| import OAI.GroupTheory.Thompson.CorrectionData |
|
|
| namespace OAI |
|
|
| /-! |
| # The local Lipschitz estimate for the explicit tube correction |
|
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| 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 |
|
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| 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 |
|
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| end OAI |
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|
