openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Profile /ProfileEntryBounds.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Profile.ProfileTaylorBound | |
| import OAI.MathematicalPhysics.DefocusingNLS.Profile.ProfileCentralEnclosures | |
| namespace OAI | |
| /-! Uniform entrywise bounds for the actual K=34 free matching product. -/ | |
| open Matrix | |
| namespace DefocusingNLS.ProfileCertificate | |
| attribute [local irreducible] profileProduct | |
| private theorem derivativeB_entry (i j : Fin 2) : | |
| ‖complexMatrix result.derivB i j‖ < 20*‖centralM‖ := by | |
| fin_cases i <;> fin_cases j <;> exact central_derivative_entries _ (by simp [Matrix2.toMatrix]) | |
| private theorem derivativeZ_entry (i j : Fin 2) : | |
| ‖complexMatrix result.derivZ i j‖ < 20*‖centralM‖ := by | |
| fin_cases i <;> fin_cases j <;> exact central_derivative_entries _ (by simp [Matrix2.toMatrix]) | |
| private theorem error_entry (i j : Fin 2) : | |
| rationalMatrix result.error i j < 50000*‖centralM‖ := by | |
| fin_cases i <;> fin_cases j <;> exact (central_error_entries _ (by simp [Matrix2.toMatrix])).2 | |
| /-- The entrywise complex Taylor remainder in `free:profile-enclosures`. -/ | |
| theorem profile_entry_remainder (b z : ℝ) | |
| (hb : |b| ≤ (radius : ℝ)) (hz : |z| ≤ (radius : ℝ)) (i j : Fin 2) : | |
| ‖profileProduct ((centerB : ℝ)+b) ((centerZ : ℝ)+z) 34 i j- | |
| complexMatrix result.value i j- | |
| (b : ℂ)*complexMatrix result.derivB i j- | |
| (z : ℂ)*complexMatrix result.derivZ i j‖ < | |
| 50000*‖centralM‖*(radius : ℝ)^2 := by | |
| have h := profileProduct_remainder b z hb hz i j | |
| have he : 0 < (radius : ℝ)^2 := by norm_num [radius] | |
| have hs := mul_lt_mul_of_pos_left (error_entry i j) he | |
| have hnorm : | |
| ‖profileProduct ((centerB : ℝ)+b) ((centerZ : ℝ)+z) 34 i j- | |
| complexMatrix result.value i j- | |
| (b : ℂ)*complexMatrix result.derivB i j- | |
| (z : ℂ)*complexMatrix result.derivZ i j‖ ≤ | |
| (radius : ℝ)^2*rationalMatrix result.error i j := by | |
| convert! h using 1 | |
| simp only [linearModel, Matrix.sub_apply, Matrix.add_apply, Matrix.smul_apply, | |
| Complex.real_smul] | |
| congr 1 | |
| ring | |
| exact hnorm.trans_lt (hs.trans_eq (by ring)) | |
| /-- The entire matrix varies by less than chi times the central denominator. -/ | |
| theorem profile_entry_variation (b z : ℝ) | |
| (hb : |b| ≤ (radius : ℝ)) (hz : |z| ≤ (radius : ℝ)) (i j : Fin 2) : | |
| ‖profileProduct ((centerB : ℝ)+b) ((centerZ : ℝ)+z) 34 i j- | |
| complexMatrix result.value i j‖ < (changeBound : ℝ)*‖centralM‖ := by | |
| let L := profileProduct ((centerB : ℝ)+b) ((centerZ : ℝ)+z) 34 i j | |
| let U := complexMatrix result.value i j | |
| let X := complexMatrix result.derivB i j | |
| let Y := complexMatrix result.derivZ i j | |
| have hδ : 0 ≤ (radius : ℝ) := by norm_num [radius] | |
| have hx : ‖(b : ℂ)*X‖ ≤ (radius : ℝ)*(20*‖centralM‖) := by | |
| rw [norm_mul, Complex.norm_real, Real.norm_eq_abs] | |
| exact mul_le_mul hb (derivativeB_entry i j).le (norm_nonneg _) hδ | |
| have hy : ‖(z : ℂ)*Y‖ ≤ (radius : ℝ)*(20*‖centralM‖) := by | |
| rw [norm_mul, Complex.norm_real, Real.norm_eq_abs] | |
| exact mul_le_mul hz (derivativeZ_entry i j).le (norm_nonneg _) hδ | |
| calc | |
| ‖L-U‖ = ‖(L-U-(b : ℂ)*X-(z : ℂ)*Y)+(b : ℂ)*X+(z : ℂ)*Y‖ := by | |
| congr 1; ring | |
| _ ≤ ‖L-U-(b : ℂ)*X-(z : ℂ)*Y‖+‖(b : ℂ)*X‖+‖(z : ℂ)*Y‖ := | |
| (norm_add_le _ _).trans (add_le_add (norm_add_le _ _) le_rfl) | |
| _ < 50000*‖centralM‖*(radius : ℝ)^2+ | |
| (radius : ℝ)*(20*‖centralM‖)+(radius : ℝ)*(20*‖centralM‖) := by | |
| exact add_lt_add_of_lt_of_le | |
| (add_lt_add_of_lt_of_le (profile_entry_remainder b z hb hz i j) hx) hy | |
| _ = _ := by unfold changeBound; push_cast; ring | |
| noncomputable def normalizedProfile (b z : ℝ) (i j : Fin 2) : ℂ := | |
| profileProduct ((centerB : ℝ)+b) ((centerZ : ℝ)+z) 34 i j/centralM | |
| noncomputable def centralNormalized (i j : Fin 2) : ℂ := | |
| complexMatrix result.value i j/centralM | |
| theorem normalizedProfile_variation (b z : ℝ) | |
| (hb : |b| ≤ (radius : ℝ)) (hz : |z| ≤ (radius : ℝ)) (i j : Fin 2) : | |
| ‖normalizedProfile b z i j-centralNormalized i j‖ < (changeBound : ℝ) := by | |
| rw [normalizedProfile, centralNormalized, ← sub_div, norm_div] | |
| exact (div_lt_iff₀ (norm_pos_iff.mpr centralM_ne_zero)).mpr | |
| (profile_entry_variation b z hb hz i j) | |
| @[simp] theorem centralNormalized_zero_one : centralNormalized 0 1 = 1 := by | |
| exact div_self centralM_ne_zero | |
| end DefocusingNLS.ProfileCertificate | |
| end OAI | |