(* Title: Sequents/ILL.ML ID: $Id$ Author: Sara Kalvala and Valeria de Paiva Copyright 1992 University of Cambridge *) open ILL; val lazy_cs = empty_pack add_safes [tensl, conjr, disjl, promote0, context2,context3] add_unsafes [identity, zerol, conjll, conjlr, disjrl, disjrr, impr, tensr, impl, derelict, weaken, promote1, promote2,context1,context4a,context4b]; fun prom_tac n = REPEAT (resolve_tac [promote0,promote1,promote2] n); fun auto x = prove_goal thy x (fn prems => [best_tac lazy_cs 1]); val aux_impl = prove_goal thy "$F, $G |- A ==> $F, !(A -o B), $G |- B" (fn prems => [rtac derelict 1 THEN rtac impl 1 THEN rtac identity 2 THEN rtac context1 1 THEN rtac (hd(prems)) 1]); val conj_lemma = prove_goal thy " $F, !A, !B, $G |- C ==> $F, !(A && B), $G |- C" (fn prems => [rtac contract 1, res_inst_tac [("A","(!A) >< (!B)")] cut 1, rtac tensr 2, rtac (auto "! (A && B) |- !A") 3, rtac (auto "! (A && B) |- !B") 3, rtac context1 2, rtac tensl 2, rtac (hd(prems)) 2, rtac context3 1, rtac context3 1, rtac context1 1]); val impr_contract = prove_goal thy "!A, !A, $G |- B ==> $G |- (!A) -o B" (fn prems => [rtac impr 1 THEN rtac contract 1 THEN rtac (hd(prems)) 1]); val impr_contr_der = prove_goal thy "A, !A, $G |- B ==> $G |- (!A) -o B" (fn prems => [rtac impr 1 THEN rtac contract 1 THEN rtac derelict 1 THEN rtac (hd(prems)) 1]); val contrad1 = prove_goal thy "$F, (!B) -o 0, $G, !B, $H |- A" (fn _ => [rtac impl 1,rtac identity 3,rtac context3 1,rtac context1 1, rtac zerol 1]); val contrad2 = prove_goal thy "$F, !B, $G, (!B) -o 0, $H |- A" (fn _ => [rtac impl 1,rtac identity 3,rtac context3 1,rtac context1 1, rtac zerol 1]); val ll_mp = prove_goal thy "A -o B, A |- B" (fn _ => [rtac impl 1 THEN rtac identity 2 THEN rtac identity 2 THEN rtac context1 1]); val mp_rule1 = prove_goal thy "$F, B, $G, $H |- C ==> $F, A, $G, A -o B, $H |- C" (fn prems => [res_inst_tac [("A","B")] cut 1, rtac ll_mp 2, rtac (hd(prems)) 2, rtac context3 1, rtac context3 1, rtac context1 1]); val mp_rule2 = prove_goal thy "$F, B, $G, $H |- C ==> $F, A -o B, $G, A, $H |- C" (fn prems => [res_inst_tac [("A","B")] cut 1, rtac ll_mp 2, rtac (hd(prems)) 2, rtac context3 1, rtac context3 1, rtac context1 1]); val or_to_and = prove_goal thy "!((!(A ++ B)) -o 0) |- !( ((!A) -o 0) && ((!B) -o 0))" (fn _ => [best_tac lazy_cs 1]); val o_a_rule = prove_goal thy "$F, !( ((!A) -o 0) && ((!B) -o 0)), $G |- C ==> \ \ $F, !((!(A ++ B)) -o 0), $G |- C" (fn prems => [rtac cut 1, rtac or_to_and 2, rtac (hd(prems)) 2, rtac context3 1, rtac context1 1]); val conj_imp = prove_goal thy "((!A) -o C) ++ ((!B) -o C) |- (!(A && B)) -o C" (fn _ => [rtac impr 1,rtac conj_lemma 1, rtac disjl 1, ALLGOALS (rtac mp_rule1 THEN' best_tac lazy_cs)]); val not_imp = auto "!A, !((!B) -o 0) |- (!((!A) -o B)) -o 0"; val a_not_a = prove_goal thy "!A -o (!A -o 0) |- !A -o 0" (fn _ => [rtac impr 1, rtac contract 1, rtac impl 1, rtac context1 1 THEN rtac identity 2, best_tac lazy_cs 1]); val a_not_a_rule = prove_goal thy "$J1, !A -o 0, $J2 |- B ==> $J1, !A -o (!A -o 0), $J2 |- B" (fn prems => [res_inst_tac [("A","!A -o 0")] cut 1, rtac a_not_a 2 THEN rtac (hd(prems)) 2 THEN best_tac lazy_cs 1]); fun thm_to_rule x y = prove_goal thy x (fn prems => [rtac cut 1, rtac y 2, rtac (hd(prems)) 2, best_tac lazy_cs 1]); val safe_cs = lazy_cs add_safes [conj_lemma, ll_mp,contrad1, contrad2, mp_rule1, mp_rule2, o_a_rule, a_not_a_rule] add_unsafes [aux_impl]; val power_cs = safe_cs add_unsafes [impr_contr_der]; fun auto1 x = prove_goal thy x (fn prems => [best_tac safe_cs 1]) ; fun auto2 x = prove_goal thy x (fn prems => [best_tac power_cs 1]) ; (* some examples from Troelstra and van Dalen auto1 "!((!A) -o ((!B) -o 0)) |- (!(A && B)) -o 0"; auto1 "!((!(A && B)) -o 0) |- !((!A) -o ((!B) -o 0))"; auto1 "!( (!((! ((!A) -o B) ) -o 0)) -o 0) |- \ \ (!A) -o ( (! ((!B) -o 0)) -o 0)"; auto2 "!( (!A) -o ( (! ((!B) -o 0)) -o 0) ) |- \ \ (!((! ((!A) -o B) ) -o 0)) -o 0"; *)