src/HOLCF/Tools/domain/domain_axioms.ML

define copy functions using combinators; add checking for failed proofs of induction rules

(*  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) -> DatatypeAux.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 = cproj (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 List.mapPartial I (List.concat(map (fn (con,args) => mapn (sdef con) 1 args) cons)) end;


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

        val reach_ax = ("reach", mk_trp(cproj (mk_fix (%%:(comp_dname^"_copy"))) eqs n
					      `%x_name === %:x_name));
        val take_def = ("take_def",%%:(dname^"_take") == mk_lam("n",cproj
	                                                                (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 FixrecPackage.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 = List.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 (thy,(dnam,axs,dfs)) =
            thy |> Sign.add_path dnam
                |> add_defs_infer dfs
                |> add_axioms_infer axs
                |> Sign.parent_path;

        val thy = Library.foldl add_one (thy', mapn (calc_axioms comp_dname eqs) 0 eqs);

    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 *)