import OAI.Algebra.Finitistic.Localization.FormalSummandAction
import OAI.Algebra.Finitistic.Localization.DiagramLifting

namespace OAI

namespace LittleFinitistic.FixedDiagram
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Idempotents
open FormalSummandAction

variable {k R : Type*} [CommRing k] [Ring R] [Algebra k R]
variable {C Q : Type*} [Category C] [Category Q] [Preadditive C] [Preadditive Q]
  [Linear k C] [Linear k Q]

/-- Algebra relations are the homogeneous quadratic diagram relations under any action. -/
theorem action_relations {X : Q} (ρ : R →ₐ[k] End X) (na nb nr : ℕ)
    (x : Fin na → R) (y : Fin nb → R) (c : Fin nr → Fin na → Fin nb → k)
    (hc : ∀ t, ∑ a, ∑ b, c t a b • (y b * x a) = 0) :
    ∀ t, ∑ a, ∑ b, c t a b • ((ρ (x a) : X ⟶ X) ≫ ρ (y b)) = 0 := by
  intro t
  have h := congrArg ρ (hc t)
  simpa only [map_sum,map_smul,map_mul,map_zero,End.mul_def] using h

variable (q : C ⥤ Q) (W : MorphismProperty C) [q.IsLocalization W]
  [W.HasRightCalculusOfFractions] [q.Additive] [q.Linear k]
variable [HasShift Q ℤ] [HasZeroObject Q]
  [∀ n : ℤ, (shiftFunctor Q n).Additive] [Pretriangulated Q]
  [∀ n : ℤ, (shiftFunctor Q n).Linear k]
local instance : HasFiniteBiproducts (Karoubi Q) := Karoubi.karoubi_hasFiniteBiproducts
local instance : HasBinaryBiproducts (Karoubi Q) :=
  hasBinaryBiproducts_of_finite_biproducts (Karoubi Q)

include W in
/-- Fixed source-category objects and arrows, with every quadratic relation exact,
represent the odd double of the selecting action in the localization. -/
theorem fixed_diagram {L : Q} (θ : R →ₐ[k] End L)
    (e : R) (he : e*e=e) (hec : ∀ r : R, e*r=r*e) (α : R →ₐ[k] R)
    (na nb nr : ℕ) (x : Fin na → R) (y : Fin nb → R)
    (c : Fin nr → Fin na → Fin nb → k)
    (hc : ∀ t, ∑ a, ∑ b, c t a b • (y b * x a) = 0) :
    ∃ (V₀ : Q) (ρ₀ : R →ₐ[k] End V₀)
      (i : (toKaroubi Q).obj V₀ ≅
        summand θ e he ⊞ OddDouble.formalShift (summand θ e he) 3)
      (D₀ D₁ D₂ : C) (φ₀ : q.obj D₀ ≅ V₀) (φ₁ : q.obj D₁ ≅ V₀)
      (φ₂ : q.obj D₂ ≅ V₀) (f : Fin na → (D₀ ⟶ D₁)) (g : Fin nb → (D₁ ⟶ D₂)),
      (∀ r : R, (toKaroubi Q).map (ρ₀ r) ≫ i.hom =
        i.hom ≫ oddDoubleAction θ e he hec α r) ∧
      (∀ a, q.map (f a) ≫ φ₁.hom = φ₀.hom ≫ ρ₀ (x a)) ∧
      (∀ b, q.map (g b) ≫ φ₂.hom = φ₁.hom ≫ ρ₀ (y b)) ∧
      (∀ t, ∑ a, ∑ b, c t a b • (f a ≫ g b) = 0) := by
  obtain ⟨V₀,ρ₀,i,hi⟩ := fixed_old_object_with_action θ e he hec α
  obtain ⟨D₀,D₁,D₂,φ₀,φ₁,φ₂,f,g,hf,hg,hrel⟩ :=
    DiagramLifting.finite_quadratic_diagram_lifting q W na nb nr V₀ V₀ V₀
      (fun a => ρ₀ (x a)) (fun b => ρ₀ (y b)) c (action_relations ρ₀ na nb nr x y c hc)
  exact ⟨V₀,ρ₀,i,D₀,D₁,D₂,φ₀,φ₁,φ₂,f,g,hi,hf,hg,hrel⟩
end LittleFinitistic.FixedDiagram

end OAI