Bitwise operations on integers

Possibly only of archaeological significance.

Recursors

• Int.bitCasesOn: Parity disjunction. Something is true/defined on ℤ if it's true/defined for even and for odd values.

def Int.div2 : ℤ → ℤ

div2 n = n/2

Equations

• Int.div2 x = match x with | Int.ofNat n => ↑(Nat.div2 n) | Int.negSucc n => Int.negSucc (Nat.div2 n)

def Int.bodd : ℤ → Bool

bodd n returns true if n is odd

Equations

• Int.bodd x = match x with | Int.ofNat n => Nat.bodd n | Int.negSucc n => !Nat.bodd n

def Int.bit (b : Bool) : ℤ → ℤ

bit b appends the digit b to the binary representation of its integer input.

Equations

• Int.bit b = bif b then bit1 else bit0

def Int.testBit : ℤ → ℕ → Bool

testBit m n returns whether the (n+1)ˢᵗ least significant bit is 1 or 0

Equations

• Int.testBit x✝ x = match x✝, x with | Int.ofNat m, n => Nat.testBit m n | Int.negSucc m, n => !Nat.testBit m n

def Int.natBitwise (f : Bool → Bool → Bool) (m : ℕ) (n : ℕ) : ℤ

Int.natBitwise is an auxiliary definition for Int.bitwise.

Equations

• Int.natBitwise f m n = bif f false false then Int.negSucc (Nat.bitwise (fun (x y : Bool) => !f x y) m n) else ↑(Nat.bitwise f m n)

def Int.bitwise (f : Bool → Bool → Bool) : ℤ → ℤ → ℤ

Int.bitwise applies the function f to pairs of bits in the same position in the binary representations of its inputs.

Equations

• One or more equations did not get rendered due to their size.

def Int.lnot : ℤ → ℤ

lnot flips all the bits in the binary representation of its input

Equations

• Int.lnot x = match x with | Int.ofNat m => Int.negSucc m | Int.negSucc m => ↑m

def Int.lor : ℤ → ℤ → ℤ

lor takes two integers and returns their bitwise or

Equations

• One or more equations did not get rendered due to their size.

def Int.land : ℤ → ℤ → ℤ

land takes two integers and returns their bitwise and

Equations

• One or more equations did not get rendered due to their size.

def Int.ldiff : ℤ → ℤ → ℤ

ldiff a b performs bitwise set difference. For each corresponding pair of bits taken as booleans, say aᵢ and bᵢ, it applies the boolean operation aᵢ ∧ bᵢ to obtain the iᵗʰ bit of the result.

Equations

• One or more equations did not get rendered due to their size.

def Int.xor : ℤ → ℤ → ℤ

xor computes the bitwise xor of two natural numbers

Equations

• One or more equations did not get rendered due to their size.

instance Int.instShiftLeftInt : ShiftLeft ℤ

m <<< n produces an integer whose binary representation is obtained by left-shifting the binary representation of m by n places

Equations

• One or more equations did not get rendered due to their size.

instance Int.instShiftRightInt : ShiftRight ℤ

m >>> n produces an integer whose binary representation is obtained by right-shifting the binary representation of m by n places

Equations

• Int.instShiftRightInt = { shiftRight := fun (m n : ℤ) => m <<< (-n) }

bitwise ops

@[simp]

theorem Int.bodd_zero : Int.bodd 0 = false

@[simp]

theorem Int.bodd_one : Int.bodd 1 = true

theorem Int.bodd_two : Int.bodd 2 = false

@[simp]

theorem Int.bodd_coe (n : ℕ) : Int.bodd ↑n = Nat.bodd n

@[simp]

theorem Int.bodd_subNatNat (m : ℕ) (n : ℕ) : Int.bodd (Int.subNatNat m n) = xor (Nat.bodd m) (Nat.bodd n)

