| import OAI.Probability.GammaPassage.ConnectorContact |
| import OAI.Probability.GammaPassage.ConnectorEvents |
| import OAI.Probability.GammaPassage.RareFastEvents |
|
|
| namespace OAI |
|
|
| namespace GammaFPP.CrossingBridge |
| open PlanarFPP GammaFPP.RealDirectionBridge GammaFPP.IntervalTravelBridge Set |
|
|
| def connectorWindow (b : ℝ) (X : ℤ) (Y S : ℝ) : Set Vertex := |
| {p | p.1=X ∧ |transverse b p-Y| ≤ S/4} |
|
|
| def connectorRegion (b : ℝ) (X d : ℤ) (Y S s : ℝ) : Set Vertex := |
| {p | (X : ℝ)-s ≤ p.1 ∧ (p.1 : ℝ) ≤ (X : ℝ)+d+s ∧ |
| |transverse b p-Y| ≤ 4*S} |
|
|
| theorem connectorRegion_finite (b : ℝ) (X d : ℤ) (Y S s : ℝ) : |
| (connectorRegion b X d Y S s).Finite := by |
| apply (finite_tilted_rectangle b X Y (|(d : ℝ)|+|s|) (4*S)).subset |
| intro p hp |
| refine ⟨abs_le.mpr ⟨?_,?_⟩,hp.2.2⟩ <;> |
| linarith [hp.1,hp.2.1,le_abs_self s,le_abs_self (d : ℝ),abs_nonneg (d : ℝ)] |
|
|
| noncomputable def connectorVertices (b : ℝ) (X d : ℤ) (Y S s : ℝ) : Finset Vertex := |
| (connectorRegion_finite b X d Y S s).toFinset |
|
|
| @[simp] theorem mem_connectorVertices (b : ℝ) (X d : ℤ) (Y S s : ℝ) (p : Vertex) : |
| p ∈ connectorVertices b X d Y S s ↔ p ∈ connectorRegion b X d Y S s := |
| (connectorRegion_finite b X d Y S s).mem_toFinset |
|
|
| theorem connector_transverse_displacement (b : ℝ) (X d : ℤ) (Y S σ a : ℝ) |
| (hS : 0 ≤ S) (hσ : |σ|=1) (ha : 1 ≤ a ∧ a ≤ 2) |
| (p q : Vertex) (hp : p ∈ connectorWindow b X Y S) |
| (hq : q ∈ connectorWindow b (X+d) (Y+σ*a*S) S) : |
| S/2 ≤ σ*(transverse b q-transverse b p) ∧ |
| σ*(transverse b q-transverse b p) ≤ 5*S/2 := by |
| have hsq : σ*σ=1 := by nlinarith [sq_abs σ] |
| have hep := hp.2 |
| have heq := hq.2 |
| have hp' : |σ*(transverse b p-Y)| ≤ S/4 := by simpa [abs_mul,hσ] using hep |
| have hq' : |σ*(transverse b q-(Y+σ*a*S))| ≤ S/4 := by simpa [abs_mul,hσ] using heq |
| have he : σ*(transverse b q-transverse b p) = |
| σ*(transverse b q-(Y+σ*a*S))-σ*(transverse b p-Y)+a*S := by |
| calc |
| _ = σ*(transverse b q-(Y+σ*a*S))-σ*(transverse b p-Y)+(σ*σ)*(a*S) := by ring |
| _ = _ := by rw [hsq]; ring |
| rw [he] |
| obtain ⟨hp₀,hp₁⟩ := abs_le.mp hp' |
| obtain ⟨hq₀,hq₁⟩ := abs_le.mp hq' |
| constructor <;> nlinarith [mul_le_mul_of_nonneg_right ha.1 hS, |
| mul_le_mul_of_nonneg_right ha.2 hS] |
|
|
| theorem connector_window_bounds (b : ℝ) (X d : ℤ) (Y S σ a : ℝ) |
| (hd : 0 ≤ d) (hS : 0 ≤ S) (hσ : |σ|=1) (ha : 1 ≤ a ∧ a ≤ 2) |
| (p : Vertex) (hp : p ∈ connectorWindow b X Y S ∪ |
| connectorWindow b (X+d) (Y+σ*a*S) S) : |
| (X : ℝ) ≤ p.1 ∧ (p.1 : ℝ) ≤ (X : ℝ)+d ∧ |
