theory PositionsExt
imports "SpecExt" "LexerExt"
begin
section {* Positions in Values *}
fun
at :: "val \<Rightarrow> nat list \<Rightarrow> val"
where
"at v [] = v"
| "at (Left v) (0#ps)= at v ps"
| "at (Right v) (Suc 0#ps)= at v ps"
| "at (Seq v1 v2) (0#ps)= at v1 ps"
| "at (Seq v1 v2) (Suc 0#ps)= at v2 ps"
| "at (Stars vs) (n#ps)= at (nth vs n) ps"
fun Pos :: "val \<Rightarrow> (nat list) set"
where
"Pos (Void) = {[]}"
| "Pos (Char c) = {[]}"
| "Pos (Left v) = {[]} \<union> {0#ps | ps. ps \<in> Pos v}"
| "Pos (Right v) = {[]} \<union> {1#ps | ps. ps \<in> Pos v}"
| "Pos (Seq v1 v2) = {[]} \<union> {0#ps | ps. ps \<in> Pos v1} \<union> {1#ps | ps. ps \<in> Pos v2}"
| "Pos (Stars []) = {[]}"
| "Pos (Stars (v#vs)) = {[]} \<union> {0#ps | ps. ps \<in> Pos v} \<union> {Suc n#ps | n ps. n#ps \<in> Pos (Stars vs)}"
lemma Pos_stars:
"Pos (Stars vs) = {[]} \<union> (\<Union>n < length vs. {n#ps | ps. ps \<in> Pos (vs ! n)})"
apply(induct vs)
apply(auto simp add: insert_ident less_Suc_eq_0_disj)
done
lemma Pos_empty:
shows "[] \<in> Pos v"
by (induct v rule: Pos.induct)(auto)
abbreviation
"intlen vs \<equiv> int (length vs)"
definition pflat_len :: "val \<Rightarrow> nat list => int"
where
"pflat_len v p \<equiv> (if p \<in> Pos v then intlen (flat (at v p)) else -1)"
lemma pflat_len_simps:
shows "pflat_len (Seq v1 v2) (0#p) = pflat_len v1 p"
and "pflat_len (Seq v1 v2) (Suc 0#p) = pflat_len v2 p"
and "pflat_len (Left v) (0#p) = pflat_len v p"
and "pflat_len (Left v) (Suc 0#p) = -1"
and "pflat_len (Right v) (Suc 0#p) = pflat_len v p"
and "pflat_len (Right v) (0#p) = -1"
and "pflat_len (Stars (v#vs)) (Suc n#p) = pflat_len (Stars vs) (n#p)"
and "pflat_len (Stars (v#vs)) (0#p) = pflat_len v p"
and "pflat_len v [] = intlen (flat v)"
by (auto simp add: pflat_len_def Pos_empty)
lemma pflat_len_Stars_simps:
assumes "n < length vs"
shows "pflat_len (Stars vs) (n#p) = pflat_len (vs!n) p"
using assms
apply(induct vs arbitrary: n p)
apply(auto simp add: less_Suc_eq_0_disj pflat_len_simps)
done
lemma pflat_len_outside:
assumes "p \<notin> Pos v1"
shows "pflat_len v1 p = -1 "
using assms by (simp add: pflat_len_def)
section {* Orderings *}
definition prefix_list:: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" ("_ \<sqsubseteq>pre _" [60,59] 60)
where
"ps1 \<sqsubseteq>pre ps2 \<equiv> \<exists>ps'. ps1 @ps' = ps2"
definition sprefix_list:: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" ("_ \<sqsubset>spre _" [60,59] 60)
where
"ps1 \<sqsubset>spre ps2 \<equiv> ps1 \<sqsubseteq>pre ps2 \<and> ps1 \<noteq> ps2"
inductive lex_list :: "nat list \<Rightarrow> nat list \<Rightarrow> bool" ("_ \<sqsubset>lex _" [60,59] 60)
where
"[] \<sqsubset>lex (p#ps)"
| "ps1 \<sqsubset>lex ps2 \<Longrightarrow> (p#ps1) \<sqsubset>lex (p#ps2)"
| "p1 < p2 \<Longrightarrow> (p1#ps1) \<sqsubset>lex (p2#ps2)"
lemma lex_irrfl:
fixes ps1 ps2 :: "nat list"
assumes "ps1 \<sqsubset>lex ps2"
shows "ps1 \<noteq> ps2"
using assms
by(induct rule: lex_list.induct)(auto)
lemma lex_simps [simp]:
fixes xs ys :: "nat list"
shows "[] \<sqsubset>lex ys \<longleftrightarrow> ys \<noteq> []"
and "xs \<sqsubset>lex [] \<longleftrightarrow> False"
and "(x # xs) \<sqsubset>lex (y # ys) \<longleftrightarrow> (x < y \<or> (x = y \<and> xs \<sqsubset>lex ys))"
by (auto simp add: neq_Nil_conv elim: lex_list.cases intro: lex_list.intros)
lemma lex_trans:
fixes ps1 ps2 ps3 :: "nat list"
assumes "ps1 \<sqsubset>lex ps2" "ps2 \<sqsubset>lex ps3"
shows "ps1 \<sqsubset>lex ps3"
using assms
by (induct arbitrary: ps3 rule: lex_list.induct)
(auto elim: lex_list.cases)
lemma lex_trichotomous:
fixes p q :: "nat list"
shows "p = q \<or> p \<sqsubset>lex q \<or> q \<sqsubset>lex p"
apply(induct p arbitrary: q)
apply(auto elim: lex_list.cases)
apply(case_tac q)
apply(auto)
done
section {* POSIX Ordering of Values According to Okui & Suzuki *}
definition PosOrd:: "val \<Rightarrow> nat list \<Rightarrow> val \<Rightarrow> bool" ("_ \<sqsubset>val _ _" [60, 60, 59] 60)
where
"v1 \<sqsubset>val p v2 \<equiv> pflat_len v1 p > pflat_len v2 p \<and>
(\<forall>q \<in> Pos v1 \<union> Pos v2. q \<sqsubset>lex p \<longrightarrow> pflat_len v1 q = pflat_len v2 q)"
lemma PosOrd_def2:
shows "v1 \<sqsubset>val p v2 \<longleftrightarrow>
pflat_len v1 p > pflat_len v2 p \<and>
