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b797a5b | 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 | import Mathlib
namespace OAI
noncomputable section
namespace PiExponent.FormalMatrixBridge
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))
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
end PiExponent.FormalMatrixBridge
end
end OAI
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