--- /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";
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Sequents/LK0.thy Tue Jul 27 18:52:23 1999 +0200
@@ -0,0 +1,142 @@
+(* Title: LK/LK0
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1993 University of Cambridge
+
+Classical First-Order Sequent Calculus
+
+There may be printing problems if a seqent is in expanded normal form
+ (eta-expanded, beta-contracted)
+*)
+
+LK0 = Sequents +
+
+global
+
+classes
+ term < logic
+
+default
+ term
+
+consts
+
+ Trueprop :: "two_seqi"
+ "@Trueprop" :: "two_seqe" ("((_)/ |- (_))" [6,6] 5)
+
+ (*Constant to allow definitions of SEQUENCES of formulas*)
+ "@Side" :: "seq=>(seq'=>seq')" ("<<(_)>>")
+
+ True,False :: o
+ "=" :: ['a,'a] => o (infixl 50)
+ Not :: o => o ("~ _" [40] 40)
+ "&" :: [o,o] => o (infixr 35)
+ "|" :: [o,o] => o (infixr 30)
+ "-->","<->" :: [o,o] => o (infixr 25)
+ The :: ('a => o) => 'a (binder "THE " 10)
+ All :: ('a => o) => o (binder "ALL " 10)
+ Ex :: ('a => o) => o (binder "EX " 10)
+
+syntax
+ "~=" :: ['a, 'a] => o (infixl 50)
+
+translations
+ "x ~= y" == "~ (x = y)"
+
+syntax (symbols)
+ Not :: o => o ("\\<not> _" [40] 40)
+ "op &" :: [o, o] => o (infixr "\\<and>" 35)
+ "op |" :: [o, o] => o (infixr "\\<or>" 30)
+ "op -->" :: [o, o] => o (infixr "\\<midarrow>\\<rightarrow>" 25)
+ "op <->" :: [o, o] => o (infixr "\\<leftarrow>\\<rightarrow>" 25)
+ "ALL " :: [idts, o] => o ("(3\\<forall>_./ _)" [0, 10] 10)
+ "EX " :: [idts, o] => o ("(3\\<exists>_./ _)" [0, 10] 10)
+ "EX! " :: [idts, o] => o ("(3\\<exists>!_./ _)" [0, 10] 10)
+ "op ~=" :: ['a, 'a] => o (infixl "\\<noteq>" 50)
+
+syntax (xsymbols)
+ "op -->" :: [o, o] => o (infixr "\\<longrightarrow>" 25)
+ "op <->" :: [o, o] => o (infixr "\\<longleftrightarrow>" 25)
+
+syntax (HTML output)
+ Not :: o => o ("\\<not> _" [40] 40)
+
+
+local
+
+rules
+
+ (*Structural rules: contraction, thinning, exchange [Soren Heilmann] *)
+
+ contRS "$H |- $E, $S, $S, $F ==> $H |- $E, $S, $F"
+ contLS "$H, $S, $S, $G |- $E ==> $H, $S, $G |- $E"
+
+ thinRS "$H |- $E, $F ==> $H |- $E, $S, $F"
+ thinLS "$H, $G |- $E ==> $H, $S, $G |- $E"
+
+ exchRS "$H |- $E, $R, $S, $F ==> $H |- $E, $S, $R, $F"
+ exchLS "$H, $R, $S, $G |- $E ==> $H, $S, $R, $G |- $E"
+
+ cut "[| $H |- $E, P; $H, P |- $E |] ==> $H |- $E"
+
+ (*Propositional rules*)
+
+ basic "$H, P, $G |- $E, P, $F"
+
+ conjR "[| $H|- $E, P, $F; $H|- $E, Q, $F |] ==> $H|- $E, P&Q, $F"
+ conjL "$H, P, Q, $G |- $E ==> $H, P & Q, $G |- $E"
+
+ disjR "$H |- $E, P, Q, $F ==> $H |- $E, P|Q, $F"
+ disjL "[| $H, P, $G |- $E; $H, Q, $G |- $E |] ==> $H, P|Q, $G |- $E"
+
+ impR "$H, P |- $E, Q, $F ==> $H |- $E, P-->Q, $F"
+ impL "[| $H,$G |- $E,P; $H, Q, $G |- $E |] ==> $H, P-->Q, $G |- $E"
+
+ notR "$H, P |- $E, $F ==> $H |- $E, ~P, $F"
+ notL "$H, $G |- $E, P ==> $H, ~P, $G |- $E"
+
+ FalseL "$H, False, $G |- $E"
+
+ True_def "True == False-->False"
+ iff_def "P<->Q == (P-->Q) & (Q-->P)"
+
+ (*Quantifiers*)
+
+ allR "(!!x.$H |- $E, P(x), $F) ==> $H |- $E, ALL x. P(x), $F"
+ allL "$H, P(x), $G, ALL x. P(x) |- $E ==> $H, ALL x. P(x), $G |- $E"
+
+ exR "$H |- $E, P(x), $F, EX x. P(x) ==> $H |- $E, EX x. P(x), $F"
+ exL "(!!x.$H, P(x), $G |- $E) ==> $H, EX x. P(x), $G |- $E"
+
+ (*Equality*)
+
+ refl "$H |- $E, a=a, $F"
+ subst "$H(a), $G(a) |- $E(a) ==> $H(b), a=b, $G(b) |- $E(b)"
+
+ (* Reflection *)
+
+ eq_reflection "|- x=y ==> (x==y)"
+ iff_reflection "|- P<->Q ==> (P==Q)"
+
+ (*Descriptions*)
+
+ The "[| $H |- $E, P(a), $F; !!x.$H, P(x) |- $E, x=a, $F |] ==>
+ $H |- $E, P(THE x. P(x)), $F"
+
+constdefs
+ If :: [o, 'a, 'a] => 'a ("(if (_)/ then (_)/ else (_))" 10)
+ "If(P,x,y) == THE z::'a. (P --> z=x) & (~P --> z=y)"
+
+
+setup
+ Simplifier.setup
+
+setup
+ prover_setup
+
+end
+
+ ML
+
+val parse_translation = [("@Trueprop",Sequents.two_seq_tr "Trueprop")];
+val print_translation = [("Trueprop",Sequents.two_seq_tr' "@Trueprop")];