openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /PolygonGeometry /ClosedChartOverlap.lean
| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.RectangularBound | |
| namespace OAI | |
| section | |
| section | |
| open scoped symmDiff | |
| namespace SimpleAmenable | |
| open scoped commutatorElement | |
| open scoped commutatorElement | |
| section ClosedChartOverlap | |
| theorem planar_lift_avoidsCuts {a : ℕ} (p : GenericSquare a) (t : ℝ × ℝ) | |
| (ht : p.val = (Int.fract t.1,Int.fract t.2)) : AvoidsCuts a t := by | |
| have h := avoidsCuts_translate p.property.2.2 | |
| ((⌊t.1⌋ : CutRing),(⌊t.2⌋ : CutRing)) | |
| have he : p.val+(ordinary (⌊t.1⌋ : CutRing),ordinary (⌊t.2⌋ : CutRing)) = t := by | |
| rw [ht] | |
| apply Prod.ext | |
| · change Int.fract t.1+ordinary (⌊t.1⌋ : CutRing) = t.1 | |
| rw [map_intCast] | |
| exact sub_add_cancel _ _ | |
| · change Int.fract t.2+ordinary (⌊t.2⌋ : CutRing) = t.2 | |
| rw [map_intCast] | |
| exact sub_add_cancel _ _ | |
| rwa [he] at h | |
| theorem coordinateRectangle_strict_lift {a : ℕ} (l h : Fin 2 → CutRing) | |
| (hlh : ∀ j, ordinary (l j) ≤ ordinary (h j)) | |
| (hlen : ∀ j, ordinary (h j)-ordinary (l j) < 1) | |
| (p : GenericSquare a) (hp : p ∈ (coordinateRectangle a l h).val) : | |
| ∃ t : ℝ × ℝ, p.val = (Int.fract t.1,Int.fract t.2) ∧ | |
| (ordinary (l 0) < t.1 ∧ t.1 < ordinary (h 0)) ∧ | |
| (ordinary (l 1) < t.2 ∧ t.2 < ordinary (h 1)) := by | |
| obtain ⟨t,ht,hx,hy⟩ := coordinateRectangle_lift l h hlh hlen p hp | |
| have hx' := planar_lift_avoidsCuts p t ht 0 (l 0) | |
| have hy' := planar_lift_avoidsCuts p t ht 1 (l 1) | |
| simp only [cutForm,Matrix.cons_val_zero,Matrix.cons_val_one] at hx' hy' | |
| exact ⟨t,ht,⟨lt_of_le_of_ne hx.1 (Ne.symm hx'),hx.2⟩, | |
| ⟨lt_of_le_of_ne hy.1 (Ne.symm hy'),hy.2⟩⟩ | |
| theorem clippedSlope_agree_on_closed_rectangle {a : ℕ} (r : CutRing) (j : Fin 4) | |
| (hr : 0 < ordinary r ∧ ordinary r < 1/2) | |
| (u v : CutRing × CutRing) (l h : Fin 2 → CutRing) | |
| (hlh : ∀ k, ordinary (l k) ≤ ordinary (h k)) | |
| (hlen : ∀ k, ordinary (h k)-ordinary (l k) < 1) | |
| (hu : ∀ k, -ordinary r ≤ ordinary (l k)-ordinary (pointCoordinate u k) ∧ | |
| ordinary (h k)-ordinary (pointCoordinate u k) ≤ ordinary r) | |
| (hv : ∀ k, -ordinary r ≤ ordinary (l k)-ordinary (pointCoordinate v k) ∧ | |
| ordinary (h k)-ordinary (pointCoordinate v k) ≤ ordinary r) | |
| (hline : integralCutForm a j u = integralCutForm a j v) : | |
| spatialTranslate u (clippedSlopePrimitive a r j) ⊓ coordinateRectangle a l h = | |
| spatialTranslate v (clippedSlopePrimitive a r j) ⊓ coordinateRectangle a l h := by | |
| apply Subtype.ext | |
| ext p | |
| change (p ∈ (spatialTranslate u (clippedSlopePrimitive a r j)).val ∧ | |
| p ∈ (coordinateRectangle a l h).val) ↔ | |
| (p ∈ (spatialTranslate v (clippedSlopePrimitive a r j)).val ∧ | |
| p ∈ (coordinateRectangle a l h).val) | |
| by_cases hp : p ∈ (coordinateRectangle a l h).val | |
| · obtain ⟨t,ht,hx,hy⟩ := coordinateRectangle_strict_lift l h hlh hlen p hp | |
| have hbox (w : CutRing × CutRing) | |
| (hw : ∀ k : Fin 2, -ordinary r ≤ ordinary (l k)-ordinary (pointCoordinate w k) ∧ | |
| ordinary (h k)-ordinary (pointCoordinate w k) ≤ ordinary r) : | |
| (-ordinary r < t.1-ordinary w.1 ∧ t.1-ordinary w.1 < ordinary r) ∧ | |
| (-ordinary r < t.2-ordinary w.2 ∧ t.2-ordinary w.2 < ordinary r) := by | |
