src/HOL/Library/Efficient_Nat.thy
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(*  Title:      HOL/Library/Efficient_Nat.thy
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    ID:         $Id$
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    Author:     Stefan Berghofer, Florian Haftmann, TU Muenchen
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*)
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header {* Implementation of natural numbers by target-language integers *}
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theory Efficient_Nat
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imports Main Code_Integer Code_Index
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begin
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text {*
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  When generating code for functions on natural numbers, the
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  canonical representation using @{term "0::nat"} and
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  @{term "Suc"} is unsuitable for computations involving large
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  numbers.  The efficiency of the generated code can be improved
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  drastically by implementing natural numbers by target-language
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  integers.  To do this, just include this theory.
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*}
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subsection {* Basic arithmetic *}
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text {*
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  Most standard arithmetic functions on natural numbers are implemented
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  using their counterparts on the integers:
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*}
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code_datatype number_nat_inst.number_of_nat
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lemma zero_nat_code [code, code unfold]:
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  "0 = (Numeral0 :: nat)"
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  by simp
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lemmas [code post] = zero_nat_code [symmetric]
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lemma one_nat_code [code, code unfold]:
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  "1 = (Numeral1 :: nat)"
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  by simp
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lemmas [code post] = one_nat_code [symmetric]
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lemma Suc_code [code]:
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  "Suc n = n + 1"
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  by simp
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lemma plus_nat_code [code]:
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  "n + m = nat (of_nat n + of_nat m)"
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  by simp
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lemma minus_nat_code [code]:
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  "n - m = nat (of_nat n - of_nat m)"
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  by simp
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lemma times_nat_code [code]:
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  "n * m = nat (of_nat n * of_nat m)"
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  unfolding of_nat_mult [symmetric] by simp
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text {* Specialized @{term "op div \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"} 
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  and @{term "op mod \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"} operations. *}
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definition
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  divmod_aux ::  "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat"
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where
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  [code func del]: "divmod_aux = divmod"
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lemma [code func]:
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  "divmod n m = (if m = 0 then (0, n) else divmod_aux n m)"
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  unfolding divmod_aux_def divmod_div_mod by simp
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lemma divmod_aux_code [code]:
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  "divmod_aux n m = (nat (of_nat n div of_nat m), nat (of_nat n mod of_nat m))"
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  unfolding divmod_aux_def divmod_div_mod zdiv_int [symmetric] zmod_int [symmetric] by simp
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lemma eq_nat_code [code]:
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  "n = m \<longleftrightarrow> (of_nat n \<Colon> int) = of_nat m"
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  by simp
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lemma less_eq_nat_code [code]:
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  "n \<le> m \<longleftrightarrow> (of_nat n \<Colon> int) \<le> of_nat m"
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  by simp
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lemma less_nat_code [code]:
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  "n < m \<longleftrightarrow> (of_nat n \<Colon> int) < of_nat m"
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  by simp
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subsection {* Case analysis *}
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text {*
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  Case analysis on natural numbers is rephrased using a conditional
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  expression:
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*}
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lemma [code func, code unfold]:
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  "nat_case = (\<lambda>f g n. if n = 0 then f else g (n - 1))"
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  by (auto simp add: expand_fun_eq dest!: gr0_implies_Suc)
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subsection {* Preprocessors *}
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text {*
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  In contrast to @{term "Suc n"}, the term @{term "n + (1::nat)"} is no longer
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  a constructor term. Therefore, all occurrences of this term in a position
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  where a pattern is expected (i.e.\ on the left-hand side of a recursion
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  equation or in the arguments of an inductive relation in an introduction
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  rule) must be eliminated.
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  This can be accomplished by applying the following transformation rules:
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*}
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lemma Suc_if_eq: "(\<And>n. f (Suc n) = h n) \<Longrightarrow> f 0 = g \<Longrightarrow>
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  f n = (if n = 0 then g else h (n - 1))"
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  by (case_tac n) simp_all
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lemma Suc_clause: "(\<And>n. P n (Suc n)) \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> P (n - 1) n"
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  by (case_tac n) simp_all
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text {*
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  The rules above are built into a preprocessor that is plugged into
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  the code generator. Since the preprocessor for introduction rules
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  does not know anything about modes, some of the modes that worked
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  for the canonical representation of natural numbers may no longer work.
