| import OAI.Algebra.Finitistic.Encoding.RelationSequence |
|
|
| namespace OAI |
| noncomputable section |
| open scoped TensorProduct |
| namespace LittleFinitistic.RelationBasisCoefficients |
| open QuadraticEncoding |
| universe u |
| variable {k : Type u} [Field k] {d t : ℕ} (P : Relations k d t) |
|
|
| / |
| def pathBasis : Module.Basis (Fin (d+1) × Fin (d+1)) k (Paths k d) := |
| (Pi.basisFun k (Fin (d+1))).tensorProduct (Pi.basisFun k (Fin (d+1))) |
|
|
| / |
| def coefficient (r : relationSpace P) (a b : Fin (d+1)) : k := |
| (pathBasis (k := k) (d := d)).repr r.val (b,a) |
|
|
| theorem expansion (r : relationSpace P) : |
| (∑ a, ∑ b, coefficient P r a b • (basis b ⊗ₜ[k] basis a)) = r.val := by |
| rw [Finset.sum_comm] |
| simpa only [coefficient,pathBasis,Fintype.sum_prod_type, |
| Module.Basis.tensorProduct_apply,Pi.basisFun_apply,basis] using |
| (pathBasis (k := k) (d := d)).sum_repr r.val |
|
|
| / |
| theorem algebra_relation (r : relationSpace P) : |
| (∑ a, ∑ b, coefficient P r a b • (arrow12 P b * arrow01 P a)) = 0 := by |
| have h := (pathMap_eq_zero_iff P r.val).2 r.property |
| rw [← expansion P r] at h |
| simpa only [map_sum,map_smul,← arrow_product] using h |
|
|
| abbrev RelationIndex := Fin (Module.finrank k (relationSpace P)) |
|
|
| def basisCoefficient (r : RelationIndex P) (a b : Fin (d+1)) : k := |
| coefficient P (RelationSequence.relationBasis P r) a b |
|
|
| theorem basis_relation (r : RelationIndex P) : |
| (∑ a, ∑ b, basisCoefficient P r a b • (arrow12 P b * arrow01 P a)) = 0 := |
| algebra_relation P (RelationSequence.relationBasis P r) |
| end LittleFinitistic.RelationBasisCoefficients |
|
|
| end |
|
|
| end OAI |
|
|
