--- a/src/HOL/Library/List_Prefix.thy Tue Dec 18 14:37:00 2007 +0100
+++ b/src/HOL/Library/List_Prefix.thy Tue Dec 18 16:26:46 2007 +0100
@@ -77,10 +77,10 @@
by (induct xs) simp_all
lemma same_prefix_nil [iff]: "(xs @ ys \<le> xs) = (ys = [])"
-by (metis append_Nil2 append_self_conv order_eq_iff prefixI)
+ by (metis append_Nil2 append_self_conv order_eq_iff prefixI)
lemma prefix_prefix [simp]: "xs \<le> ys ==> xs \<le> ys @ zs"
-by (metis order_le_less_trans prefixI strict_prefixE strict_prefixI)
+ by (metis order_le_less_trans prefixI strict_prefixE strict_prefixI)
lemma append_prefixD: "xs @ ys \<le> zs \<Longrightarrow> xs \<le> zs"
by (auto simp add: prefix_def)
@@ -98,41 +98,40 @@
lemma append_one_prefix:
"xs \<le> ys ==> length xs < length ys ==> xs @ [ys ! length xs] \<le> ys"
-by (unfold prefix_def)
- (metis Cons_eq_appendI append_eq_appendI append_eq_conv_conj eq_Nil_appendI nth_drop')
+ unfolding prefix_def
+ by (metis Cons_eq_appendI append_eq_appendI append_eq_conv_conj
+ eq_Nil_appendI nth_drop')
theorem prefix_length_le: "xs \<le> ys ==> length xs \<le> length ys"
by (auto simp add: prefix_def)
lemma prefix_same_cases:
"(xs\<^isub>1::'a list) \<le> ys \<Longrightarrow> xs\<^isub>2 \<le> ys \<Longrightarrow> xs\<^isub>1 \<le> xs\<^isub>2 \<or> xs\<^isub>2 \<le> xs\<^isub>1"
-by (unfold prefix_def) (metis append_eq_append_conv2)
+ unfolding prefix_def by (metis append_eq_append_conv2)
lemma set_mono_prefix: "xs \<le> ys \<Longrightarrow> set xs \<subseteq> set ys"
-by (auto simp add: prefix_def)
+ by (auto simp add: prefix_def)
lemma take_is_prefix: "take n xs \<le> xs"
-by (unfold prefix_def) (metis append_take_drop_id)
+ unfolding prefix_def by (metis append_take_drop_id)
-lemma map_prefixI:
- "xs \<le> ys \<Longrightarrow> map f xs \<le> map f ys"
-by (clarsimp simp: prefix_def)
+lemma map_prefixI: "xs \<le> ys \<Longrightarrow> map f xs \<le> map f ys"
+ by (auto simp: prefix_def)
-lemma prefix_length_less:
- "xs < ys \<Longrightarrow> length xs < length ys"
-by (clarsimp simp: strict_prefix_def prefix_def)
+lemma prefix_length_less: "xs < ys \<Longrightarrow> length xs < length ys"
+ by (auto simp: strict_prefix_def prefix_def)
lemma strict_prefix_simps [simp]:
- "xs < [] = False"
- "[] < (x # xs) = True"
- "(x # xs) < (y # ys) = (x = y \<and> xs < ys)"
-by (simp_all add: strict_prefix_def cong: conj_cong)
+ "xs < [] = False"
+ "[] < (x # xs) = True"
+ "(x # xs) < (y # ys) = (x = y \<and> xs < ys)"
+ by (simp_all add: strict_prefix_def cong: conj_cong)
lemma take_strict_prefix: "xs < ys \<Longrightarrow> take n xs < ys"
-apply (induct n arbitrary: xs ys)
- apply (case_tac ys, simp_all)[1]
-apply (metis order_less_trans strict_prefixI take_is_prefix)
-done
+ apply (induct n arbitrary: xs ys)
+ apply (case_tac ys, simp_all)[1]
+ apply (metis order_less_trans strict_prefixI take_is_prefix)
+ done
lemma not_prefix_cases:
assumes pfx: "\<not> ps \<le> ls"
@@ -141,13 +140,13 @@
| (c2) a as x xs where "ps = a#as" and "ls = x#xs" and "x = a" and "\<not> as \<le> xs"
| (c3) a as x xs where "ps = a#as" and "ls = x#xs" and "x \<noteq> a"
proof (cases ps)
- case Nil thus ?thesis using pfx by simp
+ case Nil then show ?thesis using pfx by simp
next
case (Cons a as)
- hence c: "ps = a#as" .