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*}
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(*<*)
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ML {*
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fun remove_suc thy thms =
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  let
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    val vname = Name.variant (map fst
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      (fold (Term.add_varnames o Thm.full_prop_of) thms [])) "x";
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    val cv = cterm_of thy (Var ((vname, 0), HOLogic.natT));
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    fun lhs_of th = snd (Thm.dest_comb
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      (fst (Thm.dest_comb (snd (Thm.dest_comb (cprop_of th))))));
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    fun rhs_of th = snd (Thm.dest_comb (snd (Thm.dest_comb (cprop_of th))));
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    fun find_vars ct = (case term_of ct of
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        (Const ("Suc", _) $ Var _) => [(cv, snd (Thm.dest_comb ct))]
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      | _ $ _ =>
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        let val (ct1, ct2) = Thm.dest_comb ct
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        in 
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          map (apfst (fn ct => Thm.capply ct ct2)) (find_vars ct1) @
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          map (apfst (Thm.capply ct1)) (find_vars ct2)
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        end
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      | _ => []);
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    val eqs = maps
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      (fn th => map (pair th) (find_vars (lhs_of th))) thms;
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    fun mk_thms (th, (ct, cv')) =
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      let
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        val th' =
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          Thm.implies_elim
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           (Conv.fconv_rule (Thm.beta_conversion true)
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             (Drule.instantiate'
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               [SOME (ctyp_of_term ct)] [SOME (Thm.cabs cv ct),
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                 SOME (Thm.cabs cv' (rhs_of th)), NONE, SOME cv']
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               @{thm Suc_if_eq})) (Thm.forall_intr cv' th)
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      in
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        case map_filter (fn th'' =>
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            SOME (th'', singleton
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              (Variable.trade (K (fn [th'''] => [th''' RS th'])) (Variable.thm_context th'')) th'')
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          handle THM _ => NONE) thms of
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            [] => NONE
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          | thps =>
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              let val (ths1, ths2) = split_list thps
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              in SOME (subtract Thm.eq_thm (th :: ths1) thms @ ths2) end
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      end
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  in
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    case get_first mk_thms eqs of
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      NONE => thms
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    | SOME x => remove_suc thy x
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  end;
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fun eqn_suc_preproc thy ths =
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  let
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    val dest = fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of;
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    fun contains_suc t = member (op =) (term_consts t) @{const_name Suc};
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  in
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    if forall (can dest) ths andalso
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      exists (contains_suc o dest) ths
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    then remove_suc thy ths else ths
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  end;
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fun remove_suc_clause thy thms =
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  let
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    val vname = Name.variant (map fst
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      (fold (Term.add_varnames o Thm.full_prop_of) thms [])) "x";
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    fun find_var (t as Const (@{const_name Suc}, _) $ (v as Var _)) = SOME (t, v)
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      | find_var (t $ u) = (case find_var t of NONE => find_var u | x => x)
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      | find_var _ = NONE;
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    fun find_thm th =
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      let val th' = Conv.fconv_rule ObjectLogic.atomize th
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      in Option.map (pair (th, th')) (find_var (prop_of th')) end
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  in
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    case get_first find_thm thms of
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      NONE => thms
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    | SOME ((th, th'), (Sucv, v)) =>
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        let
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          val cert = cterm_of (Thm.theory_of_thm th);
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          val th'' = ObjectLogic.rulify (Thm.implies_elim
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            (Conv.fconv_rule (Thm.beta_conversion true)
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              (Drule.instantiate' []
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                [SOME (cert (lambda v (Abs ("x", HOLogic.natT,
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                   abstract_over (Sucv,
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                     HOLogic.dest_Trueprop (prop_of th')))))),
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                 SOME (cert v)] @{thm Suc_clause}))
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            (Thm.forall_intr (cert v) th'))
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        in
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          remove_suc_clause thy (map (fn th''' =>
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            if (op = o pairself prop_of) (th''', th) then th'' else th''') thms)
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        end
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  end;
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fun clause_suc_preproc thy ths =
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  let
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    val dest = fst o HOLogic.dest_mem o HOLogic.dest_Trueprop
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  in
