(* Title: Sequents/LK.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
Axiom to express monotonicity (a variant of the deduction theorem). Makes the
link between \<turnstile> and \<Longrightarrow>, needed for instance to prove imp_cong.
Axiom left_cong allows the simplifier to use left-side formulas. Ideally it
should be derived from lower-level axioms.
CANNOT be added to LK0.thy because modal logic is built upon it, and
various modal rules would become inconsistent.
*)
theory LK
imports LK0
begin
axiomatization where
monotonic: "($H \<turnstile> P \<Longrightarrow> $H \<turnstile> Q) \<Longrightarrow> $H, P \<turnstile> Q" and
left_cong: "\<lbrakk>P == P'; \<turnstile> P' \<Longrightarrow> ($H \<turnstile> $F) \<equiv> ($H' \<turnstile> $F')\<rbrakk>
\<Longrightarrow> (P, $H \<turnstile> $F) \<equiv> (P', $H' \<turnstile> $F')"
subsection \<open>Rewrite rules\<close>
lemma conj_simps:
"\<turnstile> P \<and> True \<longleftrightarrow> P"
"\<turnstile> True \<and> P \<longleftrightarrow> P"
"\<turnstile> P \<and> False \<longleftrightarrow> False"
"\<turnstile> False \<and> P \<longleftrightarrow> False"
"\<turnstile> P \<and> P \<longleftrightarrow> P"
"\<turnstile> P \<and> P \<and> Q \<longleftrightarrow> P \<and> Q"
"\<turnstile> P \<and> \<not> P \<longleftrightarrow> False"
"\<turnstile> \<not> P \<and> P \<longleftrightarrow> False"
"\<turnstile> (P \<and> Q) \<and> R \<longleftrightarrow> P \<and> (Q \<and> R)"
by (fast add!: subst)+
lemma disj_simps:
"\<turnstile> P \<or> True \<longleftrightarrow> True"
"\<turnstile> True \<or> P \<longleftrightarrow> True"
"\<turnstile> P \<or> False \<longleftrightarrow> P"
"\<turnstile> False \<or> P \<longleftrightarrow> P"
"\<turnstile> P \<or> P \<longleftrightarrow> P"
"\<turnstile> P \<or> P \<or> Q \<longleftrightarrow> P \<or> Q"
"\<turnstile> (P \<or> Q) \<or> R \<longleftrightarrow> P \<or> (Q \<or> R)"
by (fast add!: subst)+
lemma not_simps:
"\<turnstile> \<not> False \<longleftrightarrow> True"
"\<turnstile> \<not> True \<longleftrightarrow> False"
by (fast add!: subst)+
lemma imp_simps:
"\<turnstile> (P \<longrightarrow> False) \<longleftrightarrow> \<not> P"
"\<turnstile> (P \<longrightarrow> True) \<longleftrightarrow> True"
"\<turnstile> (False \<longrightarrow> P) \<longleftrightarrow> True"
"\<turnstile> (True \<longrightarrow> P) \<longleftrightarrow> P"
"\<turnstile> (P \<longrightarrow> P) \<longleftrightarrow> True"
"\<turnstile> (P \<longrightarrow> \<not> P) \<longleftrightarrow> \<not> P"
by (fast add!: subst)+
lemma iff_simps:
"\<turnstile> (True \<longleftrightarrow> P) \<longleftrightarrow> P"
"\<turnstile> (P \<longleftrightarrow> True) \<longleftrightarrow> P"
"\<turnstile> (P \<longleftrightarrow> P) \<longleftrightarrow> True"
"\<turnstile> (False \<longleftrightarrow> P) \<longleftrightarrow> \<not> P"
"\<turnstile> (P \<longleftrightarrow> False) \<longleftrightarrow> \<not> P"
by (fast add!: subst)+
lemma quant_simps:
"\<And>P. \<turnstile> (\<forall>x. P) \<longleftrightarrow> P"
"\<And>P. \<turnstile> (\<forall>x. x = t \<longrightarrow> P(x)) \<longleftrightarrow> P(t)"
"\<And>P. \<turnstile> (\<forall>x. t = x \<longrightarrow> P(x)) \<longleftrightarrow> P(t)"
"\<And>P. \<turnstile> (\<exists>x. P) \<longleftrightarrow> P"
"\<And>P. \<turnstile> (\<exists>x. x = t \<and> P(x)) \<longleftrightarrow> P(t)"
"\<And>P. \<turnstile> (\<exists>x. t = x \<and> P(x)) \<longleftrightarrow> P(t)"
by (fast add!: subst)+
subsection \<open>Miniscoping: pushing quantifiers in\<close>
text \<open>
We do NOT distribute of \<forall> over \<and>, or dually that of \<exists> over \<or>
Baaz and Leitsch, On Skolemization and Proof Complexity (1994)
show that this step can increase proof length!
