--- a/NEWS Mon May 23 18:04:45 2016 +0200
+++ b/NEWS Mon May 23 22:43:22 2016 +0200
@@ -199,6 +199,8 @@
pave way for a possible future different type class instantiation
for polynomials over factorial rings. INCOMPATIBILITY.
+* Library/Sublist.thy: renamed prefixeq -> prefix and prefix -> strict_prefix
+
* Dropped various legacy fact bindings, whose replacements are often
of a more general type also:
lcm_left_commute_nat ~> lcm.left_commute
--- a/src/HOL/Library/Linear_Temporal_Logic_on_Streams.thy Mon May 23 18:04:45 2016 +0200
+++ b/src/HOL/Library/Linear_Temporal_Logic_on_Streams.thy Mon May 23 22:43:22 2016 +0200
@@ -13,7 +13,7 @@
lemma shift_prefix:
assumes "xl @- xs = yl @- ys" and "length xl \<le> length yl"
-shows "prefixeq xl yl"
+shows "prefix xl yl"
using assms proof(induct xl arbitrary: yl xs ys)
case (Cons x xl yl xs ys)
thus ?case by (cases yl) auto
@@ -21,7 +21,7 @@
lemma shift_prefix_cases:
assumes "xl @- xs = yl @- ys"
-shows "prefixeq xl yl \<or> prefixeq yl xl"
+shows "prefix xl yl \<or> prefix yl xl"
using shift_prefix[OF assms]
by (cases "length xl \<le> length yl") (metis, metis assms nat_le_linear shift_prefix)
@@ -297,17 +297,17 @@
moreover obtain yl ys1 where xs2: "xs = yl @- ys1" and \<psi>\<psi>: "alw \<psi> ys1"
using \<psi> by (metis ev_imp_shift)
ultimately have 0: "xl @- xs1 = yl @- ys1" by auto
- hence "prefixeq xl yl \<or> prefixeq yl xl" using shift_prefix_cases by auto
+ hence "prefix xl yl \<or> prefix yl xl" using shift_prefix_cases by auto
thus ?thesis proof
- assume "prefixeq xl yl"
- then obtain yl1 where yl: "yl = xl @ yl1" by (elim prefixeqE)
+ assume "prefix xl yl"
+ then obtain yl1 where yl: "yl = xl @ yl1" by (elim prefixE)
have xs1': "xs1 = yl1 @- ys1" using 0 unfolding yl by simp
have "alw \<phi> ys1" using \<phi>\<phi> unfolding xs1' by (metis alw_shift)
hence "alw (\<phi> aand \<psi>) ys1" using \<psi>\<psi> unfolding alw_aand by auto
thus ?thesis unfolding xs2 by (auto intro: alw_ev_shift)
next
- assume "prefixeq yl xl"
- then obtain xl1 where xl: "xl = yl @ xl1" by (elim prefixeqE)
+ assume "prefix yl xl"
+ then obtain xl1 where xl: "xl = yl @ xl1" by (elim prefixE)
have ys1': "ys1 = xl1 @- xs1" using 0 unfolding xl by simp
have "alw \<psi> xs1" using \<psi>\<psi> unfolding ys1' by (metis alw_shift)
hence "alw (\<phi> aand \<psi>) xs1" using \<phi>\<phi> unfolding alw_aand by auto
--- a/src/HOL/Library/Prefix_Order.thy Mon May 23 18:04:45 2016 +0200
+++ b/src/HOL/Library/Prefix_Order.thy Mon May 23 22:43:22 2016 +0200
@@ -11,7 +11,7 @@
instantiation list :: (type) order
begin
-definition "(xs::'a list) \<le> ys \<equiv> prefixeq xs ys"
+definition "(xs::'a list) \<le> ys \<equiv> prefix xs ys"
definition "(xs::'a list) < ys \<equiv> xs \<le> ys \<and> \<not> (ys \<le> xs)"
instance
@@ -19,23 +19,26 @@
end
-lemmas prefixI [intro?] = prefixeqI [folded less_eq_list_def]
-lemmas prefixE [elim?] = prefixeqE [folded less_eq_list_def]
-lemmas strict_prefixI' [intro?] = prefixI' [folded less_list_def]
-lemmas strict_prefixE' [elim?] = prefixE' [folded less_list_def]
