src/HOL/Library/Efficient_Nat.thy

composition of convex and measurable function is measurable

(*  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_Integer Main
begin

text {*
  When generating code for functions on natural numbers, the
  canonical representation using @{term "0::nat"} and
  @{term Suc} is unsuitable for computations involving large
  numbers.  The efficiency of the generated code can be improved
  drastically by implementing natural numbers by target-language
  integers.  To do this, just include this theory.
*}

subsection {* Basic arithmetic *}

text {*
  Most standard arithmetic functions on natural numbers are implemented
  using their counterparts on the integers:
*}

code_datatype number_nat_inst.number_of_nat

lemma zero_nat_code [code, code_unfold_post]:
  "0 = (Numeral0 :: nat)"
  by simp

lemma one_nat_code [code, code_unfold_post]:
  "1 = (Numeral1 :: nat)"
  by simp

lemma Suc_code [code]:
  "Suc n = n + 1"
  by simp

lemma plus_nat_code [code]:
  "n + m = nat (of_nat n + of_nat m)"
  by simp

lemma minus_nat_code [code]:
  "n - m = nat (of_nat n - of_nat m)"
  by simp

lemma times_nat_code [code]:
  "n * m = nat (of_nat n * of_nat m)"
  unfolding of_nat_mult [symmetric] by simp

lemma divmod_nat_code [code]:
  "divmod_nat n m = map_pair nat nat (pdivmod (of_nat n) (of_nat m))"
  by (simp add: map_pair_def split_def pdivmod_def nat_div_distrib nat_mod_distrib divmod_nat_div_mod)

lemma eq_nat_code [code]:
  "HOL.equal n m \<longleftrightarrow> HOL.equal (of_nat n \<Colon> int) (of_nat m)"
  by (simp add: equal)

lemma eq_nat_refl [code nbe]:
  "HOL.equal (n::nat) n \<longleftrightarrow> True"
  by (rule equal_refl)

lemma less_eq_nat_code [code]:
  "n \<le> m \<longleftrightarrow> (of_nat n \<Colon> int) \<le> of_nat m"
  by simp

lemma less_nat_code [code]:
  "n < m \<longleftrightarrow> (of_nat n \<Colon> int) < of_nat m"
  by simp

subsection {* Case analysis *}

text {*
  Case analysis on natural numbers is rephrased using a conditional
  expression:
*}

lemma [code, code_unfold]:
  "nat_case = (\<lambda>f g n. if n = 0 then f else g (n - 1))"
  by (auto simp add: fun_eq_iff dest!: gr0_implies_Suc)


subsection {* Preprocessors *}

text {*
  In contrast to @{term "Suc n"}, the term @{term "n + (1::nat)"} is no longer
  a constructor term. Therefore, all occurrences of this term in a position
  where a pattern is expected (i.e.\ on the left-hand side of a recursion
  equation or in the arguments of an inductive relation in an introduction
  rule) must be eliminated.
  This can be accomplished by applying the following transformation rules:
*}

lemma Suc_if_eq: "(\<And>n. f (Suc n) \<equiv> h n) \<Longrightarrow> f 0 \<equiv> g \<Longrightarrow>
  f n \<equiv> if n = 0 then g else h (n - 1)"
  by (rule eq_reflection) (cases n, simp_all)

lemma Suc_clause: "(\<And>n. P n (Suc n)) \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> P (n - 1) n"
  by (cases n) simp_all

text {*
  The rules above are built into a preprocessor that is plugged into
  the code generator. Since the preprocessor for introduction rules
  does not know anything about modes, some of the modes that worked
  for the canonical representation of natural numbers may no longer work.
*}

