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theory PositionsExt
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imports "SpecExt" "LexerExt"
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begin
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section {* Positions in Values *}
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fun
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at :: "val \<Rightarrow> nat list \<Rightarrow> val"
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where
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"at v [] = v"
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| "at (Left v) (0#ps)= at v ps"
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| "at (Right v) (Suc 0#ps)= at v ps"
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| "at (Seq v1 v2) (0#ps)= at v1 ps"
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| "at (Seq v1 v2) (Suc 0#ps)= at v2 ps"
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| "at (Stars vs) (n#ps)= at (nth vs n) ps"
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fun Pos :: "val \<Rightarrow> (nat list) set"
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where
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"Pos (Void) = {[]}"
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| "Pos (Char c) = {[]}"
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| "Pos (Left v) = {[]} \<union> {0#ps | ps. ps \<in> Pos v}"
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| "Pos (Right v) = {[]} \<union> {1#ps | ps. ps \<in> Pos v}"
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| "Pos (Seq v1 v2) = {[]} \<union> {0#ps | ps. ps \<in> Pos v1} \<union> {1#ps | ps. ps \<in> Pos v2}"
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| "Pos (Stars []) = {[]}"
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| "Pos (Stars (v#vs)) = {[]} \<union> {0#ps | ps. ps \<in> Pos v} \<union> {Suc n#ps | n ps. n#ps \<in> Pos (Stars vs)}"
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lemma Pos_stars:
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"Pos (Stars vs) = {[]} \<union> (\<Union>n < length vs. {n#ps | ps. ps \<in> Pos (vs ! n)})"
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apply(induct vs)
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apply(auto simp add: insert_ident less_Suc_eq_0_disj)
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done
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lemma Pos_empty:
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shows "[] \<in> Pos v"
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by (induct v rule: Pos.induct)(auto)
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abbreviation
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"intlen vs \<equiv> int (length vs)"
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definition pflat_len :: "val \<Rightarrow> nat list => int"
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where
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"pflat_len v p \<equiv> (if p \<in> Pos v then intlen (flat (at v p)) else -1)"
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lemma pflat_len_simps:
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shows "pflat_len (Seq v1 v2) (0#p) = pflat_len v1 p"
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and "pflat_len (Seq v1 v2) (Suc 0#p) = pflat_len v2 p"
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and "pflat_len (Left v) (0#p) = pflat_len v p"
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and "pflat_len (Left v) (Suc 0#p) = -1"
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and "pflat_len (Right v) (Suc 0#p) = pflat_len v p"
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and "pflat_len (Right v) (0#p) = -1"
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and "pflat_len (Stars (v#vs)) (Suc n#p) = pflat_len (Stars vs) (n#p)"
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and "pflat_len (Stars (v#vs)) (0#p) = pflat_len v p"
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and "pflat_len v [] = intlen (flat v)"
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by (auto simp add: pflat_len_def Pos_empty)
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lemma pflat_len_Stars_simps:
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assumes "n < length vs"
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shows "pflat_len (Stars vs) (n#p) = pflat_len (vs!n) p"
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using assms
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apply(induct vs arbitrary: n p)
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apply(auto simp add: less_Suc_eq_0_disj pflat_len_simps)
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done
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lemma pflat_len_outside:
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assumes "p \<notin> Pos v1"
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shows "pflat_len v1 p = -1 "
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using assms by (simp add: pflat_len_def)
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section {* Orderings *}
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definition prefix_list:: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" ("_ \<sqsubseteq>pre _" [60,59] 60)
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where
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"ps1 \<sqsubseteq>pre ps2 \<equiv> \<exists>ps'. ps1 @ps' = ps2"
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definition sprefix_list:: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" ("_ \<sqsubset>spre _" [60,59] 60)
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where
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"ps1 \<sqsubset>spre ps2 \<equiv> ps1 \<sqsubseteq>pre ps2 \<and> ps1 \<noteq> ps2"
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inductive lex_list :: "nat list \<Rightarrow> nat list \<Rightarrow> bool" ("_ \<sqsubset>lex _" [60,59] 60)
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where
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"[] \<sqsubset>lex (p#ps)"
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| "ps1 \<sqsubset>lex ps2 \<Longrightarrow> (p#ps1) \<sqsubset>lex (p#ps2)"
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| "p1 < p2 \<Longrightarrow> (p1#ps1) \<sqsubset>lex (p2#ps2)"
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lemma lex_irrfl:
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fixes ps1 ps2 :: "nat list"
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assumes "ps1 \<sqsubset>lex ps2"
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shows "ps1 \<noteq> ps2"
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using assms
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by(induct rule: lex_list.induct)(auto)
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lemma lex_simps [simp]:
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fixes xs ys :: "nat list"
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shows "[] \<sqsubset>lex ys \<longleftrightarrow> ys \<noteq> []"
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and "xs \<sqsubset>lex [] \<longleftrightarrow> False"
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and "(x # xs) \<sqsubset>lex (y # ys) \<longleftrightarrow> (x < y \<or> (x = y \<and> xs \<sqsubset>lex ys))"
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by (auto simp add: neq_Nil_conv elim: lex_list.cases intro: lex_list.intros)
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lemma lex_trans:
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fixes ps1 ps2 ps3 :: "nat list"
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assumes "ps1 \<sqsubset>lex ps2" "ps2 \<sqsubset>lex ps3"
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shows "ps1 \<sqsubset>lex ps3"
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using assms
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by (induct arbitrary: ps3 rule: lex_list.induct)
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(auto elim: lex_list.cases)
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lemma lex_trichotomous:
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fixes p q :: "nat list"
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shows "p = q \<or> p \<sqsubset>lex q \<or> q \<sqsubset>lex p"
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apply(induct p arbitrary: q)
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apply(auto elim: lex_list.cases)
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apply(case_tac q)
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apply(auto)
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done
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section {* POSIX Ordering of Values According to Okui \& Suzuki *}
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definition PosOrd:: "val \<Rightarrow> nat list \<Rightarrow> val \<Rightarrow> bool" ("_ \<sqsubset>val _ _" [60, 60, 59] 60)
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where
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"v1 \<sqsubset>val p v2 \<equiv> pflat_len v1 p > pflat_len v2 p \<and>
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(\<forall>q \<in> Pos v1 \<union> Pos v2. q \<sqsubset>lex p \<longrightarrow> pflat_len v1 q = pflat_len v2 q)"
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lemma PosOrd_def2:
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shows "v1 \<sqsubset>val p v2 \<longleftrightarrow>
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pflat_len v1 p > pflat_len v2 p \<and>
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(\<forall>q \<in> Pos v1. q \<sqsubset>lex p \<longrightarrow> pflat_len v1 q = pflat_len v2 q) \<and>
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(\<forall>q \<in> Pos v2. q \<sqsubset>lex p \<longrightarrow> pflat_len v1 q = pflat_len v2 q)"
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unfolding PosOrd_def
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apply(auto)
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done
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definition PosOrd_ex:: "val \<Rightarrow> val \<Rightarrow> bool" ("_ :\<sqsubset>val _" [60, 59] 60)
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where
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"v1 :\<sqsubset>val v2 \<equiv> \<exists>p. v1 \<sqsubset>val p v2"
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definition PosOrd_ex_eq:: "val \<Rightarrow> val \<Rightarrow> bool" ("_ :\<sqsubseteq>val _" [60, 59] 60)
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where
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"v1 :\<sqsubseteq>val v2 \<equiv> v1 :\<sqsubset>val v2 \<or> v1 = v2"
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lemma PosOrd_trans:
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assumes "v1 :\<sqsubset>val v2" "v2 :\<sqsubset>val v3"
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shows "v1 :\<sqsubset>val v3"
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proof -
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from assms obtain p p'
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where as: "v1 \<sqsubset>val p v2" "v2 \<sqsubset>val p' v3" unfolding PosOrd_ex_def by blast
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then have pos: "p \<in> Pos v1" "p' \<in> Pos v2" unfolding PosOrd_def pflat_len_def
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by (smt not_int_zless_negative)+
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have "p = p' \<or> p \<sqsubset>lex p' \<or> p' \<sqsubset>lex p"
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by (rule lex_trichotomous)
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moreover
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{ assume "p = p'"
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with as have "v1 \<sqsubset>val p v3" unfolding PosOrd_def pflat_len_def
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by (smt Un_iff)
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then have " v1 :\<sqsubset>val v3" unfolding PosOrd_ex_def by blast
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}
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moreover
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{ assume "p \<sqsubset>lex p'"
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with as have "v1 \<sqsubset>val p v3" unfolding PosOrd_def pflat_len_def
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by (smt Un_iff lex_trans)
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then have " v1 :\<sqsubset>val v3" unfolding PosOrd_ex_def by blast
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}
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moreover
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{ assume "p' \<sqsubset>lex p"
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with as have "v1 \<sqsubset>val p' v3" unfolding PosOrd_def
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by (smt Un_iff lex_trans pflat_len_def)
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then have "v1 :\<sqsubset>val v3" unfolding PosOrd_ex_def by blast
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}
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ultimately show "v1 :\<sqsubset>val v3" by blast
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qed
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lemma PosOrd_irrefl:
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assumes "v :\<sqsubset>val v"
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shows "False"
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using assms unfolding PosOrd_ex_def PosOrd_def
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by auto
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lemma PosOrd_assym:
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assumes "v1 :\<sqsubset>val v2"
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shows "\<not>(v2 :\<sqsubset>val v1)"
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using assms
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using PosOrd_irrefl PosOrd_trans by blast
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(*
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:\<sqsubseteq>val and :\<sqsubset>val are partial orders.
