src/HOL/subset.ML

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/subset.ML	Fri Mar 03 12:02:25 1995 +0100
@@ -0,0 +1,135 @@
+(*  Title: 	HOL/subset
+    ID:         $Id$
+    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1991  University of Cambridge
+
+Derived rules involving subsets
+Union and Intersection as lattice operations
+*)
+
+(*** insert ***)
+
+qed_goal "subset_insertI" Set.thy "B <= insert a B"
+ (fn _=> [ (rtac subsetI 1), (etac insertI2 1) ]);
+
+(*** Big Union -- least upper bound of a set  ***)
+
+val prems = goal Set.thy
+    "B:A ==> B <= Union(A)";
+by (REPEAT (ares_tac (prems@[subsetI,UnionI]) 1));
+qed "Union_upper";
+
+val [prem] = goal Set.thy
+    "[| !!X. X:A ==> X<=C |] ==> Union(A) <= C";
+br subsetI 1;
+by (REPEAT (eresolve_tac [asm_rl, UnionE, prem RS subsetD] 1));
+qed "Union_least";
+
+(** General union **)
+
+val prems = goal Set.thy
+    "a:A ==> B(a) <= (UN x:A. B(x))";
+by (REPEAT (ares_tac (prems@[UN_I RS subsetI]) 1));
+qed "UN_upper";
+
+val [prem] = goal Set.thy
+    "[| !!x. x:A ==> B(x)<=C |] ==> (UN x:A. B(x)) <= C";
+br subsetI 1;
+by (REPEAT (eresolve_tac [asm_rl, UN_E, prem RS subsetD] 1));
+qed "UN_least";
+
+goal Set.thy "B(a) <= (UN x. B(x))";
+by (REPEAT (ares_tac [UN1_I RS subsetI] 1));
+qed "UN1_upper";
+
+val [prem] = goal Set.thy "[| !!x. B(x)<=C |] ==> (UN x. B(x)) <= C";
+br subsetI 1;
+by (REPEAT (eresolve_tac [asm_rl, UN1_E, prem RS subsetD] 1));
+qed "UN1_least";
+
+
+(*** Big Intersection -- greatest lower bound of a set ***)
+
+val prems = goal Set.thy "B:A ==> Inter(A) <= B";
+br subsetI 1;
+by (REPEAT (resolve_tac prems 1 ORELSE etac InterD 1));
+qed "Inter_lower";
+
+val [prem] = goal Set.thy
+    "[| !!X. X:A ==> C<=X |] ==> C <= Inter(A)";
+br (InterI RS subsetI) 1;
+by (REPEAT (eresolve_tac [asm_rl, prem RS subsetD] 1));
+qed "Inter_greatest";
+
+val prems = goal Set.thy "a:A ==> (INT x:A. B(x)) <= B(a)";
+br subsetI 1;
+by (REPEAT (resolve_tac prems 1 ORELSE etac INT_D 1));
+qed "INT_lower";
+
+val [prem] = goal Set.thy
+    "[| !!x. x:A ==> C<=B(x) |] ==> C <= (INT x:A. B(x))";
+br (INT_I RS subsetI) 1;
+by (REPEAT (eresolve_tac [asm_rl, prem RS subsetD] 1));
+qed "INT_greatest";
+
+goal Set.thy "(INT x. B(x)) <= B(a)";
+br subsetI 1;
+by (REPEAT (resolve_tac prems 1 ORELSE etac INT1_D 1));
+qed "INT1_lower";
+
+val [prem] = goal Set.thy
+    "[| !!x. C<=B(x) |] ==> C <= (INT x. B(x))";
+br (INT1_I RS subsetI) 1;
+by (REPEAT (eresolve_tac [asm_rl, prem RS subsetD] 1));
+qed "INT1_greatest";
+
+(*** Finite Union -- the least upper bound of 2 sets ***)
+
+goal Set.thy "A <= A Un B";
+by (REPEAT (ares_tac [subsetI,UnI1] 1));
+qed "Un_upper1";
+
+goal Set.thy "B <= A Un B";
+by (REPEAT (ares_tac [subsetI,UnI2] 1));
+qed "Un_upper2";
+
+val prems = goal Set.thy "[| A<=C;  B<=C |] ==> A Un B <= C";
+by (cut_facts_tac prems 1);
+by (DEPTH_SOLVE (ares_tac [subsetI] 1 
+          ORELSE eresolve_tac [UnE,subsetD] 1));
+qed "Un_least";
+
+(*** Finite Intersection -- the greatest lower bound of 2 sets *)
+
+goal Set.thy "A Int B <= A";
+by (REPEAT (ares_tac [subsetI] 1 ORELSE etac IntE 1));
+qed "Int_lower1";
+
+goal Set.thy "A Int B <= B";
+by (REPEAT (ares_tac [subsetI] 1 ORELSE etac IntE 1));
+qed "Int_lower2";
+
+val prems = goal Set.thy "[| C<=A;  C<=B |] ==> C <= A Int B";
+by (cut_facts_tac prems 1);
+by (REPEAT (ares_tac [subsetI,IntI] 1
+     ORELSE etac subsetD 1));
+qed "Int_greatest";
+
+(*** Set difference ***)
+
+qed_goal "Diff_subset" Set.thy "A-B <= (A::'a set)"
+ (fn _ => [ (REPEAT (ares_tac [subsetI] 1 ORELSE etac DiffE 1)) ]);
+
+(*** Monotonicity ***)
+
+val [prem] = goal Set.thy "mono(f) ==> f(A) Un f(B) <= f(A Un B)";
+by (rtac Un_least 1);
+by (rtac (Un_upper1 RS (prem RS monoD)) 1);
+by (rtac (Un_upper2 RS (prem RS monoD)) 1);
+qed "mono_Un";
+
+val [prem] = goal Set.thy "mono(f) ==> f(A Int B) <= f(A) Int f(B)";
+by (rtac Int_greatest 1);
+by (rtac (Int_lower1 RS (prem RS monoD)) 1);
+by (rtac (Int_lower2 RS (prem RS monoD)) 1);
+qed "mono_Int";