renamed "emb" to "list_hembeq";
make "list_hembeq" reflexive independent of the base order;
renamed "sub" to "sublisteq";
dropped "transp_on" (state transitivity explicitly instead);
no need to hide "sub" after renaming;
replaced some ASCII symbols by proper Isabelle symbols;
NEWS
(* 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_Binary_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_Binary_Nat.dup
(SML "IntInf.*/ (2,/ (_))")
(OCaml "Big'_int.mult'_big'_int/ 2")
(Haskell "!(2 * _)")
(Scala "!(2 * _)")
(Eval "!(2 * _)")
code_const Code_Binary_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 {*
Evaluation with @{text "Quickcheck_Narrowing"} does not work yet,
since a couple of questions how to perform evaluations in Haskell are not that
clear yet. 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