src/Sequents/LK0.ML

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Sequents/LK0.ML	Tue Jul 27 18:52:23 1999 +0200
@@ -0,0 +1,169 @@
+(*  Title:      LK/LK0
+    ID:         $Id$
+    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1992  University of Cambridge
+
+Tactics and lemmas for LK (thanks also to Philippe de Groote)  
+
+Structural rules by Soren Heilmann
+*)
+
+(** Structural Rules on formulas **)
+
+(*contraction*)
+
+Goal "$H |- $E, P, P, $F ==> $H |- $E, P, $F";
+by (etac contRS 1);
+qed "contR";
+
+Goal "$H, P, P, $G |- $E ==> $H, P, $G |- $E";
+by (etac contLS 1);
+qed "contL";
+
+(*thinning*)
+
+Goal "$H |- $E, $F ==> $H |- $E, P, $F";
+by (etac thinRS 1);
+qed "thinR";
+
+Goal "$H, $G |- $E ==> $H, P, $G |- $E";
+by (etac thinLS 1);
+qed "thinL";
+
+(*exchange*)
+
+Goal "$H |- $E, Q, P, $F ==> $H |- $E, P, Q, $F";
+by (etac exchRS 1);
+qed "exchR";
+
+Goal "$H, Q, P, $G |- $E ==> $H, P, Q, $G |- $E";
+by (etac exchLS 1);
+qed "exchL";
+
+(*Cut and thin, replacing the right-side formula*)
+fun cutR_tac (sP: string) i = 
+    res_inst_tac [ ("P",sP) ] cut i  THEN  rtac thinR i;
+
+(*Cut and thin, replacing the left-side formula*)
+fun cutL_tac (sP: string) i = 
+    res_inst_tac [ ("P",sP) ] cut i  THEN  rtac thinL (i+1);
+
+
+(** If-and-only-if rules **)
+qed_goalw "iffR" thy [iff_def]
+    "[| $H,P |- $E,Q,$F;  $H,Q |- $E,P,$F |] ==> $H |- $E, P <-> Q, $F"
+ (fn prems=> [ (REPEAT (resolve_tac (prems@[conjR,impR]) 1)) ]);
+
+qed_goalw "iffL" thy [iff_def]
+   "[| $H,$G |- $E,P,Q;  $H,Q,P,$G |- $E |] ==> $H, P <-> Q, $G |- $E"
+ (fn prems=> [ (REPEAT (resolve_tac (prems@[conjL,impL,basic]) 1)) ]);
+
+qed_goalw "TrueR" thy [True_def]
+    "$H |- $E, True, $F"
+ (fn _=> [ rtac impR 1, rtac basic 1 ]);
+
+
+(** Weakened quantifier rules.  Incomplete, they let the search terminate.**)
+
+Goal "$H, P(x), $G |- $E ==> $H, ALL x. P(x), $G |- $E";
+by (rtac allL 1);
+by (etac thinL 1);
+qed "allL_thin";
+
+Goal "$H |- $E, P(x), $F ==> $H |- $E, EX x. P(x), $F";
+by (rtac exR 1);
+by (etac thinR 1);
+qed "exR_thin";
+
+
+(*The rules of LK*)
+val prop_pack = empty_pack add_safes 
+                [basic, refl, TrueR, FalseL, 
+		 conjL, conjR, disjL, disjR, impL, impR, 
+                 notL, notR, iffL, iffR];
+
+val LK_pack = prop_pack add_safes   [allR, exL] 
+                        add_unsafes [allL_thin, exR_thin];
+
+val LK_dup_pack = prop_pack add_safes   [allR, exL] 
+                            add_unsafes [allL, exR];
+
+
+thm_pack_ref() := LK_pack;
+
+fun Fast_tac st = fast_tac (thm_pack()) st;
+fun Step_tac st = step_tac (thm_pack()) st;
+fun Safe_tac st = safe_tac (thm_pack()) st;
+
+fun lemma_tac th i = 
+    rtac (thinR RS cut) i THEN REPEAT (rtac thinL i) THEN rtac th i;
+
+val [major,minor] = goal thy 
+    "[| $H |- $E, $F, P --> Q;  $H |- $E, $F, P |] ==> $H |- $E, Q, $F";
+by (rtac (thinRS RS cut) 1 THEN rtac major 1);
+by (Step_tac 1);
+by (rtac thinR 1 THEN rtac minor 1);
+qed "mp_R";
+
+val [major,minor] = goal thy 
+    "[| $H, $G |- $E, P --> Q;  $H, $G, Q |- $E |] ==> $H, P, $G |- $E";
+by (rtac (thinL RS cut) 1 THEN rtac major 1);
+by (Step_tac 1);
+by (rtac thinL 1 THEN rtac minor 1);
+qed "mp_L";
+
+
+(** Two rules to generate left- and right- rules from implications **)
+
+val [major,minor] = goal thy 
+    "[| |- P --> Q;  $H |- $E, $F, P |] ==> $H |- $E, Q, $F";
+by (rtac mp_R 1);
+by (rtac minor 2);
+by (rtac thinRS 1 THEN rtac (major RS thinLS) 1);
+qed "R_of_imp";
+
+val [major,minor] = goal thy 
+    "[| |- P --> Q;  $H, $G, Q |- $E |] ==> $H, P, $G |- $E";
+by (rtac mp_L 1);
+by (rtac minor 2);
+by (rtac thinRS 1 THEN rtac (major RS thinLS) 1);
+qed "L_of_imp";
+
+(*Can be used to create implications in a subgoal*)
+val [prem] = goal thy 
+    "[| $H, $G |- $E, $F, P --> Q |] ==> $H, P, $G |- $E, Q, $F";
+by (rtac mp_L 1);
+by (rtac basic 2);
+by (rtac thinR 1 THEN rtac prem 1);
+qed "backwards_impR";
+
+ 
+qed_goal "conjunct1" thy "|-P&Q ==> |-P"
+    (fn [major] => [lemma_tac major 1,  Fast_tac 1]);
+
+qed_goal "conjunct2" thy "|-P&Q ==> |-Q"
+    (fn [major] => [lemma_tac major 1,  Fast_tac 1]);
+
+qed_goal "spec" thy "|- (ALL x. P(x)) ==> |- P(x)"
+    (fn [major] => [lemma_tac major 1,  Fast_tac 1]);
+
+(** Equality **)
+
+Goal "|- a=b --> b=a";
+by (safe_tac (LK_pack add_safes [subst]) 1);
+qed "sym";
+
+Goal "|- a=b --> b=c --> a=c";
+by (safe_tac (LK_pack add_safes [subst]) 1);
+qed "trans";
+
+(* Symmetry of equality in hypotheses *)
+bind_thm ("symL", sym RS L_of_imp);
+
+(* Symmetry of equality in hypotheses *)
+bind_thm ("symR", sym RS R_of_imp);
+
+Goal "[| $H|- $E, $F, a=b;  $H|- $E, $F, b=c |] ==> $H|- $E, a=c, $F";
+by (rtac (trans RS R_of_imp RS mp_R) 1);
+by (ALLGOALS assume_tac);
+qed "transR";