# File FOL_lemmas1.ML

    (*  Title:      FOL/FOL_lemmas1.ML
        ID:         $Id: FOL_lemmas1.ML,v 1.3 2000/07/06 11:28:36 paulson Exp $
        Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
        Copyright   1991  University of Cambridge

    Tactics and lemmas for theory FOL (classical First-Order Logic).
    *)

    val classical = thm "classical";
    bind_thm ("ccontr", FalseE RS classical);

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

    val prems = Goal "(~Q ==> 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 "( ~(EX x. P(x)) ==> P(a)) ==> 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 "(ALL x. ~P(x) ==> P(a)) ==> 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";

    Goal"~P | P";
    by (rtac disjCI 1);
    by (assume_tac 1) ;
    qed "excluded_middle";

    (*For disjunctive case analysis*)
    fun excluded_middle_tac sP =
        res_inst_tac [("Q",sP)] (excluded_middle RS disjE);

    val [p1,p2] = Goal"[| P ==> Q; ~P ==> Q |] ==> Q";
    by (rtac (excluded_middle RS disjE) 1);
    by (etac p2 1);
    by (etac p1 1);
    qed "case_split_thm";

    (*HOL's more natural case analysis tactic*)
    fun case_tac a = res_inst_tac [("P",a)] case_split_thm;

    (*** Special elimination rules *)

    (*Classical implies (-->) elimination. *)
    val major::prems = Goal "[| P-->Q;  ~P ==> R;  Q ==> R |] ==> R";
    by (resolve_tac [excluded_middle RS disjE] 1);
    by (DEPTH_SOLVE (ares_tac (prems@[major RS mp]) 1)) ;
    qed "impCE";

    (*This version of --> elimination works on Q before P.  It works best for
      those cases in which P holds "almost everywhere".  Can't install as
      default: would break old proofs.*)
    val major::prems = Goal "[| P-->Q;  Q ==> R;  ~P ==> R |] ==> 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";
    by (rtac classical 1);
    by (etac notE 1);
    by (assume_tac 1);
    qed "notnotD";

    val [p1,p2] = Goal"[| Q; ~ P ==> ~ Q |] ==> P";
    by (rtac classical 1);
    by (dtac p2 1);
    by (etac notE 1);
    by (rtac p1 1);
    qed "contrapos2";

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

    (*Classical <-> elimination.  Proof substitutes P=Q in
        ~P ==> ~Q    and    P ==> Q  *)
    val major::prems =
    Goalw  [iff_def] "[| P<->Q;  [| P; Q |] ==> R;  [| ~P; ~Q |] ==> R |] ==> R";
    by (rtac (major RS conjE) 1);
    by (REPEAT_FIRST (etac impCE));
    by (REPEAT (DEPTH_SOLVE_1 (mp_tac 1  ORELSE  ares_tac prems 1)));
    qed "iffCE";