor a=pathVertexEmbedding q j.castSucc := by rw [←hj,hg _ (hiq j)] |
| refine Finset.mem_filter.mpr ⟨Finset.mem_inter.mpr ⟨hia.symm ▸ hip i,hja.symm ▸ hiq j⟩,?_⟩ |
| calc |
| f (anchor a) = a := by rw [hia,hi] |
| _ = g (anchor a) := by rw [hja,hj] |
|
|
| lemma route_shared_count_bound (p q : SimplePath x y) (anchor : J→DiscreteTorus K) |
| (f g : DiscreteTorus K→J) |
| (hf : ∀v∈departureSites p,anchor (f v)=v) |
| (hg : ∀v∈departureSites q,anchor (g v)=v) : |
| (routeSlots (departureEmbedding p anchor (fun i ↦ f (pathVertexEmbedding p i.castSucc)) |
| (fun i ↦ hf _ (Finset.mem_image_of_mem _ (Finset.mem_univ i))))∩ |
| routeSlots (departureEmbedding q anchor (fun i ↦ g (pathVertexEmbedding q i.castSucc)) |
| (fun i ↦ hg _ (Finset.mem_image_of_mem _ (Finset.mem_univ i))))).card≤ |
| 1+(matchingSites p q f g).card := by |
| rw [routeSlots_inter_card] |
| exact Nat.add_le_add_left (matched_labels_bound p q anchor f g hf hg) 1 |
|
|
| end HardSphere.Comparison |
|
|
| end |
| end |
|
|
| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped BigOperators ENNReal Classical |
| namespace HardSphere.Comparison |
| open _root_.OAI.HardSphere.Lattice |
| variable {K : ℕ} {x y : DiscreteTorus K} {J : Type*} [DecidableEq J] |
|
|
| lemma departureSites_subset (p : SimplePath x y) : departureSites p⊆p.val.support.toFinset := by |
| intro v hv |
| obtain ⟨i,_,rfl⟩ := Finset.mem_image.mp hv |
| exact List.mem_toFinset.mpr (p.val.getVert_mem_support i.castSucc.val) |
|
|
| lemma shared_departures_le_encounters (R : ℕ) (p q : SimplePath x y) : |
| (departureSites p∩departureSites q).card ≤ simpleEncounter R p q := by |
| let s := departureSites p∩departureSites q |
| have hi : Function.Injective (fun v : DiscreteTorus K ↦ (v,v)) := by |
| intro a b h;exact congrArg Prod.fst h |
| have hs : s.image (fun v ↦ (v,v))⊆encounterSet (torusNear K R) |
| p.val.support.toFinset q.val.support.toFinset := by |
| intro v hv |
| obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hv |
| refine (mem_encounterSet _ _ _ _ _).mpr ⟨departureSites_subset p (Finset.mem_inter.mp ha).1, |
| departureSites_subset q (Finset.mem_inter.mp ha).2,?_⟩ |
| exact torusNear_refl K R a |
| have hc := Finset.card_le_card hs |
| rw [Finset.card_image_of_injective _ hi] at hc |
| exact hc |
|
|
| lemma route_label_integral [NeZero K] (p q : SimplePath x y) (anchor : J→DiscreteTorus K) |
| (A : DiscreteTorus K→Type*) [∀v,Fintype (A v)] [∀v,Nonempty (A v)] |
| [∀v,DecidableEq (A v)] [∀v,MeasurableSpace (A v)] [∀v,MeasurableSingletonClass (A v)] |
| (decode : ∀v,A v→J) |
| (hd : ∀v∈departureSites p∩departureSites q,Function.Injective (decode v)) |
| (ha : ∀v∈departureSites p∪departureSites q,∀a,anchor (decode v a)=v) |
| (G M : ℝ) (hG : 1≤G) (hM : 0<M) |
| (hm : ∀v∈departureSites p∩departureSites q,M≤2*(Fintype.card (A v):ℝ)) : |
| (∫z:∀v,A v×A v, |
| G^(1+(((Finset.univ.image (fun i : Fin p.val.length ↦ |
| decode (pathVertexEmbedding p i.castSucc) (z (pathVertexEmbedding p i.castSucc)).1))∩ |
| (Finset.univ.image (fun i : Fin q.val.length ↦ |
| decode (pathVertexEmbedding q i.castSucc) (z (pathVertexEmbedding q i.castSucc)).2)))).card) |
| ∂Measure.pi (fun v ↦ (PMF.uniformOfFintype (A v)).toMeasure.prod |
| (PMF.uniformOfFintype (A v)).toMeasure))≤ |
| G*Real.exp ((2*(G-1)/M)*((departureSites p∩departureSites q).card:ℝ)) := by |
| let μ := Measure.pi (fun v ↦ (PMF.uniformOfFintype (A v)).toMeasure.prod |
| (PMF.uniformOfFintype (A v)).toMeasure) |
| have hdec (z : ∀v,A v×A v) : matchingSites p q (fun v ↦ decode v (z v).1) |
| (fun v ↦ decode v (z v).2)= |
| (departureSites p∩departureSites q).filter (fun v ↦ (z v).1=(z v).2) := by |
| ext v |
| simp only [matchingSites,Finset.mem_filter] |
| constructor |
| · exact fun h ↦ ⟨h.1,hd v h.1 h.2⟩ |
| · exact fun h ↦ ⟨h.1,congrArg (decode v) h.2⟩ |
| have hbound (z : ∀v,A v×A v) := matched_labels_bound p q anchor |
| (fun v ↦ decode v (z v).1) (fun v ↦ decode v (z v).2) |
| (fun v hv ↦ ha v (Finset.mem_union_left _ hv) _) (fun v hv ↦ ha v (Finset.mem_union_right _ hv) _) |
| calc |
| _ ≤ ∫z:∀v,A v×A v, |
| G*G^((departureSites p∩departureSites q).filter (fun v ↦ (z v).1=(z v).2)).card ∂μ := by |
| apply integral_mono (Integrable.of_finite) (Integrable.of_finite) |
| intro z |
| dsimp only |
| rw [←pow_succ'] |
| apply pow_le_pow_right₀ hG |
| rw [Nat.add_comm] |
| exact Nat.succ_le_succ (by simpa only [hdec z] using hbound z) |
| _ = G*(∫z:∀v,A v×A v, |
| G^((departureSites p∩departureSites q).filter (fun v ↦ (z v).1=(z v).2)).card ∂μ) := integral_const_mul _ _ |
| _ ≤ _ := mul_le_mul_of_nonneg_left (partial_label_match_moment A _ G M hG hM hm) (by linarith) |
| |
| end HardSphere.Comparison |
| |
| end |
| end |
| |
| end |
| |
| end OAI |
| |
