# File axioms.ML

    (*  Title:      HOLCF/domain/axioms.ML
        ID:         $Id: axioms.ML,v 1.24 2005/07/14 17:28:27 wenzelm Exp $
        Author:     David von Oheimb

    Syntax generator for domain section.
    *)

    structure Domain_Axioms = struct

    local

    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;

    fun infer_types thy' = map (inferT_axm (sign_of thy'));

    fun calc_axioms comp_dname (eqs : eq list) n (((dname,_),cons) : eq)=
    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 = Sign.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") ==
         foldr (uncurry /\ ) (/\x_name'((when_body cons (fn (x,y) =>
    				Bound(1+length cons+x-y)))`(dc_rep`Bound 0))) (when_funs cons));

      fun con_def outer recu m n (_,args) = let
         fun idxs z x arg = (if is_lazy arg then fn t => %%:upN`t else I)
    			(if recu andalso is_rec arg then (cproj (Bound z) eqs
    				  (rec_of arg))`Bound(z-x) else Bound(z-x));
         fun parms [] = %%:ONE_N
         |   parms vs = foldr'(fn(x,t)=> %%:spairN`x`t)(mapn (idxs(length vs))1 vs);
         fun inj y 1 _ = y
         |   inj y _ 0 = %%:sinlN`y
         |   inj y i j = %%:sinrN`(inj y (i-1) (j-1));
      in foldr /\# (outer (inj (parms args) m n)) args end;

      val copy_def = ("copy_def", %%:(dname^"_copy") == /\"f" (dc_abs oo
    	Library.foldl (op `) (%%:(dname^"_when") ,
    	              mapn (con_def I true (length cons)) 0 cons)));

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

      val con_defs = mapn (fn n => fn (con,args) => (extern_name con ^"_def",
      %%:con == con_def (fn t => dc_abs`t) false (length cons) n (con,args))) 0 cons;

      val dis_defs = let
    	fun ddef (con,_) = (dis_name con ^"_def",%%:(dis_name con) ==
    		 mk_cRep_CFun(%%:(dname^"_when"),map
    			(fn (con',args) => (foldr /\#
    			   (if con'=con then %%:TT_N else %%:FF_N) args)) cons))
    	in map ddef cons end;

      val mat_defs = let
    	fun mdef (con,_) = (mat_name con ^"_def",%%:(mat_name con) ==
    		 mk_cRep_CFun(%%:(dname^"_when"),map
    			(fn (con',args) => (foldr /\#
    			   (if con'=con
                                   then %%:returnN`(mk_ctuple (map (bound_arg args) args))
                                   else %%:failN) args)) cons))
    	in map mdef cons end;

      val sel_defs = let
    	fun sdef con n arg = Option.map (fn sel => (sel^"_def",%%:sel ==
    		 mk_cRep_CFun(%%:(dname^"_when"),map
    			(fn (con',args) => if con'<>con then UU else
    			 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 (%%:fixN`%%(comp_dname^"_copy")) eqs n
    					`%x_name === %:x_name));
      val take_def = ("take_def",%%:(dname^"_take") == mk_lam("n",cproj'
    	     (%%:iterateN $ 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 @ sel_defs @
        [take_def, finite_def])
    end; (* let *)

    fun add_axioms_i x = #1 o PureThy.add_axioms_i (map Thm.no_attributes x);
    fun add_axioms_infer axms thy = add_axioms_i (infer_types thy axms) thy;

    in (* local *)

    fun add_axioms (comp_dnam, eqs : eq list) thy' = let
      val comp_dname = Sign.full_name (sign_of 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"(foldr' cpair (map copy_app dnames)));
      val bisim_def = ("bisim_def",%%:(comp_dname^"_bisim")==mk_lam("R",
        let
          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 = foldr mk_conj (mk_conj(
    	   Bound(allargs_cnt+1)===mk_cRep_CFun(%%:con,map (bound_arg allvns) vns1),
    	   Bound(allargs_cnt+0)===mk_cRep_CFun(%%:con,map (bound_arg allvns) vns2)))
               (mapn rel_app 1 rec_args);
            in 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,
             		foldr' mk_disj (mk_conj(Bound 1 === UU,Bound 0 === UU)
    					::map one_con cons))));
        in foldr' mk_conj (mapn one_comp 0 eqs)end ));
      fun add_one (thy,(dnam,axs,dfs)) = thy
    	|> Theory.add_path dnam
    	|> add_axioms_infer dfs(*add_defs_i*)
    	|> add_axioms_infer axs
    	|> Theory.parent_path;
      val thy = Library.foldl add_one (thy', mapn (calc_axioms comp_dname eqs) 0 eqs);
    in thy |> Theory.add_path comp_dnam
           |> add_axioms_infer (bisim_def::(if length eqs>1 then [copy_def] else []))
           |> Theory.parent_path
    end;

    end; (* local *)
    end; (* struct *)