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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 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 | 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
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