src/HOLCF/Tools/Domain/domain_axioms.ML
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modernized structure Datatype_Aux
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(*  Title:      HOLCF/Tools/Domain/domain_axioms.ML
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    Author:     David von Oheimb
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Syntax generator for domain command.
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*)
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signature DOMAIN_AXIOMS =
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sig
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  val copy_of_dtyp :
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      string Symtab.table -> (int -> term) -> Datatype.dtyp -> term
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  val calc_axioms :
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      bool -> string Symtab.table ->
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      string -> Domain_Library.eq list -> int -> Domain_Library.eq ->
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      string * (string * term) list * (string * term) list
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  val add_axioms :
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      bool ->
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      bstring -> Domain_Library.eq list -> theory -> theory
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end;
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structure Domain_Axioms : DOMAIN_AXIOMS =
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struct
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open Domain_Library;
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infixr 0 ===>;infixr 0 ==>;infix 0 == ; 
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infix 1 ===; infix 1 ~= ; infix 1 <<; infix 1 ~<<;
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infix 9 `   ; infix 9 `% ; infix 9 `%%; infixr 9 oo;
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(* FIXME: use theory data for this *)
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val copy_tab : string Symtab.table =
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    Symtab.make [(@{type_name "->"}, @{const_name "cfun_map"}),
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                 (@{type_name "++"}, @{const_name "ssum_map"}),
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                 (@{type_name "**"}, @{const_name "sprod_map"}),
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                 (@{type_name "*"}, @{const_name "cprod_map"}),
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                 (@{type_name "u"}, @{const_name "u_map"})];
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fun copy_of_dtyp tab r dt =
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    if Datatype_Aux.is_rec_type dt then copy tab r dt else ID
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and copy tab r (Datatype_Aux.DtRec i) = r i
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  | copy tab r (Datatype_Aux.DtTFree a) = ID
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  | copy tab r (Datatype_Aux.DtType (c, ds)) =
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    case Symtab.lookup tab c of
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      SOME f => list_ccomb (%%:f, map (copy_of_dtyp tab r) ds)
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    | NONE => (warning ("copy_of_dtyp: unknown type constructor " ^ c); ID);
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fun calc_axioms
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    (definitional : bool)
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    (map_tab : string Symtab.table)
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    (comp_dname : string)
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    (eqs : eq list)
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    (n : int)
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    (eqn as ((dname,_),cons) : eq)
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    : string * (string * term) list * (string * term) list =
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  let
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(* ----- axioms and definitions concerning the isomorphism ------------------ *)
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    val dc_abs = %%:(dname^"_abs");
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    val dc_rep = %%:(dname^"_rep");
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    val x_name'= "x";
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    val x_name = idx_name eqs x_name' (n+1);
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    val dnam = Long_Name.base_name dname;
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    val abs_iso_ax = ("abs_iso", mk_trp(dc_rep`(dc_abs`%x_name') === %:x_name'));
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    val rep_iso_ax = ("rep_iso", mk_trp(dc_abs`(dc_rep`%x_name') === %:x_name'));
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    val when_def = ("when_def",%%:(dname^"_when") == 
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        List.foldr (uncurry /\ ) (/\x_name'((when_body cons (fn (x,y) =>
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          Bound(1+length cons+x-y)))`(dc_rep`Bound 0))) (when_funs cons));
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    val copy_def =
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      let fun r i = proj (Bound 0) eqs i;
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      in
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        ("copy_def", %%:(dname^"_copy") == /\ "f"
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          (dc_abs oo (copy_of_dtyp map_tab r (dtyp_of_eq eqn)) oo dc_rep))
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      end;
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(* -- definitions concerning the constructors, discriminators and selectors - *)
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    fun con_def m n (_,args) = let
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      fun idxs z x arg = (if is_lazy arg then mk_up else I) (Bound(z-x));
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      fun parms vs = mk_stuple (mapn (idxs(length vs)) 1 vs);
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      fun inj y 1 _ = y
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        | inj y _ 0 = mk_sinl y
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        | inj y i j = mk_sinr (inj y (i-1) (j-1));
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    in List.foldr /\# (dc_abs`(inj (parms args) m n)) args end;
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    val con_defs = mapn (fn n => fn (con,args) =>
