summary |
shortlog |
changelog |
graph |
tags |
bookmarks |
branches |
files |
changeset |
file |
latest |
revisions |
annotate |
diff |
comparison |
raw |
help

src/Sequents/LK0.thy

changeset 21426 | 87ac12bed1ab |

parent 17481 | 75166ebb619b |

child 21428 | f84cf8e9cad8 |

--- 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