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    if forall (can (dest o concl_of)) ths andalso
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      exists (fn th => member (op =) (foldr add_term_consts
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        [] (map_filter (try dest) (concl_of th :: prems_of th))) "Suc") ths
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    then remove_suc_clause thy ths else ths
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  end;
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fun lift_obj_eq f thy thms =
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  thms
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  |> try (
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    map (fn thm => thm RS @{thm meta_eq_to_obj_eq})
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    #> f thy
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    #> map (fn thm => thm RS @{thm eq_reflection})
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    #> map (Conv.fconv_rule Drule.beta_eta_conversion))
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  |> the_default thms
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*}
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setup {*
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  Codegen.add_preprocessor eqn_suc_preproc
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  #> Codegen.add_preprocessor clause_suc_preproc
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  #> Code.add_preproc ("eqn_Suc", lift_obj_eq eqn_suc_preproc)
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  #> Code.add_preproc ("clause_Suc", lift_obj_eq clause_suc_preproc)
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*}
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(*>*)
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subsection {* Target language setup *}
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text {*
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  For ML, we map @{typ nat} to target language integers, where we
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  assert that values are always non-negative.
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*}
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code_type nat
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  (SML "int")
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  (OCaml "Big'_int.big'_int")
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types_code
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  nat ("int")
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attach (term_of) {*
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val term_of_nat = HOLogic.mk_number HOLogic.natT;
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*}
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attach (test) {*
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fun gen_nat i =
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  let val n = random_range 0 i
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  in (n, fn () => term_of_nat n) end;
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*}
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text {*
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  For Haskell we define our own @{typ nat} type.  The reason
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  is that we have to distinguish type class instances
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  for @{typ nat} and @{typ int}.
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*}
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code_include Haskell "Nat" {*
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newtype Nat = Nat Integer deriving (Show, Eq);
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instance Num Nat where {
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  fromInteger k = Nat (if k >= 0 then k else 0);
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  Nat n + Nat m = Nat (n + m);
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  Nat n - Nat m = fromInteger (n - m);
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  Nat n * Nat m = Nat (n * m);
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  abs n = n;
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  signum _ = 1;
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  negate n = error "negate Nat";
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};
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instance Ord Nat where {
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  Nat n <= Nat m = n <= m;
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  Nat n < Nat m = n < m;
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};
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instance Real Nat where {
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  toRational (Nat n) = toRational n;
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};
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instance Enum Nat where {
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  toEnum k = fromInteger (toEnum k);
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  fromEnum (Nat n) = fromEnum n;
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};
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instance Integral Nat where {
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  toInteger (Nat n) = n;
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  divMod n m = quotRem n m;
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  quotRem (Nat n) (Nat m) = (Nat k, Nat l) where (k, l) = quotRem n m;
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};
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*}
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code_reserved Haskell Nat
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code_type nat
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  (Haskell "Nat")
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code_instance nat :: eq
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  (Haskell -)
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text {*
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  Natural numerals.
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*}
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lemma [code inline, symmetric, code post]:
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  "nat (number_of i) = number_nat_inst.number_of_nat i"
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  -- {* this interacts as desired with @{thm nat_number_of_def} *}
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  by (simp add: number_nat_inst.number_of_nat)
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setup {*
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  fold (Numeral.add_code @{const_name number_nat_inst.number_of_nat}
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    true false) ["SML", "OCaml", "Haskell"]
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*}
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text {*
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  Since natural numbers are implemented
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  using integers in ML, the coercion function @{const "of_nat"} of type
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  @{typ "nat \<Rightarrow> int"} is simply implemented by the identity function.
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  For the @{const "nat"} function for converting an integer to a natural
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  number, we give a specific implementation using an ML function that
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  returns its input value, provided that it is non-negative, and otherwise
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  returns @{text "0"}.
