src/HOLCF/Tools/Domain/domain_axioms.ML

domain package no longer uses cfst/csnd/cpair

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

Syntax generator for domain command.
*)

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

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

  val add_axioms :
      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_fun"}),
                 (@{type_name "++"}, @{const_name "ssum_fun"}),
                 (@{type_name "**"}, @{const_name "sprod_fun"}),
                 (@{type_name "*"}, @{const_name "cprod_fun"}),
                 (@{type_name "u"}, @{const_name "u_fun"})];

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

fun calc_axioms
      (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 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,
        [abs_iso_ax, rep_iso_ax, reach_ax],
        [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 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 thy = fold add_one (mapn (calc_axioms comp_dname eqs) 0 eqs) thy';

    in thy |> Sign.add_path comp_dnam  
           |> add_defs_infer (bisim_def::(if length eqs>1 then [copy_def] else []))
           |> Sign.parent_path
           |> fold add_matchers eqs
    end; (* let (add_axioms) *)

end; (* struct *)