File size: 3,049 Bytes
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 67 68 69 70 71 72 | import Mathlib
import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.WindowCellPartition
namespace OAI
section
section
open scoped symmDiff
namespace SimpleAmenable
open scoped commutatorElement
open scoped commutatorElement
section CellGeometry
theorem coordinateInterval_union {a : ℕ} (j : Fin 2) (u v w : CutRing)
(huv : ordinary u ≤ ordinary v) (hvw : ordinary v ≤ ordinary w)
(hlen : ordinary w-ordinary u < 1) :
coordinateInterval a j u w = coordinateInterval a j u v ⊔ coordinateInterval a j v w := by
apply Subtype.ext
ext p
change p ∈ coordinateBetween a j u w ↔ p ∈ coordinateBetween a j u v ∨ p ∈ coordinateBetween a j v w
rw [mem_coordinateBetween_iff j u w (huv.trans hvw) hlen,
mem_coordinateBetween_iff j u v huv (by linarith),
mem_coordinateBetween_iff j v w hvw (by linarith)]
constructor
· rintro ⟨k,hk,hk'⟩
by_cases hh : coordinate j p+(k:ℝ) < ordinary v
· exact Or.inl ⟨k,hk,hh⟩
· exact Or.inr ⟨k,le_of_not_gt hh,hk'⟩
· rintro (⟨k,hk,hk'⟩|⟨k,hk,hk'⟩)
· exact ⟨k,hk,hk'.trans_le hvw⟩
· exact ⟨k,huv.trans hk,hk'⟩
theorem coordinateInterval_nonempty {a : ℕ} (j : Fin 2) (u v : CutRing)
(huv : ordinary u < ordinary v) (hlen : ordinary v-ordinary u < 1) :
(coordinateInterval a j u v).val.Nonempty := by
have hd : 0 < ordinary v-ordinary u := sub_pos.mpr huv
have hp : ∃ p : GenericSquare a, coordinate j p < ordinary v-ordinary u := by
fin_cases j
· obtain ⟨p,hp,_⟩ := generic_rectangle a 0 (ordinary v-ordinary u) 0 1
(by norm_num) hlen.le hd (by norm_num) (by norm_num) (by norm_num)
exact ⟨p,hp.2⟩
· obtain ⟨p,_,hp⟩ := generic_rectangle a 0 1 0 (ordinary v-ordinary u)
(by norm_num) (by norm_num) (by norm_num) (by norm_num) hlen.le hd
exact ⟨p,hp.2⟩
obtain ⟨p,hp⟩ := hp
refine ⟨translate a (coordinateShift j u) p,?_⟩
change translate a (-coordinateShift j u) (translate a (coordinateShift j u) p) ∈ coordinateArc a j (v-u)
rw [← translate_add,neg_add_cancel,translate_zero]
change coordinate j p < Int.fract (ordinary (v-u))
rw [map_sub,Int.fract_eq_self.mpr ⟨hd.le,hlen⟩]
exact hp
theorem windowCell_nonempty {a : ℕ} (j : Fin 2) (n : ℕ) (hn : 201 ≤ n) (q : ℤ) (i : Fin n) :
(windowCell a j n q i).val.Nonempty := by
apply coordinateInterval_nonempty
· exact windowCut_strictMono n q (Fin.castSucc_lt_succ (i := i))
· have hn' : (200:ℝ) < n := by exact_mod_cast (by omega : 200 < n)
exact (windowCut_mesh n q i).trans_lt ((div_lt_one (by linarith)).mpr hn')
theorem windowCell_axis_resolved {a : ℕ} {r : CutRing} (j : Fin 2) (n : ℕ) (hn : 0 < n) (q : ℤ) (i : Fin n) :
ResolvedBy (fun z => (InitialCoverSystem.primitiveTests (a := a) (r := r)
(axisWindowPrimitives j n q) z).val) (windowCell a j n q i).val :=
axisInterval_resolved j n q _ _ (windowCut_label n q hn i.castSucc) (windowCut_label n q hn i.succ)
end CellGeometry
end SimpleAmenable
end
end
end OAI
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