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*)
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lemma PosOrd_ordering:
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shows "ordering (\<lambda>v1 v2. v1 :\<sqsubseteq>val v2) (\<lambda> v1 v2. v1 :\<sqsubset>val v2)"
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unfolding ordering_def PosOrd_ex_eq_def
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apply(auto)
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using PosOrd_irrefl apply blast
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using PosOrd_assym apply blast
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using PosOrd_trans by blast
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lemma PosOrd_order:
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shows "class.order (\<lambda>v1 v2. v1 :\<sqsubseteq>val v2) (\<lambda> v1 v2. v1 :\<sqsubset>val v2)"
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using PosOrd_ordering
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apply(simp add: class.order_def class.preorder_def class.order_axioms_def)
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unfolding ordering_def
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by blast
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lemma PosOrd_ex_eq2:
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shows "v1 :\<sqsubset>val v2 \<longleftrightarrow> (v1 :\<sqsubseteq>val v2 \<and> v1 \<noteq> v2)"
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using PosOrd_ordering
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unfolding ordering_def
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by auto
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lemma PosOrdeq_trans:
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assumes "v1 :\<sqsubseteq>val v2" "v2 :\<sqsubseteq>val v3"
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shows "v1 :\<sqsubseteq>val v3"
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using assms PosOrd_ordering
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unfolding ordering_def
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by blast
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lemma PosOrdeq_antisym:
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assumes "v1 :\<sqsubseteq>val v2" "v2 :\<sqsubseteq>val v1"
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shows "v1 = v2"
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using assms PosOrd_ordering
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unfolding ordering_def
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by blast
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lemma PosOrdeq_refl:
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shows "v :\<sqsubseteq>val v"
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unfolding PosOrd_ex_eq_def
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by auto
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lemma PosOrd_shorterE:
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assumes "v1 :\<sqsubset>val v2"
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shows "length (flat v2) \<le> length (flat v1)"
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using assms unfolding PosOrd_ex_def PosOrd_def
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apply(auto)
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apply(case_tac p)
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apply(simp add: pflat_len_simps)
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apply(drule_tac x="[]" in bspec)
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apply(simp add: Pos_empty)
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apply(simp add: pflat_len_simps)
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done
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lemma PosOrd_shorterI:
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assumes "length (flat v2) < length (flat v1)"
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shows "v1 :\<sqsubset>val v2"
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unfolding PosOrd_ex_def PosOrd_def pflat_len_def
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using assms Pos_empty by force
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lemma PosOrd_spreI:
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assumes "flat v' \<sqsubset>spre flat v"
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shows "v :\<sqsubset>val v'"
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using assms
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apply(rule_tac PosOrd_shorterI)
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unfolding prefix_list_def sprefix_list_def
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by (metis append_Nil2 append_eq_conv_conj drop_all le_less_linear)
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lemma pflat_len_inside:
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assumes "pflat_len v2 p < pflat_len v1 p"
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shows "p \<in> Pos v1"
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using assms
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unfolding pflat_len_def
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by (auto split: if_splits)
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lemma PosOrd_Left_Right:
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assumes "flat v1 = flat v2"
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shows "Left v1 :\<sqsubset>val Right v2"
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unfolding PosOrd_ex_def
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apply(rule_tac x="[0]" in exI)
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apply(auto simp add: PosOrd_def pflat_len_simps assms)
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done
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lemma PosOrd_LeftE:
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assumes "Left v1 :\<sqsubset>val Left v2" "flat v1 = flat v2"
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shows "v1 :\<sqsubset>val v2"
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using assms
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unfolding PosOrd_ex_def PosOrd_def2
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apply(auto simp add: pflat_len_simps)
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apply(frule pflat_len_inside)
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apply(auto simp add: pflat_len_simps)
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by (metis lex_simps(3) pflat_len_simps(3))
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lemma PosOrd_LeftI:
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assumes "v1 :\<sqsubset>val v2" "flat v1 = flat v2"
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shows "Left v1 :\<sqsubset>val Left v2"
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using assms
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unfolding PosOrd_ex_def PosOrd_def2
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apply(auto simp add: pflat_len_simps)
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by (metis less_numeral_extra(3) lex_simps(3) pflat_len_simps(3))
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lemma PosOrd_Left_eq:
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assumes "flat v1 = flat v2"
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shows "Left v1 :\<sqsubset>val Left v2 \<longleftrightarrow> v1 :\<sqsubset>val v2"
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using assms PosOrd_LeftE PosOrd_LeftI
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by blast
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lemma PosOrd_RightE:
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assumes "Right v1 :\<sqsubset>val Right v2" "flat v1 = flat v2"
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314 |
shows "v1 :\<sqsubset>val v2"
|
|
|
315 |
using assms
|
|
|
316 |
unfolding PosOrd_ex_def PosOrd_def2
|
|
|
317 |
apply(auto simp add: pflat_len_simps)
|
|
|
318 |
apply(frule pflat_len_inside)
|
|
|
319 |
apply(auto simp add: pflat_len_simps)
|
|
|
320 |
by (metis lex_simps(3) pflat_len_simps(5))
|
|
|
321 |
|
|
|
322 |
lemma PosOrd_RightI:
|
|
|
323 |
assumes "v1 :\<sqsubset>val v2" "flat v1 = flat v2"
|
|
|
324 |
shows "Right v1 :\<sqsubset>val Right v2"
|
|
|
325 |
using assms
|
|
|
326 |
unfolding PosOrd_ex_def PosOrd_def2
|
|
|
327 |
apply(auto simp add: pflat_len_simps)
|
|
|
328 |
by (metis lex_simps(3) nat_neq_iff pflat_len_simps(5))
|
|
|
329 |
|
|
|
330 |
|
|
|
331 |
lemma PosOrd_Right_eq:
|
|
|
332 |
assumes "flat v1 = flat v2"
|
|
|
333 |
shows "Right v1 :\<sqsubset>val Right v2 \<longleftrightarrow> v1 :\<sqsubset>val v2"
|
|
|
334 |
using assms PosOrd_RightE PosOrd_RightI
|
|
|
335 |
by blast
|
|
|
336 |
|
|
|
337 |
|
|
|
338 |
lemma PosOrd_SeqI1:
|
|
|
339 |
assumes "v1 :\<sqsubset>val w1" "flat (Seq v1 v2) = flat (Seq w1 w2)"
|
|
|
340 |
shows "Seq v1 v2 :\<sqsubset>val Seq w1 w2"
|
|
|
341 |
using assms(1)
|
|
|
342 |
apply(subst (asm) PosOrd_ex_def)
|
|
|
343 |
apply(subst (asm) PosOrd_def)
|
|
|
344 |
apply(clarify)
|
|
|
345 |
apply(subst PosOrd_ex_def)
|
|
|
346 |
apply(rule_tac x="0#p" in exI)
|
|
|
347 |
apply(subst PosOrd_def)