@[simp]

theorem Int.bodd_negOfNat (n : ℕ) : Int.bodd (Int.negOfNat n) = Nat.bodd n

@[simp]

theorem Int.bodd_neg (n : ℤ) : Int.bodd (-n) = Int.bodd n

@[simp]

theorem Int.bodd_add (m : ℤ) (n : ℤ) : Int.bodd (m + n) = xor (Int.bodd m) (Int.bodd n)

@[simp]

theorem Int.bodd_mul (m : ℤ) (n : ℤ) : Int.bodd (m * n) = (Int.bodd m && Int.bodd n)

theorem Int.bodd_add_div2 (n : ℤ) : (bif Int.bodd n then 1 else 0) + 2 * Int.div2 n = n

theorem Int.div2_val (n : ℤ) : Int.div2 n = n / 2

@[deprecated]

theorem Int.bit0_val (n : ℤ) : bit0 n = 2 * n

@[deprecated]

theorem Int.bit1_val (n : ℤ) : bit1 n = 2 * n + 1

theorem Int.bit_val (b : Bool) (n : ℤ) : Int.bit b n = 2 * n + bif b then 1 else 0

theorem Int.bit_decomp (n : ℤ) : Int.bit (Int.bodd n) (Int.div2 n) = n

def Int.bitCasesOn {C : ℤ → Sort u} (n : ℤ) (h : (b : Bool) → (n : ℤ) → C (Int.bit b n)) : C n

Defines a function from ℤ conditionally, if it is defined for odd and even integers separately using bit.

Equations

• Int.bitCasesOn n h = Eq.mpr ⋯ (h (Int.bodd n) (Int.div2 n))

@[simp]

theorem Int.bit_zero : Int.bit false 0 = 0

@[simp]

theorem Int.bit_coe_nat (b : Bool) (n : ℕ) : Int.bit b ↑n = ↑(Nat.bit b n)

@[simp]

theorem Int.bit_negSucc (b : Bool) (n : ℕ) : Int.bit b (Int.negSucc n) = Int.negSucc (Nat.bit (!b) n)

@[simp]

theorem Int.bodd_bit (b : Bool) (n : ℤ) : Int.bodd (Int.bit b n) = b

@[simp, deprecated]

theorem Int.bodd_bit0 (n : ℤ) : Int.bodd (bit0 n) = false

@[simp, deprecated]

theorem Int.bodd_bit1 (n : ℤ) : Int.bodd (bit1 n) = true

@[deprecated]

theorem Int.bit0_ne_bit1 (m : ℤ) (n : ℤ) : bit0 m ≠ bit1 n

@[deprecated]

theorem Int.bit1_ne_bit0 (m : ℤ) (n : ℤ) : bit1 m ≠ bit0 n

@[deprecated]

theorem Int.bit1_ne_zero (m : ℤ) : bit1 m ≠ 0

@[simp]

theorem Int.testBit_bit_zero (b : Bool) (n : ℤ) : Int.testBit (Int.bit b n) 0 = b

@[simp]

theorem Int.testBit_bit_succ (m : ℕ) (b : Bool) (n : ℤ) : Int.testBit (Int.bit b n) (Nat.succ m) = Int.testBit n m

theorem Int.bitwise_or : Int.bitwise or = Int.lor

theorem Int.bitwise_and : Int.bitwise and = Int.land

theorem Int.bitwise_diff : (Int.bitwise fun (a b : Bool) => a && !b) = Int.ldiff

theorem Int.bitwise_xor : Int.bitwise xor = Int.xor

@[simp]

theorem Int.bitwise_bit (f : Bool → Bool → Bool) (a : Bool) (m : ℤ) (b : Bool) (n : ℤ) : Int.bitwise f (Int.bit a m) (Int.bit b n) = Int.bit (f a b) (Int.bitwise f m n)

@[simp]

theorem Int.lor_bit (a : Bool) (m : ℤ) (b : Bool) (n : ℤ) : Int.lor (Int.bit a m) (Int.bit b n) = Int.bit (a || b) (Int.lor m n)

@[simp]

theorem Int.land_bit (a : Bool) (m : ℤ) (b : Bool) (n : ℤ) : Int.land (Int.bit a m) (Int.bit b n) = Int.bit (a && b) (Int.land m n)

