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0dd3af4 | 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 93 94 95 96 97 98 99 100 101 102 103 104 | 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)
/-- The additive change on the actual, uniquely parameterized open tube. -/
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)
/-- The actual correction inside the tube. -/
def inside (x : C.tube) : E := (x : E) + D.change x
/-- Gluing to the identity outside the open tube, including its boundary. -/
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
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