--- a/thys/Positions.thy Tue Jul 04 15:59:31 2017 +0100
+++ b/thys/Positions.thy Tue Jul 04 16:42:49 2017 +0100
@@ -868,165 +868,146 @@
section {* The Posix Value is smaller than any other Value *}
+
lemma Posix_PosOrd:
- assumes "s \<in> r \<rightarrow> v1" "v2 \<in> CPTpre r s"
+ assumes "s \<in> r \<rightarrow> v1" "v2 \<in> CPT r s"
shows "v1 :\<sqsubseteq>val v2"
using assms
proof (induct arbitrary: v2 rule: Posix.induct)
case (Posix_ONE v)
- have "v \<in> CPTpre ONE []" by fact
+ have "v \<in> CPT ONE []" by fact
+ then have "v = Void"
+ by (simp add: CPT_simps)
then show "Void :\<sqsubseteq>val v"
- by (auto simp add: CPTpre_def PosOrd_ex1_def elim: CPrf.cases)
+ by (simp add: PosOrd_ex1_def)
next
case (Posix_CHAR c v)
- have "v \<in> CPTpre (CHAR c) [c]" by fact
+ have "v \<in> CPT (CHAR c) [c]" by fact
+ then have "v = Char c"
+ by (simp add: CPT_simps)
then show "Char c :\<sqsubseteq>val v"
- by (auto simp add: CPTpre_def PosOrd_ex1_def elim: CPrf.cases)
+ by (simp add: PosOrd_ex1_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> CPTpre r1 s \<Longrightarrow> v :\<sqsubseteq>val v2" by fact
- have "v2 \<in> CPTpre (ALT r1 r2) s" by fact
- then have "\<Turnstile> v2 : ALT r1 r2" "flat v2 \<sqsubseteq>pre s"
- by(auto simp add: CPTpre_def prefix_list_def)
+ have IH: "\<And>v2. v2 \<in> CPT r1 s \<Longrightarrow> v :\<sqsubseteq>val v2" by fact
+ have "v2 \<in> CPT (ALT r1 r2) s" by fact
+ then have "\<Turnstile> v2 : ALT r1 r2" "flat v2 = s"
+ by(auto simp add: CPT_def prefix_list_def)
then consider
- (Left) v3 where "v2 = Left v3" "\<Turnstile> v3 : r1" "flat v3 \<sqsubseteq>pre s"
- | (Right) v3 where "v2 = Right v3" "\<Turnstile> v3 : r2" "flat v3 \<sqsubseteq>pre s"
+ (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: CPrf.cases)
then show "Left v :\<sqsubseteq>val v2"
proof(cases)
case (Left v3)
- have "v3 \<in> CPTpre r1 s" using Left(2,3)
- by (auto simp add: CPTpre_def prefix_list_def)
+ have "v3 \<in> CPT r1 s" using Left(2,3)
+ by (auto simp add: CPT_def prefix_list_def)
with IH have "v :\<sqsubseteq>val v3" by simp
moreover
- have "flat v3 \<sqsubset>spre flat v \<or> flat v3 = flat v" using as1 Left(3)
- by (simp add: Posix1(2) sprefix_list_def)
+ have "flat v3 = flat v" using as1 Left(3)
+ by (simp add: Posix1(2))
ultimately have "Left v :\<sqsubseteq>val Left v3"
- by (auto simp add: PosOrd_ex1_def PosOrd_LeftI PosOrd_spreI)
+ by (auto simp add: PosOrd_ex1_def PosOrd_LeftI)
then show "Left v :\<sqsubseteq>val v2" unfolding Left .
