author | wenzelm |
Thu, 30 May 2013 20:38:50 +0200 | |
changeset 52252 | 81fcc11d8c65 |
parent 50516 | ed6b40d15d1c |
child 52729 | 412c9e0381a1 |
permissions | -rw-r--r-- |
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(* Title: HOL/Library/Sublist.thy |
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Author: Tobias Nipkow and Markus Wenzel, TU Muenchen |
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Author: Christian Sternagel, JAIST |
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*) |
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header {* List prefixes, suffixes, and homeomorphic embedding *} |
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theory Sublist |
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imports Main |
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begin |
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subsection {* Prefix order on lists *} |
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definition prefixeq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" |
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where "prefixeq xs ys \<longleftrightarrow> (\<exists>zs. ys = xs @ zs)" |
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definition prefix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" |
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where "prefix xs ys \<longleftrightarrow> prefixeq xs ys \<and> xs \<noteq> ys" |
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interpretation prefix_order: order prefixeq prefix |
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by default (auto simp: prefixeq_def prefix_def) |
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interpretation prefix_bot: bot prefixeq prefix Nil |
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by default (simp add: prefixeq_def) |
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lemma prefixeqI [intro?]: "ys = xs @ zs \<Longrightarrow> prefixeq xs ys" |
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unfolding prefixeq_def by blast |
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lemma prefixeqE [elim?]: |
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assumes "prefixeq xs ys" |
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obtains zs where "ys = xs @ zs" |
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using assms unfolding prefixeq_def by blast |
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lemma prefixI' [intro?]: "ys = xs @ z # zs \<Longrightarrow> prefix xs ys" |
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unfolding prefix_def prefixeq_def by blast |
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lemma prefixE' [elim?]: |
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assumes "prefix xs ys" |
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obtains z zs where "ys = xs @ z # zs" |
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proof - |
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from `prefix xs ys` obtain us where "ys = xs @ us" and "xs \<noteq> ys" |
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unfolding prefix_def prefixeq_def by blast |
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with that show ?thesis by (auto simp add: neq_Nil_conv) |
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qed |
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lemma prefixI [intro?]: "prefixeq xs ys \<Longrightarrow> xs \<noteq> ys \<Longrightarrow> prefix xs ys" |
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unfolding prefix_def by blast |
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lemma prefixE [elim?]: |
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fixes xs ys :: "'a list" |
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assumes "prefix xs ys" |
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obtains "prefixeq xs ys" and "xs \<noteq> ys" |
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using assms unfolding prefix_def by blast |
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subsection {* Basic properties of prefixes *} |
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theorem Nil_prefixeq [iff]: "prefixeq [] xs" |
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by (simp add: prefixeq_def) |
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theorem prefixeq_Nil [simp]: "(prefixeq xs []) = (xs = [])" |
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by (induct xs) (simp_all add: prefixeq_def) |
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lemma prefixeq_snoc [simp]: "prefixeq xs (ys @ [y]) \<longleftrightarrow> xs = ys @ [y] \<or> prefixeq xs ys" |
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proof |
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assume "prefixeq xs (ys @ [y])" |
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then obtain zs where zs: "ys @ [y] = xs @ zs" .. |
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show "xs = ys @ [y] \<or> prefixeq xs ys" |
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by (metis append_Nil2 butlast_append butlast_snoc prefixeqI zs) |
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next |
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assume "xs = ys @ [y] \<or> prefixeq xs ys" |
