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ae00908 | 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.Selected.SelectedRectification
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Limits
namespace LittleFinitistic.SelectedWindow
open SelectedFixedDiagram SelectedRectification
/-- A common finite support window chosen from the fixed representatives alone. -/
structure Window where
lo : ℤ
hi : ℤ
bound0 : ∀ n, n < lo ∨ hi < n → IsZero (D0.X n)
bound1 : ∀ n, n < lo ∨ hi < n → IsZero (D1.X n)
bound2 : ∀ n, n < lo ∨ hi < n → IsZero (D2.X n)
instance : Nonempty Window := by
obtain ⟨l₀,h₀,b₀⟩ := UnderlyingRepresentatives.bounded P fixed.D0
obtain ⟨l₁,h₁,b₁⟩ := UnderlyingRepresentatives.bounded P fixed.D1
obtain ⟨l₂,h₂,b₂⟩ := UnderlyingRepresentatives.bounded P fixed.D2
refine ⟨⟨min l₀ (min l₁ l₂), max h₀ (max h₁ h₂), ?_, ?_, ?_⟩⟩
· intro n hn; apply b₀ n; omega
· intro n hn; apply b₁ n; omega
· intro n hn; apply b₂ n; omega
def window : Window := Classical.choice inferInstance
abbrev lo := window.lo
abbrev hi := window.hi
/-- Every term outside the fixed total's resulting support window is zero. -/
theorem bounded (n : ℤ) (hn : n < lo - 2 ∨ hi < n) : IsZero (bimodule.X n) :=
InducedRectification.total_bounded (ConcreteQuadraticDiagram.quadratic P)
D0 D1 D2 f g relations lo hi window.bound0 window.bound1 window.bound2 n hn
/-- The support choices retain the complete proved rectification certificate. -/
theorem verified : RectificationLemma.Verified P D0 D1 D2 f g relations lo hi :=
RectificationLemma.rectification P D0 D1 D2 f g relations lo hi
window.bound0 window.bound1 window.bound2
(UnderlyingRepresentatives.projective P fixed.D0)
(UnderlyingRepresentatives.projective P fixed.D1)
(UnderlyingRepresentatives.projective P fixed.D2)
(UnderlyingRepresentatives.finite P fixed.D0)
(UnderlyingRepresentatives.finite P fixed.D1)
(UnderlyingRepresentatives.finite P fixed.D2)
end LittleFinitistic.SelectedWindow
end
end OAI
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