| import OAI.Algebra.Finitistic.Bar.SupportedInnerCoordinates |
| import OAI.Algebra.Finitistic.Bar.LeadingTotalComparison |
|
|
| namespace OAI |
| open CategoryTheory CategoryTheory.Limits TensorProduct TrivSqZeroExt |
| open scoped TensorProduct DirectSum |
| noncomputable section |
| namespace DerivedBar.ParityBridge |
| open DerivedBar.Multibar DerivedBar.RecursiveTotal |
| universe u |
| variable (k D X : Type u) [Field k] [Ring D] [Algebra k D] |
| [AddCommGroup X] [Module k X] [Module D X] [Module Dᵐᵒᵖ X] |
| [SMulCommClass D Dᵐᵒᵖ X] [IsScalarTower k D X] [IsScalarTower k Dᵐᵒᵖ X] |
| local instance indexedPatternEmbeddingScalarModule (V : ModuleCat.{u} D) : |
| Module k V := BimoduleTensor.scalarModule k D V |
| local instance indexedPatternEmbeddingScalarTower (V : ModuleCat.{u} D) : |
| IsScalarTower k D V := |
| IsScalarTower.of_algebraMap_smul (fun _ _ => rfl) |
| local instance indexedPatternEmbeddingIndexDecidableEq (r n : ℕ) : |
| DecidableEq (Index r n) := Classical.decEq _ |
| variable (N : ModuleCat.{u} D) |
| local notation "A" => TrivSqZeroExt D X |
| local notation "XB" => BimoduleTensor.twoSidedBarFunctor k D X |
| local notation "Z" => Functor.obj (ChainComplex.single₀ (ModuleCat D)) N |
| local notation "B" => (fun _ : ℕ => XB) |
| local notation "c" => ComplexShape.down ℕ |
| local instance indexedPatternEmbeddingInflatedModule : Module A N := inflatedModule D X N |
| local instance indexedPatternEmbeddingInflatedScalarTower : IsScalarTower k A N := |
| IsScalarTower.of_algebraMap_smul |
| (fun a v => IsScalarTower.algebraMap_smul D a v) |
|
|
| theorem eqToHom_tensor_tmul (L : Type u) [AddCommGroup L] [Module k L] |
| (V W : ModuleCat.{u} k) (h : V=W) (x : L) (v : V) : |
| (eqToHom (congrArg (fun T : ModuleCat k => ModuleCat.of k (L ⊗[k] T)) h)).hom (x ⊗ₜ[k] v) = |
| x ⊗ₜ[k] (eqToHom h).hom v := by |
| subst W |
| rfl |
|
|
| theorem eqToHom_outer_tmul (V W : ModuleCat.{u} k) (h : V=W) (d : D) (v : V) : |
| (eqToHom (congrArg (fun T : ModuleCat k => ModuleCat.of D (D ⊗[k] T)) h)).hom (d ⊗ₜ[k] v) = |
| d ⊗ₜ[k] (eqToHom h).hom v := by |
| subst W |
| rfl |
|
|
| /-- The old indexed inclusion and old list-pattern embedding agree; transports |
| come solely from the proved equality of their tensor objects and their degrees. -/ |
| theorem indexedInclusion_toList {n r : ℕ} (p : Multibar.Indexed.Shape n r) |
| (v : Multibar.Indexed.tensor k D X N p) : |
| Multibar.Indexed.inclusion k D X N p v = |
| termCast k D X N (Multibar.Indexed.length_toList p) |
| (embed k D X N (Multibar.Indexed.toList p) |
| ((eqToHom (Multibar.Indexed.tensor_toList k D X N p)).hom v)) := by |
| induction p with |
| | nil => rfl |
| | @base n r p ih => |
| induction v using TensorProduct.inductionOn with |
| | tmul d v => |
| change (inl d : A) ⊗ₜ[k] Multibar.Indexed.inclusion k D X N p v = |
| termCast k D X N (show (false :: Multibar.Indexed.toList p).length=n+1 by simp) |
| (embed k D X N (false :: Multibar.Indexed.toList p) |
| ((eqToHom (congrArg (fun T : ModuleCat k => ModuleCat.of k (D ⊗[k] T)) |
