# File FOLP_lemmas.ML

    (*  Title:      FOLP/FOLP.ML
        ID:         $Id: FOLP_lemmas.ML,v 1.1 2005/09/18 12:25:48 wenzelm Exp $
        Author:     Martin D Coen, Cambridge University Computer Laboratory
        Copyright   1991  University of Cambridge
    *)

    (*** Classical introduction rules for | and EX ***)

    val prems= goal (the_context ())
       "(!!x. x:~Q ==> f(x):P) ==> ?p : P|Q";
    by (rtac classical 1);
    by (REPEAT (ares_tac (prems@[disjI1,notI]) 1));
    by (REPEAT (ares_tac (prems@[disjI2,notE]) 1)) ;
    qed "disjCI";

    (*introduction rule involving only EX*)
    val prems= goal (the_context ())
       "( !!u. u:~(EX x. P(x)) ==> f(u):P(a)) ==> ?p : EX x. P(x)";
    by (rtac classical 1);
    by (eresolve_tac (prems RL [exI]) 1) ;
    qed "ex_classical";

    (*version of above, simplifying ~EX to ALL~ *)
    val [prem]= goal (the_context ())
       "(!!u. u:ALL x. ~P(x) ==> f(u):P(a)) ==> ?p : EX x. P(x)";
    by (rtac ex_classical 1);
    by (resolve_tac [notI RS allI RS prem] 1);
    by (etac notE 1);
    by (etac exI 1) ;
    qed "exCI";

    val excluded_middle = prove_goal (the_context ()) "?p : ~P | P"
     (fn _=> [ rtac disjCI 1, assume_tac 1 ]);

    (*** Special elimination rules *)

    (*Classical implies (-->) elimination. *)
    val major::prems= goal (the_context ())
        "[| p:P-->Q;  !!x. x:~P ==> f(x):R;  !!y. y:Q ==> g(y):R |] ==> ?p : R";
    by (resolve_tac [excluded_middle RS disjE] 1);
    by (DEPTH_SOLVE (ares_tac (prems@[major RS mp]) 1)) ;
    qed "impCE";

    (*Double negation law*)
    Goal "p:~~P ==> ?p : P";
    by (rtac classical 1);
    by (etac notE 1);
    by (assume_tac 1);
    qed "notnotD";

    (*** Tactics for implication and contradiction ***)

    (*Classical <-> elimination.  Proof substitutes P=Q in
        ~P ==> ~Q    and    P ==> Q  *)
    val prems = goalw (the_context ()) [iff_def]
        "[| p:P<->Q; !!x y.[| x:P; y:Q |] ==> f(x,y):R;  \
    \                !!x y.[| x:~P; y:~Q |] ==> g(x,y):R |] ==> ?p : R";
    by (rtac conjE 1);
    by (REPEAT (DEPTH_SOLVE_1 (etac impCE 1
                   ORELSE  mp_tac 1  ORELSE  ares_tac prems 1))) ;
    qed "iffCE";

    (*Should be used as swap since ~P becomes redundant*)
    val major::prems= goal (the_context ())
       "p:~P ==> (!!x. x:~Q ==> f(x):P) ==> ?p : Q";
    by (rtac classical 1);
    by (rtac (major RS notE) 1);
    by (REPEAT (ares_tac prems 1)) ;
    qed "swap";