forked from Isabelle_DOF/Isabelle_DOF
Move some theorems in regexpinterface
This commit is contained in:
Nicolas Méric
parent
20e90f688f
commit
93ef94cddb
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@ -82,11 +82,6 @@ doc_class report =
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\<lbrakk>index\<rbrakk> ~~ \<lbrace>annex\<rbrace>\<^sup>* )"
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section\<open>Experimental\<close>
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(* switch on regexp syntax *)
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@ -96,72 +91,6 @@ notation Times (infixr "~~" 60)
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notation Atom ("\<lfloor>_\<rfloor>" 65)
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lemma regexp_sub : "a \<le> b \<Longrightarrow> L\<^sub>s\<^sub>u\<^sub>b (\<lfloor>a\<rfloor>) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (\<lfloor>b\<rfloor>)"
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using dual_order.trans by auto
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lemma regexp_seq_mono:
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"Lang(a) \<subseteq> Lang (a') \<Longrightarrow> Lang(b) \<subseteq> Lang (b') \<Longrightarrow> Lang(a ~~ b) \<subseteq> Lang(a' ~~ b')" by auto
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lemma regexp_seq_mono':
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"L\<^sub>s\<^sub>u\<^sub>b(a) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (a') \<Longrightarrow> L\<^sub>s\<^sub>u\<^sub>b(b) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (b') \<Longrightarrow> L\<^sub>s\<^sub>u\<^sub>b(a ~~ b) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b(a' ~~ b')" by auto
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lemma regexp_alt_mono :"Lang(a) \<subseteq> Lang (a') \<Longrightarrow> Lang(a || b) \<subseteq> Lang(a' || b)" by auto
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lemma regexp_alt_mono' :"L\<^sub>s\<^sub>u\<^sub>b(a) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (a') \<Longrightarrow> L\<^sub>s\<^sub>u\<^sub>b(a || b) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b(a' || b)" by auto
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lemma regexp_alt_commute : "Lang(a || b) = Lang(b || a)" by auto
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lemma regexp_alt_commute' : "L\<^sub>s\<^sub>u\<^sub>b(a || b) = L\<^sub>s\<^sub>u\<^sub>b(b || a)" by auto
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lemma regexp_unit_right : "Lang (a) = Lang (a ~~ One) " by simp
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lemma regexp_unit_right' : "L\<^sub>s\<^sub>u\<^sub>b (a) = L\<^sub>s\<^sub>u\<^sub>b (a ~~ One) " by simp
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lemma regexp_unit_left : "Lang (a) = Lang (One ~~ a) " by simp
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lemma regexp_unit_left' : "L\<^sub>s\<^sub>u\<^sub>b (a) = L\<^sub>s\<^sub>u\<^sub>b (One ~~ a) " by simp
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lemma opt_star_incl :"Lang (opt a) \<subseteq> Lang (Star a)" by (simp add: opt_def subset_iff)
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lemma opt_star_incl':"L\<^sub>s\<^sub>u\<^sub>b (opt a) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (Star a)" by (simp add: opt_def subset_iff)
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lemma rep1_star_incl:"Lang (rep1 a) \<subseteq> Lang (Star a)"
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unfolding rep1_def by(subst L_Star, subst L_Conc)(force)
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lemma rep1_star_incl':"L\<^sub>s\<^sub>u\<^sub>b (rep1 a) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (Star a)"
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unfolding rep1_def by(subst L\<^sub>s\<^sub>u\<^sub>b_Star, subst L\<^sub>s\<^sub>u\<^sub>b_Conc)(force)
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lemma cancel_rep1 : "Lang (a) \<subseteq> Lang (rep1 a)"
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unfolding rep1_def by auto
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lemma cancel_rep1' : "L\<^sub>s\<^sub>u\<^sub>b (a) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (rep1 a)"
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unfolding rep1_def by auto
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lemma seq_cancel_opt : "Lang (a) \<subseteq> Lang (c) \<Longrightarrow> Lang (a) \<subseteq> Lang (opt b ~~ c)"
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by(subst regexp_unit_left, rule regexp_seq_mono)(simp_all add: opt_def)
