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6aac220 | 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 | import OAI.GroupTheory.PolycyclicRecognition.Analysis.NormalizedPairPacking
namespace OAI
/-! # Image paths centered at their actual common starting point -/
noncomputable section
namespace PolycyclicRecognition
open scoped NNReal
variable {G W : Type*} [Group G] [MetricSpace G]
[IsTopologicalGroup G] [IsIsometricSMul G G]
[NormedAddCommGroup W] [NormedSpace ℝ W]
def basedImage (F : G → G) (a g : G) : G := (F a)⁻¹ * F g
omit [IsTopologicalGroup G] in
theorem basedImage_dist (F : G → G) (a g h : G) :
dist (basedImage F a g) (basedImage F a h) = dist (F g) (F h) :=
dist_mul_left _ _ _
omit [IsTopologicalGroup G] [IsIsometricSMul G G] [MetricSpace G] [NormedSpace ℝ W] in
theorem basedImage_height (π : G →* Multiplicative W) (F : G → G) (a g : G) :
(π (basedImage F a g)).toAdd = (π (F g)).toAdd - (π (F a)).toAdd := by
simp only [basedImage, map_mul, map_inv, toAdd_mul, toAdd_inv]
abel
/-- Centering the image gives a path based at the identity, preserves all
relative heights, and multiplies both coarse metric constants by the same
Lipschitz constant. -/
theorem basedImage_map_path (F : G → G) {K : ℝ≥0} (hF : LipschitzWith K F)
{a g : G} (γ : Path a g) {L κ : ℝ} (hγ : PathMetricBound γ L κ) :
∃ η : Path (1 : G) (basedImage F a g),
(∀ t, η t = basedImage F a (γ t)) ∧ PathMetricBound η (K * L) (K * κ) := by
have hc : Continuous (basedImage F a) := continuous_const.mul hF.continuous
let η : Path (1 : G) (basedImage F a g) := (γ.map hc).cast (by simp [basedImage]) rfl
refine ⟨η, (fun _ => rfl), ?_⟩
intro s t
change dist (basedImage F a (γ s)) (basedImage F a (γ t)) ≤ _
rw [basedImage_dist]
calc
_ ≤ (K : ℝ) * dist (γ s) (γ t) := hF.dist_le_mul _ _
_ ≤ (K : ℝ) * (L * |(s : ℝ) - t| + κ) :=
mul_le_mul_of_nonneg_left (hγ s t) K.coe_nonneg
_ = _ := by ring
omit [IsTopologicalGroup G] in
/-- A based image of a separated source set has the same cardinality and
unit-separated points, by the actual lower quasi-isometry inequality. -/
theorem basedImage_separated_card [DecidableEq G] {Λ : ℝ≥0} (p : NormalizedPair G Λ)
(hΛ : 0 < (Λ : ℝ)) (a : G) (s : Finset G)
(hsep : ∀ x ∈ s, ∀ y ∈ s, x ≠ y → (Λ : ℝ) * ((Λ : ℝ) + 1) < dist x y) :
(s.image (basedImage p.val.1 a)).card = s.card ∧
∀ x ∈ s.image (basedImage p.val.1 a),
∀ y ∈ s.image (basedImage p.val.1 a), x ≠ y → 1 ≤ dist x y := by
classical
have hd (x y : G) (hx : x ∈ s) (hy : y ∈ s) (hne : x ≠ y) :
1 < dist (basedImage p.val.1 a x) (basedImage p.val.1 a y) := by
rw [basedImage_dist]
exact normalizedPair_image_dist_gt_one p hΛ (hsep x hx y hy hne)
constructor
· apply Finset.card_image_iff.mpr
intro x hx y hy he
by_contra hne
have hh := hd x y hx hy hne
rw [he, dist_self] at hh
linarith
· intro x hx y hy hne
obtain ⟨u, hu, rfl⟩ := Finset.mem_image.mp hx
obtain ⟨v, hv, rfl⟩ := Finset.mem_image.mp hy
exact (hd u v hu hv (fun he => hne (congrArg _ he))).le
end PolycyclicRecognition
end
end OAI
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