| import Mathlib |
| import OAI.Computability.MinUncut.Model |
|
|
| namespace OAI |
|
|
| section |
| namespace MinUncutGames.Integration.BinaryCoordinates |
|
|
| abbrev Coordinates (n : Nat) := Fin n → Bool |
|
|
| def pack : {n : Nat} → Coordinates n → BitVec n |
| | 0, _ => 0 |
| | _ + 1, f => BitVec.concat (pack (fun i => f i.succ)) (f 0) |
|
|
| def unpack {n : Nat} (v : BitVec n) : Coordinates n := |
| fun i => v.getLsbD i.val |
|
|
| theorem pack_get {n : Nat} (f : Coordinates n) (i : Fin n) : |
| (pack f).getLsbD i.val = f i := by |
| induction n with |
| | zero => exact Fin.elim0 i |
| | succ n ih => |
| refine Fin.cases ?_ (fun j => ?_) i |
| · exact BitVec.getLsbD_concat_zero |
| · exact (BitVec.getLsbD_concat_succ).trans (ih (fun j => f j.succ) j) |
|
|
| theorem unpack_pack {n : Nat} (f : Coordinates n) : unpack (pack f) = f := by |
| funext i |
| exact pack_get f i |
|
|
| theorem pack_unpack {n : Nat} (v : BitVec n) : pack (unpack v) = v := by |
| apply BitVec.eq_of_getLsbD_eq |
| intro i hi |
| exact pack_get (unpack v) ⟨i, hi⟩ |
|
|
| theorem unpack_injective {n : Nat} : Function.Injective (@unpack n) := by |
| intro u v h |
| have h' := congrArg pack h |
| simpa only [pack_unpack] using h' |
|
|
| theorem pack_injective {n : Nat} : Function.Injective (@pack n) := by |
| intro u v h |
| have h' := congrArg unpack h |
| simpa only [unpack_pack] using h' |
|
|
| theorem unpack_xor {n : Nat} (u v : BitVec n) : |
| unpack (u ^^^ v) = fun i => (unpack u i ^^ unpack v i) := by |
| funext i |
| exact BitVec.getLsbD_xor |
|
|
| theorem pack_xor {n : Nat} (u v : Coordinates n) : |
| pack (fun i => (u i ^^ v i)) = pack u ^^^ pack v := by |
| apply unpack_injective |
| rw [unpack_pack, unpack_xor, unpack_pack, unpack_pack] |
|
|
| theorem unpack_zero {n : Nat} : unpack (0 : BitVec n) = fun _ => false := by |
| funext i |
| simp [unpack] |
|
|
| theorem pack_zero {n : Nat} : pack (fun _ : Fin n => false) = 0 := by |
| apply unpack_injective |
| rw [unpack_pack, unpack_zero] |
|
|
| end MinUncutGames.Integration.BinaryCoordinates |
|
|
| end |
|
|
| end OAI |
|
|