-lemmas strict_prefixI [intro?] = prefixI [folded less_list_def]
-lemmas strict_prefixE [elim?] = prefixE [folded less_list_def]
-lemmas Nil_prefix [iff] = Nil_prefixeq [folded less_eq_list_def]
-lemmas prefix_Nil [simp] = prefixeq_Nil [folded less_eq_list_def]
-lemmas prefix_snoc [simp] = prefixeq_snoc [folded less_eq_list_def]
-lemmas Cons_prefix_Cons [simp] = Cons_prefixeq_Cons [folded less_eq_list_def]
-lemmas same_prefix_prefix [simp] = same_prefixeq_prefixeq [folded less_eq_list_def]
-lemmas same_prefix_nil [iff] = same_prefixeq_nil [folded less_eq_list_def]
-lemmas prefix_prefix [simp] = prefixeq_prefixeq [folded less_eq_list_def]
-lemmas prefix_Cons = prefixeq_Cons [folded less_eq_list_def]
-lemmas prefix_length_le = prefixeq_length_le [folded less_eq_list_def]
-lemmas strict_prefix_simps [simp, code] = prefix_simps [folded less_list_def]
+lemma less_list_def': "(xs::'a list) < ys \<longleftrightarrow> strict_prefix xs ys"
+by (simp add: less_eq_list_def order.strict_iff_order prefix_order.less_le)
+
+lemmas prefixI [intro?] = prefixI [folded less_eq_list_def]
+lemmas prefixE [elim?] = prefixE [folded less_eq_list_def]
+lemmas strict_prefixI' [intro?] = strict_prefixI' [folded less_list_def']
+lemmas strict_prefixE' [elim?] = strict_prefixE' [folded less_list_def']
+lemmas strict_prefixI [intro?] = strict_prefixI [folded less_list_def']
+lemmas strict_prefixE [elim?] = strict_prefixE [folded less_list_def']
+lemmas Nil_prefix [iff] = Nil_prefix [folded less_eq_list_def]
+lemmas prefix_Nil [simp] = prefix_Nil [folded less_eq_list_def]
+lemmas prefix_snoc [simp] = prefix_snoc [folded less_eq_list_def]
+lemmas Cons_prefix_Cons [simp] = Cons_prefix_Cons [folded less_eq_list_def]
+lemmas same_prefix_prefix [simp] = same_prefix_prefix [folded less_eq_list_def]
+lemmas same_prefix_nil [iff] = same_prefix_nil [folded less_eq_list_def]
+lemmas prefix_prefix [simp] = prefix_prefix [folded less_eq_list_def]
+lemmas prefix_Cons = prefix_Cons [folded less_eq_list_def]
+lemmas prefix_length_le = prefix_length_le [folded less_eq_list_def]
+lemmas strict_prefix_simps [simp, code] = strict_prefix_simps [folded less_list_def']
lemmas not_prefix_induct [consumes 1, case_names Nil Neq Eq] =
- not_prefixeq_induct [folded less_eq_list_def]
+ not_prefix_induct [folded less_eq_list_def]
end
--- a/src/HOL/Library/Sublist.thy Mon May 23 18:04:45 2016 +0200
+++ b/src/HOL/Library/Sublist.thy Mon May 23 22:43:22 2016 +0200
@@ -11,103 +11,103 @@
subsection \<open>Prefix order on lists\<close>
-definition prefixeq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
- where "prefixeq xs ys \<longleftrightarrow> (\<exists>zs. ys = xs @ zs)"
+definition prefix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
+ where "prefix xs ys \<longleftrightarrow> (\<exists>zs. ys = xs @ zs)"
-definition prefix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
- where "prefix xs ys \<longleftrightarrow> prefixeq xs ys \<and> xs \<noteq> ys"
+definition strict_prefix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
