author | nipkow |
Sun, 29 May 2016 14:10:48 +0200 | |
changeset 63173 | 3413b1cf30cd |
parent 63155 | ea8540c71581 |
child 63649 | e690d6f2185b |
permissions | -rw-r--r-- |
49087 | 1 |
(* Title: HOL/Library/Sublist.thy |
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Author: Tobias Nipkow and Markus Wenzel, TU Muenchen |
49087 | 3 |
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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49087 | 8 |
theory Sublist |
9 |
imports Main |
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15131 | 10 |
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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63117 | 29 |
lemma prefixE [elim?]: |
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assumes "prefix xs ys" |
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obtains zs where "ys = xs @ zs" |
63117 | 32 |
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 - |
63117 | 41 |
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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63155 | 46 |
(* FIXME rm *) |
63117 | 47 |
lemma strict_prefixI [intro?]: "prefix xs ys \<Longrightarrow> xs \<noteq> ys \<Longrightarrow> strict_prefix xs ys" |
63155 | 48 |
by(fact prefix_order.le_neq_trans) |
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63117 | 50 |
lemma strict_prefixE [elim?]: |
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fixes xs ys :: "'a list" |
63117 | 52 |
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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60500 | 57 |
subsection \<open>Basic properties of prefixes\<close> |
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63155 | 59 |
(* FIXME rm *) |
63117 | 60 |
theorem Nil_prefix [iff]: "prefix [] xs" |
63155 | 61 |
by(fact prefix_bot.bot_least) |
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63155 | 63 |
(* FIXME rm *) |
63117 | 64 |
theorem prefix_Nil [simp]: "(prefix xs []) = (xs = [])" |
63155 | 65 |
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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68 |
proof |
63117 | 69 |
assume "prefix xs (ys @ [y])" |
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then obtain zs where zs: "ys @ [y] = xs @ zs" .. |
63117 | 71 |
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 |
63117 | 74 |
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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77 |
qed |
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63117 | 79 |
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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85 |
"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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90 |
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63117 | 91 |
lemma same_prefix_nil [iff]: "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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93 |
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63117 | 94 |
lemma prefix_prefix [simp]: "prefix xs ys \<Longrightarrow> prefix xs (ys @ zs)" |
95 |
by (metis prefix_order.le_less_trans prefixI strict_prefixE strict_prefixI) |
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96 |
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63117 | 97 |
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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99 |
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63117 | 100 |