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lemma seq_cancel_opt' : "L\<^sub>s\<^sub>u\<^sub>b (a) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (c) \<Longrightarrow> L\<^sub>s\<^sub>u\<^sub>b (a) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (opt b ~~ c)"
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by(subst regexp_unit_left', rule regexp_seq_mono')(simp_all add: opt_def)
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lemma seq_cancel_Star : "Lang (a) \<subseteq> Lang (c) \<Longrightarrow> Lang (a) \<subseteq> Lang (Star b ~~ c)"
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by auto
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lemma seq_cancel_Star' : "L\<^sub>s\<^sub>u\<^sub>b (a) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (c) \<Longrightarrow> L\<^sub>s\<^sub>u\<^sub>b (a) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (Star b ~~ c)"
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by auto
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lemma mono_Star : "Lang (a) \<subseteq> Lang (b) \<Longrightarrow> Lang (Star a) \<subseteq> Lang (Star b)"
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by(auto)(metis in_star_iff_concat order.trans)
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lemma mono_Star' : "L\<^sub>s\<^sub>u\<^sub>b (a) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (b) \<Longrightarrow> L\<^sub>s\<^sub>u\<^sub>b (Star a) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (Star b)"
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by(auto)(metis in_star_iff_concat order.trans)
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lemma mono_rep1_star:"Lang (a) \<subseteq> Lang (b) \<Longrightarrow> Lang (rep1 a) \<subseteq> Lang (Star b)"
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using mono_Star rep1_star_incl by blast
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lemma mono_rep1_star':"L\<^sub>s\<^sub>u\<^sub>b (a) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (b) \<Longrightarrow> L\<^sub>s\<^sub>u\<^sub>b (rep1 a) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (Star b)"
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using mono_Star' rep1_star_incl' by blast
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text\<open>Not a terribly deep theorem, but an interesting property of consistency between
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ontologies - so, a lemma that shouldn't break if the involved ontologies were changed.
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@ -236,6 +236,72 @@ end; (* local *)
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end (* struct *)
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\<close>
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lemma regexp_sub : "a \<le> b \<Longrightarrow> L\<^sub>s\<^sub>u\<^sub>b (\<lfloor>a\<rfloor>) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (\<lfloor>b\<rfloor>)"
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using dual_order.trans by auto
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lemma regexp_seq_mono:
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"Lang(a) \<subseteq> Lang (a') \<Longrightarrow> Lang(b) \<subseteq> Lang (b') \<Longrightarrow> Lang(a ~~ b) \<subseteq> Lang(a' ~~ b')" by auto
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lemma regexp_seq_mono':
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"L\<^sub>s\<^sub>u\<^sub>b(a) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (a') \<Longrightarrow> L\<^sub>s\<^sub>u\<^sub>b(b) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (b') \<Longrightarrow> L\<^sub>s\<^sub>u\<^sub>b(a ~~ b) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b(a' ~~ b')" by auto
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lemma regexp_alt_mono :"Lang(a) \<subseteq> Lang (a') \<Longrightarrow> Lang(a || b) \<subseteq> Lang(a' || b)" by auto
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lemma regexp_alt_mono' :"L\<^sub>s\<^sub>u\<^sub>b(a) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (a') \<Longrightarrow> L\<^sub>s\<^sub>u\<^sub>b(a || b) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b(a' || b)" by auto
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lemma regexp_alt_commute : "Lang(a || b) = Lang(b || a)" by auto
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lemma regexp_alt_commute' : "L\<^sub>s\<^sub>u\<^sub>b(a || b) = L\<^sub>s\<^sub>u\<^sub>b(b || a)" by auto
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lemma regexp_unit_right : "Lang (a) = Lang (a ~~ One) " by simp
