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216fb31 | 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 | 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
/-- Divide each homogenized path by s's on the target side. -/
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]
/-- The original, inhomogeneous relations hold on L in the actual quotient. -/
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
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