| import Mathlib |
| import OAI.Analysis.HardSphere.Geometry.CenteredCircleLift |
| import OAI.Analysis.HardSphere.Geometry.StarGraph |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped ENNReal Topology BigOperators |
| open scoped BigOperators Classical |
| open scoped BigOperators Classical |
| namespace HardSphere.BathSlice |
| open HardSphere.Lattice |
|
|
| lemma torusDistance_lift_le_two {L : Length} {x y : Point} {b : ℝ} (hb : 0 ≤ b) |
| (hxy : ∀ i, |x i - y i| ≤ b) : |
| torusDistance L (liftPoint L x) (liftPoint L y) ≤ 2*b := by |
| have hcoord (i : Fin 3) : dist ((x i : ℝ) : AddCircle L.val) (y i : AddCircle L.val) ≤ b := by |
| rw [dist_eq_norm, ← AddCircle.coe_sub] |
| exact (QuotientAddGroup.norm_mk_le_norm (m := x i-y i)).trans (by simpa only [Real.norm_eq_abs] using hxy i) |
| unfold torusDistance |
| apply (Real.sqrt_le_iff).mpr |
| refine ⟨by positivity, ?_⟩ |
| calc |
| _ ≤ ∑ _i : Fin 3, b^2 := Finset.sum_le_sum fun i _ ↦ |
| sq_le_sq₀ dist_nonneg hb |>.mpr (hcoord i) |
| _ ≤ (2*b)^2 := by simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]; norm_num only [Nat.cast_ofNat]; nlinarith [sq_nonneg b] |
|
|
| def gridCorner (L : Length) (h : ℝ) (z : Vertex) : Torus L := |
| liftPoint L (fun i ↦ h*(z i : ℝ)-1/2) |
|
|
| lemma microCube_dist_corner {L : Length} {h d : ℝ} (hh : 0 ≤ h) (hd : 0 ≤ d) |
| {z w : Vertex} (hw : ∀ i, h * |(w i : ℝ)-z i| ≤ d) |
| {x : Torus L} (hx : x ∈ microCube L h w) : |
| torusDistance L x (gridCorner L h z) ≤ 2*(d+h) := by |
| obtain ⟨u,hu,rfl⟩ := hx |
| apply torusDistance_lift_le_two (add_nonneg hd hh) |
| intro i |
| have hab : |h*((w i : ℝ)-z i)| ≤ d := by |
| simpa only [abs_mul, abs_of_nonneg hh] using hw i |
| have hl := hu.1 i |
| have hr := hu.2 i |
| change h*(w i : ℝ)-1/2 ≤ u i at hl |
| change u i ≤ h*((w i : ℝ)+1)-1/2 at hr |
| obtain ⟨ha,hb⟩ := abs_le.mp hab |
| apply abs_le.mpr |
| constructor <;> nlinarith |
|
|
| lemma microCube_dist_adjacent {L : Length} {h : ℝ} (hh : 0 ≤ h) |
| {z w : Vertex} (hzw : starStep z w) {x y : Torus L} |
| (hx : x ∈ microCube L h z) (hy : y ∈ microCube L h w) : |
| torusDistance L x y ≤ 4*h := by |
| obtain ⟨u,hu,rfl⟩ := hx |
| obtain ⟨v,hv,rfl⟩ := hy |
| have hc := torusDistance_lift_le_two (L := L) (x := u) (y := v) (b := 2*h) (by positivity) |
| rw [show 4*h = 2*(2*h) by ring] |
| apply hc |
| intro i |
| have he : |(z i : ℝ)-w i| ≤ 1 := by exact_mod_cast hzw.2 i |
| have hmul := mul_le_mul_of_nonneg_left he hh |
| have hab : |h*((z i : ℝ)-w i)| ≤ h := by |
| simpa only [abs_mul, abs_of_nonneg hh, mul_one] using hmul |
| obtain ⟨ha,hb⟩ := abs_le.mp hab |
| have huz := hu.1 i |
| have huz' := hu.2 i |
| have hvw := hv.1 i |
| have hvw' := hv.2 i |
| dsimp [liftedCube] at huz huz' hvw hvw' |
| apply abs_le.mpr |
