openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /Amenability /ComponentChartContainment.lean
| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.Amenability.BarrierSignalSystem | |
| import OAI.GroupTheory.SimpleAmenable.Amenability.ChartLineTransport | |
| import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.ChartBoundaryEndpoints | |
| namespace OAI | |
| section | |
| section | |
| open scoped symmDiff | |
| namespace SimpleAmenable | |
| open scoped commutatorElement | |
| open scoped commutatorElement | |
| section ComponentChartContainment | |
| open Classical Set | |
| theorem barrier_component_cut_ordinary {a : ℕ} {B : Set (ℝ×ℝ)} {j : Fin 4} {c : CutRing} | |
| (hline : ∀x ∈ squareInterior,cutForm a j x=ordinary c → x ∈ B) | |
| {p x : ℝ×ℝ} (hp : p ∈ barrierRegion B) | |
| (hx : x ∈ connectedComponentIn (barrierRegion B) p) : | |
| cutForm a j x ≠ ordinary c ∧ (cutForm a j x < ordinary c ↔ cutForm a j p < ordinary c) := by | |
| have hn (y : ℝ×ℝ) (hy : y ∈ connectedComponentIn (barrierRegion B) p) : cutForm a j y ≠ ordinary c := by | |
| intro he | |
| have hr := connectedComponentIn_subset (barrierRegion B) p hy | |
| exact hr.2 (hline y hr.1 he) | |
| refine ⟨hn x hx,?_⟩ | |
| have hh := continuous_nonzero_sign isPreconnected_connectedComponentIn | |
| ((cutForm_continuous a j).sub continuous_const).continuousOn | |
| (fun y hy => sub_ne_zero.mpr (hn y hy)) hx (mem_connectedComponentIn hp) | |
| simpa only [Pi.sub_apply,sub_lt_zero] using hh | |
| theorem barrier_component_openChart {a : ℕ} {B : Set (ℝ×ℝ)} | |
| (T : Finset PlaneCut) (p : GenericSquare a) | |
| (hp : p.val ∈ barrierRegion B) | |
| (hline : ∀l ∈ T,∀x ∈ squareInterior,cutForm a l.1 x=ordinary l.2 → x ∈ B) : | |
| connectedComponentIn (barrierRegion B) p.val ⊆ openSquareChart T p := by | |
| intro x hx | |
| refine ⟨(connectedComponentIn_subset (barrierRegion B) p.val hx).1,?_⟩ | |
| intro l | |
| obtain ⟨hne,hs⟩ := barrier_component_cut_ordinary (hline l.val l.property) hp hx | |
| by_cases hh : cutForm a l.val.1 p.val < ordinary l.val.2 | |
| · simpa [squareSign,hh] using hs.mpr hh | |
| · simpa [squareSign,hh] using lt_of_le_of_ne (le_of_not_gt (fun h => hh (hs.mp h))) hne.symm | |
| theorem barrier_component_translatedChart {a : ℕ} {B : Set (ℝ×ℝ)} | |
| (T : Finset PlaneCut) (p q : GenericSquare a) (u : CutRing×CutRing) | |
| (hqp : q.val=p.val+(ordinary u.1,ordinary u.2)) | |
| (hq : q.val ∈ barrierRegion B) | |
| (hline : ∀l ∈ T ∪ squareBoundaryCuts,∀x ∈ squareInterior, | |
| cutForm a l.1 x=ordinary (l.2+integralCutForm a l.1 u) → x ∈ B) : | |
| ∀x ∈ connectedComponentIn (barrierRegion B) q.val, | |
| x-(ordinary u.1,ordinary u.2) ∈ openSquareChart T p := by | |
| intro x hx | |
| have hsign (l : PlaneCut) (hl : l ∈ T ∪ squareBoundaryCuts) : | |
| cutForm a l.1 (x-(ordinary u.1,ordinary u.2)) ≠ ordinary l.2 ∧ | |
| (cutForm a l.1 (x-(ordinary u.1,ordinary u.2)) < ordinary l.2 ↔ | |
| cutForm a l.1 p.val < ordinary l.2) := by | |
| obtain ⟨hne,hs⟩ := barrier_component_cut_ordinary (hline l hl) hq hx | |
| rw [hqp,cutForm_add,cutForm_ordinary,map_add] at hs | |