+ note c = `ps = a#as`
show ?thesis
proof (cases ls)
- case Nil thus ?thesis by (metis append_Nil2 pfx c1 same_prefix_nil)
+ case Nil then show ?thesis by (metis append_Nil2 pfx c1 same_prefix_nil)
next
case (Cons x xs)
show ?thesis
@@ -187,16 +186,16 @@
"(xs \<parallel> ys) = (\<not> xs \<le> ys \<and> \<not> ys \<le> xs)"
lemma parallelI [intro]: "\<not> xs \<le> ys ==> \<not> ys \<le> xs ==> xs \<parallel> ys"
-unfolding parallel_def by blast
+ unfolding parallel_def by blast
lemma parallelE [elim]:
-assumes "xs \<parallel> ys"
-obtains "\<not> xs \<le> ys \<and> \<not> ys \<le> xs"
-using assms unfolding parallel_def by blast
+ assumes "xs \<parallel> ys"
+ obtains "\<not> xs \<le> ys \<and> \<not> ys \<le> xs"
+ using assms unfolding parallel_def by blast
theorem prefix_cases:
-obtains "xs \<le> ys" | "ys < xs" | "xs \<parallel> ys"
-unfolding parallel_def strict_prefix_def by blast
+ obtains "xs \<le> ys" | "ys < xs" | "xs \<parallel> ys"
+ unfolding parallel_def strict_prefix_def by blast
theorem parallel_decomp:
"xs \<parallel> ys ==> \<exists>as b bs c cs. b \<noteq> c \<and> xs = as @ b # bs \<and> ys = as @ c # cs"
@@ -213,11 +212,12 @@
show ?thesis
proof (cases ys')
assume "ys' = []"
- thus ?thesis by (metis append_Nil2 parallelE prefixI snoc.prems ys)
+ then show ?thesis by (metis append_Nil2 parallelE prefixI snoc.prems ys)
next
fix c cs assume ys': "ys' = c # cs"
- thus ?thesis
- by (metis Cons_eq_appendI eq_Nil_appendI parallelE prefixI same_prefix_prefix snoc.prems ys)
+ then show ?thesis
+ by (metis Cons_eq_appendI eq_Nil_appendI parallelE prefixI
+ same_prefix_prefix snoc.prems ys)
qed
next
assume "ys < xs" then have "ys \<le> xs @ [x]" by (simp add: strict_prefix_def)
@@ -234,15 +234,16 @@
qed
lemma parallel_append: "a \<parallel> b \<Longrightarrow> a @ c \<parallel> b @ d"
-by (rule parallelI)
- (erule parallelE, erule conjE,
- induct rule: not_prefix_induct, simp+)+
+ apply (rule parallelI)
+ apply (erule parallelE, erule conjE,
+ induct rule: not_prefix_induct, simp+)+
+ done
-lemma parallel_appendI: "\<lbrakk> xs \<parallel> ys; x = xs @ xs' ; y = ys @ ys' \<rbrakk> \<Longrightarrow> x \<parallel> y"
-by simp (rule parallel_append)
+lemma parallel_appendI: "xs \<parallel> ys \<Longrightarrow> x = xs @ xs' \<Longrightarrow> y = ys @ ys' \<Longrightarrow> x \<parallel> y"
+ by (simp add: parallel_append)
-lemma parallel_commute: "(a \<parallel> b) = (b \<parallel> a)"
-unfolding parallel_def by auto
+lemma parallel_commute: "a \<parallel> b \<longleftrightarrow> b \<parallel> a"
+ unfolding parallel_def by auto
subsection {* Postfix order on lists *}
@@ -252,12 +253,12 @@
"(xs >>= ys) = (\<exists>zs. xs = zs @ ys)"
lemma postfixI [intro?]: "xs = zs @ ys ==> xs >>= ys"
-unfolding postfix_def by blast
+ unfolding postfix_def by blast
lemma postfixE [elim?]:
-assumes "xs >>= ys"
-obtains zs where "xs = zs @ ys"
-using assms unfolding postfix_def by blast
+ assumes "xs >>= ys"
+ obtains zs where "xs = zs @ ys"
+ using assms unfolding postfix_def by blast
lemma postfix_refl [iff]: "xs >>= xs"
by (auto simp add: postfix_def)
@@ -311,35 +312,37 @@
qed
lemma distinct_postfix: "distinct xs \<Longrightarrow> xs >>= ys \<Longrightarrow> distinct ys"
-by (clarsimp elim!: postfixE)
+ by (clarsimp elim!: postfixE)
lemma postfix_map: "xs >>= ys \<Longrightarrow> map f xs >>= map f ys"
-by (auto elim!: postfixE intro: postfixI)
+ by (auto elim!: postfixE intro: postfixI)
lemma postfix_drop: "as >>= drop n as"
-unfolding postfix_def
-by (rule exI [where x = "take n as"]) simp
+ unfolding postfix_def
+ apply (rule exI [where x = "take n as"])
+ apply simp
+ done
lemma postfix_take: "xs >>= ys \<Longrightarrow> xs = take (length xs - length ys) xs @ ys"
-by (clarsimp elim!: postfixE)
+ by (clarsimp elim!: postfixE)
lemma parallelD1: "x \<parallel> y \<Longrightarrow> \<not> x \<le> y"
-by blast
+ by blast
lemma parallelD2: "x \<parallel> y \<Longrightarrow> \<not> y \<le> x"
-by blast
+ by blast
lemma parallel_Nil1 [simp]: "\<not> x \<parallel> []"
-unfolding parallel_def by simp
+ unfolding parallel_def by simp
lemma parallel_Nil2 [simp]: "\<not> [] \<parallel> x"
-unfolding parallel_def by simp
+ unfolding parallel_def by simp
lemma Cons_parallelI1: "a \<noteq> b \<Longrightarrow> a # as \<parallel> b # bs"
-by auto
+ by auto
lemma Cons_parallelI2: "\<lbrakk> a = b; as \<parallel> bs \<rbrakk> \<Longrightarrow> a # as \<parallel> b # bs"
-by (metis Cons_prefix_Cons parallelE parallelI)
+ by (metis Cons_prefix_Cons parallelE parallelI)
lemma not_equal_is_parallel:
assumes neq: "xs \<noteq> ys"