File size: 3,508 Bytes
9b6e014 | 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 73 74 75 76 77 78 79 80 81 82 83 | import OAI.Probability.ContinuumTransition.GridBoundary
namespace OAI
section
open scoped BigOperators
open MeasureTheory Set
namespace ContinuumTemperature
open Classical
noncomputable def gridIndices (m q : ℕ) : Finset Lattice :=
Fintype.piFinset (fun _ : Fin 3 => indexInterval m q)
noncomputable def boundaryIndices (m q H : ℕ) : Finset Lattice :=
Finset.univ.biUnion (fun i : Fin 3 => Fintype.piFinset
(fun j : Fin 3 => if j = i then edgeIndices m q H else indexInterval m q))
theorem card_gridIndices (m q : ℕ) : (gridIndices m q).card = (2*m*q+1)^3 := by
simp only [gridIndices, Fintype.card_piFinset, card_indexInterval, Finset.prod_const,
Finset.card_univ, Fintype.card_fin]
theorem card_boundaryIndices_le (m q H : ℕ) (hq : 0 < q) :
(boundaryIndices m q H).card ≤ 3*((2*m+1)*(2*H))*(2*m*q+1)^2 := by
have hc (i : Fin 3) : (Fintype.piFinset (fun j : Fin 3 =>
if j = i then edgeIndices m q H else indexInterval m q)).card ≤
((2*m+1)*(2*H))*(2*m*q+1)^2 := by
rw [Fintype.card_piFinset]
have he : (∏ j : Fin 3, (if j = i then edgeIndices m q H else indexInterval m q).card) =
(edgeIndices m q H).card*(indexInterval m q).card^2 := by
fin_cases i <;> simp [Fin.prod_univ_succ] <;> ring
rw [he, card_indexInterval]
exact Nat.mul_le_mul_right _ (card_edgeIndices_le m q H hq)
have hh := Finset.card_biUnion_le (s := (Finset.univ : Finset (Fin 3)))
(t := fun i : Fin 3 => Fintype.piFinset
(fun j : Fin 3 => if j = i then edgeIndices m q H else indexInterval m q))
have hh' := hh.trans (Finset.sum_le_sum (fun i (_ : i ∈ (Finset.univ : Finset (Fin 3))) => hc i))
simpa only [boundaryIndices, Finset.sum_const, Finset.card_univ, Fintype.card_fin,
smul_eq_mul, mul_assoc] using hh'
theorem mem_boundaryIndices {m q H : ℕ} {k : Lattice} (hk : k ∈ gridIndices m q)
(he : ∃ i, k i ∈ edgeIndices m q H) : k ∈ boundaryIndices m q H := by
obtain ⟨i, hi⟩ := he
apply Finset.mem_biUnion.mpr ⟨i, Finset.mem_univ i, ?_⟩
apply Fintype.mem_piFinset.mpr
intro j
by_cases hj : j = i
· simpa only [ite_eq_left hj, hj, ite_true] using hi
· simpa only [ite_eq_right hj] using (Fintype.mem_piFinset.mp hk j)
theorem cellIndex_mem_gridIndices {m q : ℕ} {x : Space}
(hx : x ∈ cube ((m : ℝ)*(q : ℝ)/2)) : cellIndex x ∈ gridIndices m q := by
apply Fintype.mem_piFinset.mpr
intro i
apply Finset.mem_Icc.mpr
have hxi := hx i
change -↑m*↑q ≤ ⌊4*x i⌋ ∧ ⌊4*x i⌋ ≤ ↑m*↑q
constructor
· apply Int.le_floor.mpr
push_cast
change -((m : ℝ)*(q : ℝ)/2)/2 ≤ x i ∧ x i ≤ ((m : ℝ)*(q : ℝ)/2)/2 at hxi
linarith [hxi.1]
· apply Int.floor_le_iff.mpr
push_cast
change -((m : ℝ)*(q : ℝ)/2)/2 ≤ x i ∧ x i ≤ ((m : ℝ)*(q : ℝ)/2)/2 at hxi
linarith [hxi.2]
theorem boundary_of_close_crossing {q H : ℕ} (hq : 0 < q) {m : ℕ} {x y : Space}
(hx : x ∈ cube ((m : ℝ)*(q : ℝ)/2))
(hd : ∀ i, |cellIndex x i-cellIndex y i| ≤ (H : ℤ))
(hc : gridIndex ((q : ℝ)/4) x ≠ gridIndex ((q : ℝ)/4) y) :
cellIndex x ∈ boundaryIndices m q H := by
have hk := cellIndex_mem_gridIndices hx
apply mem_boundaryIndices hk
rw [gridIndex_quarter_nat, gridIndex_quarter_nat] at hc
obtain ⟨i, hi⟩ := Function.ne_iff.mp hc
refine ⟨i, Finset.mem_filter.mpr ⟨Fintype.mem_piFinset.mp hk i, ?_⟩⟩
exact edgeResidue_of_crossing hq (hd i) hi
end ContinuumTemperature
end
end OAI
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