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                                    (extern_name con ^"_def", %%:con == con_def (length cons) n (con,args))) 0 cons;
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    val dis_defs = let
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      fun ddef (con,_) = (dis_name con ^"_def",%%:(dis_name con) == 
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                                              list_ccomb(%%:(dname^"_when"),map 
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                                                                              (fn (con',args) => (List.foldr /\#
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      (if con'=con then TT else FF) args)) cons))
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    in map ddef cons end;
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    val mat_defs =
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      let
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        fun mdef (con,_) =
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          let
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            val k = Bound 0
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            val x = Bound 1
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            fun one_con (con', args') =
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                if con'=con then k else List.foldr /\# mk_fail args'
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            val w = list_ccomb(%%:(dname^"_when"), map one_con cons)
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            val rhs = /\ "x" (/\ "k" (w ` x))
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          in (mat_name con ^"_def", %%:(mat_name con) == rhs) end
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      in map mdef cons end;
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    val pat_defs =
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      let
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        fun pdef (con,args) =
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          let
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            val ps = mapn (fn n => fn _ => %:("pat" ^ string_of_int n)) 1 args;
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            val xs = map (bound_arg args) args;
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            val r = Bound (length args);
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            val rhs = case args of [] => mk_return HOLogic.unit
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                                 | _ => mk_ctuple_pat ps ` mk_ctuple xs;
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            fun one_con (con',args') = List.foldr /\# (if con'=con then rhs else mk_fail) args';
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          in (pat_name con ^"_def", list_comb (%%:(pat_name con), ps) == 
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                                              list_ccomb(%%:(dname^"_when"), map one_con cons))
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          end
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      in map pdef cons end;
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    val sel_defs = let
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      fun sdef con n arg = Option.map (fn sel => (sel^"_def",%%:sel == 
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                                                            list_ccomb(%%:(dname^"_when"),map 
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                                                                                            (fn (con',args) => if con'<>con then UU else
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                                                                                                               List.foldr /\# (Bound (length args - n)) args) cons))) (sel_of arg);
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    in map_filter I (maps (fn (con,args) => mapn (sdef con) 1 args) cons) end;
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(* ----- axiom and definitions concerning induction ------------------------- *)
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    val reach_ax = ("reach", mk_trp(proj (mk_fix (%%:(comp_dname^"_copy"))) eqs n
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                                         `%x_name === %:x_name));
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    val take_def =
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        ("take_def",
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         %%:(dname^"_take") ==
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            mk_lam("n",proj
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                         (mk_iterate (Bound 0, %%:(comp_dname^"_copy"), UU)) eqs n));
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    val finite_def =
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        ("finite_def",
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         %%:(dname^"_finite") ==
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            mk_lam(x_name,
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                   mk_ex("n",(%%:(dname^"_take") $ Bound 0)`Bound 1 === Bound 1)));
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  in (dnam,
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      (if definitional then [] else [abs_iso_ax, rep_iso_ax, reach_ax]),
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      (if definitional then [when_def] else [when_def, copy_def]) @
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      con_defs @ dis_defs @ mat_defs @ pat_defs @ sel_defs @
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      [take_def, finite_def])
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  end; (* let (calc_axioms) *)
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(* legacy type inference *)
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fun legacy_infer_term thy t =
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    singleton (Syntax.check_terms (ProofContext.init thy)) (Sign.intern_term thy t);
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fun legacy_infer_prop thy t = legacy_infer_term thy (TypeInfer.constrain propT t);
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fun infer_props thy = map (apsnd (legacy_infer_prop thy));
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fun add_axioms_i x = snd o PureThy.add_axioms (map (Thm.no_attributes o apfst Binding.name) x);
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fun add_axioms_infer axms thy = add_axioms_i (infer_props thy axms) thy;
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fun add_defs_i x = snd o (PureThy.add_defs false) (map (Thm.no_attributes o apfst Binding.name) x);
