author | eberlm <eberlm@in.tum.de> |
Thu, 18 May 2017 12:02:21 +0200 | |
changeset 65869 | a6ed757b8585 |
parent 64886 | cea327ecb8e3 |
child 65954 | 431024edc9cf |
permissions | -rw-r--r-- |
49087 | 1 |
(* 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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section \<open>List prefixes, suffixes, and homeomorphic embedding\<close> |
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theory Sublist |
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imports Main |
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begin |
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subsection \<open>Prefix order on lists\<close> |
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definition prefix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" |
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where "prefix xs ys \<longleftrightarrow> (\<exists>zs. ys = xs @ zs)" |
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63117 | 17 |
definition strict_prefix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" |
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where "strict_prefix xs ys \<longleftrightarrow> prefix xs ys \<and> xs \<noteq> ys" |
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63117 | 20 |
interpretation prefix_order: order prefix strict_prefix |
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by standard (auto simp: prefix_def strict_prefix_def) |
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63117 | 23 |
interpretation prefix_bot: order_bot Nil prefix strict_prefix |
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by standard (simp add: prefix_def) |
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63117 | 26 |
lemma prefixI [intro?]: "ys = xs @ zs \<Longrightarrow> prefix xs ys" |
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unfolding prefix_def by blast |
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lemma prefixE [elim?]: |
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assumes "prefix xs ys" |
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obtains zs where "ys = xs @ zs" |
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using assms unfolding prefix_def by blast |
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63117 | 34 |
lemma strict_prefixI' [intro?]: "ys = xs @ z # zs \<Longrightarrow> strict_prefix xs ys" |
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unfolding strict_prefix_def prefix_def by blast |
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63117 | 37 |
lemma strict_prefixE' [elim?]: |
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assumes "strict_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 \<open>strict_prefix xs ys\<close> obtain us where "ys = xs @ us" and "xs \<noteq> ys" |
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unfolding strict_prefix_def prefix_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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(* FIXME rm *) |
63117 | 47 |
lemma strict_prefixI [intro?]: "prefix xs ys \<Longrightarrow> xs \<noteq> ys \<Longrightarrow> strict_prefix xs ys" |
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by(fact prefix_order.le_neq_trans) |
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lemma strict_prefixE [elim?]: |
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fixes xs ys :: "'a list" |
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assumes "strict_prefix xs ys" |
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obtains "prefix xs ys" and "xs \<noteq> ys" |
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using assms unfolding strict_prefix_def by blast |
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subsection \<open>Basic properties of prefixes\<close> |
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(* FIXME rm *) |
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theorem Nil_prefix [simp]: "prefix [] xs" |
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by (fact prefix_bot.bot_least) |
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(* FIXME rm *) |
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theorem prefix_Nil [simp]: "(prefix xs []) = (xs = [])" |
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by (fact prefix_bot.bot_unique) |
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63117 | 67 |
lemma prefix_snoc [simp]: "prefix xs (ys @ [y]) \<longleftrightarrow> xs = ys @ [y] \<or> prefix xs ys" |
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proof |
63117 | 69 |
assume "prefix 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> prefix xs ys" |
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by (metis append_Nil2 butlast_append butlast_snoc prefixI zs) |
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next |
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assume "xs = ys @ [y] \<or> prefix xs ys" |
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then show "prefix xs (ys @ [y])" |
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by (metis prefix_order.eq_iff prefix_order.order_trans prefixI) |
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qed |
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lemma Cons_prefix_Cons [simp]: "prefix (x # xs) (y # ys) = (x = y \<and> prefix xs ys)" |
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by (auto simp add: prefix_def) |
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63117 | 82 |
lemma prefix_code [code]: |
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"prefix [] xs \<longleftrightarrow> True" |
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"prefix (x # xs) [] \<longleftrightarrow> False" |
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"prefix (x # xs) (y # ys) \<longleftrightarrow> x = y \<and> prefix xs ys" |
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by simp_all |
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63117 | 88 |
lemma same_prefix_prefix [simp]: "prefix (xs @ ys) (xs @ zs) = prefix ys zs" |
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by (induct xs) simp_all |
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lemma same_prefix_nil [simp]: "prefix (xs @ ys) xs = (ys = [])" |
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by (metis append_Nil2 append_self_conv prefix_order.eq_iff prefixI) |
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lemma prefix_prefix [simp]: "prefix xs ys \<Longrightarrow> prefix xs (ys @ zs)" |
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unfolding prefix_def by fastforce |
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lemma append_prefixD: "prefix (xs @ ys) zs \<Longrightarrow> prefix xs zs" |
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by (auto simp add: prefix_def) |
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63117 | 100 |
theorem prefix_Cons: "prefix xs (y # ys) = (xs = [] \<or> (\<exists>zs. xs = y # zs \<and> prefix zs ys))" |
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by (cases xs) (auto simp add: prefix_def) |
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63117 | 103 |
theorem prefix_append: |
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"prefix xs (ys @ zs) = (prefix xs ys \<or> (\<exists>us. xs = ys @ us \<and> prefix 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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63117 | 111 |
lemma append_one_prefix: |
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"prefix xs ys \<Longrightarrow> length xs < length ys \<Longrightarrow> prefix (xs @ [ys ! length xs]) ys" |
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proof (unfold prefix_def) |
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assume a1: "\<exists>zs. ys = xs @ zs" |
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then obtain sk :: "'a list" where sk: "ys = xs @ sk" by fastforce |
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assume a2: "length xs < length ys" |
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have f1: "\<And>v. ([]::'a list) @ v = v" using append_Nil2 by simp |
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have "[] \<noteq> sk" using a1 a2 sk less_not_refl by force |
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hence "\<exists>v. xs @ hd sk # v = ys" using sk by (metis hd_Cons_tl) |
