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0dd3af4 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 | import Mathlib
import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.VaryingCellClassification
import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.SupportedCellPatching
namespace OAI
section
section
open scoped symmDiff
namespace SimpleAmenable
open scoped commutatorElement
open scoped commutatorElement
section ConcurrentCellDecisions
namespace ConcurrentGeometry
variable {a : ℕ} {r : CutRing} (C : ConcurrentGeometry a r)
theorem cell_axis_eq (q : Fin 2 → ℤ) (cell : Fin 2 → Fin C.mesh)
(j : Fin 2) (v : CutRing × CutRing) (c : CutRing) (positive : Bool)
(hc : ∀ p ∈ (windowRectangle a C.mesh q cell).val,
(p ∈ (spatialTranslate v (coordinatePrimitive a j)).val ↔
if positive then ordinary c≤realCoordinate (windowPlanarLift q p) j
else realCoordinate (windowPlanarLift q p) j<ordinary c))
(t : VertexType (commonVertexDenominator a)) (u : CutRing × CutRing)
(haxis : integralCutForm a (axisDirection j) (u+C.anchors t (axisDirection j))=c)
(hmargin : ∀ p : ℝ × ℝ,
(∀ k, ordinary (windowCut C.mesh (q k) (cell k).castSucc)≤realCoordinate p k ∧
realCoordinate p k≤ordinary (windowCut C.mesh (q k) (cell k).succ)) →
∀ k, ordinary ((C.margins t).lower k+pointCoordinate u k)<realCoordinate p k ∧
realCoordinate p k<ordinary ((C.margins t).upper k+pointCoordinate u k))
(hnear : ∀ p : ℝ × ℝ,
(∀ k, ordinary ((C.margins t).lower k+pointCoordinate u k)≤realCoordinate p k ∧
realCoordinate p k≤ordinary ((C.margins t).upper k+pointCoordinate u k)) →
∀ k, |realCoordinate p k-ordinary (pointCoordinate (u+C.anchors t (axisDirection j)) k)|<3*C.epsilon) :
spatialTranslate v (coordinatePrimitive a j) ⊓ windowRectangle a C.mesh q cell =
(if positive then spatialTranslate (u+C.anchors t (axisDirection j)) (coordinatePrimitive a j)
else (spatialTranslate (u+C.anchors t (axisDirection j)) (coordinatePrimitive a j))ᶜ) ⊓
windowRectangle a C.mesh q cell := by
apply Subtype.ext
ext p
change (p ∈ (spatialTranslate v (coordinatePrimitive a j)).val ∧
p ∈ (windowRectangle a C.mesh q cell).val) ↔ _
by_cases hp : p ∈ (windowRectangle a C.mesh q cell).val
· have hz := hmargin (windowPlanarLift q p) (fun k =>
⟨((windowRectangle_mem C.mesh C.mesh_large q cell p).mp hp k).1,
((windowRectangle_mem C.mesh C.mesh_large q cell p).mp hp k).2.le⟩)
have hh := C.axis_on_margin j u (u+C.anchors t (axisDirection j)) t hnear p
(windowPlanarLift q p) (windowPlanarLift_spec q p) (fun k => ⟨(hz k).1.le,(hz k).2.le⟩)
rw [integralCutForm_axis] at haxis
rw [haxis] at hh
cases positive <;> simp only [Bool.false_eq_true,↓reduceIte] at hc ⊢
· change (_ ∧ _) ↔ (¬ p ∈ (spatialTranslate (u+C.anchors t (axisDirection j)) (coordinatePrimitive a j)).val ∧ _)
exact and_congr ((hc p hp).trans ((not_le).symm.trans (not_congr hh).symm)) Iff.rfl
· change (_ ∧ _) ↔ (p ∈ (spatialTranslate (u+C.anchors t (axisDirection j)) (coordinatePrimitive a j)).val ∧ _)
exact and_congr ((hc p hp).trans hh.symm) Iff.rfl
· change (_ ∧ _) ↔ (_ ∧ _)
simp only [hp,and_false]
end ConcurrentGeometry
end ConcurrentCellDecisions
end SimpleAmenable
end
end
end OAI
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