(\<forall>q \<in> Pos v1. q \<sqsubset>lex p \<longrightarrow> pflat_len v1 q = pflat_len v2 q) \<and>
(\<forall>q \<in> Pos v2. q \<sqsubset>lex p \<longrightarrow> pflat_len v1 q = pflat_len v2 q)"
unfolding PosOrd_def
apply(auto)
done
definition PosOrd_ex:: "val \<Rightarrow> val \<Rightarrow> bool" ("_ :\<sqsubset>val _" [60, 59] 60)
where
"v1 :\<sqsubset>val v2 \<equiv> \<exists>p. v1 \<sqsubset>val p v2"
definition PosOrd_ex_eq:: "val \<Rightarrow> val \<Rightarrow> bool" ("_ :\<sqsubseteq>val _" [60, 59] 60)
where
"v1 :\<sqsubseteq>val v2 \<equiv> v1 :\<sqsubset>val v2 \<or> v1 = v2"
lemma PosOrd_trans:
assumes "v1 :\<sqsubset>val v2" "v2 :\<sqsubset>val v3"
shows "v1 :\<sqsubset>val v3"
proof -
from assms obtain p p'
where as: "v1 \<sqsubset>val p v2" "v2 \<sqsubset>val p' v3" unfolding PosOrd_ex_def by blast
then have pos: "p \<in> Pos v1" "p' \<in> Pos v2" unfolding PosOrd_def pflat_len_def
by (smt not_int_zless_negative)+
have "p = p' \<or> p \<sqsubset>lex p' \<or> p' \<sqsubset>lex p"
by (rule lex_trichotomous)
moreover
{ assume "p = p'"
with as have "v1 \<sqsubset>val p v3" unfolding PosOrd_def pflat_len_def
by (smt Un_iff)
then have " v1 :\<sqsubset>val v3" unfolding PosOrd_ex_def by blast
}
moreover
{ assume "p \<sqsubset>lex p'"
with as have "v1 \<sqsubset>val p v3" unfolding PosOrd_def pflat_len_def
by (smt Un_iff lex_trans)
then have " v1 :\<sqsubset>val v3" unfolding PosOrd_ex_def by blast
}
moreover
{ assume "p' \<sqsubset>lex p"
with as have "v1 \<sqsubset>val p' v3" unfolding PosOrd_def
by (smt Un_iff lex_trans pflat_len_def)
then have "v1 :\<sqsubset>val v3" unfolding PosOrd_ex_def by blast
}
ultimately show "v1 :\<sqsubset>val v3" by blast
qed
lemma PosOrd_irrefl:
assumes "v :\<sqsubset>val v"
shows "False"
using assms unfolding PosOrd_ex_def PosOrd_def
by auto
lemma PosOrd_assym:
assumes "v1 :\<sqsubset>val v2"
shows "\<not>(v2 :\<sqsubset>val v1)"
using assms
using PosOrd_irrefl PosOrd_trans by blast
text {*
:\<sqsubseteq>val and :\<sqsubset>val are partial orders.
*}
lemma PosOrd_ordering:
shows "ordering (\<lambda>v1 v2. v1 :\<sqsubseteq>val v2) (\<lambda> v1 v2. v1 :\<sqsubset>val v2)"
unfolding ordering_def PosOrd_ex_eq_def
apply(auto)
using PosOrd_irrefl apply blast
using PosOrd_assym apply blast
using PosOrd_trans by blast
lemma PosOrd_order:
shows "class.order (\<lambda>v1 v2. v1 :\<sqsubseteq>val v2) (\<lambda> v1 v2. v1 :\<sqsubset>val v2)"
using PosOrd_ordering
apply(simp add: class.order_def class.preorder_def class.order_axioms_def)
unfolding ordering_def
by blast
lemma PosOrd_ex_eq2:
shows "v1 :\<sqsubset>val v2 \<longleftrightarrow> (v1 :\<sqsubseteq>val v2 \<and> v1 \<noteq> v2)"
using PosOrd_ordering
unfolding ordering_def
by auto
lemma PosOrdeq_trans:
assumes "v1 :\<sqsubseteq>val v2" "v2 :\<sqsubseteq>val v3"
shows "v1 :\<sqsubseteq>val v3"
using assms PosOrd_ordering
unfolding ordering_def
by blast
lemma PosOrdeq_antisym:
assumes "v1 :\<sqsubseteq>val v2" "v2 :\<sqsubseteq>val v1"
shows "v1 = v2"
using assms PosOrd_ordering
unfolding ordering_def
by blast
lemma PosOrdeq_refl:
shows "v :\<sqsubseteq>val v"
unfolding PosOrd_ex_eq_def
by auto
lemma PosOrd_shorterE:
assumes "v1 :\<sqsubset>val v2"
shows "length (flat v2) \<le> length (flat v1)"
using assms unfolding PosOrd_ex_def PosOrd_def
apply(auto)
apply(case_tac p)
apply(simp add: pflat_len_simps)
apply(drule_tac x="[]" in bspec)
apply(simp add: Pos_empty)
apply(simp add: pflat_len_simps)
done
lemma PosOrd_shorterI:
assumes "length (flat v2) < length (flat v1)"
shows "v1 :\<sqsubset>val v2"
unfolding PosOrd_ex_def PosOrd_def pflat_len_def
using assms Pos_empty by force
lemma PosOrd_spreI:
assumes "flat v' \<sqsubset>spre flat v"
shows "v :\<sqsubset>val v'"
using assms
apply(rule_tac PosOrd_shorterI)
unfolding prefix_list_def sprefix_list_def
by (metis append_Nil2 append_eq_conv_conj drop_all le_less_linear)
lemma pflat_len_inside:
assumes "pflat_len v2 p < pflat_len v1 p"
shows "p \<in> Pos v1"
using assms
unfolding pflat_len_def
by (auto split: if_splits)
lemma PosOrd_Left_Right:
assumes "flat v1 = flat v2"
shows "Left v1 :\<sqsubset>val Right v2"
unfolding PosOrd_ex_def
apply(rule_tac x="[0]" in exI)
apply(auto simp add: PosOrd_def pflat_len_simps assms)
done
lemma PosOrd_LeftE:
assumes "Left v1 :\<sqsubset>val Left v2" "flat v1 = flat v2"
shows "v1 :\<sqsubset>val v2"
