--- a/src/HOL/Library/List_Prefix.thy Mon Sep 03 11:54:21 2012 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,382 +0,0 @@
-(* Title: HOL/Library/List_Prefix.thy
- Author: Tobias Nipkow and Markus Wenzel, TU Muenchen
-*)
-
-header {* List prefixes and postfixes *}
-
-theory List_Prefix
-imports List Main
-begin
-
-subsection {* Prefix order on lists *}
-
-instantiation list :: (type) "{order, bot}"
-begin
-
-definition
- prefix_def: "xs \<le> ys \<longleftrightarrow> (\<exists>zs. ys = xs @ zs)"
-
-definition
- strict_prefix_def: "xs < ys \<longleftrightarrow> xs \<le> ys \<and> xs \<noteq> (ys::'a list)"
-
-definition
- "bot = []"
-
-instance proof
-qed (auto simp add: prefix_def strict_prefix_def bot_list_def)
-
-end
-
-lemma prefixI [intro?]: "ys = xs @ zs ==> xs \<le> ys"
- unfolding prefix_def by blast
-
-lemma prefixE [elim?]:
- assumes "xs \<le> ys"
- obtains zs where "ys = xs @ zs"
- using assms unfolding prefix_def by blast
-
-lemma strict_prefixI' [intro?]: "ys = xs @ z # zs ==> xs < ys"
- unfolding strict_prefix_def prefix_def by blast
-
-lemma strict_prefixE' [elim?]:
- assumes "xs < ys"
- obtains z zs where "ys = xs @ z # zs"
-proof -
- from `xs < ys` 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 strict_prefixI [intro?]: "xs \<le> ys ==> xs \<noteq> ys ==> xs < (ys::'a list)"
- unfolding strict_prefix_def by blast
-
-lemma strict_prefixE [elim?]:
- fixes xs ys :: "'a list"
- assumes "xs < ys"
- obtains "xs \<le> ys" and "xs \<noteq> ys"
- using assms unfolding strict_prefix_def by blast
-
-
-subsection {* Basic properties of prefixes *}
-
-theorem Nil_prefix [iff]: "[] \<le> xs"
- by (simp add: prefix_def)
-
-theorem prefix_Nil [simp]: "(xs \<le> []) = (xs = [])"
- by (induct xs) (simp_all add: prefix_def)
-
-lemma prefix_snoc [simp]: "(xs \<le> ys @ [y]) = (xs = ys @ [y] \<or> xs \<le> ys)"
-proof
- assume "xs \<le> ys @ [y]"
- then obtain zs where zs: "ys @ [y] = xs @ zs" ..
- show "xs = ys @ [y] \<or> xs \<le> ys"
- by (metis append_Nil2 butlast_append butlast_snoc prefixI zs)
-next
- assume "xs = ys @ [y] \<or> xs \<le> ys"
- then show "xs \<le> ys @ [y]"
- by (metis order_eq_iff order_trans prefixI)
-qed
-
-lemma Cons_prefix_Cons [simp]: "(x # xs \<le> y # ys) = (x = y \<and> xs \<le> ys)"
- by (auto simp add: prefix_def)
-
-lemma less_eq_list_code [code]:
- "([]\<Colon>'a\<Colon>{equal, ord} list) \<le> xs \<longleftrightarrow> True"
- "(x\<Colon>'a\<Colon>{equal, ord}) # xs \<le> [] \<longleftrightarrow> False"
- "(x\<Colon>'a\<Colon>{equal, ord}) # xs \<le> y # ys \<longleftrightarrow> x = y \<and> xs \<le> ys"
- by simp_all
-
-lemma same_prefix_prefix [simp]: "(xs @ ys \<le> xs @ zs) = (ys \<le> zs)"
- 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)
-
-lemma prefix_prefix [simp]: "xs \<le> ys ==> xs \<le> ys @ zs"
- 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)
-
-theorem prefix_Cons: "(xs \<le> y # ys) = (xs = [] \<or> (\<exists>zs. xs = y # zs \<and> zs \<le> ys))"
- by (cases xs) (auto simp add: prefix_def)
-
-theorem prefix_append:
- "(xs \<le> ys @ zs) = (xs \<le> ys \<or> (\<exists>us. xs = ys @ us \<and> us \<le> 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_prefix:
