# File subset.ML

    (*  Title:      CCL/subsetI
        ID:         $Id: subset.ML,v 1.9 2005/09/17 15:35:30 wenzelm Exp $

    Derived rules involving subsets.
    Union and Intersection as lattice operations.
    *)

    (*** Big Union -- least upper bound of a set  ***)

    val prems = goal (the_context ())
        "B:A ==> B <= Union(A)";
    by (REPEAT (ares_tac (prems@[subsetI,UnionI]) 1));
    qed "Union_upper";

    val prems = goal (the_context ())
        "[| !!X. X:A ==> X<=C |] ==> Union(A) <= C";
    by (REPEAT (ares_tac [subsetI] 1
         ORELSE eresolve_tac ([UnionE] @ (prems RL [subsetD])) 1));
    qed "Union_least";

    (*** Big Intersection -- greatest lower bound of a set ***)

    val prems = goal (the_context ())
        "B:A ==> Inter(A) <= B";
    by (REPEAT (resolve_tac (prems@[subsetI]) 1
         ORELSE etac InterD 1));
    qed "Inter_lower";

    val prems = goal (the_context ())
        "[| !!X. X:A ==> C<=X |] ==> C <= Inter(A)";
    by (REPEAT (ares_tac [subsetI,InterI] 1
         ORELSE eresolve_tac (prems RL [subsetD]) 1));
    qed "Inter_greatest";

    (*** Finite Union -- the least upper bound of 2 sets ***)

    goal (the_context ()) "A <= A Un B";
    by (REPEAT (ares_tac [subsetI,UnI1] 1));
    qed "Un_upper1";

    goal (the_context ()) "B <= A Un B";
    by (REPEAT (ares_tac [subsetI,UnI2] 1));
    qed "Un_upper2";

    val prems = goal (the_context ()) "[| 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 (the_context ()) "A Int B <= A";
    by (REPEAT (ares_tac [subsetI] 1 ORELSE etac IntE 1));
    qed "Int_lower1";

    goal (the_context ()) "A Int B <= B";
    by (REPEAT (ares_tac [subsetI] 1 ORELSE etac IntE 1));
    qed "Int_lower2";

    val prems = goal (the_context ()) "[| 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";

    (*** Monotonicity ***)

    val [prem] = goalw (the_context ()) [mono_def]
        "[| !!A B. A <= B ==> f(A) <= f(B) |] ==> mono(f)";
    by (REPEAT (ares_tac [allI, impI, prem] 1));
    qed "monoI";

    val [major,minor] = goalw (the_context ()) [mono_def]
        "[| mono(f);  A <= B |] ==> f(A) <= f(B)";
    by (rtac (major RS spec RS spec RS mp) 1);
    by (rtac minor 1);
    qed "monoD";

    val [prem] = goal (the_context ()) "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 (the_context ()) "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";

    (****)

    val set_cs = FOL_cs
        addSIs [ballI, subsetI, InterI, INT_I, CollectI,
                ComplI, IntI, UnCI, singletonI]
        addIs  [bexI, UnionI, UN_I]
        addSEs [bexE, UnionE, UN_E,
                CollectE, ComplE, IntE, UnE, emptyE, singletonE]
        addEs  [ballE, InterD, InterE, INT_D, INT_E, subsetD, subsetCE];

    fun cfast_tac prems = cut_facts_tac prems THEN' fast_tac set_cs;

    fun prover s = prove_goal (the_context ()) s (fn _=>[fast_tac set_cs 1]);

    val mem_rews = [trivial_set,empty_eq] @ (map prover
     [ "(a : A Un B)   <->  (a:A | a:B)",
       "(a : A Int B)  <->  (a:A & a:B)",
       "(a : Compl(B)) <->  (~a:B)",
       "(a : {b})      <->  (a=b)",
       "(a : {})       <->   False",
       "(a : {x. P(x)}) <->  P(a)" ]);

    val set_congs = [ball_cong, bex_cong, INT_cong, UN_cong];

    val set_ss = FOL_ss addcongs set_congs
                        addsimps mem_rews;