\<close>
text \<open>existential miniscoping\<close>
lemma ex_simps:
"\<And>P Q. \<turnstile> (\<exists>x. P(x) \<and> Q) \<longleftrightarrow> (\<exists>x. P(x)) \<and> Q"
"\<And>P Q. \<turnstile> (\<exists>x. P \<and> Q(x)) \<longleftrightarrow> P \<and> (\<exists>x. Q(x))"
"\<And>P Q. \<turnstile> (\<exists>x. P(x) \<or> Q) \<longleftrightarrow> (\<exists>x. P(x)) \<or> Q"
"\<And>P Q. \<turnstile> (\<exists>x. P \<or> Q(x)) \<longleftrightarrow> P \<or> (\<exists>x. Q(x))"
"\<And>P Q. \<turnstile> (\<exists>x. P(x) \<longrightarrow> Q) \<longleftrightarrow> (\<forall>x. P(x)) \<longrightarrow> Q"
"\<And>P Q. \<turnstile> (\<exists>x. P \<longrightarrow> Q(x)) \<longleftrightarrow> P \<longrightarrow> (\<exists>x. Q(x))"
by (fast add!: subst)+
text \<open>universal miniscoping\<close>
lemma all_simps:
"\<And>P Q. \<turnstile> (\<forall>x. P(x) \<and> Q) \<longleftrightarrow> (\<forall>x. P(x)) \<and> Q"
"\<And>P Q. \<turnstile> (\<forall>x. P \<and> Q(x)) \<longleftrightarrow> P \<and> (\<forall>x. Q(x))"
"\<And>P Q. \<turnstile> (\<forall>x. P(x) \<longrightarrow> Q) \<longleftrightarrow> (\<exists>x. P(x)) \<longrightarrow> Q"
"\<And>P Q. \<turnstile> (\<forall>x. P \<longrightarrow> Q(x)) \<longleftrightarrow> P \<longrightarrow> (\<forall>x. Q(x))"
"\<And>P Q. \<turnstile> (\<forall>x. P(x) \<or> Q) \<longleftrightarrow> (\<forall>x. P(x)) \<or> Q"
"\<And>P Q. \<turnstile> (\<forall>x. P \<or> Q(x)) \<longleftrightarrow> P \<or> (\<forall>x. Q(x))"
by (fast add!: subst)+
text \<open>These are NOT supplied by default!\<close>
lemma distrib_simps:
"\<turnstile> P \<and> (Q \<or> R) \<longleftrightarrow> P \<and> Q \<or> P \<and> R"
"\<turnstile> (Q \<or> R) \<and> P \<longleftrightarrow> Q \<and> P \<or> R \<and> P"
"\<turnstile> (P \<or> Q \<longrightarrow> R) \<longleftrightarrow> (P \<longrightarrow> R) \<and> (Q \<longrightarrow> R)"
by (fast add!: subst)+
lemma P_iff_F: "\<turnstile> \<not> P \<Longrightarrow> \<turnstile> (P \<longleftrightarrow> False)"
apply (erule thinR [THEN cut])
apply fast
done
lemmas iff_reflection_F = P_iff_F [THEN iff_reflection]
lemma P_iff_T: "\<turnstile> P \<Longrightarrow> \<turnstile> (P \<longleftrightarrow> True)"
apply (erule thinR [THEN cut])
apply fast
done
lemmas iff_reflection_T = P_iff_T [THEN iff_reflection]
lemma LK_extra_simps:
"\<turnstile> P \<or> \<not> P"
"\<turnstile> \<not> P \<or> P"
"\<turnstile> \<not> \<not> P \<longleftrightarrow> P"
"\<turnstile> (\<not> P \<longrightarrow> P) \<longleftrightarrow> P"
"\<turnstile> (\<not> P \<longleftrightarrow> \<not> Q) \<longleftrightarrow> (P \<longleftrightarrow> Q)"
by (fast add!: subst)+
subsection \<open>Named rewrite rules\<close>
lemma conj_commute: "\<turnstile> P \<and> Q \<longleftrightarrow> Q \<and> P"
and conj_left_commute: "\<turnstile> P \<and> (Q \<and> R) \<longleftrightarrow> Q \<and> (P \<and> R)"
by (fast add!: subst)+
lemmas conj_comms = conj_commute conj_left_commute
lemma disj_commute: "\<turnstile> P \<or> Q \<longleftrightarrow> Q \<or> P"
and disj_left_commute: "\<turnstile> P \<or> (Q \<or> R) \<longleftrightarrow> Q \<or> (P \<or> R)"
by (fast add!: subst)+
lemmas disj_comms = disj_commute disj_left_commute
lemma conj_disj_distribL: "\<turnstile> P \<and> (Q \<or> R) \<longleftrightarrow> (P \<and> Q \<or> P \<and> R)"
