| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| namespace PiExponent.FormalMatrixBridge |
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| def nestedPolynomialSwap (R : Type*) [CommRing R] (m : ℕ) : |
| MvPolynomial (Fin m) (Polynomial R) ≃ₐ[R] Polynomial (MvPolynomial (Fin m) R) := |
| (MvPolynomial.mapAlgEquiv (Fin m) (MvPolynomial.uniqueAlgEquiv R Unit).symm).trans |
| ((MvPolynomial.commAlgEquiv R (Fin m) Unit).trans |
| ((MvPolynomial.uniqueAlgEquiv (MvPolynomial (Fin m) R) Unit).restrictScalars R)) |
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| theorem coeff_commAlgEquiv {R σ τ : Type*} [CommRing R] |
| (p : MvPolynomial σ (MvPolynomial τ R)) (b : σ →₀ ℕ) (c : τ →₀ ℕ) : |
| ((MvPolynomial.commAlgEquiv R σ τ p).coeff c).coeff b = (p.coeff b).coeff c := by |
| classical |
| induction p using MvPolynomial.induction_on' with |
| | add p q hp hq => |
| simp only [map_add, AddMonoidAlgebra.coeff_add, Finsupp.add_apply, hp, hq] |
| | monomial d a => |
| induction a using MvPolynomial.induction_on' with |
| | add a e ha he => |
| simpa only [map_add, AddMonoidAlgebra.coeff_add, Finsupp.add_apply] |
| using congrArg₂ (· + ·) ha he |
| | monomial e r => |
| have hs : MvPolynomial.commAlgEquiv R σ τ |
| (MvPolynomial.monomial d (MvPolynomial.monomial e r)) = |
| MvPolynomial.monomial e (MvPolynomial.monomial d r) := |
| AddMonoidAlgebra.commAlgEquiv_single_single (R := R) d e r |
| rw [hs] |
| simp only [MvPolynomial.coeff_monomial] |
| split_ifs <;> simp_all |
| |
| theorem coeff_nestedPolynomialSwap {R : Type*} [CommRing R] {m : ℕ} |
| (p : MvPolynomial (Fin m) (Polynomial R)) (s : ℕ) (b : Fin m →₀ ℕ) : |
| ((nestedPolynomialSwap R m p).coeff s).coeff b = (p.coeff b).coeff s := by |
| change ((MvPolynomial.uniqueAlgEquiv (MvPolynomial (Fin m) R) Unit |
| (MvPolynomial.commAlgEquiv R (Fin m) Unit |
| (MvPolynomial.mapAlgEquiv (Fin m) (MvPolynomial.uniqueAlgEquiv R Unit).symm p))).coeff s).coeff b = _ |
| rw [MvPolynomial.coeff_uniqueAlgEquiv, coeff_commAlgEquiv] |
| rw [MvPolynomial.mapAlgEquiv_apply, MvPolynomial.coeff_map] |
| change (((MvPolynomial.uniqueAlgEquiv R Unit).symm (p.coeff b)).coeff |
| (Finsupp.single () s)) = _ |
| rw [MvPolynomial.coeff_uniqueAlgEquiv_symm] |
| simp |
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| end PiExponent.FormalMatrixBridge |
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| end |
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| end OAI |
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|
