| import Mathlib |
| import OAI.Geometry.RecorderBoxes.Placement |
|
|
| namespace OAI |
|
|
| /-! Separated physical rectangles and their reciprocal affine maps. -/ |
|
|
| namespace Solenoidal |
| namespace Bridge |
| variable {M : Machine} |
| noncomputable def sourceRectangle {m : ℕ} (e : Recorder.Letter M ≃ Fin (m + 1)) |
| (i : Branch M) : Rectangle := physicalBox i.src (branchSource e i) |
|
|
| noncomputable def targetRectangle {m : ℕ} (e : Recorder.Letter M ≃ Fin (m + 1)) |
| (i : Branch M) : Rectangle := physicalBox i.dst (branchTarget e i) |
|
|
| theorem common_gap_le_kappa (m : ℕ) : kappa M / Radix.base m ^ 2 ≤ kappa M := by |
| apply (div_le_self (kappa_pos M).le) |
| have := Radix.base_three_le m |
| nlinarith |
|
|
| |
| theorem source_rectangles_separated {m : ℕ} (e : Recorder.Letter M ≃ Fin (m + 1)) |
| {i j : Branch M} (hij : i ≠ j) : |
| (sourceRectangle e i).SeparatedBy (kappa M / Radix.base m ^ 2) (sourceRectangle e j) := by |
| by_cases hs : i.src = j.src |
| · have h := Rectangle.gap_mono (branch_source_gap e hij hs) (inverse_square_le_inverse m) |
| have hp := physical_gap_same i.src h |
| simpa only [sourceRectangle, hs, mul_one_div] using hp |
| · exact Rectangle.gap_mono |
| (physical_gap_different hs (source_in_unit _ _) (source_in_unit _ _)) |
| (common_gap_le_kappa m) |
|
|
| |
|
|
| theorem target_rectangles_separated {m : ℕ} (e : Recorder.Letter M ≃ Fin (m + 1)) |
| {i j : Branch M} (hij : i ≠ j) : |
| (targetRectangle e i).SeparatedBy (kappa M / Radix.base m ^ 2) (targetRectangle e j) := by |
| by_cases hs : i.dst = j.dst |
| · have hp := physical_gap_same i.dst (branch_target_gap e hij hs) |
| simpa only [targetRectangle, hs, mul_one_div] using hp |
| · exact Rectangle.gap_mono |
| (physical_gap_different hs (target_in_unit _ _ _) (target_in_unit _ _ _)) |
| (common_gap_le_kappa m) |
|
|
| |
| noncomputable def affine {m : ℕ} (e : Recorder.Letter M ≃ Fin (m + 1)) |
| (i : Branch M) (p : ℝ × ℝ) : ℝ × ℝ := |
| place i.dst (Radix.update (e i.read) (e i.written) (e i.left) i.move (unplace i.src p)) |
|
|
| |
| theorem affine_image {m : ℕ} (e : Recorder.Letter M ≃ Fin (m + 1)) (i : Branch M) : |
| affine e i '' (sourceRectangle e i).carrier = (targetRectangle e i).carrier := by |
| rw [sourceRectangle, targetRectangle, physicalBox_carrier, physicalBox_carrier] |
| rw [Set.image_image] |
| have he : (affine e i ∘ place i.src) = |
| place i.dst ∘ Radix.update (e i.read) (e i.written) (e i.left) i.move := by |
| funext p |
| simp [affine] |
| change (affine e i ∘ place i.src) '' (branchSource e i).carrier = _ |
| rw [he, Set.image_comp] |
| congr 1 |
| exact (Radix.full_image (e i.read) (e i.written) (e i.left) i.move).trans |
| (targetBox_carrier _ _ _).symm |
|
|
| |
| theorem affine_linear_part {m : ℕ} (e : Recorder.Letter M ≃ Fin (m + 1)) (i : Branch M) |
| (p q : ℝ × ℝ) : |
| affine e i p - affine e i q = |
| (Radix.scale m i.move * (p.1 - q.1), (Radix.scale m i.move)⁻¹ * (p.2 - q.2)) := by |
| have hh := Radix.reciprocal_linear_part (e i.read) (e i.written) (e i.left) i.move |
| (unplace i.src p) (unplace i.src q) |
| apply Prod.ext |
| · calc |
| _ = kappa M * (Radix.update (e i.read) (e i.written) (e i.left) i.move (unplace i.src p) - |
| Radix.update (e i.read) (e i.written) (e i.left) i.move (unplace i.src q)).1 := by |
| dsimp [affine, place]; ring |
| _ = kappa M * (Radix.scale m i.move * ((unplace i.src p).1 - (unplace i.src q).1)) := by |
| rw [hh] |
| _ = _ := by dsimp [unplace]; field_simp [(kappa_pos M).ne']; ring |
| · calc |
| _ = kappa M * (Radix.update (e i.read) (e i.written) (e i.left) i.move (unplace i.src p) - |
| Radix.update (e i.read) (e i.written) (e i.left) i.move (unplace i.src q)).2 := by |
| dsimp [affine, place]; ring |
| _ = kappa M * ((Radix.scale m i.move)⁻¹ * ((unplace i.src p).2 - (unplace i.src q).2)) := by |
| rw [hh] |
| _ = _ := by dsimp [unplace]; field_simp [(kappa_pos M).ne']; ring |
| end Bridge |
| end Solenoidal |
| |
| end OAI |
| |
