| import OAI.Probability.InvariantIsing.Haar.HaarPolynomialGamma |
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| namespace OAI |
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| /-! 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 |
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| 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 |
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|
| 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] |
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| 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] |
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| 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 |
|
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| end InvariantIsing |
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| end |
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| end OAI |
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