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then show "prefixeq xs (ys @ [y])" |
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by (metis prefix_order.eq_iff prefix_order.order_trans prefixeqI) |
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qed |
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lemma Cons_prefixeq_Cons [simp]: "prefixeq (x # xs) (y # ys) = (x = y \<and> prefixeq xs ys)" |
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by (auto simp add: prefixeq_def) |
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lemma prefixeq_code [code]: |
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"prefixeq [] xs \<longleftrightarrow> True" |
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"prefixeq (x # xs) [] \<longleftrightarrow> False" |
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"prefixeq (x # xs) (y # ys) \<longleftrightarrow> x = y \<and> prefixeq xs ys" |
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by simp_all |
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lemma same_prefixeq_prefixeq [simp]: "prefixeq (xs @ ys) (xs @ zs) = prefixeq ys zs" |
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by (induct xs) simp_all |
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lemma same_prefixeq_nil [iff]: "prefixeq (xs @ ys) xs = (ys = [])" |
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by (metis append_Nil2 append_self_conv prefix_order.eq_iff prefixeqI) |
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lemma prefixeq_prefixeq [simp]: "prefixeq xs ys \<Longrightarrow> prefixeq xs (ys @ zs)" |
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by (metis prefix_order.le_less_trans prefixeqI prefixE prefixI) |
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lemma append_prefixeqD: "prefixeq (xs @ ys) zs \<Longrightarrow> prefixeq xs zs" |
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by (auto simp add: prefixeq_def) |
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theorem prefixeq_Cons: "prefixeq xs (y # ys) = (xs = [] \<or> (\<exists>zs. xs = y # zs \<and> prefixeq zs ys))" |
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by (cases xs) (auto simp add: prefixeq_def) |
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theorem prefixeq_append: |
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"prefixeq xs (ys @ zs) = (prefixeq xs ys \<or> (\<exists>us. xs = ys @ us \<and> prefixeq us zs))" |
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apply (induct zs rule: rev_induct) |
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apply force |
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apply (simp del: append_assoc add: append_assoc [symmetric]) |
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apply (metis append_eq_appendI) |
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done |
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lemma append_one_prefixeq: |
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"prefixeq xs ys \<Longrightarrow> length xs < length ys \<Longrightarrow> prefixeq (xs @ [ys ! length xs]) ys" |
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unfolding prefixeq_def |
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by (metis Cons_eq_appendI append_eq_appendI append_eq_conv_conj |
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eq_Nil_appendI nth_drop') |
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theorem prefixeq_length_le: "prefixeq xs ys \<Longrightarrow> length xs \<le> length ys" |
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by (auto simp add: prefixeq_def) |
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lemma prefixeq_same_cases: |
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"prefixeq (xs\<^isub>1::'a list) ys \<Longrightarrow> prefixeq xs\<^isub>2 ys \<Longrightarrow> prefixeq xs\<^isub>1 xs\<^isub>2 \<or> prefixeq xs\<^isub>2 xs\<^isub>1" |
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unfolding prefixeq_def by (metis append_eq_append_conv2) |
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lemma set_mono_prefixeq: "prefixeq xs ys \<Longrightarrow> set xs \<subseteq> set ys" |
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by (auto simp add: prefixeq_def) |
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lemma take_is_prefixeq: "prefixeq (take n xs) xs" |
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unfolding prefixeq_def by (metis append_take_drop_id) |
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lemma map_prefixeqI: "prefixeq xs ys \<Longrightarrow> prefixeq (map f xs) (map f ys)" |
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by (auto simp: prefixeq_def) |
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lemma prefixeq_length_less: "prefix xs ys \<Longrightarrow> length xs < length ys" |
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by (auto simp: prefix_def prefixeq_def) |
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lemma prefix_simps [simp, code]: |
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"prefix xs [] \<longleftrightarrow> False" |
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"prefix [] (x # xs) \<longleftrightarrow> True" |
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"prefix (x # xs) (y # ys) \<longleftrightarrow> x = y \<and> prefix xs ys" |