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lemma regexp_unit_right' : "L\<^sub>s\<^sub>u\<^sub>b (a) = L\<^sub>s\<^sub>u\<^sub>b (a ~~ One) " by simp
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lemma regexp_unit_left : "Lang (a) = Lang (One ~~ a) " by simp
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lemma regexp_unit_left' : "L\<^sub>s\<^sub>u\<^sub>b (a) = L\<^sub>s\<^sub>u\<^sub>b (One ~~ a) " by simp
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lemma opt_star_incl :"Lang (opt a) \<subseteq> Lang (Star a)" by (simp add: opt_def subset_iff)
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lemma opt_star_incl':"L\<^sub>s\<^sub>u\<^sub>b (opt a) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (Star a)" by (simp add: opt_def subset_iff)
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lemma rep1_star_incl:"Lang (rep1 a) \<subseteq> Lang (Star a)"
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unfolding rep1_def by(subst L_Star, subst L_Conc)(force)
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lemma rep1_star_incl':"L\<^sub>s\<^sub>u\<^sub>b (rep1 a) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (Star a)"
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unfolding rep1_def by(subst L\<^sub>s\<^sub>u\<^sub>b_Star, subst L\<^sub>s\<^sub>u\<^sub>b_Conc)(force)
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lemma cancel_rep1 : "Lang (a) \<subseteq> Lang (rep1 a)"
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unfolding rep1_def by auto
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lemma cancel_rep1' : "L\<^sub>s\<^sub>u\<^sub>b (a) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (rep1 a)"
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unfolding rep1_def by auto
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lemma seq_cancel_opt : "Lang (a) \<subseteq> Lang (c) \<Longrightarrow> Lang (a) \<subseteq> Lang (opt b ~~ c)"
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by(subst regexp_unit_left, rule regexp_seq_mono)(simp_all add: opt_def)
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lemma seq_cancel_opt' : "L\<^sub>s\<^sub>u\<^sub>b (a) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (c) \<Longrightarrow> L\<^sub>s\<^sub>u\<^sub>b (a) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (opt b ~~ c)"
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by(subst regexp_unit_left', rule regexp_seq_mono')(simp_all add: opt_def)
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lemma seq_cancel_Star : "Lang (a) \<subseteq> Lang (c) \<Longrightarrow> Lang (a) \<subseteq> Lang (Star b ~~ c)"
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by auto
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lemma seq_cancel_Star' : "L\<^sub>s\<^sub>u\<^sub>b (a) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (c) \<Longrightarrow> L\<^sub>s\<^sub>u\<^sub>b (a) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (Star b ~~ c)"
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by auto
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lemma mono_Star : "Lang (a) \<subseteq> Lang (b) \<Longrightarrow> Lang (Star a) \<subseteq> Lang (Star b)"
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by(auto)(metis in_star_iff_concat order.trans)
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lemma mono_Star' : "L\<^sub>s\<^sub>u\<^sub>b (a) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (b) \<Longrightarrow> L\<^sub>s\<^sub>u\<^sub>b (Star a) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (Star b)"
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by(auto)(metis in_star_iff_concat order.trans)
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lemma mono_rep1_star:"Lang (a) \<subseteq> Lang (b) \<Longrightarrow> Lang (rep1 a) \<subseteq> Lang (Star b)"
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using mono_Star rep1_star_incl by blast
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lemma mono_rep1_star':"L\<^sub>s\<^sub>u\<^sub>b (a) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (b) \<Longrightarrow> L\<^sub>s\<^sub>u\<^sub>b (rep1 a) \<subseteq> L\<^sub>s\<^sub>u\<^sub>b (Star b)"
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using mono_Star' rep1_star_incl' by blast
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no_notation Star ("\<lbrace>(_)\<rbrace>\<^sup>*" [0]100)
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no_notation Plus (infixr "||" 55)
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no_notation Times (infixr "~~" 60)
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