mono.ML

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/mono.ML	Thu Sep 16 12:21:07 1993 +0200
@@ -0,0 +1,72 @@
+(*  Title: 	HOL/mono
+    ID:         $Id$
+    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1991  University of Cambridge
+
+Monotonicity of various operations
+*)
+
+val [prem] = goal Set.thy
+    "[| !!x. P(x) ==> Q(x) |] ==> Collect(P) <= Collect(Q)";
+by (fast_tac (set_cs addIs [prem]) 1);
+val Collect_mono = result();
+
+goal Set.thy "!!A B. A<=B ==> f``A <= f``B";
+by (fast_tac set_cs 1);
+val image_mono = result();
+
+goal Set.thy "!!A B. A<=B ==> Union(A) <= Union(B)";
+by (fast_tac set_cs 1);
+val Union_mono = result();
+
+goal Set.thy "!!A B. B<=A ==> Inter(A) <= Inter(B)";
+by (fast_tac set_cs 1);
+val Inter_anti_mono = result();
+
+val prems = goal Set.thy
+    "[| A<=B;  !!x. x:A ==> f(x)<=g(x) |] ==> \
+\    (UN x:A. f(x)) <= (UN x:B. g(x))";
+by (fast_tac (set_cs addIs (prems RL [subsetD])) 1);
+val UN_mono = result();
+
+val [prem] = goal Set.thy
+    "[| !!x. f(x)<=g(x) |] ==> (UN x. f(x)) <= (UN x. g(x))";
+by (fast_tac (set_cs addIs [prem RS subsetD]) 1);
+val UN1_mono = result();
+
+val prems = goal Set.thy
+    "[| B<=A;  !!x. x:A ==> f(x)<=g(x) |] ==> \
+\    (INT x:A. f(x)) <= (INT x:A. g(x))";
+by (fast_tac (set_cs addIs (prems RL [subsetD])) 1);
+val INT_anti_mono = result();
+
+(*The inclusion is POSITIVE! *)
+val [prem] = goal Set.thy
+    "[| !!x. f(x)<=g(x) |] ==> (INT x. f(x)) <= (INT x. g(x))";
+by (fast_tac (set_cs addIs [prem RS subsetD]) 1);
+val INT1_mono = result();
+
+goal Set.thy "!!A B. [| A<=C;  B<=D |] ==> A Un B <= C Un D";
+by (fast_tac set_cs 1);
+val Un_mono = result();
+
+goal Set.thy "!!A B. [| A<=C;  B<=D |] ==> A Int B <= C Int D";
+by (fast_tac set_cs 1);
+val Int_mono = result();
+
+goal Set.thy "!!A::'a set. [| A<=C;  D<=B |] ==> A-B <= C-D";
+by (fast_tac set_cs 1);
+val Diff_mono = result();
+
+goal Set.thy "!!A B. A<=B ==> Compl(B) <= Compl(A)";
+by (fast_tac set_cs 1);
+val Compl_anti_mono = result();
+
+val prems = goal Prod.thy
+    "[| A<=C;  !!x. x:A ==> B<=D |] ==> Sigma(A,%x.B) <= Sigma(C,%x.D)";
+by (cut_facts_tac prems 1);
+by (fast_tac (set_cs addIs (prems RL [subsetD]) 
+                     addSIs [SigmaI] 
+                     addSEs [SigmaE]) 1);
+val Sigma_mono = result();
+