| 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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|
