File size: 3,195 Bytes
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 | import OAI.Algebra.Finitistic.Bar.PatternPrefixCoordinates
import OAI.Algebra.Finitistic.Bar.BarCoordinateNaturality
namespace OAI
open CategoryTheory TensorProduct TrivSqZeroExt
open scoped TensorProduct
noncomputable section
namespace DerivedBar.ParityBridge
open DerivedBar.Multibar
universe u
variable (k D X N : 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]
[AddCommGroup N] [Module k N] [Module D N] [IsScalarTower k D N]
local notation "A" => TrivSqZeroExt D X
/-- The categorical X-bar pattern coordinates and the suffix formula have
exactly the same embedding into the original A bar, including the degree cast. -/
theorem embed_twoSidedBarPatternIso (i : ℕ) (p : List Bool)
(v : ((BimoduleTensor.twoSidedBarFunctor k D X).obj
(ModuleCat.of D (patternTerm k D X N p))).X i) :
embed k D X N (true :: (List.replicate i false ++ p))
((twoSidedBarPatternIso k D X N i p).hom.hom v) =
termCast k D X N
(show (p.length+i)+1=(true :: (List.replicate i false ++ p)).length by simp; omega)
(idealSuffixEmbed k D X N (patternTerm k D X N p) p.length i (embed k D X N p)
((twoSidedBarObjIso k D X (patternTerm k D X N p) i).hom.hom v)) := by
let e := (termPatternIso k D X N i p).toLinearEquiv.restrictScalars k
suffices h : ∀ w : X ⊗[k] term k D (patternTerm k D X N p) i,
embed k D X N (true :: (List.replicate i false ++ p))
(AlgebraTensorModule.congr (LinearEquiv.refl D X) e w) =
termCast k D X N
(show (p.length+i)+1=(true :: (List.replicate i false ++ p)).length by simp; omega)
(idealSuffixEmbed k D X N (patternTerm k D X N p) p.length i (embed k D X N p) w) from
h ((twoSidedBarObjIso k D X (patternTerm k D X N p) i).hom.hom v)
intro w
induction w using TensorProduct.inductionOn with
| tmul x w =>
change (inr x : A) ⊗ₜ[k]
embed k D X N (List.replicate i false ++ p) ((termPatternIso k D X N i p).hom.hom w) = _
rw [embed_termPatternIso, idealSuffixEmbed_tmul]
exact (termCast_tmul k D X N
(show p.length+i=(List.replicate i false ++ p).length by simp; omega)
(show (p.length+i)+1=(List.replicate i false ++ p).length+1 by simp; omega)
(inr x) (suffixEmbed k D X N _ p.length (embed k D X N p) i w)).symm
| add v w hv hw => simp only [map_add, hv, hw]
/-- The same compatibility for the final free D bar, before projecting its exterior slot. -/
theorem embed_freeBarPatternIso (i : ℕ) (p : List Bool)
(v : ((barFunctor k D).obj (ModuleCat.of D (patternTerm k D X N p))).X i) :
embed k D X N (List.replicate (i+1) false ++ p)
((freeBarPatternIso k D X N i p).hom.hom v) =
termCast k D X N
(show p.length+(i+1)=(List.replicate (i+1) false ++ p).length by simp; omega)
(suffixEmbed k D X N (patternTerm k D X N p) p.length (embed k D X N p) (i+1)
((freeBarObjIso k D (patternTerm k D X N p) i).hom.hom v)) :=
embed_termPatternIso k D X N (i+1) p
((freeBarObjIso k D (patternTerm k D X N p) i).hom.hom v)
end DerivedBar.ParityBridge
end
end OAI
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