| (Multibar.Indexed.tensor_toList k D X N p))).hom (d ⊗ₜ[k] v))) |
| rw [eqToHom_tensor_tmul k D (Multibar.Indexed.tensor k D X N p) (patternTerm k D X N (Multibar.Indexed.toList p)) (Multibar.Indexed.tensor_toList k D X N p) d v] |
| change (inl d : A) ⊗ₜ[k] Multibar.Indexed.inclusion k D X N p v = |
| termCast k D X N (show (false :: Multibar.Indexed.toList p).length=n+1 by simp) ((inl d : A) ⊗ₜ[k] _) |
| rw [ih] |
| exact (termCast_tmul k D X N (Multibar.Indexed.length_toList p) |
| (by simp) _ _).symm |
| | add v w hv hw => simp only [map_add, hv, hw] |
| | @ideal n r p ih => |
| induction v using TensorProduct.inductionOn with |
| | tmul x v => |
| change (inr x : A) ⊗ₜ[k] Multibar.Indexed.inclusion k D X N p v = |
| termCast k D X N (show (true :: Multibar.Indexed.toList p).length=n+1 by simp) |
| (embed k D X N (true :: Multibar.Indexed.toList p) |
| ((eqToHom (congrArg (fun T : ModuleCat k => ModuleCat.of k (X ⊗[k] T)) |
| (Multibar.Indexed.tensor_toList k D X N p))).hom (x ⊗ₜ[k] v))) |
| rw [eqToHom_tensor_tmul k X (Multibar.Indexed.tensor k D X N p) (patternTerm k D X N (Multibar.Indexed.toList p)) (Multibar.Indexed.tensor_toList k D X N p) x v] |
| change (inr x : A) ⊗ₜ[k] Multibar.Indexed.inclusion k D X N p v = |
| termCast k D X N (show (true :: Multibar.Indexed.toList p).length=n+1 by simp) ((inr x : A) ⊗ₜ[k] _) |
| rw [ih] |
| exact (termCast_tmul k D X N (Multibar.Indexed.length_toList p) |
| (by simp) _ _).symm |
| | add v w hv hw => simp only [map_add, hv, hw] |
|
|
| theorem outerInclusion_toList {n r : ℕ} (p : Multibar.Indexed.Shape n r) |
| (v : Multibar.Indexed.outerTensor k D X N p) : |
| Multibar.Indexed.outerInclusion k D X N p v = |
| outerCast k D X N (Multibar.Indexed.length_toList p) |
| (AlgebraTensorModule.lTensor D D (embed k D X N (Multibar.Indexed.toList p)) |
| ((eqToHom (congrArg (fun T : ModuleCat k => ModuleCat.of D (D ⊗[k] T)) |
| (Multibar.Indexed.tensor_toList k D X N p))).hom v)) := by |
| induction v using TensorProduct.inductionOn with |
| | tmul d v => |
| rw [eqToHom_outer_tmul k D (Multibar.Indexed.tensor k D X N p) (patternTerm k D X N (Multibar.Indexed.toList p)) (Multibar.Indexed.tensor_toList k D X N p) d v] |
| change d ⊗ₜ[k] Multibar.Indexed.inclusion k D X N p v = |
| d ⊗ₜ[k] termCast k D X N _ |
| (embed k D X N (Multibar.Indexed.toList p) |
| ((eqToHom (Multibar.Indexed.tensor_toList k D X N p)).hom v)) |
| rw [indexedInclusion_toList] |
| | add v w hv hw => simp only [map_add, hv, hw] |
|
|
| theorem outerInclusion_pattern {n r : ℕ} (s : Multibar.Indexed.Shape n r) |
| (p : List Bool) (hp : Multibar.Indexed.toList s=p) |
| (ht : Multibar.Indexed.tensor k D X N s=patternTerm k D X N p) |
| (hl : p.length=n) (v : Multibar.Indexed.outerTensor k D X N s) : |
| Multibar.Indexed.outerInclusion k D X N s v = |
| outerCast k D X N hl (AlgebraTensorModule.lTensor D D (embed k D X N p) |
| ((eqToHom (congrArg (fun T : ModuleCat k => ModuleCat.of D (D ⊗[k] T)) ht)).hom v)) := by |
| subst p |
| exact outerInclusion_toList k D X N s v |
| end DerivedBar.ParityBridge |
|
|
| end |
|
|
| end OAI |
|
|