+ where "strict_prefix xs ys \<longleftrightarrow> prefix xs ys \<and> xs \<noteq> ys"
-interpretation prefix_order: order prefixeq prefix
- by standard (auto simp: prefixeq_def prefix_def)
+interpretation prefix_order: order prefix strict_prefix
+ by standard (auto simp: prefix_def strict_prefix_def)
-interpretation prefix_bot: order_bot Nil prefixeq prefix
- by standard (simp add: prefixeq_def)
+interpretation prefix_bot: order_bot Nil prefix strict_prefix
+ by standard (simp add: prefix_def)
-lemma prefixeqI [intro?]: "ys = xs @ zs \<Longrightarrow> prefixeq xs ys"
- unfolding prefixeq_def by blast
+lemma prefixI [intro?]: "ys = xs @ zs \<Longrightarrow> prefix xs ys"
+ unfolding prefix_def by blast
-lemma prefixeqE [elim?]:
- assumes "prefixeq xs ys"
+lemma prefixE [elim?]:
+ assumes "prefix xs ys"
obtains zs where "ys = xs @ zs"
- using assms unfolding prefixeq_def by blast
+ using assms unfolding prefix_def by blast
-lemma prefixI' [intro?]: "ys = xs @ z # zs \<Longrightarrow> prefix xs ys"
- unfolding prefix_def prefixeq_def by blast
+lemma strict_prefixI' [intro?]: "ys = xs @ z # zs \<Longrightarrow> strict_prefix xs ys"
+ unfolding strict_prefix_def prefix_def by blast
-lemma prefixE' [elim?]:
- assumes "prefix xs ys"
+lemma strict_prefixE' [elim?]:
+ assumes "strict_prefix xs ys"
obtains z zs where "ys = xs @ z # zs"
proof -
- from \<open>prefix xs ys\<close> obtain us where "ys = xs @ us" and "xs \<noteq> ys"
- unfolding prefix_def prefixeq_def by blast
+ from \<open>strict_prefix xs ys\<close> obtain us where "ys = xs @ us" and "xs \<noteq> ys"
+ unfolding strict_prefix_def prefix_def by blast
with that show ?thesis by (auto simp add: neq_Nil_conv)
qed
-lemma prefixI [intro?]: "prefixeq xs ys \<Longrightarrow> xs \<noteq> ys \<Longrightarrow> prefix xs ys"
- unfolding prefix_def by blast
+lemma strict_prefixI [intro?]: "prefix xs ys \<Longrightarrow> xs \<noteq> ys \<Longrightarrow> strict_prefix xs ys"
+ unfolding strict_prefix_def by blast
-lemma prefixE [elim?]:
+lemma strict_prefixE [elim?]:
fixes xs ys :: "'a list"
- assumes "prefix xs ys"
- obtains "prefixeq xs ys" and "xs \<noteq> ys"
- using assms unfolding prefix_def by blast
+ assumes "strict_prefix xs ys"
+ obtains "prefix xs ys" and "xs \<noteq> ys"
+ using assms unfolding strict_prefix_def by blast
subsection \<open>Basic properties of prefixes\<close>
-theorem Nil_prefixeq [iff]: "prefixeq [] xs"
- by (simp add: prefixeq_def)
+theorem Nil_prefix [iff]: "prefix [] xs"
+ by (simp add: prefix_def)
-theorem prefixeq_Nil [simp]: "(prefixeq xs []) = (xs = [])"
- by (induct xs) (simp_all add: prefixeq_def)
+theorem prefix_Nil [simp]: "(prefix xs []) = (xs = [])"
+ by (induct xs) (simp_all add: prefix_def)
-lemma prefixeq_snoc [simp]: "prefixeq xs (ys @ [y]) \<longleftrightarrow> xs = ys @ [y] \<or> prefixeq xs ys"
+lemma prefix_snoc [simp]: "prefix xs (ys @ [y]) \<longleftrightarrow> xs = ys @ [y] \<or> prefix xs ys"
proof
- assume "prefixeq xs (ys @ [y])"
+ assume "prefix xs (ys @ [y])"
then obtain zs where zs: "ys @ [y] = xs @ zs" ..
