(*  Title:      FOL/FOL.ML

     ID:         $Id$

     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

     Copyright   1991  University of Cambridge

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

 *)

 open FOL;

 val ccontr = FalseE RS classical;

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

 qed_goal "disjCI" FOL.thy 

    "(~Q ==> P) ==> P|Q"

  (fn prems=>

   [ (rtac classical 1),

     (REPEAT (ares_tac (prems@[disjI1,notI]) 1)),

     (REPEAT (ares_tac (prems@[disjI2,notE]) 1)) ]);

 (*introduction rule involving only EX*)

 qed_goal "ex_classical" FOL.thy 

    "( ~(EX x. P(x)) ==> P(a)) ==> EX x. P(x)"

  (fn prems=>

   [ (rtac classical 1),

     (eresolve_tac (prems RL [exI]) 1) ]);

 (*version of above, simplifying ~EX to ALL~ *)

 qed_goal "exCI" FOL.thy 

    "(ALL x. ~P(x) ==> P(a)) ==> EX x. P(x)"

  (fn [prem]=>

   [ (rtac ex_classical 1),

     (resolve_tac [notI RS allI RS prem] 1),

     (etac notE 1),

     (etac exI 1) ]);

 qed_goal "excluded_middle" FOL.thy "~P | P"

  (fn _=> [ rtac disjCI 1, assume_tac 1 ]);

 (*For disjunctive case analysis*)

 fun excluded_middle_tac sP =

     res_inst_tac [("Q",sP)] (excluded_middle RS disjE);

 (*** Special elimination rules *)

 (*Classical implies (-->) elimination. *)

 qed_goal "impCE" FOL.thy 

     "[| P-->Q;  ~P ==> R;  Q ==> R |] ==> R"

  (fn major::prems=>

   [ (resolve_tac [excluded_middle RS disjE] 1),

     (DEPTH_SOLVE (ares_tac (prems@[major RS mp]) 1)) ]);

 (*Double negation law*)

 qed_goal "notnotD" FOL.thy "~~P ==> P"

  (fn [major]=>

   [ (rtac classical 1), (eresolve_tac [major RS notE] 1) ]);

 qed_goal "contrapos2" FOL.thy "[| Q; ~ P ==> ~ Q |] ==> P" (fn [p1,p2] => [

         rtac classical 1,

         dtac p2 1,

         etac notE 1,

         rtac p1 1]);

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

 (*Classical <-> elimination.  Proof substitutes P=Q in 

     ~P ==> ~Q    and    P ==> Q  *)

 qed_goalw "iffCE" FOL.thy [iff_def]

     "[| P<->Q;  [| P; Q |] ==> R;  [| ~P; ~Q |] ==> R |] ==> R"

  (fn prems =>

   [ (rtac conjE 1),

     (REPEAT (DEPTH_SOLVE_1 

         (etac impCE 1  ORELSE  mp_tac 1  ORELSE  ares_tac prems 1))) ]);