using assms
unfolding PosOrd_ex_def PosOrd_def2
apply(auto simp add: pflat_len_simps)
apply(frule pflat_len_inside)
apply(auto simp add: pflat_len_simps)
by (metis lex_simps(3) pflat_len_simps(3))
lemma PosOrd_LeftI:
assumes "v1 :\<sqsubset>val v2" "flat v1 = flat v2"
shows "Left v1 :\<sqsubset>val Left v2"
using assms
unfolding PosOrd_ex_def PosOrd_def2
apply(auto simp add: pflat_len_simps)
by (metis less_numeral_extra(3) lex_simps(3) pflat_len_simps(3))
lemma PosOrd_Left_eq:
assumes "flat v1 = flat v2"
shows "Left v1 :\<sqsubset>val Left v2 \<longleftrightarrow> v1 :\<sqsubset>val v2"
using assms PosOrd_LeftE PosOrd_LeftI
by blast
lemma PosOrd_RightE:
assumes "Right v1 :\<sqsubset>val Right v2" "flat v1 = flat v2"
shows "v1 :\<sqsubset>val v2"
using assms
unfolding PosOrd_ex_def PosOrd_def2
apply(auto simp add: pflat_len_simps)
apply(frule pflat_len_inside)
apply(auto simp add: pflat_len_simps)
by (metis lex_simps(3) pflat_len_simps(5))
lemma PosOrd_RightI:
assumes "v1 :\<sqsubset>val v2" "flat v1 = flat v2"
shows "Right v1 :\<sqsubset>val Right v2"
using assms
unfolding PosOrd_ex_def PosOrd_def2
apply(auto simp add: pflat_len_simps)
by (metis lex_simps(3) nat_neq_iff pflat_len_simps(5))
lemma PosOrd_Right_eq:
assumes "flat v1 = flat v2"
shows "Right v1 :\<sqsubset>val Right v2 \<longleftrightarrow> v1 :\<sqsubset>val v2"
using assms PosOrd_RightE PosOrd_RightI
by blast
lemma PosOrd_SeqI1:
assumes "v1 :\<sqsubset>val w1" "flat (Seq v1 v2) = flat (Seq w1 w2)"
shows "Seq v1 v2 :\<sqsubset>val Seq w1 w2"
using assms(1)
apply(subst (asm) PosOrd_ex_def)
apply(subst (asm) PosOrd_def)
apply(clarify)
apply(subst PosOrd_ex_def)
apply(rule_tac x="0#p" in exI)
apply(subst PosOrd_def)
apply(rule conjI)
apply(simp add: pflat_len_simps)
apply(rule ballI)
apply(rule impI)
apply(simp only: Pos.simps)
apply(auto)[1]
apply(simp add: pflat_len_simps)
apply(auto simp add: pflat_len_simps)
using assms(2)
apply(simp)
apply(metis length_append of_nat_add)
done
lemma PosOrd_SeqI2:
assumes "v2 :\<sqsubset>val w2" "flat v2 = flat w2"
shows "Seq v v2 :\<sqsubset>val Seq v w2"
using assms(1)
apply(subst (asm) PosOrd_ex_def)
apply(subst (asm) PosOrd_def)
apply(clarify)
apply(subst PosOrd_ex_def)
apply(rule_tac x="Suc 0#p" in exI)
apply(subst PosOrd_def)
apply(rule conjI)
apply(simp add: pflat_len_simps)
apply(rule ballI)
apply(rule impI)
apply(simp only: Pos.simps)
apply(auto)[1]
apply(simp add: pflat_len_simps)
using assms(2)
apply(simp)
apply(auto simp add: pflat_len_simps)
done
lemma PosOrd_Seq_eq:
assumes "flat v2 = flat w2"
shows "(Seq v v2) :\<sqsubset>val (Seq v w2) \<longleftrightarrow> v2 :\<sqsubset>val w2"
using assms
apply(auto)
prefer 2
apply(simp add: PosOrd_SeqI2)
apply(simp add: PosOrd_ex_def)
apply(auto)
apply(case_tac p)
apply(simp add: PosOrd_def pflat_len_simps)
apply(case_tac a)
apply(simp add: PosOrd_def pflat_len_simps)
apply(clarify)
apply(case_tac nat)
prefer 2
apply(simp add: PosOrd_def pflat_len_simps pflat_len_outside)
apply(rule_tac x="list" in exI)
apply(auto simp add: PosOrd_def2 pflat_len_simps)
apply(smt Collect_disj_eq lex_list.intros(2) mem_Collect_eq pflat_len_simps(2))
apply(smt Collect_disj_eq lex_list.intros(2) mem_Collect_eq pflat_len_simps(2))
done
lemma PosOrd_StarsI:
assumes "v1 :\<sqsubset>val v2" "flats (v1#vs1) = flats (v2#vs2)"
shows "Stars (v1#vs1) :\<sqsubset>val Stars (v2#vs2)"
using assms(1)
apply(subst (asm) PosOrd_ex_def)
apply(subst (asm) PosOrd_def)
apply(clarify)
apply(subst PosOrd_ex_def)
apply(subst PosOrd_def)
apply(rule_tac x="0#p" in exI)
apply(simp add: pflat_len_Stars_simps pflat_len_simps)
using assms(2)
apply(simp add: pflat_len_simps)
apply(auto simp add: pflat_len_Stars_simps pflat_len_simps)
by (metis length_append of_nat_add)
lemma PosOrd_StarsI2:
assumes "Stars vs1 :\<sqsubset>val Stars vs2" "flats vs1 = flats vs2"
shows "Stars (v#vs1) :\<sqsubset>val Stars (v#vs2)"
using assms(1)
apply(subst (asm) PosOrd_ex_def)
apply(subst (asm) PosOrd_def)
apply(clarify)
apply(subst PosOrd_ex_def)
apply(subst PosOrd_def)
apply(case_tac p)
apply(simp add: pflat_len_simps)
apply(rule_tac x="Suc a#list" in exI)
apply(auto simp add: pflat_len_Stars_simps pflat_len_simps assms(2))
done
lemma PosOrd_Stars_appendI:
assumes "Stars vs1 :\<sqsubset>val Stars vs2" "flat (Stars vs1) = flat (Stars vs2)"