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by (simp_all add: prefix_def cong: conj_cong) |
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lemma take_prefix: "prefix xs ys \<Longrightarrow> prefix (take n xs) ys" |
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apply (induct n arbitrary: xs ys) |
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apply (case_tac ys, simp_all)[1] |
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apply (metis prefix_order.less_trans prefixI take_is_prefixeq) |
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done |
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lemma not_prefixeq_cases: |
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assumes pfx: "\<not> prefixeq ps ls" |
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obtains |
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(c1) "ps \<noteq> []" and "ls = []" |
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| (c2) a as x xs where "ps = a#as" and "ls = x#xs" and "x = a" and "\<not> prefixeq as xs" |
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| (c3) a as x xs where "ps = a#as" and "ls = x#xs" and "x \<noteq> a" |
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proof (cases ps) |
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case Nil |
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then show ?thesis using pfx by simp |
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next |
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case (Cons a as) |
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note c = `ps = a#as` |
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show ?thesis |
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proof (cases ls) |
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case Nil then show ?thesis by (metis append_Nil2 pfx c1 same_prefixeq_nil) |
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next |
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case (Cons x xs) |
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show ?thesis |
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proof (cases "x = a") |
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case True |
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have "\<not> prefixeq as xs" using pfx c Cons True by simp |
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with c Cons True show ?thesis by (rule c2) |
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next |
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case False |
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with c Cons show ?thesis by (rule c3) |
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qed |
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qed |
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qed |
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lemma not_prefixeq_induct [consumes 1, case_names Nil Neq Eq]: |
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assumes np: "\<not> prefixeq ps ls" |
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and base: "\<And>x xs. P (x#xs) []" |
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and r1: "\<And>x xs y ys. x \<noteq> y \<Longrightarrow> P (x#xs) (y#ys)" |
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and r2: "\<And>x xs y ys. \<lbrakk> x = y; \<not> prefixeq xs ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys)" |
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shows "P ps ls" using np |
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proof (induct ls arbitrary: ps) |
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case Nil then show ?case |
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by (auto simp: neq_Nil_conv elim!: not_prefixeq_cases intro!: base) |
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next |
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case (Cons y ys) |
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then have npfx: "\<not> prefixeq ps (y # ys)" by simp |
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then obtain x xs where pv: "ps = x # xs" |
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by (rule not_prefixeq_cases) auto |
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show ?case by (metis Cons.hyps Cons_prefixeq_Cons npfx pv r1 r2) |
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qed |
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subsection {* Parallel lists *} |
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definition parallel :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infixl "\<parallel>" 50) |
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where "(xs \<parallel> ys) = (\<not> prefixeq xs ys \<and> \<not> prefixeq ys xs)" |
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lemma parallelI [intro]: "\<not> prefixeq xs ys \<Longrightarrow> \<not> prefixeq ys xs \<Longrightarrow> xs \<parallel> ys" |
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unfolding parallel_def by blast |
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lemma parallelE [elim]: |
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assumes "xs \<parallel> ys" |
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obtains "\<not> prefixeq xs ys \<and> \<not> prefixeq ys xs" |
