| import OAI.Algebra.Finitistic.Localization.QuotientRelations |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory |
| namespace LittleFinitistic.QuadraticEncoding |
| universe u |
| variable {k : Type u} [Field k] {d t : ℕ} (P : Relations k d t) |
|
|
| lemma quotient_arrow12 (h : Fin d) : |
| arrowQ12 P h.succ = inv (arrowQ01 P 0) ≫ arrowQ01 P h.succ ≫ arrowQ12 P 0 := by |
| have h := congrArg (fun f => inv (arrowQ01 P 0) ≫ f) (quotient_commutation P h) |
| simpa using h |
|
|
| / |
| def normalizePath : (vertexQ P 0 ⟶ vertexQ P 2) →ₗ[k] End (vertexQ P 0) where |
| toFun f := f ≫ inv (arrowQ12 P 0) ≫ inv (arrowQ01 P 0) |
| map_add' f g := by simp [Preadditive.add_comp] |
| map_smul' c f := by simp [Linear.smul_comp] |
|
|
| lemma normalizePath_quadratic (h l : Fin d) : |
| normalizePath P (arrowQ01 P l.succ ≫ arrowQ12 P h.succ) = |
| normalizedGenerator P h * normalizedGenerator P l := by |
| change (arrowQ01 P l.succ ≫ arrowQ12 P h.succ) ≫ inv (arrowQ12 P 0) ≫ inv (arrowQ01 P 0) = |
| (arrowQ01 P l.succ ≫ inv (arrowQ01 P 0)) ≫ (arrowQ01 P h.succ ≫ inv (arrowQ01 P 0)) |
| rw [quotient_arrow12] |
| simp [Category.assoc] |
|
|
| lemma normalizePath_linear (h : Fin d) : |
| normalizePath P (arrowQ01 P h.succ ≫ arrowQ12 P 0) = normalizedGenerator P h := by |
| change (arrowQ01 P h.succ ≫ arrowQ12 P 0) ≫ inv (arrowQ12 P 0) ≫ inv (arrowQ01 P 0) = _ |
| simp [Category.assoc,normalizedGenerator] |
|
|
| lemma normalizePath_constant : |
| normalizePath P (arrowQ01 P 0 ≫ arrowQ12 P 0) = (1 : End (vertexQ P 0)) := by |
| change (arrowQ01 P 0 ≫ arrowQ12 P 0) ≫ inv (arrowQ12 P 0) ≫ inv (arrowQ01 P 0) = 𝟙 _ |
| simp [Category.assoc] |
|
|
| / |
| theorem normalizedGenerator_satisfies : P.Satisfies (normalizedGenerator P) := by |
| intro j |
| have h := congrArg (normalizePath P) (quotient_homogenized P j) |
| simpa only [map_add,map_sum,map_smul,map_zero,normalizePath_quadratic, |
| normalizePath_linear,normalizePath_constant] using h |
| end LittleFinitistic.QuadraticEncoding |
|
|
| end |
|
|
| end OAI |
|
|