|
|
|
348 |
apply(rule conjI)
|
|
|
349 |
apply(simp add: pflat_len_simps)
|
|
|
350 |
apply(rule ballI)
|
|
|
351 |
apply(rule impI)
|
|
|
352 |
apply(simp only: Pos.simps)
|
|
|
353 |
apply(auto)[1]
|
|
|
354 |
apply(simp add: pflat_len_simps)
|
|
|
355 |
apply(auto simp add: pflat_len_simps)
|
|
|
356 |
using assms(2)
|
|
|
357 |
apply(simp)
|
|
|
358 |
apply(metis length_append of_nat_add)
|
|
|
359 |
done
|
|
|
360 |
|
|
|
361 |
lemma PosOrd_SeqI2:
|
|
|
362 |
assumes "v2 :\<sqsubset>val w2" "flat v2 = flat w2"
|
|
|
363 |
shows "Seq v v2 :\<sqsubset>val Seq v w2"
|
|
|
364 |
using assms(1)
|
|
|
365 |
apply(subst (asm) PosOrd_ex_def)
|
|
|
366 |
apply(subst (asm) PosOrd_def)
|
|
|
367 |
apply(clarify)
|
|
|
368 |
apply(subst PosOrd_ex_def)
|
|
|
369 |
apply(rule_tac x="Suc 0#p" in exI)
|
|
|
370 |
apply(subst PosOrd_def)
|
|
|
371 |
apply(rule conjI)
|
|
|
372 |
apply(simp add: pflat_len_simps)
|
|
|
373 |
apply(rule ballI)
|
|
|
374 |
apply(rule impI)
|
|
|
375 |
apply(simp only: Pos.simps)
|
|
|
376 |
apply(auto)[1]
|
|
|
377 |
apply(simp add: pflat_len_simps)
|
|
|
378 |
using assms(2)
|
|
|
379 |
apply(simp)
|
|
|
380 |
apply(auto simp add: pflat_len_simps)
|
|
|
381 |
done
|
|
|
382 |
|
|
|
383 |
lemma PosOrd_Seq_eq:
|
|
|
384 |
assumes "flat v2 = flat w2"
|
|
|
385 |
shows "(Seq v v2) :\<sqsubset>val (Seq v w2) \<longleftrightarrow> v2 :\<sqsubset>val w2"
|
|
|
386 |
using assms
|
|
|
387 |
apply(auto)
|
|
|
388 |
prefer 2
|
|
|
389 |
apply(simp add: PosOrd_SeqI2)
|
|
|
390 |
apply(simp add: PosOrd_ex_def)
|
|
|
391 |
apply(auto)
|
|
|
392 |
apply(case_tac p)
|
|
|
393 |
apply(simp add: PosOrd_def pflat_len_simps)
|
|
|
394 |
apply(case_tac a)
|
|
|
395 |
apply(simp add: PosOrd_def pflat_len_simps)
|
|
|
396 |
apply(clarify)
|
|
|
397 |
apply(case_tac nat)
|
|
|
398 |
prefer 2
|
|
|
399 |
apply(simp add: PosOrd_def pflat_len_simps pflat_len_outside)
|
|
|
400 |
apply(rule_tac x="list" in exI)
|
|
|
401 |
apply(auto simp add: PosOrd_def2 pflat_len_simps)
|
|
|
402 |
apply(smt Collect_disj_eq lex_list.intros(2) mem_Collect_eq pflat_len_simps(2))
|
|
|
403 |
apply(smt Collect_disj_eq lex_list.intros(2) mem_Collect_eq pflat_len_simps(2))
|
|
|
404 |
done
|
|
|
405 |
|
|
|
406 |
|
|
|
407 |
|
|
|
408 |
lemma PosOrd_StarsI:
|
|
|
409 |
assumes "v1 :\<sqsubset>val v2" "flats (v1#vs1) = flats (v2#vs2)"
|
|
|
410 |
shows "Stars (v1#vs1) :\<sqsubset>val Stars (v2#vs2)"
|
|
|
411 |
using assms(1)
|
|
|
412 |
apply(subst (asm) PosOrd_ex_def)
|
|
|
413 |
apply(subst (asm) PosOrd_def)
|
|
|
414 |
apply(clarify)
|
|
|
415 |
apply(subst PosOrd_ex_def)
|
|
|
416 |
apply(subst PosOrd_def)
|
|
|
417 |
apply(rule_tac x="0#p" in exI)
|
|
|
418 |
apply(simp add: pflat_len_Stars_simps pflat_len_simps)
|
|
|
419 |
using assms(2)
|
|
|
420 |
apply(simp add: pflat_len_simps)
|
|
|
421 |
apply(auto simp add: pflat_len_Stars_simps pflat_len_simps)
|
|
|
422 |
by (metis length_append of_nat_add)
|
|
|
423 |
|
|
|
424 |
lemma PosOrd_StarsI2:
|
|
|
425 |
assumes "Stars vs1 :\<sqsubset>val Stars vs2" "flats vs1 = flats vs2"
|
|
|
426 |
shows "Stars (v#vs1) :\<sqsubset>val Stars (v#vs2)"
|
|
|
427 |
using assms(1)
|
|
|
428 |
apply(subst (asm) PosOrd_ex_def)
|
|
|
429 |
apply(subst (asm) PosOrd_def)
|
|
|
430 |
apply(clarify)
|
|
|
431 |
apply(subst PosOrd_ex_def)
|
|
|
432 |
apply(subst PosOrd_def)
|
|
|
433 |
apply(case_tac p)
|
|
|
434 |
apply(simp add: pflat_len_simps)
|
|
|
435 |
apply(rule_tac x="Suc a#list" in exI)
|
|
|
436 |
apply(auto simp add: pflat_len_Stars_simps pflat_len_simps assms(2))
|
|
|
437 |
done
|
|
|
438 |
|
|
|
439 |
lemma PosOrd_Stars_appendI:
|
|
|
440 |
assumes "Stars vs1 :\<sqsubset>val Stars vs2" "flat (Stars vs1) = flat (Stars vs2)"
|
|
|
441 |
shows "Stars (vs @ vs1) :\<sqsubset>val Stars (vs @ vs2)"
|
|
|
442 |
using assms
|
|
|
443 |
apply(induct vs)
|
|
|
444 |
apply(simp)
|
|
|
445 |
apply(simp add: PosOrd_StarsI2)
|
|
|
446 |
done
|
|
|
447 |
|
|
|
448 |
lemma PosOrd_eq_Stars_zipI:
|
|
|
449 |
assumes "\<forall>(v1, v2) \<in> set (zip vs1 vs2). v1 :\<sqsubseteq>val v2"
|
|
|
450 |
"length vs1 = length vs2" "flats vs1 = flats vs2"
|
|
|
451 |
shows "Stars vs1 :\<sqsubseteq>val Stars vs2"
|
|
|
452 |
using assms
|
|
|
453 |
apply(induct vs1 arbitrary: vs2)
|
|
|
454 |
apply(case_tac vs2)
|
|
|
455 |
apply(simp add: PosOrd_ex_eq_def)
|
|
|
456 |
apply(simp)
|
|
|
457 |
apply(case_tac vs2)
|
|
|
458 |
apply(simp)
|
|
|
459 |
apply(simp)
|
|
|
460 |
apply(auto)
|
|
|
461 |
apply(subst (asm) (2)PosOrd_ex_eq_def)
|
|
|
462 |
apply(auto)
|
|
|
463 |
apply(subst PosOrd_ex_eq_def)
|
|
|
464 |
apply(rule disjI1)
|
|
|
465 |
apply(rule PosOrd_StarsI)
|
|
|
466 |
apply(simp)
|
|
|
467 |
apply(simp)
|
|
|
468 |
using PosOrd_StarsI2 PosOrd_ex_eq_def by fastforce
|
|
|
469 |
|
|
|
470 |
lemma PosOrd_StarsE2:
|
|
|
471 |
assumes "Stars (v # vs1) :\<sqsubset>val Stars (v # vs2)"
|
|
|
472 |
shows "Stars vs1 :\<sqsubset>val Stars vs2"
|
|
|
473 |
using assms
|
|
|
474 |
apply(subst (asm) PosOrd_ex_def)
|
|
|
475 |
apply(erule exE)
|
|
|
476 |
apply(case_tac p)
|
|
|
477 |
apply(simp)
|
|
|
478 |
apply(simp add: PosOrd_def pflat_len_simps)
|
|
|
479 |
apply(subst PosOrd_ex_def)
|
|
|
480 |
apply(rule_tac x="[]" in exI)
|
|
|
481 |
apply(simp add: PosOrd_def pflat_len_simps Pos_empty)
|
|
|
482 |
apply(simp)
|
|
|
483 |
apply(case_tac a)
|
|
|
484 |
apply(clarify)
|
|
|
485 |
apply(auto simp add: pflat_len_simps PosOrd_def pflat_len_def split: if_splits)[1]
|
|
|
486 |
apply(clarify)
|
|
|
487 |
apply(simp add: PosOrd_ex_def)
|
|
|
488 |
apply(rule_tac x="nat#list" in exI)
|
|
|
489 |
apply(auto simp add: PosOrd_def pflat_len_simps)[1]
|
|
|
490 |
apply(case_tac q)
|
|
|
491 |
apply(simp add: PosOrd_def pflat_len_simps)
|
|
|
492 |
apply(clarify)
|
|
|
493 |
apply(drule_tac x="Suc a # lista" in bspec)
|
|
|
494 |
apply(simp)
|
|
|
495 |
apply(auto simp add: PosOrd_def pflat_len_simps)[1]
|
|
|
496 |
apply(case_tac q)
|
|
|
497 |
apply(simp add: PosOrd_def pflat_len_simps)
|
|
|
498 |
apply(clarify)
|
|
|
499 |
apply(drule_tac x="Suc a # lista" in bspec)
|
|
|
500 |
apply(simp)
|
|
|
501 |
apply(auto simp add: PosOrd_def pflat_len_simps)[1]
|
|
|
502 |
done
|
|
|
503 |
|
|
|
504 |
lemma PosOrd_Stars_appendE:
|
|
|
505 |
assumes "Stars (vs @ vs1) :\<sqsubset>val Stars (vs @ vs2)"
|
|
|
506 |
shows "Stars vs1 :\<sqsubset>val Stars vs2"
|
|
|
507 |
using assms
|
|
|
508 |
apply(induct vs)
|
|
|
509 |
apply(simp)
|
|
|
510 |
apply(simp add: PosOrd_StarsE2)
|
|
|
511 |
done
|
|
|
512 |
|
|
|
513 |
lemma PosOrd_Stars_append_eq:
|
|
|
514 |
assumes "flats vs1 = flats vs2"
|
|
|
515 |
shows "Stars (vs @ vs1) :\<sqsubset>val Stars (vs @ vs2) \<longleftrightarrow> Stars vs1 :\<sqsubset>val Stars vs2"
|
|
|
516 |
using assms
|
|
|
517 |
apply(rule_tac iffI)
|
|
|
518 |
apply(erule PosOrd_Stars_appendE)
|
|
|
519 |
apply(rule PosOrd_Stars_appendI)
|
|
|
520 |
apply(auto)
|
|
|
521 |
done
|
|
|
522 |
|
|
|
523 |
lemma PosOrd_almost_trichotomous:
|
|
|
524 |
shows "v1 :\<sqsubset>val v2 \<or> v2 :\<sqsubset>val v1 \<or> (length (flat v1) = length (flat v2))"
|
|
|
525 |
apply(auto simp add: PosOrd_ex_def)
|
|
|
526 |
apply(auto simp add: PosOrd_def)
|
|
|
527 |
apply(rule_tac x="[]" in exI)
|
|
|
528 |
apply(auto simp add: Pos_empty pflat_len_simps)
|
|
|
529 |
apply(drule_tac x="[]" in spec)
|
|
|
530 |
apply(auto simp add: Pos_empty pflat_len_simps)
|
|
|
531 |
done
|
|
|
532 |
|
|
|
533 |
|
|
|
534 |
|
|
|
535 |
section {* The Posix Value is smaller than any other Value *}
|
|
|
536 |
|
|
|
537 |
|
|
|
538 |
lemma Posix_PosOrd:
|
|
|
539 |
assumes "s \<in> r \<rightarrow> v1" "v2 \<in> LV r s"
|
|
|
540 |
shows "v1 :\<sqsubseteq>val v2"
|
|
|
541 |
using assms
|
|
|
542 |
proof (induct arbitrary: v2 rule: Posix.induct)
|
|
|
543 |
case (Posix_ONE v)
|
|
|
544 |
have "v \<in> LV ONE []" by fact
|
|
|
545 |
then have "v = Void"
|
|
|
546 |
by (simp add: LV_simps)
|
|
|
547 |
then show "Void :\<sqsubseteq>val v"
|
|
|
548 |
by (simp add: PosOrd_ex_eq_def)
|
|
|
549 |
next
|
|
|
550 |
case (Posix_CHAR c v)
|
|
|
551 |
have "v \<in> LV (CHAR c) [c]" by fact
|
|
|
552 |
then have "v = Char c"
|
|
|
553 |
by (simp add: LV_simps)
|
|
|
554 |
then show "Char c :\<sqsubseteq>val v"
|
|
|
555 |
by (simp add: PosOrd_ex_eq_def)
|
|
|
556 |
next
|
|
|
557 |
case (Posix_ALT1 s r1 v r2 v2)
|
|
|
558 |
have as1: "s \<in> r1 \<rightarrow> v" by fact
|
|
|
559 |
have IH: "\<And>v2. v2 \<in> LV r1 s \<Longrightarrow> v :\<sqsubseteq>val v2" by fact
|
|
|
560 |
have "v2 \<in> LV (ALT r1 r2) s" by fact
|
|
|
561 |
then have "\<Turnstile> v2 : ALT r1 r2" "flat v2 = s"
|
|
|
562 |
by(auto simp add: LV_def prefix_list_def)
|
|
|
563 |
then consider
|
|
|
564 |
(Left) v3 where "v2 = Left v3" "\<Turnstile> v3 : r1" "flat v3 = s"
|
|
|
565 |
| (Right) v3 where "v2 = Right v3" "\<Turnstile> v3 : r2" "flat v3 = s"
|
|
|
566 |
by (auto elim: Prf.cases)
|
|
|
567 |
then show "Left v :\<sqsubseteq>val v2"
|
|
|
568 |
proof(cases)
|
|
|
569 |
case (Left v3)
|
|
|
570 |
have "v3 \<in> LV r1 s" using Left(2,3)
|
|
|
571 |
by (auto simp add: LV_def prefix_list_def)
|
|
|
572 |
with IH have "v :\<sqsubseteq>val v3" by simp
|
|
|
573 |
moreover
|
|
|
574 |
have "flat v3 = flat v" using as1 Left(3)
|
|
|
575 |
by (simp add: Posix1(2))
|
|
|
576 |
ultimately have "Left v :\<sqsubseteq>val Left v3"
|
|
|
577 |
by (simp add: PosOrd_ex_eq_def PosOrd_Left_eq)
|
|
|
578 |
then show "Left v :\<sqsubseteq>val v2" unfolding Left .