| constructor <;> nlinarith |
|
|
| lemma adjacent_witnesses_close {n : ℕ} {L : Length} {h r : ℝ} (hh : 0 ≤ h) (hr : h ≤ r) |
| (Y : Configuration n L) {z w : Vertex} (hzw : starStep z w) {i j : Fin n} |
| (hi : ∃ x ∈ microCube L h z, torusDistance L x (Y i) ≤ r) |
| (hj : ∃ y ∈ microCube L h w, torusDistance L y (Y j) ≤ r) : |
| torusDistance L (Y i) (Y j) ≤ 6*r := by |
| obtain ⟨x,hx,hxi⟩ := hi |
| obtain ⟨y,hy,hyj⟩ := hj |
| have hc := microCube_dist_adjacent hh hzw hx hy |
| have ht := torusDistance_triangle' L (Y i) x (Y j) |
| have ht' := torusDistance_triangle' L x y (Y j) |
| rw [torusDistance_comm L (Y i) x] at ht |
| linarith |
|
|
| end HardSphere.BathSlice |
|
|
| namespace SimpleGraph.Walk |
| open scoped _root_.SimpleGraph _root_.SimpleGraph.Walk |
|
|
| lemma label_boundary {α β : Type*} {G : SimpleGraph α} {u v : α} (p : G.Walk u v) |
| (f : α → β) {a : β} (hu : f u = a) (hv : f v ≠ a) : |
| ∃ x ∈ p.support, ∃ y ∈ p.support, G.Adj x y ∧ f x = a ∧ f y ≠ a := by |
| induction p with |
| | nil => exact (hv hu).elim |
| | @cons u v w huv p ih => |
| by_cases he : f v = a |
| · obtain ⟨x,hx,y,hy,hxy,hfx,hfy⟩ := ih he hv |
| exact ⟨x,by simp [hx],y,by simp [hy],hxy,hfx,hfy⟩ |
| · exact ⟨u,by simp,v,by simp [p.start_mem_support],huv,hu,he⟩ |
|
|
| lemma each_label_has_boundary {α β : Type*} [DecidableEq α] [DecidableEq β] |
| {G : SimpleGraph α} {u v : α} (p : G.Walk u v) (f : α → β) |
| (hc : 1 < (p.support.toFinset.image f).card) {a : β} |
| (ha : a ∈ p.support.toFinset.image f) : |
| ∃ x ∈ p.support, ∃ y ∈ p.support, G.Adj x y ∧ f x = a ∧ f y ≠ a := by |
| obtain ⟨x,hx,hxa⟩ := Finset.mem_image.mp ha |
| obtain ⟨b,hb,c,hc',hbc⟩ := Finset.one_lt_card.mp hc |
| have hd : ∃ d ∈ p.support.toFinset.image f, d ≠ a := by |
| by_cases he : b = a |
| · exact ⟨c,hc',by simpa [he] using Ne.symm hbc⟩ |
| · exact ⟨b,hb,he⟩ |
| obtain ⟨d,hd,hda⟩ := hd |
| obtain ⟨y,hy,hyd⟩ := Finset.mem_image.mp hd |
| have hx' : x ∈ p.support := List.mem_toFinset.mp hx |
| have hy' : y ∈ p.support := List.mem_toFinset.mp hy |
| obtain ⟨q⟩ := p.connected_induce_support.preconnected ⟨x,hx'⟩ ⟨y,hy'⟩ |
| obtain ⟨s,_,t,_,hst,hsa,hta⟩ := _root_.OAI.SimpleGraph.Walk.label_boundary q |
| (fun z ↦ f z.val) hxa (by simpa [hyd] using hda) |
| exact ⟨s.val,s.property,t.val,t.property,hst,hsa,hta⟩ |
|
|
| end SimpleGraph.Walk |
|
|
| namespace HardSphere.BathSlice |
| open HardSphere.Lattice |
|
|
| theorem first_exit_witnesses {n : ℕ} {L : Length} {J : ℕ} {h r : ℝ} |
| (hh : 0 < h) (hhr : h ≤ r) (hrh : r ≤ 2*h) (hr : r ≤ 1/4096) |
| (hperiod : L.val = h * J) (hL : 10 < L.val) |
| (Y : Configuration n L) {z w : Vertex} |
| (hz : z ∈ bad h r Y) (hp : withinReach starStep (bad h r Y) z w) |
| (hw : ∃ i, (1:ℝ)/4 ≤ h * |(w i : ℝ)-z i|) : |
