openAI-math-dataset / lean /OAI /Probability /BinarySweep /MatrixBounds /TraceDensityComparison.lean
| import Mathlib | |
| import OAI.Probability.BinarySweep.MatrixBounds.Whitening | |
| namespace OAI | |
| noncomputable section | |
| open scoped BigOperators Classical ComplexOrder MatrixOrder | |
| open Matrix | |
| namespace BinaryCoordinateSweeps.Density | |
| variable {I W : Type*} [Fintype I] [DecidableEq I] [Fintype W] | |
| omit [DecidableEq I] in | |
| lemma trace_re_real_smul (a : ℝ) (X : Matrix I I ℂ) : | |
| ((a : ℂ) • X).trace.re = a * X.trace.re := by | |
| simp [Matrix.trace_smul,Complex.mul_re] | |
| omit [DecidableEq I] in | |
| lemma trace_re_nat_smul (a : ℕ) (X : Matrix I I ℂ) : | |
| ((a : ℂ) • X).trace.re = a * X.trace.re := by | |
| simpa only [Complex.ofReal_natCast] using trace_re_real_smul (a:ℝ) X | |
| lemma trace_density_whitening (R Q P C : Matrix I I ℂ) (d a : ℝ) | |
| (V : W → Matrix I I ℂ) (hR : R.PosSemidef) (hC : C.IsHermitian) | |
| (hcap : (C*Q*C-(d:ℂ)•(Q*P)).PosSemidef) | |
| (hdom : ((a:ℂ)•∑w, V w-Q).PosSemidef) : | |
| d*(R*Q*P).trace.re ≤ a*∑w, (C*R*C*V w).trace.re := by | |
| have h1 := trace_product_re_mono hR hcap | |
| rw [Matrix.mul_smul,trace_re_real_smul,← mul_assoc R Q P] at h1 | |
| have h2 := hdom.mul_mul_conjTranspose_same C | |
| rw [hC.eq,Matrix.mul_sub,Matrix.sub_mul,Matrix.mul_smul,Matrix.smul_mul, | |
| Matrix.mul_sum,Matrix.sum_mul] at h2 | |
| have h3 := trace_product_re_mono hR h2 | |
| rw [Matrix.mul_smul,trace_re_real_smul,Matrix.mul_sum,Matrix.trace_sum, | |
| Complex.re_sum] at h3 | |
| have he (w : W) : (R*(C*V w*C)).trace = (C*R*C*V w).trace := by | |
| rw [← mul_assoc R (C*V w) C,Matrix.trace_mul_cycle] | |
| congr 1 | |
| noncomm_ring | |
| simp only [he] at h3 | |
| exact h1.trans h3 | |
| lemma trace_right_density_domination (R S : Matrix I I ℂ) (a : ℝ) | |
| (U : W → Matrix I I ℂ) (hS : S.PosSemidef) | |
| (hdom : ((a:ℂ)•∑w, U w-R).PosSemidef) : | |
| (R*S).trace.re ≤ a * ∑w, (U w*S).trace.re := by | |
| have ht := trace_product_re_mono hS hdom | |
| rw [Matrix.mul_smul,trace_re_real_smul,Matrix.mul_sum,Matrix.trace_sum,Complex.re_sum, | |
| Matrix.trace_mul_comm S R] at ht | |
| simpa only [Matrix.trace_mul_comm S] using ht | |
| end BinaryCoordinateSweeps.Density | |
| end | |
| end OAI | |