| import Mathlib.Data.Matrix.Basic |
| import Mathlib.Data.Rat.Defs |
|
|
| namespace OAI |
|
|
| namespace NCHitting |
|
|
| abbrev Word (n : ℕ) := List (Fin n) |
| abbrev WordSeries (F : Type*) (n : ℕ) := Word n → F |
|
|
| def convolution {F : Type*} {n : ℕ} [Semiring F] |
| (f g : WordSeries F n) : Word n → F |
| | [] => f [] * g [] |
| | i :: w => f [] * g (i :: w) + convolution (fun v => f (i :: v)) g w |
|
|
| inductive Formula (F : Type*) (n : ℕ) where |
| | scalar : F → Formula F n |
| | var : Fin n → Formula F n |
| | add : Formula F n → Formula F n → Formula F n |
| | mul : Formula F n → Formula F n → Formula F n |
|
|
| namespace Formula |
|
|
| def size {F : Type*} {n : ℕ} : Formula F n → ℕ |
| | scalar _ => 1 |
| | var _ => 1 |
| | add f g => f.size + g.size + 1 |
| | mul f g => f.size + g.size + 1 |
|
|
| noncomputable def coeff {F : Type*} {n : ℕ} [Semiring F] : |
| Formula F n → WordSeries F n |
| | scalar a => fun w => if w = [] then a else 0 |
| | var i => fun w => if w = [i] then 1 else 0 |
| | add f g => fun w => coeff f w + coeff g w |
| | mul f g => convolution (coeff f) (coeff g) |
|
|
| def eval {F A : Type*} {n : ℕ} [Semiring A] |
| (scalarMap : F → A) (T : Fin n → A) : Formula F n → A |
| | scalar a => scalarMap a |
| | var i => T i |
| | add f g => eval scalarMap T f + eval scalarMap T g |
| | mul f g => eval scalarMap T f * eval scalarMap T g |
|
|
| end Formula |
|
|
| def dimension (n s : ℕ) : ℕ := if n = 1 ∧ s = 1 then 1 else 2 * n * s ^ 2 |
|
|
| def matrixEntry (i r q : ℕ) : ℚ := |
| if q < r then (-1 : ℚ) ^ (r - q - 1) / ((r : ℚ) * (i : ℚ) ^ (r - q)) |
| else 0 |
|
|
| def generator (n s : ℕ) : Fin n → Matrix (Fin (dimension n s)) (Fin (dimension n s)) ℚ := |
| fun i r q => if n = 1 ∧ s = 1 then 1 else matrixEntry (i.val + 1) r.val q.val |
|
|
| end NCHitting |
|
|
| end OAI |
|
|
