| import Mathlib.Data.ZMod.Basic |
| import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas |
| import OAI.Computability.UniqueGames.Model |
| import OAI.Computability.UniqueGames.Reduction.BinaryCoordinates |
|
|
| namespace OAI |
|
|
| section |
|
|
| /-! Exact transport from XOR bit vectors to vector spaces over `ZMod 2`. -/ |
| namespace UniqueGamesTheorem.Integration.BinaryLinear |
|
|
| open BinaryCoordinates |
|
|
| def ofBit (b : Bool) : F2 := if b then 1 else 0 |
| def toBit (a : F2) : Bool := decide (a = 1) |
|
|
| theorem scalar_cases : ∀ a : F2, a = 0 ∨ a = 1 := by decide |
| theorem ofBit_toBit : ∀ a : F2, ofBit (toBit a) = a := by decide |
| theorem toBit_ofBit : ∀ b : Bool, toBit (ofBit b) = b := by decide |
| theorem ofBit_xor : ∀ a b : Bool, ofBit (a ^^ b) = ofBit a + ofBit b := by decide |
| theorem toBit_add : ∀ a b : F2, toBit (a + b) = (toBit a ^^ toBit b) := by decide |
|
|
| def toVector {n : Nat} (v : BitVec n) : Vector n := fun i => ofBit (unpack v i) |
| def fromVector {n : Nat} (v : Vector n) : BitVec n := pack (fun i => toBit (v i)) |
|
|
| theorem toVector_fromVector {n : Nat} (v : Vector n) : toVector (fromVector v) = v := by |
| funext i |
| simp only [toVector, fromVector, unpack_pack, ofBit_toBit] |
|
|
| theorem fromVector_toVector {n : Nat} (v : BitVec n) : fromVector (toVector v) = v := by |
| simp only [fromVector, toVector, toBit_ofBit] |
| exact pack_unpack v |
|
|
| def coordinatesEquiv (n : Nat) : BitVec n ≃ Vector n where |
| toFun := toVector |
| invFun := fromVector |
| left_inv := fromVector_toVector |
| right_inv := toVector_fromVector |
|
|
| theorem toVector_xor {n : Nat} (u v : BitVec n) : |
| toVector (u ^^^ v) = toVector u + toVector v := by |
| funext i |
| simp only [toVector, unpack_xor, ofBit_xor, Pi.add_apply] |
|
|
| theorem fromVector_add {n : Nat} (u v : Vector n) : |
| fromVector (u + v) = fromVector u ^^^ fromVector v := by |
| simp only [fromVector, Pi.add_apply, toBit_add, pack_xor] |
|
|
| @[simp] theorem toVector_zero {n : Nat} : toVector (0 : BitVec n) = 0 := by |
| unfold toVector |
| rw [unpack_zero] |
| rfl |
|
|
| @[simp] theorem fromVector_zero {n : Nat} : fromVector (0 : Vector n) = 0 := by |
| simp [fromVector, toBit, pack_zero] |
|
|
| / |
| def liftLinear {n m : Nat} (f : BitVec n → BitVec m) |
| (hzero : f 0 = 0) (hadd : ∀ x y, f (x ^^^ y) = f x ^^^ f y) : |
| Vector n →ₗ[F2] Vector m where |
| toFun x := toVector (f (fromVector x)) |
| map_add' x y := by rw [fromVector_add, hadd, toVector_xor] |
| map_smul' c x := by |
| rcases scalar_cases c with rfl | rfl |
| · change toVector (f (fromVector (0 • x))) = 0 • toVector (f (fromVector x)) |
| rw [zero_smul, fromVector_zero, hzero, toVector_zero, zero_smul] |
| · simp |
|
|
| theorem liftLinear_apply {n m : Nat} (f : BitVec n → BitVec m) |
| (hzero : f 0 = 0) (hadd : ∀ x y, f (x ^^^ y) = f x ^^^ f y) |
| (x : Vector n) : |
| liftLinear f hzero hadd x = toVector (f (fromVector x)) := rfl |
|
|
| theorem finrank_vector (n : Nat) : Module.finrank F2 (Vector n) = n := by |
| simp [Vector] |
|
|
| end UniqueGamesTheorem.Integration.BinaryLinear |
|
|
| end |
|
|
| end OAI |
|
|
