openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /PolygonGeometry /ChartBoundaryEndpoints.lean
| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.OpenAffineCharts | |
| namespace OAI | |
| section | |
| section | |
| open scoped symmDiff | |
| namespace SimpleAmenable | |
| open scoped commutatorElement | |
| open scoped commutatorElement | |
| section ChartBoundaryEndpoints | |
| open Classical Set | |
| def squareBoundaryCuts : Finset PlaneCut := {(0,0),(0,1),(1,0),(1,1)} | |
| theorem interval_endpoints_closed {X : Type*} [TopologicalSpace X] {f : ℝ → X} | |
| (hf : Continuous f) {A : Set X} (hA : IsClosed A) {L U : ℝ} (hLU : L < U) | |
| (h : ∀r ∈ Ioo L U,f r ∈ A) : f L ∈ A ∧ f U ∈ A := by | |
| have hs : closure (Ioo L U) ⊆ f ⁻¹' A := (hA.preimage hf).closure_subset_iff.mpr h | |
| rw [closure_Ioo hLU.ne] at hs | |
| exact ⟨hs (left_mem_Icc.mpr hLU.le),hs (right_mem_Icc.mpr hLU.le)⟩ | |
| theorem squareBoundaryCuts_of_closed_not_open {a : ℕ} {p : ℝ×ℝ} | |
| (hp : p ∈ Icc (0:ℝ) 1 ×ˢ Icc (0:ℝ) 1) (hn : p ∉ squareInterior) : | |
| ∃l ∈ squareBoundaryCuts,cutForm a l.1 p=ordinary l.2 := by | |
| by_cases hx₀ : p.1=0 | |
| · exact ⟨(0,0),by simp [squareBoundaryCuts],by simp [cutForm,hx₀]⟩ | |
| by_cases hx₁ : p.1=1 | |
| · exact ⟨(0,1),by simp [squareBoundaryCuts],by simp [cutForm,hx₁]⟩ | |
| by_cases hy₀ : p.2=0 | |
| · exact ⟨(1,0),by simp [squareBoundaryCuts],by simp [cutForm,hy₀]⟩ | |
| by_cases hy₁ : p.2=1 | |
| · exact ⟨(1,1),by simp [squareBoundaryCuts],by simp [cutForm,hy₁]⟩ | |
| exact False.elim (hn ⟨⟨lt_of_le_of_ne hp.1.1 (Ne.symm hx₀),lt_of_le_of_ne hp.1.2 hx₁⟩, | |
| ⟨lt_of_le_of_ne hp.2.1 (Ne.symm hy₀),lt_of_le_of_ne hp.2.2 hy₁⟩⟩) | |
| theorem squareBoundaryCuts_nonzero_on_interior {a : ℕ} {p : ℝ×ℝ} | |
| (hp : p ∈ squareInterior) {l : PlaneCut} (hl : l ∈ squareBoundaryCuts) : | |
| cutForm a l.1 p ≠ ordinary l.2 := by | |
| simp only [squareBoundaryCuts,Finset.mem_insert,Finset.mem_singleton] at hl | |
| rcases hl with rfl|rfl|rfl|rfl | |
| · simpa [cutForm] using ne_of_gt hp.1.1 | |
| · simpa [cutForm] using ne_of_lt hp.1.2 | |
| · simpa [cutForm] using ne_of_gt hp.2.1 | |
| · simpa [cutForm] using ne_of_lt hp.2.2 | |
| theorem barrier_geometric_endpoint_cut {a : ℕ} {j : Fin 4} {c : CutRing} {N r : ℝ} | |
| (hc : c ∈ barrierCandidate a j N) | |
| (hr : r=barrierLineLower a j c ∨ r=barrierLineUpper a j c) : | |
| ∃l ∈ squareBoundaryCuts,j ≠ l.1 ∧ cutForm a l.1 (barrierLinePoint a j c r)=ordinary l.2 := by | |
| have hgap := barrierLineLower_lt_upper hc | |
| have hh := interval_endpoints_closed (barrierLinePoint_continuous a j c) | |
| (isClosed_Icc.prod isClosed_Icc) hgap | |
| (fun t ht => let hp := (barrierLinePoint_interior_iff hc t).mpr ht | |
| show barrierLinePoint a j c t ∈ Icc (0:ℝ) 1 ×ˢ Icc (0:ℝ) 1 from | |
| ⟨⟨hp.1.1.le,hp.1.2.le⟩,⟨hp.2.1.le,hp.2.2.le⟩⟩) | |
| have hclosed : barrierLinePoint a j c r ∈ Icc (0:ℝ) 1 ×ˢ Icc (0:ℝ) 1 := | |
| hr.elim (fun h => h ▸ hh.1) (fun h => h ▸ hh.2) | |
| have hn : barrierLinePoint a j c r ∉ squareInterior := by | |
| rw [barrierLinePoint_interior_iff hc r] | |
| rcases hr with rfl|rfl <;> simp | |
| obtain ⟨l,hl,hcut⟩ := squareBoundaryCuts_of_closed_not_open hclosed hn | |
| refine ⟨l,hl,?_,hcut⟩ | |
| intro he | |
| obtain ⟨p,hp,hpc⟩ := hc.2 | |
| have hjc := cutForm_barrierLinePoint a j c r | |
| rw [←he] at hcut | |
| have hp' := hpc.trans (hjc.symm.trans hcut) | |
| rw [he] at hp' | |
| exact squareBoundaryCuts_nonzero_on_interior hp hl hp' | |
| theorem chart_line_endpoints_forced {a : ℕ} (T : Finset PlaneCut) (p : GenericSquare a) | |
| {j : Fin 4} {c : CutRing} {N x : ℝ} (hc : c ∈ barrierCandidate a j N) | |
| (hx : barrierLinePoint a j c x ∈ openSquareChart T p) : | |
| ∃L U : ℝ,x ∈ Ioo L U ∧ | |
| L ∈ Icc (barrierLineLower a j c) (barrierLineUpper a j c) ∧ | |
| U ∈ Icc (barrierLineLower a j c) (barrierLineUpper a j c) ∧ | |
| (∀r ∈ Ioo L U,barrierLinePoint a j c r ∈ openSquareChart T p) ∧ | |
| (∀r ∈ ({L,U} : Finset ℝ),∃l ∈ T ∪ squareBoundaryCuts,j ≠ l.1 ∧ | |
| cutForm a l.1 (barrierLinePoint a j c r)=ordinary l.2) := by | |
| obtain ⟨L,U,hx,hL,hU,hchart,hend⟩ := openSquareChart_line_interval T p hc hx | |
| refine ⟨L,U,hx,hL,hU,hchart,?_⟩ | |
| intro r hr | |
| rcases hend r hr with h | h | h | |
| · obtain ⟨l,hl,hj,hc⟩ := barrier_geometric_endpoint_cut hc (Or.inl h) | |
| exact ⟨l,Finset.mem_union_right _ hl,hj,hc⟩ | |
| · obtain ⟨l,hl,hj,hc⟩ := barrier_geometric_endpoint_cut hc (Or.inr h) | |
| exact ⟨l,Finset.mem_union_right _ hl,hj,hc⟩ | |
| · obtain ⟨l,hl,hj,hc⟩ := h | |
| exact ⟨l,Finset.mem_union_left _ hl,hj,hc⟩ | |
| end ChartBoundaryEndpoints | |
| end SimpleAmenable | |
| end | |
| end | |
| end OAI | |