--- a/src/Sequents/LK.thy Wed Jun 11 15:41:08 2008 +0200
+++ b/src/Sequents/LK.thy Wed Jun 11 15:41:33 2008 +0200
@@ -18,13 +18,231 @@
uses ("simpdata.ML")
begin
-axioms
-
- monotonic: "($H |- P ==> $H |- Q) ==> $H, P |- Q"
+axiomatization where
+ monotonic: "($H |- P ==> $H |- Q) ==> $H, P |- Q" and
left_cong: "[| P == P'; |- P' ==> ($H |- $F) == ($H' |- $F') |]
==> (P, $H |- $F) == (P', $H' |- $F')"
+
+subsection {* Rewrite rules *}
+
+lemma conj_simps:
+ "|- P & True <-> P"
+ "|- True & P <-> P"
+ "|- P & False <-> False"
+ "|- False & P <-> False"
+ "|- P & P <-> P"
+ "|- P & P & Q <-> P & Q"
+ "|- P & ~P <-> False"
+ "|- ~P & P <-> False"
+ "|- (P & Q) & R <-> P & (Q & R)"
+ apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *})
+ done
+
+lemma disj_simps:
+ "|- P | True <-> True"
+ "|- True | P <-> True"
+ "|- P | False <-> P"
+ "|- False | P <-> P"
+ "|- P | P <-> P"
+ "|- P | P | Q <-> P | Q"
+ "|- (P | Q) | R <-> P | (Q | R)"
+ apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *})
+ done
+
+lemma not_simps:
+ "|- ~ False <-> True"
+ "|- ~ True <-> False"
+ apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *})
+ done
+
+lemma imp_simps:
+ "|- (P --> False) <-> ~P"
+ "|- (P --> True) <-> True"
+ "|- (False --> P) <-> True"
+ "|- (True --> P) <-> P"
+ "|- (P --> P) <-> True"
+ "|- (P --> ~P) <-> ~P"
+ apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *})
+ done
+
+lemma iff_simps:
+ "|- (True <-> P) <-> P"
+ "|- (P <-> True) <-> P"
+ "|- (P <-> P) <-> True"
+ "|- (False <-> P) <-> ~P"
+ "|- (P <-> False) <-> ~P"
+ apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *})
+ done
+
+lemma quant_simps:
+ "!!P. |- (ALL x. P) <-> P"
+ "!!P. |- (ALL x. x=t --> P(x)) <-> P(t)"
+ "!!P. |- (ALL x. t=x --> P(x)) <-> P(t)"
+ "!!P. |- (EX x. P) <-> P"
+ "!!P. |- (EX x. x=t & P(x)) <-> P(t)"
+ "!!P. |- (EX x. t=x & P(x)) <-> P(t)"
+ apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *})
+ done
+
+
+subsection {* Miniscoping: pushing quantifiers in *}
+
+text {*
+ We do NOT distribute of ALL over &, or dually that of EX over |
+ Baaz and Leitsch, On Skolemization and Proof Complexity (1994)
+ show that this step can increase proof length!
+*}
+
+text {*existential miniscoping*}
+lemma ex_simps:
+ "!!P Q. |- (EX x. P(x) & Q) <-> (EX x. P(x)) & Q"
+ "!!P Q. |- (EX x. P & Q(x)) <-> P & (EX x. Q(x))"
+ "!!P Q. |- (EX x. P(x) | Q) <-> (EX x. P(x)) | Q"
+ "!!P Q. |- (EX x. P | Q(x)) <-> P | (EX x. Q(x))"
+ "!!P Q. |- (EX x. P(x) --> Q) <-> (ALL x. P(x)) --> Q"
+ "!!P Q. |- (EX x. P --> Q(x)) <-> P --> (EX x. Q(x))"
+ apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *})
+ done
+
+text {*universal miniscoping*}
+lemma all_simps:
+ "!!P Q. |- (ALL x. P(x) & Q) <-> (ALL x. P(x)) & Q"
+ "!!P Q. |- (ALL x. P & Q(x)) <-> P & (ALL x. Q(x))"
+ "!!P Q. |- (ALL x. P(x) --> Q) <-> (EX x. P(x)) --> Q"
+ "!!P Q. |- (ALL x. P --> Q(x)) <-> P --> (ALL x. Q(x))"
+ "!!P Q. |- (ALL x. P(x) | Q) <-> (ALL x. P(x)) | Q"
+ "!!P Q. |- (ALL x. P | Q(x)) <-> P | (ALL x. Q(x))"
+ apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *})
+ done
+
+text {*These are NOT supplied by default!*}
+lemma distrib_simps:
+ "|- P & (Q | R) <-> P&Q | P&R"
+ "|- (Q | R) & P <-> Q&P | R&P"
+ "|- (P | Q --> R) <-> (P --> R) & (Q --> R)"
+ apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *})
+ done
+
+lemma P_iff_F: "|- ~P ==> |- (P <-> False)"
+ apply (erule thinR [THEN cut])
+ apply (tactic {* fast_tac LK_pack 1 *})
