src/FOL/FOL_lemmas1.ML

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/FOL/FOL_lemmas1.ML	Wed Aug 25 20:45:19 1999 +0200
@@ -0,0 +1,92 @@
+(*  Title:      FOL/FOL_lemmas1.ML
+    ID:         $Id$
+    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";
+val ccontr = FalseE RS classical;
+
+
+(*** Classical introduction rules for | and EX ***)
+
+qed_goal "disjCI" (the_context ()) 
+   "(~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" (the_context ()) 
+   "( ~(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" (the_context ()) 
+   "(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" (the_context ()) "~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);
+
+qed_goal "case_split_thm" (the_context ()) "[| P ==> Q; ~P ==> Q |] ==> Q"
+  (fn [p1,p2] => [rtac (excluded_middle RS disjE) 1,
+                  etac p2 1, etac p1 1]);
+
+(*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. *)
+qed_goal "impCE" (the_context ()) 
+    "[| 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)) ]);
+
+(*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.*)
+qed_goal "impCE'" (the_context ()) 
+    "[| P-->Q;  Q ==> R;  ~P ==> 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" (the_context ()) "~~P ==> P"
+ (fn [major]=>
+  [ (rtac classical 1), (eresolve_tac [major RS notE] 1) ]);
+
+qed_goal "contrapos2" (the_context ()) "[| 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" (the_context ()) [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))) ]);