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namespace OAI
/-!
# The local Lipschitz estimate for the explicit tube correction
The radius is the distance to the curve, so its Lipschitz constant is one.
The residual itself has local Lipschitz constant eight. Keeping these two
estimates separate gives the stated constant for the two cutoff terms.
-/
noncomputable section
universe uE
namespace ThompsonNonamenability.Analytic
variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
private theorem norm_smul_sub_smul_bound (a b : ℝ) (u v : E) :
‖a • u - b • v‖ ≤ |a| * ‖u - v‖ + |a - b| * ‖v‖ := by
have hid : a • u - b • v = a • (u - v) + (a - b) • v := by
simp only [smul_sub, sub_smul]
abel
rw [hid]
exact (norm_add_le _ _).trans_eq (by simp only [norm_smul, Real.norm_eq_abs])
namespace RegularCurveData
variable (C : RegularCurveData E)
/-- The residual length is the actual distance to the curve. -/
theorem residual_radius_lipschitz (x y : C.tube) :
|‖C.residual x‖ - ‖C.residual y‖| ≤ ‖(x : E) - (y : E)‖ := by
have hxy := Metric.infDist_le_infDist_add_dist
(x := (x : E)) (y := (y : E)) (s := Set.range C.curve)
have hyx := Metric.infDist_le_infDist_add_dist
(x := (y : E)) (y := (x : E)) (s := Set.range C.curve)
rw [dist_eq_norm] at hxy hyx
rw [norm_sub_rev (y : E) (x : E)] at hyx
rw [C.residual_norm, C.residual_norm, abs_le]
constructor <;> linarith
end RegularCurveData
namespace CorrectionData
variable {C : RegularCurveData E} (D : CorrectionData C)
include D in
/-- Subtracting the nearest curve points costs at most another seven times
the distance, on the local parameter-comparison scale. -/
theorem residual_local_bound (x y : C.tube)
(hxy : ‖(x : E) - (y : E)‖ < C.rho) :
‖C.residual x - C.residual y‖ ≤ 8 * ‖(x : E) - (y : E)‖ := by
have ht := C.parameter_local_bound16 x y hxy
have hc := D.curve_lipschitz (C.parameter y) (C.parameter x)
have hc' : ‖C.curve (C.parameter x) - C.curve (C.parameter y)‖ ≤
7 * ‖(x : E) - (y : E)‖ := by
linarith
have hid : C.residual x - C.residual y =
((x : E) - (y : E)) -
(C.curve (C.parameter x) - C.curve (C.parameter y)) := by
simp only [RegularCurveData.residual]
abel
rw [hid]
have hn := norm_sub_le ((x : E) - (y : E))
(C.curve (C.parameter x) - C.curve (C.parameter y))
linarith
/-- Local Lipschitz control of the actual corrected point. -/
theorem inside_local_bound (x y : C.tube)
(hxy : ‖(x : E) - (y : E)‖ < C.rho) :
‖D.inside x - D.inside y‖ ≤
(30 + 7 / (4 * C.rho) + 3808 * C.rho) * ‖(x : E) - (y : E)‖ := by
let d : ℝ := ‖(x : E) - (y : E)‖
let s : ℝ := C.parameter x
let t : ℝ := C.parameter y
let e : E := C.residual x
let f : E := C.residual y
have hd : 0 ≤ d := norm_nonneg _
have hrho : 0 < C.rho := C.rho_pos
have he : ‖e‖ ≤ C.rho := (C.residual_norm_lt x).le
have hf : ‖f‖ ≤ C.rho := (C.residual_norm_lt y).le
have hst : |s - t| ≤ 16 * d := C.parameter_local_bound16 x y hxy
have hef : ‖e - f‖ ≤ 8 * d := D.residual_local_bound x y hxy
have hr : |‖e‖ - ‖f‖| ≤ d := C.residual_radius_lipschitz x y
have hq : |D.q s - D.q t| ≤ 2 * d := by
have h := D.q_lipschitz t s
linarith
have hv : ‖D.v s - D.v t‖ ≤ 16 * d :=
(D.v_lipschitz t s).trans hst
have hqv : ‖D.q s • D.v s - D.q t • D.v t‖ ≤ 9 * d := by
calc
‖D.q s • D.v s - D.q t • D.v t‖
≤ |D.q s| * ‖D.v s - D.v t‖ +
|D.q s - D.q t| * ‖D.v t‖ := norm_smul_sub_smul_bound _ _ _ _
_ ≤ (7 / 16) * (16 * d) + (2 * d) * 1 :=
add_le_add
(mul_le_mul (D.q_bound s) hv (norm_nonneg _) (by norm_num))
(mul_le_mul hq (le_of_eq (D.norm_v t)) (norm_nonneg _) (by positivity))
_ = 9 * d := by ring
have hqvt : ‖D.q t • D.v t‖ ≤ 7 / 16 := by
simpa only [norm_smul, Real.norm_eq_abs, D.norm_v, mul_one] using D.q_bound t
have hbeta : |beta C.rho ‖e‖ - beta C.rho ‖f‖| ≤ (4 / C.rho) * d :=
