| import Mathlib.LinearAlgebra.Pi |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace ArExplicit |
| universe u v w z |
| variable {k : Type u} [Field k] {ι : Type v} {κ : Type w} |
| variable (M : κ → Type z) [∀ j, AddCommGroup (M j)] [∀ j, Module k (M j)] |
| variable (f : ι → κ) |
|
|
| def supportedRestriction : (∀ j, M j) →ₗ[k] (∀ i, M (f i)) where |
| toFun x i := x (f i) |
| map_add' _ _ := rfl |
| map_smul' _ _ := rfl |
|
|
| theorem supportedRestriction_bijective (hf : Function.Injective f) |
| (hz : ∀ j, (¬ ∃ i, f i = j) → ∀ x : M j, x = 0) : |
| Function.Bijective (supportedRestriction (k := k) M f) := by |
| constructor |
| · intro x y hxy |
| funext j |
| by_cases hj : ∃ i, f i = j |
| · obtain ⟨i, rfl⟩ := hj |
| exact congrFun hxy i |
| · exact (hz j hj (x j)).trans (hz j hj (y j)).symm |
| · intro y |
| classical |
| let x : ∀ j, M j := fun j => if hj : ∃ i, f i = j then |
| hj.choose_spec ▸ y hj.choose else 0 |
| refine ⟨x, ?_⟩ |
| funext i |
| change x (f i) = y i |
| have hi : ∃ a, f a = f i := ⟨i, rfl⟩ |
| simp only [x, dite_eq_left hi] |
| have hs (a : ι) (ha : f a = f i) : ha ▸ y a = y i := by |
| have hai : a = i := hf ha |
| subst a |
| rfl |
| exact hs _ _ |
|
|
| def supportedPiEquiv (hf : Function.Injective f) |
| (hz : ∀ j, (¬ ∃ i, f i = j) → ∀ x : M j, x = 0) : |
| (∀ j, M j) ≃ₗ[k] (∀ i, M (f i)) := |
| LinearEquiv.ofBijective (supportedRestriction M f) |
| (supportedRestriction_bijective M f hf hz) |
|
|
| end ArExplicit |
|
|
| end |
|
|
| end OAI |
|
|
