| import OAI.Algebra.Finitistic.Encoding.PathBasisIso | |
| namespace OAI | |
| noncomputable section | |
| open scoped TensorProduct | |
| namespace LittleFinitistic.RelationPathCoordinates | |
| open QuadraticEncoding RelationBasisCoefficients | |
| universe u | |
| variable {k : Type u} [Field k] {d t : ℕ} (P : Relations k d t) | |
| abbrev coordinates : Paths k d ≃ₗ[k] (Fin (d+1) → Arrows k d) := | |
| TensorProduct.piScalarRight k k (Arrows k d) (Fin (d+1)) | |
| /-- The coefficient matrix of an actual relation, grouped by its first traversed arrow. -/ | |
| theorem relation_coordinate (r : relationSpace P) (a : Fin (d+1)) : | |
| coordinates (r : Paths k d) a = ∑ b, coefficient P r a b • basis b := by | |
| have h := congrArg (fun z : Paths k d => coordinates z a) (expansion P r) | |
| simpa [coordinates,TensorProduct.piScalarRightHom_tmul,Finset.sum_apply, | |
| basis,Pi.single_apply] using h.symm | |
| /-- The actual independent relation basis has precisely the manuscript's q coefficients. -/ | |
| theorem basis_coordinate (r : RelationIndex P) (a : Fin (d+1)) : | |
| coordinates ((RelationSequence.relationBasis P r : relationSpace P) : Paths k d) a = | |
| ∑ b, basisCoefficient P r a b • basis b := | |
| relation_coordinate P (RelationSequence.relationBasis P r) a | |
| end LittleFinitistic.RelationPathCoordinates | |
| end | |
| end OAI | |