|
|
|
579 |
next
|
|
|
580 |
case (Right v3)
|
|
|
581 |
have "flat v3 = flat v" using as1 Right(3)
|
|
|
582 |
by (simp add: Posix1(2))
|
|
|
583 |
then have "Left v :\<sqsubseteq>val Right v3"
|
|
|
584 |
unfolding PosOrd_ex_eq_def
|
|
|
585 |
by (simp add: PosOrd_Left_Right)
|
|
|
586 |
then show "Left v :\<sqsubseteq>val v2" unfolding Right .
|
|
|
587 |
qed
|
|
|
588 |
next
|
|
|
589 |
case (Posix_ALT2 s r2 v r1 v2)
|
|
|
590 |
have as1: "s \<in> r2 \<rightarrow> v" by fact
|
|
|
591 |
have as2: "s \<notin> L r1" by fact
|
|
|
592 |
have IH: "\<And>v2. v2 \<in> LV r2 s \<Longrightarrow> v :\<sqsubseteq>val v2" by fact
|
|
|
593 |
have "v2 \<in> LV (ALT r1 r2) s" by fact
|
|
|
594 |
then have "\<Turnstile> v2 : ALT r1 r2" "flat v2 = s"
|
|
|
595 |
by(auto simp add: LV_def prefix_list_def)
|
|
|
596 |
then consider
|
|
|
597 |
(Left) v3 where "v2 = Left v3" "\<Turnstile> v3 : r1" "flat v3 = s"
|
|
|
598 |
| (Right) v3 where "v2 = Right v3" "\<Turnstile> v3 : r2" "flat v3 = s"
|
|
|
599 |
by (auto elim: Prf.cases)
|
|
|
600 |
then show "Right v :\<sqsubseteq>val v2"
|
|
|
601 |
proof (cases)
|
|
|
602 |
case (Right v3)
|
|
|
603 |
have "v3 \<in> LV r2 s" using Right(2,3)
|
|
|
604 |
by (auto simp add: LV_def prefix_list_def)
|
|
|
605 |
with IH have "v :\<sqsubseteq>val v3" by simp
|
|
|
606 |
moreover
|
|
|
607 |
have "flat v3 = flat v" using as1 Right(3)
|
|
|
608 |
by (simp add: Posix1(2))
|
|
|
609 |
ultimately have "Right v :\<sqsubseteq>val Right v3"
|
|
|
610 |
by (auto simp add: PosOrd_ex_eq_def PosOrd_RightI)
|
|
|
611 |
then show "Right v :\<sqsubseteq>val v2" unfolding Right .
|
|
|
612 |
next
|
|
|
613 |
case (Left v3)
|
|
|
614 |
have "v3 \<in> LV r1 s" using Left(2,3) as2
|
|
|
615 |
by (auto simp add: LV_def prefix_list_def)
|
|
|
616 |
then have "flat v3 = flat v \<and> \<Turnstile> v3 : r1" using as1 Left(3)
|
|
|
617 |
by (simp add: Posix1(2) LV_def)
|
|
|
618 |
then have "False" using as1 as2 Left
|
|
|
619 |
by (auto simp add: Posix1(2) L_flat_Prf1)
|
|
|
620 |
then show "Right v :\<sqsubseteq>val v2" by simp
|
|
|
621 |
qed
|
|
|
622 |
next
|
|
|
623 |
case (Posix_SEQ s1 r1 v1 s2 r2 v2 v3)
|
|
|
624 |
have "s1 \<in> r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2" by fact+
|
|
|
625 |
then have as1: "s1 = flat v1" "s2 = flat v2" by (simp_all add: Posix1(2))
|
|
|
626 |
have IH1: "\<And>v3. v3 \<in> LV r1 s1 \<Longrightarrow> v1 :\<sqsubseteq>val v3" by fact
|
|
|
627 |
have IH2: "\<And>v3. v3 \<in> LV r2 s2 \<Longrightarrow> v2 :\<sqsubseteq>val v3" by fact
|
|
|
628 |
have cond: "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r1 \<and> s\<^sub>4 \<in> L r2)" by fact
|
|
|
629 |
have "v3 \<in> LV (SEQ r1 r2) (s1 @ s2)" by fact
|
|
|
630 |
then obtain v3a v3b where eqs:
|
|
|
631 |
"v3 = Seq v3a v3b" "\<Turnstile> v3a : r1" "\<Turnstile> v3b : r2"
|
|
|
632 |
"flat v3a @ flat v3b = s1 @ s2"
|
|
|
633 |
by (force simp add: prefix_list_def LV_def elim: Prf.cases)
|
|
|
634 |
with cond have "flat v3a \<sqsubseteq>pre s1" unfolding prefix_list_def
|
|
|
635 |
by (smt L_flat_Prf1 append_eq_append_conv2 append_self_conv)
|
|
|
636 |
then have "flat v3a \<sqsubset>spre s1 \<or> (flat v3a = s1 \<and> flat v3b = s2)" using eqs
|
|
|
637 |
by (simp add: sprefix_list_def append_eq_conv_conj)
|
|
|
638 |
then have q2: "v1 :\<sqsubset>val v3a \<or> (flat v3a = s1 \<and> flat v3b = s2)"
|
|
|
639 |
using PosOrd_spreI as1(1) eqs by blast
|
|
|
640 |
then have "v1 :\<sqsubset>val v3a \<or> (v3a \<in> LV r1 s1 \<and> v3b \<in> LV r2 s2)" using eqs(2,3)
|
|
|
641 |
by (auto simp add: LV_def)
|
|
|
642 |
then have "v1 :\<sqsubset>val v3a \<or> (v1 :\<sqsubseteq>val v3a \<and> v2 :\<sqsubseteq>val v3b)" using IH1 IH2 by blast
|
|
|
643 |
then have "Seq v1 v2 :\<sqsubseteq>val Seq v3a v3b" using eqs q2 as1
|
|
|
644 |
unfolding PosOrd_ex_eq_def by (auto simp add: PosOrd_SeqI1 PosOrd_Seq_eq)
|
|
|
645 |
then show "Seq v1 v2 :\<sqsubseteq>val v3" unfolding eqs by blast
|
|
|
646 |
next
|
|
|
647 |
case (Posix_STAR1 s1 r v s2 vs v3)
|
|
|
648 |
have "s1 \<in> r \<rightarrow> v" "s2 \<in> STAR r \<rightarrow> Stars vs" by fact+
|
|
|
649 |
then have as1: "s1 = flat v" "s2 = flat (Stars vs)" by (auto dest: Posix1(2))
|
|
|
650 |
have IH1: "\<And>v3. v3 \<in> LV r s1 \<Longrightarrow> v :\<sqsubseteq>val v3" by fact
|
|
|
651 |
have IH2: "\<And>v3. v3 \<in> LV (STAR r) s2 \<Longrightarrow> Stars vs :\<sqsubseteq>val v3" by fact
|
|
|
652 |
have cond: "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))" by fact
|
|
|
653 |
have cond2: "flat v \<noteq> []" by fact
|
|
|
654 |
have "v3 \<in> LV (STAR r) (s1 @ s2)" by fact
|
|
|
655 |
then consider
|
|
|
656 |
(NonEmpty) v3a vs3 where "v3 = Stars (v3a # vs3)"
|
|
|
657 |
"\<Turnstile> v3a : r" "\<Turnstile> Stars vs3 : STAR r"
|
|
|
658 |
"flat (Stars (v3a # vs3)) = s1 @ s2"
|
|
|
659 |
| (Empty) "v3 = Stars []"
|
|
|
660 |
unfolding LV_def
|
|
|
661 |
apply(auto)
|
|
|
662 |
apply(erule Prf.cases)
|
|
|
663 |
apply(auto)
|
|
|
664 |
apply(case_tac vs)
|
|
|
665 |
apply(auto intro: Prf.intros)
|
|
|
666 |
done
|
|
|
667 |
then show "Stars (v # vs) :\<sqsubseteq>val v3"
|
|
|
668 |
proof (cases)
|
|
|
669 |
case (NonEmpty v3a vs3)
|
|
|
670 |
have "flat (Stars (v3a # vs3)) = s1 @ s2" using NonEmpty(4) .
