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a89d282 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 | 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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