| import OAI.Dynamics.StandardMap.StablePrefix |
|
|
| namespace OAI |
|
|
| open MeasureTheory Set |
| open scoped ENNReal BigOperators |
|
|
| open MeasureTheory Set Filter Metric |
| open scoped Topology ENNReal |
| namespace StandardMapEntropy |
| lemma real_axes_decomposition (s : ℂ) : s=s.re • (1:ℂ)+s.im • Complex.I := by |
| apply Complex.ext <;> simp |
| lemma row_axes_decomposition (row : ℂ →L[ℝ] ℝ) (s : ℂ) : |
| row s=s.re*row 1+s.im*row Complex.I := by |
| conv_lhs => rw [real_axes_decomposition s] |
| simp only [map_add,map_smul,smul_eq_mul] |
|
|
| lemma normalized_kernel_from_stable (row : ℂ →L[ℝ] ℝ) (s : ℂ) (D : ℝ) |
| (hD : 0 < D) (hu : ‖s‖=1) (hs : |s.re| ≤ 1/4) |
| (hrow : D/3 ≤ |row 1|) (hsmall : |row s| ≤ D⁻¹) : |
| ∃ (u : ℂ) (e : ℝ), u.im=1 ∧ row u=0 ∧ |
| u=(s.im)⁻¹ • s+e • (1:ℂ) ∧ |e| ≤ 6/D^2 := by |
| have him := vertical_im_lower s hu hs |
| have hsim : s.im ≠ 0 := abs_pos.mp (by linarith : 0 < |s.im|) |
| have hr1 : row 1 ≠ 0 := abs_pos.mp (by linarith : 0 < |row 1|) |
| let u : ℂ := (-row Complex.I/row 1) • (1:ℂ)+Complex.I |
| let e := -row s/(row 1*s.im) |
| refine ⟨u,e,?_,?_,?_,?_⟩ |
| · simp [u] |
| · simp only [u,map_add,map_smul,smul_eq_mul] |
| field_simp |
| ring |
| · have haxes:=row_axes_decomposition row s |
| apply Complex.ext |
| · simp only [u,Complex.add_re,Complex.smul_re,Complex.one_re,smul_eq_mul,mul_one,Complex.I_re,add_zero] |
| dsimp [e] |
| field_simp |
| nlinarith |
| · simp only [u,Complex.add_im,Complex.smul_im,Complex.one_im,smul_eq_mul,mul_zero,Complex.I_im,zero_add,add_zero] |
| exact (inv_mul_cancel₀ hsim).symm |
| · have hden : 0 < |row 1| *|s.im| := mul_pos (abs_pos.mpr hr1) (abs_pos.mpr hsim) |
| dsimp [e] |
| rw [abs_div,abs_neg,abs_mul] |
| apply (div_le_iff₀ hden).mpr |
| have hprod : D/6 ≤ |row 1| *|s.im| := by nlinarith |
| have hh := mul_le_mul_of_nonneg_left hprod (by positivity : 0 ≤ 6/D^2) |
| have he : (6/D^2)*(D/6)=D⁻¹ := by field_simp |
| rw [he] at hh |
| exact hsmall.trans hh |
| lemma eligible_row_contraction (A : ℕ → ℂ →L[ℝ] ℂ) (s u : ℂ) (e M D : ℝ) |
| (B l : ℕ) (_hl : 1 ≤ l) (hlB : l ≤ B) (hM : 1 < M) |
| (hu : ‖s‖=1) (hs : |s.re| ≤ 1/4) |
| (heq : u=(s.im)⁻¹ • s+e • (1:ℂ)) (he : |e| ≤ 6/D^2) |
| (hD : M^((49/50:ℝ)*(B:ℝ)) ≤ D) |
| (hstable : ‖A l s‖ ≤ 3*M^(-(47/50:ℝ)*(l:ℝ))) |
| (hup : ‖A l‖ ≤ M^(l:ℝ)) : |
| ‖A l u‖ ≤ 12*M^(-(47/50:ℝ)*(l:ℝ)) := by |
| have hp : 0 < M := by linarith |
| have hDpos : 0 < D := (Real.rpow_pos_of_pos hp _).trans_le hD |
| have him:=vertical_im_lower s hu hs |
| have hi : |(s.im)⁻¹| ≤ 2 := by |
| rw [abs_inv] |
| rw [inv_eq_one_div] |
| apply (div_le_iff₀ (by linarith : 0 < |s.im|)).mpr |
| linarith |
| have he' : |e| ≤ 6*M^(-(49/25:ℝ)*(B:ℝ)) := by |
| calc |
| _ ≤ 6/D^2 := he |
| _ ≤ 6/(M^((49/50:ℝ)*(B:ℝ)))^2 := by gcongr |
| _ = _ := by |
| rw [← Real.rpow_natCast,← Real.rpow_mul hp.le,div_eq_mul_inv,← Real.rpow_neg hp.le] |
| congr 2 |
| ring |
| have herror : |e| * M^(l:ℝ) ≤ 6*M^(-(47/50:ℝ)*(l:ℝ)) := by |
| calc |
| _ ≤ 6*M^(-(49/25:ℝ)*(B:ℝ))*M^(l:ℝ) := mul_le_mul_of_nonneg_right he' (Real.rpow_nonneg hp.le _) |
| _ = 6*M^(-(49/25:ℝ)*(B:ℝ)+(l:ℝ)) := by rw [Real.rpow_add hp]; ring |
| _ ≤ _ := by |
| apply mul_le_mul_of_nonneg_left (Real.rpow_le_rpow_of_exponent_le hM.le ?_) (by norm_num) |
| have hlR : (l:ℝ) ≤ B := by exact_mod_cast hlB |
| have hl0 : (0:ℝ) ≤ l := Nat.cast_nonneg l |
| linarith |
| rw [heq,map_add,map_smul,map_smul] |
| calc |
| _ ≤ ‖(s.im)⁻¹ • A l s‖+‖e • A l 1‖ := norm_add_le _ _ |
| _ = |(s.im)⁻¹| *‖A l s‖+|e| *‖A l 1‖ := by simp only [norm_smul,Real.norm_eq_abs] |
| _ ≤ 2*(3*M^(-(47/50:ℝ)*(l:ℝ)))+|e| *M^(l:ℝ) := by |
| apply add_le_add (mul_le_mul hi hstable (norm_nonneg _) (by norm_num)) |
| apply mul_le_mul_of_nonneg_left _ (abs_nonneg _) |
| exact ((A l).le_opNorm 1).trans (by simpa using hup) |
| _ ≤ _ := by linarith |
| end StandardMapEntropy |
|
|
| end OAI |
|
|