| have hw0 := hw 0 | |
| have hw1 := hw 1 | |
| simp only [pointCoordinate,ite_true,Fin.isValue,show (1:Fin 2) ≠ 0 by decide,ite_false] at hw0 hw1 | |
| constructor <;> constructor <;> linarith | |
| rw [translated_clippedSlope_lift r j hr u p t ht (hbox u hu), | |
| translated_clippedSlope_lift r j hr v p t ht (hbox v hv),hline] | |
| · simp only [hp,and_false] | |
| end ClosedChartOverlap | |
| section FullTangentOverlap | |
| noncomputable def orderedCutMax (x y : CutRing) : CutRing := | |
| if ordinary x ≤ ordinary y then y else x | |
| noncomputable def orderedCutMin (x y : CutRing) : CutRing := | |
| if ordinary x ≤ ordinary y then x else y | |
| @[simp] theorem ordinary_orderedCutMax (x y : CutRing) : | |
| ordinary (orderedCutMax x y) = max (ordinary x) (ordinary y) := by | |
| unfold orderedCutMax | |
| split_ifs with h | |
| · exact (max_eq_right h).symm | |
| · exact (max_eq_left (le_of_not_ge h)).symm | |
| @[simp] theorem ordinary_orderedCutMin (x y : CutRing) : | |
| ordinary (orderedCutMin x y) = min (ordinary x) (ordinary y) := by | |
| unfold orderedCutMin | |
| split_ifs with h | |
| · exact (min_eq_left h).symm | |
| · exact (min_eq_right (le_of_not_ge h)).symm | |
| noncomputable def clippingOverlapLower (r : CutRing) (t : CutRing × CutRing) (j : Fin 2) : CutRing := | |
| orderedCutMax (-r) (pointCoordinate t j-r) | |
| noncomputable def clippingOverlapUpper (r : CutRing) (t : CutRing × CutRing) (j : Fin 2) : CutRing := | |
| orderedCutMin r (pointCoordinate t j+r) | |
| theorem clippingOverlap_bounds (r : CutRing) (t : CutRing × CutRing) | |
| (hr : 0 < ordinary r ∧ ordinary r < 1/2) | |
| (ht : ∀ j, |ordinary (pointCoordinate t j)| ≤ 2*ordinary r) (j : Fin 2) : | |
| ordinary (clippingOverlapLower r t j) ≤ ordinary (clippingOverlapUpper r t j) ∧ | |
| ordinary (clippingOverlapUpper r t j)-ordinary (clippingOverlapLower r t j) < 1 ∧ | |
| (-ordinary r ≤ ordinary (clippingOverlapLower r t j) ∧ | |
| ordinary (clippingOverlapUpper r t j) ≤ ordinary r) ∧ | |
| (-ordinary r ≤ ordinary (clippingOverlapLower r t j)-ordinary (pointCoordinate t j) ∧ | |
| ordinary (clippingOverlapUpper r t j)-ordinary (pointCoordinate t j) ≤ ordinary r) := by | |
| have hh := abs_le.mp (ht j) | |
| simp only [clippingOverlapLower,clippingOverlapUpper,ordinary_orderedCutMax, | |
| ordinary_orderedCutMin,map_sub,map_neg,map_add] | |
| have h1 := le_max_left (-ordinary r) (ordinary (pointCoordinate t j)-ordinary r) | |
| have h2 := le_max_right (-ordinary r) (ordinary (pointCoordinate t j)-ordinary r) | |
| have h3 := min_le_left (ordinary r) (ordinary (pointCoordinate t j)+ordinary r) | |
| have h4 := min_le_right (ordinary r) (ordinary (pointCoordinate t j)+ordinary r) | |
| refine ⟨max_le (le_min (by linarith [hr.1]) (by linarith)) | |
| (le_min (by linarith) (by linarith [hr.1])),by linarith [hr.2],⟨h1,h3⟩,?_,?_⟩ <;> linarith | |
| theorem exists_full_overlap_window (a : ℕ) (r u : CutRing) : | |
| ∃ k : ℕ, ∃ p : Fin 2 → ℤ, ∀ d : Fin 2, ∀ e : Fin 5, ∀ j : Fin 2, | |
| (p j ≤ endpointLabel (clippingOverlapLower r (tangentOffset a d (signedShortSteps u e)) j) ∧ | |
| endpointLabel (clippingOverlapLower r (tangentOffset a d (signedShortSteps u e)) j) < p j+k) ∧ | |
| (p j ≤ endpointLabel (clippingOverlapUpper r (tangentOffset a d (signedShortSteps u e)) j) ∧ | |
| endpointLabel (clippingOverlapUpper r (tangentOffset a d (signedShortSteps u e)) j) < p j+k) := by | |
