openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Spectral /PositiveResolventContraction.lean
| import OAI.Probability.InvariantIsing.Spectral.PositiveResolventSpectrum | |
| namespace OAI | |
| /-! A dimension-free quadratic contraction for positive resolvents. -/ | |
| noncomputable section | |
| open Matrix | |
| open scoped BigOperators | |
| namespace InvariantIsing | |
| lemma positiveResolvent_quadratic_contraction {N : ℕ} {t : ℝ} (ht : 0 < t) | |
| {B : Matrix (Fin N) (Fin N) ℝ} (hB : B.PosSemidef) (x : Fin N → ℝ) : | |
| t*((positiveResolvent t B *ᵥ x) ⬝ᵥ (positiveResolvent t B *ᵥ x)) ≤ | |
| x ⬝ᵥ (positiveResolvent t B *ᵥ x) := by | |
| let y := positiveResolvent t B *ᵥ x | |
| have hinv : (t • (1 : Matrix (Fin N) (Fin N) ℝ)+B)*positiveResolvent t B = 1 := | |
| Matrix.mul_nonsing_inv _ (isUnit_iff_ne_zero.mpr | |
| (((Matrix.PosDef.one.smul ht).add_posSemidef hB).det_pos.ne')) | |
| have hy : (t • (1 : Matrix (Fin N) (Fin N) ℝ)+B) *ᵥ y = x := by | |
| change (t • (1 : Matrix (Fin N) (Fin N) ℝ)+B) *ᵥ (positiveResolvent t B *ᵥ x) = x | |
| rw [Matrix.mulVec_mulVec,hinv,Matrix.one_mulVec] | |
| have hp : 0 ≤ y ⬝ᵥ (B *ᵥ y) := by | |
| simpa only [star_trivial] using hB.dotProduct_mulVec_nonneg y | |
| have hh := congrArg (fun z => y ⬝ᵥ z) hy | |
| rw [Matrix.add_mulVec,Matrix.smul_mulVec,Matrix.one_mulVec, | |
| dotProduct_add,dotProduct_smul] at hh | |
| simp only [smul_eq_mul] at hh | |
| rw [dotProduct_comm x y] | |
| change t*(y ⬝ᵥ y) ≤ y ⬝ᵥ x | |
| linarith | |
| end InvariantIsing | |
| end | |
| end OAI | |