(*<*)
setup {*
let

fun remove_suc thy thms =
  let
    val vname = Name.variant (map fst
      (fold (Term.add_var_names o Thm.full_prop_of) thms [])) "n";
    val cv = cterm_of thy (Var ((vname, 0), HOLogic.natT));
    fun lhs_of th = snd (Thm.dest_comb
      (fst (Thm.dest_comb (cprop_of th))));
    fun rhs_of th = snd (Thm.dest_comb (cprop_of th));
    fun find_vars ct = (case term_of ct of
        (Const (@{const_name Suc}, _) $ Var _) => [(cv, snd (Thm.dest_comb ct))]
      | _ $ _ =>
        let val (ct1, ct2) = Thm.dest_comb ct
        in 
          map (apfst (fn ct => Thm.capply ct ct2)) (find_vars ct1) @
          map (apfst (Thm.capply ct1)) (find_vars ct2)
        end
      | _ => []);
    val eqs = maps
      (fn th => map (pair th) (find_vars (lhs_of th))) thms;
    fun mk_thms (th, (ct, cv')) =
      let
        val th' =
          Thm.implies_elim
           (Conv.fconv_rule (Thm.beta_conversion true)
             (Drule.instantiate'
               [SOME (ctyp_of_term ct)] [SOME (Thm.cabs cv ct),
                 SOME (Thm.cabs cv' (rhs_of th)), NONE, SOME cv']
               @{thm Suc_if_eq})) (Thm.forall_intr cv' th)
      in
        case map_filter (fn th'' =>
            SOME (th'', singleton
              (Variable.trade (K (fn [th'''] => [th''' RS th']))
                (Variable.global_thm_context th'')) th'')
          handle THM _ => NONE) thms of
            [] => NONE
          | thps =>
              let val (ths1, ths2) = split_list thps
              in SOME (subtract Thm.eq_thm (th :: ths1) thms @ ths2) end
      end
  in get_first mk_thms eqs end;

fun eqn_suc_base_preproc thy thms =
  let
    val dest = fst o Logic.dest_equals o prop_of;
    val contains_suc = exists_Const (fn (c, _) => c = @{const_name Suc});
  in
    if forall (can dest) thms andalso exists (contains_suc o dest) thms
      then thms |> perhaps_loop (remove_suc thy) |> (Option.map o map) Drule.zero_var_indexes
       else NONE
  end;

val eqn_suc_preproc = Code_Preproc.simple_functrans eqn_suc_base_preproc;

fun remove_suc_clause thy thms =
  let
    val vname = Name.variant (map fst
      (fold (Term.add_var_names o Thm.full_prop_of) thms [])) "x";
    fun find_var (t as Const (@{const_name Suc}, _) $ (v as Var _)) = SOME (t, v)
      | find_var (t $ u) = (case find_var t of NONE => find_var u | x => x)
      | find_var _ = NONE;
    fun find_thm th =
      let val th' = Conv.fconv_rule Object_Logic.atomize th
      in Option.map (pair (th, th')) (find_var (prop_of th')) end
  in
    case get_first find_thm thms of
      NONE => thms
    | SOME ((th, th'), (Sucv, v)) =>
        let
          val cert = cterm_of (Thm.theory_of_thm th);
          val th'' = Object_Logic.rulify (Thm.implies_elim
            (Conv.fconv_rule (Thm.beta_conversion true)
              (Drule.instantiate' []
                [SOME (cert (lambda v (Abs ("x", HOLogic.natT,
                   abstract_over (Sucv,
                     HOLogic.dest_Trueprop (prop_of th')))))),
                 SOME (cert v)] @{thm Suc_clause}))
            (Thm.forall_intr (cert v) th'))
        in
          remove_suc_clause thy (map (fn th''' =>
            if (op = o pairself prop_of) (th''', th) then th'' else th''') thms)
        end
  end;

fun clause_suc_preproc thy ths =
  let
    val dest = fst o HOLogic.dest_mem o HOLogic.dest_Trueprop
  in
    if forall (can (dest o concl_of)) ths andalso
      exists (fn th => exists (exists_Const (fn (c, _) => c = @{const_name Suc}))
        (map_filter (try dest) (concl_of th :: prems_of th))) ths
    then remove_suc_clause thy ths else ths
  end;
in