next
case (Right v3)
- have "flat v3 \<sqsubset>spre flat v \<or> flat v3 = flat v" using as1 Right(3)
- by (simp add: Posix1(2) sprefix_list_def)
+ have "flat v3 = flat v" using as1 Right(3)
+ by (simp add: Posix1(2))
then have "Left v :\<sqsubseteq>val Right v3" using Right(3) as1
- by (auto simp add: PosOrd_ex1_def PosOrd_Left_Right PosOrd_spreI)
+ by (auto simp add: PosOrd_ex1_def 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> CPTpre r2 s \<Longrightarrow> v :\<sqsubseteq>val v2" by fact
- have "v2 \<in> CPTpre (ALT r1 r2) s" by fact
- then have "\<Turnstile> v2 : ALT r1 r2" "flat v2 \<sqsubseteq>pre s"
- by(auto simp add: CPTpre_def prefix_list_def)
+ have IH: "\<And>v2. v2 \<in> CPT r2 s \<Longrightarrow> v :\<sqsubseteq>val v2" by fact
+ have "v2 \<in> CPT (ALT r1 r2) s" by fact
+ then have "\<Turnstile> v2 : ALT r1 r2" "flat v2 = s"
+ by(auto simp add: CPT_def prefix_list_def)
then consider
- (Left) v3 where "v2 = Left v3" "\<Turnstile> v3 : r1" "flat v3 \<sqsubseteq>pre s"
- | (Right) v3 where "v2 = Right v3" "\<Turnstile> v3 : r2" "flat v3 \<sqsubseteq>pre s"
+ (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: CPrf.cases)
then show "Right v :\<sqsubseteq>val v2"
proof (cases)
case (Right v3)
- have "v3 \<in> CPTpre r2 s" using Right(2,3)
- by (auto simp add: CPTpre_def prefix_list_def)
+ have "v3 \<in> CPT r2 s" using Right(2,3)
+ by (auto simp add: CPT_def prefix_list_def)
with IH have "v :\<sqsubseteq>val v3" by simp
moreover
- have "flat v3 \<sqsubset>spre flat v \<or> flat v3 = flat v" using as1 Right(3)
- by (simp add: Posix1(2) sprefix_list_def)
+ 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_ex1_def PosOrd_RightI PosOrd_spreI)
+ by (auto simp add: PosOrd_ex1_def PosOrd_RightI)
then show "Right v :\<sqsubseteq>val v2" unfolding Right .
next
case (Left v3)
- have w: "v3 \<in> CPTpre r1 s" using Left(2,3) as2
- by (auto simp add: CPTpre_def prefix_list_def)
- have "flat v3 \<sqsubset>spre flat v \<or> flat v3 = flat v" using as1 Left(3)
- by (simp add: Posix1(2) sprefix_list_def)
- then have "flat v3 \<sqsubset>spre flat v \<or> \<Turnstile> v3 : r1" using w
- by(auto simp add: CPTpre_def)
- then have "flat v3 \<sqsubset>spre flat v" using as1 as2 Left
- by (auto simp add: prefix_list_def sprefix_list_def Posix1(2) L_flat_Prf1 Prf_CPrf)
- then have "Right v :\<sqsubseteq>val Left v3"
- by (simp add: PosOrd_ex1_def PosOrd_spreI)
- then show "Right v :\<sqsubseteq>val v2" unfolding Left .