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fun add_defs_infer defs thy = add_defs_i (infer_props thy defs) thy;
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fun add_matchers (((dname,_),cons) : eq) thy =
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    let
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      val con_names = map fst cons;
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      val mat_names = map mat_name con_names;
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      fun qualify n = Sign.full_name thy (Binding.name n);
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      val ms = map qualify con_names ~~ map qualify mat_names;
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    in Fixrec.add_matchers ms thy end;
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fun add_axioms definitional comp_dnam (eqs : eq list) thy' =
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  let
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    val comp_dname = Sign.full_bname thy' comp_dnam;
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    val dnames = map (fst o fst) eqs;
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    val x_name = idx_name dnames "x"; 
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    fun copy_app dname = %%:(dname^"_copy")`Bound 0;
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    val copy_def = ("copy_def" , %%:(comp_dname^"_copy") ==
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                                 /\ "f"(mk_ctuple (map copy_app dnames)));
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    fun one_con (con,args) =
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      let
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        val nonrec_args = filter_out is_rec args;
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        val    rec_args = filter is_rec args;
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        val    recs_cnt = length rec_args;
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        val allargs     = nonrec_args @ rec_args
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                          @ map (upd_vname (fn s=> s^"'")) rec_args;
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        val allvns      = map vname allargs;
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        fun vname_arg s arg = if is_rec arg then vname arg^s else vname arg;
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        val vns1        = map (vname_arg "" ) args;
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        val vns2        = map (vname_arg "'") args;
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        val allargs_cnt = length nonrec_args + 2*recs_cnt;
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        val rec_idxs    = (recs_cnt-1) downto 0;
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        val nonlazy_idxs = map snd (filter_out (fn (arg,_) => is_lazy arg)
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                                               (allargs~~((allargs_cnt-1) downto 0)));
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        fun rel_app i ra = proj (Bound(allargs_cnt+2)) eqs (rec_of ra) $ 
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                                Bound (2*recs_cnt-i) $ Bound (recs_cnt-i);
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        val capps =
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          List.foldr
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            mk_conj
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            (mk_conj(
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             Bound(allargs_cnt+1)===list_ccomb(%%:con,map (bound_arg allvns) vns1),
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             Bound(allargs_cnt+0)===list_ccomb(%%:con,map (bound_arg allvns) vns2)))
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            (mapn rel_app 1 rec_args);
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      in
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        List.foldr
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          mk_ex
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          (Library.foldr mk_conj
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                         (map (defined o Bound) nonlazy_idxs,capps)) allvns
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      end;
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    fun one_comp n (_,cons) =
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        mk_all (x_name(n+1),
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        mk_all (x_name(n+1)^"'",
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        mk_imp (proj (Bound 2) eqs n $ Bound 1 $ Bound 0,
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        foldr1 mk_disj (mk_conj(Bound 1 === UU,Bound 0 === UU)
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                        ::map one_con cons))));
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    val bisim_def =
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        ("bisim_def", %%:(comp_dname^"_bisim") ==
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                         mk_lam("R", foldr1 mk_conj (mapn one_comp 0 eqs)));
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    fun add_one (dnam, axs, dfs) =
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        Sign.add_path dnam
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          #> add_defs_infer dfs
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          #> add_axioms_infer axs
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          #> Sign.parent_path;
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    val map_tab = Domain_Isomorphism.get_map_tab thy';
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    val thy = thy'
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      |> fold add_one (mapn (calc_axioms definitional map_tab comp_dname eqs) 0 eqs);
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    val use_copy_def = length eqs>1 andalso not definitional;
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  in
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    thy
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    |> Sign.add_path comp_dnam  
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    |> add_defs_infer (bisim_def::(if use_copy_def then [copy_def] else []))
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    |> Sign.parent_path
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    |> fold add_matchers eqs
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  end; (* let (add_axioms) *)
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end; (* struct *)