--- a/src/Sequents/LK0.thy Mon Nov 20 21:23:12 2006 +0100
+++ b/src/Sequents/LK0.thy Mon Nov 20 23:47:10 2006 +0100
@@ -132,8 +132,262 @@
setup
prover_setup
-ML {* use_legacy_bindings (the_context ()) *}
+
+(** Structural Rules on formulas **)
+
+(*contraction*)
+
+lemma contR: "$H |- $E, P, P, $F ==> $H |- $E, P, $F"
+ by (rule contRS)
+
+lemma contL: "$H, P, P, $G |- $E ==> $H, P, $G |- $E"
+ by (rule contLS)
+
+(*thinning*)
+
+lemma thinR: "$H |- $E, $F ==> $H |- $E, P, $F"
+ by (rule thinRS)
+
+lemma thinL: "$H, $G |- $E ==> $H, P, $G |- $E"
+ by (rule thinLS)
+
+(*exchange*)
+
+lemma exchR: "$H |- $E, Q, P, $F ==> $H |- $E, P, Q, $F"
+ by (rule exchRS)
+
+lemma exchL: "$H, Q, P, $G |- $E ==> $H, P, Q, $G |- $E"
+ by (rule exchLS)
+
+ML {*
+local
+ val thinR = thm "thinR"
+ val thinL = thm "thinL"
+ val cut = thm "cut"
+in
+
+(*Cut and thin, replacing the right-side formula*)
+fun cutR_tac s i =
+ res_inst_tac [ ("P", s) ] cut i THEN rtac thinR i
+
+(*Cut and thin, replacing the left-side formula*)
+fun cutL_tac s i =
+ res_inst_tac [("P", s)] cut i THEN rtac thinL (i+1)
end
+*}
+
+
+(** If-and-only-if rules **)
+lemma iffR:
+ "[| $H,P |- $E,Q,$F; $H,Q |- $E,P,$F |] ==> $H |- $E, P <-> Q, $F"
+ apply (unfold iff_def)
+ apply (assumption | rule conjR impR)+
+ done
+
+lemma iffL:
+ "[| $H,$G |- $E,P,Q; $H,Q,P,$G |- $E |] ==> $H, P <-> Q, $G |- $E"
+ apply (unfold iff_def)
+ apply (assumption | rule conjL impL basic)+
+ done
+
+lemma iff_refl: "$H |- $E, (P <-> P), $F"
+ apply (rule iffR basic)+
+ done
+
+lemma TrueR: "$H |- $E, True, $F"
+ apply (unfold True_def)
+ apply (rule impR)
+ apply (rule basic)
+ done
+
+(*Descriptions*)
+lemma the_equality:
+ assumes p1: "$H |- $E, P(a), $F"
+ and p2: "!!x. $H, P(x) |- $E, x=a, $F"
+ shows "$H |- $E, (THE x. P(x)) = a, $F"
+ apply (rule cut)
+ apply (rule_tac [2] p2)
+ apply (rule The, rule thinR, rule exchRS, rule p1)
+ apply (rule thinR, rule exchRS, rule p2)
+ done
+
+
+(** Weakened quantifier rules. Incomplete, they let the search terminate.**)
+
+lemma allL_thin: "$H, P(x), $G |- $E ==> $H, ALL x. P(x), $G |- $E"
+ apply (rule allL)
+ apply (erule thinL)
+ done
+
+lemma exR_thin: "$H |- $E, P(x), $F ==> $H |- $E, EX x. P(x), $F"
+ apply (rule exR)
+ apply (erule thinR)
+ done
+
+(*The rules of LK*)
+
+ML {*
+val prop_pack = empty_pack add_safes
+ [thm "basic", thm "refl", thm "TrueR", thm "FalseL",
+ thm "conjL", thm "conjR", thm "disjL", thm "disjR", thm "impL", thm "impR",
+ thm "notL", thm "notR", thm "iffL", thm "iffR"];
+
+val LK_pack = prop_pack add_safes [thm "allR", thm "exL"]
+ add_unsafes [thm "allL_thin", thm "exR_thin", thm "the_equality"];
+
+val LK_dup_pack = prop_pack add_safes [thm "allR", thm "exL"]
+ add_unsafes [thm "allL", thm "exR", thm "the_equality"];
+
+
+pack_ref() := LK_pack;
+
+local
+ val thinR = thm "thinR"
+ val thinL = thm "thinL"
+ val cut = thm "cut"
+in
+
+fun lemma_tac th i =
+ rtac (thinR RS cut) i THEN REPEAT (rtac thinL i) THEN rtac th i;
+
+end;
+*}
+
+method_setup fast_prop =
+ {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (fast_tac prop_pack)) *}
+ "propositional reasoning"
+
+method_setup fast =
+ {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (fast_tac LK_pack)) *}