+ done
+
+lemmas iff_reflection_F = P_iff_F [THEN iff_reflection, standard]
+
+lemma P_iff_T: "|- P ==> |- (P <-> True)"
+ apply (erule thinR [THEN cut])
+ apply (tactic {* fast_tac LK_pack 1 *})
+ done
+
+lemmas iff_reflection_T = P_iff_T [THEN iff_reflection, standard]
+
+
+lemma LK_extra_simps:
+ "|- P | ~P"
+ "|- ~P | P"
+ "|- ~ ~ P <-> P"
+ "|- (~P --> P) <-> P"
+ "|- (~P <-> ~Q) <-> (P<->Q)"
+ apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *})
+ done
+
+
+subsection {* Named rewrite rules *}
+
+lemma conj_commute: "|- P&Q <-> Q&P"
+ and conj_left_commute: "|- P&(Q&R) <-> Q&(P&R)"
+ apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *})
+ done
+
+lemmas conj_comms = conj_commute conj_left_commute
+
+lemma disj_commute: "|- P|Q <-> Q|P"
+ and disj_left_commute: "|- P|(Q|R) <-> Q|(P|R)"
+ apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *})
+ done
+
+lemmas disj_comms = disj_commute disj_left_commute
+
+lemma conj_disj_distribL: "|- P&(Q|R) <-> (P&Q | P&R)"
+ and conj_disj_distribR: "|- (P|Q)&R <-> (P&R | Q&R)"
+
+ and disj_conj_distribL: "|- P|(Q&R) <-> (P|Q) & (P|R)"
+ and disj_conj_distribR: "|- (P&Q)|R <-> (P|R) & (Q|R)"
+
+ and imp_conj_distrib: "|- (P --> (Q&R)) <-> (P-->Q) & (P-->R)"
+ and imp_conj: "|- ((P&Q)-->R) <-> (P --> (Q --> R))"
+ and imp_disj: "|- (P|Q --> R) <-> (P-->R) & (Q-->R)"
+
+ and imp_disj1: "|- (P-->Q) | R <-> (P-->Q | R)"
+ and imp_disj2: "|- Q | (P-->R) <-> (P-->Q | R)"
+
+ and de_Morgan_disj: "|- (~(P | Q)) <-> (~P & ~Q)"
+ and de_Morgan_conj: "|- (~(P & Q)) <-> (~P | ~Q)"
+
+ and not_iff: "|- ~(P <-> Q) <-> (P <-> ~Q)"
+ apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *})
+ done
+
+lemma imp_cong:
+ assumes p1: "|- P <-> P'"
+ and p2: "|- P' ==> |- Q <-> Q'"
+ shows "|- (P-->Q) <-> (P'-->Q')"
+ apply (tactic {* lemma_tac @{thm p1} 1 *})
+ apply (tactic {* safe_tac LK_pack 1 *})
+ apply (tactic {*
+ REPEAT (rtac @{thm cut} 1 THEN
+ DEPTH_SOLVE_1
+ (resolve_tac [@{thm thinL}, @{thm thinR}, @{thm p2} COMP @{thm monotonic}] 1) THEN
+ safe_tac LK_pack 1) *})
+ done
+
+lemma conj_cong:
+ assumes p1: "|- P <-> P'"
+ and p2: "|- P' ==> |- Q <-> Q'"
+ shows "|- (P&Q) <-> (P'&Q')"
+ apply (tactic {* lemma_tac @{thm p1} 1 *})
+ apply (tactic {* safe_tac LK_pack 1 *})
+ apply (tactic {*
+ REPEAT (rtac @{thm cut} 1 THEN
+ DEPTH_SOLVE_1
+ (resolve_tac [@{thm thinL}, @{thm thinR}, @{thm p2} COMP @{thm monotonic}] 1) THEN
+ safe_tac LK_pack 1) *})
+ done
+
+lemma eq_sym_conv: "|- (x=y) <-> (y=x)"
+ apply (tactic {* fast_tac (LK_pack add_safes @{thms subst}) 1 *})
+ done
+
use "simpdata.ML"
+
+text {* To create substition rules *}
+
+lemma eq_imp_subst: "|- a=b ==> $H, A(a), $G |- $E, A(b), $F"
+ apply (tactic {* asm_simp_tac LK_basic_ss 1 *})
+ done
+
+lemma split_if: "|- P(if Q then x else y) <-> ((Q --> P(x)) & (~Q --> P(y)))"
+ apply (rule_tac P = Q in cut)
+ apply (tactic {* simp_tac (simpset () addsimps @{thms if_P}) 2 *})
+ apply (rule_tac P = "~Q" in cut)
+ apply (tactic {* simp_tac (simpset() addsimps @{thms if_not_P}) 2 *})
+ apply (tactic {* fast_tac LK_pack 1 *})
+ done
+
+lemma if_cancel: "|- (if P then x else x) = x"
+ apply (tactic {* lemma_tac @{thm split_if} 1 *})
+ apply (tactic {* fast_tac LK_pack 1 *})
+ done
+
+lemma if_eq_cancel: "|- (if x=y then y else x) = x"
+ apply (tactic {* lemma_tac @{thm split_if} 1 *})
+ apply (tactic {* safe_tac LK_pack 1 *})
+ apply (rule symL)
+ apply (rule basic)
+ done
+
end