  Code_Preproc.add_functrans ("eqn_Suc", eqn_suc_preproc)
  #> Codegen.add_preprocessor clause_suc_preproc

end;
*}
(*>*)


subsection {* Target language setup *}

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")

types_code
  nat ("int")
attach (term_of) {*
val term_of_nat = HOLogic.mk_number HOLogic.natT;
*}
attach (test) {*
fun gen_nat i =
  let val n = random_range 0 i
  in (n, fn () => term_of_nat n) end;
*}

text {*
  For Haskell ans 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 -)

text {*
  Natural numerals.
*}

lemma [code_unfold_post]:
  "nat (number_of i) = number_nat_inst.number_of_nat i"
  -- {* this interacts as desired with @{thm nat_number_of_def} *}
  by (simp add: number_nat_inst.number_of_nat)

setup {*
  fold (Numeral.add_code @{const_name number_nat_inst.number_of_nat}
    false Code_Printer.literal_positive_numeral) ["SML", "OCaml", "Haskell", "Scala"]
*}

text {*
  Since natural numbers are implemented
  using integers in ML, the coercion function @{const "of_nat"} 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 function that
  returns its input value, provided that it is non-negative, and otherwise
  returns @{text "0"}.
*}

definition int :: "nat \<Rightarrow> int" where
  [code del]: "int = of_nat"

lemma int_code' [code]:
  "int (number_of l) = (if neg (number_of l \<Colon> int) then 0 else number_of l)"
  unfolding int_nat_number_of [folded int_def] ..

lemma nat_code' [code]:
  "nat (number_of l) = (if neg (number_of l \<Colon> int) then 0 else number_of l)"
  unfolding nat_number_of_def number_of_is_id neg_def by simp

lemma of_nat_int [code_unfold_post]:
  "of_nat = int" by (simp add: int_def)

lemma of_nat_aux_int [code_unfold]:
  "of_nat_aux (\<lambda>i. i + 1) k 0 = int k"
  by (simp add: int_def Nat.of_nat_code)

code_const int
  (SML "_")
  (OCaml "_")

consts_code
  int ("(_)")
  nat ("\<module>nat")
attach {*
fun nat i = if i < 0 then 0 else i;
*}

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

text {* Conversion from and to code numerals. *}

code_const Code_Numeral.of_nat
  (SML "IntInf.toInt")
  (OCaml "_")
  (Haskell "!(fromInteger/ ./ toInteger)")
  (Scala "!Natural(_.as'_BigInt)")
  (Eval "_")

code_const Code_Numeral.nat_of
  (SML "IntInf.fromInt")
  (OCaml "_")
  (Haskell "!(fromInteger/ ./ toInteger)")
  (Scala "!Nat(_.as'_BigInt)")
  (Eval "_")

text {* Using target language arithmetic operations whenever appropriate *}

code_const "op + \<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 "op - \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"
  (Haskell infixl 6 "-")
  (Scala infixl 7 "-")

code_const "op * \<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 "op \<le> \<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 "op < \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
  (SML "IntInf.< ((_), (_))")
  (OCaml "Big'_int.lt'_big'_int")
  (Haskell infix 4 "<")
  (Scala infixl 4 "<")
  (Eval infixl 6 "<")

consts_code
  "0::nat"                     ("0")
  "1::nat"                     ("1")
  Suc                          ("(_ +/ 1)")
  "op + \<Colon>  nat \<Rightarrow> nat \<Rightarrow> nat"   ("(_ +/ _)")
  "op * \<Colon>  nat \<Rightarrow> nat \<Rightarrow> nat"   ("(_ */ _)")
  "op \<le> \<Colon>  nat \<Rightarrow> nat \<Rightarrow> bool"  ("(_ <=/ _)")
  "op < \<Colon>  nat \<Rightarrow> nat \<Rightarrow> bool"  ("(_ </ _)")


text {* 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 {* Module names *}

code_modulename SML
  Efficient_Nat Arith

code_modulename OCaml
  Efficient_Nat Arith

code_modulename Haskell
  Efficient_Nat Arith

hide_const int

end