shows "Stars (vs @ vs1) :\<sqsubset>val Stars (vs @ vs2)"
using assms
apply(induct vs)
apply(simp)
apply(simp add: PosOrd_StarsI2)
done
lemma PosOrd_eq_Stars_zipI:
assumes "\<forall>(v1, v2) \<in> set (zip vs1 vs2). v1 :\<sqsubseteq>val v2"
"length vs1 = length vs2" "flats vs1 = flats vs2"
shows "Stars vs1 :\<sqsubseteq>val Stars vs2"
using assms
apply(induct vs1 arbitrary: vs2)
apply(case_tac vs2)
apply(simp add: PosOrd_ex_eq_def)
apply(simp)
apply(case_tac vs2)
apply(simp)
apply(simp)
apply(auto)
apply(subst (asm) (2)PosOrd_ex_eq_def)
apply(auto)
apply(subst PosOrd_ex_eq_def)
apply(rule disjI1)
apply(rule PosOrd_StarsI)
apply(simp)
apply(simp)
using PosOrd_StarsI2 PosOrd_ex_eq_def by fastforce
lemma PosOrd_StarsE2:
assumes "Stars (v # vs1) :\<sqsubset>val Stars (v # vs2)"
shows "Stars vs1 :\<sqsubset>val Stars vs2"
using assms
apply(subst (asm) PosOrd_ex_def)
apply(erule exE)
apply(case_tac p)
apply(simp)
apply(simp add: PosOrd_def pflat_len_simps)
apply(subst PosOrd_ex_def)
apply(rule_tac x="[]" in exI)
apply(simp add: PosOrd_def pflat_len_simps Pos_empty)
apply(simp)
apply(case_tac a)
apply(clarify)
apply(auto simp add: pflat_len_simps PosOrd_def pflat_len_def split: if_splits)[1]
apply(clarify)
apply(simp add: PosOrd_ex_def)
apply(rule_tac x="nat#list" in exI)
apply(auto simp add: PosOrd_def pflat_len_simps)[1]
apply(case_tac q)
apply(simp add: PosOrd_def pflat_len_simps)
apply(clarify)
apply(drule_tac x="Suc a # lista" in bspec)
apply(simp)
apply(auto simp add: PosOrd_def pflat_len_simps)[1]
apply(case_tac q)
apply(simp add: PosOrd_def pflat_len_simps)
apply(clarify)
apply(drule_tac x="Suc a # lista" in bspec)
apply(simp)
apply(auto simp add: PosOrd_def pflat_len_simps)[1]
done
lemma PosOrd_Stars_appendE:
assumes "Stars (vs @ vs1) :\<sqsubset>val Stars (vs @ vs2)"
shows "Stars vs1 :\<sqsubset>val Stars vs2"
using assms
apply(induct vs)
apply(simp)
apply(simp add: PosOrd_StarsE2)
done
lemma PosOrd_Stars_append_eq:
assumes "flats vs1 = flats vs2"
shows "Stars (vs @ vs1) :\<sqsubset>val Stars (vs @ vs2) \<longleftrightarrow> Stars vs1 :\<sqsubset>val Stars vs2"
using assms
apply(rule_tac iffI)
apply(erule PosOrd_Stars_appendE)
apply(rule PosOrd_Stars_appendI)
apply(auto)
done
lemma PosOrd_almost_trichotomous:
shows "v1 :\<sqsubset>val v2 \<or> v2 :\<sqsubset>val v1 \<or> (length (flat v1) = length (flat v2))"
apply(auto simp add: PosOrd_ex_def)
apply(auto simp add: PosOrd_def)
apply(rule_tac x="[]" in exI)
apply(auto simp add: Pos_empty pflat_len_simps)
apply(drule_tac x="[]" in spec)
apply(auto simp add: Pos_empty pflat_len_simps)
done
section {* The Posix Value is smaller than any other Value *}
lemma Posix_PosOrd:
assumes "s \<in> r \<rightarrow> v1" "v2 \<in> LV r s"
shows "v1 :\<sqsubseteq>val v2"
using assms
proof (induct arbitrary: v2 rule: Posix.induct)
case (Posix_ONE v)
have "v \<in> LV ONE []" by fact
then have "v = Void"
by (simp add: LV_simps)
then show "Void :\<sqsubseteq>val v"
by (simp add: PosOrd_ex_eq_def)
next
case (Posix_CHAR c v)
have "v \<in> LV (CHAR c) [c]" by fact
then have "v = Char c"
by (simp add: LV_simps)
then show "Char c :\<sqsubseteq>val v"
by (simp add: PosOrd_ex_eq_def)
next
case (Posix_ALT1 s r1 v r2 v2)
have as1: "s \<in> r1 \<rightarrow> v" by fact
have IH: "\<And>v2. v2 \<in> LV r1 s \<Longrightarrow> v :\<sqsubseteq>val v2" by fact
have "v2 \<in> LV (ALT r1 r2) s" by fact
then have "\<Turnstile> v2 : ALT r1 r2" "flat v2 = s"
by(auto simp add: LV_def prefix_list_def)
then consider
(Left) v3 where "v2 = Left v3" "\<Turnstile> v3 : r1" "flat v3 = s"
| (Right) v3 where "v2 = Right v3" "\<Turnstile> v3 : r2" "flat v3 = s"
by (auto elim: Prf.cases)
then show "Left v :\<sqsubseteq>val v2"
proof(cases)
case (Left v3)
have "v3 \<in> LV r1 s" using Left(2,3)
by (auto simp add: LV_def prefix_list_def)
with IH have "v :\<sqsubseteq>val v3" by simp
moreover
have "flat v3 = flat v" using as1 Left(3)
by (simp add: Posix1(2))
ultimately have "Left v :\<sqsubseteq>val Left v3"
by (simp add: PosOrd_ex_eq_def PosOrd_Left_eq)
then show "Left v :\<sqsubseteq>val v2" unfolding Left .
next
case (Right v3)
have "flat v3 = flat v" using as1 Right(3)
by (simp add: Posix1(2))
then have "Left v :\<sqsubseteq>val Right v3"
unfolding PosOrd_ex_eq_def
by (simp add: PosOrd_Left_Right)
then show "Left v :\<sqsubseteq>val v2" unfolding Right .
