--- a/src/CCL/ex/Stream.ML Mon Jul 17 18:42:38 2006 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,110 +0,0 @@
-(* Title: CCL/ex/Stream.ML
- ID: $Id$
- 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";