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thus "\<exists>zs. ys = (xs @ [ys ! length xs]) @ zs" using f1 by fastforce |
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qed |
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63117 | 123 |
theorem prefix_length_le: "prefix xs ys \<Longrightarrow> length xs \<le> length ys" |
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by (auto simp add: prefix_def) |
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63117 | 126 |
lemma prefix_same_cases: |
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"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" |
|
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unfolding prefix_def by (force simp: append_eq_append_conv2) |
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63173 | 130 |
lemma prefix_length_prefix: |
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"prefix ps xs \<Longrightarrow> prefix qs xs \<Longrightarrow> length ps \<le> length qs \<Longrightarrow> prefix ps qs" |
|
132 |
by (auto simp: prefix_def) (metis append_Nil2 append_eq_append_conv_if) |
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133 |
||
63117 | 134 |
lemma set_mono_prefix: "prefix xs ys \<Longrightarrow> set xs \<subseteq> set ys" |
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by (auto simp add: prefix_def) |
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63117 | 137 |
lemma take_is_prefix: "prefix (take n xs) xs" |
138 |
unfolding prefix_def by (metis append_take_drop_id) |
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139 |
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63155 | 140 |
lemma prefixeq_butlast: "prefix (butlast xs) xs" |
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by (simp add: butlast_conv_take take_is_prefix) |
|
142 |
||
63117 | 143 |
lemma map_prefixI: "prefix xs ys \<Longrightarrow> prefix (map f xs) (map f ys)" |
144 |
by (auto simp: prefix_def) |
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145 |
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63117 | 146 |
lemma prefix_length_less: "strict_prefix xs ys \<Longrightarrow> length xs < length ys" |
147 |
by (auto simp: strict_prefix_def prefix_def) |
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148 |
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63155 | 149 |
lemma prefix_snocD: "prefix (xs@[x]) ys \<Longrightarrow> strict_prefix xs ys" |
150 |
by (simp add: strict_prefixI' prefix_order.dual_order.strict_trans1) |
|
151 |
||
63117 | 152 |
lemma strict_prefix_simps [simp, code]: |
153 |
"strict_prefix xs [] \<longleftrightarrow> False" |
|
154 |
"strict_prefix [] (x # xs) \<longleftrightarrow> True" |
|
155 |
"strict_prefix (x # xs) (y # ys) \<longleftrightarrow> x = y \<and> strict_prefix xs ys" |
|
156 |
by (simp_all add: strict_prefix_def cong: conj_cong) |
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157 |
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63117 | 158 |
lemma take_strict_prefix: "strict_prefix xs ys \<Longrightarrow> strict_prefix (take n xs) ys" |
63649 | 159 |
proof (induct n arbitrary: xs ys) |
160 |
case 0 |
|
161 |
then show ?case by (cases ys) simp_all |
|
162 |
next |
|
163 |
case (Suc n) |
|
164 |
then show ?case by (metis prefix_order.less_trans strict_prefixI take_is_prefix) |
|
165 |
qed |
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166 |
|
63117 | 167 |
lemma not_prefix_cases: |
168 |
assumes pfx: "\<not> prefix ps ls" |
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169 |
obtains |
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170 |
(c1) "ps \<noteq> []" and "ls = []" |
63117 | 171 |
| (c2) a as x xs where "ps = a#as" and "ls = x#xs" and "x = a" and "\<not> prefix as xs" |
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172 |
| (c3) a as x xs where "ps = a#as" and "ls = x#xs" and "x \<noteq> a" |
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173 |
proof (cases ps) |
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174 |
case Nil |
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175 |
then show ?thesis using pfx by simp |
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176 |
next |
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177 |
case (Cons a as) |
60500 | 178 |
note c = \<open>ps = a#as\<close> |
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179 |
show ?thesis |
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180 |
proof (cases ls) |
63117 | 181 |
case Nil then show ?thesis by (metis append_Nil2 pfx c1 same_prefix_nil) |
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182 |
next |
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|
183 |
case (Cons x xs) |
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184 |
show ?thesis |
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185 |
proof (cases "x = a") |
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186 |
case True |
63117 | 187 |
have "\<not> prefix as xs" using pfx c Cons True by simp |
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188 |
with c Cons True show ?thesis by (rule c2) |
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|
189 |
next |
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|
190 |
case False |
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191 |
with c Cons show ?thesis by (rule c3) |
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|
192 |
qed |
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|
193 |
qed |
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|
194 |
qed |
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195 |
|
63117 | 196 |
lemma not_prefix_induct [consumes 1, case_names Nil Neq Eq]: |
197 |
assumes np: "\<not> prefix ps ls" |
|
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198 |
and base: "\<And>x xs. P (x#xs) []" |
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|
199 |
and r1: "\<And>x xs y ys. x \<noteq> y \<Longrightarrow> P (x#xs) (y#ys)" |
63117 | 200 |
and r2: "\<And>x xs y ys. \<lbrakk> x = y; \<not> prefix xs ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys)" |
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|
201 |
shows "P ps ls" using np |
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|
202 |
proof (induct ls arbitrary: ps) |
63649 | 203 |
case Nil |
204 |
then show ?case |
|
63117 | 205 |
by (auto simp: neq_Nil_conv elim!: not_prefix_cases intro!: base) |
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|
206 |
next |
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|
207 |
case (Cons y ys) |
63117 | 208 |
then have npfx: "\<not> prefix ps (y # ys)" by simp |
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|
209 |
then obtain x xs where pv: "ps = x # xs" |
63117 | 210 |
by (rule not_prefix_cases) auto |
211 |
show ?case by (metis Cons.hyps Cons_prefix_Cons npfx pv r1 r2) |
|
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|
212 |
qed |
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|
213 |
|
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|
214 |
|
63155 | 215 |
subsection \<open>Prefixes\<close> |
216 |
||
217 |
fun prefixes where |
|
218 |
"prefixes [] = [[]]" | |
|
219 |
"prefixes (x#xs) = [] # map (op # x) (prefixes xs)" |
|
220 |
||
221 |
lemma in_set_prefixes[simp]: "xs \<in> set (prefixes ys) \<longleftrightarrow> prefix xs ys" |
|
63649 | 222 |
proof (induct xs arbitrary: ys) |
223 |
case Nil |
|
224 |
then show ?case by (cases ys) auto |
|
225 |
next |
|
226 |
case (Cons a xs) |
|
227 |
then show ?case by (cases ys) auto |
|
228 |
qed |
|
63155 | 229 |
|
230 |
lemma length_prefixes[simp]: "length (prefixes xs) = length xs+1" |
|
65869 | 231 |
by (induction xs) auto |
232 |
||
233 |
lemma distinct_prefixes [intro]: "distinct (prefixes xs)" |
|
234 |
by (induction xs) (auto simp: distinct_map) |
|
235 |
||
236 |
lemma prefixes_snoc [simp]: "prefixes (xs@[x]) = prefixes xs @ [xs@[x]]" |
|
237 |
by (induction xs) auto |
|
238 |
||
239 |
lemma prefixes_not_Nil [simp]: "prefixes xs \<noteq> []" |
|
240 |
by (cases xs) auto |
|
63155 | 241 |
|
65869 | 242 |
lemma hd_prefixes [simp]: "hd (prefixes xs) = []" |
243 |
by (cases xs) simp_all |
|
63155 | 244 |
|
65869 | 245 |
lemma last_prefixes [simp]: "last (prefixes xs) = xs" |
246 |
by (induction xs) (simp_all add: last_map) |
|
247 |
||
248 |