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using assms unfolding parallel_def by blast |
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theorem prefixeq_cases: |
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obtains "prefixeq xs ys" | "prefix ys xs" | "xs \<parallel> ys" |
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unfolding parallel_def prefix_def by blast |
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theorem parallel_decomp: |
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"xs \<parallel> ys \<Longrightarrow> \<exists>as b bs c cs. b \<noteq> c \<and> xs = as @ b # bs \<and> ys = as @ c # cs" |
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proof (induct xs rule: rev_induct) |
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case Nil |
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then have False by auto |
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then show ?case .. |
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next |
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case (snoc x xs) |
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show ?case |
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proof (rule prefixeq_cases) |
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assume le: "prefixeq xs ys" |
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then obtain ys' where ys: "ys = xs @ ys'" .. |
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show ?thesis |
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proof (cases ys') |
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assume "ys' = []" |
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then show ?thesis by (metis append_Nil2 parallelE prefixeqI snoc.prems ys) |
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next |
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fix c cs assume ys': "ys' = c # cs" |
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then show ?thesis |
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by (metis Cons_eq_appendI eq_Nil_appendI parallelE prefixeqI |
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same_prefixeq_prefixeq snoc.prems ys) |
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qed |
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next |
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assume "prefix ys xs" |
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then have "prefixeq ys (xs @ [x])" by (simp add: prefix_def) |
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with snoc have False by blast |
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then show ?thesis .. |
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next |
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assume "xs \<parallel> ys" |
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with snoc obtain as b bs c cs where neq: "(b::'a) \<noteq> c" |
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and xs: "xs = as @ b # bs" and ys: "ys = as @ c # cs" |
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by blast |
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from xs have "xs @ [x] = as @ b # (bs @ [x])" by simp |
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with neq ys show ?thesis by blast |
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qed |
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qed |
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lemma parallel_append: "a \<parallel> b \<Longrightarrow> a @ c \<parallel> b @ d" |
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apply (rule parallelI) |
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apply (erule parallelE, erule conjE, |
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induct rule: not_prefixeq_induct, simp+)+ |
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done |
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lemma parallel_appendI: "xs \<parallel> ys \<Longrightarrow> x = xs @ xs' \<Longrightarrow> y = ys @ ys' \<Longrightarrow> x \<parallel> y" |
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by (simp add: parallel_append) |
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lemma parallel_commute: "a \<parallel> b \<longleftrightarrow> b \<parallel> a" |
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unfolding parallel_def by auto |
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subsection {* Suffix order on lists *} |
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definition suffixeq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" |
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where "suffixeq xs ys = (\<exists>zs. ys = zs @ xs)" |
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definition suffix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" |
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where "suffix xs ys \<longleftrightarrow> (\<exists>us. ys = us @ xs \<and> us \<noteq> [])" |
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lemma suffix_imp_suffixeq: |
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"suffix xs ys \<Longrightarrow> suffixeq xs ys" |
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by (auto simp: suffixeq_def suffix_def) |