- "xs \<le> ys ==> length xs < length ys ==> xs @ [ys ! length xs] \<le> ys"
- 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"
- 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)
-
-lemma take_is_prefix: "take n xs \<le> xs"
- unfolding prefix_def by (metis append_take_drop_id)
-
-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 (auto simp: strict_prefix_def prefix_def)
-
-lemma strict_prefix_simps [simp, code]:
- "xs < [] \<longleftrightarrow> False"
- "[] < x # xs \<longleftrightarrow> True"
- "x # xs < y # ys \<longleftrightarrow> 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
-
-lemma not_prefix_cases:
- assumes pfx: "\<not> ps \<le> 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> 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 then show ?thesis using pfx by simp
-next
- case (Cons a as)
- note c = `ps = a#as`
- show ?thesis
- proof (cases ls)
- 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> as \<le> xs" using pfx c Cons True by simp
- with c Cons True show ?thesis by (rule c2)
- next
- case False
- with c Cons show ?thesis by (rule c3)
- qed
- qed
-qed
-
-lemma not_prefix_induct [consumes 1, case_names Nil Neq Eq]:
- assumes np: "\<not> ps \<le> 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> xs \<le> 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_prefix_cases intro!: base)
-next
- case (Cons y ys)
- then have npfx: "\<not> ps \<le> (y # ys)" by simp
- then obtain x xs where pv: "ps = x # xs"
- by (rule not_prefix_cases) auto
- show ?case by (metis Cons.hyps Cons_prefix_Cons npfx pv r1 r2)
-qed
-
-
-subsection {* Parallel lists *}
-
-definition
- parallel :: "'a list => 'a list => bool" (infixl "\<parallel>" 50) where
- "(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
-
-lemma parallelE [elim]:
- 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
-
-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"
-proof (induct xs rule: rev_induct)
- case Nil
- then have False by auto
- then show ?case ..
-next
- case (snoc x xs)
- show ?case
- proof (rule prefix_cases)
- assume le: "xs \<le> ys"
- then obtain ys' where ys: "ys = xs @ ys'" ..
- show ?thesis
- proof (cases ys')
- assume "ys' = []"
- then show ?thesis by (metis append_Nil2 parallelE prefixI snoc.prems ys)
- next
- fix c cs assume ys': "ys' = c # cs"
- 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)
- with snoc have False by blast
- then show ?thesis ..
- next
- assume "xs \<parallel> ys"
- with snoc obtain as b bs c cs where neq: "(b::'a) \<noteq> c"
- and xs: "xs = as @ b # bs" and ys: "ys = as @ c # cs"
- by blast
- from xs have "xs @ [x] = as @ b # (bs @ [x])" by simp
- with neq ys show ?thesis by blast
- qed
-qed
-
-lemma parallel_append: "a \<parallel> b \<Longrightarrow> a @ c \<parallel> b @ d"
- apply (rule parallelI)
- apply (erule parallelE, erule conjE,
- 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"
- by (simp add: parallel_append)
-
-lemma parallel_commute: "a \<parallel> b \<longleftrightarrow> b \<parallel> a"
- unfolding parallel_def by auto
-
-
-subsection {* Postfix order on lists *}
-
-definition
- postfix :: "'a list => 'a list => bool" ("(_/ >>= _)" [51, 50] 50) where
- "(xs >>= ys) = (\<exists>zs. xs = zs @ ys)"
-
