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695bd70 | 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 84 85 86 87 88 89 90 91 92 | 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
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