theorem prefix_Cons: "prefix xs (y # ys) = (xs = [] \<or> (\<exists>zs. xs = y # zs \<and> prefix zs ys))" |
101 |
by (cases xs) (auto simp add: prefix_def) |
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102 |
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63117 | 103 |
theorem prefix_append: |
104 |
"prefix xs (ys @ zs) = (prefix xs ys \<or> (\<exists>us. xs = ys @ us \<and> prefix us zs))" |
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105 |
apply (induct zs rule: rev_induct) |
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106 |
apply force |
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apply (simp del: append_assoc add: append_assoc [symmetric]) |
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108 |
apply (metis append_eq_appendI) |
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109 |
done |
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110 |
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63117 | 111 |
lemma append_one_prefix: |
112 |
"prefix xs ys \<Longrightarrow> length xs < length ys \<Longrightarrow> prefix (xs @ [ys ! length xs]) ys" |
|
113 |
proof (unfold prefix_def) |
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114 |
assume a1: "\<exists>zs. ys = xs @ zs" |
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115 |
then obtain sk :: "'a list" where sk: "ys = xs @ sk" by fastforce |
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116 |
assume a2: "length xs < length ys" |
61076 | 117 |
have f1: "\<And>v. ([]::'a list) @ v = v" using append_Nil2 by simp |
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118 |
have "[] \<noteq> sk" using a1 a2 sk less_not_refl by force |
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119 |
hence "\<exists>v. xs @ hd sk # v = ys" using sk by (metis hd_Cons_tl) |
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120 |
thus "\<exists>zs. ys = (xs @ [ys ! length xs]) @ zs" using f1 by fastforce |
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121 |
qed |
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122 |
|
63117 | 123 |
theorem prefix_length_le: "prefix xs ys \<Longrightarrow> length xs \<le> length ys" |
124 |
by (auto simp add: prefix_def) |
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125 |
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63117 | 126 |
lemma prefix_same_cases: |
127 |
"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" |
|
128 |
unfolding prefix_def by (force simp: append_eq_append_conv2) |
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129 |
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63173 | 130 |
lemma prefix_length_prefix: |
131 |
"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) |
|
133 |
||
63117 | 134 |
lemma set_mono_prefix: "prefix xs ys \<Longrightarrow> set xs \<subseteq> set ys" |
135 |
by (auto simp add: prefix_def) |
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136 |
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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" |
141 |
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 |
|
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 |
|
63117 | 158 |
lemma take_strict_prefix: "strict_prefix xs ys \<Longrightarrow> strict_prefix (take n xs) ys" |
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|
159 |
apply (induct n arbitrary: xs ys) |
59997 | 160 |
apply (case_tac ys; simp) |
63117 | 161 |
apply (metis prefix_order.less_trans strict_prefixI take_is_prefix) |
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|
