(*  Title:      HOL/Library/EfficientNat.thy

    ID:         $Id$

    Author:     Stefan Berghofer, TU Muenchen

*)

header {* Implementation of natural numbers by integers *}

theory EfficientNat

imports 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

integers. To do this, just include this theory.

*}

subsection {* Basic functions *}

text {*

The implementation of @{term "0::nat"} and @{term "Suc"} using the

ML integers is straightforward. Since natural numbers are implemented

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

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

For the @{term "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"}.

*}

types_code

  nat ("int")

attach (term_of) {*

fun term_of_nat 0 = Const ("0", HOLogic.natT)

  | term_of_nat 1 = Const ("1", HOLogic.natT)

  | term_of_nat i = HOLogic.number_of_const HOLogic.natT $

      HOLogic.mk_bin (IntInf.fromInt i);

*}

attach (test) {*

fun gen_nat i = random_range 0 i;

*}

consts_code

  0 :: nat ("0")

  Suc ("(_ + 1)")

  nat ("\<module>nat")

attach {*

fun nat i = if i < 0 then 0 else i;

*}

  int ("(_)")

(* code_syntax_tyco nat

  ml (atom "InfInt.int")

  haskell (atom "Integer")

code_syntax_const 0 :: nat

  ml ("0 : InfInt.int")

  haskell (atom "0")

code_syntax_const Suc

  ml (infixl 8 "_ + 1")

  haskell (infixl 6 "_ + 1")

code_primconst nat

ml {*

fun nat i = if i < 0 then 0 else i;

*}

haskell {*

nat i = if i < 0 then 0 else i

*} *)

text {*

Case analysis on natural numbers is rephrased using a conditional

expression:

*}

lemma [code unfold]: "nat_case \<equiv> (\<lambda>f g n. if n = 0 then f else g (n - 1))"

  apply (rule eq_reflection ext)+

  apply (case_tac n)

  apply simp_all

  done

text {*

Most standard arithmetic functions on natural numbers are implemented

using their counterparts on the integers:

*}

lemma [code]: "m - n = nat (int m - int n)" by arith

lemma [code]: "m + n = nat (int m + int n)" by arith

lemma [code]: "m * n = nat (int m * int n)"

  by (simp add: zmult_int)

lemma [code]: "m div n = nat (int m div int n)"

  by (simp add: zdiv_int [symmetric])

lemma [code]: "m mod n = nat (int m mod int n)"

  by (simp add: zmod_int [symmetric])

lemma [code]: "(m < n) = (int m < int n)"

  by simp

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:

*}

theorem Suc_if_eq: "(\<And>n. f (Suc n) = h n) \<Longrightarrow> f 0 = g \<Longrightarrow>

  f n = (if n = 0 then g else h (n - 1))"

  by (case_tac n) simp_all

theorem Suc_clause: "(\<And>n. P n (Suc n)) \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> P (n - 1) n"

  by (case_tac 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.

*}

(*<*)

ML {*

val Suc_if_eq = thm "Suc_if_eq";

fun remove_suc thy thms =

  let

    val Suc_if_eq' = Thm.transfer thy Suc_if_eq;

    val vname = variant (map fst

      (fold (add_term_varnames o Thm.full_prop_of) thms [])) "x";

    val cv = cterm_of Main.thy (Var ((vname, 0), HOLogic.natT));

    fun lhs_of th = snd (Thm.dest_comb

      (fst (Thm.dest_comb (snd (Thm.dest_comb (cprop_of th))))));

    fun rhs_of th = snd (Thm.dest_comb (snd (Thm.dest_comb (cprop_of th))));

    fun find_vars ct = (case term_of ct of

        (Const ("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 = List.concat (map

      (fn th => map (pair th) (find_vars (lhs_of th))) thms);

    fun mk_thms (th, (ct, cv')) =

      let

        val th' =

          Thm.implies_elim

           (Drule.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']

               Suc_if_eq')) (Thm.forall_intr cv' th)

      in

        case List.mapPartial (fn th'' =>

            SOME (th'', standard (Drule.freeze_all th'' RS th'))

          handle THM _ => NONE) thms of

            [] => NONE

          | thps =>

              let val (ths1, ths2) = ListPair.unzip thps

              in SOME (gen_rems eq_thm (thms, th :: ths1) @ ths2) end

      end

  in

    case Library.get_first mk_thms eqs of

      NONE => thms

    | SOME x => remove_suc thy x

  end;

fun eqn_suc_preproc thy ths =

  let val dest = fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of

  in

    if forall (can dest) ths andalso

      exists (fn th => "Suc" mem term_consts (dest th)) ths

    then remove_suc thy ths else ths

  end;

val Suc_clause = thm "Suc_clause";

fun remove_suc_clause thy thms =

  let

    val Suc_clause' = Thm.transfer thy Suc_clause;

    val vname = variant (map fst

      (fold (add_term_varnames o Thm.full_prop_of) thms [])) "x";

    fun find_var (t as Const ("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' = ObjectLogic.atomize_thm 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'' = ObjectLogic.rulify (Thm.implies_elim

            (Drule.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)] Suc_clause'))

            (Thm.forall_intr (cert v) th'))

        in

          remove_suc_clause thy (map (fn th''' =>

            if 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 => "Suc" mem foldr add_term_consts

        [] (List.mapPartial (try dest) (concl_of th :: prems_of th))) ths

    then remove_suc_clause thy ths else ths

  end;

val suc_preproc_setup =

  [Codegen.add_preprocessor eqn_suc_preproc,

   Codegen.add_preprocessor clause_suc_preproc];

*}

setup suc_preproc_setup

(*>*)

end