Documentation

Init.Data.Int.DivMod

Quotient and remainder #

There are three main conventions for integer division, referred here as the E, F, T rounding conventions. All three pairs satisfy the identity x % y + (x / y) * y = x unconditionally, and satisfy x / 0 = 0 and x % 0 = x.

T-rounding division #

@[extern lean_int_div]
def Int.div :
IntIntInt

div uses the ["T-rounding"][t-rounding] (Truncation-rounding) convention, meaning that it rounds toward zero. Also note that division by zero is defined to equal zero.

The relation between integer division and modulo is found in Int.mod_add_div which states that a % b + b * (a / b) = a, unconditionally.

[t-rounding]: https://dl.acm.org/doi/pdf/10.1145/128861.128862 [theo mod_add_div]: https://leanprover-community.github.io/mathlib4_docs/find/?pattern=Int.mod_add_div#doc

Examples:

#eval (7 : Int) / (0 : Int) -- 0
#eval (0 : Int) / (7 : Int) -- 0

#eval (12 : Int) / (6 : Int) -- 2
#eval (12 : Int) / (-6 : Int) -- -2
#eval (-12 : Int) / (6 : Int) -- -2
#eval (-12 : Int) / (-6 : Int) -- 2

#eval (12 : Int) / (7 : Int) -- 1
#eval (12 : Int) / (-7 : Int) -- -1
#eval (-12 : Int) / (7 : Int) -- -1
#eval (-12 : Int) / (-7 : Int) -- 1

Implemented by efficient native code.

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Instances For
    @[extern lean_int_mod]
    def Int.mod :
    IntIntInt

    Integer modulo. This function uses the "T-rounding" (Truncation-rounding) convention to pair with Int.div, meaning that a % b + b * (a / b) = a unconditionally (see Int.mod_add_div). In particular, a % 0 = a.

    Examples:

    #eval (7 : Int) % (0 : Int) -- 7
    #eval (0 : Int) % (7 : Int) -- 0
    
    #eval (12 : Int) % (6 : Int) -- 0
    #eval (12 : Int) % (-6 : Int) -- 0
    #eval (-12 : Int) % (6 : Int) -- 0
    #eval (-12 : Int) % (-6 : Int) -- 0
    
    #eval (12 : Int) % (7 : Int) -- 5
    #eval (12 : Int) % (-7 : Int) -- 5
    #eval (-12 : Int) % (7 : Int) -- 2
    #eval (-12 : Int) % (-7 : Int) -- 2
    

    Implemented by efficient native code.

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    Instances For

      F-rounding division #

      This pair satisfies fdiv x y = floor (x / y).

      def Int.fdiv :
      IntIntInt

      Integer division. This version of division uses the F-rounding convention (flooring division), in which Int.fdiv x y satisfies fdiv x y = floor (x / y) and Int.fmod is the unique function satisfying fmod x y + (fdiv x y) * y = x.

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      Instances For
        def Int.fmod :
        IntIntInt

        Integer modulus. This version of Int.mod uses the F-rounding convention (flooring division), in which Int.fdiv x y satisfies fdiv x y = floor (x / y) and Int.fmod is the unique function satisfying fmod x y + (fdiv x y) * y = x.

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          E-rounding division #

          This pair satisfies 0 ≤ mod x y < natAbs y for y ≠ 0.

          @[extern lean_int_ediv]
          def Int.ediv :
          IntIntInt

          Integer division. This version of Int.div uses the E-rounding convention (euclidean division), in which Int.emod x y satisfies 0 ≤ mod x y < natAbs y for y ≠ 0 and Int.ediv is the unique function satisfying emod x y + (ediv x y) * y = x.

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          Instances For
            @[extern lean_int_emod]
            def Int.emod :
            IntIntInt

            Integer modulus. This version of Int.mod uses the E-rounding convention (euclidean division), in which Int.emod x y satisfies 0 ≤ emod x y < natAbs y for y ≠ 0 and Int.ediv is the unique function satisfying emod x y + (ediv x y) * y = x.

            Equations
            Instances For
              instance Int.instDiv :

              The Div and Mod syntax uses ediv and emod for compatibility with SMTLIb and mathematical reasoning tends to be easier.

              Equations
              instance Int.instMod :
              Equations
              @[simp]
              theorem Int.ofNat_ediv (m : Nat) (n : Nat) :
              (m / n) = m / n
              theorem Int.ofNat_div (m : Nat) (n : Nat) :
              (m / n) = (↑m).div n
              theorem Int.ofNat_fdiv (m : Nat) (n : Nat) :
              (m / n) = (↑m).fdiv n

              bmod ("balanced" mod) #

              Balanced mod (and balanced div) are a division and modulus pair such that b * (Int.bdiv a b) + Int.bmod a b = a and b/2 ≤ Int.bmod a b < b/2 for all a : Int and b > 0.

              This is used in Omega as well as signed bitvectors.

              def Int.bmod (x : Int) (m : Nat) :

              Balanced modulus. This version of Integer modulus uses the balanced rounding convention, which guarantees that m/2 ≤ bmod x m < m/2 for m ≠ 0 and bmod x m is congruent to x modulo m.

              If m = 0, then bmod x m = x.

              Equations
              • x.bmod m = let r := x % m; if r < (m + 1) / 2 then r else r - m
              Instances For
                def Int.bdiv (x : Int) (m : Nat) :

                Balanced division. This returns the unique integer so that b * (Int.bdiv a b) + Int.bmod a b = a.

                Equations
                • x.bdiv m = if m = 0 then 0 else let q := x / m; let r := x % m; if r < (m + 1) / 2 then q else q + 1
                Instances For