(*  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_Index 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 inline]:

   "0 = (Numeral0 :: nat)"

   by simp

 lemmas [code post] = zero_nat_code [symmetric]

 lemma one_nat_code [code, code inline]:

   "1 = (Numeral1 :: nat)"

   by simp

 lemmas [code post] = one_nat_code [symmetric]

 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

 text {* Specialized @{term "op div \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"} 

   and @{term "op mod \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"} operations. *}

 definition divmod_aux ::  "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where

   [code del]: "divmod_aux = Divides.divmod"

 lemma [code]:

   "Divides.divmod n m = (if m = 0 then (0, n) else divmod_aux n m)"

   unfolding divmod_aux_def divmod_div_mod by simp

 lemma divmod_aux_code [code]:

   "divmod_aux n m = (nat (of_nat n div of_nat m), nat (of_nat n mod of_nat m))"

   unfolding divmod_aux_def divmod_div_mod zdiv_int [symmetric] zmod_int [symmetric] by simp

 lemma eq_nat_code [code]:

   "eq_class.eq n m \<longleftrightarrow> eq_class.eq (of_nat n \<Colon> int) (of_nat m)"

   by (simp add: eq)

 lemma eq_nat_refl [code nbe]:

   "eq_class.eq (n::nat) n \<longleftrightarrow> True"

   by (rule HOL.eq_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: expand_fun_eq 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) = h n) \<Longrightarrow> f 0 = g \<Longrightarrow>

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

   by (cases n) simp_all

 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, rule Suc_if_eq')

     (rule meta_eq_to_obj_eq, assumption,

      rule meta_eq_to_obj_eq, assumption)

 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 gen_remove_suc Suc_if_eq dest_judgement 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 (dest_judgement (cprop_of th)))));

     fun rhs_of th = snd (Thm.dest_comb (dest_judgement (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']

                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.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 gen_eqn_suc_preproc Suc_if_eq dest_judgement dest_lhs thy thms =

   let

     val dest = dest_lhs 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 perhaps_loop (gen_remove_suc Suc_if_eq dest_judgement thy) thms

        else NONE

   end;

 val eqn_suc_preproc = Code.simple_functrans (gen_eqn_suc_preproc

   @{thm Suc_if_eq} I (fst o Logic.dest_equals));

 fun eqn_suc_preproc' thy thms = gen_eqn_suc_preproc

   @{thm Suc_if_eq'} (snd o Thm.dest_comb) (fst o HOLogic.dest_eq o HOLogic.dest_Trueprop) thy thms

   |> the_default thms;

 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 ObjectLogic.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'' = ObjectLogic.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

   Codegen.add_preprocessor eqn_suc_preproc'

   #> Codegen.add_preprocessor clause_suc_preproc

   #> Code.add_functrans ("eqn_Suc", eqn_suc_preproc)

 end;

 *}

 (*>*)

 subsection {* Target language setup *}

 text {*

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

   assert that values are always non-negative.

 *}

 code_type nat

   (SML "IntInf.int")

   (OCaml "Big'_int.big'_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 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 (Show, Eq);

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

 };

 *}

 code_reserved Haskell Nat

 code_type nat

   (Haskell "Nat.Nat")

 code_instance nat :: eq

   (Haskell -)

 text {*

   Natural numerals.

 *}

 lemma [code inline, symmetric, code 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}

     true false) ["SML", "OCaml", "Haskell"]

 *}

 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]:

   "of_nat = int" by (simp add: int_def)

 declare of_nat_int [symmetric, code post]

 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")

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

 code_const int and nat

   (Haskell "toInteger" and "fromInteger")

 text {* Conversion from and to indices. *}

 code_const Code_Index.of_nat

   (SML "IntInf.toInt")

   (OCaml "Big'_int.int'_of'_big'_int")

   (Haskell "fromEnum")

 code_const Code_Index.nat_of

   (SML "IntInf.fromInt")

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

   (Haskell "toEnum")

 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 "+")

 code_const "op * \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"

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

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

   (Haskell infixl 7 "*")

 code_const divmod_aux

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

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

   (Haskell "divMod")

 code_const "eq_class.eq \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"

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

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

   (Haskell infixl 4 "==")

 code_const "op \<le> \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"

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

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

   (Haskell infix 4 "<=")

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

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

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

   (Haskell infix 4 "<")

 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_Eval.term_of \<Colon> nat \<Rightarrow> term) = Code_Eval.term_of" ..

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

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

 text {* Module names *}

 code_modulename SML

   Nat Integer

   Divides Integer

   Ring_and_Field Integer

   Efficient_Nat Integer

 code_modulename OCaml

   Nat Integer

   Divides Integer

   Ring_and_Field Integer

   Efficient_Nat Integer

 code_modulename Haskell

   Nat Integer

   Divides Integer

   Ring_and_Field Integer

   Efficient_Nat Integer

 hide const int

 end