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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 | import OAI.Algebra.Finitistic.Iteration.BinomialEvenIteration
namespace OAI
open CategoryTheory CategoryTheory.Limits
noncomputable section
namespace ExternalTensor.Binomial.Objectwise
universe v u w
variable {C : Type u} [Category.{v} C] [Preadditive C]
[HasFiniteBiproducts C] [HasBinaryBiproducts C]
[HasShift C ℤ] [∀ n : ℤ, (shiftFunctor C n).Additive]
variable {S : Type w}
/-- Exact generic Section 4 statement, assuming only existence of each objectwise split. -/
theorem section4_objectwise (Phi : C ⥤ C) [Phi.Additive] [Phi.CommShift ℤ]
(M : S → C) (H : S → S)
(split : ∀ Y, Nonempty (Phi.obj (M Y) ≅ M (H Y) ⊞ (M (H Y))⟦(3 : ℤ)⟧))
(j : ℕ) (Y : S) :
Nonempty ((Phi.obj^[j]) (M Y) ≅
⨁ fun q : Fin (j+1) => ⨁ fun _ : Fin (Nat.choose j q.val) =>
(M ((H^[j]) Y))⟦(3 : ℤ)*(q.val : ℤ)⟧) :=
⟨section4Iteration Phi M H (fun Y => Classical.choice (split Y)) j Y⟩
/-- Exact generic Section 6 formula, assuming only existence of each two-step split. -/
theorem section6_objectwise (F : C ⥤ C) [F.Additive] [F.CommShift ℤ]
(M : S → C) (H : S → S) (b : ℤ)
(split : ∀ Y, Nonempty (F.obj (F.obj (M Y)) ≅
(M (H Y))⟦b⟧ ⊞ (M (H Y))⟦b+3⟧))
(j : ℕ) (Y : S) :
Nonempty ((F.obj^[2*j]) (M Y) ≅
⨁ fun q : Fin (j+1) => ⨁ fun _ : Fin (Nat.choose j q.val) =>
(M ((H^[j]) Y))⟦(j : ℤ)*b + 3*(q.val : ℤ)⟧) :=
⟨evenIteration F M H b (fun Y => Classical.choice (split Y)) j Y⟩
/-- Exact generic zero detection for the two-step formula, with all hypotheses exposed. -/
theorem section6_zero_detection (F : C ⥤ C) [F.Additive] [F.CommShift ℤ]
(M : S → C) (H : S → S) (b : ℤ)
(split : ∀ Y, Nonempty (F.obj (F.obj (M Y)) ≅
(M (H Y))⟦b⟧ ⊞ (M (H Y))⟦b+3⟧))
(j : ℕ) (Y : S) :
IsZero ((F.obj^[2*j]) (M Y)) ↔ IsZero (M ((H^[j]) Y)) :=
evenIteration_isZero_iff F M H b (fun Y => Classical.choice (split Y)) j Y
end ExternalTensor.Binomial.Objectwise
end
end OAI
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