(beta_abs_sub_le hrho _ _).trans
(mul_le_mul_of_nonneg_left hr (by positivity))
have hchi : |chi C.rho ‖e‖ - chi C.rho ‖f‖| ≤ (2 / C.rho) * d :=
(chi_abs_sub_le hrho _ _).trans
(mul_le_mul_of_nonneg_left hr (by positivity))
have hbeta_abs : |beta C.rho ‖e‖| ≤ 1 := by
rw [abs_of_nonneg (beta_nonneg _ _)]
exact beta_le_one _ _
have hchi_abs : |chi C.rho ‖e‖| ≤ 1 := by
rw [abs_of_nonneg (chi_nonneg _ _)]
exact chi_le_one _ _
have hradial :
‖(beta C.rho ‖e‖ * D.q s) • D.v s -
(beta C.rho ‖f‖ * D.q t) • D.v t‖ ≤
(9 + 7 / (4 * C.rho)) * d := by
rw [mul_smul, mul_smul]
calc
‖beta C.rho ‖e‖ • (D.q s • D.v s) -
beta C.rho ‖f‖ • (D.q t • D.v t)‖
≤ |beta C.rho ‖e‖| * ‖D.q s • D.v s - D.q t • D.v t‖ +
|beta C.rho ‖e‖ - beta C.rho ‖f‖| * ‖D.q t • D.v t‖ :=
norm_smul_sub_smul_bound _ _ _ _
_ ≤ 1 * (9 * d) + ((4 / C.rho) * d) * (7 / 16) :=
add_le_add
(mul_le_mul hbeta_abs hqv (norm_nonneg _) zero_le_one)
(mul_le_mul hbeta hqvt (norm_nonneg _) (by positivity))
_ = (9 + 7 / (4 * C.rho)) * d := by
field_simp [ne_of_gt hrho] ; ring
have hrotation : ‖D.R s f - D.R t f‖ ≤ 3808 * C.rho * d := by
calc
‖D.R s f - D.R t f‖ ≤ 238 * |s - t| * ‖f‖ := D.R_lipschitz t s f
_ ≤ (238 * (16 * d)) * C.rho :=
mul_le_mul (mul_le_mul_of_nonneg_left hst (by norm_num)) hf
(norm_nonneg _) (by positivity)
_ = 3808 * C.rho * d := by ring
have hdefect_norm : ‖D.R t f - f‖ ≤ 2 * C.rho := by
have h := norm_sub_le (D.R t f) f
rw [D.norm_R] at h
linarith
have hdefect : ‖(D.R s e - e) - (D.R t f - f)‖ ≤
(16 + 3808 * C.rho) * d := by
have hid : (D.R s e - e) - (D.R t f - f) =
(D.R s (e - f) + (D.R s f - D.R t f)) - (e - f) := by
rw [map_sub]
abel
calc
‖(D.R s e - e) - (D.R t f - f)‖
= ‖(D.R s (e - f) + (D.R s f - D.R t f)) - (e - f)‖ := by rw [hid]
_ ≤ ‖D.R s (e - f) + (D.R s f - D.R t f)‖ + ‖e - f‖ := norm_sub_le _ _
_ ≤ (‖D.R s (e - f)‖ + ‖D.R s f - D.R t f‖) + ‖e - f‖ :=
add_le_add (norm_add_le _ _) le_rfl
_ = (‖e - f‖ + ‖D.R s f - D.R t f‖) + ‖e - f‖ := by rw [D.norm_R]
_ ≤ (8 * d + 3808 * C.rho * d) + 8 * d :=
add_le_add (add_le_add hef hrotation) hef
_ = (16 + 3808 * C.rho) * d := by ring
have hangular :
‖chi C.rho ‖e‖ • (D.R s e - e) - chi C.rho ‖f‖ • (D.R t f - f)‖ ≤
(20 + 3808 * C.rho) * d := by
calc
‖chi C.rho ‖e‖ • (D.R s e - e) - chi C.rho ‖f‖ • (D.R t f - f)‖
≤ |chi C.rho ‖e‖| * ‖(D.R s e - e) - (D.R t f - f)‖ +
|chi C.rho ‖e‖ - chi C.rho ‖f‖| * ‖D.R t f - f‖ :=
norm_smul_sub_smul_bound _ _ _ _
_ ≤ 1 * ((16 + 3808 * C.rho) * d) + ((2 / C.rho) * d) * (2 * C.rho) :=
add_le_add
(mul_le_mul hchi_abs hdefect (norm_nonneg _) zero_le_one)
(mul_le_mul hchi hdefect_norm (norm_nonneg _) (by positivity))
_ = (20 + 3808 * C.rho) * d := by
field_simp [ne_of_gt hrho] ; ring
have hchange : ‖D.change x - D.change y‖ ≤
(29 + 7 / (4 * C.rho) + 3808 * C.rho) * d := by
have hid : D.change x - D.change y =
((beta C.rho ‖e‖ * D.q s) • D.v s -
(beta C.rho ‖f‖ * D.q t) • D.v t) +
(chi C.rho ‖e‖ • (D.R s e - e) -
chi C.rho ‖f‖ • (D.R t f - f)) := by
dsimp [change, s, t, e, f]
abel
rw [hid]
exact (norm_add_le _ _).trans ((add_le_add hradial hangular).trans_eq (by ring))
have hid : D.inside x - D.inside y =
((x : E) - (y : E)) + (D.change x - D.change y) := by
simp only [inside]
abel
rw [hid]
calc
‖((x : E) - (y : E)) + (D.change x - D.change y)‖
≤ d + ‖D.change x - D.change y‖ := norm_add_le _ _
_ ≤ d + (29 + 7 / (4 * C.rho) + 3808 * C.rho) * d := add_le_add le_rfl hchange
_ = (30 + 7 / (4 * C.rho) + 3808 * C.rho) * ‖(x : E) - (y : E)‖ := by
change d + (29 + 7 / (4 * C.rho) + 3808 * C.rho) * d =
(30 + 7 / (4 * C.rho) + 3808 * C.rho) * d
ring
end CorrectionData
end ThompsonNonamenability.Analytic
end
end OAI
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