qed
next
case (Posix_ALT2 s r2 v r1 v2)
have as1: "s \<in> r2 \<rightarrow> v" by fact
have as2: "s \<notin> L r1" by fact
have IH: "\<And>v2. v2 \<in> LV r2 s \<Longrightarrow> v :\<sqsubseteq>val v2" by fact
have "v2 \<in> LV (ALT r1 r2) s" by fact
then have "\<Turnstile> v2 : ALT r1 r2" "flat v2 = s"
by(auto simp add: LV_def prefix_list_def)
then consider
(Left) v3 where "v2 = Left v3" "\<Turnstile> v3 : r1" "flat v3 = s"
| (Right) v3 where "v2 = Right v3" "\<Turnstile> v3 : r2" "flat v3 = s"
by (auto elim: Prf.cases)
then show "Right v :\<sqsubseteq>val v2"
proof (cases)
case (Right v3)
have "v3 \<in> LV r2 s" using Right(2,3)
by (auto simp add: LV_def prefix_list_def)
with IH have "v :\<sqsubseteq>val v3" by simp
moreover
have "flat v3 = flat v" using as1 Right(3)
by (simp add: Posix1(2))
ultimately have "Right v :\<sqsubseteq>val Right v3"
by (auto simp add: PosOrd_ex_eq_def PosOrd_RightI)
then show "Right v :\<sqsubseteq>val v2" unfolding Right .
next
case (Left v3)
have "v3 \<in> LV r1 s" using Left(2,3) as2
by (auto simp add: LV_def prefix_list_def)
then have "flat v3 = flat v \<and> \<Turnstile> v3 : r1" using as1 Left(3)
by (simp add: Posix1(2) LV_def)
then have "False" using as1 as2 Left
by (auto simp add: Posix1(2) L_flat_Prf1)
then show "Right v :\<sqsubseteq>val v2" by simp
qed
next
case (Posix_SEQ s1 r1 v1 s2 r2 v2 v3)
have "s1 \<in> r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2" by fact+
then have as1: "s1 = flat v1" "s2 = flat v2" by (simp_all add: Posix1(2))
have IH1: "\<And>v3. v3 \<in> LV r1 s1 \<Longrightarrow> v1 :\<sqsubseteq>val v3" by fact
have IH2: "\<And>v3. v3 \<in> LV r2 s2 \<Longrightarrow> v2 :\<sqsubseteq>val v3" by fact
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
have "v3 \<in> LV (SEQ r1 r2) (s1 @ s2)" by fact
then obtain v3a v3b where eqs:
"v3 = Seq v3a v3b" "\<Turnstile> v3a : r1" "\<Turnstile> v3b : r2"
"flat v3a @ flat v3b = s1 @ s2"
by (force simp add: prefix_list_def LV_def elim: Prf.cases)
with cond have "flat v3a \<sqsubseteq>pre s1" unfolding prefix_list_def
by (smt L_flat_Prf1 append_eq_append_conv2 append_self_conv)
then have "flat v3a \<sqsubset>spre s1 \<or> (flat v3a = s1 \<and> flat v3b = s2)" using eqs
by (simp add: sprefix_list_def append_eq_conv_conj)
then have q2: "v1 :\<sqsubset>val v3a \<or> (flat v3a = s1 \<and> flat v3b = s2)"
using PosOrd_spreI as1(1) eqs by blast
then have "v1 :\<sqsubset>val v3a \<or> (v3a \<in> LV r1 s1 \<and> v3b \<in> LV r2 s2)" using eqs(2,3)
by (auto simp add: LV_def)
then have "v1 :\<sqsubset>val v3a \<or> (v1 :\<sqsubseteq>val v3a \<and> v2 :\<sqsubseteq>val v3b)" using IH1 IH2 by blast
then have "Seq v1 v2 :\<sqsubseteq>val Seq v3a v3b" using eqs q2 as1
unfolding PosOrd_ex_eq_def by (auto simp add: PosOrd_SeqI1 PosOrd_Seq_eq)
then show "Seq v1 v2 :\<sqsubseteq>val v3" unfolding eqs by blast
next
case (Posix_STAR1 s1 r v s2 vs v3)
have "s1 \<in> r \<rightarrow> v" "s2 \<in> STAR r \<rightarrow> Stars vs" by fact+
then have as1: "s1 = flat v" "s2 = flat (Stars vs)" by (auto dest: Posix1(2))
have IH1: "\<And>v3. v3 \<in> LV r s1 \<Longrightarrow> v :\<sqsubseteq>val v3" by fact
have IH2: "\<And>v3. v3 \<in> LV (STAR r) s2 \<Longrightarrow> Stars vs :\<sqsubseteq>val v3" by fact
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
have cond2: "flat v \<noteq> []" by fact
have "v3 \<in> LV (STAR r) (s1 @ s2)" by fact
then consider
(NonEmpty) v3a vs3 where "v3 = Stars (v3a # vs3)"
"\<Turnstile> v3a : r" "\<Turnstile> Stars vs3 : STAR r"
"flat (Stars (v3a # vs3)) = s1 @ s2"
| (Empty) "v3 = Stars []"
unfolding LV_def
apply(auto)
apply(erule Prf.cases)
apply(auto)
apply(case_tac vs)
apply(auto intro: Prf.intros)
done
then show "Stars (v # vs) :\<sqsubseteq>val v3"
proof (cases)
case (NonEmpty v3a vs3)
have "flat (Stars (v3a # vs3)) = s1 @ s2" using NonEmpty(4) .