lemma prefixes_append: |
|
249 |
"prefixes (xs @ ys) = prefixes xs @ map (\<lambda>ys'. xs @ ys') (tl (prefixes ys))" |
|
250 |
proof (induction xs) |
|
251 |
case Nil |
|
252 |
thus ?case by (cases ys) auto |
|
253 |
qed simp_all |
|
254 |
||
255 |
lemma prefixes_eq_snoc: |
|
63155 | 256 |
"prefixes ys = xs @ [x] \<longleftrightarrow> |
257 |
(ys = [] \<and> xs = [] \<or> (\<exists>z zs. ys = zs@[z] \<and> xs = prefixes zs)) \<and> x = ys" |
|
65869 | 258 |
by (cases ys rule: rev_cases) auto |
259 |
||
260 |
lemma prefixes_tailrec [code]: |
|
261 |
"prefixes xs = rev (snd (foldl (\<lambda>(acc1, acc2) x. (x#acc1, rev (x#acc1)#acc2)) ([],[[]]) xs))" |
|
262 |
proof - |
|
263 |
have "foldl (\<lambda>(acc1, acc2) x. (x#acc1, rev (x#acc1)#acc2)) (ys, rev ys # zs) xs = |
|
264 |
(rev xs @ ys, rev (map (\<lambda>as. rev ys @ as) (prefixes xs)) @ zs)" for ys zs |
|
265 |
proof (induction xs arbitrary: ys zs) |
|
266 |
case (Cons x xs ys zs) |
|
267 |
from Cons.IH[of "x # ys" "rev ys # zs"] |
|
268 |
show ?case by (simp add: o_def) |
|
269 |
qed simp_all |
|
270 |
from this [of "[]" "[]"] show ?thesis by simp |
|
271 |
qed |
|
272 |
||
273 |
lemma set_prefixes_eq: "set (prefixes xs) = {ys. prefix ys xs}" |
|
274 |
by auto |
|
275 |
||
276 |
lemma card_set_prefixes [simp]: "card (set (prefixes xs)) = Suc (length xs)" |
|
277 |
by (subst distinct_card) auto |
|
278 |
||
279 |
lemma set_prefixes_append: |
|
280 |
"set (prefixes (xs @ ys)) = set (prefixes xs) \<union> {xs @ ys' |ys'. ys' \<in> set (prefixes ys)}" |
|
281 |
by (subst prefixes_append, cases ys) auto |
|
63155 | 282 |
|
283 |
||
63173 | 284 |
subsection \<open>Longest Common Prefix\<close> |
285 |
||
286 |
definition Longest_common_prefix :: "'a list set \<Rightarrow> 'a list" where |
|
287 |
"Longest_common_prefix L = (GREATEST ps WRT length. \<forall>xs \<in> L. prefix ps xs)" |
|
288 |
||
289 |
lemma Longest_common_prefix_ex: "L \<noteq> {} \<Longrightarrow> |
|
290 |
\<exists>ps. (\<forall>xs \<in> L. prefix ps xs) \<and> (\<forall>qs. (\<forall>xs \<in> L. prefix qs xs) \<longrightarrow> size qs \<le> size ps)" |
|
291 |
(is "_ \<Longrightarrow> \<exists>ps. ?P L ps") |
|
292 |
proof(induction "LEAST n. \<exists>xs \<in>L. n = length xs" arbitrary: L) |
|
293 |
case 0 |
|
294 |
have "[] : L" using "0.hyps" LeastI[of "\<lambda>n. \<exists>xs\<in>L. n = length xs"] \<open>L \<noteq> {}\<close> |
|
295 |
by auto |
|
296 |
hence "?P L []" by(auto) |
|
297 |
thus ?case .. |
|
298 |
next |
|
299 |
case (Suc n) |
|
300 |
let ?EX = "\<lambda>n. \<exists>xs\<in>L. n = length xs" |
|
301 |
obtain x xs where xxs: "x#xs \<in> L" "size xs = n" using Suc.prems Suc.hyps(2) |
|
302 |
by(metis LeastI_ex[of ?EX] Suc_length_conv ex_in_conv) |
|
303 |
hence "[] \<notin> L" using Suc.hyps(2) by auto |
|
304 |
show ?case |
|
305 |
proof (cases "\<forall>xs \<in> L. \<exists>ys. xs = x#ys") |
|
306 |
case True |
|
307 |
let ?L = "{ys. x#ys \<in> L}" |
|
308 |
have 1: "(LEAST n. \<exists>xs \<in> ?L. n = length xs) = n" |
|
309 |
using xxs Suc.prems Suc.hyps(2) Least_le[of "?EX"] |
|
310 |
by - (rule Least_equality, fastforce+) |
|
311 |
have 2: "?L \<noteq> {}" using \<open>x # xs \<in> L\<close> by auto |
|
312 |
from Suc.hyps(1)[OF 1[symmetric] 2] obtain ps where IH: "?P ?L ps" .. |
|
313 |
{ fix qs |
|
314 |
assume "\<forall>qs. (\<forall>xa. x # xa \<in> L \<longrightarrow> prefix qs xa) \<longrightarrow> length qs \<le> length ps" |
|
315 |
and "\<forall>xs\<in>L. prefix qs xs" |
|
316 |
hence "length (tl qs) \<le> length ps" |
|
317 |
by (metis Cons_prefix_Cons hd_Cons_tl list.sel(2) Nil_prefix) |
|
318 |
hence "length qs \<le> Suc (length ps)" by auto |
|
319 |
} |
|
320 |
hence "?P L (x#ps)" using True IH by auto |
|
321 |
thus ?thesis .. |
|
322 |
next |
|
323 |
case False |
|
324 |
then obtain y ys where yys: "x\<noteq>y" "y#ys \<in> L" using \<open>[] \<notin> L\<close> |
|
325 |
by (auto) (metis list.exhaust) |
|
326 |
have "\<forall>qs. (\<forall>xs\<in>L. prefix qs xs) \<longrightarrow> qs = []" using yys \<open>x#xs \<in> L\<close> |
|
327 |
by auto (metis Cons_prefix_Cons prefix_Cons) |
|
328 |
hence "?P L []" by auto |
|
329 |
thus ?thesis .. |
|
330 |
qed |
|
331 |
qed |
|
332 |
||
333 |
lemma Longest_common_prefix_unique: "L \<noteq> {} \<Longrightarrow> |
|
334 |
\<exists>! ps. (\<forall>xs \<in> L. prefix ps xs) \<and> (\<forall>qs. (\<forall>xs \<in> L. prefix qs xs) \<longrightarrow> size qs \<le> size ps)" |
|
335 |
by(rule ex_ex1I[OF Longest_common_prefix_ex]; |
|
336 |
meson equals0I prefix_length_prefix prefix_order.antisym) |
|
337 |
||
338 |
lemma Longest_common_prefix_eq: |
|
339 |
"\<lbrakk> L \<noteq> {}; \<forall>xs \<in> L. prefix ps xs; |
|
340 |
\<forall>qs. (\<forall>xs \<in> L. prefix qs xs) \<longrightarrow> size qs \<le> size ps \<rbrakk> |
|
341 |
\<Longrightarrow> Longest_common_prefix L = ps" |
|
342 |
unfolding Longest_common_prefix_def GreatestM_def |
|
343 |
by(rule some1_equality[OF Longest_common_prefix_unique]) auto |
|
344 |
||
345 |
lemma Longest_common_prefix_prefix: |
|
346 |
"xs \<in> L \<Longrightarrow> prefix (Longest_common_prefix L) xs" |
|
347 |
unfolding Longest_common_prefix_def GreatestM_def |
|
348 |
by(rule someI2_ex[OF Longest_common_prefix_ex]) auto |
|
349 |
||
350 |
lemma Longest_common_prefix_longest: |
|
351 |
"L \<noteq> {} \<Longrightarrow> \<forall>xs\<in>L. prefix ps xs \<Longrightarrow> length ps \<le> length(Longest_common_prefix L)" |
|
352 |
unfolding Longest_common_prefix_def GreatestM_def |
|
353 |
by(rule someI2_ex[OF Longest_common_prefix_ex]) auto |
|
354 |
||
355 |
lemma Longest_common_prefix_max_prefix: |
|
356 |
"L \<noteq> {} \<Longrightarrow> \<forall>xs\<in>L. prefix ps xs \<Longrightarrow> prefix ps (Longest_common_prefix L)" |
|
357 |
by(metis Longest_common_prefix_prefix Longest_common_prefix_longest |
|
358 |
prefix_length_prefix ex_in_conv) |
|
359 |
||
360 |
lemma Longest_common_prefix_Nil: "[] \<in> L \<Longrightarrow> Longest_common_prefix L = []" |
|
361 |
using Longest_common_prefix_prefix prefix_Nil by blast |
|
362 |
||
363 |
lemma Longest_common_prefix_image_Cons: "L \<noteq> {} \<Longrightarrow> |
|
364 |
Longest_common_prefix (op # x ` L) = x # Longest_common_prefix L" |
|
365 |
apply(rule Longest_common_prefix_eq) |
|
366 |
apply(simp) |
|
367 |
apply (simp add: Longest_common_prefix_prefix) |
|
368 |
apply simp |
|
369 |
by(metis Longest_common_prefix_longest[of L] Cons_prefix_Cons Nitpick.size_list_simp(2) |
|
370 |
Suc_le_mono hd_Cons_tl order.strict_implies_order zero_less_Suc) |
|
371 |
||
372 |
lemma Longest_common_prefix_eq_Cons: assumes "L \<noteq> {}" "[] \<notin> L" "\<forall>xs\<in>L. hd xs = x" |
|
373 |
shows "Longest_common_prefix L = x # Longest_common_prefix {ys. x#ys \<in> L}" |
|
374 |
proof - |
|
375 |
have "L = op # x ` {ys. x#ys \<in> L}" using assms(2,3) |
|
376 |
by (auto simp: image_def)(metis hd_Cons_tl) |
|
377 |
thus ?thesis |
|
378 |
by (metis Longest_common_prefix_image_Cons image_is_empty assms(1)) |
|
379 |
qed |
|
380 |
||
381 |
lemma Longest_common_prefix_eq_Nil: |
|
382 |
"\<lbrakk>x#ys \<in> L; y#zs \<in> L; x \<noteq> y \<rbrakk> \<Longrightarrow> Longest_common_prefix L = []" |
|
383 |
by (metis Longest_common_prefix_prefix list.inject prefix_Cons) |
|
384 |
||
385 |
||
386 |
fun longest_common_prefix :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where |
|
387 |
"longest_common_prefix (x#xs) (y#ys) = |
|
388 |
(if x=y then x # longest_common_prefix xs ys else [])" | |
|
389 |
"longest_common_prefix _ _ = []" |
|
390 |
||
391 |
lemma longest_common_prefix_prefix1: |
|
392 |
"prefix (longest_common_prefix xs ys) xs" |
|
393 |
by(induction xs ys rule: longest_common_prefix.induct) auto |
|
394 |
||
395 |
lemma longest_common_prefix_prefix2: |
|
396 |
"prefix (longest_common_prefix xs ys) ys" |
|
397 |
by(induction xs ys rule: longest_common_prefix.induct) auto |
|
398 |
||
399 |
lemma longest_common_prefix_max_prefix: |
|
400 |
"\<lbrakk> prefix ps xs; prefix ps ys \<rbrakk> |
|
401 |
\<Longrightarrow> prefix ps (longest_common_prefix xs ys)" |
|
402 |
by(induction xs ys arbitrary: ps rule: longest_common_prefix.induct) |
|
403 |
(auto simp: prefix_Cons) |
|
404 |
||
405 |
||
60500 | 406 |
subsection \<open>Parallel lists\<close> |
10389 | 407 |
|
50516 | 408 |
definition parallel :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infixl "\<parallel>" 50) |