| |transverse b p-Y| ≤ 9*S/4 := by |
| have hd' : (0 : ℝ) ≤ d := by exact_mod_cast hd |
| rcases hp with hp | hp |
| · refine ⟨by simp [hp.1],?_,?_⟩ |
| · simp only [hp.1]; linarith |
| · linarith [hp.2] |
| · refine ⟨?_,by simp [hp.1],?_⟩ |
| · simp only [hp.1,Int.cast_add]; linarith |
| · have hh := abs_add_le (transverse b p-(Y+σ*a*S)) (σ*a*S) |
| have he : transverse b p-(Y+σ*a*S)+σ*a*S=transverse b p-Y := by ring |
| rw [he,abs_mul,abs_mul,hσ,one_mul,abs_of_nonneg (by linarith : 0 ≤ a),abs_of_nonneg hS] at hh |
| nlinarith [hp.2,mul_le_mul_of_nonneg_right ha.2 hS] |
|
|
| theorem transverse_plane_lipschitz (b : ℝ) (hb : |b| ≤ 1) (z : Plane) : |
| |rCoord b z| ≤ 2*‖z‖ := by |
| have hh := abs_sub_le (z 1) 0 (b*z 0) |
| simp only [sub_zero,zero_sub,abs_neg,abs_mul] at hh |
| have hc₀ := plane_coordinate_le_norm z 0 |
| have hc₁ := plane_coordinate_le_norm z 1 |
| have hm := mul_le_mul_of_nonneg_right hb (abs_nonneg (z 0)) |
| change |z 1-b*z 0| ≤ _ |
| nlinarith |
|
|
| theorem plane_norm_le_shear (b : ℝ) (hb : |b| ≤ 1) (z : Plane) : |
| ‖z‖ ≤ 2*|xCoord z|+|rCoord b z| := by |
| have hh := abs_add_le (rCoord b z) (b*z 0) |
| have he : rCoord b z+b*z 0=z 1 := by simp [rCoord_apply] |
| rw [he,abs_mul] at hh |
| have hm := mul_le_mul_of_nonneg_right hb (abs_nonneg (z 0)) |
| have hn := plane_norm_le_coordinates z |
| change ‖z‖ ≤ 2*|z 0|+|rCoord b z| |
| linarith |
|
|
| theorem connector_union_diameter (b : ℝ) (hb : |b| ≤ 1) |
| (X d : ℤ) (Y S σ a : ℝ) (hd : 0 ≤ d) (hS : 0 ≤ S) |
| (hσ : |σ|=1) (ha : 1 ≤ a ∧ a ≤ 2) |
| (p q : Vertex) |
| (hp : p ∈ connectorWindow b X Y S ∪ connectorWindow b (X+d) (Y+σ*a*S) S) |
| (hq : q ∈ connectorWindow b X Y S ∪ connectorWindow b (X+d) (Y+σ*a*S) S) : |
| ‖latticeEmbed (q-p)‖ ≤ 2*(d : ℝ)+9*S/2 := by |
| obtain ⟨hp₀,hp₁,hp₂⟩ := connector_window_bounds b X d Y S σ a hd hS hσ ha p hp |
| obtain ⟨hq₀,hq₁,hq₂⟩ := connector_window_bounds b X d Y S σ a hd hS hσ ha q hq |
| have hx : |xCoord (latticeEmbed (q-p))| ≤ (d : ℝ) := by |
| change |((q-p).1 : ℝ)| ≤ _ |
| simp only [Prod.fst_sub,Int.cast_sub] |
| exact abs_le.mpr ⟨by linarith,by linarith⟩ |
| have hr : |rCoord b (latticeEmbed (q-p))| ≤ 9*S/2 := by |
| have hh := abs_sub_le (transverse b q-Y) 0 (transverse b p-Y) |
| simp only [sub_zero,zero_sub,abs_neg] at hh |
| have he : rCoord b (latticeEmbed (q-p))=(transverse b q-Y)-(transverse b p-Y) := by |
| simp [rCoord_apply,latticeEmbed,transverse]; ring |
| rw [he] |
| linarith |
| exact (plane_norm_le_shear b hb _).trans (by linarith) |
|
|
| theorem connector_tube_containment (b : ℝ) (hb : |b| ≤ 1) |