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lemma suffixeqI [intro?]: "ys = zs @ xs \<Longrightarrow> suffixeq xs ys" |
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unfolding suffixeq_def by blast |
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lemma suffixeqE [elim?]: |
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assumes "suffixeq xs ys" |
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obtains zs where "ys = zs @ xs" |
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using assms unfolding suffixeq_def by blast |
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lemma suffixeq_refl [iff]: "suffixeq xs xs" |
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by (auto simp add: suffixeq_def) |
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lemma suffix_trans: |
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"suffix xs ys \<Longrightarrow> suffix ys zs \<Longrightarrow> suffix xs zs" |
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by (auto simp: suffix_def) |
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lemma suffixeq_trans: "\<lbrakk>suffixeq xs ys; suffixeq ys zs\<rbrakk> \<Longrightarrow> suffixeq xs zs" |
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by (auto simp add: suffixeq_def) |
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lemma suffixeq_antisym: "\<lbrakk>suffixeq xs ys; suffixeq ys xs\<rbrakk> \<Longrightarrow> xs = ys" |
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by (auto simp add: suffixeq_def) |
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lemma suffixeq_tl [simp]: "suffixeq (tl xs) xs" |
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by (induct xs) (auto simp: suffixeq_def) |
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lemma suffix_tl [simp]: "xs \<noteq> [] \<Longrightarrow> suffix (tl xs) xs" |
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by (induct xs) (auto simp: suffix_def) |
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lemma Nil_suffixeq [iff]: "suffixeq [] xs" |
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by (simp add: suffixeq_def) |
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lemma suffixeq_Nil [simp]: "(suffixeq xs []) = (xs = [])" |
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by (auto simp add: suffixeq_def) |
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lemma suffixeq_ConsI: "suffixeq xs ys \<Longrightarrow> suffixeq xs (y # ys)" |
49087 | 301 |
by (auto simp add: suffixeq_def) |
49107 | 302 |
lemma suffixeq_ConsD: "suffixeq (x # xs) ys \<Longrightarrow> suffixeq xs ys" |
49087 | 303 |
by (auto simp add: suffixeq_def) |
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changeset
|
304 |
|
49087 | 305 |
lemma suffixeq_appendI: "suffixeq xs ys \<Longrightarrow> suffixeq xs (zs @ ys)" |
306 |
by (auto simp add: suffixeq_def) |
|
307 |
lemma suffixeq_appendD: "suffixeq (zs @ xs) ys \<Longrightarrow> suffixeq xs ys" |
|
308 |
by (auto simp add: suffixeq_def) |
|
309 |
||
310 |
lemma suffix_set_subset: |
|
311 |
"suffix xs ys \<Longrightarrow> set xs \<subseteq> set ys" by (auto simp: suffix_def) |
|
14538
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oheimb
parents:
14300
diff
changeset
|
312 |
|
49087 | 313 |
lemma suffixeq_set_subset: |
314 |
"suffixeq xs ys \<Longrightarrow> set xs \<subseteq> set ys" by (auto simp: suffixeq_def) |
|
315 |
||
49107 | 316 |
lemma suffixeq_ConsD2: "suffixeq (x # xs) (y # ys) \<Longrightarrow> suffixeq xs ys" |
21305 | 317 |
proof - |
49107 | 318 |
assume "suffixeq (x # xs) (y # ys)" |
319 |
then obtain zs where "y # ys = zs @ x # xs" .. |
|
49087 | 320 |
then show ?thesis |
321 |
by (induct zs) (auto intro!: suffixeq_appendI suffixeq_ConsI) |
|
21305 | 322 |
qed |
14538
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oheimb
parents:
14300
diff
changeset
|
323 |
|
49087 | 324 |
lemma suffixeq_to_prefixeq [code]: "suffixeq xs ys \<longleftrightarrow> prefixeq (rev xs) (rev ys)" |
325 |
proof |
|
326 |
assume "suffixeq xs ys" |
|
327 |
then obtain zs where "ys = zs @ xs" .. |
|
328 |
then have "rev ys = rev xs @ rev zs" by simp |
|
329 |
then show "prefixeq (rev xs) (rev ys)" .. |
|
330 |
next |
|
331 |
assume "prefixeq (rev xs) (rev ys)" |
|
332 |
then obtain zs where "rev ys = rev xs @ zs" .. |
|
333 |
then have "rev (rev ys) = rev zs @ rev (rev xs)" by simp |
|
334 |
then have "ys = rev zs @ xs" by simp |
|
335 |
then show "suffixeq xs ys" .. |
|
21305 | 336 |
qed |
14538
1d9d75a8efae
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oheimb
parents:
14300
diff
changeset
|
337 |
|
49087 | 338 |
lemma distinct_suffixeq: "distinct ys \<Longrightarrow> suffixeq xs ys \<Longrightarrow> distinct xs" |
339 |
by (clarsimp elim!: suffixeqE) |
|
17201 | 340 |
|
49087 | 341 |
lemma suffixeq_map: "suffixeq xs ys \<Longrightarrow> suffixeq (map f xs) (map f ys)" |
342 |
by (auto elim!: suffixeqE intro: suffixeqI) |
|
25299 | 343 |
|
49087 | 344 |
lemma suffixeq_drop: "suffixeq (drop n as) as" |
345 |
unfolding suffixeq_def |
|
25692 | 346 |
apply (rule exI [where x = "take n as"]) |
347 |
apply simp |
|
348 |
done |
|
25299 | 349 |
|
49087 | 350 |
lemma suffixeq_take: "suffixeq xs ys \<Longrightarrow> ys = take (length ys - length xs) ys @ xs" |
49107 | 351 |
by (auto elim!: suffixeqE) |
25299 | 352 |
|
49107 | 353 |
