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9b6e014 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 | import Mathlib
import OAI.Probability.BinarySweep.GridBounds.EmptyGrid
namespace OAI
noncomputable section
section
open scoped Topology
open Set Complex Complex.HadamardThreeLines
namespace BinaryCoordinateSweeps
lemma cosh_norm_le (z : ℂ) : ‖Complex.cosh z‖ ≤ Real.cosh z.re := by
rw [Complex.cosh, norm_div]
norm_num only [Complex.norm_ofNat]
rw [Real.cosh_eq]
have h := norm_add_le (Complex.exp z) (Complex.exp (-z))
rw [Complex.norm_exp,Complex.norm_exp,Complex.neg_re] at h
linarith
def ellipseMap (v : ℝ) (z : ℂ) : ℂ := (v:ℂ)/2*(Complex.cosh z+1)
lemma ellipseMap_norm {v L : ℝ} (hv : 0 ≤ v) (hL : 0 ≤ L) (z : ℂ)
(hz : z ∈ verticalClosedStrip 0 L) :
‖ellipseMap v z‖ ≤ v/2*(Real.cosh L+1) := by
change 0 ≤ z.re ∧ z.re ≤ L at hz
have hc : Real.cosh z.re ≤ Real.cosh L := by
rw [Real.cosh_le_cosh,abs_of_nonneg hz.1,abs_of_nonneg hL]
exact hz.2
have hn : ‖Complex.cosh z+1‖ ≤ Real.cosh L+1 := by
have := norm_add_le (Complex.cosh z) (1:ℂ)
norm_num only [norm_one] at this
linarith [cosh_norm_le z]
simpa only [ellipseMap,norm_mul,norm_div,Complex.norm_real,Real.norm_eq_abs,
abs_of_nonneg hv,Complex.norm_ofNat] using mul_le_mul_of_nonneg_left hn (by positivity : 0 ≤ v/2)
lemma ellipseMap_left {v : ℝ} (hv : 0 ≤ v) (z : ℂ) (hz : z.re=0) :
∃ t : ℝ, 0 ≤ t ∧ t ≤ v ∧ ellipseMap v z=(t:ℂ) := by
have he : z=(z.im:ℂ)*Complex.I := by apply Complex.ext <;> simp [hz]
refine ⟨v/2*(Real.cos z.im+1),?_,?_,?_⟩
· have := Real.neg_one_le_cos z.im
exact mul_nonneg (div_nonneg hv (by norm_num)) (by linarith)
· have := Real.cos_le_one z.im
nlinarith
· have hc : Complex.cosh z=(Real.cos z.im:ℂ) := by
calc Complex.cosh z = Complex.cosh ((z.im:ℂ)*Complex.I) := congrArg _ he
_ = (Real.cos z.im:ℂ) := by rw [Complex.cosh_mul_I,Complex.ofReal_cos]
rw [ellipseMap,hc]
push_cast
rfl
theorem analytic_segment_transfer (R a : ℝ) (hR : 1 < R) (ha : 0 < a) :
∃ θ : ℝ, 0 < θ ∧ θ ≤ 1 ∧
∀ E : Type*, ∀ [NormedAddCommGroup E] [NormedSpace ℂ E],
∀ f : ℂ → E, Differentiable ℂ f → ∀ ε : ℝ, 0 ≤ ε →
(∀ z : ℂ, ‖z‖ ≤ R → ‖f z‖ ≤ 1) →
(∀ t : ℝ, 0 ≤ t → t ≤ a → ‖f (t:ℂ)‖ ≤ ε) → ‖f 1‖ ≤ ε^θ := by
let v := min a (1/2)
have hv : 0 < v := lt_min ha (by norm_num)
have hva : v ≤ a := min_le_left _ _
have hv1 : v < 1 := (min_le_right _ _).trans_lt (by norm_num)
let x := Real.arcosh (2/v-1)
let L := Real.arcosh (2*R/v-1)
have hargx : 1 < 2/v-1 := by
have : (2:ℝ) < 2/v := (lt_div_iff₀ hv).mpr (by linarith)
linarith
have hargL : 2/v-1 < 2*R/v-1 := by
apply sub_lt_sub_right
apply (div_lt_div_iff_of_pos_right hv).2
linarith
have hx : 0 < x := Real.arcosh_pos hargx
have hxL : x < L := (Real.arcosh_lt_arcosh (by linarith) (by linarith)).mpr hargL
have hL : 0 < L := hx.trans hxL
