(*  Title: 	ZF/list.ML

     ID:         $Id$

     Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory

     Copyright   1993  University of Cambridge

 Datatype definition of Lists

 *)

 structure List = Datatype_Fun

  (val thy = Univ.thy;

   val rec_specs = 

       [("list", "univ(A)",

 	  [(["Nil"],	"i"), 

 	   (["Cons"],	"[i,i]=>i")])];

   val rec_styp = "i=>i";

   val ext = None

   val sintrs = 

       ["Nil : list(A)",

        "[| a: A;  l: list(A) |] ==> Cons(a,l) : list(A)"];

   val monos = [];

   val type_intrs = data_typechecks

   val type_elims = []);

 val [NilI, ConsI] = List.intrs;

 (*An elimination rule, for type-checking*)

 val ConsE = List.mk_cases List.con_defs "Cons(a,l) : list(A)";

 (*Proving freeness results*)

 val Cons_iff     = List.mk_free "Cons(a,l)=Cons(a',l') <-> a=a' & l=l'";

 val Nil_Cons_iff = List.mk_free "~ Nil=Cons(a,l)";

 (*Perform induction on l, then prove the major premise using prems. *)

 fun list_ind_tac a prems i = 

     EVERY [res_inst_tac [("x",a)] List.induct i,

 	   rename_last_tac a ["1"] (i+2),

 	   ares_tac prems i];

 (**  Lemmas to justify using "list" in other recursive type definitions **)

 goalw List.thy List.defs "!!A B. A<=B ==> list(A) <= list(B)";

 by (rtac lfp_mono 1);

 by (REPEAT (rtac List.bnd_mono 1));

 by (REPEAT (ares_tac (univ_mono::basic_monos) 1));

 val list_mono = result();

 (*There is a similar proof by list induction.*)

 goalw List.thy (List.defs@List.con_defs) "list(univ(A)) <= univ(A)";

 by (rtac lfp_lowerbound 1);

 by (rtac (A_subset_univ RS univ_mono) 2);

 by (fast_tac (ZF_cs addSIs [zero_in_univ, Inl_in_univ, Inr_in_univ,

 			    Pair_in_univ]) 1);

 val list_univ = result();

 val list_subset_univ = standard

     (list_mono RS (list_univ RSN (2,subset_trans)));

 (*****

 val major::prems = goal List.thy

     "[| l: list(A);    \

 \       c: C(0);       \

 \       !!x y. [| x: A;  y: list(A) |] ==> h(x,y): C(<x,y>)  \

 \    |] ==> list_case(l,c,h) : C(l)";

 by (rtac (major RS list_induct) 1);

 by (ALLGOALS (asm_simp_tac (ZF_ss addsimps

 			    ([list_case_0,list_case_Pair]@prems))));

 val list_case_type = result();

 ****)

 (** For recursion **)

 goalw List.thy List.con_defs "rank(a) : rank(Cons(a,l))";

 by (simp_tac rank_ss 1);

 val rank_Cons1 = result();

 goalw List.thy List.con_defs "rank(l) : rank(Cons(a,l))";

 by (simp_tac rank_ss 1);

 val rank_Cons2 = result();