openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Profile /ProfileDeterminantBound.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Profile.ProfileCertificateComparisons | |
| import OAI.MathematicalPhysics.DefocusingNLS.Profile.ProfileMatrixProduct | |
| namespace OAI | |
| /-! The scalar determinant bound holds for the actual parameter-dependent product. -/ | |
| open Matrix BigOperators | |
| namespace DefocusingNLS.ProfileCertificate | |
| attribute [local irreducible] backwardProduct profileProduct | |
| private theorem backward_det_normSq (b Z : ℝ) (K : ℕ) : | |
| Complex.normSq (backwardProduct 5 (Complex.I*Z) (-Complex.I*b) K).det = | |
| ∏ n ∈ Finset.range K, (((n : ℝ)^2+b^2)*(((n : ℝ)-5)^2+b^2)) := by | |
| induction K with | |
| | zero => simp [backwardProduct] | |
| | succ K ih => | |
| rw [backwardProduct, Matrix.det_mul, Complex.normSq_mul, ih, | |
| det_backwardMatrix, Finset.prod_range_succ] | |
| congr 1 | |
| rw [Complex.normSq_mul] | |
| simp [Complex.normSq_apply, Complex.mul_re, Complex.mul_im] | |
| ring | |
| private theorem profile_det_normSq (b Z : ℝ) (K : ℕ) : | |
| Complex.normSq (profileProduct b Z K).det = | |
| (∏ n ∈ Finset.range K, (((n : ℝ)^2+b^2)*(((n : ℝ)-5)^2+b^2)))/ | |
| (K.factorial : ℝ)^4 := by | |
| rw [profileProduct, Matrix.det_smul, Fintype.card_fin, Complex.normSq_mul, | |
| map_pow, Complex.normSq_inv, Complex.normSq_natCast, backward_det_normSq] | |
| simp only [div_eq_mul_inv, _root_.mul_inv_rev, mul_pow] | |
| ring | |
| private theorem profile_det_bound (b Z B : ℝ) (hB : |b| ≤ B) (K : ℕ) : | |
| Complex.normSq (profileProduct b Z K).det ≤ | |
| (∏ n ∈ Finset.range K, (((n : ℝ)^2+B^2)*(((n : ℝ)-5)^2+B^2)))/ | |
| (K.factorial : ℝ)^4 := by | |
| rw [profile_det_normSq] | |
| have hBB : 0 ≤ B := (abs_nonneg b).trans hB | |
| have hb : b^2 ≤ B^2 := by | |
| have h := mul_self_le_mul_self (abs_nonneg b) hB | |
| nlinarith [sq_abs b] | |
| apply div_le_div_of_nonneg_right _ (by positivity) | |
| apply Finset.prod_le_prod₀ | |
| · intro n _; positivity | |
| · intro n _ | |
| apply mul_le_mul <;> nlinarith [sq_nonneg (n : ℝ), sq_nonneg ((n : ℝ)-5)] | |
| private theorem certificate_as_real : (determinantSqBound : ℝ) = | |
| (∏ n ∈ Finset.range 34, | |
| (((n : ℝ)^2+((centerB : ℝ)+(radius : ℝ))^2)* | |
| (((n : ℝ)-5)^2+((centerB : ℝ)+(radius : ℝ))^2)))/(Nat.factorial 34 : ℝ)^4 := by | |
| unfold determinantSqBound | |
| simp only [bind_pure_comp, List.map_eq_map, List.map_map, Function.comp_def] | |
| rw [← List.prod_toFinset _ List.nodup_range] | |
| simp only [List.toFinset_range] | |
| push_cast | |
| rfl | |
| /-- Uniform determinant modulus bound throughout the certificate square. -/ | |
| theorem actual_profile_determinant_bound (b z : ℝ) (hb : |b| ≤ (radius : ℝ)) : | |
| ‖(profileProduct ((centerB : ℝ)+b) ((centerZ : ℝ)+z) 34).det‖ < | |
| (62/10000000000000 : ℝ)*‖RationalComplex.toComplex result.value.b‖^2 := by | |
| have hcb : 0 ≤ (centerB : ℝ) := by norm_num [centerB] | |
| have hB : |(centerB : ℝ)+b| ≤ (centerB : ℝ)+(radius : ℝ) := by | |
| exact (abs_add_le _ _).trans (by | |
| simpa only [abs_of_nonneg hcb] using add_le_add le_rfl hb) | |
| have hu := profile_det_bound ((centerB : ℝ)+b) ((centerZ : ℝ)+z) | |
| ((centerB : ℝ)+(radius : ℝ)) hB 34 | |
| rw [← certificate_as_real] at hu | |
| have hc := (Rat.cast_lt (K := ℝ)).mpr determinant_bound | |
| push_cast at hc | |
| rw [← RationalComplex.normSq_toComplex, Complex.normSq_eq_norm_sq] at hc | |
| have hs := hu.trans_lt hc | |
| rw [Complex.normSq_eq_norm_sq] at hs | |
| nlinarith [norm_nonneg ((profileProduct ((centerB : ℝ)+b) ((centerZ : ℝ)+z) 34).det), | |
| sq_nonneg ‖RationalComplex.toComplex result.value.b‖] | |
| end DefocusingNLS.ProfileCertificate | |
| end OAI | |