- show "xs = ys @ [y] \<or> prefixeq xs ys"
- by (metis append_Nil2 butlast_append butlast_snoc prefixeqI zs)
+ show "xs = ys @ [y] \<or> prefix xs ys"
+ by (metis append_Nil2 butlast_append butlast_snoc prefixI zs)
next
- assume "xs = ys @ [y] \<or> prefixeq xs ys"
- then show "prefixeq xs (ys @ [y])"
- by (metis prefix_order.eq_iff prefix_order.order_trans prefixeqI)
+ assume "xs = ys @ [y] \<or> prefix xs ys"
+ then show "prefix xs (ys @ [y])"
+ by (metis prefix_order.eq_iff prefix_order.order_trans prefixI)
qed
-lemma Cons_prefixeq_Cons [simp]: "prefixeq (x # xs) (y # ys) = (x = y \<and> prefixeq xs ys)"
- by (auto simp add: prefixeq_def)
+lemma Cons_prefix_Cons [simp]: "prefix (x # xs) (y # ys) = (x = y \<and> prefix xs ys)"
+ by (auto simp add: prefix_def)
-lemma prefixeq_code [code]:
- "prefixeq [] xs \<longleftrightarrow> True"
- "prefixeq (x # xs) [] \<longleftrightarrow> False"
- "prefixeq (x # xs) (y # ys) \<longleftrightarrow> x = y \<and> prefixeq xs ys"
+lemma prefix_code [code]:
+ "prefix [] xs \<longleftrightarrow> True"
+ "prefix (x # xs) [] \<longleftrightarrow> False"
+ "prefix (x # xs) (y # ys) \<longleftrightarrow> x = y \<and> prefix xs ys"
by simp_all
-lemma same_prefixeq_prefixeq [simp]: "prefixeq (xs @ ys) (xs @ zs) = prefixeq ys zs"
+lemma same_prefix_prefix [simp]: "prefix (xs @ ys) (xs @ zs) = prefix ys zs"
by (induct xs) simp_all
-lemma same_prefixeq_nil [iff]: "prefixeq (xs @ ys) xs = (ys = [])"
- by (metis append_Nil2 append_self_conv prefix_order.eq_iff prefixeqI)
+lemma same_prefix_nil [iff]: "prefix (xs @ ys) xs = (ys = [])"
+ by (metis append_Nil2 append_self_conv prefix_order.eq_iff prefixI)
-lemma prefixeq_prefixeq [simp]: "prefixeq xs ys \<Longrightarrow> prefixeq xs (ys @ zs)"
- by (metis prefix_order.le_less_trans prefixeqI prefixE prefixI)
+lemma prefix_prefix [simp]: "prefix xs ys \<Longrightarrow> prefix xs (ys @ zs)"
+ by (metis prefix_order.le_less_trans prefixI strict_prefixE strict_prefixI)
-lemma append_prefixeqD: "prefixeq (xs @ ys) zs \<Longrightarrow> prefixeq xs zs"
- by (auto simp add: prefixeq_def)
+lemma append_prefixD: "prefix (xs @ ys) zs \<Longrightarrow> prefix xs zs"
+ by (auto simp add: prefix_def)
-theorem prefixeq_Cons: "prefixeq xs (y # ys) = (xs = [] \<or> (\<exists>zs. xs = y # zs \<and> prefixeq zs ys))"
- by (cases xs) (auto simp add: prefixeq_def)
+theorem prefix_Cons: "prefix xs (y # ys) = (xs = [] \<or> (\<exists>zs. xs = y # zs \<and> prefix zs ys))"
+ by (cases xs) (auto simp add: prefix_def)
-theorem prefixeq_append:
- "prefixeq xs (ys @ zs) = (prefixeq xs ys \<or> (\<exists>us. xs = ys @ us \<and> prefixeq us zs))"
+theorem prefix_append:
+ "prefix xs (ys @ zs) = (prefix xs ys \<or> (\<exists>us. xs = ys @ us \<and> prefix us zs))"
apply (induct zs rule: rev_induct)
apply force
apply (simp del: append_assoc add: append_assoc [symmetric])
apply (metis append_eq_appendI)
done
-lemma append_one_prefixeq:
- "prefixeq xs ys \<Longrightarrow> length xs < length ys \<Longrightarrow> prefixeq (xs @ [ys ! length xs]) ys"
- proof (unfold prefixeq_def)
+lemma append_one_prefix:
+ "prefix xs ys \<Longrightarrow> length xs < length ys \<Longrightarrow> prefix (xs @ [ys ! length xs]) ys"