+ have "v3 \<in> CPT r1 s" using Left(2,3) as2
+ by (auto simp add: CPT_def prefix_list_def)
+ then have "flat v3 = flat v \<and> \<Turnstile> v3 : r1" using as1 Left(3)
+ by (simp add: Posix1(2) CPT_def)
+ then have "False" using as1 as2 Left
+ by (auto simp add: Posix1(2) L_flat_Prf1 Prf_CPrf)
+ then show "Right v :\<sqsubseteq>val v2" by simp
qed
next
case (Posix_SEQ s1 r1 v1 s2 r2 v2 v3)
have as1: "s1 \<in> r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2" by fact+
- have IH1: "\<And>v3. v3 \<in> CPTpre r1 s1 \<Longrightarrow> v1 :\<sqsubseteq>val v3" by fact
- have IH2: "\<And>v3. v3 \<in> CPTpre r2 s2 \<Longrightarrow> v2 :\<sqsubseteq>val v3" by fact
+ have IH1: "\<And>v3. v3 \<in> CPT r1 s1 \<Longrightarrow> v1 :\<sqsubseteq>val v3" by fact
+ have IH2: "\<And>v3. v3 \<in> CPT 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> CPTpre (SEQ r1 r2) (s1 @ s2)" by fact
+ have "v3 \<in> CPT (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 \<sqsubseteq>pre s1 @ s2"
- by (force simp add: prefix_list_def CPTpre_def elim: CPrf.cases)
- then have "flat v3a @ flat v3b \<sqsubset>spre s1 @ s2 \<or> flat v3a @ flat v3b = s1 @ s2"
- by (simp add: sprefix_list_def)
- moreover
- { assume "flat v3a @ flat v3b \<sqsubset>spre s1 @ s2"
- then have "Seq v1 v2 :\<sqsubseteq>val Seq v3a v3b" using as1
- by (auto simp add: PosOrd_ex1_def PosOrd_spreI Posix1(2))
- }
- moreover
- { assume q1: "flat v3a @ flat v3b = s1 @ s2"
- then have "flat v3a \<sqsubseteq>pre s1" using eqs(2,3) cond
- unfolding prefix_list_def
- by (smt L_flat_Prf1 Prf_CPrf append_eq_append_conv2 append_self_conv)
- then have "flat v3a \<sqsubset>spre s1 \<or> (flat v3a = s1 \<and> flat v3b = s2)" using q1
- 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 Posix1(2) as1(1) q1 by blast
- then have "v1 :\<sqsubset>val v3a \<or> (v3a \<in> CPTpre r1 s1 \<and> v3b \<in> CPTpre r2 s2)" using eqs(2,3)
- by (auto simp add: CPTpre_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 q1 q2 as1
- unfolding PosOrd_ex1_def
- by (metis PosOrd_SeqI1 PosOrd_SeqI2 Posix1(2) flat.simps(5))
- }
- ultimately show "Seq v1 v2 :\<sqsubseteq>val v3" unfolding eqs by blast
+ "flat v3a @ flat v3b = s1 @ s2"
+ by (force simp add: prefix_list_def CPT_def elim: CPrf.cases)
+ with cond have "flat v3a \<sqsubseteq>pre s1" unfolding prefix_list_def
+ by (smt L_flat_Prf1 Prf_CPrf 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 Posix1(2) as1(1) eqs by blast
+ then have "v1 :\<sqsubset>val v3a \<or> (v3a \<in> CPT r1 s1 \<and> v3b \<in> CPT r2 s2)" using eqs(2,3)
+ by (auto simp add: CPT_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_ex1_def
+ by (metis PosOrd_SeqI1 PosOrd_SeqI2 Posix1(2) flat.simps(5))
+ then show "Seq v1 v2 :\<sqsubseteq>val v3" unfolding eqs by blast
next
case (Posix_STAR1 s1 r v s2 vs v3)
have as1: "s1 \<in> r \<rightarrow> v" "s2 \<in> STAR r \<rightarrow> Stars vs" by fact+
- have IH1: "\<And>v3. v3 \<in> CPTpre r s1 \<Longrightarrow> v :\<sqsubseteq>val v3" by fact
- have IH2: "\<And>v3. v3 \<in> CPTpre (STAR r) s2 \<Longrightarrow> Stars vs :\<sqsubseteq>val v3" by fact
+ have IH1: "\<And>v3. v3 \<in> CPT r s1 \<Longrightarrow> v :\<sqsubseteq>val v3" by fact