| let q (i : Fin 2 × Fin 5 × Bool) (j : Fin 2) : ℤ := | |
| endpointLabel (if i.2.2 then clippingOverlapUpper r (tangentOffset a i.1 (signedShortSteps u i.2.1)) j | |
| else clippingOverlapLower r (tangentOffset a i.1 (signedShortSteps u i.2.1)) j) | |
| obtain ⟨k,p,hp⟩ := finite_coordinate_windows_container (fun _ : Fin 2 × Fin 5 × Bool => 1) q | |
| refine ⟨k,p,fun d e j => ?_⟩ | |
| have hL := hp (d,e,false) j | |
| have hU := hp (d,e,true) j | |
| dsimp [q] at hL hU | |
| omega | |
| namespace InitialCoverSystem | |
| variable {a m M : ℕ} {r : CutRing} {hm : 2 ≤ m} | |
| (B : InitialCoverSystem a r m hm M) | |
| [Group.IsPerfect (alternatingGroup (Fin (m+1)))] | |
| theorem tangentSign_step_action_closed (hlarge : 20 ≤ m+1) | |
| (hr : 0 < ordinary r ∧ ordinary r < 1/2) | |
| (k : ℕ) (p : Fin 2 → ℤ) (u : CutRing) (h : B.TangentChartLaws k p u) | |
| (d : Fin 2) (e : Fin 5) (z : CutRing × CutRing) | |
| (l v : Fin 2 → CutRing) | |
| (hl : ∀ j, p j ≤ endpointLabel (l j) ∧ endpointLabel (l j) < p j+k) | |
| (hv : ∀ j, p j ≤ endpointLabel (v j) ∧ endpointLabel (v j) < p j+k) | |
| (hlv : ∀ j, ordinary (l j) ≤ ordinary (v j)) | |
| (hlen : ∀ j, ordinary (v j)-ordinary (l j) < 1) | |
| (hbox : ∀ j, -ordinary r ≤ ordinary (l j) ∧ ordinary (v j) ≤ ordinary r) | |
| (hbox' : ∀ j, -ordinary r ≤ ordinary (l j)-ordinary | |
| (pointCoordinate (tangentOffset a d (signedShortSteps u e)) j) ∧ | |
| ordinary (v j)-ordinary | |
| (pointCoordinate (tangentOffset a d (signedShortSteps u e)) j) ≤ ordinary r) | |
| (f : TrackStar (Fin (m+1)) →* BoundedRelationCover M (alternatingGenerator a r m hm)) | |
| (hf : B.AlignedSmallSupported f) | |
| (hc : SmallControlled B.c f (B.fullGeometricSector (by omega) | |
| (translatedTemplate (tangentChartTemplate a k p d (signedShortSteps u e)) z) | |
| (h d e z) (spatialTranslate z (coordinateRectangle a l v)))) | |
| (I : ControlAlphabet (Fin (m+1))) (s : UniversalExtension (alternatingGroup I.val)) : | |
| ∀ x ∈ f.range, | |
| B.tangentSign (by omega) k p u h d z true (universalMap (subtypeAlternatingHom I.val) s)*x* | |
| (B.tangentSign (by omega) k p u h d z true (universalMap (subtypeAlternatingHom I.val) s))⁻¹ = | |
| B.tangentSign (by omega) k p u h d (z+tangentOffset a d (signedShortSteps u e)) true | |
| (universalMap (subtypeAlternatingHom I.val) s)*x* | |
| (B.tangentSign (by omega) k p u h d (z+tangentOffset a d (signedShortSteps u e)) true | |
| (universalMap (subtypeAlternatingHom I.val) s))⁻¹ := by | |
| rw [B.tangentSign_first (by omega) k p u h d z e true, | |
| B.tangentSign_second (by omega) k p u h d z e true] | |
| apply B.chart_action_transfer hlarge _ (h d e z) _ _ _ | |
| (tangentChart_first_resolved p d _ z) (tangentChart_second_resolved p d _ z) | |
| (tangentChart_rectangle_resolved p d _ z l v hl hv) _ f hf hc I s | |
| have he := clippedSlope_agree_on_closed_rectangle (a := a) r (slopeDirection d) hr | |
| (0 : CutRing × CutRing) (tangentOffset a d (signedShortSteps u e)) l v hlv hlen | |
| (by simpa [pointCoordinate] using hbox) hbox' | |
| (by simp only [integralCutForm_zero,integralCutForm_tangentOffset]) | |
| have he' := congrArg (spatialTranslate z) he | |
| simpa only [spatialTranslate_inter,← spatialTranslate_add,add_zero] using he' | |
| end InitialCoverSystem | |
| end FullTangentOverlap | |
| end SimpleAmenable | |
| end | |
| end | |
| end OAI | |