63117 | 409 |
where "(xs \<parallel> ys) = (\<not> prefix xs ys \<and> \<not> prefix ys xs)" |
10389 | 410 |
|
63117 | 411 |
lemma parallelI [intro]: "\<not> prefix xs ys \<Longrightarrow> \<not> prefix ys xs \<Longrightarrow> xs \<parallel> ys" |
25692 | 412 |
unfolding parallel_def by blast |
10330
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"List prefixes" library theory (replaces old Lex/Prefix);
wenzelm
parents:
diff
changeset
|
413 |
|
10389 | 414 |
lemma parallelE [elim]: |
25692 | 415 |
assumes "xs \<parallel> ys" |
63117 | 416 |
obtains "\<not> prefix xs ys \<and> \<not> prefix ys xs" |
25692 | 417 |
using assms unfolding parallel_def by blast |
10330
4362e906b745
"List prefixes" library theory (replaces old Lex/Prefix);
wenzelm
parents:
diff
changeset
|
418 |
|
63117 | 419 |
theorem prefix_cases: |
420 |
obtains "prefix xs ys" | "strict_prefix ys xs" | "xs \<parallel> ys" |
|
421 |
unfolding parallel_def strict_prefix_def by blast |
|
10330
4362e906b745
"List prefixes" library theory (replaces old Lex/Prefix);
wenzelm
parents:
diff
changeset
|
422 |
|
10389 | 423 |
theorem parallel_decomp: |
50516 | 424 |
"xs \<parallel> ys \<Longrightarrow> \<exists>as b bs c cs. b \<noteq> c \<and> xs = as @ b # bs \<and> ys = as @ c # cs" |
10408 | 425 |
proof (induct xs rule: rev_induct) |
11987 | 426 |
case Nil |
23254 | 427 |
then have False by auto |
428 |
then show ?case .. |
|
10408 | 429 |
next |
11987 | 430 |
case (snoc x xs) |
431 |
show ?case |
|
63117 | 432 |
proof (rule prefix_cases) |
433 |
assume le: "prefix xs ys" |
|
10408 | 434 |
then obtain ys' where ys: "ys = xs @ ys'" .. |
435 |
show ?thesis |
|
436 |
proof (cases ys') |
|
25564 | 437 |
assume "ys' = []" |
63117 | 438 |
then show ?thesis by (metis append_Nil2 parallelE prefixI snoc.prems ys) |
10389 | 439 |
next |
10408 | 440 |
fix c cs assume ys': "ys' = c # cs" |
54483 | 441 |
have "x \<noteq> c" using snoc.prems ys ys' by fastforce |
442 |
thus "\<exists>as b bs c cs. b \<noteq> c \<and> xs @ [x] = as @ b # bs \<and> ys = as @ c # cs" |
|
443 |
using ys ys' by blast |
|
10389 | 444 |
qed |
10408 | 445 |
next |
63117 | 446 |
assume "strict_prefix ys xs" |
447 |
then have "prefix ys (xs @ [x])" by (simp add: strict_prefix_def) |
|
11987 | 448 |
with snoc have False by blast |
23254 | 449 |
then show ?thesis .. |
10408 | 450 |
next |
451 |
assume "xs \<parallel> ys" |
|
11987 | 452 |
with snoc obtain as b bs c cs where neq: "(b::'a) \<noteq> c" |
10408 | 453 |
and xs: "xs = as @ b # bs" and ys: "ys = as @ c # cs" |
454 |
by blast |
|
455 |
from xs have "xs @ [x] = as @ b # (bs @ [x])" by simp |
|
456 |
with neq ys show ?thesis by blast |
|
10389 | 457 |
qed |
458 |
qed |
|
10330
4362e906b745
"List prefixes" library theory (replaces old Lex/Prefix);
wenzelm
parents:
diff
changeset
|
459 |
|
25564 | 460 |
lemma parallel_append: "a \<parallel> b \<Longrightarrow> a @ c \<parallel> b @ d" |
25692 | 461 |
apply (rule parallelI) |
462 |
apply (erule parallelE, erule conjE, |
|
63117 | 463 |
induct rule: not_prefix_induct, simp+)+ |
25692 | 464 |
done |
25299 | 465 |
|
25692 | 466 |
lemma parallel_appendI: "xs \<parallel> ys \<Longrightarrow> x = xs @ xs' \<Longrightarrow> y = ys @ ys' \<Longrightarrow> x \<parallel> y" |
467 |
by (simp add: parallel_append) |
|
25299 | 468 |
|
25692 | 469 |
lemma parallel_commute: "a \<parallel> b \<longleftrightarrow> b \<parallel> a" |
470 |
unfolding parallel_def by auto |
|
14538
1d9d75a8efae
removed o2l and fold_rel; moved postfix to Library/List_Prefix.thy
oheimb
parents:
14300
diff
changeset
|
471 |
|
25356 | 472 |
|
60500 | 473 |
subsection \<open>Suffix order on lists\<close> |
17201 | 474 |
|
63149 | 475 |
definition suffix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" |
476 |
where "suffix xs ys = (\<exists>zs. ys = zs @ xs)" |
|
49087 | 477 |
|
63149 | 478 |
definition strict_suffix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" |
65869 | 479 |
where "strict_suffix xs ys \<longleftrightarrow> suffix xs ys \<and> xs \<noteq> ys" |
14538
1d9d75a8efae
removed o2l and fold_rel; moved postfix to Library/List_Prefix.thy
oheimb
parents:
14300
diff
changeset
|
480 |
|
65869 | 481 |
interpretation suffix_order: order suffix strict_suffix |
482 |
by standard (auto simp: suffix_def strict_suffix_def) |
|
483 |
||
484 |
interpretation suffix_bot: order_bot Nil suffix strict_suffix |
|
485 |
by standard (simp add: suffix_def) |
|
49087 | 486 |
|
63149 | 487 |
lemma suffixI [intro?]: "ys = zs @ xs \<Longrightarrow> suffix xs ys" |
488 |
unfolding suffix_def by blast |
|
21305 | 489 |
|
63149 | 490 |
lemma suffixE [elim?]: |
491 |
assumes "suffix xs ys" |
|
49087 | 492 |
obtains zs where "ys = zs @ xs" |
63149 | 493 |
using assms unfolding suffix_def by blast |
21305 | 494 |
|
63149 | 495 |
lemma suffix_tl [simp]: "suffix (tl xs) xs" |
49087 | 496 |
by (induct xs) (auto simp: suffix_def) |
14538
1d9d75a8efae
removed o2l and fold_rel; moved postfix to Library/List_Prefix.thy
oheimb
parents:
14300
diff
changeset
|
497 |
|
63149 | 498 |
lemma strict_suffix_tl [simp]: "xs \<noteq> [] \<Longrightarrow> strict_suffix (tl xs) xs" |
65869 | 499 |
by (induct xs) (auto simp: strict_suffix_def suffix_def) |
63149 | 500 |
|
65869 | 501 |
lemma Nil_suffix [simp]: "suffix [] xs" |
63149 | 502 |
by (simp add: suffix_def) |
49087 | 503 |
|
63149 | 504 |
lemma suffix_Nil [simp]: "(suffix xs []) = (xs = [])" |
505 |
by (auto simp add: suffix_def) |
|
506 |
||
507 |
lemma suffix_ConsI: "suffix xs ys \<Longrightarrow> suffix xs (y # ys)" |
|
508 |
by (auto simp add: suffix_def) |
|
509 |
||
510 |
lemma suffix_ConsD: "suffix (x # xs) ys \<Longrightarrow> suffix xs ys" |
|
511 |
by (auto simp add: suffix_def) |
|
14538
1d9d75a8efae
removed o2l and fold_rel; moved postfix to Library/List_Prefix.thy
oheimb
parents:
14300
diff
changeset
|
512 |
|
63149 | 513 |
lemma suffix_appendI: "suffix xs ys \<Longrightarrow> suffix xs (zs @ ys)" |
514 |
by (auto simp add: suffix_def) |
|
515 |
||
516 |
lemma suffix_appendD: "suffix (zs @ xs) ys \<Longrightarrow> suffix xs ys" |
|
517 |
by (auto simp add: suffix_def) |
|
49087 | 518 |
|
63149 | 519 |
lemma strict_suffix_set_subset: "strict_suffix xs ys \<Longrightarrow> set xs \<subseteq> set ys" |
65869 | 520 |
by (auto simp: strict_suffix_def suffix_def) |
14538
1d9d75a8efae
removed o2l and fold_rel; moved postfix to Library/List_Prefix.thy
oheimb
parents:
14300
diff
changeset
|
521 |
|
63149 | 522 |
lemma suffix_set_subset: "suffix xs ys \<Longrightarrow> set xs \<subseteq> set ys" |
65869 | 523 |
by (auto simp: suffix_def) |
49087 | 524 |
|
63149 | 525 |
lemma suffix_ConsD2: "suffix (x # xs) (y # ys) \<Longrightarrow> suffix xs ys" |
21305 | 526 |
proof - |
63149 | 527 |
assume "suffix (x # xs) (y # ys)" |
49107 | 528 |
then obtain zs where "y # ys = zs @ x # xs" .. |
49087 | 529 |
then show ?thesis |
63149 | 530 |
by (induct zs) (auto intro!: suffix_appendI suffix_ConsI) |
21305 | 531 |
qed |
14538
1d9d75a8efae
removed o2l and fold_rel; moved postfix to Library/List_Prefix.thy
oheimb
parents:
14300
diff
changeset
|
532 |
|
63149 | 533 |
lemma suffix_to_prefix [code]: "suffix xs ys \<longleftrightarrow> prefix (rev xs) (rev ys)" |
49087 | 534 |
proof |
63149 | 535 |
assume "suffix xs ys" |
49087 | 536 |
then obtain zs where "ys = zs @ xs" .. |
537 |
then have "rev ys = rev xs @ rev zs" by simp |
|
63117 | 538 |
then show "prefix (rev xs) (rev ys)" .. |
49087 | 539 |
next |
63117 | 540 |
assume "prefix (rev xs) (rev ys)" |
49087 | 541 |
then obtain zs where "rev ys = rev xs @ zs" .. |
542 |
then have "rev (rev ys) = rev zs @ rev (rev xs)" by simp |
|
543 |
then have "ys = rev zs @ xs" by simp |
|
63149 | 544 |
then show "suffix xs ys" .. |
21305 | 545 |
qed |
65869 | 546 |
|
547 |
lemma strict_suffix_to_prefix [code]: "strict_suffix xs ys \<longleftrightarrow> strict_prefix (rev xs) (rev ys)" |
|
548 |
by (auto simp: suffix_to_prefix strict_suffix_def strict_prefix_def) |
|
14538
1d9d75a8efae
removed o2l and fold_rel; moved postfix to Library/List_Prefix.thy
oheimb
parents:
14300
diff
changeset
|
549 |
|
63149 | 550 |
lemma distinct_suffix: "distinct ys \<Longrightarrow> suffix xs ys \<Longrightarrow> distinct xs" |
551 |
by (clarsimp elim!: suffixE) |
|
17201 | 552 |
|
63149 | 553 |
lemma suffix_map: "suffix xs ys \<Longrightarrow> suffix (map f xs) (map f ys)" |
