# File Lfp.ML

    (*  Title:      CCL/Lfp.ML
        ID:         $Id: Lfp.ML,v 1.5 2005/09/17 15:35:28 wenzelm Exp $
    *)

    (*** Proof of Knaster-Tarski Theorem ***)

    (* lfp(f) is the greatest lower bound of {u. f(u) <= u} *)

    val prems = goalw (the_context ()) [lfp_def] "[| f(A) <= A |] ==> lfp(f) <= A";
    by (rtac (CollectI RS Inter_lower) 1);
    by (resolve_tac prems 1);
    qed "lfp_lowerbound";

    val prems = goalw (the_context ()) [lfp_def]
        "[| !!u. f(u) <= u ==> A<=u |] ==> A <= lfp(f)";
    by (REPEAT (ares_tac ([Inter_greatest]@prems) 1));
    by (etac CollectD 1);
    qed "lfp_greatest";

    val [mono] = goal (the_context ()) "mono(f) ==> f(lfp(f)) <= lfp(f)";
    by (EVERY1 [rtac lfp_greatest, rtac subset_trans,
                rtac (mono RS monoD), rtac lfp_lowerbound, atac, atac]);
    qed "lfp_lemma2";

    val [mono] = goal (the_context ()) "mono(f) ==> lfp(f) <= f(lfp(f))";
    by (EVERY1 [rtac lfp_lowerbound, rtac (mono RS monoD),
                rtac lfp_lemma2, rtac mono]);
    qed "lfp_lemma3";

    val [mono] = goal (the_context ()) "mono(f) ==> lfp(f) = f(lfp(f))";
    by (REPEAT (resolve_tac [equalityI,lfp_lemma2,lfp_lemma3,mono] 1));
    qed "lfp_Tarski";

    (*** General induction rule for least fixed points ***)

    val [lfp,mono,indhyp] = goal (the_context ())
        "[| a: lfp(f);  mono(f);                            \
    \       !!x. [| x: f(lfp(f) Int {x. P(x)}) |] ==> P(x)   \
    \    |] ==> P(a)";
    by (res_inst_tac [("a","a")] (Int_lower2 RS subsetD RS CollectD) 1);
    by (rtac (lfp RSN (2, lfp_lowerbound RS subsetD)) 1);
    by (EVERY1 [rtac Int_greatest, rtac subset_trans,
                rtac (Int_lower1 RS (mono RS monoD)),
                rtac (mono RS lfp_lemma2),
                rtac (CollectI RS subsetI), rtac indhyp, atac]);
    qed "induct";

    (** Definition forms of lfp_Tarski and induct, to control unfolding **)

    val [rew,mono] = goal (the_context ()) "[| h==lfp(f);  mono(f) |] ==> h = f(h)";
    by (rewtac rew);
    by (rtac (mono RS lfp_Tarski) 1);
    qed "def_lfp_Tarski";

    val rew::prems = goal (the_context ())
        "[| A == lfp(f);  a:A;  mono(f);                    \
    \       !!x. [| x: f(A Int {x. P(x)}) |] ==> P(x)        \
    \    |] ==> P(a)";
    by (EVERY1 [rtac induct,        (*backtracking to force correct induction*)
                REPEAT1 o (ares_tac (map (rewrite_rule [rew]) prems))]);
    qed "def_induct";

    (*Monotonicity of lfp!*)
    val prems = goal (the_context ())
        "[| mono(g);  !!Z. f(Z)<=g(Z) |] ==> lfp(f) <= lfp(g)";
    by (rtac lfp_lowerbound 1);
    by (rtac subset_trans 1);
    by (resolve_tac prems 1);
    by (rtac lfp_lemma2 1);
    by (resolve_tac prems 1);
    qed "lfp_mono";