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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 | import OAI.Probability.InvariantIsing.Haar.HaarPolynomialGamma
namespace OAI
/-! First-order product and square-gradient identities for the rotation carré du champ. -/
noncomputable section
open Matrix MvPolynomial
open scoped BigOperators
namespace InvariantIsing
lemma haarPolynomialGamma_mul {N : ℕ} (p q r : MatrixPolynomial N) :
haarPolynomialGamma p (q*r) = q*haarPolynomialGamma p r+r*haarPolynomialGamma p q := by
simp only [haarPolynomialGamma,Derivation.leibniz,smul_eq_mul,
Finset.mul_sum,← Finset.sum_add_distrib]
apply Finset.sum_congr rfl
intro i _
apply Finset.sum_congr rfl
intro j _
ring
lemma haarPolynomialGamma_sq {N : ℕ} (p q : MatrixPolynomial N) :
haarPolynomialGamma p (q^2) = 2*q*haarPolynomialGamma p q := by
rw [pow_two,haarPolynomialGamma_mul]
ring
lemma haarPolynomialGamma_sub {N : ℕ} (p q r : MatrixPolynomial N) :
haarPolynomialGamma p (q-r) = haarPolynomialGamma p q-haarPolynomialGamma p r := by
simp only [haarPolynomialGamma,map_sub,mul_sub,Finset.sum_sub_distrib]
lemma haarPolynomialGamma_smul {N : ℕ} (p q : MatrixPolynomial N) (c : ℝ) :
haarPolynomialGamma p (c • q) = c • haarPolynomialGamma p q := by
simp only [haarPolynomialGamma,Derivation.map_smul_of_tower,mul_smul_comm,
Finset.smul_sum]
lemma haarPolynomialDerivation_gamma_sq {N : ℕ} (p : MatrixPolynomial N)
(A : Matrix (Fin N) (Fin N) ℝ) :
matrixPolynomialDerivation A (haarPolynomialGamma p p) =
∑ i, ∑ j, 2*(matrixPolynomialDerivation (planeGenerator i j) p*
matrixPolynomialDerivation A (matrixPolynomialDerivation (planeGenerator i j) p)) := by
simp only [haarPolynomialGamma,map_sum,Derivation.leibniz,smul_eq_mul]
apply Finset.sum_congr rfl
intro i _
apply Finset.sum_congr rfl
intro j _
ring
end InvariantIsing
end
end OAI
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