| import Mathlib.Basic.Complex.Basic |
| import OAI.Algebra.Finitistic.Selection.MatrixModel |
|
|
| namespace OAI |
| namespace LittleFinitistic |
|
|
| noncomputable def parityValue (a : ZMod 2) : ℂˣ := if a = 0 then 1 else -1 |
|
|
| @[simp] theorem parityValue_zero : parityValue 0 = 1 := by simp [parityValue] |
| @[simp] theorem parityValue_one : parityValue 1 = -1 := by norm_num [parityValue] |
|
|
| theorem parityValue_add (a b : ZMod 2) : parityValue (a+b) = parityValue a * parityValue b := by |
| have hc : ∀ a : ZMod 2, a = 0 ∨ a = 1 := by decide |
| rcases hc a with rfl | rfl <;> rcases hc b with rfl | rfl |
| · simp |
| · simp |
| · simp |
| · rw [show (1 : ZMod 2) + 1 = 0 from rfl,parityValue_zero,parityValue_one] |
| simp |
|
|
| def lastIndex (m : ℕ) [NeZero m] : Fin m := ⟨m-1,by have := NeZero.ne m; omega⟩ |
|
|
| noncomputable def selectionCharacter (m : ℕ) [NeZero m] : |
| Multiplicative (Fin m → ZMod 2) →* ℂˣ where |
| toFun c := parityValue (c.toAdd (lastIndex m)) |
| map_one' := parityValue_zero |
| map_mul' first second := parityValue_add (first.toAdd (lastIndex m)) |
| (second.toAdd (lastIndex m)) |
|
|
| @[simp] theorem selectionCharacter_basis (m : ℕ) [NeZero m] (i : Fin m) : |
| selectionCharacter m (Multiplicative.ofAdd (Pi.single i 1)) = |
| if i = lastIndex m then -1 else 1 := by |
| classical |
| change parityValue ((Pi.single i (1 : ZMod 2) : Fin m → ZMod 2) (lastIndex m)) = _ |
| by_cases h : i = lastIndex m |
| · subst i |
| simp |
| · simp [h,Ne.symm h] |
|
|
| end LittleFinitistic |
|
|
| end OAI |
|
|