| rw [map_add] at hne | |
| rw [cutForm_sub,cutForm_ordinary] | |
| refine ⟨fun h => hne (by linarith),?_⟩ | |
| constructor | |
| · intro h | |
| have hh := hs.mp (by linarith : cutForm a l.1 x < ordinary l.2+ordinary (integralCutForm a l.1 u)) | |
| linarith | |
| · intro h | |
| have hh := hs.mpr (by linarith : cutForm a l.1 p.val+ordinary (integralCutForm a l.1 u) < ordinary l.2+ordinary (integralCutForm a l.1 u)) | |
| linarith | |
| have hp := generic_mem_squareInterior p | |
| have hx₀ := hsign (0,0) (Finset.mem_union_right _ (by simp [squareBoundaryCuts])) | |
| have hx₁ := hsign (0,1) (Finset.mem_union_right _ (by simp [squareBoundaryCuts])) | |
| have hy₀ := hsign (1,0) (Finset.mem_union_right _ (by simp [squareBoundaryCuts])) | |
| have hy₁ := hsign (1,1) (Finset.mem_union_right _ (by simp [squareBoundaryCuts])) | |
| simp only [cutForm,Matrix.cons_val_zero,Matrix.cons_val_one, | |
| map_zero,map_one,Prod.fst_sub,Prod.snd_sub] at hx₀ hx₁ hy₀ hy₁ | |
| refine ⟨⟨⟨?_,?_⟩,⟨?_,?_⟩⟩,?_⟩ | |
| · exact lt_of_le_of_ne (le_of_not_gt (fun h => not_lt_of_gt hp.1.1 (hx₀.2.mp h))) hx₀.1.symm | |
| · exact hx₁.2.mpr hp.1.2 | |
| · exact lt_of_le_of_ne (le_of_not_gt (fun h => not_lt_of_gt hp.2.1 (hy₀.2.mp h))) hy₀.1.symm | |
| · exact hy₁.2.mpr hp.2.2 | |
| · intro l | |
| obtain ⟨hne,hs⟩ := hsign l.val (Finset.mem_union_left _ l.property) | |
| by_cases hh : cutForm a l.val.1 p.val < ordinary l.val.2 | |
| · simpa [squareSign,hh] using hs.mpr hh | |
| · simpa [squareSign,hh] using lt_of_le_of_ne (le_of_not_gt (fun h => hh (hs.mp h))) hne.symm | |
| end ComponentChartContainment | |
| section ComponentTransport | |
| open Classical Set | |
| namespace BarrierSignalSystem | |
| variable {a : ℕ} {ha : 0 < a} {Q B : ℝ} (S : BarrierSignalSystem a ha Q B) | |
| theorem component_transport {m : ℕ} (g : polygonFullGroup a m) (i k : Fin m) | |
| (T : Finset PlaneCut) | |
| (hT : ∀p q,(∀l ∈ T,squareSign l p=squareSign l q) → | |
| fullGroupAffineData g i p=fullGroupAffineData g i q) (p q : GenericSquare a) | |
| (u : CutRing×CutRing) (hdata : fullGroupAffineData g i p=(k,u)) | |
| (hqp : q.val=p.val+(ordinary u.1,ordinary u.2)) | |
| (hQ : 0 ≤ Q) (huQ : ∀j,|conjugate (integralCutForm a j u)| ≤ Q) | |
| (hB : ∀l ∈ T ∪ squareBoundaryCuts,|conjugate l.2| ≤ B ∧ | |
| |conjugate (l.2+integralCutForm a l.1 u)| ≤ B) | |
| {N : ℝ} (hN : 1 ≤ N) (hQN : Q < N) (hBN : B ≤ N) | |
| (b₁ b₂ b₁' b₂' : ∀j,FlagSite a m (commonVertexDenominator a) (barrierFlagDirection ha j) → Bool) | |
| (hs : ∀j,signalSuccess (flagMeanSignal S.first.signal N) (b₁ j)) | |
| (hs' : ∀j,signalSuccess (flagMeanSignal S.first.signal N) (b₁' j)) | |
| (ht : ∀j,signalSuccess (flagMeanSignal (S.second j).signal N) (b₂ j)) | |
| (ht' : ∀j,signalSuccess (flagMeanSignal (S.second j).signal N) (b₂' j)) | |
| (hb₁ : ∀j z,b₁' j (flagSiteAction (commonVertexDenominator_pos ha) g z)=b₁ j z) | |
| (hb₂ : ∀j z,b₂' j (flagSiteAction (commonVertexDenominator_pos ha) g z)=b₂ j z) : | |
| (fun x => x+(ordinary u.1,ordinary u.2)) '' connectedComponentIn | |
| (barrierRegion (sampledBarriers ha i N | |