| ∃ A : Finset (Fin n), |
| (1:ℝ)/1372 ≤ r * A.card ∧ |
| (∀ i ∈ A, torusDistance L (Y i) (gridCorner L h z) ≤ 1) ∧ |
| (∀ i ∈ A, ∃ j ∈ A, j ≠ i ∧ torusDistance L (Y i) (Y j) ≤ 6*r) := by |
| classical |
| obtain ⟨v,p,hlocal,ih,hspan⟩ := star_first_exit hh (by norm_num : (0:ℝ)<1/4) hz hp hw |
| let S := p.support.toFinset |
| have hS : ∀ x ∈ S, x ∈ bad h r Y := fun x hx ↦ (hlocal x (List.mem_toFinset.mp hx)).1 |
| obtain ⟨x0,hx0,j0,hj0⟩ := hz |
| have hex (x : Vertex) : ∃ j : Fin n, x ∈ S → |
| ∃ u ∈ microCube L h x, torusDistance L u (Y j) ≤ r := by |
| by_cases hx : x ∈ S |
| · obtain ⟨u,hu,j,hj⟩ := hS x hx |
| exact ⟨j,fun _ ↦ ⟨u,hu,hj⟩⟩ |
| · exact ⟨j0,fun h ↦ (hx h).elim⟩ |
| choose f hf using hex |
| let A := S.image f |
| have hinj : Set.InjOn (torusProjection J) (↑S : Set Vertex) := by |
| apply (local_projection_injective hh (d := (1:ℝ)/4+h) (z := z) ?_).mono |
| · intro x hx |
| exact (hlocal x (List.mem_toFinset.mp hx)).2 |
| · linarith |
| have hcount : S.card ≤ 343*A.card := by |
| apply Finset.card_le_mul_card_image S 343 |
| intro j _ |
| apply card_witness_sites_le hh hperiod hrh Y _ (hinj.mono (Finset.filter_subset _ _)) j |
| intro x hx |
| obtain ⟨hxS,hfx⟩ := Finset.mem_filter.mp hx |
| simpa only [hfx] using hf x hxS |
| have hpSpan : |(v ih : ℝ)-z ih| ≤ (S.card : ℝ)-1 := by |
| have he := star_path_span p ih |
| rw [abs_sub_comm] at he |
| exact_mod_cast he |
| have hcR : (S.card : ℝ) ≤ 343*(A.card : ℝ) := by exact_mod_cast hcount |
| have hlarge : (1:ℝ)/1372 ≤ r*A.card := by |
| have ht := mul_le_mul_of_nonneg_left hpSpan hh.le |
| have hcount' := mul_le_mul_of_nonneg_left hcR hh.le |
| have hhr' := mul_le_mul_of_nonneg_right hhr (Nat.cast_nonneg A.card : (0:ℝ)≤A.card) |
| nlinarith |
| have htwo : 1 < A.card := by |
| by_contra hn |
| have hc : (A.card : ℝ) ≤ 1 := by exact_mod_cast (show A.card ≤ 1 by omega) |
| have h0 : 0 ≤ r := hh.le.trans hhr |
| have ht := mul_le_mul_of_nonneg_left hc h0 |
| nlinarith |
| refine ⟨A,hlarge,?_,?_⟩ |
| · intro i hi |
| obtain ⟨x,hx,hxi⟩ := Finset.mem_image.mp hi |
| obtain ⟨u,hu,huf⟩ := hf x hx |
| rw [hxi] at huf |
| have hd := microCube_dist_corner hh.le (show (0:ℝ)≤1/4+h by positivity) |
| (hlocal x (List.mem_toFinset.mp hx)).2 hu |
| have ht := torusDistance_triangle' L (Y i) u (gridCorner L h z) |
| rw [torusDistance_comm L (Y i) u] at ht |
| linarith |
| · intro i hi |
| obtain ⟨x,hx,y,hy,hxy,hfi,hfj⟩ := _root_.OAI.SimpleGraph.Walk.each_label_has_boundary p f htwo hi |
| refine ⟨f y,Finset.mem_image.mpr ⟨y,List.mem_toFinset.mpr hy,rfl⟩,hfj,?_⟩ |
| rw [← hfi] |
| exact adjacent_witnesses_close hh.le hhr Y hxy (hf x (List.mem_toFinset.mpr hx)) |
| (hf y (List.mem_toFinset.mpr hy)) |
|
|
| end HardSphere.BathSlice |
|
|
| end |
|
|
| end OAI |
|
|