with cond have "flat v3a \<sqsubseteq>pre s1" using NonEmpty(2,3)
unfolding prefix_list_def
by (smt L_flat_Prf1 append_Nil2 append_eq_append_conv2 flat.simps(7))
then have "flat v3a \<sqsubset>spre s1 \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)" using NonEmpty(4)
by (simp add: sprefix_list_def append_eq_conv_conj)
then have q2: "v :\<sqsubset>val v3a \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)"
using PosOrd_spreI as1(1) NonEmpty(4) by blast
then have "v :\<sqsubset>val v3a \<or> (v3a \<in> LV r s1 \<and> Stars vs3 \<in> LV (STAR r) s2)"
using NonEmpty(2,3) by (auto simp add: LV_def)
then have "v :\<sqsubset>val v3a \<or> (v :\<sqsubseteq>val v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)" using IH1 IH2 by blast
then have "v :\<sqsubset>val v3a \<or> (v = v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)"
unfolding PosOrd_ex_eq_def by auto
then have "Stars (v # vs) :\<sqsubseteq>val Stars (v3a # vs3)" using NonEmpty(4) q2 as1
unfolding PosOrd_ex_eq_def
using PosOrd_StarsI PosOrd_StarsI2 by auto
then show "Stars (v # vs) :\<sqsubseteq>val v3" unfolding NonEmpty by blast
next
case Empty
have "v3 = Stars []" by fact
then show "Stars (v # vs) :\<sqsubseteq>val v3"
unfolding PosOrd_ex_eq_def using cond2
by (simp add: PosOrd_shorterI)
qed
next
case (Posix_STAR2 r v2)
have "v2 \<in> LV (STAR r) []" by fact
then have "v2 = Stars []"
unfolding LV_def by (auto elim: Prf.cases)
then show "Stars [] :\<sqsubseteq>val v2"
by (simp add: PosOrd_ex_eq_def)
next
case (Posix_NTIMES1 s1 r v s2 n vs v3)
have "s1 \<in> r \<rightarrow> v" "s2 \<in> NTIMES r (n - 1) \<rightarrow> Stars vs" by fact+
then have as1: "s1 = flat v" "s2 = flats vs" by (auto dest: Posix1(2))
have IH1: "\<And>v3. v3 \<in> LV r s1 \<Longrightarrow> v :\<sqsubseteq>val v3" by fact
have IH2: "\<And>v3. v3 \<in> LV (NTIMES r (n - 1)) s2 \<Longrightarrow> Stars vs :\<sqsubseteq>val v3" by fact
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
have cond2: "flat v \<noteq> []" by fact
have "v3 \<in> LV (NTIMES r n) (s1 @ s2)" by fact
then consider
(NonEmpty) v3a vs3 where "v3 = Stars (v3a # vs3)"
"\<Turnstile> v3a : r" "\<Turnstile> Stars vs3 : NTIMES r (n - 1)"
"flats (v3a # vs3) = s1 @ s2"
| (Empty) "v3 = Stars []"
unfolding LV_def
apply(auto)
apply(erule Prf.cases)
apply(auto)
apply(case_tac vs1)
apply(auto intro: Prf.intros)
apply(case_tac vs2)
apply(auto intro: Prf.intros)
apply (simp add: as1(1) cond2 flats_empty)
by (simp add: Prf.intros(8))
then show "Stars (v # vs) :\<sqsubseteq>val v3"
proof (cases)
case (NonEmpty v3a vs3)
have "flats (v3a # vs3) = s1 @ s2" using NonEmpty(4) .
with cond have "flat v3a \<sqsubseteq>pre s1" using NonEmpty(2,3)
unfolding prefix_list_def
by (smt L_flat_Prf1 append_Nil2 append_eq_append_conv2 flat.simps(7) flat_Stars)
then have "flat v3a \<sqsubset>spre s1 \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)" using NonEmpty(4)
by (simp add: sprefix_list_def append_eq_conv_conj)
then have q2: "v :\<sqsubset>val v3a \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)"
using PosOrd_spreI as1(1) NonEmpty(4) by blast
then have "v :\<sqsubset>val v3a \<or> (v3a \<in> LV r s1 \<and> Stars vs3 \<in> LV (NTIMES r (n - 1)) s2)"
using NonEmpty(2,3) by (auto simp add: LV_def)
then have "v :\<sqsubset>val v3a \<or> (v :\<sqsubseteq>val v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)" using IH1 IH2 by blast
then have "v :\<sqsubset>val v3a \<or> (v = v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)"
unfolding PosOrd_ex_eq_def by auto
then have "Stars (v # vs) :\<sqsubseteq>val Stars (v3a # vs3)" using NonEmpty(4) q2 as1
unfolding PosOrd_ex_eq_def
using PosOrd_StarsI PosOrd_StarsI2 by auto
then show "Stars (v # vs) :\<sqsubseteq>val v3" unfolding NonEmpty by blast
next
case Empty
have "v3 = Stars []" by fact
then show "Stars (v # vs) :\<sqsubseteq>val v3"
unfolding PosOrd_ex_eq_def using cond2
by (simp add: PosOrd_shorterI)
qed
next
case (Posix_NTIMES2 vs r n v2)
then show "Stars vs :\<sqsubseteq>val v2"
apply(simp add: LV_def)
apply(auto)
apply(erule Prf_elims)
apply(auto)
apply(rule PosOrd_eq_Stars_zipI)
prefer 2
apply(simp)
prefer 2
apply (metis Posix1(2) flats_empty)
apply(auto)
by (meson in_set_zipE)
next
case (Posix_UPNTIMES2 r n v2)
then show "Stars [] :\<sqsubseteq>val v2"
apply(simp add: LV_def)
apply(auto)
apply(erule Prf_elims)
apply(auto)
unfolding PosOrd_ex_eq_def by simp
next
case (Posix_UPNTIMES1 s1 r v s2 n vs v3)
have "s1 \<in> r \<rightarrow> v" "s2 \<in> UPNTIMES r (n - 1) \<rightarrow> Stars vs" by fact+
then have as1: "s1 = flat v" "s2 = flat (Stars vs)" by (auto dest: Posix1(2))
have IH1: "\<And>v3. v3 \<in> LV r s1 \<Longrightarrow> v :\<sqsubseteq>val v3" by fact
have IH2: "\<And>v3. v3 \<in> LV (UPNTIMES r (n - 1)) s2 \<Longrightarrow> Stars vs :\<sqsubseteq>val v3" by fact
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
have cond2: "flat v \<noteq> []" by fact
have "v3 \<in> LV (UPNTIMES r n) (s1 @ s2)" by fact
then consider
(NonEmpty) v3a vs3 where "v3 = Stars (v3a # vs3)"
"\<Turnstile> v3a : r" "\<Turnstile> Stars vs3 : UPNTIMES r (n - 1)"
"flats (v3a # vs3) = s1 @ s2"
| (Empty) "v3 = Stars []"
unfolding LV_def
apply(auto)
apply(erule Prf.cases)
apply(auto)
apply(case_tac vs)
apply(auto intro: Prf.intros)
by (simp add: Prf.intros(7) as1(1) cond2)
then show "Stars (v # vs) :\<sqsubseteq>val v3"
proof (cases)
case (NonEmpty v3a vs3)
have "flats (v3a # vs3) = s1 @ s2" using NonEmpty(4) .
