(*  Title:      HOL/Library/Efficient_Nat.thy

     Author:     Stefan Berghofer, Florian Haftmann, TU Muenchen

 *)

 header {* Implementation of natural numbers by target-language integers *}

 theory Efficient_Nat

 imports Code_Nat Code_Integer Main

 begin

 text {*

   The efficiency of the generated code for natural numbers can be improved

   drastically by implementing natural numbers by target-language

   integers.  To do this, just include this theory.

 *}

 subsection {* Target language fundamentals *}

 text {*

   For ML, we map @{typ nat} to target language integers, where we

   ensure that values are always non-negative.

 *}

 code_type nat

   (SML "IntInf.int")

   (OCaml "Big'_int.big'_int")

   (Eval "int")

 text {*

   For Haskell and Scala we define our own @{typ nat} type.  The reason

   is that we have to distinguish type class instances for @{typ nat}

   and @{typ int}.

 *}

 code_include Haskell "Nat"

 {*newtype Nat = Nat Integer deriving (Eq, Show, Read);

 instance Num Nat where {

   fromInteger k = Nat (if k >= 0 then k else 0);

   Nat n + Nat m = Nat (n + m);

   Nat n - Nat m = fromInteger (n - m);

   Nat n * Nat m = Nat (n * m);

   abs n = n;

   signum _ = 1;

   negate n = error "negate Nat";

 };

 instance Ord Nat where {

   Nat n <= Nat m = n <= m;

   Nat n < Nat m = n < m;

 };

 instance Real Nat where {

   toRational (Nat n) = toRational n;

 };

 instance Enum Nat where {

   toEnum k = fromInteger (toEnum k);

   fromEnum (Nat n) = fromEnum n;

 };

 instance Integral Nat where {

   toInteger (Nat n) = n;

   divMod n m = quotRem n m;

   quotRem (Nat n) (Nat m)

     | (m == 0) = (0, Nat n)

     | otherwise = (Nat k, Nat l) where (k, l) = quotRem n m;

 };

 *}

 code_reserved Haskell Nat

 code_include Scala "Nat"

 {*object Nat {

   def apply(numeral: BigInt): Nat = new Nat(numeral max 0)

   def apply(numeral: Int): Nat = Nat(BigInt(numeral))

   def apply(numeral: String): Nat = Nat(BigInt(numeral))

 }

 class Nat private(private val value: BigInt) {

   override def hashCode(): Int = this.value.hashCode()

   override def equals(that: Any): Boolean = that match {

     case that: Nat => this equals that

     case _ => false

   }

   override def toString(): String = this.value.toString

   def equals(that: Nat): Boolean = this.value == that.value

   def as_BigInt: BigInt = this.value

   def as_Int: Int = if (this.value >= scala.Int.MinValue && this.value <= scala.Int.MaxValue)

       this.value.intValue

     else error("Int value out of range: " + this.value.toString)

   def +(that: Nat): Nat = new Nat(this.value + that.value)

   def -(that: Nat): Nat = Nat(this.value - that.value)

   def *(that: Nat): Nat = new Nat(this.value * that.value)

   def /%(that: Nat): (Nat, Nat) = if (that.value == 0) (new Nat(0), this)

     else {

       val (k, l) = this.value /% that.value

       (new Nat(k), new Nat(l))

     }

   def <=(that: Nat): Boolean = this.value <= that.value

   def <(that: Nat): Boolean = this.value < that.value

 }

 *}

 code_reserved Scala Nat

 code_type nat

   (Haskell "Nat.Nat")

   (Scala "Nat")

 code_instance nat :: equal

   (Haskell -)

 setup {*

   fold (Numeral.add_code @{const_name nat_of_num}

     false Code_Printer.literal_positive_numeral) ["SML", "OCaml", "Haskell", "Scala"]

 *}

 code_const "0::nat"

   (SML "0")

   (OCaml "Big'_int.zero'_big'_int")

   (Haskell "0")

   (Scala "Nat(0)")

 subsection {* Conversions *}

 text {*

   Since natural numbers are implemented

   using integers in ML, the coercion function @{term "int"} of type

   @{typ "nat \<Rightarrow> int"} is simply implemented by the identity function.

   For the @{const nat} function for converting an integer to a natural

   number, we give a specific implementation using an ML expression that

   returns its input value, provided that it is non-negative, and otherwise

   returns @{text "0"}.

