src/FOL/FOL.ML
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(*  Title:      FOL/FOL.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 FOL.thy (classical First-Order Logic)
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*)
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open FOL;
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val ccontr = FalseE RS classical;
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(*** Classical introduction rules for | and EX ***)
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qed_goal "disjCI" FOL.thy 
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   "(~Q ==> P) ==> P|Q"
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 (fn prems=>
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  [ (rtac classical 1),
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    (REPEAT (ares_tac (prems@[disjI1,notI]) 1)),
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    (REPEAT (ares_tac (prems@[disjI2,notE]) 1)) ]);
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(*introduction rule involving only EX*)
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qed_goal "ex_classical" FOL.thy 
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   "( ~(EX x. P(x)) ==> P(a)) ==> EX x. P(x)"
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 (fn prems=>
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  [ (rtac classical 1),
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    (eresolve_tac (prems RL [exI]) 1) ]);
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(*version of above, simplifying ~EX to ALL~ *)
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qed_goal "exCI" FOL.thy 
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   "(ALL x. ~P(x) ==> P(a)) ==> EX x. P(x)"
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 (fn [prem]=>
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  [ (rtac ex_classical 1),
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    (resolve_tac [notI RS allI RS prem] 1),
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    (etac notE 1),
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    (etac exI 1) ]);
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qed_goal "excluded_middle" FOL.thy "~P | P"
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 (fn _=> [ rtac disjCI 1, assume_tac 1 ]);
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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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qed_goal "case_split_thm" FOL.thy "[| P ==> Q; ~P ==> Q |] ==> Q"
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  (fn [p1,p2] => [rtac (excluded_middle RS disjE) 1,
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                  etac p2 1, etac p1 1]);
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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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qed_goal "impCE" FOL.thy 
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    "[| P-->Q;  ~P ==> R;  Q ==> R |] ==> R"
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 (fn major::prems=>
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  [ (resolve_tac [excluded_middle RS disjE] 1),
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    (DEPTH_SOLVE (ares_tac (prems@[major RS mp]) 1)) ]);
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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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qed_goal "impCE'" thy 
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    "[| P-->Q;  Q ==> R;  ~P ==> R |] ==> R"
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 (fn major::prems=>
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  [ (resolve_tac [excluded_middle RS disjE] 1),
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    (DEPTH_SOLVE (ares_tac (prems@[major RS mp]) 1)) ]);
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(*Double negation law*)
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qed_goal "notnotD" FOL.thy "~~P ==> P"
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 (fn [major]=>
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  [ (rtac classical 1), (eresolve_tac [major RS notE] 1) ]);
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qed_goal "contrapos2" FOL.thy "[| Q; ~ P ==> ~ Q |] ==> P" (fn [p1,p2] => [
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        rtac classical 1,
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        dtac p2 1,
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        etac notE 1,
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        rtac p1 1]);
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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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qed_goalw "iffCE" FOL.thy [iff_def]
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    "[| P<->Q;  [| P; Q |] ==> R;  [| ~P; ~Q |] ==> R |] ==> R"
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 (fn prems =>
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  [ (rtac conjE 1),
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    (REPEAT (DEPTH_SOLVE_1 
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        (etac impCE 1  ORELSE  mp_tac 1  ORELSE  ares_tac prems 1))) ]);
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