import Mathlib
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