+ proof (unfold prefix_def)
assume a1: "\<exists>zs. ys = xs @ zs"
then obtain sk :: "'a list" where sk: "ys = xs @ sk" by fastforce
assume a2: "length xs < length ys"
@@ -117,42 +117,42 @@
thus "\<exists>zs. ys = (xs @ [ys ! length xs]) @ zs" using f1 by fastforce
qed
-theorem prefixeq_length_le: "prefixeq xs ys \<Longrightarrow> length xs \<le> length ys"
- by (auto simp add: prefixeq_def)
+theorem prefix_length_le: "prefix xs ys \<Longrightarrow> length xs \<le> length ys"
+ by (auto simp add: prefix_def)
-lemma prefixeq_same_cases:
- "prefixeq (xs\<^sub>1::'a list) ys \<Longrightarrow> prefixeq xs\<^sub>2 ys \<Longrightarrow> prefixeq xs\<^sub>1 xs\<^sub>2 \<or> prefixeq xs\<^sub>2 xs\<^sub>1"
- unfolding prefixeq_def by (force simp: append_eq_append_conv2)
+lemma prefix_same_cases:
+ "prefix (xs\<^sub>1::'a list) ys \<Longrightarrow> prefix xs\<^sub>2 ys \<Longrightarrow> prefix xs\<^sub>1 xs\<^sub>2 \<or> prefix xs\<^sub>2 xs\<^sub>1"
+ unfolding prefix_def by (force simp: append_eq_append_conv2)
-lemma set_mono_prefixeq: "prefixeq xs ys \<Longrightarrow> set xs \<subseteq> set ys"
- by (auto simp add: prefixeq_def)
+lemma set_mono_prefix: "prefix xs ys \<Longrightarrow> set xs \<subseteq> set ys"
+ by (auto simp add: prefix_def)
-lemma take_is_prefixeq: "prefixeq (take n xs) xs"
- unfolding prefixeq_def by (metis append_take_drop_id)
+lemma take_is_prefix: "prefix (take n xs) xs"
+ unfolding prefix_def by (metis append_take_drop_id)
-lemma map_prefixeqI: "prefixeq xs ys \<Longrightarrow> prefixeq (map f xs) (map f ys)"
- by (auto simp: prefixeq_def)
+lemma map_prefixI: "prefix xs ys \<Longrightarrow> prefix (map f xs) (map f ys)"
+ by (auto simp: prefix_def)
-lemma prefixeq_length_less: "prefix xs ys \<Longrightarrow> length xs < length ys"
- by (auto simp: prefix_def prefixeq_def)
+lemma prefix_length_less: "strict_prefix xs ys \<Longrightarrow> length xs < length ys"
+ by (auto simp: strict_prefix_def prefix_def)
-lemma prefix_simps [simp, code]:
- "prefix xs [] \<longleftrightarrow> False"
- "prefix [] (x # xs) \<longleftrightarrow> True"
- "prefix (x # xs) (y # ys) \<longleftrightarrow> x = y \<and> prefix xs ys"
- by (simp_all add: prefix_def cong: conj_cong)
+lemma strict_prefix_simps [simp, code]:
+ "strict_prefix xs [] \<longleftrightarrow> False"
+ "strict_prefix [] (x # xs) \<longleftrightarrow> True"
+ "strict_prefix (x # xs) (y # ys) \<longleftrightarrow> x = y \<and> strict_prefix xs ys"
+ by (simp_all add: strict_prefix_def cong: conj_cong)
-lemma take_prefix: "prefix xs ys \<Longrightarrow> prefix (take n xs) ys"
+lemma take_strict_prefix: "strict_prefix xs ys \<Longrightarrow> strict_prefix (take n xs) ys"
apply (induct n arbitrary: xs ys)
apply (case_tac ys; simp)
- apply (metis prefix_order.less_trans prefixI take_is_prefixeq)
+ apply (metis prefix_order.less_trans strict_prefixI take_is_prefix)
done
-lemma not_prefixeq_cases:
- assumes pfx: "\<not> prefixeq ps ls"
+lemma not_prefix_cases:
+ assumes pfx: "\<not> prefix ps ls"
obtains
(c1) "ps \<noteq> []" and "ls = []"
- | (c2) a as x xs where "ps = a#as" and "ls = x#xs" and "x = a" and "\<not> prefixeq as xs"
+ | (c2) a as x xs where "ps = a#as" and "ls = x#xs" and "x = a" and "\<not> prefix as xs"
