| import OAI.GroupTheory.Thompson.AnalyticTube |
| import OAI.GroupTheory.Thompson.AnalyticCutoffs |
| import Mathlib.Tactic.Abel |
| import Mathlib.Tactic.FieldSimp |
| import Mathlib.Tactic.Ring |
|
|
| namespace OAI |
|
|
| /-! |
| # Elementary inputs and explicit formula for the tube correction |
|
|
| Every field below is a scalar, curve, or rotation identity/estimate. In |
| particular, no property of the corrected map is included in the inputs. |
| -/ |
|
|
| noncomputable section |
|
|
| universe uE |
|
|
| namespace ThompsonNonamenability.Analytic |
|
|
| open scoped InnerProductSpace |
|
|
| variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
|
|
| structure CorrectionData (C : RegularCurveData E) where |
| a : ℝ → ℝ |
| v : ℝ → E |
| q : ℝ → ℝ |
| R : ℝ → E →L[ℝ] E |
| curve_eq : ∀ t, C.curve t = a t • v t |
| a_zero : a 0 = 0 |
| a_upper : ∀ t, a t ≤ 5 / 16 |
| a_late_lower : ∀ t, 1 < t → 1 / 8 ≤ a t |
| q_eq : ∀ t, q t = min (a t) (-1 / 8) - a t |
| q_bound : ∀ t, |q t| ≤ 7 / 16 |
| q_lipschitz : ∀ s t, |q t - q s| ≤ (1 / 8) * |t - s| |
| norm_v : ∀ t, ‖v t‖ = 1 |
| v_lipschitz : ∀ s t, ‖v t - v s‖ ≤ |t - s| |
| curve_lipschitz : ∀ s t, ‖C.curve t - C.curve s‖ ≤ (7 / 16) * |t - s| |
| norm_R : ∀ t e, ‖R t e‖ = ‖e‖ |
| R_perp : ∀ t e, ⟪e, C.tangent t⟫_ℝ = 0 → ⟪R t e, v t⟫_ℝ = 0 |
| R_lipschitz : ∀ s t e, ‖R t e - R s e‖ ≤ 238 * |t - s| * ‖e‖ |
| R_early : ∀ t, t ≤ 1 → ∀ e, R t e = e |
| tangent_early : ∀ t, t ≤ 1 → C.tangent t = v t |
|
|
| namespace CorrectionData |
|
|
| variable {C : RegularCurveData E} (D : CorrectionData C) |
|
|
| / |
| def change (x : C.tube) : E := |
| (beta C.rho ‖C.residual x‖ * D.q (C.parameter x)) • D.v (C.parameter x) + |
| chi C.rho ‖C.residual x‖ • (D.R (C.parameter x) (C.residual x) - C.residual x) |
|
|
| / |
| def inside (x : C.tube) : E := (x : E) + D.change x |
|
|
| / |
| def G (x : E) : E := by |
| classical |
| exact if hx : x ∈ C.tube then D.inside ⟨x, hx⟩ else x |
|
|
| theorem G_of_mem (x : C.tube) : D.G (x : E) = D.inside x := by |
| simp only [G, dite_eq_left x.property] |
|
|
| theorem G_of_not_mem {x : E} (hx : x ∉ C.tube) : D.G x = x := by |
| simp only [G, dite_eq_right hx] |
|
|
| theorem inside_radial (x : C.tube) : |
| D.inside x = |
| (D.a (C.parameter x) + beta C.rho ‖C.residual x‖ * D.q (C.parameter x)) • |
| D.v (C.parameter x) + |
| ((1 - chi C.rho ‖C.residual x‖) • C.residual x + |
| chi C.rho ‖C.residual x‖ • D.R (C.parameter x) (C.residual x)) := by |
| have hx := C.curve_add_residual x |
| rw [D.curve_eq] at hx |
| unfold inside change |
| rw [← hx] |
| simp only [add_smul, sub_smul, one_smul, smul_sub] |
| abel |
|
|
| include D in |
| theorem curve_zero : C.curve 0 = 0 := by |
| rw [D.curve_eq, D.a_zero, zero_smul] |
|
|
| theorem norm_curve (t : ℝ) : ‖C.curve t‖ = |D.a t| := by |
| rw [D.curve_eq, norm_smul, Real.norm_eq_abs, D.norm_v, mul_one] |
|
|
| theorem q_eq_zero {t : ℝ} (h : D.a t ≤ -1 / 8) : D.q t = 0 := by |
| rw [D.q_eq, min_eq_left h, sub_self] |
|
|
| theorem radial_eq_min (t : ℝ) : D.a t + D.q t = min (D.a t) (-1 / 8) := by |
| rw [D.q_eq] |
| ring |
|
|
| end CorrectionData |
|
|
| end ThompsonNonamenability.Analytic |
|
|
| end |
|
|
| end OAI |
|
|
