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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 | 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]
/-- Negative radial normalization, with the field's usual value at zero. -/
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]
/-- A scalar identity used to avoid any estimate depending on the larger norm. -/
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']
/-- Radial normalization is `2 / ε`-Lipschitz where the norm is at least `ε`. -/
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 _)
/-- Normalize the given nonvanishing map and retain the closed-ball carrier. -/
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)))
/-- The outer identity and unit normalization give a displacement of at least `1/2`. -/
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
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