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*}
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definition
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  int :: "nat \<Rightarrow> int"
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where
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  [code func del]: "int = of_nat"
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lemma int_code' [code func]:
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  "int (number_of l) = (if neg (number_of l \<Colon> int) then 0 else number_of l)"
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  unfolding int_nat_number_of [folded int_def] ..
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lemma nat_code' [code func]:
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  "nat (number_of l) = (if neg (number_of l \<Colon> int) then 0 else number_of l)"
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  by auto
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lemma of_nat_int [code unfold]:
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  "of_nat = int" by (simp add: int_def)
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declare of_nat_int [symmetric, code post]
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code_const int
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  (SML "_")
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  (OCaml "_")
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consts_code
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  int ("(_)")
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  nat ("\<module>nat")
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attach {*
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fun nat i = if i < 0 then 0 else i;
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*}
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code_const nat
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  (SML "IntInf.max/ (/0,/ _)")
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  (OCaml "Big'_int.max'_big'_int/ Big'_int.zero'_big'_int")
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   361
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text {* For Haskell, things are slightly different again. *}
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   363
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code_const int and nat
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  (Haskell "toInteger" and "fromInteger")
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   366
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text {* Conversion from and to indices. *}
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code_const index_of_nat
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   370
  (SML "IntInf.toInt")
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   371
  (OCaml "Big'_int.int'_of'_big'_int")
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  (Haskell "toEnum")
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code_const nat_of_index
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  (SML "IntInf.fromInt")
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  (OCaml "Big'_int.big'_int'_of'_int")
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  (Haskell "fromEnum")
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text {* Using target language arithmetic operations whenever appropriate *}
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code_const "op + \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"
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  (SML "IntInf.+ ((_), (_))")
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  (OCaml "Big'_int.add'_big'_int")
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  (Haskell infixl 6 "+")
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code_const "op * \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"
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  (SML "IntInf.* ((_), (_))")
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  (OCaml "Big'_int.mult'_big'_int")
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   389
  (Haskell infixl 7 "*")
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   390
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code_const divmod_aux
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  (SML "IntInf.divMod/ ((_),/ (_))")
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  (OCaml "Big'_int.quomod'_big'_int")
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  (Haskell "divMod")
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code_const "op = \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
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  (SML "!((_ : IntInf.int) = _)")
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   398
  (OCaml "Big'_int.eq'_big'_int")
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   399
  (Haskell infixl 4 "==")
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code_const "op \<le> \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
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   402
  (SML "IntInf.<= ((_), (_))")
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   403
  (OCaml "Big'_int.le'_big'_int")
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   404
  (Haskell infix 4 "<=")
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   405
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code_const "op < \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
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   407
  (SML "IntInf.< ((_), (_))")
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   408
  (OCaml "Big'_int.lt'_big'_int")
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   409
  (Haskell infix 4 "<")
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   410
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consts_code
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  0                            ("0")
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  Suc                          ("(_ +/ 1)")
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   414
  "op + \<Colon>  nat \<Rightarrow> nat \<Rightarrow> nat"   ("(_ +/ _)")
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   415
  "op * \<Colon>  nat \<Rightarrow> nat \<Rightarrow> nat"   ("(_ */ _)")
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   416
  "op \<le> \<Colon>  nat \<Rightarrow> nat \<Rightarrow> bool"  ("(_ <=/ _)")
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   417
  "op < \<Colon>  nat \<Rightarrow> nat \<Rightarrow> bool"  ("(_ </ _)")
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   418
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   419
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text {* Module names *}
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code_modulename SML
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   423
  Nat Integer
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  Divides Integer
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  Efficient_Nat Integer
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   426
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   427
code_modulename OCaml
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   428
  Nat Integer
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   429
  Divides Integer
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   430
  Efficient_Nat Integer
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   431
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   432
code_modulename Haskell
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   433
  Nat Integer
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   434
  Divides Integer
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   435
  Efficient_Nat Integer
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   436
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   437
hide const int
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   438
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   439
end