(* 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";