554 |
by (auto elim!: suffixE intro: suffixI) |
|
25299 | 555 |
|
63149 | 556 |
lemma suffix_drop: "suffix (drop n as) as" |
65869 | 557 |
unfolding suffix_def by (rule exI [where x = "take n as"]) simp |
25299 | 558 |
|
63149 | 559 |
lemma suffix_take: "suffix xs ys \<Longrightarrow> ys = take (length ys - length xs) ys @ xs" |
560 |
by (auto elim!: suffixE) |
|
25299 | 561 |
|
63149 | 562 |
lemma strict_suffix_reflclp_conv: "strict_suffix\<^sup>=\<^sup>= = suffix" |
65869 | 563 |
by (intro ext) (auto simp: suffix_def strict_suffix_def) |
63149 | 564 |
|
565 |
lemma suffix_lists: "suffix xs ys \<Longrightarrow> ys \<in> lists A \<Longrightarrow> xs \<in> lists A" |
|
566 |
unfolding suffix_def by auto |
|
49087 | 567 |
|
65869 | 568 |
lemma suffix_snoc [simp]: "suffix xs (ys @ [y]) \<longleftrightarrow> xs = [] \<or> (\<exists>zs. xs = zs @ [y] \<and> suffix zs ys)" |
569 |
by (cases xs rule: rev_cases) (auto simp: suffix_def) |
|
570 |
||
571 |
lemma snoc_suffix_snoc [simp]: "suffix (xs @ [x]) (ys @ [y]) = (x = y \<and> suffix xs ys)" |
|
572 |
by (auto simp add: suffix_def) |
|
573 |
||
574 |
lemma same_suffix_suffix [simp]: "suffix (ys @ xs) (zs @ xs) = suffix ys zs" |
|
575 |
by (simp add: suffix_to_prefix) |
|
576 |
||
577 |
lemma same_suffix_nil [simp]: "suffix (ys @ xs) xs = (ys = [])" |
|
578 |
by (simp add: suffix_to_prefix) |
|
579 |
||
580 |
theorem suffix_Cons: "suffix xs (y # ys) \<longleftrightarrow> xs = y # ys \<or> suffix xs ys" |
|
581 |
unfolding suffix_def by (auto simp: Cons_eq_append_conv) |
|
582 |
||
583 |
theorem suffix_append: |
|
584 |
"suffix xs (ys @ zs) \<longleftrightarrow> suffix xs zs \<or> (\<exists>xs'. xs = xs' @ zs \<and> suffix xs' ys)" |
|
585 |
by (auto simp: suffix_def append_eq_append_conv2) |
|
586 |
||
587 |
theorem suffix_length_le: "suffix xs ys \<Longrightarrow> length xs \<le> length ys" |
|
588 |
by (auto simp add: suffix_def) |
|
589 |
||
590 |
lemma suffix_same_cases: |
|
591 |
"suffix (xs\<^sub>1::'a list) ys \<Longrightarrow> suffix xs\<^sub>2 ys \<Longrightarrow> suffix xs\<^sub>1 xs\<^sub>2 \<or> suffix xs\<^sub>2 xs\<^sub>1" |
|
592 |
unfolding suffix_def by (force simp: append_eq_append_conv2) |
|
593 |
||
594 |
lemma suffix_length_suffix: |
|
595 |
"suffix ps xs \<Longrightarrow> suffix qs xs \<Longrightarrow> length ps \<le> length qs \<Longrightarrow> suffix ps qs" |
|
596 |
by (auto simp: suffix_to_prefix intro: prefix_length_prefix) |
|
597 |
||
598 |
lemma suffix_length_less: "strict_suffix xs ys \<Longrightarrow> length xs < length ys" |
|
599 |
by (auto simp: strict_suffix_def suffix_def) |
|
600 |
||
601 |
lemma suffix_ConsD': "suffix (x#xs) ys \<Longrightarrow> strict_suffix xs ys" |
|
602 |
by (auto simp: strict_suffix_def suffix_def) |
|
603 |
||
604 |
lemma drop_strict_suffix: "strict_suffix xs ys \<Longrightarrow> strict_suffix (drop n xs) ys" |
|
605 |
proof (induct n arbitrary: xs ys) |
|
606 |
case 0 |
|
607 |
then show ?case by (cases ys) simp_all |
|
608 |
next |
|
609 |
case (Suc n) |
|
610 |
then show ?case |
|
611 |
by (cases xs) (auto intro: Suc dest: suffix_ConsD' suffix_order.less_imp_le) |
|
612 |
qed |
|
613 |
||
614 |
lemma not_suffix_cases: |
|
615 |
assumes pfx: "\<not> suffix ps ls" |
|
616 |
obtains |
|
617 |
(c1) "ps \<noteq> []" and "ls = []" |
|
618 |
| (c2) a as x xs where "ps = as@[a]" and "ls = xs@[x]" and "x = a" and "\<not> suffix as xs" |
|
619 |
| (c3) a as x xs where "ps = as@[a]" and "ls = xs@[x]" and "x \<noteq> a" |
|
620 |
proof (cases ps rule: rev_cases) |
|
621 |
case Nil |
|
622 |
then show ?thesis using pfx by simp |
|
623 |
next |
|
624 |
case (snoc as a) |
|
625 |
note c = \<open>ps = as@[a]\<close> |
|
626 |
show ?thesis |
|
627 |
proof (cases ls rule: rev_cases) |
|
628 |
case Nil then show ?thesis by (metis append_Nil2 pfx c1 same_suffix_nil) |
|
629 |
next |
|
630 |
case (snoc xs x) |
|
631 |
show ?thesis |
|
632 |
proof (cases "x = a") |
|
633 |
case True |
|
634 |
have "\<not> suffix as xs" using pfx c snoc True by simp |
|
635 |
with c snoc True show ?thesis by (rule c2) |
|
636 |
next |
|
637 |
case False |
|
638 |
with c snoc show ?thesis by (rule c3) |
|
639 |
qed |
|
640 |
qed |
|
641 |
qed |
|
642 |
||
643 |
lemma not_suffix_induct [consumes 1, case_names Nil Neq Eq]: |
|
644 |
assumes np: "\<not> suffix ps ls" |
|
645 |
and base: "\<And>x xs. P (xs@[x]) []" |
|
646 |
and r1: "\<And>x xs y ys. x \<noteq> y \<Longrightarrow> P (xs@[x]) (ys@[y])" |
|
647 |
and r2: "\<And>x xs y ys. \<lbrakk> x = y; \<not> suffix xs ys; P xs ys \<rbrakk> \<Longrightarrow> P (xs@[x]) (ys@[y])" |
|
648 |
shows "P ps ls" using np |
|
649 |
proof (induct ls arbitrary: ps rule: rev_induct) |
|
650 |
case Nil |
|
651 |
then show ?case by (cases ps rule: rev_cases) (auto intro: base) |
|
652 |
next |
|
653 |
case (snoc y ys ps) |
|
654 |
then have npfx: "\<not> suffix ps (ys @ [y])" by simp |
|
655 |
then obtain x xs where pv: "ps = xs @ [x]" |
|
656 |
by (rule not_suffix_cases) auto |
|
657 |
show ?case by (metis snoc.hyps snoc_suffix_snoc npfx pv r1 r2) |
|
658 |
qed |
|
659 |
||
660 |
||
63117 | 661 |
lemma parallelD1: "x \<parallel> y \<Longrightarrow> \<not> prefix x y" |
25692 | 662 |
by blast |
25299 | 663 |
|
63117 | 664 |
lemma parallelD2: "x \<parallel> y \<Longrightarrow> \<not> prefix y x" |
25692 | 665 |
by blast |
25355 | 666 |
|
667 |
lemma parallel_Nil1 [simp]: "\<not> x \<parallel> []" |
|
25692 | 668 |
unfolding parallel_def by simp |
25355 | 669 |
|
25299 | 670 |
lemma parallel_Nil2 [simp]: "\<not> [] \<parallel> x" |
25692 | 671 |
unfolding parallel_def by simp |
25299 | 672 |
|
25564 | 673 |
lemma Cons_parallelI1: "a \<noteq> b \<Longrightarrow> a # as \<parallel> b # bs" |
25692 | 674 |
by auto |
25299 | 675 |
|
25564 | 676 |
lemma Cons_parallelI2: "\<lbrakk> a = b; as \<parallel> bs \<rbrakk> \<Longrightarrow> a # as \<parallel> b # bs" |
63117 | 677 |
by (metis Cons_prefix_Cons parallelE parallelI) |
25665 | 678 |
|
25299 | 679 |
lemma not_equal_is_parallel: |
680 |
assumes neq: "xs \<noteq> ys" |
|
25356 | 681 |
and len: "length xs = length ys" |
682 |
shows "xs \<parallel> ys" |
|
25299 | 683 |
using len neq |
25355 | 684 |
proof (induct rule: list_induct2) |
26445 | 685 |
case Nil |
25356 | 686 |
then show ?case by simp |
25299 | 687 |
next |
26445 | 688 |
case (Cons a as b bs) |
25355 | 689 |
have ih: "as \<noteq> bs \<Longrightarrow> as \<parallel> bs" by fact |
25299 | 690 |
show ?case |
691 |
proof (cases "a = b") |
|
25355 | 692 |
case True |
26445 | 693 |
then have "as \<noteq> bs" using Cons by simp |
25355 | 694 |
then show ?thesis by (rule Cons_parallelI2 [OF True ih]) |
25299 | 695 |
next |
696 |
case False |
|
25355 | 697 |
then show ?thesis by (rule Cons_parallelI1) |
25299 | 698 |
qed |
699 |
qed |
|
22178 | 700 |
|
65869 | 701 |
subsection \<open>Suffixes\<close> |
702 |
||
703 |
fun suffixes where |
|
704 |
"suffixes [] = [[]]" |
|
705 |
| "suffixes (x#xs) = suffixes xs @ [x # xs]" |
|
706 |
||
707 |
lemma in_set_suffixes [simp]: "xs \<in> set (suffixes ys) \<longleftrightarrow> suffix xs ys" |
|
708 |
by (induction ys) (auto simp: suffix_def Cons_eq_append_conv) |
|
709 |
||
710 |
lemma distinct_suffixes [intro]: "distinct (suffixes xs)" |
|
711 |
by (induction xs) (auto simp: suffix_def) |
|
712 |
||
713 |
lemma length_suffixes [simp]: "length (suffixes xs) = Suc (length xs)" |
|
714 |
by (induction xs) auto |
|
715 |
||
716 |
lemma suffixes_snoc [simp]: "suffixes (xs @ [x]) = [] # map (\<lambda>ys. ys @ [x]) (suffixes xs)" |
|
717 |
by (induction xs) auto |
|
718 |
||
719 |
lemma suffixes_not_Nil [simp]: "suffixes xs \<noteq> []" |
|
720 |
by (cases xs) auto |
|
721 |
||
722 |
lemma hd_suffixes [simp]: "hd (suffixes xs) = []" |
|
723 |
by (induction xs) simp_all |
|
724 |
||
725 |
lemma last_suffixes [simp]: "last (suffixes xs) = xs" |
|
726 |
by (cases xs) simp_all |
|
727 |
||
728 |
lemma suffixes_append: |
|
729 |
"suffixes (xs @ ys) = suffixes ys @ map (\<lambda>xs'. xs' @ ys) (tl (suffixes xs))" |
|
730 |
proof (induction ys rule: rev_induct) |
|
731 |
case Nil |
|
732 |
thus ?case by (cases xs rule: rev_cases) auto |
|
733 |
next |
|
734 |
case (snoc y ys) |
|
735 |
show ?case |
|
736 |
by (simp only: append.assoc [symmetric] suffixes_snoc snoc.IH) simp |
|
737 |
qed |
|
738 |
||
739 |
lemma suffixes_eq_snoc: |
|
740 |