|
|
|
671 |
with cond have "flat v3a \<sqsubseteq>pre s1" using NonEmpty(2,3)
|
|
|
672 |
unfolding prefix_list_def
|
|
|
673 |
by (smt L_flat_Prf1 append_Nil2 append_eq_append_conv2 flat.simps(7))
|
|
|
674 |
then have "flat v3a \<sqsubset>spre s1 \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)" using NonEmpty(4)
|
|
|
675 |
by (simp add: sprefix_list_def append_eq_conv_conj)
|
|
|
676 |
then have q2: "v :\<sqsubset>val v3a \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)"
|
|
|
677 |
using PosOrd_spreI as1(1) NonEmpty(4) by blast
|
|
|
678 |
then have "v :\<sqsubset>val v3a \<or> (v3a \<in> LV r s1 \<and> Stars vs3 \<in> LV (STAR r) s2)"
|
|
|
679 |
using NonEmpty(2,3) by (auto simp add: LV_def)
|
|
|
680 |
then have "v :\<sqsubset>val v3a \<or> (v :\<sqsubseteq>val v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)" using IH1 IH2 by blast
|
|
|
681 |
then have "v :\<sqsubset>val v3a \<or> (v = v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)"
|
|
|
682 |
unfolding PosOrd_ex_eq_def by auto
|
|
|
683 |
then have "Stars (v # vs) :\<sqsubseteq>val Stars (v3a # vs3)" using NonEmpty(4) q2 as1
|
|
|
684 |
unfolding PosOrd_ex_eq_def
|
|
|
685 |
using PosOrd_StarsI PosOrd_StarsI2 by auto
|
|
|
686 |
then show "Stars (v # vs) :\<sqsubseteq>val v3" unfolding NonEmpty by blast
|
|
|
687 |
next
|
|
|
688 |
case Empty
|
|
|
689 |
have "v3 = Stars []" by fact
|
|
|
690 |
then show "Stars (v # vs) :\<sqsubseteq>val v3"
|
|
|
691 |
unfolding PosOrd_ex_eq_def using cond2
|
|
|
692 |
by (simp add: PosOrd_shorterI)
|
|
|
693 |
qed
|
|
|
694 |
next
|
|
|
695 |
case (Posix_STAR2 r v2)
|
|
|
696 |
have "v2 \<in> LV (STAR r) []" by fact
|
|
|
697 |
then have "v2 = Stars []"
|
|
|
698 |
unfolding LV_def by (auto elim: Prf.cases)
|
|
|
699 |
then show "Stars [] :\<sqsubseteq>val v2"
|
|
|
700 |
by (simp add: PosOrd_ex_eq_def)
|
|
|
701 |
next
|
|
|
702 |
case (Posix_NTIMES1 s1 r v s2 n vs v3)
|
|
|
703 |
have "s1 \<in> r \<rightarrow> v" "s2 \<in> NTIMES r (n - 1) \<rightarrow> Stars vs" by fact+
|
|
|
704 |
then have as1: "s1 = flat v" "s2 = flats vs" by (auto dest: Posix1(2))
|
|
|
705 |
have IH1: "\<And>v3. v3 \<in> LV r s1 \<Longrightarrow> v :\<sqsubseteq>val v3" by fact
|
|
|
706 |
have IH2: "\<And>v3. v3 \<in> LV (NTIMES r (n - 1)) s2 \<Longrightarrow> Stars vs :\<sqsubseteq>val v3" by fact
|
|
|
707 |
have cond: "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (NTIMES r (n - 1)))" by fact
|
|
|
708 |
have cond2: "flat v \<noteq> []" by fact
|
|
|
709 |
have "v3 \<in> LV (NTIMES r n) (s1 @ s2)" by fact
|
|
|
710 |
then consider
|
|
|
711 |
(NonEmpty) v3a vs3 where "v3 = Stars (v3a # vs3)"
|
|
|
712 |
"\<Turnstile> v3a : r" "\<Turnstile> Stars vs3 : NTIMES r (n - 1)"
|
|
|
713 |
"flats (v3a # vs3) = s1 @ s2"
|
|
|
714 |
| (Empty) "v3 = Stars []"
|
|
|
715 |
unfolding LV_def
|
|
|
716 |
apply(auto)
|
|
|
717 |
apply(erule Prf.cases)
|
|
|
718 |
apply(auto)
|
|
|
719 |
apply(case_tac vs1)
|
|
|
720 |
apply(auto intro: Prf.intros)
|
|
|
721 |
apply(case_tac vs2)
|
|
|
722 |
apply(auto intro: Prf.intros)
|
|
|
723 |
apply (simp add: as1(1) cond2 flats_empty)
|
|
|
724 |
by (simp add: Prf.intros(8))
|
|
|
725 |
then show "Stars (v # vs) :\<sqsubseteq>val v3"
|
|
|
726 |
proof (cases)
|
|
|
727 |
case (NonEmpty v3a vs3)
|
|
|
728 |
have "flats (v3a # vs3) = s1 @ s2" using NonEmpty(4) .
|
|
|
729 |
with cond have "flat v3a \<sqsubseteq>pre s1" using NonEmpty(2,3)
|
|
|
730 |
unfolding prefix_list_def
|
|
|
731 |
by (smt L_flat_Prf1 append_Nil2 append_eq_append_conv2 flat.simps(7) flat_Stars)
|
|
|
732 |
then have "flat v3a \<sqsubset>spre s1 \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)" using NonEmpty(4)
|
|
|
733 |
by (simp add: sprefix_list_def append_eq_conv_conj)
|
|
|
734 |
then have q2: "v :\<sqsubset>val v3a \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)"
|
|
|
735 |
using PosOrd_spreI as1(1) NonEmpty(4) by blast
|
|
|
736 |
then have "v :\<sqsubset>val v3a \<or> (v3a \<in> LV r s1 \<and> Stars vs3 \<in> LV (NTIMES r (n - 1)) s2)"
|
|
|
737 |
using NonEmpty(2,3) by (auto simp add: LV_def)
|
|
|
738 |
then have "v :\<sqsubset>val v3a \<or> (v :\<sqsubseteq>val v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)" using IH1 IH2 by blast
|
|
|
739 |
then have "v :\<sqsubset>val v3a \<or> (v = v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)"
|
|
|
740 |
unfolding PosOrd_ex_eq_def by auto
|
|
|
741 |
then have "Stars (v # vs) :\<sqsubseteq>val Stars (v3a # vs3)" using NonEmpty(4) q2 as1
|
|
|
742 |
unfolding PosOrd_ex_eq_def
|
|
|
743 |
using PosOrd_StarsI PosOrd_StarsI2 by auto
|
|
|
744 |
then show "Stars (v # vs) :\<sqsubseteq>val v3" unfolding NonEmpty by blast
|
|
|
745 |
next
|
|
|
746 |
case Empty
|
|
|
747 |
have "v3 = Stars []" by fact
|
|
|
748 |
then show "Stars (v # vs) :\<sqsubseteq>val v3"
|
|
|
749 |
unfolding PosOrd_ex_eq_def using cond2
|
|
|
750 |
by (simp add: PosOrd_shorterI)
|
|
|
751 |
qed
|
|
|
752 |
next
|
|
|
753 |
case (Posix_NTIMES2 vs r n v2)
|
|
|
754 |
then show "Stars vs :\<sqsubseteq>val v2"
|
|
|
755 |
apply(simp add: LV_def)
|
|
|
756 |
apply(auto)
|
|
|
757 |
apply(erule Prf_elims)
|
|
|
758 |
apply(auto)
|
|
|
759 |
apply(rule PosOrd_eq_Stars_zipI)
|
|
|
760 |
prefer 2
|
|
|
761 |
apply(simp)
|
|
|
762 |
prefer 2
|
|
|
763 |
apply (metis Posix1(2) flats_empty)
|
|
|
764 |
apply(auto)
|
|
|
765 |
by (meson in_set_zipE)
|
|
|
766 |
next
|
|
|
767 |
case (Posix_UPNTIMES2 r n v2)
|
|
|
768 |
then show "Stars [] :\<sqsubseteq>val v2"
|
|
|
769 |
apply(simp add: LV_def)
|
|
|
770 |
apply(auto)
|
|
|
771 |
apply(erule Prf_elims)
|
|
|
772 |
apply(auto)
|
|
|
773 |
unfolding PosOrd_ex_eq_def by simp
|
|
|
774 |
next
|
|
|
775 |
case (Posix_UPNTIMES1 s1 r v s2 n vs v3)
|
|
|
776 |
have "s1 \<in> r \<rightarrow> v" "s2 \<in> UPNTIMES r (n - 1) \<rightarrow> Stars vs" by fact+
|
|
|
777 |
then have as1: "s1 = flat v" "s2 = flat (Stars vs)" by (auto dest: Posix1(2))
|
|
|
778 |
have IH1: "\<And>v3. v3 \<in> LV r s1 \<Longrightarrow> v :\<sqsubseteq>val v3" by fact
|
|
|
779 |
have IH2: "\<And>v3. v3 \<in> LV (UPNTIMES r (n - 1)) s2 \<Longrightarrow> Stars vs :\<sqsubseteq>val v3" by fact
|
|
|
780 |
have cond: "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (UPNTIMES r (n - 1)))" by fact
|
|
|
781 |
have cond2: "flat v \<noteq> []" by fact
|
|
|
782 |
have "v3 \<in> LV (UPNTIMES r n) (s1 @ s2)" by fact
|
|
|
783 |
then consider
|
|
|
784 |
(NonEmpty) v3a vs3 where "v3 = Stars (v3a # vs3)"
|
|
|
785 |
"\<Turnstile> v3a : r" "\<Turnstile> Stars vs3 : UPNTIMES r (n - 1)"
|
|
|
786 |
"flats (v3a # vs3) = s1 @ s2"
|
|
|
787 |
| (Empty) "v3 = Stars []"
|
|
|
788 |
unfolding LV_def
|
|
|
789 |
apply(auto)
|
|
|
790 |
apply(erule Prf.cases)
|
|
|
791 |
apply(auto)
|
|
|
792 |
apply(case_tac vs)
|
|
|
793 |
apply(auto intro: Prf.intros)
|
|
|
794 |
by (simp add: Prf.intros(7) as1(1) cond2)
|
|
|
795 |
then show "Stars (v # vs) :\<sqsubseteq>val v3"
|
|
|
796 |
proof (cases)
|
|
|
797 |
case (NonEmpty v3a vs3)
|
|
|
798 |
have "flats (v3a # vs3) = s1 @ s2" using NonEmpty(4) .