| (c3) a as x xs where "ps = a#as" and "ls = x#xs" and "x \<noteq> a"
proof (cases ps)
case Nil
@@ -162,13 +162,13 @@
note c = \<open>ps = a#as\<close>
show ?thesis
proof (cases ls)
- case Nil then show ?thesis by (metis append_Nil2 pfx c1 same_prefixeq_nil)
+ case Nil then show ?thesis by (metis append_Nil2 pfx c1 same_prefix_nil)
next
case (Cons x xs)
show ?thesis
proof (cases "x = a")
case True
- have "\<not> prefixeq as xs" using pfx c Cons True by simp
+ have "\<not> prefix as xs" using pfx c Cons True by simp
with c Cons True show ?thesis by (rule c2)
next
case False
@@ -177,40 +177,40 @@
qed
qed
-lemma not_prefixeq_induct [consumes 1, case_names Nil Neq Eq]:
- assumes np: "\<not> prefixeq ps ls"
+lemma not_prefix_induct [consumes 1, case_names Nil Neq Eq]:
+ assumes np: "\<not> prefix ps ls"
and base: "\<And>x xs. P (x#xs) []"
and r1: "\<And>x xs y ys. x \<noteq> y \<Longrightarrow> P (x#xs) (y#ys)"
- and r2: "\<And>x xs y ys. \<lbrakk> x = y; \<not> prefixeq xs ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys)"
+ and r2: "\<And>x xs y ys. \<lbrakk> x = y; \<not> prefix xs ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys)"
shows "P ps ls" using np
proof (induct ls arbitrary: ps)
case Nil then show ?case
- by (auto simp: neq_Nil_conv elim!: not_prefixeq_cases intro!: base)
+ by (auto simp: neq_Nil_conv elim!: not_prefix_cases intro!: base)
next
case (Cons y ys)
- then have npfx: "\<not> prefixeq ps (y # ys)" by simp
+ then have npfx: "\<not> prefix ps (y # ys)" by simp
then obtain x xs where pv: "ps = x # xs"
- by (rule not_prefixeq_cases) auto
- show ?case by (metis Cons.hyps Cons_prefixeq_Cons npfx pv r1 r2)
+ by (rule not_prefix_cases) auto
+ show ?case by (metis Cons.hyps Cons_prefix_Cons npfx pv r1 r2)
qed
subsection \<open>Parallel lists\<close>
definition parallel :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infixl "\<parallel>" 50)
- where "(xs \<parallel> ys) = (\<not> prefixeq xs ys \<and> \<not> prefixeq ys xs)"
+ where "(xs \<parallel> ys) = (\<not> prefix xs ys \<and> \<not> prefix ys xs)"
-lemma parallelI [intro]: "\<not> prefixeq xs ys \<Longrightarrow> \<not> prefixeq ys xs \<Longrightarrow> xs \<parallel> ys"
+lemma parallelI [intro]: "\<not> prefix xs ys \<Longrightarrow> \<not> prefix ys xs \<Longrightarrow> xs \<parallel> ys"
unfolding parallel_def by blast
lemma parallelE [elim]:
assumes "xs \<parallel> ys"
- obtains "\<not> prefixeq xs ys \<and> \<not> prefixeq ys xs"
+ obtains "\<not> prefix xs ys \<and> \<not> prefix ys xs"
using assms unfolding parallel_def by blast
-theorem prefixeq_cases:
- obtains "prefixeq xs ys" | "prefix ys xs" | "xs \<parallel> ys"
- unfolding parallel_def prefix_def by blast
+theorem prefix_cases:
+ obtains "prefix xs ys" | "strict_prefix ys xs" | "xs \<parallel> ys"
+ unfolding parallel_def strict_prefix_def by blast
theorem parallel_decomp:
"xs \<parallel> ys \<Longrightarrow> \<exists>as b bs c cs. b \<noteq> c \<and> xs = as @ b # bs \<and> ys = as @ c # cs"
@@ -221,13 +221,13 @@
next
case (snoc x xs)
show ?case
- proof (rule prefixeq_cases)
- assume le: "prefixeq xs ys"
+ proof (rule prefix_cases)
+ assume le: "prefix xs ys"
then obtain ys' where ys: "ys = xs @ ys'" ..