lemma suffixeq_suffix_reflclp_conv: "suffixeq = suffix\<^sup>=\<^sup>=" |
49087 | 354 |
proof (intro ext iffI) |
355 |
fix xs ys :: "'a list" |
|
356 |
assume "suffixeq xs ys" |
|
357 |
show "suffix\<^sup>=\<^sup>= xs ys" |
|
358 |
proof |
|
359 |
assume "xs \<noteq> ys" |
|
49107 | 360 |
with `suffixeq xs ys` show "suffix xs ys" |
361 |
by (auto simp: suffixeq_def suffix_def) |
|
49087 | 362 |
qed |
363 |
next |
|
364 |
fix xs ys :: "'a list" |
|
365 |
assume "suffix\<^sup>=\<^sup>= xs ys" |
|
49107 | 366 |
then show "suffixeq xs ys" |
49087 | 367 |
proof |
49107 | 368 |
assume "suffix xs ys" then show "suffixeq xs ys" |
369 |
by (rule suffix_imp_suffixeq) |
|
49087 | 370 |
next |
49107 | 371 |
assume "xs = ys" then show "suffixeq xs ys" |
372 |
by (auto simp: suffixeq_def) |
|
49087 | 373 |
qed |
374 |
qed |
|
375 |
||
376 |
lemma parallelD1: "x \<parallel> y \<Longrightarrow> \<not> prefixeq x y" |
|
25692 | 377 |
by blast |
25299 | 378 |
|
49087 | 379 |
lemma parallelD2: "x \<parallel> y \<Longrightarrow> \<not> prefixeq y x" |
25692 | 380 |
by blast |
25355 | 381 |
|
382 |
lemma parallel_Nil1 [simp]: "\<not> x \<parallel> []" |
|
25692 | 383 |
unfolding parallel_def by simp |
25355 | 384 |
|
25299 | 385 |
lemma parallel_Nil2 [simp]: "\<not> [] \<parallel> x" |
25692 | 386 |
unfolding parallel_def by simp |
25299 | 387 |
|
25564 | 388 |
lemma Cons_parallelI1: "a \<noteq> b \<Longrightarrow> a # as \<parallel> b # bs" |
25692 | 389 |
by auto |
25299 | 390 |
|
25564 | 391 |
lemma Cons_parallelI2: "\<lbrakk> a = b; as \<parallel> bs \<rbrakk> \<Longrightarrow> a # as \<parallel> b # bs" |
49087 | 392 |
by (metis Cons_prefixeq_Cons parallelE parallelI) |
25665 | 393 |
|
25299 | 394 |
lemma not_equal_is_parallel: |
395 |
assumes neq: "xs \<noteq> ys" |
|
25356 | 396 |
and len: "length xs = length ys" |
397 |
shows "xs \<parallel> ys" |
|
25299 | 398 |
using len neq |
25355 | 399 |
proof (induct rule: list_induct2) |
26445 | 400 |
case Nil |
25356 | 401 |
then show ?case by simp |
25299 | 402 |
next |
26445 | 403 |
case (Cons a as b bs) |
25355 | 404 |
have ih: "as \<noteq> bs \<Longrightarrow> as \<parallel> bs" by fact |
25299 | 405 |
show ?case |
406 |
proof (cases "a = b") |
|
25355 | 407 |
case True |
26445 | 408 |
then have "as \<noteq> bs" using Cons by simp |
25355 | 409 |
then show ?thesis by (rule Cons_parallelI2 [OF True ih]) |
25299 | 410 |
next |
411 |
case False |
|
25355 | 412 |
then show ?thesis by (rule Cons_parallelI1) |
25299 | 413 |
qed |
414 |
qed |
|
22178 | 415 |
|
49107 | 416 |
lemma suffix_reflclp_conv: "suffix\<^sup>=\<^sup>= = suffixeq" |
49087 | 417 |
by (intro ext) (auto simp: suffixeq_def suffix_def) |
418 |
||
49107 | 419 |
lemma suffix_lists: "suffix xs ys \<Longrightarrow> ys \<in> lists A \<Longrightarrow> xs \<in> lists A" |
49087 | 420 |
unfolding suffix_def by auto |
421 |
||
422 |
||
50516 | 423 |
subsection {* Homeomorphic embedding on lists *} |
49087 | 424 |
|
50516 | 425 |
inductive list_hembeq :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" |
49087 | 426 |
for P :: "('a \<Rightarrow> 'a \<Rightarrow> bool)" |
427 |
where |
|
50516 | 428 |
list_hembeq_Nil [intro, simp]: "list_hembeq P [] ys" |
429 |
| list_hembeq_Cons [intro] : "list_hembeq P xs ys \<Longrightarrow> list_hembeq P xs (y#ys)" |
|
430 |
| list_hembeq_Cons2 [intro]: "P\<^sup>=\<^sup>= x y \<Longrightarrow> list_hembeq P xs ys \<Longrightarrow> list_hembeq P (x#xs) (y#ys)" |
|
431 |
||
432 |
lemma list_hembeq_Nil2 [simp]: |
|
433 |
assumes "list_hembeq P xs []" shows "xs = []" |
|
434 |
using assms by (cases rule: list_hembeq.cases) auto |
|
49087 | 435 |
|
50516 | 436 |
lemma list_hembeq_refl [simp, intro!]: |
437 |
"list_hembeq P xs xs" |
|
438 |
by (induct xs) auto |
|
49087 | 439 |
|
50516 | 440 |
lemma list_hembeq_Cons_Nil [simp]: "list_hembeq P (x#xs) [] = False" |
49087 | 441 |
proof - |
50516 | 442 |
{ assume "list_hembeq P (x#xs) []" |
443 |
from list_hembeq_Nil2 [OF this] have False by simp |
|
49087 | 444 |
} moreover { |
445 |
assume False |
|
50516 | 446 |
then have "list_hembeq P (x#xs) []" by simp |
49087 | 447 |
} ultimately show ?thesis by blast |
448 |
qed |
|
449 |
||
50516 | 450 |
lemma list_hembeq_append2 [intro]: "list_hembeq P xs ys \<Longrightarrow> list_hembeq P xs (zs @ ys)" |
49087 | 451 |
by (induct zs) auto |
452 |
||
50516 | 453 |
lemma list_hembeq_prefix [intro]: |
454 |
assumes "list_hembeq P xs ys" shows "list_hembeq P xs (ys @ zs)" |
|
49087 | 455 |
using assms |
456 |
by (induct arbitrary: zs) auto |
|
457 |
||
50516 | 458 |
lemma list_hembeq_ConsD: |
459 |
assumes "list_hembeq P (x#xs) ys" |
|
460 |
shows "\<exists>us v vs. ys = us @ v # vs \<and> P\<^sup>=\<^sup>= x v \<and> list_hembeq P xs vs" |
|
49087 | 461 |
using assms |
49107 | 462 |
proof (induct x \<equiv> "x # xs" ys arbitrary: x xs) |
50516 | 463 |
case list_hembeq_Cons |
49107 | 464 |
then show ?case by (metis append_Cons) |
49087 | 465 |
next |
50516 | 466 |
case (list_hembeq_Cons2 x y xs ys) |
49107 | 467 |
then show ?case by (cases xs) (auto, blast+) |
49087 | 468 |
qed |
469 |
||
50516 | 470 |
lemma list_hembeq_appendD: |
471 |
assumes "list_hembeq P (xs @ ys) zs" |
|
472 |
shows "\<exists>us vs. zs = us @ vs \<and> list_hembeq P xs us \<and> list_hembeq P ys vs" |
|
49087 | 473 |
using assms |
474 |
proof (induction xs arbitrary: ys zs) |