| (fun j => S.marks (N:=N) (lt_of_lt_of_le zero_lt_one hN) j (b₁ j)) b₂)) p.val = | |
| connectedComponentIn (barrierRegion (sampledBarriers ha k N | |
| (fun j => S.marks (N:=N) (lt_of_lt_of_le zero_lt_one hN) j (b₁' j)) b₂')) q.val := by | |
| let v := (ordinary u.1,ordinary u.2) | |
| let A₀ := sampledBarriers ha i N | |
| (fun j => S.marks (N:=N) (lt_of_lt_of_le zero_lt_one hN) j (b₁ j)) b₂ | |
| let A₁ := sampledBarriers ha k N | |
| (fun j => S.marks (N:=N) (lt_of_lt_of_le zero_lt_one hN) j (b₁' j)) b₂' | |
| have hp : p.val ∈ barrierRegion A₀ := generic_mem_barrierRegion (sampledBarriers_supported ha i N _ b₂) p | |
| have hq : q.val ∈ barrierRegion A₁ := generic_mem_barrierRegion (sampledBarriers_supported ha k N _ b₂') q | |
| have hsource : connectedComponentIn (barrierRegion A₀) p.val ⊆ openSquareChart T p := by | |
| apply barrier_component_openChart T p hp | |
| intro l hl x hx hxc | |
| have hband := ((hB l (Finset.mem_union_left _ hl)).1).trans hBN | |
| exact S.central_line_barrier i hN b₁ b₂ hs ht | |
| ⟨by linarith, x,hx,hxc⟩ hband x hx hxc | |
| have htarget : ∀x ∈ connectedComponentIn (barrierRegion A₁) q.val,x-v ∈ openSquareChart T p := by | |
| apply barrier_component_translatedChart T p q u hqp hq | |
| intro l hl x hx hxc | |
| have hband := ((hB l hl).2).trans hBN | |
| exact S.central_line_barrier k hN b₁' b₂' hs' ht' | |
| ⟨by linarith,x,hx,hxc⟩ hband x hx hxc | |
| have hbar (x : ℝ×ℝ) (hx : x ∈ openSquareChart T p) : x ∈ A₀ ↔ x+v ∈ A₁ := | |
| S.chart_barrier_transport g i k T hT p u hdata hQ huQ hB hN hQN | |
| b₁ b₂ b₁' b₂' hs hs' ht ht' hb₁ hb₂ hx | |
| have hforward : (fun x => x+v) '' connectedComponentIn (barrierRegion A₀) p.val ⊆ | |
| connectedComponentIn (barrierRegion A₁) q.val := by | |
| apply (isPreconnected_connectedComponentIn.image _ (continuous_id.add continuous_const).continuousOn).subset_connectedComponentIn | |
| · exact ⟨p.val,mem_connectedComponentIn hp,hqp.symm⟩ | |
| · rintro x ⟨y,hy,rfl⟩ | |
| have hyC := hsource hy | |
| have hyreg := connectedComponentIn_subset (barrierRegion A₀) p.val hy | |
| refine ⟨?_,fun h => hyreg.2 ((hbar y hyC).mpr h)⟩ | |
| have himage := fullGroupAffineChart_square g i T hT p | |
| rw [hdata] at himage | |
| exact himage ⟨y,hyC,rfl⟩ | |
| have hbackward : (fun x => x-v) '' connectedComponentIn (barrierRegion A₁) q.val ⊆ | |
| connectedComponentIn (barrierRegion A₀) p.val := by | |
| apply (isPreconnected_connectedComponentIn.image _ (continuous_id.sub continuous_const).continuousOn).subset_connectedComponentIn | |
| · refine ⟨q.val,mem_connectedComponentIn hq,?_⟩ | |
| change q.val-v=p.val | |
| rw [hqp,add_sub_cancel_right] | |
| · rintro x ⟨y,hy,rfl⟩ | |
| have hyC := htarget y hy | |
| have hyreg := connectedComponentIn_subset (barrierRegion A₁) q.val hy | |
| refine ⟨hyC.1,?_⟩ | |
| intro h | |
| have hh := (hbar (y-v) hyC).mp h | |
| rw [sub_add_cancel] at hh | |
| exact hyreg.2 hh | |
| apply Subset.antisymm hforward | |
| intro y hy | |
| exact ⟨y-v,hbackward ⟨y,hy,rfl⟩,sub_add_cancel y v⟩ | |
| end BarrierSignalSystem | |
| end ComponentTransport | |
| end SimpleAmenable | |
| end | |
| end | |
| end OAI | |