| import OAI.Probability.InvariantIsing.Haar.HaarPolynomialEntropyBound |
|
|
| namespace OAI |
|
|
| /-! Positive polynomial truncations of an exponential observable. -/ |
| noncomputable section |
| open Matrix MvPolynomial |
| open scoped BigOperators |
| namespace InvariantIsing |
|
|
| def haarExpPolynomial {N : ℕ} (n : ℕ) (p : MatrixPolynomial N) : MatrixPolynomial N := |
| ∑ k ∈ Finset.range (n+1), ((Nat.factorial k : ℝ)⁻¹) • p^k |
|
|
| lemma haarExpPolynomial_eval {N : ℕ} (n : ℕ) (p : MatrixPolynomial N) |
| (M : Matrix (Fin N) (Fin N) ℝ) : |
| matrixPolynomialEval M (haarExpPolynomial n p) = |
| ∑ k ∈ Finset.range (n+1), (matrixPolynomialEval M p)^k/(Nat.factorial k : ℝ) := by |
| simp only [haarExpPolynomial,map_sum,map_smul,map_pow,smul_eq_mul,div_eq_mul_inv,mul_comm] |
|
|
| lemma haarExpPolynomial_derivation {N : ℕ} (n : ℕ) (p : MatrixPolynomial N) |
| (A : Matrix (Fin N) (Fin N) ℝ) : |
| matrixPolynomialDerivation A (haarExpPolynomial (n+1) p) = |
| haarExpPolynomial n p*matrixPolynomialDerivation A p := by |
| unfold haarExpPolynomial |
| rw [Finset.sum_range_succ'] |
| simp only [map_add,map_sum,Derivation.map_smul_of_tower,pow_zero, |
| Derivation.map_one_eq_zero,smul_zero,add_zero] |
| rw [Finset.sum_mul] |
| apply Finset.sum_congr rfl |
| intro k _ |
| have hk : (k:ℝ)+1 ≠ 0 := by positivity |
| have hf : (Nat.factorial k : ℝ) ≠ 0 := by exact_mod_cast Nat.factorial_ne_zero k |
| have hcoef : ((Nat.factorial (k+1) : ℝ)⁻¹)*((k+1:ℕ):ℝ) = |
| (Nat.factorial k : ℝ)⁻¹ := by |
| rw [Nat.factorial_succ,Nat.cast_mul,Nat.cast_add,Nat.cast_one] |
| field_simp |
| rw [Derivation.leibniz_pow] |
| simp only [Nat.add_sub_cancel] |
| rw [← Nat.cast_smul_eq_nsmul ℝ] |
| change (Nat.factorial (k+1) : ℝ)⁻¹ • |
| (((k+1:ℕ):ℝ) • (p^k*matrixPolynomialDerivation A p)) = _ |
| rw [smul_smul,hcoef,smul_mul_assoc] |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
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|
