Evaluation of reconstruction agrees with the representation throughout the interval

@[reducible, inline]

abbrev MagnitudeConjecture.Graded.FiniteGradedModule.PrincipalIntervalCategory {k A : Type u} [Field k] [Ring A] [Algebra k A] [FiniteDimensional k A] (R : VectorGrading k A) (hmul : ∀ {i j : ℤ} {a b : A}, a ∈ R.component i → b ∈ R.component j → a * b ∈ R.component (i + j)) {ι : Type} (e : ι → A) (he0 : ∀ (i : ι), e i ∈ R.component 0) (m : ℕ) : Type

The finite full category of selected projectives with degrees in [0,m].

def MagnitudeConjecture.Graded.FiniteGradedModule.principalIntervalInclusion {k A : Type u} [Field k] [Ring A] [Algebra k A] [FiniteDimensional k A] (R : VectorGrading k A) (hmul : ∀ {i j : ℤ} {a b : A}, a ∈ R.component i → b ∈ R.component j → a * b ∈ R.component (i + j)) {ι : Type} (e : ι → A) (he0 : ∀ (i : ι), e i ∈ R.component 0) (m : ℕ) : CategoryTheory.Functor (PrincipalIntervalCategory R ⋯ e he0 m) (PrincipalDegreeCategory R ⋯ e he0)

Include the interval-labelled projectives in the whole degree category.

noncomputable def MagnitudeConjecture.Graded.FiniteGradedModule.intervalEvaluationReconstructionIso {k A : Type u} [Field k] [Ring A] [Algebra k A] [FiniteDimensional k A] (R : VectorGrading k A) (hmul : ∀ {i j : ℤ} {a b : A}, a ∈ R.component i → b ∈ R.component j → a * b ∈ R.component (i + j)) {ι : Type} [Fintype ι] (e : ι → A) (he0 : ∀ (i : ι), e i ∈ R.component 0) (he : ∀ (i : ι), e i * e i = e i) (F : CategoryTheory.Functor (PrincipalDegreeCategory R ⋯ e he0)ᵒᵖ (ModuleCat k)) [F.Additive] [CategoryTheory.Functor.Linear k F] (hneg : ∀ d < 0, R.component d = ⊥) (h1 : 1 ∈ R.component 0) (hsum : ∑ i : ι, e i = 1) (horth : Pairwise fun (i j : ι) => e i * e j = 0) (hfinite : ∀ (p : (PrincipalDegreeCategory R ⋯ e he0)ᵒᵖ), FiniteDimensional k ↑(F.obj p)) (m : ℕ) : (principalIntervalInclusion R ⋯ e he0 m).op.comp (CoveringHom.restrictedLinearYoneda (principalDegreeInclusion R ⋯ e he0) (intervalReconstructedSupportedObject R ⋯ e he0 he F hneg h1 hsum horth hfinite m).obj) ≅ (principalIntervalInclusion R ⋯ e he0 m).op.comp F

The coordinate comparisons assemble into a natural isomorphism on the finite interval.