openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /PolygonGeometry /ExteriorPhaseGeometry.lean
| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.ClippedGerms | |
| import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.FarClippingGeometry | |
| import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.LocalClippedBoxFormula | |
| namespace OAI | |
| section | |
| section | |
| open scoped symmDiff | |
| namespace SimpleAmenable | |
| open scoped commutatorElement | |
| open scoped commutatorElement | |
| section ExteriorPhaseGeometry | |
| namespace ClippedGerm | |
| variable {a : ℕ} {r : CutRing} {slope : Fin 4} {u : CutRing × CutRing} {z : ℝ × ℝ} | |
| (G : ClippedGerm a r slope u z) | |
| theorem gate_box_iff {ι : Type*} {C : ConcurrentGeometry a r} | |
| {j : ι → Fin 4} {c : ι → CutRing} (T : C.InwardChart j c z) | |
| (hbox : G.BoxValid) (lo hi : Fin 2 → ι) | |
| (hlo : ∀ k, j (lo k)=axisDirection k ∧ c (lo k)=pointCoordinate G.offset k-r) | |
| (hhi : ∀ k, j (hi k)=axisDirection k ∧ c (hi k)=pointCoordinate G.offset k+r) | |
| (σ : Fin 2 × Bool → Bool) (p : GenericSquare a) | |
| (hp : p ∈ (C.coordinateGate T.vertex T.offset z Prod.fst G.axialCut σ).val) | |
| (hpnear : dist p.val z<G.radius) | |
| (hlocal : ∀ i, p ∈ (C.localDecision T.vertex T.offset z (j i) (c i)).val ↔ | |
| p ∈ (cutPolygon a (j i) (c i)).val) : | |
| p ∈ (spatialTranslate u (coordinateRectangle a (fun _ => -r) (fun _ => r))).val ↔ | |
| ∀ k, σ (k,false)=false ∧ σ (k,true)=true := by | |
| classical | |
| rw [hbox p hpnear] | |
| apply forall_congr' | |
| intro k | |
| have hL := hlocal (lo k) | |
| have hH := hlocal (hi k) | |
| rw [(hlo k).1,(hlo k).2] at hL | |
| rw [(hhi k).1,(hhi k).2] at hH | |
| have hsL : decide (p ∈ (C.localDecision T.vertex T.offset z (axisDirection k) | |
| (pointCoordinate G.offset k-r)).val)=σ (k,false) := congrFun hp.2 (k,false) | |
| have hsH : decide (p ∈ (C.localDecision T.vertex T.offset z (axisDirection k) | |
| (pointCoordinate G.offset k+r)).val)=σ (k,true) := congrFun hp.2 (k,true) | |
| change (p ∈ (C.localDecision T.vertex T.offset z (axisDirection k) (pointCoordinate G.offset k-r)).val ↔ | |
| cutForm a (axisDirection k) p.val<ordinary (pointCoordinate G.offset k-r)) at hL | |
| change (p ∈ (C.localDecision T.vertex T.offset z (axisDirection k) (pointCoordinate G.offset k+r)).val ↔ | |
| cutForm a (axisDirection k) p.val<ordinary (pointCoordinate G.offset k+r)) at hH | |
| rw [cutForm_axis] at hL hH | |
| have hL' : σ (k,false)=false ↔ ordinary (pointCoordinate G.offset k-r)≤realCoordinate p.val k := by | |
| rw [← hsL,decide_eq_false_iff_not,hL,not_lt] | |
| have hH' : σ (k,true)=true ↔ realCoordinate p.val k<ordinary (pointCoordinate G.offset k+r) := by | |
| rw [← hsH,decide_eq_true_eq,hH] | |
| exact and_congr hL'.symm hH'.symm | |
| theorem exterior_old_cell {ι : Type*} {C : ConcurrentGeometry a r} | |
| {j : ι → Fin 4} {c : ι → CutRing} (T : C.InwardChart j c z) | |
| (hr : 0<ordinary r ∧ ordinary r<1/2) (hbox : G.BoxValid) | |
| (lo hi : Fin 2 → ι) | |
| (hlo : ∀ k, j (lo k)=axisDirection k ∧ c (lo k)=pointCoordinate G.offset k-r) | |
| (hhi : ∀ k, j (hi k)=axisDirection k ∧ c (hi k)=pointCoordinate G.offset k+r) | |
| (σ : Fin 2 × Bool → Bool) (hσ : ¬ ∀ k, σ (k,false)=false ∧ σ (k,true)=true) | |
| (p : GenericSquare a) | |
| (hp : p ∈ (C.coordinateGate T.vertex T.offset z Prod.fst G.axialCut σ).val) | |
| (hpnear : dist p.val z<G.radius) | |
| (hlocal : ∀ i, p ∈ (C.localDecision T.vertex T.offset z (j i) (c i)).val ↔ | |
| p ∈ (cutPolygon a (j i) (c i)).val) | |
| (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) | |
| (hinitial : p ∈ (spatialTranslate u (windowRectangle a n q cell)).val) : | |
| Disjoint (windowRectangle a n q cell) (coordinateRectangle a (fun _ => -r) (fun _ => r)) := by | |
| have hout : p ∉ (spatialTranslate u (coordinateRectangle a (fun _ => -r) (fun _ => r))).val := | |
| fun hh => hσ ((G.gate_box_iff T hbox lo hi hlo hhi σ p hp hpnear hlocal).mp hh) | |
| exact windowRectangle_exterior_of_point r hr n hn q cell hb he (translate a (-u) p) hinitial hout | |
| end ClippedGerm | |
| end ExteriorPhaseGeometry | |
| end SimpleAmenable | |
| end | |
| end | |
| end OAI | |