--- a/src/HOL/Library/Efficient_Nat.thy Fri Feb 15 11:36:34 2013 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,305 +0,0 @@
-(* 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
-