(* 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;
(*** Classical introduction rules for | and EX ***)
qed_goal "disjCI" FOL.thy
"(~Q ==> P) ==> P|Q"
(fn prems=>
[ (resolve_tac [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=>
[ (resolve_tac [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]=>
[ (resolve_tac [ex_classical] 1),
(resolve_tac [notI RS allI RS prem] 1),
(eresolve_tac [notE] 1),
(eresolve_tac [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]=>
[ (resolve_tac [classical] 1), (eresolve_tac [major RS notE] 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 =>
[ (resolve_tac [conjE] 1),
(REPEAT (DEPTH_SOLVE_1
(etac impCE 1 ORELSE mp_tac 1 ORELSE ares_tac prems 1))) ]);