|
|
|
799 |
with cond have "flat v3a \<sqsubseteq>pre s1" using NonEmpty(2,3)
|
|
|
800 |
unfolding prefix_list_def
|
|
|
801 |
apply(simp)
|
|
|
802 |
apply(simp add: append_eq_append_conv2)
|
|
|
803 |
apply(auto)
|
|
|
804 |
by (metis L_flat_Prf1 One_nat_def cond flat_Stars)
|
|
|
805 |
then have "flat v3a \<sqsubset>spre s1 \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)" using NonEmpty(4)
|
|
|
806 |
by (simp add: sprefix_list_def append_eq_conv_conj)
|
|
|
807 |
then have q2: "v :\<sqsubset>val v3a \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)"
|
|
|
808 |
using PosOrd_spreI as1(1) NonEmpty(4) by blast
|
|
|
809 |
then have "v :\<sqsubset>val v3a \<or> (v3a \<in> LV r s1 \<and> Stars vs3 \<in> LV (UPNTIMES r (n - 1)) s2)"
|
|
|
810 |
using NonEmpty(2,3) by (auto simp add: LV_def)
|
|
|
811 |
then have "v :\<sqsubset>val v3a \<or> (v :\<sqsubseteq>val v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)" using IH1 IH2 by blast
|
|
|
812 |
then have "v :\<sqsubset>val v3a \<or> (v = v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)"
|
|
|
813 |
unfolding PosOrd_ex_eq_def by auto
|
|
|
814 |
then have "Stars (v # vs) :\<sqsubseteq>val Stars (v3a # vs3)" using NonEmpty(4) q2 as1
|
|
|
815 |
unfolding PosOrd_ex_eq_def
|
|
|
816 |
using PosOrd_StarsI PosOrd_StarsI2 by auto
|
|
|
817 |
then show "Stars (v # vs) :\<sqsubseteq>val v3" unfolding NonEmpty by blast
|
|
|
818 |
next
|
|
|
819 |
case Empty
|
|
|
820 |
have "v3 = Stars []" by fact
|
|
|
821 |
then show "Stars (v # vs) :\<sqsubseteq>val v3"
|
|
|
822 |
unfolding PosOrd_ex_eq_def using cond2
|
|
|
823 |
by (simp add: PosOrd_shorterI)
|
|
|
824 |
qed
|
|
|
825 |
next
|
|
|
826 |
case (Posix_FROMNTIMES2 vs r n v2)
|
|
|
827 |
then show "Stars vs :\<sqsubseteq>val v2"
|
|
|
828 |
apply(simp add: LV_def)
|
|
|
829 |
apply(auto)
|
|
|
830 |
apply(erule Prf_elims)
|
|
|
831 |
apply(auto)
|
|
|
832 |
apply(rule PosOrd_eq_Stars_zipI)
|
|
|
833 |
prefer 2
|
|
|
834 |
apply(simp)
|
|
|
835 |
prefer 2
|
|
|
836 |
apply (metis Posix1(2) flats_empty)
|
|
|
837 |
apply(auto)
|
|
|
838 |
by (meson in_set_zipE)
|
|
|
839 |
next
|
|
|
840 |
case (Posix_FROMNTIMES1 s1 r v s2 n vs v3)
|
|
|
841 |
have "s1 \<in> r \<rightarrow> v" "s2 \<in> FROMNTIMES r (n - 1) \<rightarrow> Stars vs" by fact+
|
|
|
842 |
then have as1: "s1 = flat v" "s2 = flats vs" by (auto dest: Posix1(2))
|
|
|
843 |
have IH1: "\<And>v3. v3 \<in> LV r s1 \<Longrightarrow> v :\<sqsubseteq>val v3" by fact
|
|
|
844 |
have IH2: "\<And>v3. v3 \<in> LV (FROMNTIMES r (n - 1)) s2 \<Longrightarrow> Stars vs :\<sqsubseteq>val v3" by fact
|
|
|
845 |
have cond: "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (FROMNTIMES r (n - 1)))" by fact
|
|
|
846 |
have cond2: "flat v \<noteq> []" by fact
|
|
|
847 |
have "v3 \<in> LV (FROMNTIMES r n) (s1 @ s2)" by fact
|
|
|
848 |
then consider
|
|
|
849 |
(NonEmpty) v3a vs3 where "v3 = Stars (v3a # vs3)"
|
|
|
850 |
"\<Turnstile> v3a : r" "\<Turnstile> Stars vs3 : FROMNTIMES r (n - 1)"
|
|
|
851 |
"flats (v3a # vs3) = s1 @ s2"
|
|
|
852 |
| (Empty) "v3 = Stars []"
|
|
|
853 |
unfolding LV_def
|
|
|
854 |
apply(auto)
|
|
|
855 |
apply(erule Prf.cases)
|
|
|
856 |
apply(auto)
|
|
|
857 |
apply(case_tac vs1)
|
|
|
858 |
apply(auto intro: Prf.intros)
|
|
|
859 |
apply(case_tac vs2)
|
|
|
860 |
apply(auto intro: Prf.intros)
|
|
|
861 |
apply (simp add: as1(1) cond2 flats_empty)
|
|
|
862 |
apply (simp add: Prf.intros)
|
|
|
863 |
apply(case_tac vs)
|
|
|
864 |
apply(auto)
|
|
|
865 |
using Posix_FROMNTIMES1.hyps(6) Prf.intros(10) by auto
|
|
|
866 |
then show "Stars (v # vs) :\<sqsubseteq>val v3"
|
|
|
867 |
proof (cases)
|
|
|
868 |
case (NonEmpty v3a vs3)
|
|
|
869 |
have "flats (v3a # vs3) = s1 @ s2" using NonEmpty(4) .
|
|
|
870 |
with cond have "flat v3a \<sqsubseteq>pre s1" using NonEmpty(2,3)
|
|
|
871 |
unfolding prefix_list_def
|
|
|
872 |
by (smt L_flat_Prf1 append_Nil2 append_eq_append_conv2 flat.simps(7) flat_Stars)
|
|
|
873 |
then have "flat v3a \<sqsubset>spre s1 \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)" using NonEmpty(4)
|
|
|
874 |
by (simp add: sprefix_list_def append_eq_conv_conj)
|
|
|
875 |
then have q2: "v :\<sqsubset>val v3a \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)"
|
|
|
876 |
using PosOrd_spreI as1(1) NonEmpty(4) by blast
|
|
|
877 |
then have "v :\<sqsubset>val v3a \<or> (v3a \<in> LV r s1 \<and> Stars vs3 \<in> LV (FROMNTIMES r (n - 1)) s2)"
|
|
|
878 |
using NonEmpty(2,3) by (auto simp add: LV_def)
|
|
|
879 |
then have "v :\<sqsubset>val v3a \<or> (v :\<sqsubseteq>val v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)" using IH1 IH2 by blast
|
|
|
880 |
then have "v :\<sqsubset>val v3a \<or> (v = v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)"
|
|
|
881 |
unfolding PosOrd_ex_eq_def by auto
|
|
|
882 |
then have "Stars (v # vs) :\<sqsubseteq>val Stars (v3a # vs3)" using NonEmpty(4) q2 as1
|
|
|
883 |
unfolding PosOrd_ex_eq_def
|
|
|
884 |
using PosOrd_StarsI PosOrd_StarsI2 by auto
|
|
|
885 |
then show "Stars (v # vs) :\<sqsubseteq>val v3" unfolding NonEmpty by blast
|
|
|
886 |
next
|
|
|
887 |
case Empty
|
|
|
888 |
have "v3 = Stars []" by fact
|
|
|
889 |
then show "Stars (v # vs) :\<sqsubseteq>val v3"
|
|
|
890 |
unfolding PosOrd_ex_eq_def using cond2
|
|
|
891 |
by (simp add: PosOrd_shorterI)
|
|
|
892 |
qed
|
|
|
893 |
next
|
|
|
894 |
case (Posix_FROMNTIMES3 s1 r v s2 vs v3)
|
|
|
895 |
have "s1 \<in> r \<rightarrow> v" "s2 \<in> STAR r \<rightarrow> Stars vs" by fact+
|
|
|
896 |
then have as1: "s1 = flat v" "s2 = flat (Stars vs)" by (auto dest: Posix1(2))
|
|
|
897 |
have IH1: "\<And>v3. v3 \<in> LV r s1 \<Longrightarrow> v :\<sqsubseteq>val v3" by fact
|
|
|
898 |
have IH2: "\<And>v3. v3 \<in> LV (STAR r) s2 \<Longrightarrow> Stars vs :\<sqsubseteq>val v3" by fact
|
|
|
899 |
have cond: "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))" by fact
|
|
|
900 |
have cond2: "flat v \<noteq> []" by fact
|
|
|
901 |
have "v3 \<in> LV (FROMNTIMES r 0) (s1 @ s2)" by fact
|
|
|
902 |
then consider
|
|
|
903 |
(NonEmpty) v3a vs3 where "v3 = Stars (v3a # vs3)"
|
|
|
904 |
"\<Turnstile> v3a : r" "\<Turnstile> Stars vs3 : STAR r"
|
|
|
905 |
"flat (Stars (v3a # vs3)) = s1 @ s2"
|
|
|
906 |
| (Empty) "v3 = Stars []"
|
|
|
907 |
unfolding LV_def
|
|
|
908 |
apply(auto)
|
|
|
909 |
apply(erule Prf.cases)
|
|
|
910 |
apply(auto)
|
|
|
911 |
apply(case_tac vs)
|
|
|
912 |
apply(auto intro: Prf.intros)
|
|
|
913 |
done
|
|
|
914 |
then show "Stars (v # vs) :\<sqsubseteq>val v3"
|
|
|
915 |
proof (cases)
|
|
|
916 |
case (NonEmpty v3a vs3)
|
|
|
917 |
have "flat (Stars (v3a # vs3)) = s1 @ s2" using NonEmpty(4) .
