| import Mathlib |
| import OAI.GroupTheory.SimpleAmenable.Amenability.BarrierChartRefinement |
| import OAI.GroupTheory.SimpleAmenable.Amenability.LocalBarrierTransport |
| import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.ChartBoundaryEndpoints |
|
|
| namespace OAI |
|
|
| section |
| section |
| open scoped symmDiff |
| namespace SimpleAmenable |
| open scoped commutatorElement |
| open scoped commutatorElement |
| section ChartLineTransport |
| open Classical Set |
| namespace BarrierSignalSystem |
| variable {a : ℕ} {ha : 0 < a} {Q B : ℝ} (S : BarrierSignalSystem a ha Q B) |
| |
| theorem chart_line_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 : GenericSquare a) |
| (u : CutRing×CutRing) (hdata : fullGroupAffineData g i p=(k,u)) |
| (hQ : 0 ≤ Q) |
| (hB : ∀l ∈ T ∪ squareBoundaryCuts,|conjugate l.2| ≤ B ∧ |
| |conjugate (l.2+integralCutForm a l.1 u)| ≤ B) |
| {N : ℝ} (hN : 1 ≤ N) (j : Fin 4) (c : CutRing) |
| (hc : c ∈ barrierCandidate a j N) |
| (hc' : c+integralCutForm a j u ∈ barrierCandidate a j N) |
| (b₁ b₂ b₁' b₂' : FlagSite a m (commonVertexDenominator a) (barrierFlagDirection ha j) → Bool) |
| (hs : signalSuccess (flagMeanSignal S.first.signal N) b₁) |
| (hs' : signalSuccess (flagMeanSignal S.first.signal N) b₁') |
| (hb₁ : ∀z,b₁' (flagSiteAction (commonVertexDenominator_pos ha) g z)=b₁ z) |
| (hb₂ : ∀z,b₂' (flagSiteAction (commonVertexDenominator_pos ha) g z)=b₂ z) |
| {x : ℝ} (hx : barrierLinePoint a j c x ∈ openSquareChart T p) : |
| x ∈ subdivisionBarrier (barrierSubdivision ha i j c |
| (S.marks (N:=N) (lt_of_lt_of_le zero_lt_one hN) j b₁)) (barrierActivation ha i j c b₂) ↔ |
| x+barrierLineParameter j (ordinary u.1,ordinary u.2) ∈ |
| subdivisionBarrier (barrierSubdivision ha k j (c+integralCutForm a j u) |
| (S.marks (N:=N) (lt_of_lt_of_le zero_lt_one hN) j b₁')) |
| (barrierActivation ha k j (c+integralCutForm a j u) b₂') := by |
| let d := barrierLineParameter j (ordinary u.1,ordinary u.2) |
| obtain ⟨L,U,hxLU,hL,hU,hchart,hend⟩ := chart_line_endpoints_forced T p hc hx |
| have hLU : L < U := hxLU.1.trans hxLU.2 |
| have htrans : ∀r ∈ Ioo L U,barrierLinePoint a j (c+integralCutForm a j u) (r+d) ∈ squareInterior := by |
| intro r hr |
| rw [barrierLinePoint_translate] |
| have hh := fullGroupAffineChart_square g i T hT p |
| rw [hdata] at hh |
| exact hh ⟨barrierLinePoint a j c r,hchart r hr,rfl⟩ |
| have hbounds := interval_endpoints_closed (continuous_id.add continuous_const) |
| isClosed_Icc hLU (fun r hr => let hp := (barrierLinePoint_interior_iff hc' (r+d)).mp (htrans r hr) |
| show r+d ∈ Icc (barrierLineLower a j (c+integralCutForm a j u)) |
| (barrierLineUpper a j (c+integralCutForm a j u)) from ⟨hp.1.le,hp.2.le⟩) |
| have hsource (r : ℝ) (hr : r ∈ ({L,U} : Finset ℝ)) |
| (hrb : r ∈ Icc (barrierLineLower a j c) (barrierLineUpper a j c)) : |
| r ∈ barrierSubdivision ha i j c (S.marks (N:=N) (lt_of_lt_of_le zero_lt_one hN) j b₁) := by |
| obtain ⟨l,hl,hjl,hzero⟩ := hend r hr |
| exact S.crossing_in_subdivision i hN hjl hc (by linarith [hc.1]) (hB l hl).1 b₁ hs hrb hzero |
