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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 | import Mathlib
import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.CompactFineGrid
import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.RefinedEndpointDecisions
namespace OAI
section
section
open scoped symmDiff
namespace SimpleAmenable
open scoped commutatorElement
open scoped commutatorElement
section CompactCommonRefinement
def closedRefinedBox (N : ℕ) (e : Fin (N+1) → CutRing) (cell : Fin 2 → Fin N) : Set (ℝ × ℝ) :=
{p | ∀ j, ordinary (e (cell j).castSucc)≤realCoordinate p j ∧
realCoordinate p j≤ordinary (e (cell j).succ)}
theorem compact_common_refining_grid {ι κ : Type*} [Fintype ι]
(S : Finset CutRing) (size : ι → ℕ) (hs : ∀ i, 201 ≤ size i)
(q : ι → Fin 2 → ℤ) (v : ι → CutRing × CutRing)
(O : κ → Set (ℝ × ℝ)) (hO : ∀ i, IsOpen (O i))
(hcover : Set.Icc (0:ℝ) 1 ×ˢ Set.Icc (0:ℝ) 1 ⊆ ⋃ i, O i)
(ε : κ → ℝ) (hε : ∀ i, 0<ε i) :
∃ N : ℕ, 0<N ∧ ∃ e : Fin (N+1) → CutRing,
StrictMono (fun i => ordinary (e i)) ∧ e 0=0 ∧ e (Fin.last N)=1 ∧
(∀ i : Fin N, ordinary (e i.succ)-ordinary (e i.castSucc)<1) ∧
(∀ c ∈ S, cutFraction c ∈ Set.range e) ∧
(∀ i, ∀ j : Fin 2, ∀ k : Fin (size i+1),
cutFraction (windowCut (size i) (q i j) k+pointCoordinate (v i) j) ∈ Set.range e) ∧
(∀ cell : Fin 2 → Fin N, ∃ k, closedRefinedBox N e cell ⊆ O k ∧
∀ j, ordinary (e (cell j).succ)-ordinary (e (cell j).castSucc)<ε k) := by
obtain ⟨n,hn,hfine⟩ := compact_local_bounds_fine_grid O hO hcover ε hε
obtain ⟨N,hN,e,he,hzero,hlast,hmesh,hS,hcuts,href⟩ := common_translated_grid S n hn size hs q v
have hnpos : (200:ℝ)<n := by exact_mod_cast (by omega : 200<n)
refine ⟨N,hN,e,he,hzero,hlast,fun i => (hmesh i).trans_lt
((div_lt_one (by linarith)).mpr hnpos),hS,hcuts,?_⟩
intro cell
have hh (j : Fin 2) := href (cell j)
choose old hold using hh
obtain ⟨k,hδ,hk⟩ := hfine n le_rfl old
refine ⟨k,fun p hp => hk (fun j => ⟨(hold j).1.trans (hp j).1,(hp j).2.trans (hold j).2⟩),?_⟩
intro j
exact (hmesh (cell j)).trans_lt hδ
end CompactCommonRefinement
end SimpleAmenable
end
end
end OAI
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