|
|
|
918 |
with cond have "flat v3a \<sqsubseteq>pre s1" using NonEmpty(2,3)
|
|
|
919 |
unfolding prefix_list_def
|
|
|
920 |
by (smt L_flat_Prf1 append_Nil2 append_eq_append_conv2 flat.simps(7))
|
|
|
921 |
then have "flat v3a \<sqsubset>spre s1 \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)" using NonEmpty(4)
|
|
|
922 |
by (simp add: sprefix_list_def append_eq_conv_conj)
|
|
|
923 |
then have q2: "v :\<sqsubset>val v3a \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)"
|
|
|
924 |
using PosOrd_spreI as1(1) NonEmpty(4) by blast
|
|
|
925 |
then have "v :\<sqsubset>val v3a \<or> (v3a \<in> LV r s1 \<and> Stars vs3 \<in> LV (STAR r) s2)"
|
|
|
926 |
using NonEmpty(2,3) by (auto simp add: LV_def)
|
|
|
927 |
then have "v :\<sqsubset>val v3a \<or> (v :\<sqsubseteq>val v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)" using IH1 IH2 by blast
|
|
|
928 |
then have "v :\<sqsubset>val v3a \<or> (v = v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)"
|
|
|
929 |
unfolding PosOrd_ex_eq_def by auto
|
|
|
930 |
then have "Stars (v # vs) :\<sqsubseteq>val Stars (v3a # vs3)" using NonEmpty(4) q2 as1
|
|
|
931 |
unfolding PosOrd_ex_eq_def
|
|
|
932 |
using PosOrd_StarsI PosOrd_StarsI2 by auto
|
|
|
933 |
then show "Stars (v # vs) :\<sqsubseteq>val v3" unfolding NonEmpty by blast
|
|
|
934 |
next
|
|
|
935 |
case Empty
|
|
|
936 |
have "v3 = Stars []" by fact
|
|
|
937 |
then show "Stars (v # vs) :\<sqsubseteq>val v3"
|
|
|
938 |
unfolding PosOrd_ex_eq_def using cond2
|
|
|
939 |
by (simp add: PosOrd_shorterI)
|
|
|
940 |
qed
|
|
|
941 |
next
|
|
|
942 |
case (Posix_NMTIMES2 vs r n m v2)
|
|
|
943 |
then show "Stars vs :\<sqsubseteq>val v2"
|
|
|
944 |
apply(auto simp add: LV_def)
|
|
|
945 |
apply(erule Prf_elims)
|
|
|
946 |
apply(simp)
|
|
|
947 |
apply(rule PosOrd_eq_Stars_zipI)
|
|
|
948 |
apply(auto)
|
|
|
949 |
apply (meson in_set_zipE)
|
|
|
950 |
by (metis Posix1(2) flats_empty)
|
|
|
951 |
next
|
|
|
952 |
case (Posix_NMTIMES1 s1 r v s2 n m vs v3)
|
|
|
953 |
have "s1 \<in> r \<rightarrow> v" "s2 \<in> NMTIMES r (n - 1) (m - 1) \<rightarrow> Stars vs" by fact+
|
|
|
954 |
then have as1: "s1 = flat v" "s2 = flats vs" by (auto dest: Posix1(2))
|
|
|
955 |
have IH1: "\<And>v3. v3 \<in> LV r s1 \<Longrightarrow> v :\<sqsubseteq>val v3" by fact
|
|
|
956 |
have IH2: "\<And>v3. v3 \<in> LV (NMTIMES r (n - 1) (m - 1)) s2 \<Longrightarrow> Stars vs :\<sqsubseteq>val v3" by fact
|
|
|
957 |
have cond: "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (NMTIMES r (n - 1) (m - 1)))" by fact
|
|
|
958 |
have cond2: "flat v \<noteq> []" by fact
|
|
|
959 |
have "v3 \<in> LV (NMTIMES r n m) (s1 @ s2)" by fact
|
|
|
960 |
then consider
|
|
|
961 |
(NonEmpty) v3a vs3 where "v3 = Stars (v3a # vs3)"
|
|
|
962 |
"\<Turnstile> v3a : r" "\<Turnstile> Stars vs3 : NMTIMES r (n - 1) (m - 1)"
|
|
|
963 |
"flats (v3a # vs3) = s1 @ s2"
|
|
|
964 |
| (Empty) "v3 = Stars []"
|
|
|
965 |
unfolding LV_def
|
|
|
966 |
apply(auto)
|
|
|
967 |
apply(erule Prf.cases)
|
|
|
968 |
apply(auto)
|
|
|
969 |
apply(case_tac n)
|
|
|
970 |
apply(auto intro: Prf.intros)
|
|
|
971 |
apply(case_tac vs1)
|
|
|
972 |
apply(auto intro: Prf.intros)
|
|
|
973 |
apply (simp add: as1(1) cond2 flats_empty)
|
|
|
974 |
apply (simp add: Prf.intros(11))
|
|
|
975 |
apply(case_tac n)
|
|
|
976 |
apply(simp)
|
|
|
977 |
using Posix_NMTIMES1.hyps(6) apply blast
|
|
|
978 |
apply(simp)
|
|
|
979 |
apply(case_tac vs)
|
|
|
980 |
apply(auto)
|
|
|
981 |
by (simp add: Prf.intros(12))
|
|
|
982 |
then show "Stars (v # vs) :\<sqsubseteq>val v3"
|
|
|
983 |
proof (cases)
|
|
|
984 |
case (NonEmpty v3a vs3)
|
|
|
985 |
have "flats (v3a # vs3) = s1 @ s2" using NonEmpty(4) .
|
|
|
986 |
with cond have "flat v3a \<sqsubseteq>pre s1" using NonEmpty(2,3)
|
|
|
987 |
unfolding prefix_list_def
|
|
|
988 |
by (smt L_flat_Prf1 append_Nil2 append_eq_append_conv2 flat.simps(7) flat_Stars)
|
|
|
989 |
then have "flat v3a \<sqsubset>spre s1 \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)" using NonEmpty(4)
|
|
|
990 |
by (simp add: sprefix_list_def append_eq_conv_conj)
|
|
|
991 |
then have q2: "v :\<sqsubset>val v3a \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)"
|
|
|
992 |
using PosOrd_spreI as1(1) NonEmpty(4) by blast
|
|
|
993 |
then have "v :\<sqsubset>val v3a \<or> (v3a \<in> LV r s1 \<and> Stars vs3 \<in> LV (NMTIMES r (n - 1) (m - 1)) s2)"
|
|
|
994 |
using NonEmpty(2,3) by (auto simp add: LV_def)
|
|
|
995 |
then have "v :\<sqsubset>val v3a \<or> (v :\<sqsubseteq>val v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)" using IH1 IH2 by blast
|
|
|
996 |
then have "v :\<sqsubset>val v3a \<or> (v = v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)"
|
|
|
997 |
unfolding PosOrd_ex_eq_def by auto
|
|
|
998 |
then have "Stars (v # vs) :\<sqsubseteq>val Stars (v3a # vs3)" using NonEmpty(4) q2 as1
|
|
|
999 |
unfolding PosOrd_ex_eq_def
|
|
|
1000 |
using PosOrd_StarsI PosOrd_StarsI2 by auto
|
|
|
1001 |
then show "Stars (v # vs) :\<sqsubseteq>val v3" unfolding NonEmpty by blast
|
|
|
1002 |
next
|
|
|
1003 |
case Empty
|
|
|
1004 |
have "v3 = Stars []" by fact
|
|
|
1005 |
then show "Stars (v # vs) :\<sqsubseteq>val v3"
|
|
|
1006 |
unfolding PosOrd_ex_eq_def using cond2
|
|
|
1007 |
by (simp add: PosOrd_shorterI)
|
|
|
1008 |
qed
|
|
|
1009 |
next
|
|
|
1010 |
case (Posix_NMTIMES3 s1 r v s2 m vs v3)
|
|
|
1011 |
have "s1 \<in> r \<rightarrow> v" "s2 \<in> UPNTIMES r (m - 1) \<rightarrow> Stars vs" by fact+
|
|
|
1012 |
then have as1: "s1 = flat v" "s2 = flat (Stars vs)" by (auto dest: Posix1(2))
|
|
|
1013 |
have IH1: "\<And>v3. v3 \<in> LV r s1 \<Longrightarrow> v :\<sqsubseteq>val v3" by fact
|
|
|
1014 |
have IH2: "\<And>v3. v3 \<in> LV (UPNTIMES r (m - 1)) s2 \<Longrightarrow> Stars vs :\<sqsubseteq>val v3" by fact
|
|
|
1015 |
have cond: "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (UPNTIMES r (m - 1)))" by fact
|
|
|
1016 |
have cond2: "flat v \<noteq> []" by fact
|
|
|
1017 |
have "v3 \<in> LV (NMTIMES r 0 m) (s1 @ s2)" by fact
|
|
|
1018 |
then consider
|
|
|
1019 |
(NonEmpty) v3a vs3 where "v3 = Stars (v3a # vs3)"
|
|
|
1020 |
"\<Turnstile> v3a : r" "\<Turnstile> Stars vs3 : UPNTIMES r (m - 1)"
|
|
|
1021 |
"flats (v3a # vs3) = s1 @ s2"
|
|
|
1022 |
| (Empty) "v3 = Stars []"
|
|
|
1023 |
unfolding LV_def
|
|
|
1024 |
apply(auto)
|
|
|
1025 |
apply(erule Prf.cases)
|
|
|
1026 |
apply(auto)
|
|
|
1027 |
apply(case_tac vs)
|
|
|
1028 |
apply(auto intro: Prf.intros)
|
|
|
1029 |
by (simp add: Prf.intros(7) as1(1) cond2)
|
|
|
1030 |
then show "Stars (v # vs) :\<sqsubseteq>val v3"
|
|
|
1031 |
proof (cases)
|
|
|
1032 |
case (NonEmpty v3a vs3)
|
|
|
1033 |
have "flats (v3a # vs3) = s1 @ s2" using NonEmpty(4) .