162 |
done |
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|
163 |
|
63117 | 164 |
lemma not_prefix_cases: |
165 |
assumes pfx: "\<not> prefix ps ls" |
|
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166 |
obtains |
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167 |
(c1) "ps \<noteq> []" and "ls = []" |
63117 | 168 |
| (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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169 |
| (c3) a as x xs where "ps = a#as" and "ls = x#xs" and "x \<noteq> a" |
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|
170 |
proof (cases ps) |
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|
171 |
case Nil |
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172 |
then show ?thesis using pfx by simp |
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|
173 |
next |
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|
174 |
case (Cons a as) |
60500 | 175 |
note c = \<open>ps = a#as\<close> |
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176 |
show ?thesis |
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|
177 |
proof (cases ls) |
63117 | 178 |
case Nil then show ?thesis by (metis append_Nil2 pfx c1 same_prefix_nil) |
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|
179 |
next |
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180 |
case (Cons x xs) |
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181 |
show ?thesis |
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182 |
proof (cases "x = a") |
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183 |
case True |
63117 | 184 |
have "\<not> prefix as xs" using pfx c Cons True by simp |
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185 |
with c Cons True show ?thesis by (rule c2) |
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|
186 |
next |
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|
187 |
case False |
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188 |
with c Cons show ?thesis by (rule c3) |
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|
189 |
qed |
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|
190 |
qed |
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|
191 |
qed |
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192 |
|
63117 | 193 |
lemma not_prefix_induct [consumes 1, case_names Nil Neq Eq]: |
194 |
assumes np: "\<not> prefix ps ls" |
|
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195 |
and base: "\<And>x xs. P (x#xs) []" |
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196 |
and r1: "\<And>x xs y ys. x \<noteq> y \<Longrightarrow> P (x#xs) (y#ys)" |
63117 | 197 |
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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|
198 |
shows "P ps ls" using np |
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199 |
proof (induct ls arbitrary: ps) |
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200 |
case Nil then show ?case |
63117 | 201 |
by (auto simp: neq_Nil_conv elim!: not_prefix_cases intro!: base) |
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|
202 |
next |
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|
203 |
case (Cons y ys) |
63117 | 204 |
then have npfx: "\<not> prefix ps (y # ys)" by simp |
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|
205 |
then obtain x xs where pv: "ps = x # xs" |
63117 | 206 |
by (rule not_prefix_cases) auto |
207 |
show ?case by (metis Cons.hyps Cons_prefix_Cons npfx pv r1 r2) |
|
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|
