src/CCL/ex/stream.ML
author clasohm
Thu Sep 16 12:20:38 1993 +0200 (1993-09-16)
changeset 0 a5a9c433f639
child 8 c3d2c6dcf3f0
permissions -rw-r--r--
Initial revision
     1 (*  Title: 	CCL/ex/stream
     2     ID:         $Id$
     3     Author: 	Martin Coen, Cambridge University Computer Laboratory
     4     Copyright   1993  University of Cambridge
     5 
     6 For stream.thy.
     7 
     8 Proving properties about infinite lists using coinduction:
     9     Lists(A)  is the set of all finite and infinite lists of elements of A.
    10     ILists(A) is the set of infinite lists of elements of A.
    11 *)
    12 
    13 open Stream;
    14 
    15 (*** Map of composition is composition of maps ***)
    16 
    17 val prems = goal Stream.thy "l:Lists(A) ==> map(f o g,l) = map(f,map(g,l))";
    18 by (eq_coinduct3_tac 
    19        "{p. EX x y.p=<x,y> & (EX l:Lists(A).x=map(f o g,l) & y=map(f,map(g,l)))}"  1);
    20 by (fast_tac (ccl_cs addSIs prems) 1);
    21 by (safe_tac type_cs);
    22 be (XH_to_E ListsXH) 1;
    23 by (EQgen_tac list_ss [] 1);
    24 by (SIMP_TAC list_ss 1);
    25 by (fast_tac ccl_cs 1);
    26 val map_comp = result();
    27 
    28 (*** Mapping the identity function leaves a list unchanged ***)
    29 
    30 val prems = goal Stream.thy "l:Lists(A) ==> map(%x.x,l) = l";
    31 by (eq_coinduct3_tac 
    32        "{p. EX x y.p=<x,y> & (EX l:Lists(A).x=map(%x.x,l) & y=l)}"  1);
    33 by (fast_tac (ccl_cs addSIs prems) 1);
    34 by (safe_tac type_cs);
    35 be (XH_to_E ListsXH) 1;
    36 by (EQgen_tac list_ss [] 1);
    37 by (fast_tac ccl_cs 1);
    38 val map_id = result();
    39 
    40 (*** Mapping distributes over append ***)
    41 
    42 val prems = goal Stream.thy 
    43         "[| l:Lists(A); m:Lists(A) |] ==> map(f,l@m) = map(f,l) @ map(f,m)";
    44 by (eq_coinduct3_tac "{p. EX x y.p=<x,y> & (EX l:Lists(A).EX m:Lists(A). \
    45 \                                           x=map(f,l@m) & y=map(f,l) @ map(f,m))}"  1);
    46 by (fast_tac (ccl_cs addSIs prems) 1);
    47 by (safe_tac type_cs);
    48 be (XH_to_E ListsXH) 1;
    49 by (EQgen_tac list_ss [] 1);
    50 be (XH_to_E ListsXH) 1;
    51 by (EQgen_tac list_ss [] 1);
    52 by (fast_tac ccl_cs 1);
    53 val map_append = result();
    54 
    55 (*** Append is associative ***)
    56 
    57 val prems = goal Stream.thy 
    58         "[| k:Lists(A); l:Lists(A); m:Lists(A) |] ==> k @ l @ m = (k @ l) @ m";
    59 by (eq_coinduct3_tac "{p. EX x y.p=<x,y> & (EX k:Lists(A).EX l:Lists(A).EX m:Lists(A). \
    60 \                                                   x=k @ l @ m & y=(k @ l) @ m)}"  1);
    61 by (fast_tac (ccl_cs addSIs prems) 1);
    62 by (safe_tac type_cs);
    63 be (XH_to_E ListsXH) 1;
    64 by (EQgen_tac list_ss [] 1);
    65 be (XH_to_E ListsXH) 1;back();
    66 by (EQgen_tac list_ss [] 1);
    67 be (XH_to_E ListsXH) 1;
    68 by (EQgen_tac list_ss [] 1);
    69 by (fast_tac ccl_cs 1);
    70 val append_assoc = result();
    71 
    72 (*** Appending anything to an infinite list doesn't alter it ****)
    73 
    74 val prems = goal Stream.thy "l:ILists(A) ==> l @ m = l";
    75 by (eq_coinduct3_tac "{p. EX x y.p=<x,y> & (EX l:ILists(A).EX m.x=l@m & y=l)}" 1);
    76 by (fast_tac (ccl_cs addSIs prems) 1);
    77 by (safe_tac set_cs);
    78 be (XH_to_E IListsXH) 1;
    79 by (EQgen_tac list_ss [] 1);
    80 by (fast_tac ccl_cs 1);
    81 val ilist_append = result();
    82 
    83 (*** The equivalance of two versions of an iteration function       ***)
    84 (*                                                                    *)
    85 (*        fun iter1(f,a) = a.iter1(f,f(a))                            *)
    86 (*        fun iter2(f,a) = a.map(f,iter2(f,a))                        *)
    87 
    88 goalw Stream.thy [iter1_def] "iter1(f,a) = a.iter1(f,f(a))";
    89 br (letrecB RS trans) 1;
    90 by (SIMP_TAC term_ss 1);
    91 val iter1B = result();
    92 
    93 goalw Stream.thy [iter2_def] "iter2(f,a) = a . map(f,iter2(f,a))";
    94 br (letrecB RS trans) 1;
    95 br refl 1;
    96 val iter2B = result();
    97 
    98 val [prem] =goal Stream.thy
    99    "n:Nat ==> map(f) ^ n ` iter2(f,a) = f ^ n ` a . map(f) ^ n ` map(f,iter2(f,a))";
   100 br (iter2B RS ssubst) 1;back();back();
   101 by (SIMP_TAC (list_ss addrews [prem RS nmapBcons]) 1);
   102 val iter2Blemma = result();
   103 
   104 goal Stream.thy "iter1(f,a) = iter2(f,a)";
   105 by (eq_coinduct3_tac 
   106     "{p. EX x y.p=<x,y> & (EX n:Nat.x=iter1(f,f^n`a) & y=map(f)^n`iter2(f,a))}" 1);
   107 by (fast_tac (type_cs addSIs [napplyBzero RS sym,napplyBzero RS sym RS arg_cong]) 1);
   108 by (EQgen_tac list_ss [iter1B,iter2Blemma] 1);
   109 by (rtac (napply_f RS ssubst) 1 THEN atac 1);
   110 by (res_inst_tac [("f1","f")] (napplyBsucc RS subst) 1);
   111 by (fast_tac type_cs 1);
   112 val iter1_iter2_eq = result();