| import Mathlib.LinearAlgebra.Pi |
| import OAI.Computability.PerfectCompleteness.Foundations.FiniteRangeInverse |
|
|
| namespace OAI |
|
|
| section |
|
|
| namespace PerfectCompleteness.EvaluationMatrix |
|
|
| noncomputable section |
|
|
| variable {𝕜 Ω K : Type*} [Field 𝕜] [AddCommGroup K] [Module 𝕜 K] |
|
|
| def evaluation (H : Submodule 𝕜 (Ω → 𝕜)) (x : Ω) : Module.Dual 𝕜 H where |
| toFun h := h.val x |
| map_add' _ _ := rfl |
| map_smul' _ _ := rfl |
|
|
| @[simp] theorem evaluation_apply (H : Submodule 𝕜 (Ω → 𝕜)) (x : Ω) (h : H) : |
| evaluation H x h = h.val x := rfl |
|
|
| def evaluate (H : Submodule 𝕜 (Ω → 𝕜)) (X : Module.Dual 𝕜 H →ₗ[𝕜] K) (x : Ω) : K := |
| X (evaluation H x) |
|
|
| def shift (H : Submodule 𝕜 (Ω → 𝕜)) (X : Module.Dual 𝕜 H →ₗ[𝕜] K) |
| (a : K) (h : H) : Module.Dual 𝕜 H →ₗ[𝕜] K := |
| X + (Module.Dual.eval 𝕜 H h).smulRight a |
|
|
| @[simp] theorem evaluate_shift (H : Submodule 𝕜 (Ω → 𝕜)) |
| (X : Module.Dual 𝕜 H →ₗ[𝕜] K) (a : K) (h : H) (x : Ω) : |
| evaluate H (shift H X a h) x = evaluate H X x + h.val x • a := rfl |
|
|
| def joint {Z : Type*} (H : Submodule 𝕜 (Ω → 𝕜)) (other : Ω → Z) |
| (X : Module.Dual 𝕜 H →ₗ[𝕜] K) (x : Ω) : K × Z := |
| (evaluate H X x, other x) |
|
|
| @[simp] theorem joint_shift {Z : Type*} (H : Submodule 𝕜 (Ω → 𝕜)) |
| (other : Ω → Z) (X : Module.Dual 𝕜 H →ₗ[𝕜] K) (a : K) (h : H) (x : Ω) : |
| joint H other (shift H X a h) x = (evaluate H X x + h.val x • a, other x) := rfl |
|
|
| section Rows |
|
|
| variable (H : Submodule 𝕜 (Ω → 𝕜)) [FiniteDimensional 𝕜 H] {ℓ : Nat} |
|
|
| def rows (X : Module.Dual 𝕜 H →ₗ[𝕜] (Fin ℓ → 𝕜)) : Fin ℓ → H := |
| fun i => (Module.evalEquiv 𝕜 H).symm ((LinearMap.proj i).comp X) |
|
|
| def ofRows (v : Fin ℓ → H) : Module.Dual 𝕜 H →ₗ[𝕜] (Fin ℓ → 𝕜) := |
| LinearMap.pi (fun i => Module.Dual.eval 𝕜 H (v i)) |
|
|
| omit [FiniteDimensional 𝕜 H] in |
| @[simp] theorem ofRows_apply (v : Fin ℓ → H) (q : Module.Dual 𝕜 H) (i : Fin ℓ) : |
| ofRows H v q i = q (v i) := rfl |
|
|
| @[simp] theorem ofRows_rows (X : Module.Dual 𝕜 H →ₗ[𝕜] (Fin ℓ → 𝕜)) : |
| ofRows H (rows H X) = X := by |
| ext q i |
| exact Module.apply_evalEquiv_symm_apply 𝕜 H q ((LinearMap.proj i).comp X) |
|
|
| @[simp] theorem rows_ofRows (v : Fin ℓ → H) : rows H (ofRows H v) = v := by |
| funext i |
| exact (Module.evalEquiv 𝕜 H).symm_apply_apply (v i) |
|
|
| def rowsEquiv : (Module.Dual 𝕜 H →ₗ[𝕜] (Fin ℓ → 𝕜)) ≃ (Fin ℓ → H) where |
| toFun := rows H |
| invFun := ofRows H |
| left_inv := ofRows_rows H |
| right_inv := rows_ofRows H |
|
|
| omit [FiniteDimensional 𝕜 H] in |
| @[simp] theorem evaluate_ofRows (v : Fin ℓ → H) (x : Ω) : |
| evaluate H (ofRows H v) x = fun i => (v i).val x := rfl |
|
|
| theorem rows_apply (X : Module.Dual 𝕜 H →ₗ[𝕜] (Fin ℓ → 𝕜)) (i : Fin ℓ) (x : Ω) : |
| (rows H X i).val x = evaluate H X x i := |
| Module.apply_evalEquiv_symm_apply 𝕜 H (evaluation H x) ((LinearMap.proj i).comp X) |
|
|
| theorem evaluate_injective : |
| Function.Injective (evaluate H : (Module.Dual 𝕜 H →ₗ[𝕜] (Fin ℓ → 𝕜)) → (Ω → Fin ℓ → 𝕜)) := by |
| intro X X' h |
| apply (rowsEquiv H).injective |
| funext i |
| apply Subtype.ext |
| funext x |
| change (rows H X i).val x = (rows H X' i).val x |
| rw [rows_apply, rows_apply] |
| exact congrFun (congrFun h x) i |
|
|
| end Rows |
| end |
| end PerfectCompleteness.EvaluationMatrix |
|
|
| end |
|
|
| end OAI |
|
|