208 |
qed |
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|
209 |
|
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|
210 |
|
63155 | 211 |
subsection \<open>Prefixes\<close> |
212 |
||
213 |
fun prefixes where |
|
214 |
"prefixes [] = [[]]" | |
|
215 |
"prefixes (x#xs) = [] # map (op # x) (prefixes xs)" |
|
216 |
||
217 |
lemma in_set_prefixes[simp]: "xs \<in> set (prefixes ys) \<longleftrightarrow> prefix xs ys" |
|
218 |
by (induction "xs" arbitrary: "ys"; rename_tac "ys", case_tac "ys"; auto) |
|
219 |
||
220 |
lemma length_prefixes[simp]: "length (prefixes xs) = length xs+1" |
|
221 |
by (induction xs) auto |
|
222 |
||
223 |
lemma prefixes_snoc[simp]: |
|
224 |
"prefixes (xs@[x]) = prefixes xs @ [xs@[x]]" |
|
225 |
by (induction xs) auto |
|
226 |
||
227 |
lemma prefixes_eq_Snoc: |
|
228 |
"prefixes ys = xs @ [x] \<longleftrightarrow> |
|
229 |
(ys = [] \<and> xs = [] \<or> (\<exists>z zs. ys = zs@[z] \<and> xs = prefixes zs)) \<and> x = ys" |
|
230 |
by (cases ys rule: rev_cases) auto |
|
231 |
||
232 |
||
63173 | 233 |
subsection \<open>Longest Common Prefix\<close> |
234 |
||
235 |
definition Longest_common_prefix :: "'a list set \<Rightarrow> 'a list" where |
|
236 |
"Longest_common_prefix L = (GREATEST ps WRT length. \<forall>xs \<in> L. prefix ps xs)" |
|
237 |
||
238 |
lemma Longest_common_prefix_ex: "L \<noteq> {} \<Longrightarrow> |
|
239 |
\<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)" |
|
240 |
(is "_ \<Longrightarrow> \<exists>ps. ?P L ps") |
|
241 |
proof(induction "LEAST n. \<exists>xs \<in>L. n = length xs" arbitrary: L) |
|
242 |
case 0 |
|
243 |
have "[] : L" using "0.hyps" LeastI[of "\<lambda>n. \<exists>xs\<in>L. n = length xs"] \<open>L \<noteq> {}\<close> |
|
244 |
by auto |
|
245 |
hence "?P L []" by(auto) |
|
246 |
thus ?case .. |
|
247 |
next |
|
248 |
case (Suc n) |
|
249 |
let ?EX = "\<lambda>n. \<exists>xs\<in>L. n = length xs" |
|
250 |
obtain x xs where xxs: "x#xs \<in> L" "size xs = n" using Suc.prems Suc.hyps(2) |
|
251 |
by(metis LeastI_ex[of ?EX] Suc_length_conv ex_in_conv) |
|
252 |
hence "[] \<notin> L" using Suc.hyps(2) by auto |
|
253 |
show ?case |
|
254 |
proof (cases "\<forall>xs \<in> L. \<exists>ys. xs = x#ys") |
|
255 |
case True |
|
256 |
let ?L = "{ys. x#ys \<in> L}" |
|
257 |
have 1: "(LEAST n. \<exists>xs \<in> ?L. n = length xs) = n" |
|
258 |
using xxs Suc.prems Suc.hyps(2) Least_le[of "?EX"] |
|
259 |
by - (rule Least_equality, fastforce+) |
|
260 |
have 2: "?L \<noteq> {}" using \<open>x # xs \<in> L\<close> by auto |
|
261 |
from Suc.hyps(1)[OF 1[symmetric] 2] obtain ps where IH: "?P ?L ps" .. |
|
262 |
{ fix qs |
|
263 |
assume "\<forall>qs. (\<forall>xa. x # xa \<in> L \<longrightarrow> prefix qs xa) \<longrightarrow> length qs \<le> length ps" |
|
264 |
and "\<forall>xs\<in>L. prefix qs xs" |
|
265 |
hence "length (tl qs) \<le> length ps" |
|
266 |
by (metis Cons_prefix_Cons hd_Cons_tl list.sel(2) Nil_prefix) |
|
267 |
hence "length qs \<le> Suc (length ps)" by auto |
|
268 |
} |
|
269 |
hence "?P L (x#ps)" using True IH by auto |
|
270 |
thus ?thesis .. |
|
271 |
next |
|
272 |
case False |
|
273 |
then obtain y ys where yys: "x\<noteq>y" "y#ys \<in> L" using \<open>[] \<notin> L\<close> |
|
274 |
by (auto) (metis list.exhaust) |
|
275 |
have "\<forall>qs. (\<forall>xs\<in>L. prefix qs xs) \<longrightarrow> qs = []" using yys \<open>x#xs \<in> L\<close> |
|
276 |
by auto (metis Cons_prefix_Cons prefix_Cons) |
|
277 |
hence "?P L []" by auto |
|