-lemma postfixI [intro?]: "xs = zs @ ys ==> xs >>= ys"
- unfolding postfix_def by blast
-
-lemma postfixE [elim?]:
- 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)
-lemma postfix_trans: "\<lbrakk>xs >>= ys; ys >>= zs\<rbrakk> \<Longrightarrow> xs >>= zs"
- by (auto simp add: postfix_def)
-lemma postfix_antisym: "\<lbrakk>xs >>= ys; ys >>= xs\<rbrakk> \<Longrightarrow> xs = ys"
- by (auto simp add: postfix_def)
-
-lemma Nil_postfix [iff]: "xs >>= []"
- by (simp add: postfix_def)
-lemma postfix_Nil [simp]: "([] >>= xs) = (xs = [])"
- by (auto simp add: postfix_def)
-
-lemma postfix_ConsI: "xs >>= ys \<Longrightarrow> x#xs >>= ys"
- by (auto simp add: postfix_def)
-lemma postfix_ConsD: "xs >>= y#ys \<Longrightarrow> xs >>= ys"
- by (auto simp add: postfix_def)
-
-lemma postfix_appendI: "xs >>= ys \<Longrightarrow> zs @ xs >>= ys"
- by (auto simp add: postfix_def)
-lemma postfix_appendD: "xs >>= zs @ ys \<Longrightarrow> xs >>= ys"
- by (auto simp add: postfix_def)
-
-lemma postfix_is_subset: "xs >>= ys ==> set ys \<subseteq> set xs"
-proof -
- assume "xs >>= ys"
- then obtain zs where "xs = zs @ ys" ..
- then show ?thesis by (induct zs) auto
-qed
-
-lemma postfix_ConsD2: "x#xs >>= y#ys ==> xs >>= ys"
-proof -
- assume "x#xs >>= y#ys"
- then obtain zs where "x#xs = zs @ y#ys" ..
- then show ?thesis
- by (induct zs) (auto intro!: postfix_appendI postfix_ConsI)
-qed
-
-lemma postfix_to_prefix [code]: "xs >>= ys \<longleftrightarrow> rev ys \<le> rev xs"
-proof
- assume "xs >>= ys"
- then obtain zs where "xs = zs @ ys" ..
- then have "rev xs = rev ys @ rev zs" by simp
- then show "rev ys <= rev xs" ..
-next
- assume "rev ys <= rev xs"
- then obtain zs where "rev xs = rev ys @ zs" ..
- then have "rev (rev xs) = rev zs @ rev (rev ys)" by simp
- then have "xs = rev zs @ ys" by simp
- then show "xs >>= ys" ..
-qed
-
-lemma distinct_postfix: "distinct xs \<Longrightarrow> xs >>= ys \<Longrightarrow> distinct ys"
- by (clarsimp elim!: postfixE)
-
-lemma postfix_map: "xs >>= ys \<Longrightarrow> map f xs >>= map f ys"
- by (auto elim!: postfixE intro: postfixI)
-
-lemma postfix_drop: "as >>= drop n as"
- 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)
-
-lemma parallelD1: "x \<parallel> y \<Longrightarrow> \<not> x \<le> y"
- by blast
-
-lemma parallelD2: "x \<parallel> y \<Longrightarrow> \<not> y \<le> x"
- by blast
-
-lemma parallel_Nil1 [simp]: "\<not> x \<parallel> []"
- unfolding parallel_def by simp
-
-lemma parallel_Nil2 [simp]: "\<not> [] \<parallel> x"
- unfolding parallel_def by simp
-
-lemma Cons_parallelI1: "a \<noteq> b \<Longrightarrow> a # as \<parallel> b # bs"
- 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)
-
-lemma not_equal_is_parallel:
- assumes neq: "xs \<noteq> ys"
- and len: "length xs = length ys"
- shows "xs \<parallel> ys"
- using len neq
-proof (induct rule: list_induct2)
- case Nil
- then show ?case by simp
-next
- case (Cons a as b bs)
- have ih: "as \<noteq> bs \<Longrightarrow> as \<parallel> bs" by fact
- show ?case
- proof (cases "a = b")
- case True
- then have "as \<noteq> bs" using Cons by simp
- then show ?thesis by (rule Cons_parallelI2 [OF True ih])
- next
- case False
- then show ?thesis by (rule Cons_parallelI1)
- qed
-qed
-
-end