src/HOLCF/Tools/Domain/domain_axioms.ML

domain_isomorphism package defines combined copy function

(*  Title:      HOLCF/Tools/Domain/domain_axioms.ML
    Author:     David von Oheimb

Syntax generator for domain command.
*)

signature DOMAIN_AXIOMS =
sig
  val copy_of_dtyp :
      string Symtab.table -> (int -> term) -> Datatype.dtyp -> term

  val calc_axioms :
      bool -> string Symtab.table ->
      string -> Domain_Library.eq list -> int -> Domain_Library.eq ->
      string * (string * term) list * (string * term) list

  val add_axioms :
      bool ->
      bstring -> Domain_Library.eq list -> theory -> theory
end;


structure Domain_Axioms : DOMAIN_AXIOMS =
struct

open Domain_Library;

infixr 0 ===>;infixr 0 ==>;infix 0 == ; 
infix 1 ===; infix 1 ~= ; infix 1 <<; infix 1 ~<<;
infix 9 `   ; infix 9 `% ; infix 9 `%%; infixr 9 oo;

(* FIXME: use theory data for this *)
val copy_tab : string Symtab.table =
    Symtab.make [(@{type_name "->"}, @{const_name "cfun_map"}),
                 (@{type_name "++"}, @{const_name "ssum_map"}),
                 (@{type_name "**"}, @{const_name "sprod_map"}),
                 (@{type_name "*"}, @{const_name "cprod_map"}),
                 (@{type_name "u"}, @{const_name "u_map"})];

fun copy_of_dtyp tab r dt =
    if DatatypeAux.is_rec_type dt then copy tab r dt else ID
and copy tab r (DatatypeAux.DtRec i) = r i
  | copy tab r (DatatypeAux.DtTFree a) = ID
  | copy tab r (DatatypeAux.DtType (c, ds)) =
    case Symtab.lookup tab c of
      SOME f => list_ccomb (%%:f, map (copy_of_dtyp tab r) ds)
    | NONE => (warning ("copy_of_dtyp: unknown type constructor " ^ c); ID);

fun calc_axioms
    (definitional : bool)
    (map_tab : string Symtab.table)
    (comp_dname : string)
    (eqs : eq list)
    (n : int)
    (eqn as ((dname,_),cons) : eq)
    : string * (string * term) list * (string * term) list =
  let

(* ----- axioms and definitions concerning the isomorphism ------------------ *)

    val dc_abs = %%:(dname^"_abs");
    val dc_rep = %%:(dname^"_rep");
    val x_name'= "x";
    val x_name = idx_name eqs x_name' (n+1);
    val dnam = Long_Name.base_name dname;

    val abs_iso_ax = ("abs_iso", mk_trp(dc_rep`(dc_abs`%x_name') === %:x_name'));
    val rep_iso_ax = ("rep_iso", mk_trp(dc_abs`(dc_rep`%x_name') === %:x_name'));

    val when_def = ("when_def",%%:(dname^"_when") == 
        List.foldr (uncurry /\ ) (/\x_name'((when_body cons (fn (x,y) =>
          Bound(1+length cons+x-y)))`(dc_rep`Bound 0))) (when_funs cons));

    val copy_def =
      let fun r i = proj (Bound 0) eqs i;
      in
        ("copy_def", %%:(dname^"_copy") == /\ "f"
          (dc_abs oo (copy_of_dtyp map_tab r (dtyp_of_eq eqn)) oo dc_rep))
      end;

(* -- definitions concerning the constructors, discriminators and selectors - *)

    fun con_def m n (_,args) = let
      fun idxs z x arg = (if is_lazy arg then mk_up else I) (Bound(z-x));
      fun parms vs = mk_stuple (mapn (idxs(length vs)) 1 vs);
      fun inj y 1 _ = y
        | inj y _ 0 = mk_sinl y
        | inj y i j = mk_sinr (inj y (i-1) (j-1));
    in List.foldr /\# (dc_abs`(inj (parms args) m n)) args end;
          
    val con_defs = mapn (fn n => fn (con,args) =>
                                    (extern_name con ^"_def", %%:con == con_def (length cons) n (con,args))) 0 cons;
          
    val dis_defs = let
      fun ddef (con,_) = (dis_name con ^"_def",%%:(dis_name con) == 
                                              list_ccomb(%%:(dname^"_when"),map 
                                                                              (fn (con',args) => (List.foldr /\#
      (if con'=con then TT else FF) args)) cons))
    in map ddef cons end;

    val mat_defs =
      let
        fun mdef (con,_) =
          let
            val k = Bound 0
            val x = Bound 1
            fun one_con (con', args') =
                if con'=con then k else List.foldr /\# mk_fail args'
            val w = list_ccomb(%%:(dname^"_when"), map one_con cons)
            val rhs = /\ "x" (/\ "k" (w ` x))
          in (mat_name con ^"_def", %%:(mat_name con) == rhs) end
      in map mdef cons end;

    val pat_defs =
      let
        fun pdef (con,args) =
          let
            val ps = mapn (fn n => fn _ => %:("pat" ^ string_of_int n)) 1 args;
            val xs = map (bound_arg args) args;
            val r = Bound (length args);
            val rhs = case args of [] => mk_return HOLogic.unit
                                 | _ => mk_ctuple_pat ps ` mk_ctuple xs;
            fun one_con (con',args') = List.foldr /\# (if con'=con then rhs else mk_fail) args';
          in (pat_name con ^"_def", list_comb (%%:(pat_name con), ps) == 
                                              list_ccomb(%%:(dname^"_when"), map one_con cons))
          end
      in map pdef cons end;

    val sel_defs = let
      fun sdef con n arg = Option.map (fn sel => (sel^"_def",%%:sel == 
                                                            list_ccomb(%%:(dname^"_when"),map 
                                                                                            (fn (con',args) => if con'<>con then UU else
                                                                                                               List.foldr /\# (Bound (length args - n)) args) cons))) (sel_of arg);
    in map_filter I (maps (fn (con,args) => mapn (sdef con) 1 args) cons) end;


