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ae00908 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 | 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
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