# File Stream.ML

    (*  Title:      CCL/ex/Stream.ML
        ID:         $Id: Stream.ML,v 1.11 2005/09/17 15:35:32 wenzelm Exp $
        Author:     Martin Coen, Cambridge University Computer Laboratory
        Copyright   1993  University of Cambridge

    Proving properties about infinite lists using coinduction:
        Lists(A)  is the set of all finite and infinite lists of elements of A.
        ILists(A) is the set of infinite lists of elements of A.
    *)

    (*** Map of composition is composition of maps ***)

    val prems = goal (the_context ()) "l:Lists(A) ==> map(f o g,l) = map(f,map(g,l))";
    by (eq_coinduct3_tac
           "{p. EX x y. p=<x,y> & (EX l:Lists(A).x=map(f o g,l) & y=map(f,map(g,l)))}"  1);
    by (fast_tac (ccl_cs addSIs prems) 1);
    by (safe_tac type_cs);
    by (etac (XH_to_E ListsXH) 1);
    by (EQgen_tac list_ss [] 1);
    by (simp_tac list_ss 1);
    by (fast_tac ccl_cs 1);
    qed "map_comp";

    (*** Mapping the identity function leaves a list unchanged ***)

    val prems = goal (the_context ()) "l:Lists(A) ==> map(%x. x,l) = l";
    by (eq_coinduct3_tac
           "{p. EX x y. p=<x,y> & (EX l:Lists(A).x=map(%x. x,l) & y=l)}"  1);
    by (fast_tac (ccl_cs addSIs prems) 1);
    by (safe_tac type_cs);
    by (etac (XH_to_E ListsXH) 1);
    by (EQgen_tac list_ss [] 1);
    by (fast_tac ccl_cs 1);
    qed "map_id";

    (*** Mapping distributes over append ***)

    val prems = goal (the_context ())
            "[| l:Lists(A); m:Lists(A) |] ==> map(f,l@m) = map(f,l) @ map(f,m)";
    by (eq_coinduct3_tac "{p. EX x y. p=<x,y> & (EX l:Lists(A).EX m:Lists(A). \
    \                                           x=map(f,l@m) & y=map(f,l) @ map(f,m))}"  1);
    by (fast_tac (ccl_cs addSIs prems) 1);
    by (safe_tac type_cs);
    by (etac (XH_to_E ListsXH) 1);
    by (EQgen_tac list_ss [] 1);
    by (etac (XH_to_E ListsXH) 1);
    by (EQgen_tac list_ss [] 1);
    by (fast_tac ccl_cs 1);
    qed "map_append";

    (*** Append is associative ***)

    val prems = goal (the_context ())
            "[| k:Lists(A); l:Lists(A); m:Lists(A) |] ==> k @ l @ m = (k @ l) @ m";
    by (eq_coinduct3_tac
        "{p. EX x y. p=<x,y> & (EX k:Lists(A).EX l:Lists(A).EX m:Lists(A). \
    \                          x=k @ l @ m & y=(k @ l) @ m)}"  1);
    by (fast_tac (ccl_cs addSIs prems) 1);
    by (safe_tac type_cs);
    by (etac (XH_to_E ListsXH) 1);
    by (EQgen_tac list_ss [] 1);
    by (fast_tac ccl_cs 2);
    by (DEPTH_SOLVE (etac (XH_to_E ListsXH) 1 THEN EQgen_tac list_ss [] 1));
    qed "append_assoc";

    (*** Appending anything to an infinite list doesn't alter it ****)

    val prems = goal (the_context ()) "l:ILists(A) ==> l @ m = l";
    by (eq_coinduct3_tac
        "{p. EX x y. p=<x,y> & (EX l:ILists(A).EX m. x=l@m & y=l)}" 1);
    by (fast_tac (ccl_cs addSIs prems) 1);
    by (safe_tac set_cs);
    by (etac (XH_to_E IListsXH) 1);
    by (EQgen_tac list_ss [] 1);
    by (fast_tac ccl_cs 1);
    qed "ilist_append";

    (*** The equivalance of two versions of an iteration function       ***)
    (*                                                                    *)
    (*        fun iter1(f,a) = a$iter1(f,f(a))                            *)
    (*        fun iter2(f,a) = a$map(f,iter2(f,a))                        *)

    Goalw [iter1_def] "iter1(f,a) = a$iter1(f,f(a))";
    by (rtac (letrecB RS trans) 1);
    by (simp_tac term_ss 1);
    qed "iter1B";

    Goalw [iter2_def] "iter2(f,a) = a $ map(f,iter2(f,a))";
    by (rtac (letrecB RS trans) 1);
    by (rtac refl 1);
    qed "iter2B";

    val [prem] =goal (the_context ())
       "n:Nat ==> \
    \   map(f) ^ n ` iter2(f,a) = (f ^ n ` a) $ (map(f) ^ n ` map(f,iter2(f,a)))";
    by (res_inst_tac [("P", "%x. ?lhs(x) = ?rhs")] (iter2B RS ssubst) 1);
    by (simp_tac (list_ss addsimps [prem RS nmapBcons]) 1);
    qed "iter2Blemma";

    Goal "iter1(f,a) = iter2(f,a)";
    by (eq_coinduct3_tac
        "{p. EX x y. p=<x,y> & (EX n:Nat. x=iter1(f,f^n`a) & y=map(f)^n`iter2(f,a))}"
        1);
    by (fast_tac (type_cs addSIs [napplyBzero RS sym,
                                  napplyBzero RS sym RS arg_cong]) 1);
    by (EQgen_tac list_ss [iter1B,iter2Blemma] 1);
    by (stac napply_f 1 THEN atac 1);
    by (res_inst_tac [("f1","f")] (napplyBsucc RS subst) 1);
    by (fast_tac type_cs 1);
    qed "iter1_iter2_eq";