| import OAI.Probability.ContinuumTransition.GridBoundary |
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|
| 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) |
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| 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 |
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| end OAI |
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