"suffixes ys = xs @ [x] \<longleftrightarrow> |
|
741 |
(ys = [] \<and> xs = [] \<or> (\<exists>z zs. ys = z#zs \<and> xs = suffixes zs)) \<and> x = ys" |
|
742 |
by (cases ys) auto |
|
743 |
||
744 |
lemma suffixes_tailrec [code]: |
|
745 |
"suffixes xs = rev (snd (foldl (\<lambda>(acc1, acc2) x. (x#acc1, (x#acc1)#acc2)) ([],[[]]) (rev xs)))" |
|
746 |
proof - |
|
747 |
have "foldl (\<lambda>(acc1, acc2) x. (x#acc1, (x#acc1)#acc2)) (ys, ys # zs) (rev xs) = |
|
748 |
(xs @ ys, rev (map (\<lambda>as. as @ ys) (suffixes xs)) @ zs)" for ys zs |
|
749 |
proof (induction xs arbitrary: ys zs) |
|
750 |
case (Cons x xs ys zs) |
|
751 |
from Cons.IH[of ys zs] |
|
752 |
show ?case by (simp add: o_def case_prod_unfold) |
|
753 |
qed simp_all |
|
754 |
from this [of "[]" "[]"] show ?thesis by simp |
|
755 |
qed |
|
756 |
||
757 |
lemma set_suffixes_eq: "set (suffixes xs) = {ys. suffix ys xs}" |
|
758 |
by auto |
|
759 |
||
760 |
lemma card_set_suffixes [simp]: "card (set (suffixes xs)) = Suc (length xs)" |
|
761 |
by (subst distinct_card) auto |
|
762 |
||
763 |
lemma set_suffixes_append: |
|
764 |
"set (suffixes (xs @ ys)) = set (suffixes ys) \<union> {xs' @ ys |xs'. xs' \<in> set (suffixes xs)}" |
|
765 |
by (subst suffixes_append, cases xs rule: rev_cases) auto |
|
766 |
||
767 |
||
768 |
lemma suffixes_conv_prefixes: "suffixes xs = map rev (prefixes (rev xs))" |
|
769 |
by (induction xs) auto |
|
770 |
||
771 |
lemma prefixes_conv_suffixes: "prefixes xs = map rev (suffixes (rev xs))" |
|
772 |
by (induction xs) auto |
|
773 |
||
774 |
lemma prefixes_rev: "prefixes (rev xs) = map rev (suffixes xs)" |
|
775 |
by (induction xs) auto |
|
776 |
||
777 |
lemma suffixes_rev: "suffixes (rev xs) = map rev (prefixes xs)" |
|
778 |
by (induction xs) auto |
|
779 |
||
49087 | 780 |
|
60500 | 781 |
subsection \<open>Homeomorphic embedding on lists\<close> |
49087 | 782 |
|
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
783 |
inductive list_emb :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" |
49087 | 784 |
for P :: "('a \<Rightarrow> 'a \<Rightarrow> bool)" |
785 |
where |
|
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
786 |
list_emb_Nil [intro, simp]: "list_emb P [] ys" |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
787 |
| list_emb_Cons [intro] : "list_emb P xs ys \<Longrightarrow> list_emb P xs (y#ys)" |
57498
ea44ec62a574
no built-in reflexivity of list embedding (which is more standard; now embedding is reflexive whenever the base-order is)
Christian Sternagel
parents:
57497
diff
changeset
|
788 |
| list_emb_Cons2 [intro]: "P x y \<Longrightarrow> list_emb P xs ys \<Longrightarrow> list_emb P (x#xs) (y#ys)" |
50516 | 789 |
|
57499
7e22776f2d32
added monotonicity lemma for list embedding
Christian Sternagel
parents:
57498
diff
changeset
|
790 |
lemma list_emb_mono: |
7e22776f2d32
added monotonicity lemma for list embedding
Christian Sternagel
parents:
57498
diff
changeset
|
791 |
assumes "\<And>x y. P x y \<longrightarrow> Q x y" |
7e22776f2d32
added monotonicity lemma for list embedding
Christian Sternagel
parents:
57498
diff
changeset
|
792 |
shows "list_emb P xs ys \<longrightarrow> list_emb Q xs ys" |
7e22776f2d32
added monotonicity lemma for list embedding
Christian Sternagel
parents:
57498
diff
changeset
|
793 |
proof |
7e22776f2d32
added monotonicity lemma for list embedding
Christian Sternagel
parents:
57498
diff
changeset
|
794 |
assume "list_emb P xs ys" |
7e22776f2d32
added monotonicity lemma for list embedding
Christian Sternagel
parents:
57498
diff
changeset
|
795 |
then show "list_emb Q xs ys" by (induct) (auto simp: assms) |
7e22776f2d32
added monotonicity lemma for list embedding
Christian Sternagel
parents:
57498
diff
changeset
|
796 |
qed |
7e22776f2d32
added monotonicity lemma for list embedding
Christian Sternagel
parents:
57498
diff
changeset
|
797 |
|
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
798 |
lemma list_emb_Nil2 [simp]: |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
799 |
assumes "list_emb P xs []" shows "xs = []" |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
800 |
using assms by (cases rule: list_emb.cases) auto |
49087 | 801 |
|
57498
ea44ec62a574
no built-in reflexivity of list embedding (which is more standard; now embedding is reflexive whenever the base-order is)
Christian Sternagel
parents:
57497
diff
changeset
|
802 |
lemma list_emb_refl: |
ea44ec62a574
no built-in reflexivity of list embedding (which is more standard; now embedding is reflexive whenever the base-order is)
Christian Sternagel
parents:
57497
diff
changeset
|
803 |
assumes "\<And>x. x \<in> set xs \<Longrightarrow> P x x" |
ea44ec62a574
no built-in reflexivity of list embedding (which is more standard; now embedding is reflexive whenever the base-order is)
Christian Sternagel
parents:
57497
diff
changeset
|
804 |
shows "list_emb P xs xs" |
ea44ec62a574
no built-in reflexivity of list embedding (which is more standard; now embedding is reflexive whenever the base-order is)
Christian Sternagel
parents:
57497
diff
changeset
|
805 |
using assms by (induct xs) auto |
49087 | 806 |
|
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
807 |
lemma list_emb_Cons_Nil [simp]: "list_emb P (x#xs) [] = False" |
49087 | 808 |
proof - |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
809 |
{ assume "list_emb P (x#xs) []" |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
810 |
from list_emb_Nil2 [OF this] have False by simp |
49087 | 811 |
} moreover { |
812 |
assume False |
|
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
813 |
then have "list_emb P (x#xs) []" by simp |
49087 | 814 |
} ultimately show ?thesis by blast |
815 |
qed |
|
816 |
||
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
817 |
lemma list_emb_append2 [intro]: "list_emb P xs ys \<Longrightarrow> list_emb P xs (zs @ ys)" |
49087 | 818 |
by (induct zs) auto |
819 |
||
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
820 |
lemma list_emb_prefix [intro]: |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
821 |
assumes "list_emb P xs ys" shows "list_emb P xs (ys @ zs)" |
49087 | 822 |
using assms |
823 |
by (induct arbitrary: zs) auto |
|
824 |
||
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
825 |
lemma list_emb_ConsD: |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
826 |
assumes "list_emb P (x#xs) ys" |
57498
ea44ec62a574
no built-in reflexivity of list embedding (which is more standard; now embedding is reflexive whenever the base-order is)
Christian Sternagel
parents:
57497
diff
changeset
|
827 |
shows "\<exists>us v vs. ys = us @ v # vs \<and> P x v \<and> list_emb P xs vs" |
49087 | 828 |
using assms |
49107 | 829 |
proof (induct x \<equiv> "x # xs" ys arbitrary: x xs) |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
830 |
case list_emb_Cons |
49107 | 831 |
then show ?case by (metis append_Cons) |
49087 | 832 |
next |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
833 |
case (list_emb_Cons2 x y xs ys) |
54483 | 834 |
then show ?case by blast |
49087 | 835 |
qed |
836 |
||
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
837 |
lemma list_emb_appendD: |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
838 |
assumes "list_emb P (xs @ ys) zs" |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
839 |
shows "\<exists>us vs. zs = us @ vs \<and> list_emb P xs us \<and> list_emb P ys vs" |
49087 | 840 |
using assms |
841 |
proof (induction xs arbitrary: ys zs) |
|
49107 | 842 |
case Nil then show ?case by auto |
49087 | 843 |
next |
844 |
case (Cons x xs) |
|
54483 | 845 |
then obtain us v vs where |
57498
ea44ec62a574
no built-in reflexivity of list embedding (which is more standard; now embedding is reflexive whenever the base-order is)
Christian Sternagel
parents:
57497
diff
changeset
|
846 |
zs: "zs = us @ v # vs" and p: "P x v" and lh: "list_emb P (xs @ ys) vs" |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
847 |
by (auto dest: list_emb_ConsD) |
54483 | 848 |
obtain sk\<^sub>0 :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" and sk\<^sub>1 :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
849 |
sk: "\<forall>x\<^sub>0 x\<^sub>1. \<not> list_emb P (xs @ x\<^sub>0) x\<^sub>1 \<or> sk\<^sub>0 x\<^sub>0 x\<^sub>1 @ sk\<^sub>1 x\<^sub>0 x\<^sub>1 = x\<^sub>1 \<and> list_emb P xs (sk\<^sub>0 x\<^sub>0 x\<^sub>1) \<and> list_emb P x\<^sub>0 (sk\<^sub>1 x\<^sub>0 x\<^sub>1)" |
