| import OAI.Algebra.Finitistic.Rectification.VertexModuleAssembly |
| import OAI.Algebra.Finitistic.Encoding.EncodingLinearity |
| import OAI.Algebra.Finitistic.Rectification.CanonicalArrowCompatibility |
|
|
| namespace OAI |
| noncomputable section |
| namespace LittleFinitistic.EncodingModuleAssembly |
| open QuadraticEncoding VertexDecomposition VertexModuleAssembly |
| universe u v w |
| variable {k : Type u} [Field k] {d t : ℕ} (P : Relations k d t) |
| variable {M : Type v} {N : Type w} [AddCommGroup M] [AddCommGroup N] |
| [Module k M] [Module k N] [Module (B P) M] [Module (B P) N] |
| [IsScalarTower k (B P) M] [IsScalarTower k (B P) N] |
|
|
| abbrev system := ThreeVertexAlgebra.directedSystem (composition P) |
|
|
| /-- Reconstruct a full module isomorphism from the three actual vertex-space |
| isomorphisms and the two finite arrow families. Its underlying vector-space map |
| is precisely the direct-sum assembly of the supplied vertex isomorphisms. -/ |
| def equiv (E : ∀ i, Space (system P) M i ≃ₗ[k] Space (system P) N i) |
| (h01 : ∀ a x, E 1 (arrow (system P) (i := 0) (j := 1) (arrow01 P a) |
| (CanonicalArrowCompatibility.arrow01_target P a) x) = |
| arrow (system P) (i := 0) (j := 1) (arrow01 P a) |
| (CanonicalArrowCompatibility.arrow01_target P a) (E 0 x)) |
| (h12 : ∀ b x, E 2 (arrow (system P) (i := 1) (j := 2) (arrow12 P b) |
| (CanonicalArrowCompatibility.arrow12_target P b) x) = |
| arrow (system P) (i := 1) (j := 2) (arrow12 P b) |
| (CanonicalArrowCompatibility.arrow12_target P b) (E 1 x)) : M ≃ₗ[B P] N := |
| EncodingLinearity.linearEquiv P (assembled (system P) E) |
| (vertex_smul (system P) E) |
| (fun a => arrow_smul (system P) E (arrow01 P a) |
| (CanonicalArrowCompatibility.arrow01_target P a) |
| ((ConcreteQuadraticDiagram.quadratic P).ha a) (h01 a)) |
| (fun b => arrow_smul (system P) E (arrow12 P b) |
| (CanonicalArrowCompatibility.arrow12_target P b) |
| ((ConcreteQuadraticDiagram.quadratic P).hb b) (h12 b)) |
| end LittleFinitistic.EncodingModuleAssembly |
|
|
| end |
|
|
| end OAI |
|
|