278 |
thus ?thesis .. |
|
279 |
qed |
|
280 |
qed |
|
281 |
||
282 |
lemma Longest_common_prefix_unique: "L \<noteq> {} \<Longrightarrow> |
|
283 |
\<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)" |
|
284 |
by(rule ex_ex1I[OF Longest_common_prefix_ex]; |
|
285 |
meson equals0I prefix_length_prefix prefix_order.antisym) |
|
286 |
||
287 |
lemma Longest_common_prefix_eq: |
|
288 |
"\<lbrakk> L \<noteq> {}; \<forall>xs \<in> L. prefix ps xs; |
|
289 |
\<forall>qs. (\<forall>xs \<in> L. prefix qs xs) \<longrightarrow> size qs \<le> size ps \<rbrakk> |
|
290 |
\<Longrightarrow> Longest_common_prefix L = ps" |
|
291 |
unfolding Longest_common_prefix_def GreatestM_def |
|
292 |
by(rule some1_equality[OF Longest_common_prefix_unique]) auto |
|
293 |
||
294 |
lemma Longest_common_prefix_prefix: |
|
295 |
"xs \<in> L \<Longrightarrow> prefix (Longest_common_prefix L) xs" |
|
296 |
unfolding Longest_common_prefix_def GreatestM_def |
|
297 |
by(rule someI2_ex[OF Longest_common_prefix_ex]) auto |
|
298 |
||
299 |
lemma Longest_common_prefix_longest: |
|
300 |
"L \<noteq> {} \<Longrightarrow> \<forall>xs\<in>L. prefix ps xs \<Longrightarrow> length ps \<le> length(Longest_common_prefix L)" |
|
301 |
unfolding Longest_common_prefix_def GreatestM_def |
|
302 |
by(rule someI2_ex[OF Longest_common_prefix_ex]) auto |
|
303 |
||
304 |
lemma Longest_common_prefix_max_prefix: |
|
305 |
"L \<noteq> {} \<Longrightarrow> \<forall>xs\<in>L. prefix ps xs \<Longrightarrow> prefix ps (Longest_common_prefix L)" |
|
306 |
by(metis Longest_common_prefix_prefix Longest_common_prefix_longest |
|
307 |
prefix_length_prefix ex_in_conv) |
|
308 |
||
309 |
lemma Longest_common_prefix_Nil: "[] \<in> L \<Longrightarrow> Longest_common_prefix L = []" |
|
310 |
using Longest_common_prefix_prefix prefix_Nil by blast |
|
311 |
||
312 |
lemma Longest_common_prefix_image_Cons: "L \<noteq> {} \<Longrightarrow> |
|
313 |
Longest_common_prefix (op # x ` L) = x # Longest_common_prefix L" |
|
314 |
apply(rule Longest_common_prefix_eq) |
|
315 |
apply(simp) |
|
316 |
apply (simp add: Longest_common_prefix_prefix) |
|
317 |
apply simp |
|
318 |
by(metis Longest_common_prefix_longest[of L] Cons_prefix_Cons Nitpick.size_list_simp(2) |
|
319 |
Suc_le_mono hd_Cons_tl order.strict_implies_order zero_less_Suc) |
|
320 |
||
321 |
lemma Longest_common_prefix_eq_Cons: assumes "L \<noteq> {}" "[] \<notin> L" "\<forall>xs\<in>L. hd xs = x" |
|
322 |
shows "Longest_common_prefix L = x # Longest_common_prefix {ys. x#ys \<in> L}" |
|
323 |
proof - |
|
324 |
have "L = op # x ` {ys. x#ys \<in> L}" using assms(2,3) |
|
325 |
by (auto simp: image_def)(metis hd_Cons_tl) |
|
326 |
thus ?thesis |
|
327 |
by (metis Longest_common_prefix_image_Cons image_is_empty assms(1)) |
|
328 |
qed |
|
329 |
||
330 |
lemma Longest_common_prefix_eq_Nil: |
|
331 |
"\<lbrakk>x#ys \<in> L; y#zs \<in> L; x \<noteq> y \<rbrakk> \<Longrightarrow> Longest_common_prefix L = []" |
|
332 |
by (metis Longest_common_prefix_prefix list.inject prefix_Cons) |
|
333 |
||
334 |
||
335 |
fun longest_common_prefix :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where |
|
336 |
"longest_common_prefix (x#xs) (y#ys) = |
|
337 |
(if x=y then x # longest_common_prefix xs ys else [])" | |
|
338 |
"longest_common_prefix _ _ = []" |
|
339 |
||
340 |
lemma longest_common_prefix_prefix1: |
|
341 |
"prefix (longest_common_prefix xs ys) xs" |
|
342 |
by(induction xs ys rule: longest_common_prefix.induct) auto |
|
343 |
||
344 |
lemma longest_common_prefix_prefix2: |