with cond have "flat v3a \<sqsubseteq>pre s1" using NonEmpty(2,3)
unfolding prefix_list_def
apply(simp)
apply(simp add: append_eq_append_conv2)
apply(auto)
by (metis L_flat_Prf1 One_nat_def cond flat_Stars)
then have "flat v3a \<sqsubset>spre s1 \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)" using NonEmpty(4)
by (simp add: sprefix_list_def append_eq_conv_conj)
then have q2: "v :\<sqsubset>val v3a \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)"
using PosOrd_spreI as1(1) NonEmpty(4) by blast
then have "v :\<sqsubset>val v3a \<or> (v3a \<in> LV r s1 \<and> Stars vs3 \<in> LV (UPNTIMES r (n - 1)) s2)"
using NonEmpty(2,3) by (auto simp add: LV_def)
then have "v :\<sqsubset>val v3a \<or> (v :\<sqsubseteq>val v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)" using IH1 IH2 by blast
then have "v :\<sqsubset>val v3a \<or> (v = v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)"
unfolding PosOrd_ex_eq_def by auto
then have "Stars (v # vs) :\<sqsubseteq>val Stars (v3a # vs3)" using NonEmpty(4) q2 as1
unfolding PosOrd_ex_eq_def
using PosOrd_StarsI PosOrd_StarsI2 by auto
then show "Stars (v # vs) :\<sqsubseteq>val v3" unfolding NonEmpty by blast
next
case Empty
have "v3 = Stars []" by fact
then show "Stars (v # vs) :\<sqsubseteq>val v3"
unfolding PosOrd_ex_eq_def using cond2
by (simp add: PosOrd_shorterI)
qed
next
case (Posix_FROMNTIMES2 vs r n v2)
then show "Stars vs :\<sqsubseteq>val v2"
apply(simp add: LV_def)
apply(auto)
apply(erule Prf_elims)
apply(auto)
apply(rule PosOrd_eq_Stars_zipI)
prefer 2
apply(simp)
prefer 2
apply (metis Posix1(2) flats_empty)
apply(auto)
by (meson in_set_zipE)
next
case (Posix_FROMNTIMES1 s1 r v s2 n vs v3)
have "s1 \<in> r \<rightarrow> v" "s2 \<in> FROMNTIMES r (n - 1) \<rightarrow> Stars vs" by fact+
then have as1: "s1 = flat v" "s2 = flats vs" by (auto dest: Posix1(2))
have IH1: "\<And>v3. v3 \<in> LV r s1 \<Longrightarrow> v :\<sqsubseteq>val v3" by fact
have IH2: "\<And>v3. v3 \<in> LV (FROMNTIMES r (n - 1)) s2 \<Longrightarrow> Stars vs :\<sqsubseteq>val v3" by fact
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
have cond2: "flat v \<noteq> []" by fact
have "v3 \<in> LV (FROMNTIMES r n) (s1 @ s2)" by fact
then consider
(NonEmpty) v3a vs3 where "v3 = Stars (v3a # vs3)"
"\<Turnstile> v3a : r" "\<Turnstile> Stars vs3 : FROMNTIMES r (n - 1)"
"flats (v3a # vs3) = s1 @ s2"
| (Empty) "v3 = Stars []"
unfolding LV_def
apply(auto)
apply(erule Prf.cases)
apply(auto)
apply(case_tac vs1)
apply(auto intro: Prf.intros)
apply(case_tac vs2)
apply(auto intro: Prf.intros)
apply (simp add: as1(1) cond2 flats_empty)
apply (simp add: Prf.intros)
apply(case_tac vs)
apply(auto)
using Posix_FROMNTIMES1.hyps(6) Prf.intros(10) by auto
then show "Stars (v # vs) :\<sqsubseteq>val v3"
proof (cases)
case (NonEmpty v3a vs3)
have "flats (v3a # vs3) = s1 @ s2" using NonEmpty(4) .
with cond have "flat v3a \<sqsubseteq>pre s1" using NonEmpty(2,3)
unfolding prefix_list_def
by (smt L_flat_Prf1 append_Nil2 append_eq_append_conv2 flat.simps(7) flat_Stars)
then have "flat v3a \<sqsubset>spre s1 \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)" using NonEmpty(4)
by (simp add: sprefix_list_def append_eq_conv_conj)
then have q2: "v :\<sqsubset>val v3a \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)"
using PosOrd_spreI as1(1) NonEmpty(4) by blast
then have "v :\<sqsubset>val v3a \<or> (v3a \<in> LV r s1 \<and> Stars vs3 \<in> LV (FROMNTIMES r (n - 1)) s2)"
using NonEmpty(2,3) by (auto simp add: LV_def)
then have "v :\<sqsubset>val v3a \<or> (v :\<sqsubseteq>val v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)" using IH1 IH2 by blast
then have "v :\<sqsubset>val v3a \<or> (v = v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)"
unfolding PosOrd_ex_eq_def by auto
then have "Stars (v # vs) :\<sqsubseteq>val Stars (v3a # vs3)" using NonEmpty(4) q2 as1
unfolding PosOrd_ex_eq_def
using PosOrd_StarsI PosOrd_StarsI2 by auto
then show "Stars (v # vs) :\<sqsubseteq>val v3" unfolding NonEmpty by blast
next
case Empty
have "v3 = Stars []" by fact
then show "Stars (v # vs) :\<sqsubseteq>val v3"
unfolding PosOrd_ex_eq_def using cond2
by (simp add: PosOrd_shorterI)
qed
next
case (Posix_FROMNTIMES3 s1 r v s2 vs v3)
have "s1 \<in> r \<rightarrow> v" "s2 \<in> STAR r \<rightarrow> Stars vs" by fact+
then have as1: "s1 = flat v" "s2 = flat (Stars vs)" by (auto dest: Posix1(2))
have IH1: "\<And>v3. v3 \<in> LV r s1 \<Longrightarrow> v :\<sqsubseteq>val v3" by fact
have IH2: "\<And>v3. v3 \<in> LV (STAR r) s2 \<Longrightarrow> Stars vs :\<sqsubseteq>val v3" by fact
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
have cond2: "flat v \<noteq> []" by fact
have "v3 \<in> LV (FROMNTIMES r 0) (s1 @ s2)" by fact
then consider
(NonEmpty) v3a vs3 where "v3 = Stars (v3a # vs3)"
"\<Turnstile> v3a : r" "\<Turnstile> Stars vs3 : STAR r"
"flat (Stars (v3a # vs3)) = s1 @ s2"
| (Empty) "v3 = Stars []"
unfolding LV_def
apply(auto)
apply(erule Prf.cases)
apply(auto)
apply(case_tac vs)
apply(auto intro: Prf.intros)
done
then show "Stars (v # vs) :\<sqsubseteq>val v3"
proof (cases)
case (NonEmpty v3a vs3)
have "flat (Stars (v3a # vs3)) = s1 @ s2" using NonEmpty(4) .