54483 | 850 |
using Cons(1) by (metis (no_types)) |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
851 |
hence "\<forall>x\<^sub>2. list_emb P (x # xs) (x\<^sub>2 @ v # sk\<^sub>0 ys vs)" using p lh by auto |
54483 | 852 |
thus ?case using lh zs sk by (metis (no_types) append_Cons append_assoc) |
49087 | 853 |
qed |
854 |
||
63149 | 855 |
lemma list_emb_strict_suffix: |
856 |
assumes "list_emb P xs ys" and "strict_suffix ys zs" |
|
857 |
shows "list_emb P xs zs" |
|
65869 | 858 |
using assms(2) and list_emb_append2 [OF assms(1)] by (auto simp: strict_suffix_def suffix_def) |
63149 | 859 |
|
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
860 |
lemma list_emb_suffix: |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
861 |
assumes "list_emb P xs ys" and "suffix ys zs" |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
862 |
shows "list_emb P xs zs" |
63149 | 863 |
using assms and list_emb_strict_suffix |
864 |
unfolding strict_suffix_reflclp_conv[symmetric] by auto |
|
49087 | 865 |
|
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
866 |
lemma list_emb_length: "list_emb P xs ys \<Longrightarrow> length xs \<le> length ys" |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
867 |
by (induct rule: list_emb.induct) auto |
49087 | 868 |
|
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
869 |
lemma list_emb_trans: |
57500
5a8b3e9d82a4
weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents:
57499
diff
changeset
|
870 |
assumes "\<And>x y z. \<lbrakk>x \<in> set xs; y \<in> set ys; z \<in> set zs; P x y; P y z\<rbrakk> \<Longrightarrow> P x z" |
5a8b3e9d82a4
weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents:
57499
diff
changeset
|
871 |
shows "\<lbrakk>list_emb P xs ys; list_emb P ys zs\<rbrakk> \<Longrightarrow> list_emb P xs zs" |
50516 | 872 |
proof - |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
873 |
assume "list_emb P xs ys" and "list_emb P ys zs" |
57500
5a8b3e9d82a4
weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents:
57499
diff
changeset
|
874 |
then show "list_emb P xs zs" using assms |
49087 | 875 |
proof (induction arbitrary: zs) |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
876 |
case list_emb_Nil show ?case by blast |
49087 | 877 |
next |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
878 |
case (list_emb_Cons xs ys y) |
60500 | 879 |
from list_emb_ConsD [OF \<open>list_emb P (y#ys) zs\<close>] obtain us v vs |
57500
5a8b3e9d82a4
weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents:
57499
diff
changeset
|
880 |
where zs: "zs = us @ v # vs" and "P\<^sup>=\<^sup>= y v" and "list_emb P ys vs" by blast |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
881 |
then have "list_emb P ys (v#vs)" by blast |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
882 |
then have "list_emb P ys zs" unfolding zs by (rule list_emb_append2) |
57500
5a8b3e9d82a4
weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents:
57499
diff
changeset
|
883 |
from list_emb_Cons.IH [OF this] and list_emb_Cons.prems show ?case by auto |
49087 | 884 |
next |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
885 |
case (list_emb_Cons2 x y xs ys) |
60500 | 886 |
from list_emb_ConsD [OF \<open>list_emb P (y#ys) zs\<close>] obtain us v vs |
57498
ea44ec62a574
no built-in reflexivity of list embedding (which is more standard; now embedding is reflexive whenever the base-order is)
Christian Sternagel
parents:
57497
diff
changeset
|
887 |
where zs: "zs = us @ v # vs" and "P y v" and "list_emb P ys vs" by blast |
57500
5a8b3e9d82a4
weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents:
57499
diff
changeset
|
888 |
with list_emb_Cons2 have "list_emb P xs vs" by auto |
57498
ea44ec62a574
no built-in reflexivity of list embedding (which is more standard; now embedding is reflexive whenever the base-order is)
Christian Sternagel
parents:
57497
diff
changeset
|
889 |
moreover have "P x v" |
49087 | 890 |
proof - |
57500
5a8b3e9d82a4
weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents:
57499
diff
changeset
|
891 |
from zs have "v \<in> set zs" by auto |
5a8b3e9d82a4
weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents:
57499
diff
changeset
|
892 |
moreover have "x \<in> set (x#xs)" and "y \<in> set (y#ys)" by simp_all |
50516 | 893 |
ultimately show ?thesis |
60500 | 894 |
using \<open>P x y\<close> and \<open>P y v\<close> and list_emb_Cons2 |
50516 | 895 |
by blast |
49087 | 896 |
qed |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
897 |
ultimately have "list_emb P (x#xs) (v#vs)" by blast |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
898 |
then show ?case unfolding zs by (rule list_emb_append2) |
49087 | 899 |
qed |
900 |
qed |
|
901 |
||
57500
5a8b3e9d82a4
weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents:
57499
diff
changeset
|
902 |
lemma list_emb_set: |
5a8b3e9d82a4
weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents:
57499
diff
changeset
|
903 |
assumes "list_emb P xs ys" and "x \<in> set xs" |
5a8b3e9d82a4
weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents:
57499
diff
changeset
|
904 |
obtains y where "y \<in> set ys" and "P x y" |
5a8b3e9d82a4
weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents:
57499
diff
changeset
|
905 |
using assms by (induct) auto |
5a8b3e9d82a4
weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents:
57499
diff
changeset
|
906 |
|
65869 | 907 |
lemma list_emb_Cons_iff1 [simp]: |
908 |
assumes "P x y" |
|
909 |
shows "list_emb P (x#xs) (y#ys) \<longleftrightarrow> list_emb P xs ys" |
|
910 |
using assms by (subst list_emb.simps) (auto dest: list_emb_ConsD) |
|
911 |
||
912 |
lemma list_emb_Cons_iff2 [simp]: |
|
913 |
assumes "\<not>P x y" |
|
914 |
shows "list_emb P (x#xs) (y#ys) \<longleftrightarrow> list_emb P (x#xs) ys" |
|
915 |
using assms by (subst list_emb.simps) auto |
|
916 |
||
917 |
lemma list_emb_code [code]: |
|
918 |
"list_emb P [] ys \<longleftrightarrow> True" |
|
919 |
"list_emb P (x#xs) [] \<longleftrightarrow> False" |
|
920 |
"list_emb P (x#xs) (y#ys) \<longleftrightarrow> (if P x y then list_emb P xs ys else list_emb P (x#xs) ys)" |
|
921 |
by simp_all |
|
922 |
||
923 |
||
49087 | 924 |
|
60500 | 925 |
subsection \<open>Sublists (special case of homeomorphic embedding)\<close> |
49087 | 926 |
|
50516 | 927 |
abbreviation sublisteq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
928 |
where "sublisteq xs ys \<equiv> list_emb (op =) xs ys" |
65869 | 929 |
|
930 |
definition strict_sublist where "strict_sublist xs ys \<longleftrightarrow> xs \<noteq> ys \<and> sublisteq xs ys" |
|
49087 | 931 |
|
50516 | 932 |
lemma sublisteq_Cons2: "sublisteq xs ys \<Longrightarrow> sublisteq (x#xs) (x#ys)" by auto |
49087 | 933 |
|
50516 | 934 |
lemma sublisteq_same_length: |
935 |
assumes "sublisteq xs ys" and "length xs = length ys" shows "xs = ys" |
|
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
936 |
using assms by (induct) (auto dest: list_emb_length) |
49087 | 937 |
|
50516 | 938 |
lemma not_sublisteq_length [simp]: "length ys < length xs \<Longrightarrow> \<not> sublisteq xs ys" |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
939 |
by (metis list_emb_length linorder_not_less) |
49087 | 940 |
|
50516 | 941 |
lemma sublisteq_Cons': "sublisteq (x#xs) ys \<Longrightarrow> sublisteq xs ys" |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
942 |
by (induct xs, simp, blast dest: list_emb_ConsD) |
49087 | 943 |
|
50516 | 944 |
lemma sublisteq_Cons2': |
945 |
assumes "sublisteq (x#xs) (x#ys)" shows "sublisteq xs ys" |
|
946 |
using assms by (cases) (rule sublisteq_Cons') |
|
49087 | 947 |
|
50516 | 948 |
lemma sublisteq_Cons2_neq: |
949 |
assumes "sublisteq (x#xs) (y#ys)" |
|
950 |
shows "x \<noteq> y \<Longrightarrow> sublisteq (x#xs) ys" |
|
49087 | 951 |
using assms by (cases) auto |
952 |
||
65869 | 953 |
lemma sublisteq_Cons2_iff [simp]: |
50516 | 954 |
"sublisteq (x#xs) (y#ys) = (if x = y then sublisteq xs ys else sublisteq (x#xs) ys)" |
65869 | 955 |
by simp |
49087 | 956 |
|
50516 | 957 |
lemma sublisteq_append': "sublisteq (zs @ xs) (zs @ ys) \<longleftrightarrow> sublisteq xs ys" |
49087 | 958 |
by (induct zs) simp_all |
65869 | 959 |
|
960 |
interpretation sublist_order: order sublisteq strict_sublist |
|
961 |
proof |
|
962 |
fix xs ys :: "'a list" |
|
963 |
{ |
|
964 |
