File size: 1,933 Bytes
ea76aac | 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 59 60 61 62 63 64 65 66 67 68 69 | 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
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