openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /PolygonGeometry /InactiveQuadrantAction.lean
| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.RectangularAtlas | |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.GlobalSignControl | |
| namespace OAI | |
| open scoped symmDiff | |
| namespace SimpleAmenable | |
| open scoped commutatorElement | |
| theorem signQuadrant_bounds (s : CutRing) (hs : 0 ≤ ordinary s) | |
| (d : Fin 2) (positive : Bool) (j : Fin 2) : | |
| -ordinary s ≤ ordinary (signQuadrantLower s d positive j) ∧ | |
| ordinary (signQuadrantUpper s d positive j) ≤ ordinary s := by | |
| cases positive <;> by_cases h : j=d <;> | |
| simp [signQuadrantLower,signQuadrantUpper,h] <;> linarith | |
| namespace InitialCoverSystem | |
| namespace RectangularAtlas | |
| variable {a m M : ℕ} {r : CutRing} {hm : 2 ≤ m} | |
| {B : InitialCoverSystem a r m hm M} | |
| [Group.IsPerfect (alternatingGroup (Fin (m+1)))] (A : B.RectangularAtlas) | |
| theorem inactive_quadrant_action (hlarge : 20 ≤ m+1) | |
| (hr : 0 < ordinary r ∧ ordinary r < 1/2) | |
| (d : Fin 2) (z z' : CutRing × CutRing) (positive : Bool) | |
| (hline : integralCutForm a (slopeDirection d) z=integralCutForm a (slopeDirection d) z') | |
| (L V : Fin 2 → CutRing) (hLV : ∀ j, ordinary (L j) ≤ ordinary (V j)) | |
| (hz : ∀ j, -ordinary r ≤ ordinary (L j)-ordinary (pointCoordinate z j) ∧ | |
| ordinary (V j)-ordinary (pointCoordinate z j) ≤ ordinary r) | |
| (hlo : ∀ j, ordinary (pointCoordinate z' j)+ordinary (signQuadrantLower A.radius d positive j) ≤ ordinary (L j)) | |
| (hup : ∀ j, ordinary (V j) ≤ ordinary (pointCoordinate z' j)+ordinary (signQuadrantUpper A.radius d positive j)) | |
| (n : ℕ) (q : Fin 2 → ℤ) | |
| (hW : ResolvedBy (fun i => (primitiveTests (a := a) (r := r) | |
| (coordinateWindowPrimitives n q) i).val) (coordinateRectangle a L V).val) | |
| (f : TrackStar (Fin (m+1)) →* BoundedRelationCover M (alternatingGenerator a r m hm)) | |
| (hf : B.AlignedSmallSupported f) | |
| (hc : SmallControlled B.c f (B.windowSector (by omega) n (A.rectangles n) q (coordinateRectangle a L V))) | |
| (I : ControlAlphabet (Fin (m+1))) (s : UniversalExtension (alternatingGroup I.val)) | |
| (t : BoundedRelationCover M (alternatingGenerator a r m hm)) (ht : t ∈ f.range) : | |
| A.slope (by omega) d z (universalMap (subtypeAlternatingHom I.val) s)*t* | |
| (A.slope (by omega) d z (universalMap (subtypeAlternatingHom I.val) s))⁻¹ = | |
| (if positive then B.c (subtypeAlternatingHom I.val (universalProjection (alternatingGroup I.val) s)) else 1)*t* | |
| (if positive then B.c (subtypeAlternatingHom I.val (universalProjection (alternatingGroup I.val) s)) else 1)⁻¹ := by | |
| have hbound (j : Fin 2) := signQuadrant_bounds A.radius A.radius_pos.le d positive j | |
| have hdiff (j : Fin 2) : | |
| |ordinary (L j)-ordinary (pointCoordinate z' j)| ≤ ordinary A.radius ∧ | |
| |ordinary (V j)-ordinary (pointCoordinate z' j)| ≤ ordinary A.radius := by | |
| have hb := hbound j | |
| have hl := hlo j | |
| have hu := hup j | |
| have hlv := hLV j | |
| constructor <;> apply abs_le.mpr <;> constructor <;> linarith | |
| have hz' (j : Fin 2) : -ordinary r ≤ ordinary (L j)-ordinary (pointCoordinate z' j) ∧ | |
| ordinary (V j)-ordinary (pointCoordinate z' j) ≤ ordinary r := by | |
| have hh := hdiff j | |
| have h₁ := (abs_le.mp hh.1).1 | |
| have h₂ := (abs_le.mp hh.2).2 | |
| constructor <;> linarith [A.radius_small] | |
| have he := A.slope_line_action hlarge hr d z z' hline L V hLV hz hz' | |
| (ordinary z'.1,ordinary z'.2) (by rw [cutForm_ordinary,hline]) | |
| (fun j => by | |
| have hh := hdiff j | |
| have hh' : realCoordinate (ordinary z'.1,ordinary z'.2) j=ordinary (pointCoordinate z' j) := by | |
| fin_cases j <;> rfl | |
| rw [hh'] | |
| constructor <;> linarith [A.radius_small]) n q hW f hf hc I s t ht | |
| rw [he] | |
| have hlab (j : Fin 2) : | |
| symmetricWindowStart A.radius j ≤ endpointLabel (signQuadrantLower A.radius d positive j) ∧ | |
| endpointLabel (signQuadrantLower A.radius d positive j) < | |
| symmetricWindowStart A.radius j+symmetricWindowLength A.radius := by | |
| unfold signQuadrantLower | |
| split <;> split <;> first | exact (symmetricWindow_labels A.radius j).1 | exact (symmetricWindow_labels A.radius j).2.1 | |
| have hvab (j : Fin 2) : | |
| symmetricWindowStart A.radius j ≤ endpointLabel (signQuadrantUpper A.radius d positive j) ∧ | |
| endpointLabel (signQuadrantUpper A.radius d positive j) < | |
| symmetricWindowStart A.radius j+symmetricWindowLength A.radius := by | |
| unfold signQuadrantUpper | |
| split <;> split <;> first | exact (symmetricWindow_labels A.radius j).2.1 | exact (symmetricWindow_labels A.radius j).2.2 | |
| have hside := signQuadrant_side (a := a) r A.radius hr | |
| ⟨A.radius_pos,by linarith [A.radius_small]⟩ d positive | |
| have hinc : coordinateRectangle a L V ≤ spatialTranslate z' | |
| (coordinateRectangle a (signQuadrantLower A.radius d positive) (signQuadrantUpper A.radius d positive)) := by | |
| rw [spatialTranslate_coordinateRectangle] | |
| apply coordinateRectangle_mono L V _ _ | |
| · intro j; simpa only [map_add,pointCoordinate,add_comm] using hlo j | |
| · exact hLV | |
| · intro j; simpa only [map_add,pointCoordinate,add_comm] using hup j | |
| · intro j | |
| have hb := hbound j | |
| simp only [map_add,add_sub_add_right_eq_sub] | |
| linarith [A.radius_small,hr.2] | |
| have hc' := B.all_rectangle_control_to_sign hlarge A.rectangles _ _ A.u A.quadrants | |
| d z' positive _ _ hlab hvab hside n q _ hW hinc f hf hc | |
| have ht' := B.tangentSign_constant_action hlarge _ _ A.u A.quadrants d z' positive f hf hc' I s t ht | |
| have hs := B.tangentSign_eq (by omega) A.k _ A.p _ A.u A.u A.charts A.quadrants d z' true | |
| simpa only [slope,hs] using ht' | |
| end RectangularAtlas | |
| end InitialCoverSystem | |
| end SimpleAmenable | |
| end OAI | |