| import OAI.Probability.InvariantIsing.Haar.HaarLogCurvature |
| import OAI.Probability.InvariantIsing.Haar.HaarMinimumGradient |
|
|
| namespace OAI |
|
|
| /-! The pointwise minimum inequality for a logarithmic heat-gradient barrier. -/ |
| noncomputable section |
| open Matrix MvPolynomial |
| namespace InvariantIsing |
|
|
| def haarLogBarrier {N : ℕ} (p : MatrixPolynomial N) (a : ℝ) : MatrixPolynomial N := |
| a • p^2-haarPolynomialGamma p p |
|
|
| lemma haarLogBarrier_eval {N : ℕ} (p : MatrixPolynomial N) (a : ℝ) |
| (M : Matrix (Fin N) (Fin N) ℝ) : |
| matrixPolynomialEval M (haarLogBarrier p a) = |
| a*(matrixPolynomialEval M p)^2-matrixPolynomialEval M (haarPolynomialGamma p p) := by |
| simp only [haarLogBarrier,map_sub,map_smul,map_pow,smul_eq_mul] |
|
|
| lemma haarLogBarrier_laplacian {N : ℕ} (p : MatrixPolynomial N) (a : ℝ) |
| (M : Matrix (Fin N) (Fin N) ℝ) : |
| matrixPolynomialEval M (haarPolynomialLaplacian N (haarLogBarrier p a)) = |
| 2*a*(matrixPolynomialEval M p*matrixPolynomialEval M (haarPolynomialLaplacian N p)+ |
| matrixPolynomialEval M (haarPolynomialGamma p p))- |
| matrixPolynomialEval M (haarPolynomialLaplacian N (haarPolynomialGamma p p)) := by |
| simp only [haarLogBarrier,map_sub,map_smul,haarPolynomialLaplacian_sq, |
| map_mul,map_ofNat,map_add,smul_eq_mul] |
| ring |
|
|
| lemma haarLogBarrier_gamma {N : ℕ} (p : MatrixPolynomial N) (a : ℝ) |
| (M : Matrix (Fin N) (Fin N) ℝ) : |
| matrixPolynomialEval M (haarPolynomialGamma p (haarLogBarrier p a)) = |
| 2*a*matrixPolynomialEval M p*matrixPolynomialEval M (haarPolynomialGamma p p)- |
| matrixPolynomialEval M (haarPolynomialGamma p (haarPolynomialGamma p p)) := by |
| simp only [haarLogBarrier,haarPolynomialGamma_sub,haarPolynomialGamma_smul, |
| haarPolynomialGamma_sq,map_sub,map_smul,map_mul,map_ofNat,smul_eq_mul] |
| ring |
|
|
| lemma haarLogBarrier_min_derivative_lower {N : ℕ} (p : MatrixPolynomial N) (a : ℝ) |
| (U : SpecialOrthogonal N) (hu : haarPolynomialValue p U ≠ 0) |
| (hmin : ∀ V : SpecialOrthogonal N, |
| haarPolynomialValue (haarLogBarrier p a) U ≤ haarPolynomialValue (haarLogBarrier p a) V) : |
| -2*((N:ℝ)-2)*a*(haarPolynomialValue p U)^2+ |
| 2*a*haarPolynomialValue p U*haarPolynomialValue (haarPolynomialLaplacian N p) U- |
| 2*haarPolynomialValue (haarPolynomialGamma p (haarPolynomialLaplacian N p)) U ≥ |
| (-2*((N:ℝ)-2)+2*haarPolynomialValue (haarPolynomialGamma p p) U/ |
| (haarPolynomialValue p U)^2)*haarPolynomialValue (haarLogBarrier p a) U := by |
| let M : Matrix (Fin N) (Fin N) ℝ := U |
| have hc := haarLogCurvature p M (1/matrixPolynomialEval M p) |
| have hl := haarPolynomialLaplacian_nonneg_at_min (haarLogBarrier p a) U hmin |
| have hg := haarPolynomialGamma_zero_at_min (haarLogBarrier p a) p U hmin |
| change 0 ≤ matrixPolynomialEval M (haarPolynomialLaplacian N (haarLogBarrier p a)) at hl |
| change matrixPolynomialEval M (haarPolynomialGamma p (haarLogBarrier p a)) = 0 at hg |
| rw [haarLogBarrier_laplacian] at hl |
| rw [haarLogBarrier_gamma] at hg |
| change matrixPolynomialEval M p ≠ 0 at hu |
| change -2*((N:ℝ)-2)*a*(matrixPolynomialEval M p)^2+ |
| 2*a*matrixPolynomialEval M p*matrixPolynomialEval M (haarPolynomialLaplacian N p)- |
| 2*matrixPolynomialEval M (haarPolynomialGamma p (haarPolynomialLaplacian N p)) ≥ _ |
| rw [show haarPolynomialValue (haarLogBarrier p a) U = |
| matrixPolynomialEval M (haarLogBarrier p a) from rfl,haarLogBarrier_eval] |
| have hq : matrixPolynomialEval M (haarPolynomialGamma p (haarPolynomialGamma p p)) / |
| matrixPolynomialEval M p = 2*a*matrixPolynomialEval M (haarPolynomialGamma p p) := by |
| rw [div_eq_iff hu] |
| linear_combination -hg |
| rw [div_eq_mul_inv] at hq |
| have hce : ((N:ℝ)-2)*matrixPolynomialEval M (haarPolynomialGamma p p) ≤ |
| matrixPolynomialEval M (haarPolynomialLaplacian N (haarPolynomialGamma p p))/2- |
| matrixPolynomialEval M (haarPolynomialGamma p (haarPolynomialLaplacian N p))- |
| 2*a*matrixPolynomialEval M (haarPolynomialGamma p p)+ |
| (matrixPolynomialEval M (haarPolynomialGamma p p))^2/(matrixPolynomialEval M p)^2 := by |
| convert hc using 1 |
| simp only [one_div] |
| rw [mul_comm ((matrixPolynomialEval M p)⁻¹),hq] |
| ring |
| have hid : |
| (-2*((N:ℝ)-2)+2*matrixPolynomialEval M (haarPolynomialGamma p p)/(matrixPolynomialEval M p)^2)* |
| (a*(matrixPolynomialEval M p)^2-matrixPolynomialEval M (haarPolynomialGamma p p)) = |
| -2*((N:ℝ)-2)*a*(matrixPolynomialEval M p)^2+ |
| 2*((N:ℝ)-2)*matrixPolynomialEval M (haarPolynomialGamma p p)+ |
| 2*a*matrixPolynomialEval M (haarPolynomialGamma p p)- |
| 2*(matrixPolynomialEval M (haarPolynomialGamma p p))^2/(matrixPolynomialEval M p)^2 := by |
| field_simp |
| ring |
| change _ ≥ (-2*((N:ℝ)-2)+2*matrixPolynomialEval M (haarPolynomialGamma p p)/ |
| (matrixPolynomialEval M p)^2)*(a*(matrixPolynomialEval M p)^2- |
| matrixPolynomialEval M (haarPolynomialGamma p p)) |
| rw [hid] |
| linear_combination hl+2*hce |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
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|
