openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /PolygonGeometry /FarClippingGeometry.lean
| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.FinePatchAtlas | |
| namespace OAI | |
| section | |
| section | |
| open scoped symmDiff | |
| namespace SimpleAmenable | |
| open scoped commutatorElement | |
| open scoped commutatorElement | |
| section FarClippingGeometry | |
| theorem windowRectangle_clipping_chart_at_lift {a : ℕ} (r : CutRing) | |
| (n : ℕ) (hn : 201≤n) (q : Fin 2 → ℤ) (cell : Fin 2 → Fin n) | |
| (hb : ∀ d, q d≤endpointLabel (-r) ∧ endpointLabel (-r)<q d+n) | |
| (he : ∀ d, q d≤endpointLabel r ∧ endpointLabel r<q d+n) | |
| (p : GenericSquare a) (hp : p ∈ (windowRectangle a n q cell).val) | |
| (z : ℝ × ℝ) (hz : p.val=(Int.fract z.1,Int.fract z.2)) | |
| (hclip : ∀ d, -ordinary r≤realCoordinate z d ∧ realCoordinate z d<ordinary r) : | |
| ∃ k : Fin 2 → ℤ, | |
| (∀ d, -ordinary r≤ordinary (windowCut n (q d) (cell d).castSucc)+(k d:ℝ) ∧ | |
| ordinary (windowCut n (q d) (cell d).succ)+(k d:ℝ)≤ordinary r) ∧ | |
| windowPlanarLift q p+((k 0:ℝ),(k 1:ℝ))=z := by | |
| have hs := windowPlanarLift_spec q p | |
| have hd (d : Fin 2) : ∃ k : ℤ, realCoordinate z d-realCoordinate (windowPlanarLift q p) d=(k:ℝ) := by | |
| apply Int.fract_eq_fract.mp | |
| fin_cases d | |
| · exact (congrArg Prod.fst hz).symm.trans (congrArg Prod.fst hs) | |
| · exact (congrArg Prod.snd hz).symm.trans (congrArg Prod.snd hs) | |
| choose k hk using hd | |
| refine ⟨k,?_,?_⟩ | |
| · intro d | |
| have h := window_cell_enclosure n (q d) (cell d) (-r) r (hb d) (he d) | |
| (realCoordinate (windowPlanarLift q p) d) (k d) | |
| ((windowRectangle_mem n hn q cell p).mp hp d) | |
| (by simp only [map_neg]; constructor <;> linarith [(hk d),(hclip d).1,(hclip d).2]) | |
| simpa only [map_neg] using h | |
| · apply Prod.ext | |
| · change (windowPlanarLift q p).1+(k 0:ℝ)=z.1 | |
| have h := hk 0 | |
| change z.1-(windowPlanarLift q p).1=(k 0:ℝ) at h | |
| linarith | |
| · change (windowPlanarLift q p).2+(k 1:ℝ)=z.2 | |
| have h := hk 1 | |
| change z.2-(windowPlanarLift q p).2=(k 1:ℝ) at h | |
| linarith | |
| theorem far_cut_neighborhood (a : ℕ) (j : Fin 4) (z : ℝ × ℝ) (c R E : ℝ) | |
| (hR : 0<R) (hE : E<R/4) (hfar : R≤|cutForm a j z-c|) : | |
| ∃ δ : ℝ, 0<δ ∧ ∀ p : ℝ × ℝ, dist p z<δ → | |
| E < |cutForm a j p-c| ∧ (0≤cutForm a j p-c ↔ c<cutForm a j z) := by | |
| let K := 1+|ordinary (cutTau^a)| | |
| have hK : 0<K := by dsimp [K]; positivity | |
| refine ⟨R/(4*K),by positivity,?_⟩ | |
| intro p hp | |
| have hdist : K*dist p z<R/4 := by | |
| have hh := (lt_div_iff₀ (by positivity : 0<4*K)).mp hp | |
| nlinarith | |
| have hd : |cutForm a j p-cutForm a j z|<R/4 := | |
| (cutForm_dist_bound a j p z).trans_lt hdist | |
| rcases le_total c (cutForm a j z) with hz | hz | |
| · rw [abs_of_nonneg (sub_nonneg.mpr hz)] at hfar | |
| have hs : 0<cutForm a j p-c := by linarith [(abs_lt.mp hd).1] | |
| refine ⟨?_,iff_of_true hs.le (by linarith)⟩ | |
| rw [abs_of_pos hs] | |
| linarith [(abs_lt.mp hd).1] | |
