openAI-math-dataset / lean /OAI /NumberTheory /DukePrimeDegree /Geometry /CompactPhysicalRectangles.lean
| import OAI.NumberTheory.DukePrimeDegree.Geometry.RootCompactTransversals | |
| namespace OAI | |
| /-! # Physical rectangles containing prescribed compact orbit pieces -/ | |
| noncomputable section | |
| open Matrix Set | |
| open scoped Topology | |
| namespace DukePrimeDegree | |
| open PrimeDegreePackets | |
| theorem exists_shifted_root_rectangle_covering_compact | |
| {T B : Type*} [TopologicalSpace T] [TopologicalSpace B] | |
| {D : Set (T × B)} (hD : IsOpen D) | |
| {n : ℕ} {a b k : Fin n} (hab : a ≠ b) | |
| (e : D → RootSlice a b k) (he : Topology.IsOpenEmbedding e) | |
| (τ : D → ℝ) (hτ : Continuous τ) (s : T) | |
| (hsD : {s} ×ˢ (univ : Set B) ⊆ D) | |
| (hinj : Set.InjOn (fun z : D × ℝ => rootQuotientParametrization hab (e z.1,z.2)) | |
| ({z : D | z.val.1 = s} ×ˢ univ)) | |
| (K : Set B) (hK : IsCompact K) (I : Set ℝ) (hI : IsCompact I) (hne : I.Nonempty) | |
| (V : Set B) (hV : IsOpen V) (hKV : K ⊆ V) | |
| (I₀ : Set ℝ) (hI₀ : IsOpen I₀) (hII₀ : I ⊆ I₀) : | |
| ∃ (Q : RootTransverseRectangle D) (A : RootBox hab k) (R : ℝ) (J : Set ℝ), | |
| IsOpen J ∧ s ∈ Q.first ∧ K ⊆ Q.second ∧ I ⊆ J ∧ | |
| Q.second ⊆ V ∧ J ⊆ I₀ ∧ | |
| A.transverse = range (e ∘ Q.inclusion) ∧ A.time ⊆ rootLeafWindow R 0 ∧ | |
| ∀ w : Q.first × Q.second, (Homeomorph.addRight (τ (Q.inclusion w))) '' J ⊆ A.time := by | |
| let f : K → D := fun w => ⟨(s,w.val),hsD ⟨rfl,mem_univ _⟩⟩ | |
| have hf : Continuous f := | |
| (continuous_const.prodMk continuous_subtype_val).subtype_mk _ | |
| let : CompactSpace K := isCompact_iff_compactSpace.mp hK | |
| let P : Set D := range f | |
| have hP : IsCompact P := isCompact_range hf | |
| have hS : IsCompact (e '' P) := hP.image he.continuous | |
| have hi : Set.InjOn (rootQuotientParametrization hab) ((e '' P) ×ˢ univ) := by | |
| rintro ⟨x,t⟩ ⟨⟨z,⟨w,rfl⟩,rfl⟩,_⟩ ⟨y,u⟩ ⟨⟨z',⟨w',rfl⟩,rfl⟩,_⟩ h | |
| have H := hinj (x₁ := (f w,t)) (x₂ := (f w',u)) | |
| ⟨rfl,mem_univ _⟩ ⟨rfl,mem_univ _⟩ h | |
| exact congrArg (fun z : D × ℝ => (e z.1,z.2)) H | |
| let F : D × ℝ → RootSlice a b k × ℝ := fun z => (e z.1,z.2 + τ z.1) | |
| have hF : Continuous F := (he.continuous.comp continuous_fst).prodMk | |
| (continuous_snd.add (hτ.comp continuous_fst)) | |
| have hb : IsCompact ((fun z : D × ℝ => z.2 + τ z.1) '' (P ×ˢ I)) := | |