with cond have "flat v3a \<sqsubseteq>pre s1" using NonEmpty(2,3)
unfolding prefix_list_def
by (smt L_flat_Prf1 append_Nil2 append_eq_append_conv2 flat.simps(7))
then have "flat v3a \<sqsubset>spre s1 \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)" using NonEmpty(4)
by (simp add: sprefix_list_def append_eq_conv_conj)
then have q2: "v :\<sqsubset>val v3a \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)"
using PosOrd_spreI as1(1) NonEmpty(4) by blast
then have "v :\<sqsubset>val v3a \<or> (v3a \<in> LV r s1 \<and> Stars vs3 \<in> LV (STAR r) s2)"
using NonEmpty(2,3) by (auto simp add: LV_def)
then have "v :\<sqsubset>val v3a \<or> (v :\<sqsubseteq>val v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)" using IH1 IH2 by blast
then have "v :\<sqsubset>val v3a \<or> (v = v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)"
unfolding PosOrd_ex_eq_def by auto
then have "Stars (v # vs) :\<sqsubseteq>val Stars (v3a # vs3)" using NonEmpty(4) q2 as1
unfolding PosOrd_ex_eq_def
using PosOrd_StarsI PosOrd_StarsI2 by auto
then show "Stars (v # vs) :\<sqsubseteq>val v3" unfolding NonEmpty by blast
next
case Empty
have "v3 = Stars []" by fact
then show "Stars (v # vs) :\<sqsubseteq>val v3"
unfolding PosOrd_ex_eq_def using cond2
by (simp add: PosOrd_shorterI)
qed
next
case (Posix_NMTIMES2 vs r n m v2)
then show "Stars vs :\<sqsubseteq>val v2" sorry
next
case (Posix_NMTIMES1 s1 r v s2 n m vs v2)
then show "Stars (v # vs) :\<sqsubseteq>val v2" sorry
qed
lemma Posix_PosOrd_reverse:
assumes "s \<in> r \<rightarrow> v1"
shows "\<not>(\<exists>v2 \<in> LV r s. v2 :\<sqsubset>val v1)"
using assms
by (metis Posix_PosOrd less_irrefl PosOrd_def
PosOrd_ex_eq_def PosOrd_ex_def PosOrd_trans)
lemma PosOrd_Posix:
assumes "v1 \<in> LV r s" "\<forall>v\<^sub>2 \<in> LV r s. \<not> v\<^sub>2 :\<sqsubset>val v1"
shows "s \<in> r \<rightarrow> v1"
proof -
have "s \<in> L r" using assms(1) unfolding LV_def
using L_flat_Prf1 by blast
then obtain vposix where vp: "s \<in> r \<rightarrow> vposix"
using lexer_correct_Some by blast
with assms(1) have "vposix :\<sqsubseteq>val v1" by (simp add: Posix_PosOrd)
then have "vposix = v1 \<or> vposix :\<sqsubset>val v1" unfolding PosOrd_ex_eq2 by auto
moreover
{ assume "vposix :\<sqsubset>val v1"
moreover
have "vposix \<in> LV r s" using vp
using Posix_LV by blast
ultimately have "False" using assms(2) by blast
}
ultimately show "s \<in> r \<rightarrow> v1" using vp by blast
qed
lemma Least_existence:
assumes "LV r s \<noteq> {}"
shows " \<exists>vmin \<in> LV r s. \<forall>v \<in> LV r s. vmin :\<sqsubseteq>val v"
proof -
from assms
obtain vposix where "s \<in> r \<rightarrow> vposix"
unfolding LV_def
using L_flat_Prf1 lexer_correct_Some by blast
then have "\<forall>v \<in> LV r s. vposix :\<sqsubseteq>val v"
by (simp add: Posix_PosOrd)
then show "\<exists>vmin \<in> LV r s. \<forall>v \<in> LV r s. vmin :\<sqsubseteq>val v"
using Posix_LV \<open>s \<in> r \<rightarrow> vposix\<close> by blast
qed
lemma Least_existence1:
assumes "LV r s \<noteq> {}"
shows " \<exists>!vmin \<in> LV r s. \<forall>v \<in> LV r s. vmin :\<sqsubseteq>val v"
using Least_existence[OF assms] assms
using PosOrdeq_antisym by blast
lemma Least_existence1_pre:
assumes "LV r s \<noteq> {}"
shows " \<exists>!vmin \<in> LV r s. \<forall>v \<in> (LV r s \<union> {v'. flat v' \<sqsubset>spre s}). vmin :\<sqsubseteq>val v"
using Least_existence[OF assms] assms
apply -
apply(erule bexE)
apply(rule_tac a="vmin" in ex1I)
apply(auto)[1]
apply (metis PosOrd_Posix PosOrd_ex_eq2 PosOrd_spreI PosOrdeq_antisym Posix1(2))
apply(auto)[1]
apply(simp add: PosOrdeq_antisym)
done
lemma
shows "partial_order_on UNIV {(v1, v2). v1 :\<sqsubseteq>val v2}"
apply(simp add: partial_order_on_def)
apply(simp add: preorder_on_def refl_on_def)
apply(simp add: PosOrdeq_refl)
apply(auto)
apply(rule transI)
apply(auto intro: PosOrdeq_trans)[1]
apply(rule antisymI)
apply(simp add: PosOrdeq_antisym)
done
lemma
"wf {(v1, v2). v1 :\<sqsubset>val v2 \<and> v1 \<in> LV r s \<and> v2 \<in> LV r s}"
apply(rule finite_acyclic_wf)
prefer 2
apply(simp add: acyclic_def)
apply(induct_tac rule: trancl.induct)
apply(auto)[1]
oops
unused_thms
end