+ "classical reasoning"
+
+method_setup fast_dup =
+ {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (fast_tac LK_dup_pack)) *}
+ "classical reasoning"
+
+method_setup best =
+ {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (best_tac LK_pack)) *}
+ "classical reasoning"
+
+method_setup best_dup =
+ {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (best_tac LK_dup_pack)) *}
+ "classical reasoning"
+lemma mp_R:
+ assumes major: "$H |- $E, $F, P --> Q"
+ and minor: "$H |- $E, $F, P"
+ shows "$H |- $E, Q, $F"
+ apply (rule thinRS [THEN cut], rule major)
+ apply (tactic "step_tac LK_pack 1")
+ apply (rule thinR, rule minor)
+ done
+
+lemma mp_L:
+ assumes major: "$H, $G |- $E, P --> Q"
+ and minor: "$H, $G, Q |- $E"
+ shows "$H, P, $G |- $E"
+ apply (rule thinL [THEN cut], rule major)
+ apply (tactic "step_tac LK_pack 1")
+ apply (rule thinL, rule minor)
+ done
+
+
+(** Two rules to generate left- and right- rules from implications **)
+
+lemma R_of_imp:
+ assumes major: "|- P --> Q"
+ and minor: "$H |- $E, $F, P"
+ shows "$H |- $E, Q, $F"
+ apply (rule mp_R)
+ apply (rule_tac [2] minor)
+ apply (rule thinRS, rule major [THEN thinLS])
+ done
+
+lemma L_of_imp:
+ assumes major: "|- P --> Q"
+ and minor: "$H, $G, Q |- $E"
+ shows "$H, P, $G |- $E"
+ apply (rule mp_L)
+ apply (rule_tac [2] minor)
+ apply (rule thinRS, rule major [THEN thinLS])
+ done
+
+(*Can be used to create implications in a subgoal*)
+lemma backwards_impR:
+ assumes prem: "$H, $G |- $E, $F, P --> Q"
+ shows "$H, P, $G |- $E, Q, $F"
+ apply (rule mp_L)
+ apply (rule_tac [2] basic)
+ apply (rule thinR, rule prem)
+ done
+
+lemma conjunct1: "|-P&Q ==> |-P"
+ apply (erule thinR [THEN cut])
+ apply fast
+ done
+
+lemma conjunct2: "|-P&Q ==> |-Q"
+ apply (erule thinR [THEN cut])
+ apply fast
+ done
+
+lemma spec: "|- (ALL x. P(x)) ==> |- P(x)"
+ apply (erule thinR [THEN cut])
+ apply fast
+ done
+
+
+(** Equality **)
+
+lemma sym: "|- a=b --> b=a"
+ by (tactic {* safe_tac (LK_pack add_safes [thm "subst"]) 1 *})
+
+lemma trans: "|- a=b --> b=c --> a=c"
+ by (tactic {* safe_tac (LK_pack add_safes [thm "subst"]) 1 *})
+
+(* Symmetry of equality in hypotheses *)
+lemmas symL = sym [THEN L_of_imp, standard]
+
+(* Symmetry of equality in hypotheses *)
+lemmas symR = sym [THEN R_of_imp, standard]
+
+lemma transR: "[| $H|- $E, $F, a=b; $H|- $E, $F, b=c |] ==> $H|- $E, a=c, $F"
+ by (rule trans [THEN R_of_imp, THEN mp_R])
+
+(* Two theorms for rewriting only one instance of a definition:
+ the first for definitions of formulae and the second for terms *)
+
+lemma def_imp_iff: "(A == B) ==> |- A <-> B"
+ apply unfold
+ apply (rule iff_refl)
+ done
+
+lemma meta_eq_to_obj_eq: "(A == B) ==> |- A = B"
+ apply unfold
+ apply (rule refl)
+ done
+
+
+(** if-then-else rules **)
+
+lemma if_True: "|- (if True then x else y) = x"
+ unfolding If_def by fast
+
+lemma if_False: "|- (if False then x else y) = y"
+ unfolding If_def by fast
+
+lemma if_P: "|- P ==> |- (if P then x else y) = x"
+ apply (unfold If_def)
+ apply (erule thinR [THEN cut])
+ apply fast
+ done
+
+lemma if_not_P: "|- ~P ==> |- (if P then x else y) = y";
+ apply (unfold If_def)
+ apply (erule thinR [THEN cut])
+ apply fast
+ done
+
+end