|
49107 | 475 |
case Nil then show ?case by auto |
49087 | 476 |
next |
477 |
case (Cons x xs) |
|
478 |
then obtain us v vs where "zs = us @ v # vs" |
|
50516 | 479 |
and "P\<^sup>=\<^sup>= x v" and "list_hembeq P (xs @ ys) vs" by (auto dest: list_hembeq_ConsD) |
480 |
with Cons show ?case by (metis append_Cons append_assoc list_hembeq_Cons2 list_hembeq_append2) |
|
49087 | 481 |
qed |
482 |
||
50516 | 483 |
lemma list_hembeq_suffix: |
484 |
assumes "list_hembeq P xs ys" and "suffix ys zs" |
|
485 |
shows "list_hembeq P xs zs" |
|
486 |
using assms(2) and list_hembeq_append2 [OF assms(1)] by (auto simp: suffix_def) |
|
49087 | 487 |
|
50516 | 488 |
lemma list_hembeq_suffixeq: |
489 |
assumes "list_hembeq P xs ys" and "suffixeq ys zs" |
|
490 |
shows "list_hembeq P xs zs" |
|
491 |
using assms and list_hembeq_suffix unfolding suffixeq_suffix_reflclp_conv by auto |
|
49087 | 492 |
|
50516 | 493 |
lemma list_hembeq_length: "list_hembeq P xs ys \<Longrightarrow> length xs \<le> length ys" |
494 |
by (induct rule: list_hembeq.induct) auto |
|
49087 | 495 |
|
50516 | 496 |
lemma list_hembeq_trans: |
497 |
assumes "\<And>x y z. \<lbrakk>x \<in> A; y \<in> A; z \<in> A; P x y; P y z\<rbrakk> \<Longrightarrow> P x z" |
|
498 |
shows "\<And>xs ys zs. \<lbrakk>xs \<in> lists A; ys \<in> lists A; zs \<in> lists A; |
|
499 |
list_hembeq P xs ys; list_hembeq P ys zs\<rbrakk> \<Longrightarrow> list_hembeq P xs zs" |
|
500 |
proof - |
|
49087 | 501 |
fix xs ys zs |
50516 | 502 |
assume "list_hembeq P xs ys" and "list_hembeq P ys zs" |
49087 | 503 |
and "xs \<in> lists A" and "ys \<in> lists A" and "zs \<in> lists A" |
50516 | 504 |
then show "list_hembeq P xs zs" |
49087 | 505 |
proof (induction arbitrary: zs) |
50516 | 506 |
case list_hembeq_Nil show ?case by blast |
49087 | 507 |
next |
50516 | 508 |
case (list_hembeq_Cons xs ys y) |
509 |
from list_hembeq_ConsD [OF `list_hembeq P (y#ys) zs`] obtain us v vs |
|
510 |
where zs: "zs = us @ v # vs" and "P\<^sup>=\<^sup>= y v" and "list_hembeq P ys vs" by blast |
|
511 |
then have "list_hembeq P ys (v#vs)" by blast |
|
512 |
then have "list_hembeq P ys zs" unfolding zs by (rule list_hembeq_append2) |
|
513 |
from list_hembeq_Cons.IH [OF this] and list_hembeq_Cons.prems show ?case by simp |
|
49087 | 514 |
next |
50516 | 515 |
case (list_hembeq_Cons2 x y xs ys) |
516 |
from list_hembeq_ConsD [OF `list_hembeq P (y#ys) zs`] obtain us v vs |
|
517 |
where zs: "zs = us @ v # vs" and "P\<^sup>=\<^sup>= y v" and "list_hembeq P ys vs" by blast |
|
518 |
with list_hembeq_Cons2 have "list_hembeq P xs vs" by simp |
|
519 |
moreover have "P\<^sup>=\<^sup>= x v" |
|
49087 | 520 |
proof - |
521 |
from zs and `zs \<in> lists A` have "v \<in> A" by auto |
|
50516 | 522 |
moreover have "x \<in> A" and "y \<in> A" using list_hembeq_Cons2 by simp_all |
523 |
ultimately show ?thesis |
|
524 |
using `P\<^sup>=\<^sup>= x y` and `P\<^sup>=\<^sup>= y v` and assms |
|
525 |
by blast |
|
49087 | 526 |
qed |
50516 | 527 |
ultimately have "list_hembeq P (x#xs) (v#vs)" by blast |
528 |
then show ?case unfolding zs by (rule list_hembeq_append2) |
|
49087 | 529 |
qed |
530 |
qed |
|
531 |
||
532 |
||
50516 | 533 |
subsection {* Sublists (special case of homeomorphic embedding) *} |
49087 | 534 |
|
50516 | 535 |
abbreviation sublisteq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" |
536 |
where "sublisteq xs ys \<equiv> list_hembeq (op =) xs ys" |
|
49087 | 537 |
|
50516 | 538 |
lemma sublisteq_Cons2: "sublisteq xs ys \<Longrightarrow> sublisteq (x#xs) (x#ys)" by auto |
49087 | 539 |
|
50516 | 540 |
lemma sublisteq_same_length: |
541 |
assumes "sublisteq xs ys" and "length xs = length ys" shows "xs = ys" |
|
542 |
using assms by (induct) (auto dest: list_hembeq_length) |
|
49087 | 543 |
|
50516 | 544 |
lemma not_sublisteq_length [simp]: "length ys < length xs \<Longrightarrow> \<not> sublisteq xs ys" |
545 |
by (metis list_hembeq_length linorder_not_less) |
|
49087 | 546 |
|
547 |
lemma [code]: |
|
50516 | 548 |
"list_hembeq P [] ys \<longleftrightarrow> True" |
549 |
"list_hembeq P (x#xs) [] \<longleftrightarrow> False" |
|
49087 | 550 |
by (simp_all) |
551 |
||
50516 | 552 |
lemma sublisteq_Cons': "sublisteq (x#xs) ys \<Longrightarrow> sublisteq xs ys" |
553 |
by (induct xs) (auto dest: list_hembeq_ConsD) |
|
49087 | 554 |
|
50516 | 555 |
lemma sublisteq_Cons2': |
556 |
assumes "sublisteq (x#xs) (x#ys)" shows "sublisteq xs ys" |
|
557 |
using assms by (cases) (rule sublisteq_Cons') |
|
49087 | 558 |
|
50516 | 559 |
lemma sublisteq_Cons2_neq: |
560 |
assumes "sublisteq (x#xs) (y#ys)" |
|
561 |
shows "x \<noteq> y \<Longrightarrow> sublisteq (x#xs) ys" |
|
49087 | 562 |
using assms by (cases) auto |
563 |
||
50516 | 564 |
lemma sublisteq_Cons2_iff [simp, code]: |
565 |
"sublisteq (x#xs) (y#ys) = (if x = y then sublisteq xs ys else sublisteq (x#xs) ys)" |
|
566 |
by (metis list_hembeq_Cons sublisteq_Cons2 sublisteq_Cons2' sublisteq_Cons2_neq) |
|
49087 | 567 |
|
50516 | 568 |
lemma sublisteq_append': "sublisteq (zs @ xs) (zs @ ys) \<longleftrightarrow> sublisteq xs ys" |
49087 | 569 |
by (induct zs) simp_all |
570 |
||
50516 | 571 |
lemma sublisteq_refl [simp, intro!]: "sublisteq xs xs" by (induct xs) simp_all |
49087 | 572 |
|
50516 | 573 |
lemma sublisteq_antisym: |
574 |
assumes "sublisteq xs ys" and "sublisteq ys xs" |