show ?thesis
proof (cases ys')
assume "ys' = []"
- then show ?thesis by (metis append_Nil2 parallelE prefixeqI snoc.prems ys)
+ then show ?thesis by (metis append_Nil2 parallelE prefixI snoc.prems ys)
next
fix c cs assume ys': "ys' = c # cs"
have "x \<noteq> c" using snoc.prems ys ys' by fastforce
@@ -235,8 +235,8 @@
using ys ys' by blast
qed
next
- assume "prefix ys xs"
- then have "prefixeq ys (xs @ [x])" by (simp add: prefix_def)
+ assume "strict_prefix ys xs"
+ then have "prefix ys (xs @ [x])" by (simp add: strict_prefix_def)
with snoc have False by blast
then show ?thesis ..
next
@@ -252,7 +252,7 @@
lemma parallel_append: "a \<parallel> b \<Longrightarrow> a @ c \<parallel> b @ d"
apply (rule parallelI)
apply (erule parallelE, erule conjE,
- induct rule: not_prefixeq_induct, simp+)+
+ induct rule: not_prefix_induct, simp+)+
done
lemma parallel_appendI: "xs \<parallel> ys \<Longrightarrow> x = xs @ xs' \<Longrightarrow> y = ys @ ys' \<Longrightarrow> x \<parallel> y"
@@ -327,14 +327,14 @@
by (induct zs) (auto intro!: suffixeq_appendI suffixeq_ConsI)
qed
-lemma suffixeq_to_prefixeq [code]: "suffixeq xs ys \<longleftrightarrow> prefixeq (rev xs) (rev ys)"
+lemma suffixeq_to_prefix [code]: "suffixeq xs ys \<longleftrightarrow> prefix (rev xs) (rev ys)"
proof
assume "suffixeq xs ys"
then obtain zs where "ys = zs @ xs" ..
then have "rev ys = rev xs @ rev zs" by simp
- then show "prefixeq (rev xs) (rev ys)" ..
+ then show "prefix (rev xs) (rev ys)" ..
next
- assume "prefixeq (rev xs) (rev ys)"
+ assume "prefix (rev xs) (rev ys)"
then obtain zs where "rev ys = rev xs @ zs" ..
then have "rev (rev ys) = rev zs @ rev (rev xs)" by simp
then have "ys = rev zs @ xs" by simp
@@ -379,10 +379,10 @@
qed
qed
-lemma parallelD1: "x \<parallel> y \<Longrightarrow> \<not> prefixeq x y"
+lemma parallelD1: "x \<parallel> y \<Longrightarrow> \<not> prefix x y"
by blast
-lemma parallelD2: "x \<parallel> y \<Longrightarrow> \<not> prefixeq y x"
+lemma parallelD2: "x \<parallel> y \<Longrightarrow> \<not> prefix y x"
by blast
lemma parallel_Nil1 [simp]: "\<not> x \<parallel> []"
@@ -395,7 +395,7 @@
by auto
lemma Cons_parallelI2: "\<lbrakk> a = b; as \<parallel> bs \<rbrakk> \<Longrightarrow> a # as \<parallel> b # bs"
- by (metis Cons_prefixeq_Cons parallelE parallelI)
+ by (metis Cons_prefix_Cons parallelE parallelI)
lemma not_equal_is_parallel:
assumes neq: "xs \<noteq> ys"
--- a/src/HOL/Unix/Unix.thy Mon May 23 18:04:45 2016 +0200
+++ b/src/HOL/Unix/Unix.thy Mon May 23 22:43:22 2016 +0200
@@ -924,7 +924,7 @@
with tr obtain opt where root': "root' = update (path_of x) opt root"
by cases auto
show ?thesis
- proof (rule prefixeq_cases)
+ proof (rule prefix_cases)
assume "path_of x \<parallel> path"
with inv root'
have "\<And>perms. access root' path user\<^sub>1 perms = access root path user\<^sub>1 perms"
@@ -932,7 +932,7 @@
with inv show "invariant root' path"
by (simp only: invariant_def)
next
- assume "prefixeq (path_of x) path"
+ assume "prefix (path_of x) path"
then obtain ys where path: "path = path_of x @ ys" ..
show ?thesis
@@ -969,7 +969,7 @@
by (simp only: invariant_def access_def)
qed
next
- assume "prefix path (path_of x)"
+ assume "strict_prefix path (path_of x)"
then obtain y ys where path: "path_of x = path @ y # ys" ..
obtain dir' where