| import OAI.GroupTheory.Thompson.UnitBall |
| import Mathlib.Tactic.Abel |
| import Mathlib.Tactic.FieldSimp |
| import Mathlib.Tactic.Linarith |
| import Mathlib.Tactic.Ring |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| universe uE |
|
|
| namespace ThompsonNonamenability.Analytic |
|
|
| variable {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] |
|
|
| / |
| def normalizeNeg (z : E) : E := -(‖z‖⁻¹ • z) |
|
|
| theorem norm_normalizeNeg {z : E} (hz : z ≠ 0) : ‖normalizeNeg z‖ = 1 := by |
| have hn : ‖z‖ ≠ 0 := norm_ne_zero_iff.mpr hz |
| rw [normalizeNeg, norm_neg, norm_smul, Real.norm_eq_abs, |
| abs_of_nonneg (inv_nonneg.mpr (norm_nonneg z)), inv_mul_cancel₀ hn] |
|
|
| / |
| private theorem abs_inv_sub_mul {a b : ℝ} (ha : 0 < a) (hb : 0 < b) : |
| |a⁻¹ - b⁻¹| * b = |b - a| / a := by |
| have hdiff : a⁻¹ - b⁻¹ = (b - a) / (a * b) := by |
| field_simp [ha.ne', hb.ne'] |
| rw [hdiff, abs_div, abs_mul, abs_of_pos ha, abs_of_pos hb] |
| field_simp [ha.ne', hb.ne'] |
|
|
| / |
| theorem normalizeNeg_lipschitz {ε : ℝ} (hε : 0 < ε) {x y : E} |
| (hx : ε ≤ ‖x‖) (hy : ε ≤ ‖y‖) : |
| ‖normalizeNeg x - normalizeNeg y‖ ≤ (2 / ε) * ‖x - y‖ := by |
| have hxp : 0 < ‖x‖ := hε.trans_le hx |
| have hyp : 0 < ‖y‖ := hε.trans_le hy |
| have hdecomp : ‖x‖⁻¹ • x - ‖y‖⁻¹ • y = |
| ‖x‖⁻¹ • (x - y) + (‖x‖⁻¹ - ‖y‖⁻¹) • y := by |
| simp only [smul_sub, sub_smul] |
| abel |
| have hneg : normalizeNeg x - normalizeNeg y = |
| -(‖x‖⁻¹ • x - ‖y‖⁻¹ • y) := by |
| simp only [normalizeNeg, neg_sub] |
| abel |
| calc |
| ‖normalizeNeg x - normalizeNeg y‖ = |
| ‖‖x‖⁻¹ • x - ‖y‖⁻¹ • y‖ := by rw [hneg, norm_neg] |
| _ ≤ ‖‖x‖⁻¹ • (x - y)‖ + ‖(‖x‖⁻¹ - ‖y‖⁻¹) • y‖ := by |
| rw [hdecomp] |
| exact norm_add_le _ _ |
| _ = ‖x‖⁻¹ * ‖x - y‖ + |‖y‖ - ‖x‖| / ‖x‖ := by |
| rw [norm_smul, norm_smul, Real.norm_eq_abs, Real.norm_eq_abs, |
| abs_of_pos (inv_pos.mpr hxp), abs_inv_sub_mul hxp hyp] |
| _ ≤ ‖x‖⁻¹ * ‖x - y‖ + ‖x - y‖ / ‖x‖ := by |
| apply add_le_add le_rfl |
| exact div_le_div_of_nonneg_right |
| (by simpa only [norm_sub_rev] using abs_norm_sub_norm_le y x) (norm_nonneg x) |
| _ = (2 / ‖x‖) * ‖x - y‖ := by ring |
| _ ≤ (2 / ε) * ‖x - y‖ := |
| mul_le_mul_of_nonneg_right |
| (div_le_div_of_nonneg_left (by norm_num) hε hx) (norm_nonneg _) |
|
|
| / |
