# File mono.ML

    (*  Title:      CCL/mono.ML
        ID:         $Id: mono.ML,v 1.4 2005/09/17 15:35:30 wenzelm Exp $

    Monotonicity of various operations.
    *)

    val prems = goal (the_context ()) "A<=B ==> Union(A) <= Union(B)";
    by (cfast_tac prems 1);
    qed "Union_mono";

    val prems = goal (the_context ()) "[| B<=A |] ==> Inter(A) <= Inter(B)";
    by (cfast_tac prems 1);
    qed "Inter_anti_mono";

    val prems = goal (the_context ())
        "[| A<=B;  !!x. x:A ==> f(x)<=g(x) |] ==> \
    \    (UN x:A. f(x)) <= (UN x:B. g(x))";
    by (REPEAT (eresolve_tac [UN_E,ssubst] 1
         ORELSE ares_tac ((prems RL [subsetD]) @ [subsetI,UN_I]) 1));
    qed "UN_mono";

    val prems = goal (the_context ())
        "[| B<=A;  !!x. x:A ==> f(x)<=g(x) |] ==> \
    \    (INT x:A. f(x)) <= (INT x:A. g(x))";
    by (REPEAT (ares_tac ((prems RL [subsetD]) @ [subsetI,INT_I]) 1
         ORELSE etac INT_D 1));
    qed "INT_anti_mono";

    val prems = goal (the_context ()) "[| A<=C;  B<=D |] ==> A Un B <= C Un D";
    by (cfast_tac prems 1);
    qed "Un_mono";

    val prems = goal (the_context ()) "[| A<=C;  B<=D |] ==> A Int B <= C Int D";
    by (cfast_tac prems 1);
    qed "Int_mono";

    val prems = goal (the_context ()) "[| A<=B |] ==> Compl(B) <= Compl(A)";
    by (cfast_tac prems 1);
    qed "Compl_anti_mono";