| (X d : ℤ) (Y S s σ a : ℝ) (hd : 0 ≤ d) (hs : 0 ≤ s) (hS : s ≤ S) |
| (hσ : |σ|=1) (ha : 1 ≤ a ∧ a ≤ 2) |
| (p q : Vertex) (hp : p ∈ connectorWindow b X Y S) |
| (hq : q ∈ connectorWindow b (X+d) (Y+σ*a*S) S) : |
| SegmentTube p q (s/2) ⊆ (connectorVertices b X d Y S s : Set Vertex) := by |
| have hS0 : 0 ≤ S := hs.trans hS |
| obtain ⟨hp₀,hp₁,hp₂⟩ := connector_window_bounds b X d Y S σ a hd hS0 hσ ha p (Or.inl hp) |
| obtain ⟨hq₀,hq₁,hq₂⟩ := connector_window_bounds b X d Y S σ a hd hS0 hσ ha q (Or.inr hq) |
| intro z hz |
| obtain ⟨t,ht₀,ht₁,ht⟩ := hz |
| have hsegx : xCoord (segmentPoint p q t)=(1-t)*(p.1 : ℝ)+t*(q.1 : ℝ) := by |
| simp [segmentPoint,latticeEmbed,xCoord_apply,PiLp.add_apply,PiLp.sub_apply,PiLp.smul_apply] |
| ring |
| have hsegr : rCoord b (segmentPoint p q t)-Y= |
| (1-t)*(transverse b p-Y)+t*(transverse b q-Y) := by |
| simp [segmentPoint,latticeEmbed,rCoord_apply,transverse,PiLp.add_apply,PiLp.sub_apply,PiLp.smul_apply] |
| ring |
| have hseg₀ : (X : ℝ) ≤ xCoord (segmentPoint p q t) := by |
| rw [hsegx] |
| nlinarith [mul_le_mul_of_nonneg_left hp₀ (by linarith : 0 ≤ 1-t), |
| mul_le_mul_of_nonneg_left hq₀ ht₀] |
| have hseg₁ : xCoord (segmentPoint p q t) ≤ (X : ℝ)+d := by |
| rw [hsegx] |
| nlinarith [mul_le_mul_of_nonneg_left hp₁ (by linarith : 0 ≤ 1-t), |
| mul_le_mul_of_nonneg_left hq₁ ht₀] |
| have hseg₂ : |rCoord b (segmentPoint p q t)-Y| ≤ 9*S/4 := by |
| rw [hsegr] |
| have hh := abs_add_le ((1-t)*(transverse b p-Y)) (t*(transverse b q-Y)) |
| rw [abs_mul,abs_mul,abs_of_nonneg ht₀,abs_of_nonneg (by linarith : 0 ≤ 1-t)] at hh |
| nlinarith [mul_le_mul_of_nonneg_left hp₂ (by linarith : 0 ≤ 1-t), |
| mul_le_mul_of_nonneg_left hq₂ ht₀] |
| have hx := (plane_coordinate_le_norm (latticeEmbed z-segmentPoint p q t) 0).trans ht |
| have hr := (transverse_plane_lipschitz b hb (latticeEmbed z-segmentPoint p q t)).trans |
| (mul_le_mul_of_nonneg_left ht (by norm_num : (0 : ℝ) ≤ 2)) |
| have hx' : |(z.1 : ℝ)-xCoord (segmentPoint p q t)| ≤ s/2 := by |
| simpa [latticeEmbed,PiLp.sub_apply] using hx |
| have hr' : |transverse b z-rCoord b (segmentPoint p q t)| ≤ s := by |
| have he : 2*(s/2)=s := by ring |
| simpa [map_sub,rCoord_apply,latticeEmbed,transverse,he] using hr |
| change z ∈ connectorVertices b X d Y S s |
| rw [mem_connectorVertices] |
| refine ⟨?_,?_,?_⟩ |
| · linarith [(abs_le.mp hx').1] |
| · linarith [(abs_le.mp hx').2] |
| · have hh := abs_add_le (transverse b z-rCoord b (segmentPoint p q t)) |
| (rCoord b (segmentPoint p q t)-Y) |
| rw [sub_add_sub_cancel] at hh |
| linarith |
|
|
| end GammaFPP.CrossingBridge |
|
|
| end OAI |
|
|