|
|
|
1034 |
with cond have "flat v3a \<sqsubseteq>pre s1" using NonEmpty(2,3)
|
|
|
1035 |
unfolding prefix_list_def
|
|
|
1036 |
apply(simp)
|
|
|
1037 |
apply(simp add: append_eq_append_conv2)
|
|
|
1038 |
apply(auto)
|
|
|
1039 |
by (metis L_flat_Prf1 One_nat_def cond flat_Stars)
|
|
|
1040 |
then have "flat v3a \<sqsubset>spre s1 \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)" using NonEmpty(4)
|
|
|
1041 |
by (simp add: sprefix_list_def append_eq_conv_conj)
|
|
|
1042 |
then have q2: "v :\<sqsubset>val v3a \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)"
|
|
|
1043 |
using PosOrd_spreI as1(1) NonEmpty(4) by blast
|
|
|
1044 |
then have "v :\<sqsubset>val v3a \<or> (v3a \<in> LV r s1 \<and> Stars vs3 \<in> LV (UPNTIMES r (m - 1)) s2)"
|
|
|
1045 |
using NonEmpty(2,3) by (auto simp add: LV_def)
|
|
|
1046 |
then have "v :\<sqsubset>val v3a \<or> (v :\<sqsubseteq>val v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)" using IH1 IH2 by blast
|
|
|
1047 |
then have "v :\<sqsubset>val v3a \<or> (v = v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)"
|
|
|
1048 |
unfolding PosOrd_ex_eq_def by auto
|
|
|
1049 |
then have "Stars (v # vs) :\<sqsubseteq>val Stars (v3a # vs3)" using NonEmpty(4) q2 as1
|
|
|
1050 |
unfolding PosOrd_ex_eq_def
|
|
|
1051 |
using PosOrd_StarsI PosOrd_StarsI2 by auto
|
|
|
1052 |
then show "Stars (v # vs) :\<sqsubseteq>val v3" unfolding NonEmpty by blast
|
|
|
1053 |
next
|
|
|
1054 |
case Empty
|
|
|
1055 |
have "v3 = Stars []" by fact
|
|
|
1056 |
then show "Stars (v # vs) :\<sqsubseteq>val v3"
|
|
|
1057 |
unfolding PosOrd_ex_eq_def using cond2
|
|
|
1058 |
by (simp add: PosOrd_shorterI)
|
|
|
1059 |
qed
|
|
|
1060 |
qed
|
|
|
1061 |
|
|
|
1062 |
|
|
|
1063 |
lemma Posix_PosOrd_reverse:
|
|
|
1064 |
assumes "s \<in> r \<rightarrow> v1"
|
|
|
1065 |
shows "\<not>(\<exists>v2 \<in> LV r s. v2 :\<sqsubset>val v1)"
|
|
|
1066 |
using assms
|
|
|
1067 |
by (metis Posix_PosOrd less_irrefl PosOrd_def
|
|
|
1068 |
PosOrd_ex_eq_def PosOrd_ex_def PosOrd_trans)
|
|
|
1069 |
|
|
|
1070 |
lemma PosOrd_Posix:
|
|
|
1071 |
assumes "v1 \<in> LV r s" "\<forall>v\<^sub>2 \<in> LV r s. \<not> v\<^sub>2 :\<sqsubset>val v1"
|
|
|
1072 |
shows "s \<in> r \<rightarrow> v1"
|
|
|
1073 |
proof -
|
|
|
1074 |
have "s \<in> L r" using assms(1) unfolding LV_def
|
|
|
1075 |
using L_flat_Prf1 by blast
|
|
|
1076 |
then obtain vposix where vp: "s \<in> r \<rightarrow> vposix"
|
|
|
1077 |
using lexer_correct_Some by blast
|
|
|
1078 |
with assms(1) have "vposix :\<sqsubseteq>val v1" by (simp add: Posix_PosOrd)
|
|
|
1079 |
then have "vposix = v1 \<or> vposix :\<sqsubset>val v1" unfolding PosOrd_ex_eq2 by auto
|
|
|
1080 |
moreover
|
|
|
1081 |
{ assume "vposix :\<sqsubset>val v1"
|
|
|
1082 |
moreover
|
|
|
1083 |
have "vposix \<in> LV r s" using vp
|
|
|
1084 |
using Posix_LV by blast
|
|
|
1085 |
ultimately have "False" using assms(2) by blast
|
|
|
1086 |
}
|
|
|
1087 |
ultimately show "s \<in> r \<rightarrow> v1" using vp by blast
|
|
|
1088 |
qed
|
|
|
1089 |
|
|
|
1090 |
lemma Least_existence:
|
|
|
1091 |
assumes "LV r s \<noteq> {}"
|
|
|
1092 |
shows " \<exists>vmin \<in> LV r s. \<forall>v \<in> LV r s. vmin :\<sqsubseteq>val v"
|
|
|
1093 |
proof -
|
|
|
1094 |
from assms
|
|
|
1095 |
obtain vposix where "s \<in> r \<rightarrow> vposix"
|
|
|
1096 |
unfolding LV_def
|
|
|
1097 |
using L_flat_Prf1 lexer_correct_Some by blast
|
|
|
1098 |
then have "\<forall>v \<in> LV r s. vposix :\<sqsubseteq>val v"
|
|
|
1099 |
by (simp add: Posix_PosOrd)
|
|
|
1100 |
then show "\<exists>vmin \<in> LV r s. \<forall>v \<in> LV r s. vmin :\<sqsubseteq>val v"
|
|
|
1101 |
using Posix_LV \<open>s \<in> r \<rightarrow> vposix\<close> by blast
|
|
|
1102 |
qed
|
|
|
1103 |
|
|
|
1104 |
lemma Least_existence1:
|
|
|
1105 |
assumes "LV r s \<noteq> {}"
|
|
|
1106 |
shows " \<exists>!vmin \<in> LV r s. \<forall>v \<in> LV r s. vmin :\<sqsubseteq>val v"
|
|
|
1107 |
using Least_existence[OF assms] assms
|
|
|
1108 |
using PosOrdeq_antisym by blast
|
|
|
1109 |
|
|
|
1110 |
|
|
|
1111 |
|
|
|
1112 |
|
|
|
1113 |
|
|
|
1114 |
lemma Least_existence1_pre:
|
|
|
1115 |
assumes "LV r s \<noteq> {}"
|
|
|
1116 |
shows " \<exists>!vmin \<in> LV r s. \<forall>v \<in> (LV r s \<union> {v'. flat v' \<sqsubset>spre s}). vmin :\<sqsubseteq>val v"
|
|
|
1117 |
using Least_existence[OF assms] assms
|
|
|
1118 |
apply -
|
|
|
1119 |
apply(erule bexE)
|
|
|
1120 |
apply(rule_tac a="vmin" in ex1I)
|
|
|
1121 |
apply(auto)[1]
|
|
|
1122 |
apply (metis PosOrd_Posix PosOrd_ex_eq2 PosOrd_spreI PosOrdeq_antisym Posix1(2))
|
|
|
1123 |
apply(auto)[1]
|
|
|
1124 |
apply(simp add: PosOrdeq_antisym)
|
|
|
1125 |
done
|
|
|
1126 |
|
|
|
1127 |
lemma
|
|
|
1128 |
shows "partial_order_on UNIV {(v1, v2). v1 :\<sqsubseteq>val v2}"
|
|
|
1129 |
apply(simp add: partial_order_on_def)
|
|
|
1130 |
apply(simp add: preorder_on_def refl_on_def)
|
|
|
1131 |
apply(simp add: PosOrdeq_refl)
|
|
|
1132 |
apply(auto)
|
|
|
1133 |
apply(rule transI)
|
|
|
1134 |
apply(auto intro: PosOrdeq_trans)[1]
|
|
|
1135 |
apply(rule antisymI)
|
|
|
1136 |
apply(simp add: PosOrdeq_antisym)
|
|
|
1137 |
done
|
|
|
1138 |
|
|
|
1139 |
lemma
|
|
|
1140 |
"wf {(v1, v2). v1 :\<sqsubset>val v2 \<and> v1 \<in> LV r s \<and> v2 \<in> LV r s}"
|
|
|
1141 |
apply(rule finite_acyclic_wf)
|
|
|
1142 |
prefer 2
|
|
|
1143 |
apply(simp add: acyclic_def)
|
|
|
1144 |
apply(induct_tac rule: trancl.induct)
|
|
|
1145 |
apply(auto)[1]
|
|
|
1146 |
prefer 3
|
|
|
1147 |
|
|
|
1148 |
oops
|
|
|
1149 |
|
|
|
1150 |
|
|
|
1151 |
unused_thms
|
|
|
1152 |
|
|
|
1153 |
end |