|
345 |
"prefix (longest_common_prefix xs ys) ys" |
|
346 |
by(induction xs ys rule: longest_common_prefix.induct) auto |
|
347 |
||
348 |
lemma longest_common_prefix_max_prefix: |
|
349 |
"\<lbrakk> prefix ps xs; prefix ps ys \<rbrakk> |
|
350 |
\<Longrightarrow> prefix ps (longest_common_prefix xs ys)" |
|
351 |
by(induction xs ys arbitrary: ps rule: longest_common_prefix.induct) |
|
352 |
(auto simp: prefix_Cons) |
|
353 |
||
354 |
||
60500 | 355 |
subsection \<open>Parallel lists\<close> |
10389 | 356 |
|
50516 | 357 |
definition parallel :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infixl "\<parallel>" 50) |
63117 | 358 |
where "(xs \<parallel> ys) = (\<not> prefix xs ys \<and> \<not> prefix ys xs)" |
10389 | 359 |
|
63117 | 360 |
lemma parallelI [intro]: "\<not> prefix xs ys \<Longrightarrow> \<not> prefix ys xs \<Longrightarrow> xs \<parallel> ys" |
25692 | 361 |
unfolding parallel_def by blast |
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|
362 |
|
10389 | 363 |
lemma parallelE [elim]: |
25692 | 364 |
assumes "xs \<parallel> ys" |
63117 | 365 |
obtains "\<not> prefix xs ys \<and> \<not> prefix ys xs" |
25692 | 366 |
using assms unfolding parallel_def by blast |
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|
367 |
|
63117 | 368 |
theorem prefix_cases: |
369 |
obtains "prefix xs ys" | "strict_prefix ys xs" | "xs \<parallel> ys" |
|
370 |
unfolding parallel_def strict_prefix_def by blast |
|
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|
371 |
|
10389 | 372 |
theorem parallel_decomp: |
50516 | 373 |
"xs \<parallel> ys \<Longrightarrow> \<exists>as b bs c cs. b \<noteq> c \<and> xs = as @ b # bs \<and> ys = as @ c # cs" |
10408 | 374 |
proof (induct xs rule: rev_induct) |
11987 | 375 |
case Nil |
23254 | 376 |
then have False by auto |
377 |
then show ?case .. |
|
10408 | 378 |
next |
11987 | 379 |
case (snoc x xs) |
380 |
show ?case |
|
63117 | 381 |
proof (rule prefix_cases) |
382 |
assume le: "prefix xs ys" |
|
10408 | 383 |
then obtain ys' where ys: "ys = xs @ ys'" .. |
384 |
show ?thesis |
|
385 |
proof (cases ys') |
|
25564 | 386 |
assume "ys' = []" |
63117 | 387 |
then show ?thesis by (metis append_Nil2 parallelE prefixI snoc.prems ys) |
10389 | 388 |
next |
10408 | 389 |
fix c cs assume ys': "ys' = c # cs" |
54483 | 390 |
have "x \<noteq> c" using snoc.prems ys ys' by fastforce |
391 |
thus "\<exists>as b bs c cs. b \<noteq> c \<and> xs @ [x] = as @ b # bs \<and> ys = as @ c # cs" |
|
392 |
using ys ys' by blast |
|
10389 | 393 |
qed |
10408 | 394 |
next |
63117 | 395 |
assume "strict_prefix ys xs" |
396 |
then have "prefix ys (xs @ [x])" by (simp add: strict_prefix_def) |
|
11987 | 397 |
with snoc have False by blast |
23254 | 398 |
then show ?thesis .. |
10408 | 399 |
next |
400 |
assume "xs \<parallel> ys" |
|
11987 | 401 |
with snoc obtain as b bs c cs where neq: "(b::'a) \<noteq> c" |
10408 | 402 |
and xs: "xs = as @ b # bs" and ys: "ys = as @ c # cs" |
403 |
by blast |
|
404 |
from xs have "xs @ [x] = as @ b # (bs @ [x])" by simp |
|
405 |
with neq ys show ?thesis by blast |
|
10389 | 406 |
qed |
407 |
qed |
|
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"List prefixes" library theory (replaces old Lex/Prefix);
wenzelm
parents:
diff
changeset
|
408 |
|
25564 | 409 |
lemma parallel_append: "a \<parallel> b \<Longrightarrow> a @ c \<parallel> b @ d" |
25692 | 410 |
apply (rule parallelI) |
411 |
apply (erule parallelE, erule conjE, |
|
63117 | 412 |
induct rule: not_prefix_induct, simp+)+ |