and conj_disj_distribR: "\<turnstile> (P \<or> Q) \<and> R \<longleftrightarrow> (P \<and> R \<or> Q \<and> R)"
and disj_conj_distribL: "\<turnstile> P \<or> (Q \<and> R) \<longleftrightarrow> (P \<or> Q) \<and> (P \<or> R)"
and disj_conj_distribR: "\<turnstile> (P \<and> Q) \<or> R \<longleftrightarrow> (P \<or> R) \<and> (Q \<or> R)"
and imp_conj_distrib: "\<turnstile> (P \<longrightarrow> (Q \<and> R)) \<longleftrightarrow> (P \<longrightarrow> Q) \<and> (P \<longrightarrow> R)"
and imp_conj: "\<turnstile> ((P \<and> Q) \<longrightarrow> R) \<longleftrightarrow> (P \<longrightarrow> (Q \<longrightarrow> R))"
and imp_disj: "\<turnstile> (P \<or> Q \<longrightarrow> R) \<longleftrightarrow> (P \<longrightarrow> R) \<and> (Q \<longrightarrow> R)"
and imp_disj1: "\<turnstile> (P \<longrightarrow> Q) \<or> R \<longleftrightarrow> (P \<longrightarrow> Q \<or> R)"
and imp_disj2: "\<turnstile> Q \<or> (P \<longrightarrow> R) \<longleftrightarrow> (P \<longrightarrow> Q \<or> R)"
and de_Morgan_disj: "\<turnstile> (\<not> (P \<or> Q)) \<longleftrightarrow> (\<not> P \<and> \<not> Q)"
and de_Morgan_conj: "\<turnstile> (\<not> (P \<and> Q)) \<longleftrightarrow> (\<not> P \<or> \<not> Q)"
and not_iff: "\<turnstile> \<not> (P \<longleftrightarrow> Q) \<longleftrightarrow> (P \<longleftrightarrow> \<not> Q)"
by (fast add!: subst)+
lemma imp_cong:
assumes p1: "\<turnstile> P \<longleftrightarrow> P'"
and p2: "\<turnstile> P' \<Longrightarrow> \<turnstile> Q \<longleftrightarrow> Q'"
shows "\<turnstile> (P \<longrightarrow> Q) \<longleftrightarrow> (P' \<longrightarrow> Q')"
apply (lem p1)
apply safe
apply (tactic \<open>
REPEAT (resolve_tac \<^context> @{thms cut} 1 THEN
DEPTH_SOLVE_1
(resolve_tac \<^context> [@{thm thinL}, @{thm thinR}, @{thm p2} COMP @{thm monotonic}] 1) THEN
Cla.safe_tac \<^context> 1)\<close>)
done
lemma conj_cong:
assumes p1: "\<turnstile> P \<longleftrightarrow> P'"
and p2: "\<turnstile> P' \<Longrightarrow> \<turnstile> Q \<longleftrightarrow> Q'"
shows "\<turnstile> (P \<and> Q) \<longleftrightarrow> (P' \<and> Q')"
apply (lem p1)
apply safe
apply (tactic \<open>
REPEAT (resolve_tac \<^context> @{thms cut} 1 THEN
DEPTH_SOLVE_1
(resolve_tac \<^context> [@{thm thinL}, @{thm thinR}, @{thm p2} COMP @{thm monotonic}] 1) THEN
Cla.safe_tac \<^context> 1)\<close>)
done
lemma eq_sym_conv: "\<turnstile> x = y \<longleftrightarrow> y = x"
by (fast add!: subst)
ML_file \<open>simpdata.ML\<close>
setup \<open>map_theory_simpset (put_simpset LK_ss)\<close>
setup \<open>Simplifier.method_setup []\<close>
text \<open>To create substitution rules\<close>
lemma eq_imp_subst: "\<turnstile> a = b \<Longrightarrow> $H, A(a), $G \<turnstile> $E, A(b), $F"
by simp
lemma split_if: "\<turnstile> P(if Q then x else y) \<longleftrightarrow> ((Q \<longrightarrow> P(x)) \<and> (\<not> Q \<longrightarrow> P(y)))"
apply (rule_tac P = Q in cut)
prefer 2
apply (simp add: if_P)
apply (rule_tac P = "\<not> Q" in cut)
prefer 2
apply (simp add: if_not_P)
apply fast
done
lemma if_cancel: "\<turnstile> (if P then x else x) = x"
apply (lem split_if)
apply fast
done
lemma if_eq_cancel: "\<turnstile> (if x = y then y else x) = x"
apply (lem split_if)
apply safe
apply (rule symL)
apply (rule basic)
done
end