@[simp]

theorem Int.ldiff_bit (a : Bool) (m : ℤ) (b : Bool) (n : ℤ) : Int.ldiff (Int.bit a m) (Int.bit b n) = Int.bit (a && !b) (Int.ldiff m n)

@[simp]

theorem Int.lxor_bit (a : Bool) (m : ℤ) (b : Bool) (n : ℤ) : Int.xor (Int.bit a m) (Int.bit b n) = Int.bit (xor a b) (Int.xor m n)

@[simp]

theorem Int.lnot_bit (b : Bool) (n : ℤ) : Int.lnot (Int.bit b n) = Int.bit (!b) (Int.lnot n)

@[simp]

theorem Int.testBit_bitwise (f : Bool → Bool → Bool) (m : ℤ) (n : ℤ) (k : ℕ) : Int.testBit (Int.bitwise f m n) k = f (Int.testBit m k) (Int.testBit n k)

@[simp]

theorem Int.testBit_lor (m : ℤ) (n : ℤ) (k : ℕ) : Int.testBit (Int.lor m n) k = (Int.testBit m k || Int.testBit n k)

@[simp]

theorem Int.testBit_land (m : ℤ) (n : ℤ) (k : ℕ) : Int.testBit (Int.land m n) k = (Int.testBit m k && Int.testBit n k)

@[simp]

theorem Int.testBit_ldiff (m : ℤ) (n : ℤ) (k : ℕ) : Int.testBit (Int.ldiff m n) k = (Int.testBit m k && !Int.testBit n k)

@[simp]

theorem Int.testBit_lxor (m : ℤ) (n : ℤ) (k : ℕ) : Int.testBit (Int.xor m n) k = xor (Int.testBit m k) (Int.testBit n k)

@[simp]

theorem Int.testBit_lnot (n : ℤ) (k : ℕ) : Int.testBit (Int.lnot n) k = !Int.testBit n k

@[simp]

theorem Int.shiftLeft_neg (m : ℤ) (n : ℤ) : m <<< (-n) = m >>> n

@[simp]

theorem Int.shiftRight_neg (m : ℤ) (n : ℤ) : m >>> (-n) = m <<< n

@[simp]

theorem Int.shiftLeft_coe_nat (m : ℕ) (n : ℕ) : ↑m <<< ↑n = ↑(m <<< n)

@[simp]

theorem Int.shiftRight_coe_nat (m : ℕ) (n : ℕ) : ↑m >>> ↑n = ↑(m >>> n)

@[simp]

theorem Int.shiftLeft_negSucc (m : ℕ) (n : ℕ) : Int.negSucc m <<< ↑n = Int.negSucc (Nat.shiftLeft' true m n)

@[simp]

theorem Int.shiftRight_negSucc (m : ℕ) (n : ℕ) : Int.negSucc m >>> ↑n = Int.negSucc (m >>> n)

theorem Int.shiftRight_add (m : ℤ) (n : ℕ) (k : ℕ) : m >>> (↑n + ↑k) = m >>> ↑n >>> ↑k

bitwise ops

theorem Int.shiftLeft_add (m : ℤ) (n : ℕ) (k : ℤ) : m <<< (↑n + k) = m <<< ↑n <<< k

theorem Int.shiftLeft_sub (m : ℤ) (n : ℕ) (k : ℤ) : m <<< (↑n - k) = m <<< ↑n >>> k

theorem Int.shiftLeft_eq_mul_pow (m : ℤ) (n : ℕ) : m <<< ↑n = m * ↑(2 ^ n)

theorem Int.shiftRight_eq_div_pow (m : ℤ) (n : ℕ) : m >>> ↑n = m / ↑(2 ^ n)

theorem Int.one_shiftLeft (n : ℕ) : 1 <<< ↑n = ↑(2 ^ n)

@[simp]

theorem Int.zero_shiftLeft (n : ℤ) : 0 <<< n = 0

@[simp]

theorem Int.zero_shiftRight (n : ℤ) : 0 >>> n = 0