25692 | 413 |
done |
25299 | 414 |
|
25692 | 415 |
lemma parallel_appendI: "xs \<parallel> ys \<Longrightarrow> x = xs @ xs' \<Longrightarrow> y = ys @ ys' \<Longrightarrow> x \<parallel> y" |
416 |
by (simp add: parallel_append) |
|
25299 | 417 |
|
25692 | 418 |
lemma parallel_commute: "a \<parallel> b \<longleftrightarrow> b \<parallel> a" |
419 |
unfolding parallel_def by auto |
|
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oheimb
parents:
14300
diff
changeset
|
420 |
|
25356 | 421 |
|
60500 | 422 |
subsection \<open>Suffix order on lists\<close> |
17201 | 423 |
|
63149 | 424 |
definition suffix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" |
425 |
where "suffix xs ys = (\<exists>zs. ys = zs @ xs)" |
|
49087 | 426 |
|
63149 | 427 |
definition strict_suffix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" |
428 |
where "strict_suffix xs ys \<longleftrightarrow> (\<exists>us. ys = us @ xs \<and> us \<noteq> [])" |
|
14538
1d9d75a8efae
removed o2l and fold_rel; moved postfix to Library/List_Prefix.thy
oheimb
parents:
14300
diff
changeset
|
429 |
|
63149 | 430 |
lemma strict_suffix_imp_suffix: |
431 |
"strict_suffix xs ys \<Longrightarrow> suffix xs ys" |
|
432 |
by (auto simp: suffix_def strict_suffix_def) |
|
49087 | 433 |
|
63149 | 434 |
lemma suffixI [intro?]: "ys = zs @ xs \<Longrightarrow> suffix xs ys" |
435 |
unfolding suffix_def by blast |
|
21305 | 436 |
|
63149 | 437 |
lemma suffixE [elim?]: |
438 |
assumes "suffix xs ys" |
|
49087 | 439 |
obtains zs where "ys = zs @ xs" |
63149 | 440 |
using assms unfolding suffix_def by blast |
21305 | 441 |
|
63149 | 442 |
lemma suffix_refl [iff]: "suffix xs xs" |
443 |
by (auto simp add: suffix_def) |
|
444 |
||
49087 | 445 |
lemma suffix_trans: |
446 |
"suffix xs ys \<Longrightarrow> suffix ys zs \<Longrightarrow> suffix xs zs" |
|
447 |
by (auto simp: suffix_def) |
|
63149 | 448 |
|
449 |
lemma strict_suffix_trans: |
|
450 |
"\<lbrakk>strict_suffix xs ys; strict_suffix ys zs\<rbrakk> \<Longrightarrow> strict_suffix xs zs" |
|
451 |
by (auto simp add: strict_suffix_def) |
|
49087 | 452 |
|
63149 | 453 |
lemma suffix_antisym: "\<lbrakk>suffix xs ys; suffix ys xs\<rbrakk> \<Longrightarrow> xs = ys" |
454 |
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
|
455 |
|
63149 | 456 |
lemma suffix_tl [simp]: "suffix (tl xs) xs" |
49087 | 457 |
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
|
458 |
|
63149 | 459 |
lemma strict_suffix_tl [simp]: "xs \<noteq> [] \<Longrightarrow> strict_suffix (tl xs) xs" |
460 |
by (induct xs) (auto simp: strict_suffix_def) |
|
461 |
||
462 |
lemma Nil_suffix [iff]: "suffix [] xs" |
|
463 |
by (simp add: suffix_def) |
|
49087 | 464 |
|
63149 | 465 |
lemma suffix_Nil [simp]: "(suffix xs []) = (xs = [])" |
466 |
by (auto simp add: suffix_def) |
|
467 |
||
468 |
lemma suffix_ConsI: "suffix xs ys \<Longrightarrow> suffix xs (y # ys)" |
|
469 |
by (auto simp add: suffix_def) |
|
470 |
||
471 |
lemma suffix_ConsD: "suffix (x # xs) ys \<Longrightarrow> suffix xs ys" |
|
472 |
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
|
473 |
|
63149 | 474 |
lemma suffix_appendI: "suffix xs ys \<Longrightarrow> suffix xs (zs @ ys)" |
475 |
by (auto simp add: suffix_def) |
|
476 |
||
477 |
lemma suffix_appendD: "suffix (zs @ xs) ys \<Longrightarrow> suffix xs ys" |
|
478 |
by (auto simp add: suffix_def) |
|
49087 | 479 |
|
63149 | 480 |
lemma strict_suffix_set_subset: "strict_suffix xs ys \<Longrightarrow> set xs \<subseteq> set ys" |
481 |
by (auto simp: strict_suffix_def) |
