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import OAI.Computability.MinUncut.Encoding.BinaryCoordinates
namespace OAI
section
namespace MinUncutGames.Integration.BinaryLinear
open BinaryCoordinates
abbrev F2 := ZMod 2
abbrev Vector (n : Nat) := Fin n → F2
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 MinUncutGames.Integration.BinaryLinear
end
end OAI
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