|
49087 | 575 |
shows "xs = ys" |
576 |
using assms |
|
577 |
proof (induct) |
|
50516 | 578 |
case list_hembeq_Nil |
579 |
from list_hembeq_Nil2 [OF this] show ?case by simp |
|
49087 | 580 |
next |
50516 | 581 |
case list_hembeq_Cons2 |
49107 | 582 |
then show ?case by simp |
49087 | 583 |
next |
50516 | 584 |
case list_hembeq_Cons |
49107 | 585 |
then show ?case |
50516 | 586 |
by (metis sublisteq_Cons' list_hembeq_length Suc_length_conv Suc_n_not_le_n) |
49087 | 587 |
qed |
588 |
||
50516 | 589 |
lemma sublisteq_trans: "sublisteq xs ys \<Longrightarrow> sublisteq ys zs \<Longrightarrow> sublisteq xs zs" |
590 |
by (rule list_hembeq_trans [of UNIV "op ="]) auto |
|
49087 | 591 |
|
50516 | 592 |
lemma sublisteq_append_le_same_iff: "sublisteq (xs @ ys) ys \<longleftrightarrow> xs = []" |
593 |
by (auto dest: list_hembeq_length) |
|
49087 | 594 |
|
50516 | 595 |
lemma list_hembeq_append_mono: |
596 |
"\<lbrakk> list_hembeq P xs xs'; list_hembeq P ys ys' \<rbrakk> \<Longrightarrow> list_hembeq P (xs@ys) (xs'@ys')" |
|
597 |
apply (induct rule: list_hembeq.induct) |
|
598 |
apply (metis eq_Nil_appendI list_hembeq_append2) |
|
599 |
apply (metis append_Cons list_hembeq_Cons) |
|
600 |
apply (metis append_Cons list_hembeq_Cons2) |
|
49107 | 601 |
done |
49087 | 602 |
|
603 |
||
604 |
subsection {* Appending elements *} |
|
605 |
||
50516 | 606 |
lemma sublisteq_append [simp]: |
607 |
"sublisteq (xs @ zs) (ys @ zs) \<longleftrightarrow> sublisteq xs ys" (is "?l = ?r") |
|
49087 | 608 |
proof |
50516 | 609 |
{ fix xs' ys' xs ys zs :: "'a list" assume "sublisteq xs' ys'" |
610 |
then have "xs' = xs @ zs & ys' = ys @ zs \<longrightarrow> sublisteq xs ys" |
|
49087 | 611 |
proof (induct arbitrary: xs ys zs) |
50516 | 612 |
case list_hembeq_Nil show ?case by simp |
49087 | 613 |
next |
50516 | 614 |
case (list_hembeq_Cons xs' ys' x) |
615 |
{ assume "ys=[]" then have ?case using list_hembeq_Cons(1) by auto } |
|
49087 | 616 |
moreover |
617 |
{ fix us assume "ys = x#us" |
|
50516 | 618 |
then have ?case using list_hembeq_Cons(2) by(simp add: list_hembeq.list_hembeq_Cons) } |
49087 | 619 |
ultimately show ?case by (auto simp:Cons_eq_append_conv) |
620 |
next |
|
50516 | 621 |
case (list_hembeq_Cons2 x y xs' ys') |
622 |
{ assume "xs=[]" then have ?case using list_hembeq_Cons2(1) by auto } |
|
49087 | 623 |
moreover |
50516 | 624 |
{ fix us vs assume "xs=x#us" "ys=x#vs" then have ?case using list_hembeq_Cons2 by auto} |
49087 | 625 |
moreover |
50516 | 626 |
{ fix us assume "xs=x#us" "ys=[]" then have ?case using list_hembeq_Cons2(2) by bestsimp } |
627 |
ultimately show ?case using `op =\<^sup>=\<^sup>= x y` by (auto simp: Cons_eq_append_conv) |
|
49087 | 628 |
qed } |
629 |
moreover assume ?l |
|
630 |
ultimately show ?r by blast |
|
631 |
next |
|
50516 | 632 |
assume ?r then show ?l by (metis list_hembeq_append_mono sublisteq_refl) |
49087 | 633 |
qed |
634 |
||
50516 | 635 |
lemma sublisteq_drop_many: "sublisteq xs ys \<Longrightarrow> sublisteq xs (zs @ ys)" |
49087 | 636 |
by (induct zs) auto |
637 |
||
50516 | 638 |
lemma sublisteq_rev_drop_many: "sublisteq xs ys \<Longrightarrow> sublisteq xs (ys @ zs)" |
639 |
by (metis append_Nil2 list_hembeq_Nil list_hembeq_append_mono) |
|
49087 | 640 |
|
641 |
||
642 |
subsection {* Relation to standard list operations *} |
|
643 |
||
50516 | 644 |
lemma sublisteq_map: |
645 |
assumes "sublisteq xs ys" shows "sublisteq (map f xs) (map f ys)" |
|
49087 | 646 |
using assms by (induct) auto |
647 |
||
50516 | 648 |
lemma sublisteq_filter_left [simp]: "sublisteq (filter P xs) xs" |
49087 | 649 |
by (induct xs) auto |
650 |
||
50516 | 651 |
lemma sublisteq_filter [simp]: |
652 |
assumes "sublisteq xs ys" shows "sublisteq (filter P xs) (filter P ys)" |
|
49087 | 653 |
using assms by (induct) auto |
654 |
||
50516 | 655 |
lemma "sublisteq xs ys \<longleftrightarrow> (\<exists>N. xs = sublist ys N)" (is "?L = ?R") |
49087 | 656 |
proof |
657 |
assume ?L |
|
49107 | 658 |
then show ?R |
49087 | 659 |
proof (induct) |
50516 | 660 |
case list_hembeq_Nil show ?case by (metis sublist_empty) |
49087 | 661 |
next |
50516 | 662 |
case (list_hembeq_Cons xs ys x) |
49087 | 663 |
then obtain N where "xs = sublist ys N" by blast |
49107 | 664 |
then have "xs = sublist (x#ys) (Suc ` N)" |
49087 | 665 |
by (clarsimp simp add:sublist_Cons inj_image_mem_iff) |
49107 | 666 |
then show ?case by blast |
49087 | 667 |
next |
50516 | 668 |
case (list_hembeq_Cons2 x y xs ys) |
49087 | 669 |
then obtain N where "xs = sublist ys N" by blast |
49107 | 670 |
then have "x#xs = sublist (x#ys) (insert 0 (Suc ` N))" |
49087 | 671 |
by (clarsimp simp add:sublist_Cons inj_image_mem_iff) |
50516 | 672 |
moreover from list_hembeq_Cons2 have "x = y" by simp |
673 |
ultimately show ?case by blast |
|
49087 | 674 |
qed |
675 |
next |
|
676 |
assume ?R |
|
677 |
then obtain N where "xs = sublist ys N" .. |
|
50516 | 678 |
moreover have "sublisteq (sublist ys N) ys" |
49107 | 679 |
proof (induct ys arbitrary: N) |
49087 | 680 |
case Nil show ?case by simp |
681 |
next |
|
49107 | 682 |
case Cons then show ?case by (auto simp: sublist_Cons) |
49087 | 683 |
qed |
684 |
ultimately show ?L by simp |
|
685 |
qed |
|
686 |
||
10330
4362e906b745
"List prefixes" library theory (replaces old Lex/Prefix);
wenzelm
parents:
diff
changeset
|
687 |
end |