| def normalizedMap (G : UnitBall E → E) {ε : ℝ} (hε : 0 < ε) |
| (hLower : ∀ x, ε ≤ ‖G x‖) : UnitBall E → UnitBall E := |
| fun x => ⟨normalizeNeg (G x), le_of_eq (norm_normalizeNeg |
| (norm_pos_iff.mp (hε.trans_le (hLower x))))⟩ |
|
|
| @[simp] theorem coe_normalizedMap (G : UnitBall E → E) {ε : ℝ} (hε : 0 < ε) |
| (hLower : ∀ x, ε ≤ ‖G x‖) (x : UnitBall E) : |
| (normalizedMap G hε hLower x : E) = normalizeNeg (G x) := rfl |
|
|
| theorem norm_normalizedMap (G : UnitBall E → E) {ε : ℝ} (hε : 0 < ε) |
| (hLower : ∀ x, ε ≤ ‖G x‖) (x : UnitBall E) : |
| ‖(normalizedMap G hε hLower x : E)‖ = 1 := |
| norm_normalizeNeg (norm_pos_iff.mp (hε.trans_le (hLower x))) |
|
|
| / |
| theorem normalizedMap_displacement (G : UnitBall E → E) {ε : ℝ} (hε : 0 < ε) |
| (hLower : ∀ x, ε ≤ ‖G x‖) |
| (hOuter : ∀ x : UnitBall E, (1 / 2 : ℝ) ≤ ‖(x : E)‖ → G x = (x : E)) |
| (x : UnitBall E) : |
| (1 / 2 : ℝ) ≤ ‖(normalizedMap G hε hLower x : E) - (x : E)‖ := by |
| by_cases hx : (1 / 2 : ℝ) ≤ ‖(x : E)‖ |
| · have hxp : 0 < ‖(x : E)‖ := by linarith |
| have hi : 0 ≤ ‖(x : E)‖⁻¹ + 1 := by |
| have := inv_nonneg.mpr (norm_nonneg (x : E)) |
| linarith |
| have heq : normalizeNeg (x : E) - (x : E) = |
| -((‖(x : E)‖⁻¹ + 1) • (x : E)) := by |
| simp only [normalizeNeg, add_smul, one_smul, neg_add_rev] |
| abel |
| rw [coe_normalizedMap, hOuter x hx, heq, norm_neg, norm_smul, |
| Real.norm_eq_abs, abs_of_nonneg hi, add_mul, one_mul, |
| inv_mul_cancel₀ hxp.ne'] |
| linarith |
| · have hrev := norm_sub_norm_le (normalizedMap G hε hLower x : E) (x : E) |
| rw [norm_normalizedMap] at hrev |
| linarith |
| |
| /-- The concrete normalized map inherits the advertised Lipschitz estimate. -/ |
| theorem normalizedMap_lipschitz (G : UnitBall E → E) {ε C : ℝ} (hε : 0 < ε) |
| (hLower : ∀ x, ε ≤ ‖G x‖) |
| (hLip : ∀ x y : UnitBall E, ‖G x - G y‖ ≤ C * ‖(x : E) - (y : E)‖) |
| (x y : UnitBall E) : |
| ‖(normalizedMap G hε hLower x : E) - (normalizedMap G hε hLower y : E)‖ ≤ |
| (2 * C / ε) * ‖(x : E) - (y : E)‖ := by |
| calc |
| _ ≤ (2 / ε) * ‖G x - G y‖ := normalizeNeg_lipschitz hε (hLower x) (hLower y) |
| _ ≤ (2 / ε) * (C * ‖(x : E) - (y : E)‖) := |
| mul_le_mul_of_nonneg_left (hLip x y) (div_nonneg (by norm_num) hε.le) |
| _ = _ := by ring |
| |
| end ThompsonNonamenability.Analytic |
| |
| end |
| |
| end OAI |
| |