| have htarget (r : ℝ) (hr : r ∈ ({L,U} : Finset ℝ)) |
| (hrb : r+d ∈ Icc (barrierLineLower a j (c+integralCutForm a j u)) |
| (barrierLineUpper a j (c+integralCutForm a j u))) : |
| r+d ∈ barrierSubdivision ha k j (c+integralCutForm a j u) |
| (S.marks (N:=N) (lt_of_lt_of_le zero_lt_one hN) j b₁') := by |
| obtain ⟨l,hl,hjl,hzero⟩ := hend r hr |
| apply S.crossing_in_subdivision k hN hjl hc' (by linarith [hc'.1]) (hB l hl).2 b₁' hs' hrb |
| rw [barrierLinePoint_translate,cutForm_add,hzero,cutForm_ordinary,map_add] |
| apply S.local_line_transport g i k (lt_of_lt_of_le zero_lt_one hN) j c u hc hc' b₁ b₂ b₁' b₂' |
| hs hs' hb₁ hb₂ L U (fun r hr => ⟨(hchart r hr).1,htrans r hr⟩) hU.2 |
| (hsource L (by simp) hL) (hsource U (by simp) hU) |
| (htarget L (by simp) hbounds.1) (htarget U (by simp) hbounds.2) _ hxLU |
| intro r hr z hzi hzp |
| exact (flagAffineData_barrier_chart ha g i j c T hT p hLU hr |
| (fun l hl r hr => openSquareChart_signs (hchart r hr) hl) z.val.2 hzp).trans hdata |
| |
| end BarrierSignalSystem |
| end ChartLineTransport |
| |
| section ChartBarrierTransport |
| open Classical Set |
| |
| theorem mem_sampledBarriers_iff {a m : ℕ} (ha : 0 < a) (i : Fin m) (N : ℝ) |
| (M : ∀j,Finset (FlagSite a m (commonVertexDenominator a) (barrierFlagDirection ha j))) |
| (b : ∀j,FlagSite a m (commonVertexDenominator a) (barrierFlagDirection ha j) → Bool) |
| (p : ℝ×ℝ) : p ∈ sampledBarriers ha i N M b ↔ |
| ∃j c,c ∈ barrierCandidate a j N ∧ cutForm a j p=ordinary c ∧ |
| barrierLineParameter j p ∈ subdivisionBarrier (barrierSubdivision ha i j c (M j)) |
| (barrierActivation ha i j c (b j)) := by |
| constructor |
| · rintro ⟨⟨j,c,hc⟩,st,hst,x,hx,rfl⟩ |
| refine ⟨j,c,hc,cutForm_barrierLinePoint a j c x,?_⟩ |
| rw [barrierLineParameter_point] |
| exact (mem_barrierActiveIntervals_image_iff ha i j c (M j) (b j) x).mp ⟨st,hst,hx⟩ |
| · rintro ⟨j,c,hc,hpc,hp⟩ |
| obtain ⟨st,hst,hx⟩ := (mem_barrierActiveIntervals_image_iff ha i j c (M j) (b j) _).mpr hp |
| exact ⟨⟨j,⟨c,hc⟩⟩,st,hst,barrierLineParameter j p,hx,barrierLinePoint_parameter hpc⟩ |
| |
| namespace BarrierSignalSystem |
| variable {a : ℕ} {ha : 0 < a} {Q B : ℝ} (S : BarrierSignalSystem a ha Q B) |
| |
| theorem active_line_small {m : ℕ} (i : Fin m) {N : ℝ} (hN : 0 < N) (j : Fin 4) |
| (c : CutRing) (hc : c ∈ barrierCandidate a j N) |
| (M : Finset (FlagSite a m (commonVertexDenominator a) (barrierFlagDirection ha j))) |
| (b : FlagSite a m (commonVertexDenominator a) (barrierFlagDirection ha j) → Bool) |
| (hs : signalSuccess (flagMeanSignal (S.second j).signal N) b) |
| {x : ℝ} (hx : x ∈ subdivisionBarrier (barrierSubdivision ha i j c M) (barrierActivation ha i j c b)) : |
| |conjugate c| < 3*N := by |
| by_contra hn |
| have he := secondSignal_negative_empty ha i hN hc M (S.second j).signal b |
| (S.second_neg j) hs (le_of_not_gt hn) |
| obtain ⟨st,hst,_⟩ := (mem_barrierActiveIntervals_image_iff ha i j c M b x).mpr hx |
| rw [he] at hst |
| exact Finset.notMem_empty st hst |
| |