 *}

 definition int :: "nat \<Rightarrow> int" where

   [code_abbrev]: "int = of_nat"

 code_const int

   (SML "_")

   (OCaml "_")

 code_const nat

   (SML "IntInf.max/ (0,/ _)")

   (OCaml "Big'_int.max'_big'_int/ Big'_int.zero'_big'_int")

   (Eval "Integer.max/ 0")

 text {* For Haskell and Scala, things are slightly different again. *}

 code_const int and nat

   (Haskell "Prelude.toInteger" and "Prelude.fromInteger")

   (Scala "!_.as'_BigInt" and "Nat")

 text {* Alternativ implementation for @{const of_nat} *}

 lemma [code]:

   "of_nat n = (if n = 0 then 0 else

      let

        (q, m) = divmod_nat n 2;

        q' = 2 * of_nat q

      in if m = 0 then q' else q' + 1)"

 proof -

   from mod_div_equality have *: "of_nat n = of_nat (n div 2 * 2 + n mod 2)" by simp

   show ?thesis

     apply (simp add: Let_def divmod_nat_div_mod mod_2_not_eq_zero_eq_one_nat

       of_nat_mult

       of_nat_add [symmetric])

     apply (auto simp add: of_nat_mult)

     apply (simp add: * of_nat_mult add_commute mult_commute)

     done

 qed

 text {* Conversion from and to code numerals *}

 code_const Code_Numeral.of_nat

   (SML "IntInf.toInt")

   (OCaml "_")

   (Haskell "!(Prelude.fromInteger/ ./ Prelude.toInteger)")

   (Scala "!Natural(_.as'_BigInt)")

   (Eval "_")

 code_const Code_Numeral.nat_of

   (SML "IntInf.fromInt")

   (OCaml "_")

   (Haskell "!(Prelude.fromInteger/ ./ Prelude.toInteger)")

   (Scala "!Nat(_.as'_BigInt)")

   (Eval "_")

 subsection {* Target language arithmetic *}

 code_const "plus \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"

   (SML "IntInf.+/ ((_),/ (_))")

   (OCaml "Big'_int.add'_big'_int")

   (Haskell infixl 6 "+")

   (Scala infixl 7 "+")

   (Eval infixl 8 "+")

 code_const "minus \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"

   (SML "IntInf.max/ (0, IntInf.-/ ((_),/ (_)))")

   (OCaml "Big'_int.max'_big'_int/ Big'_int.zero'_big'_int/ (Big'_int.sub'_big'_int/ _/ _)")

   (Haskell infixl 6 "-")

   (Scala infixl 7 "-")

   (Eval "Integer.max/ 0/ (_ -/ _)")

 code_const Code_Nat.dup

   (SML "IntInf.*/ (2,/ (_))")

   (OCaml "Big'_int.mult'_big'_int/ 2")

   (Haskell "!(2 * _)")

   (Scala "!(2 * _)")

   (Eval "!(2 * _)")

 code_const Code_Nat.sub

   (SML "!(raise/ Fail/ \"sub\")")

   (OCaml "failwith/ \"sub\"")

   (Haskell "error/ \"sub\"")

   (Scala "!sys.error(\"sub\")")

 code_const "times \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"

   (SML "IntInf.*/ ((_),/ (_))")

   (OCaml "Big'_int.mult'_big'_int")

   (Haskell infixl 7 "*")

   (Scala infixl 8 "*")

   (Eval infixl 9 "*")

 code_const divmod_nat

   (SML "IntInf.divMod/ ((_),/ (_))")

   (OCaml "Big'_int.quomod'_big'_int")

   (Haskell "divMod")

   (Scala infixl 8 "/%")

   (Eval "Integer.div'_mod")

 code_const "HOL.equal \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"

   (SML "!((_ : IntInf.int) = _)")

   (OCaml "Big'_int.eq'_big'_int")

   (Haskell infix 4 "==")

   (Scala infixl 5 "==")

   (Eval infixl 6 "=")

 code_const "less_eq \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"

   (SML "IntInf.<=/ ((_),/ (_))")

   (OCaml "Big'_int.le'_big'_int")

   (Haskell infix 4 "<=")

   (Scala infixl 4 "<=")

   (Eval infixl 6 "<=")

 code_const "less \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"

   (SML "IntInf.</ ((_),/ (_))")

   (OCaml "Big'_int.lt'_big'_int")

   (Haskell infix 4 "<")

   (Scala infixl 4 "<")

   (Eval infixl 6 "<")

 code_const Num.num_of_nat

   (SML "!(raise/ Fail/ \"num'_of'_nat\")")

   (OCaml "failwith/ \"num'_of'_nat\"")

   (Haskell "error/ \"num'_of'_nat\"")

   (Scala "!sys.error(\"num'_of'_nat\")")

 subsection {* Evaluation *}

 lemma [code, code del]:

   "(Code_Evaluation.term_of \<Colon> nat \<Rightarrow> term) = Code_Evaluation.term_of" ..

 code_const "Code_Evaluation.term_of \<Colon> nat \<Rightarrow> term"

   (SML "HOLogic.mk'_number/ HOLogic.natT")

 text {*

   FIXME -- Evaluation with @{text "Quickcheck_Narrowing"} does not work, as

   @{text "code_module"} is very aggressive leading to bad Haskell code.

   Therefore, we simply deactivate the narrowing-based quickcheck from here on.

 *}

 declare [[quickcheck_narrowing_active = false]] 

 code_modulename SML

   Efficient_Nat Arith

 code_modulename OCaml

   Efficient_Nat Arith

 code_modulename Haskell

   Efficient_Nat Arith

 hide_const (open) int

 end