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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 | import OAI.Algebra.Finitistic.Encoding.RelationBasisCoefficients
namespace OAI
noncomputable section
open scoped TensorProduct
namespace LittleFinitistic.EncodingGenerators
open QuadraticEncoding
universe u
variable {k : Type u} [Field k] {d t : ℕ} (P : Relations k d t)
/-- The entire first arrow space inside the encoding algebra. -/
def first : Arrows k d →ₗ[k] B P where
toFun x := (0,0,0,x,0,0)
map_add' x y := by ext <;> simp
map_smul' c x := by ext <;> simp
/-- The entire second arrow space inside the encoding algebra. -/
def second : Arrows k d →ₗ[k] B P where
toFun x := (0,0,0,0,x,0)
map_add' x y := by ext <;> simp
map_smul' c x := by ext <;> simp
lemma first_mem (S : Subalgebra k (B P)) (h : ∀ a, arrow01 P a ∈ S)
(x : Arrows k d) : first P x ∈ S := by
have hx := congrArg (first P) ((Pi.basisFun k (Fin (d+1))).sum_repr x)
rw [map_sum] at hx
rw [← hx]
exact S.sum_mem fun a _ => by
simpa only [map_smul, Pi.basisFun_apply, first, LinearMap.coe_mk,
AddHom.coe_mk, arrow01, basis] using S.smul_mem (h a) (((Pi.basisFun k (Fin (d+1))).repr x) a)
lemma second_mem (S : Subalgebra k (B P)) (h : ∀ a, arrow12 P a ∈ S)
(x : Arrows k d) : second P x ∈ S := by
have hx := congrArg (second P) ((Pi.basisFun k (Fin (d+1))).sum_repr x)
rw [map_sum] at hx
rw [← hx]
exact S.sum_mem fun a _ => by
simpa only [map_smul, Pi.basisFun_apply, second, LinearMap.coe_mk,
AddHom.coe_mk, arrow12, basis] using S.smul_mem (h a) (((Pi.basisFun k (Fin (d+1))).repr x) a)
lemma path_mem (S : Subalgebra k (B P))
(h01 : ∀ a, arrow01 P a ∈ S) (h12 : ∀ a, arrow12 P a ∈ S)
(x : Paths k d) : pathMap P x ∈ S := by
have hx := congrArg (pathMap P)
((RelationBasisCoefficients.pathBasis (k := k) (d := d)).sum_repr x)
rw [map_sum] at hx
rw [← hx]
apply S.sum_mem
intro ⟨b,a⟩ _
rw [map_smul]
apply S.smul_mem
simp only [RelationBasisCoefficients.pathBasis, Module.Basis.tensorProduct_apply,
Pi.basisFun_apply]
change pathMap P (basis b ⊗ₜ[k] basis a) ∈ S
rw [← arrow_product]
exact S.mul_mem (h12 b) (h01 a)
/-- Vertices and the specified finite arrow list generate the full encoding,
including the quotient length-two path component. -/
theorem eq_top (S : Subalgebra k (B P))
(hv : ∀ i, ThreeVertexAlgebra.vertex (composition P) i ∈ S)
(h01 : ∀ a, arrow01 P a ∈ S) (h12 : ∀ a, arrow12 P a ∈ S) : S = ⊤ := by
apply top_unique
intro x _
rcases x with ⟨c0,c1,c2,v,w,z⟩
obtain ⟨r,hr⟩ := Submodule.Quotient.mk_surjective (relationSpace P) z
have hz : lengthTwo P z ∈ S := by
rw [← hr]
exact path_mem P S h01 h12 r
have h := S.add_mem (S.add_mem (S.add_mem (S.add_mem (S.add_mem
(S.smul_mem (hv 0) c0) (S.smul_mem (hv 1) c1)) (S.smul_mem (hv 2) c2))
(first_mem P S h01 v)) (second_mem P S h12 w)) hz
simpa [ThreeVertexAlgebra.vertex, ThreeVertexAlgebra.add_eq,
ThreeVertexAlgebra.smul_eq, first, second, lengthTwo] using h
end LittleFinitistic.EncodingGenerators
end
end OAI
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