| theorem chart_barrier_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 : GenericSquare a) |
| (u : CutRing×CutRing) (hdata : fullGroupAffineData g i p=(k,u)) |
| (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) |
| (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) |
| {x : ℝ×ℝ} (hx : x ∈ openSquareChart T p) : |
| x ∈ sampledBarriers ha i N (fun j => S.marks (N:=N) (lt_of_lt_of_le zero_lt_one hN) j (b₁ j)) b₂ ↔ |
| x+(ordinary u.1,ordinary u.2) ∈ sampledBarriers ha k N |
| (fun j => S.marks (N:=N) (lt_of_lt_of_le zero_lt_one hN) j (b₁' j)) b₂' := by |
| have hN₀ : 0 < N := lt_of_lt_of_le zero_lt_one hN |
| have hx' : x+(ordinary u.1,ordinary u.2) ∈ squareInterior := by |
| have hh := fullGroupAffineChart_square g i T hT p |
| rw [hdata] at hh |
| exact hh ⟨x,hx,rfl⟩ |
| rw [mem_sampledBarriers_iff,mem_sampledBarriers_iff] |
| constructor |
| · rintro ⟨j,c,hc,hline,hactive⟩ |
| have hsmall := S.active_line_small i hN₀ j c hc _ (b₂ j) (ht j) hactive |
| have hc' : c+integralCutForm a j u ∈ barrierCandidate a j N := by |
| refine ⟨?_,x+(ordinary u.1,ordinary u.2),hx',?_⟩ |
| · rw [map_add] |
| calc |
| |conjugate c+conjugate (integralCutForm a j u)| ≤ |conjugate c|+|conjugate (integralCutForm a j u)| := abs_add_le _ _ |
| _ ≤ 4*N := by linarith [huQ j] |
| · rw [cutForm_add,hline,cutForm_ordinary,map_add] |
| have hxline : barrierLinePoint a j c (barrierLineParameter j x) ∈ openSquareChart T p := by |
| rwa [barrierLinePoint_parameter hline] |
| have htransport := S.chart_line_transport g i k T hT p u hdata hQ hB hN j c hc hc' |
| (b₁ j) (b₂ j) (b₁' j) (b₂' j) (hs j) (hs' j) (hb₁ j) (hb₂ j) hxline |
| refine ⟨j,c+integralCutForm a j u,hc',?_,?_⟩ |
| · rw [cutForm_add,hline,cutForm_ordinary,map_add] |
| · rw [barrierLineParameter_add] |
| exact htransport.mp hactive |
| · rintro ⟨j,c',hc',hline',hactive'⟩ |
| let c := c'-integralCutForm a j u |
| have hce : c+integralCutForm a j u=c' := sub_add_cancel _ _ |
| have hline : cutForm a j x=ordinary c := by |
| rw [cutForm_add,cutForm_ordinary] at hline' |
| dsimp [c] |
| rw [map_sub] |
| linarith |
| have hsmall := S.active_line_small k hN₀ j c' hc' _ (b₂' j) (ht' j) hactive' |
| have hc : c ∈ barrierCandidate a j N := by |
| refine ⟨?_,x,hx.1,hline⟩ |
| change |conjugate (c'-integralCutForm a j u)| ≤ _ |
| rw [map_sub] |
| calc |
| |conjugate c'-conjugate (integralCutForm a j u)| ≤ |conjugate c'|+|conjugate (integralCutForm a j u)| := abs_sub _ _ |
| _ ≤ 4*N := by linarith [huQ j] |
| have hxline : barrierLinePoint a j c (barrierLineParameter j x) ∈ openSquareChart T p := by |
| rwa [barrierLinePoint_parameter hline] |
| have htransport := S.chart_line_transport g i k T hT p u hdata hQ hB hN j c hc |
| (hce.symm ▸ hc') (b₁ j) (b₂ j) (b₁' j) (b₂' j) (hs j) (hs' j) (hb₁ j) (hb₂ j) hxline |
| rw [hce] at htransport |
| refine ⟨j,c,hc,hline,htransport.mpr ?_⟩ |
| simpa only [barrierLineParameter_add] using hactive' |
|
|
| end BarrierSignalSystem |
| end ChartBarrierTransport |
|
|
| end SimpleAmenable |
| end |
| end |
|
|
| end OAI |
|
|