(* ----- axiom and definitions concerning induction ------------------------- *)

    val reach_ax = ("reach", mk_trp(proj (mk_fix (%%:(comp_dname^"_copy"))) eqs n
                                         `%x_name === %:x_name));
    val take_def =
        ("take_def",
         %%:(dname^"_take") ==
            mk_lam("n",proj
                         (mk_iterate (Bound 0, %%:(comp_dname^"_copy"), UU)) eqs n));
    val finite_def =
        ("finite_def",
         %%:(dname^"_finite") ==
            mk_lam(x_name,
                   mk_ex("n",(%%:(dname^"_take") $ Bound 0)`Bound 1 === Bound 1)));

  in (dnam,
      (if definitional then [reach_ax] else [abs_iso_ax, rep_iso_ax, reach_ax]),
      (if definitional then [when_def] else [when_def, copy_def]) @
      con_defs @ dis_defs @ mat_defs @ pat_defs @ sel_defs @
      [take_def, finite_def])
  end; (* let (calc_axioms) *)


(* legacy type inference *)

fun legacy_infer_term thy t =
    singleton (Syntax.check_terms (ProofContext.init thy)) (Sign.intern_term thy t);

fun legacy_infer_prop thy t = legacy_infer_term thy (TypeInfer.constrain propT t);

fun infer_props thy = map (apsnd (legacy_infer_prop thy));


fun add_axioms_i x = snd o PureThy.add_axioms (map (Thm.no_attributes o apfst Binding.name) x);
fun add_axioms_infer axms thy = add_axioms_i (infer_props thy axms) thy;

fun add_defs_i x = snd o (PureThy.add_defs false) (map (Thm.no_attributes o apfst Binding.name) x);
fun add_defs_infer defs thy = add_defs_i (infer_props thy defs) thy;

fun add_matchers (((dname,_),cons) : eq) thy =
    let
      val con_names = map fst cons;
      val mat_names = map mat_name con_names;
      fun qualify n = Sign.full_name thy (Binding.name n);
      val ms = map qualify con_names ~~ map qualify mat_names;
    in Fixrec.add_matchers ms thy end;

fun add_axioms definitional comp_dnam (eqs : eq list) thy' =
  let
    val comp_dname = Sign.full_bname thy' comp_dnam;
    val dnames = map (fst o fst) eqs;
    val x_name = idx_name dnames "x"; 
    fun copy_app dname = %%:(dname^"_copy")`Bound 0;
    val copy_def = ("copy_def" , %%:(comp_dname^"_copy") ==
                                 /\ "f"(mk_ctuple (map copy_app dnames)));

    fun one_con (con,args) =
      let
        val nonrec_args = filter_out is_rec args;
        val    rec_args = filter is_rec args;
        val    recs_cnt = length rec_args;
        val allargs     = nonrec_args @ rec_args
                          @ map (upd_vname (fn s=> s^"'")) rec_args;
        val allvns      = map vname allargs;
        fun vname_arg s arg = if is_rec arg then vname arg^s else vname arg;
        val vns1        = map (vname_arg "" ) args;
        val vns2        = map (vname_arg "'") args;
        val allargs_cnt = length nonrec_args + 2*recs_cnt;
        val rec_idxs    = (recs_cnt-1) downto 0;
        val nonlazy_idxs = map snd (filter_out (fn (arg,_) => is_lazy arg)
                                               (allargs~~((allargs_cnt-1) downto 0)));
        fun rel_app i ra = proj (Bound(allargs_cnt+2)) eqs (rec_of ra) $ 
                                Bound (2*recs_cnt-i) $ Bound (recs_cnt-i);
        val capps =
          List.foldr
            mk_conj
            (mk_conj(
             Bound(allargs_cnt+1)===list_ccomb(%%:con,map (bound_arg allvns) vns1),
             Bound(allargs_cnt+0)===list_ccomb(%%:con,map (bound_arg allvns) vns2)))
            (mapn rel_app 1 rec_args);
      in
        List.foldr
          mk_ex
          (Library.foldr mk_conj
                         (map (defined o Bound) nonlazy_idxs,capps)) allvns
      end;
    fun one_comp n (_,cons) =
        mk_all (x_name(n+1),
        mk_all (x_name(n+1)^"'",
        mk_imp (proj (Bound 2) eqs n $ Bound 1 $ Bound 0,
        foldr1 mk_disj (mk_conj(Bound 1 === UU,Bound 0 === UU)
                        ::map one_con cons))));
    val bisim_def =
        ("bisim_def", %%:(comp_dname^"_bisim") ==
                         mk_lam("R", foldr1 mk_conj (mapn one_comp 0 eqs)));

    fun add_one (dnam, axs, dfs) =
        Sign.add_path dnam
          #> add_defs_infer dfs
          #> add_axioms_infer axs
          #> Sign.parent_path;

    val map_tab = Domain_Isomorphism.get_map_tab thy';

    val thy = thy'
      |> fold add_one (mapn (calc_axioms definitional map_tab comp_dname eqs) 0 eqs);

    val use_copy_def = length eqs>1 andalso not definitional;
  in
    thy
    |> Sign.add_path comp_dnam  
    |> add_defs_infer (bisim_def::(if use_copy_def then [copy_def] else []))
    |> Sign.parent_path
    |> fold add_matchers eqs
  end; (* let (add_axioms) *)

end; (* struct *)