src/FOL/FOL_lemmas1.ML
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(*  Title:      FOL/FOL_lemmas1.ML
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1991  University of Cambridge
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Tactics and lemmas for theory FOL (classical First-Order Logic).
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*)
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val classical = thm "classical";
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bind_thm ("ccontr", FalseE RS classical);
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(*** Classical introduction rules for | and EX ***)
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val prems = Goal "(~Q ==> P) ==> P|Q";
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by (rtac classical 1);
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by (REPEAT (ares_tac (prems@[disjI1,notI]) 1));
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by (REPEAT (ares_tac (prems@[disjI2,notE]) 1)) ;
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qed "disjCI";
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(*introduction rule involving only EX*)
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val prems = Goal "( ~(EX x. P(x)) ==> P(a)) ==> EX x. P(x)";
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by (rtac classical 1);
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by (eresolve_tac (prems RL [exI]) 1) ;
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qed "ex_classical";
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(*version of above, simplifying ~EX to ALL~ *)
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val [prem]= Goal "(ALL x. ~P(x) ==> P(a)) ==> EX x. P(x)";
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by (rtac ex_classical 1);
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by (resolve_tac [notI RS allI RS prem] 1);
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by (etac notE 1);
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by (etac exI 1) ;
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qed "exCI";
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Goal"~P | P";
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by (rtac disjCI 1);
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by (assume_tac 1) ;
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qed "excluded_middle";
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(*For disjunctive case analysis*)
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fun excluded_middle_tac sP =
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    res_inst_tac [("Q",sP)] (excluded_middle RS disjE);
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val [p1,p2] = Goal"[| P ==> Q; ~P ==> Q |] ==> Q";
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by (rtac (excluded_middle RS disjE) 1);
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by (etac p2 1);
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by (etac p1 1);
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qed "case_split_thm";
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(*HOL's more natural case analysis tactic*)
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fun case_tac a = res_inst_tac [("P",a)] case_split_thm;
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(*** Special elimination rules *)
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(*Classical implies (-->) elimination. *)
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val major::prems = Goal "[| P-->Q;  ~P ==> R;  Q ==> R |] ==> R";
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by (resolve_tac [excluded_middle RS disjE] 1);
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by (DEPTH_SOLVE (ares_tac (prems@[major RS mp]) 1)) ;
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qed "impCE";
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(*This version of --> elimination works on Q before P.  It works best for
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  those cases in which P holds "almost everywhere".  Can't install as
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  default: would break old proofs.*)
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val major::prems = Goal "[| P-->Q;  Q ==> R;  ~P ==> R |] ==> R";
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by (resolve_tac [excluded_middle RS disjE] 1);
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by (DEPTH_SOLVE (ares_tac (prems@[major RS mp]) 1)) ;
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qed "impCE'";
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(*Double negation law*)
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Goal"~~P ==> P";
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by (rtac classical 1);
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by (etac notE 1);
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by (assume_tac 1);
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qed "notnotD";
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val [p1,p2] = Goal"[| Q; ~ P ==> ~ Q |] ==> P";
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by (rtac classical 1);
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by (dtac p2 1);
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by (etac notE 1);
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by (rtac p1 1);
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qed "contrapos2";
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(*** Tactics for implication and contradiction ***)
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(*Classical <-> elimination.  Proof substitutes P=Q in 
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    ~P ==> ~Q    and    P ==> Q  *)
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val major::prems = 
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Goalw  [iff_def] "[| P<->Q;  [| P; Q |] ==> R;  [| ~P; ~Q |] ==> R |] ==> R";
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by (rtac (major RS conjE) 1);
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by (REPEAT_FIRST (etac impCE));
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by (REPEAT (DEPTH_SOLVE_1 (mp_tac 1  ORELSE  ares_tac prems 1)));
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qed "iffCE";
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