+ have IH2: "\<And>v3. v3 \<in> CPT (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> CPTpre (STAR r) (s1 @ s2)" by fact
+ have "v3 \<in> CPT (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 v3a @ flat (Stars vs3) \<sqsubseteq>pre s1 @ s2"
+ "flat (Stars (v3a # vs3)) = s1 @ s2"
| (Empty) "v3 = Stars []"
- by (force simp add: CPTpre_def prefix_list_def elim: CPrf.cases)
+ by (force simp add: CPT_def elim: CPrf.cases)
then show "Stars (v # vs) :\<sqsubseteq>val v3"
proof (cases)
case (NonEmpty v3a vs3)
- then have "flat (Stars (v3a # vs3)) \<sqsubset>spre s1 @ s2 \<or> flat (Stars (v3a # vs3)) = s1 @ s2"
- by (simp add: sprefix_list_def)
- moreover
- { assume "flat (Stars (v3a # vs3)) \<sqsubset>spre s1 @ s2"
- then have "Stars (v # vs) :\<sqsubseteq>val Stars (v3a # vs3)" using as1
- by (metis PosOrd_ex1_def PosOrd_spreI Posix1(2) flat.simps(7))
- }
- moreover
- { assume q1: "flat (Stars (v3a # vs3)) = s1 @ s2"
- then have "flat v3a \<sqsubseteq>pre s1" using NonEmpty(2,3) cond
- unfolding prefix_list_def
- by (smt L_flat_Prf1 Prf_CPrf 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 q1
- 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 Posix1(2) as1(1) q1 by blast
- then have "v :\<sqsubset>val v3a \<or> (v3a \<in> CPTpre r s1 \<and> Stars vs3 \<in> CPTpre (STAR r) s2)"
- using NonEmpty(2,3) by (auto simp add: CPTpre_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 "Stars (v # vs) :\<sqsubseteq>val Stars (v3a # vs3)" using q1 q2 as1
- unfolding PosOrd_ex1_def
- by (metis PosOrd_StarsI PosOrd_StarsI2 Posix1(2) flat.simps(7) val.inject(5))
- }
- ultimately show "Stars (v # vs) :\<sqsubseteq>val v3" unfolding NonEmpty by blast
+ 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 Prf_CPrf 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 Posix1(2) as1(1) NonEmpty(4) by blast
+ then have "v :\<sqsubset>val v3a \<or> (v3a \<in> CPT r s1 \<and> Stars vs3 \<in> CPT (STAR r) s2)"
+ using NonEmpty(2,3) by (auto simp add: CPT_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 "Stars (v # vs) :\<sqsubseteq>val Stars (v3a # vs3)" using NonEmpty(4) q2 as1
+ unfolding PosOrd_ex1_def
+ by (metis PosOrd_StarsI PosOrd_StarsI2 Posix1(2) flat.simps(7) val.inject(5))
+ then show "Stars (v # vs) :\<sqsubseteq>val v3" unfolding NonEmpty by blast
next
case Empty
have "v3 = Stars []" by fact
@@ -1036,25 +1017,18 @@
qed
next
case (Posix_STAR2 r v2)
- have "v2 \<in> CPTpre (STAR r) []" by fact
- then have "v2 = Stars []" using CPTpre_subsets by auto
+ have "v2 \<in> CPT (STAR r) []" by fact
+ then have "v2 = Stars []"
+ unfolding CPT_def by (auto elim: CPrf.cases)
then show "Stars [] :\<sqsubseteq>val v2"
by (simp add: PosOrd_ex1_def)
qed
-
-lemma Posix_PosOrd_stronger:
- assumes "s \<in> r \<rightarrow> v1" "v2 \<in> CPT r s"
- shows "v1 :\<sqsubseteq>val v2"
-using assms Posix_PosOrd
-using CPT_CPTpre_subset by blast
-
-
lemma Posix_PosOrd_reverse:
assumes "s \<in> r \<rightarrow> v1"
shows "\<not>(\<exists>v2 \<in> CPT r s. v2 :\<sqsubset>val v1)"
using assms
-by (metis Posix_PosOrd_stronger less_irrefl PosOrd_def
+by (metis Posix_PosOrd less_irrefl PosOrd_def
PosOrd_ex1_def PosOrd_ex_def PosOrd_trans)