| (hP.prod hI).image (continuous_snd.add (hτ.comp continuous_fst)) | |
| obtain ⟨R,_,hR⟩ := hb.isBounded.exists_pos_norm_lt | |
| have hRI : ∀ z ∈ P ×ˢ I, z.2 + τ z.1 ∈ Ioo (-R) R := by | |
| intro z hz | |
| exact abs_lt.mp (by simpa only [Real.norm_eq_abs] using hR _ ⟨z,hz,rfl⟩) | |
| obtain ⟨A,hSA,hIA⟩ := rootBox_contains_compact_transverse hab (e '' P) hS hi | |
| (Icc (-R) R) isCompact_Icc | |
| let W := A.time ∩ Ioo (-R) R | |
| have hW : IsOpen W := A.time_open.inter isOpen_Ioo | |
| have hPW : P ×ˢ I ⊆ F ⁻¹' (A.transverse ×ˢ W) := by | |
| intro z hz | |
| exact ⟨hSA ⟨z.1,hz.1,rfl⟩,hIA (Ioo_subset_Icc_self (hRI z hz)),hRI z hz⟩ | |
| obtain ⟨Z,J,hZ,hJ,hPZ,hIJ,hZJ⟩ := generalized_tube_lemma hP hI | |
| ((A.transverse_open.prod hW).preimage hF) hPW | |
| have hZO : IsOpen ((Subtype.val '' Z) ∩ (univ ×ˢ V)) := | |
| (hD.isOpenEmbedding_subtypeVal.isOpenMap Z hZ).inter (isOpen_univ.prod hV) | |
| have hsKO : {s} ×ˢ K ⊆ (Subtype.val '' Z) ∩ (univ ×ˢ V) := by | |
| rintro ⟨u,w⟩ ⟨hu,hw⟩ | |
| have hu' : u=s := hu | |
| subst u | |
| exact ⟨⟨f ⟨w,hw⟩,hPZ ⟨⟨w,hw⟩,rfl⟩,rfl⟩,mem_univ _,hKV hw⟩ | |
| obtain ⟨U,V',hU,hV',hsU,hKV',hUV'⟩ := generalized_tube_lemma | |
| (isCompact_singleton : IsCompact ({s} : Set T)) hK hZO hsKO | |
| let Q : RootTransverseRectangle D := ⟨U,V',hU,hV',by | |
| intro z hz | |
| obtain ⟨v,_,hv⟩ := (hUV' hz).1 | |
| exact hv ▸ v.prop⟩ | |
| have hQZ : ∀ w : Q.first × Q.second, Q.inclusion w ∈ Z := by | |
| intro w | |
| obtain ⟨v,hv,heq⟩ := (hUV' | |
| (show (w.1.val,w.2.val) ∈ U ×ˢ V' from ⟨w.1.prop,w.2.prop⟩)).1 | |
| have hv' : v=Q.inclusion w := Subtype.ext heq | |
| exact hv' ▸ hv | |
| obtain ⟨t,ht⟩ := hne | |
| have hsub : range (e ∘ Q.inclusion) ⊆ A.transverse := by | |
| rintro _ ⟨w,rfl⟩ | |
| exact (hZJ (show (Q.inclusion w,t) ∈ Z ×ˢ J from ⟨hQZ w,hIJ ht⟩)).1 | |
| let A' := A.shrink (range (e ∘ Q.inclusion)) W | |
| (he.comp (Q.inclusion_isOpenEmbedding hD)).isOpen_range hW hsub inter_subset_left | |
| refine ⟨Q,A',R,J ∩ I₀,hJ.inter hI₀,hsU rfl,hKV', | |
| fun _ ht => ⟨hIJ ht,hII₀ ht⟩,?_,inter_subset_right,rfl,?_,?_⟩ | |
| · intro w hw | |
| exact (hUV' (show (s,w) ∈ U ×ˢ V' from ⟨hsU (Set.mem_singleton s),hw⟩)).2.2 | |
| · intro u hu | |
| simpa only [rootLeafWindow,Nat.cast_zero,add_zero] using hu.2 | |
| · intro w u hu | |
| obtain ⟨r,hr,rfl⟩ := hu | |
| exact (hZJ (show (Q.inclusion w,r) ∈ Z ×ˢ J from ⟨hQZ w,hr.1⟩)).2 | |
| end DukePrimeDegree | |
| end | |
| end OAI | |