have hcoshx : Real.cosh x=2/v-1 := Real.cosh_arcosh hargx.le
have hcoshL : Real.cosh L=2*R/v-1 := Real.cosh_arcosh (hargx.trans hargL).le
have houter : v/2*(Real.cosh L+1)=R := by rw [hcoshL]; field_simp; ring
have hat : ellipseMap v (x:ℂ)=1 := by
rw [ellipseMap,←Complex.ofReal_cosh,hcoshx]
push_cast
have hvC : (v:ℂ) ≠ 0 := by exact_mod_cast hv.ne'
field_simp [hvC]
ring
refine ⟨1-x/L,by have := (div_lt_one hL).mpr hxL; linarith,
by have := div_nonneg hx.le hL.le; linarith,?_⟩
intro E _ _ f hf ε hε hbound hsmall
let F : ℂ → E := fun z => f (ellipseMap v z)
have hmap : Differentiable ℂ (ellipseMap v) :=
(Complex.differentiable_cosh.add_const 1).const_mul ((v:ℂ)/2)
have hFd : Differentiable ℂ F := hf.comp hmap
have hF1 : ∀z ∈ verticalClosedStrip 0 L, ‖F z‖ ≤ 1 := by
intro z hz
exact hbound _ ((ellipseMap_norm hv.le hL.le z hz).trans_eq houter)
have hc : BddAbove ((norm ∘ F) '' verticalClosedStrip 0 L) := by
use 1
rintro y ⟨z,hz,rfl⟩
exact hF1 z hz
have hleft : ∀z ∈ Complex.re ⁻¹' {(0:ℝ)}, ‖F z‖ ≤ ε := by
intro z hz
obtain ⟨t,ht,htv,he⟩ := ellipseMap_left hv.le z hz
dsimp [F]
rw [he]
exact hsmall t ht (htv.trans hva)
have hright : ∀z ∈ Complex.re ⁻¹' {L}, ‖F z‖ ≤ 1 := by
intro z hz
apply hF1
change 0 ≤ z.re ∧ z.re ≤ L
change z.re=L at hz
rw [hz]
exact ⟨hL.le,le_rfl⟩
have ht := norm_le_interp_of_mem_verticalClosedStrip' hL
(show (x:ℂ) ∈ verticalClosedStrip 0 L from ⟨hx.le,hxL.le⟩)
hFd.diffContOnCl hc hleft hright
simpa only [F,hat,Complex.ofReal_re,sub_zero,Real.one_rpow,mul_one] using ht
end BinaryCoordinateSweeps
end
open scoped BigOperators Classical
namespace BinaryCoordinateSweeps
theorem grid_analytic_contraction : ∃ r : ℕ, 0<r ∧ ∃ g : ℝ, 0<g ∧
∀b : ℕ, 1≤b → ∀bits : Fin b → ℕ,
(∀j,r≤bits j ∧ bits j≤2*r) → ∀D : ℕ,
∀ρ : Representation ℂ (Equiv.Perm (GridSlot bits)) (RepSpace D),
ρ.IsIrreducible → IsUnitaryRep ρ →
‖complexGridAverage bits ρ 1‖ ≤ (D:ℝ)^(-g) := by
obtain ⟨r,hr,q,hq,a,ha,hreal⟩ := real_grid_bound
obtain ⟨R,hR,hdisk⟩ := complex_grid_disk (2*r)
obtain ⟨θ,hθ,hθ1,htrans⟩ := analytic_segment_transfer R a hR ha
refine ⟨r,hr,(c0/(2*q))*θ,mul_pos (div_pos (by norm_num [c0]) (by positivity)) hθ,?_⟩
intro b hb bits hbits D ρ hi hρ
have := hi
have : Nontrivial (RepSpace D) :=
Module.nontrivial_of_finrank_pos (Signed.irreducible_finrank_pos ρ)
have ht := htrans (RepSpace D →L[ℂ] RepSpace D) (complexGridAverage bits ρ)
(complexGridAverage_differentiable bits ρ) ((D:ℝ)^(-c0/(2*q))) (by positivity)
(hdisk b bits (fun j => (hbits j).2) (RepSpace D) ρ hρ)
(hreal b hb bits hbits D ρ hi hρ)
rw [←Real.rpow_mul (Nat.cast_nonneg D)] at ht
have he : (-c0/(2*(q:ℝ)))*θ = -(c0/(2*q)*θ) := by ring
simpa only [he] using ht
end BinaryCoordinateSweeps
end
end OAI
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