assume "sublisteq xs ys" and "sublisteq ys xs" |
|
965 |
thus "xs = ys" |
|
966 |
proof (induct) |
|
967 |
case list_emb_Nil |
|
968 |
from list_emb_Nil2 [OF this] show ?case by simp |
|
969 |
next |
|
970 |
case list_emb_Cons2 |
|
971 |
thus ?case by simp |
|
972 |
next |
|
973 |
case list_emb_Cons |
|
974 |
hence False using sublisteq_Cons' by fastforce |
|
975 |
thus ?case .. |
|
976 |
qed |
|
977 |
} |
|
978 |
thus "strict_sublist xs ys \<longleftrightarrow> (sublisteq xs ys \<and> \<not>sublisteq ys xs)" |
|
979 |
by (auto simp: strict_sublist_def) |
|
980 |
qed (auto simp: list_emb_refl intro: list_emb_trans) |
|
49087 | 981 |
|
65869 | 982 |
lemma in_set_sublists [simp]: "xs \<in> set (sublists ys) \<longleftrightarrow> sublisteq xs ys" |
983 |
proof |
|
984 |
assume "xs \<in> set (sublists ys)" |
|
985 |
thus "sublisteq xs ys" |
|
986 |
by (induction ys arbitrary: xs) (auto simp: Let_def) |
|
49087 | 987 |
next |
65869 | 988 |
have [simp]: "[] \<in> set (sublists ys)" for ys :: "'a list" |
989 |
by (induction ys) (auto simp: Let_def) |
|
990 |
assume "sublisteq xs ys" |
|
991 |
thus "xs \<in> set (sublists ys)" |
|
992 |
by (induction xs ys rule: list_emb.induct) (auto simp: Let_def) |
|
49087 | 993 |
qed |
994 |
||
65869 | 995 |
lemma set_sublists_eq: "set (sublists ys) = {xs. sublisteq xs ys}" |
996 |
by auto |
|
49087 | 997 |
|
50516 | 998 |
lemma sublisteq_append_le_same_iff: "sublisteq (xs @ ys) ys \<longleftrightarrow> xs = []" |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
999 |
by (auto dest: list_emb_length) |
49087 | 1000 |
|
64886 | 1001 |
lemma sublisteq_singleton_left: "sublisteq [x] ys \<longleftrightarrow> x \<in> set ys" |
1002 |
by (fastforce dest: list_emb_ConsD split_list_last) |
|
1003 |
||
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
1004 |
lemma list_emb_append_mono: |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
1005 |
"\<lbrakk> list_emb P xs xs'; list_emb P ys ys' \<rbrakk> \<Longrightarrow> list_emb P (xs@ys) (xs'@ys')" |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
1006 |
apply (induct rule: list_emb.induct) |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
1007 |
apply (metis eq_Nil_appendI list_emb_append2) |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
1008 |
apply (metis append_Cons list_emb_Cons) |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
1009 |
apply (metis append_Cons list_emb_Cons2) |
49107 | 1010 |
done |
49087 | 1011 |
|
1012 |
||
60500 | 1013 |
subsection \<open>Appending elements\<close> |
49087 | 1014 |
|
50516 | 1015 |
lemma sublisteq_append [simp]: |
1016 |
"sublisteq (xs @ zs) (ys @ zs) \<longleftrightarrow> sublisteq xs ys" (is "?l = ?r") |
|
49087 | 1017 |
proof |
50516 | 1018 |
{ fix xs' ys' xs ys zs :: "'a list" assume "sublisteq xs' ys'" |
1019 |
then have "xs' = xs @ zs & ys' = ys @ zs \<longrightarrow> sublisteq xs ys" |
|
49087 | 1020 |
proof (induct arbitrary: xs ys zs) |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
1021 |
case list_emb_Nil show ?case by simp |
49087 | 1022 |
next |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
1023 |
case (list_emb_Cons xs' ys' x) |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
1024 |
{ assume "ys=[]" then have ?case using list_emb_Cons(1) by auto } |
49087 | 1025 |
moreover |
1026 |
{ fix us assume "ys = x#us" |
|
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
1027 |
then have ?case using list_emb_Cons(2) by(simp add: list_emb.list_emb_Cons) } |
49087 | 1028 |
ultimately show ?case by (auto simp:Cons_eq_append_conv) |
1029 |
next |
|
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
1030 |
case (list_emb_Cons2 x y xs' ys') |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
1031 |
{ assume "xs=[]" then have ?case using list_emb_Cons2(1) by auto } |
49087 | 1032 |
moreover |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
1033 |
{ fix us vs assume "xs=x#us" "ys=x#vs" then have ?case using list_emb_Cons2 by auto} |
49087 | 1034 |
moreover |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
1035 |
{ fix us assume "xs=x#us" "ys=[]" then have ?case using list_emb_Cons2(2) by bestsimp } |
60500 | 1036 |
ultimately show ?case using \<open>op = x y\<close> by (auto simp: Cons_eq_append_conv) |
49087 | 1037 |
qed } |
1038 |
moreover assume ?l |
|
1039 |
ultimately show ?r by blast |
|
1040 |
next |
|
65869 | 1041 |
assume ?r then show ?l by (metis list_emb_append_mono sublist_order.order_refl) |
49087 | 1042 |
qed |
1043 |
||
65869 | 1044 |
lemma sublisteq_append_iff: |
1045 |
"sublisteq xs (ys @ zs) \<longleftrightarrow> (\<exists>xs1 xs2. xs = xs1 @ xs2 \<and> sublisteq xs1 ys \<and> sublisteq xs2 zs)" |
|
1046 |
(is "?lhs = ?rhs") |
|
1047 |
proof |
|
1048 |
assume ?lhs thus ?rhs |
|
1049 |
proof (induction xs "ys @ zs" arbitrary: ys zs rule: list_emb.induct) |
|
1050 |
case (list_emb_Cons xs ws y ys zs) |
|
1051 |
from list_emb_Cons(2)[of "tl ys" zs] and list_emb_Cons(2)[of "[]" "tl zs"] and list_emb_Cons(1,3) |
|
1052 |
show ?case by (cases ys) auto |
|
1053 |
next |
|
1054 |
case (list_emb_Cons2 x y xs ws ys zs) |
|
1055 |
from list_emb_Cons2(3)[of "tl ys" zs] and list_emb_Cons2(3)[of "[]" "tl zs"] |
|
1056 |
and list_emb_Cons2(1,2,4) |
|
1057 |
show ?case by (cases ys) (auto simp: Cons_eq_append_conv) |
|
1058 |
qed auto |
|
1059 |
qed (auto intro: list_emb_append_mono) |
|
1060 |
||
1061 |
lemma sublisteq_appendE [case_names append]: |
|
1062 |
assumes "sublisteq xs (ys @ zs)" |
|
1063 |
obtains xs1 xs2 where "xs = xs1 @ xs2" "sublisteq xs1 ys" "sublisteq xs2 zs" |
|
1064 |
using assms by (subst (asm) sublisteq_append_iff) auto |
|
1065 |
||
50516 | 1066 |
lemma sublisteq_drop_many: "sublisteq xs ys \<Longrightarrow> sublisteq xs (zs @ ys)" |
49087 | 1067 |
by (induct zs) auto |
1068 |
||
50516 | 1069 |
lemma sublisteq_rev_drop_many: "sublisteq xs ys \<Longrightarrow> sublisteq xs (ys @ zs)" |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
1070 |
by (metis append_Nil2 list_emb_Nil list_emb_append_mono) |
49087 | 1071 |
|
1072 |
||
60500 | 1073 |
subsection \<open>Relation to standard list operations\<close> |
49087 | 1074 |
|
50516 | 1075 |
lemma sublisteq_map: |
1076 |
assumes "sublisteq xs ys" shows "sublisteq (map f xs) (map f ys)" |
|
49087 | 1077 |
using assms by (induct) auto |
1078 |
||
50516 | 1079 |
lemma sublisteq_filter_left [simp]: "sublisteq (filter P xs) xs" |
49087 | 1080 |
by (induct xs) auto |
1081 |
||
50516 | 1082 |
lemma sublisteq_filter [simp]: |
1083 |
assumes "sublisteq xs ys" shows "sublisteq (filter P xs) (filter P ys)" |
|
54483 | 1084 |
using assms by induct auto |
49087 | 1085 |
|
50516 | 1086 |
lemma "sublisteq xs ys \<longleftrightarrow> (\<exists>N. xs = sublist ys N)" (is "?L = ?R") |
49087 | 1087 |
proof |
1088 |
assume ?L |
|
49107 | 1089 |
then show ?R |
49087 | 1090 |
proof (induct) |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
1091 |
case list_emb_Nil show ?case by (metis sublist_empty) |
49087 | 1092 |
next |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
1093 |
case (list_emb_Cons xs ys x) |
49087 | 1094 |
then obtain N where "xs = sublist ys N" by blast |
49107 | 1095 |
then have "xs = sublist (x#ys) (Suc ` N)" |
49087 | 1096 |
by (clarsimp simp add:sublist_Cons inj_image_mem_iff) |
49107 | 1097 |
then show ?case by blast |
49087 | 1098 |
next |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
1099 |
case (list_emb_Cons2 x y xs ys) |
49087 | 1100 |
then obtain N where "xs = sublist ys N" by blast |
49107 | 1101 |
then have "x#xs = sublist (x#ys) (insert 0 (Suc ` N))" |
49087 | 1102 |
by (clarsimp simp add:sublist_Cons inj_image_mem_iff) |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
1103 |
moreover from list_emb_Cons2 have "x = y" by simp |
50516 | 1104 |
ultimately show ?case by blast |
49087 | 1105 |
qed |
1106 |
next |
|
1107 |
assume ?R |
|
1108 |
then obtain N where "xs = sublist ys N" .. |
|
50516 | 1109 |
moreover have "sublisteq (sublist ys N) ys" |
49107 | 1110 |
proof (induct ys arbitrary: N) |
49087 | 1111 |
case Nil show ?case by simp |
1112 |
next |
|
49107 | 1113 |
case Cons then show ?case by (auto simp: sublist_Cons) |
49087 | 1114 |
qed |
1115 |
ultimately show ?L by simp |
|
1116 |
qed |
|
1117 |
||
10330
4362e906b745
"List prefixes" library theory (replaces old Lex/Prefix);
wenzelm
parents:
diff
changeset
|
1118 |
end |