| import OAI.Algebra.Finitistic.Encoding.RelationBasisCoefficients | |
| import OAI.Algebra.Finitistic.Selection.SelectionEncoding | |
| namespace OAI | |
| noncomputable section | |
| open scoped TensorProduct | |
| namespace LittleFinitistic.RelationBasisAction | |
| open QuadraticEncoding RelationBasisCoefficients | |
| universe u v | |
| variable {k : Type u} [Field k] {R : Type v} [Ring R] [Algebra k R] | |
| variable {d t : ℕ} (P : Relations k d t) (x : Fin d → R) | |
| /-- The distinguished arrow acts by one, followed by the original generators. -/ | |
| def arrow : Fin (d+1) → R := Fin.cases 1 x | |
| /-- Evaluate a length-two path b a in the original algebra. -/ | |
| def pathValue : Paths k d →ₗ[k] R := | |
| (pathBasis (k := k) (d := d)).constr k (fun ba => arrow x ba.1 * arrow x ba.2) | |
| @[simp] theorem pathValue_basis (b a : Fin (d+1)) : | |
| pathValue x (basis b ⊗ₜ[k] basis a) = arrow x b * arrow x a := by | |
| have hb : (pathBasis (k := k) (d := d)) (b,a) = basis b ⊗ₜ[k] basis a := by | |
| simp [pathBasis,basis] | |
| rw [← hb] | |
| exact (pathBasis (k := k) (d := d)).constr_basis k _ (b,a) | |
| variable (hx : ∀ j, (∑ h, ∑ l, P.quadratic j h l • (x h * x l)) + | |
| (∑ h, P.linear j h • x h) + P.constant j • (1 : R) = 0) | |
| include hx in | |
| /-- The entire relation space vanishes, including every chosen independent basis vector. -/ | |
| theorem relationSpace_le_kernel : relationSpace P ≤ (pathValue x).ker := by | |
| apply Submodule.span_le.2 | |
| rintro z (⟨h,rfl⟩ | ⟨j,rfl⟩) | |
| · change pathValue x (commutation h) = 0 | |
| simp [commutation,arrow] | |
| · change pathValue x (homogenized P j) = 0 | |
| simpa only [homogenized,map_add,map_sum,map_smul,pathValue_basis,arrow, | |
| Fin.cases_zero,Fin.cases_succ,one_mul,mul_one] using hx j | |
| include hx in | |
| /-- Exact homogeneous coefficient equations for the fixed-diagram lifting theorem. -/ | |
| theorem basis_action_relation (r : RelationIndex P) : | |
| (∑ a, ∑ b, basisCoefficient P r a b • (arrow x b * arrow x a)) = 0 := by | |
| have hz := relationSpace_le_kernel P x hx (RelationSequence.relationBasis P r).property | |
| change pathValue x (RelationSequence.relationBasis P r).val = 0 at hz | |
| rw [← expansion P (RelationSequence.relationBasis P r)] at hz | |
| simpa only [map_sum,map_smul,pathValue_basis,basisCoefficient] using hz | |
| /-- The concrete selected algebra satisfies all equations of the independent relation basis. -/ | |
| theorem encoding_basis_action (E : SelectionEncoding.Encoding k R) (r : RelationIndex E.relations) : | |
| (∑ a, ∑ b, basisCoefficient E.relations r a b • | |
| (arrow (SelectionEncoding.generator E) b * arrow (SelectionEncoding.generator E) a)) = 0 := | |
| basis_action_relation E.relations (SelectionEncoding.generator E) | |
| (SelectionEncoding.generator_relations E) r | |
| end LittleFinitistic.RelationBasisAction | |
| end | |
| end OAI | |