| · rw [abs_of_nonpos (sub_nonpos.mpr hz)] at hfar | |
| have hs : cutForm a j p-c<0 := by linarith [(abs_lt.mp hd).2] | |
| refine ⟨?_,iff_of_false (not_le_of_gt hs) (not_lt_of_ge hz)⟩ | |
| rw [abs_of_neg hs] | |
| linarith [(abs_lt.mp hd).2] | |
| end FarClippingGeometry | |
| section WindowBoxDichotomy | |
| theorem windowRectangle_le_clipping_box {a : ℕ} (r : CutRing) | |
| (hr : 0<ordinary r ∧ ordinary r<1/2) | |
| (n : ℕ) (hn : 201≤n) (q : Fin 2 → ℤ) (cell : Fin 2 → Fin n) | |
| (k : Fin 2 → ℤ) | |
| (hk : ∀ d, -ordinary r≤ordinary (windowCut n (q d) (cell d).castSucc)+(k d:ℝ) ∧ | |
| ordinary (windowCut n (q d) (cell d).succ)+(k d:ℝ)≤ordinary r) : | |
| windowRectangle a n q cell ≤ coordinateRectangle a (fun _ => -r) (fun _ => r) := by | |
| intro p hp | |
| have hb (d : Fin 2) : p ∈ coordinateBetween a d (-r) r := by | |
| apply (mem_coordinateBetween_iff d (-r) r (by rw [map_neg]; linarith [hr.1]) | |
| (by rw [map_neg]; linarith [hr.2]) p).mpr | |
| refine ⟨k d-⌊coordinate d p-ordinary ((q d:CutRing)*cutTau)⌋,?_⟩ | |
| have hh := (windowRectangle_mem n hn q cell p).mp hp d | |
| simp only [windowPlanarLift_coordinate,liftedCoordinate] at hh | |
| simp only [map_neg,Int.cast_sub] | |
| constructor <;> linarith [(hk d).1,(hk d).2,hh.1,hh.2] | |
| exact ⟨hb 0,hb 1⟩ | |
| theorem windowRectangle_clipping_dichotomy {a : ℕ} (r : CutRing) | |
| (hr : 0<ordinary r ∧ ordinary r<1/2) | |
| (n : ℕ) (hn : 201≤n) (q : Fin 2 → ℤ) (cell : Fin 2 → Fin n) | |
| (hb : ∀ d, q d≤endpointLabel (-r) ∧ endpointLabel (-r)<q d+n) | |
| (he : ∀ d, q d≤endpointLabel r ∧ endpointLabel r<q d+n) : | |
| Disjoint (windowRectangle a n q cell) (coordinateRectangle a (fun _ => -r) (fun _ => r)) ∨ | |
| windowRectangle a n q cell ≤ coordinateRectangle a (fun _ => -r) (fun _ => r) := by | |
| classical | |
| by_cases h : ∃ p : GenericSquare a, p ∈ (windowRectangle a n q cell).val ∧ | |
| p ∈ (coordinateRectangle a (fun _ => -r) (fun _ => r)).val | |
| · obtain ⟨p,hp,hbox⟩ := h | |
| obtain ⟨k,hk⟩ := windowRectangle_clipping_chart r hr n hn q cell hb he p hp hbox | |
| exact Or.inr (windowRectangle_le_clipping_box r hr n hn q cell k hk) | |
| · left | |
| rw [disjoint_iff_inf_le] | |
| exact fun p hp => h ⟨p,hp.1,hp.2⟩ | |
| theorem windowRectangle_exterior_of_point {a : ℕ} (r : CutRing) | |
| (hr : 0<ordinary r ∧ ordinary r<1/2) | |
| (n : ℕ) (hn : 201≤n) (q : Fin 2 → ℤ) (cell : Fin 2 → Fin n) | |
| (hb : ∀ d, q d≤endpointLabel (-r) ∧ endpointLabel (-r)<q d+n) | |
| (he : ∀ d, q d≤endpointLabel r ∧ endpointLabel r<q d+n) | |
| (p : GenericSquare a) (hp : p ∈ (windowRectangle a n q cell).val) | |
| (hout : p ∉ (coordinateRectangle a (fun _ => -r) (fun _ => r)).val) : | |
| Disjoint (windowRectangle a n q cell) (coordinateRectangle a (fun _ => -r) (fun _ => r)) := by | |
| rcases windowRectangle_clipping_dichotomy r hr n hn q cell hb he with h | h | |
| · exact h | |
| · exact False.elim (hout (h hp)) | |
| end WindowBoxDichotomy | |
| end SimpleAmenable | |
| end | |
| end | |
| end OAI | |