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src/Sequents/LK0.ML

author | wenzelm |

Tue, 13 Jun 2006 23:41:39 +0200 | |

changeset 19876 | 11d447d5d68c |

parent 17481 | 75166ebb619b |

permissions | -rw-r--r-- |

tuned;

(* Title: LK/LK0.ML 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 **) Goalw [iff_def] "[| $H,P |- $E,Q,$F; $H,Q |- $E,P,$F |] ==> $H |- $E, P <-> Q, $F"; by (REPEAT (ares_tac [conjR,impR] 1)); qed "iffR"; Goalw [iff_def] "[| $H,$G |- $E,P,Q; $H,Q,P,$G |- $E |] ==> $H, P <-> Q, $G |- $E"; by (REPEAT (ares_tac [conjL,impL,basic] 1)); qed "iffL"; Goal "$H |- $E, (P <-> P), $F"; by (REPEAT (resolve_tac [iffR,basic] 1)); qed "iff_refl"; Goalw [True_def] "$H |- $E, True, $F"; by (rtac impR 1); by (rtac basic 1); qed "TrueR"; (*Descriptions*) val [p1,p2] = Goal "[| $H |- $E, P(a), $F; !!x. $H, P(x) |- $E, x=a, $F |] \ \ ==> $H |- $E, (THE x. P(x)) = a, $F"; by (rtac cut 1); by (rtac p2 2); by (rtac The 1 THEN rtac thinR 1 THEN rtac exchRS 1 THEN rtac p1 1); by (rtac thinR 1 THEN rtac exchRS 1 THEN rtac p2 1); qed "the_equality"; (** Weakened quantifier rules. Incomplete, they let the search terminate.**) Goal "$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, the_equality]; val LK_dup_pack = prop_pack add_safes [allR, exL] add_unsafes [allL, exR, the_equality]; pack_ref() := LK_pack; fun lemma_tac th i = rtac (thinR RS cut) i THEN REPEAT (rtac thinL i) THEN rtac th i; val [major,minor] = goal (the_context ()) "[| $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 (the_context ()) "[| $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 (the_context ()) "[| |- 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 (the_context ()) "[| |- 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 (the_context ()) "[| $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"; Goal "|-P&Q ==> |-P"; by (etac (thinR RS cut) 1); by (Fast_tac 1); qed "conjunct1"; Goal "|-P&Q ==> |-Q"; by (etac (thinR RS cut) 1); by (Fast_tac 1); qed "conjunct2"; Goal "|- (ALL x. P(x)) ==> |- P(x)"; by (etac (thinR RS cut) 1); by (Fast_tac 1); qed "spec"; (** 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"; (* Two theorms for rewriting only one instance of a definition: the first for definitions of formulae and the second for terms *) val prems = goal (the_context ()) "(A == B) ==> |- A <-> B"; by (rewrite_goals_tac prems); by (rtac iff_refl 1); qed "def_imp_iff"; val prems = goal (the_context ()) "(A == B) ==> |- A = B"; by (rewrite_goals_tac prems); by (rtac refl 1); qed "meta_eq_to_obj_eq";