|
14538
1d9d75a8efae
removed o2l and fold_rel; moved postfix to Library/List_Prefix.thy
oheimb
parents:
14300
diff
changeset
|
482 |
|
63149 | 483 |
lemma suffix_set_subset: "suffix xs ys \<Longrightarrow> set xs \<subseteq> set ys" |
484 |
by (auto simp: suffix_def) |
|
49087 | 485 |
|
63149 | 486 |
lemma suffix_ConsD2: "suffix (x # xs) (y # ys) \<Longrightarrow> suffix xs ys" |
21305 | 487 |
proof - |
63149 | 488 |
assume "suffix (x # xs) (y # ys)" |
49107 | 489 |
then obtain zs where "y # ys = zs @ x # xs" .. |
49087 | 490 |
then show ?thesis |
63149 | 491 |
by (induct zs) (auto intro!: suffix_appendI suffix_ConsI) |
21305 | 492 |
qed |
14538
1d9d75a8efae
removed o2l and fold_rel; moved postfix to Library/List_Prefix.thy
oheimb
parents:
14300
diff
changeset
|
493 |
|
63149 | 494 |
lemma suffix_to_prefix [code]: "suffix xs ys \<longleftrightarrow> prefix (rev xs) (rev ys)" |
49087 | 495 |
proof |
63149 | 496 |
assume "suffix xs ys" |
49087 | 497 |
then obtain zs where "ys = zs @ xs" .. |
498 |
then have "rev ys = rev xs @ rev zs" by simp |
|
63117 | 499 |
then show "prefix (rev xs) (rev ys)" .. |
49087 | 500 |
next |
63117 | 501 |
assume "prefix (rev xs) (rev ys)" |
49087 | 502 |
then obtain zs where "rev ys = rev xs @ zs" .. |
503 |
then have "rev (rev ys) = rev zs @ rev (rev xs)" by simp |
|
504 |
then have "ys = rev zs @ xs" by simp |
|
63149 | 505 |
then show "suffix xs ys" .. |
21305 | 506 |
qed |
14538
1d9d75a8efae
removed o2l and fold_rel; moved postfix to Library/List_Prefix.thy
oheimb
parents:
14300
diff
changeset
|
507 |
|
63149 | 508 |
lemma distinct_suffix: "distinct ys \<Longrightarrow> suffix xs ys \<Longrightarrow> distinct xs" |
509 |
by (clarsimp elim!: suffixE) |
|
17201 | 510 |
|
63149 | 511 |
lemma suffix_map: "suffix xs ys \<Longrightarrow> suffix (map f xs) (map f ys)" |
512 |
by (auto elim!: suffixE intro: suffixI) |
|
25299 | 513 |
|
63149 | 514 |
lemma suffix_drop: "suffix (drop n as) as" |
515 |
unfolding suffix_def |
|
25692 | 516 |
apply (rule exI [where x = "take n as"]) |
517 |
apply simp |
|
518 |
done |
|
25299 | 519 |
|
63149 | 520 |
lemma suffix_take: "suffix xs ys \<Longrightarrow> ys = take (length ys - length xs) ys @ xs" |
521 |
by (auto elim!: suffixE) |
|
25299 | 522 |
|
63149 | 523 |
lemma strict_suffix_reflclp_conv: "strict_suffix\<^sup>=\<^sup>= = suffix" |
524 |
by (intro ext) (auto simp: suffix_def strict_suffix_def) |
|
525 |
||
526 |
lemma suffix_lists: "suffix xs ys \<Longrightarrow> ys \<in> lists A \<Longrightarrow> xs \<in> lists A" |
|
527 |
unfolding suffix_def by auto |
|
49087 | 528 |
|
63117 | 529 |
lemma parallelD1: "x \<parallel> y \<Longrightarrow> \<not> prefix x y" |
25692 | 530 |
by blast |
25299 | 531 |
|
63117 | 532 |
lemma parallelD2: "x \<parallel> y \<Longrightarrow> \<not> prefix y x" |
25692 | 533 |
by blast |
25355 | 534 |
|
535 |
lemma parallel_Nil1 [simp]: "\<not> x \<parallel> []" |
|
25692 | 536 |
unfolding parallel_def by simp |
25355 | 537 |
|
25299 | 538 |
lemma parallel_Nil2 [simp]: "\<not> [] \<parallel> x" |
25692 | 539 |
unfolding parallel_def by simp |
25299 | 540 |
|
25564 | 541 |
lemma Cons_parallelI1: "a \<noteq> b \<Longrightarrow> a # as \<parallel> b # bs" |
25692 | 542 |
by auto |
25299 | 543 |
|
25564 | 544 |
lemma Cons_parallelI2: "\<lbrakk> a = b; as \<parallel> bs \<rbrakk> \<Longrightarrow> a # as \<parallel> b # bs" |
63117 | 545 |
by (metis Cons_prefix_Cons parallelE parallelI) |
25665 | 546 |
|
25299 | 547 |
lemma not_equal_is_parallel: |
548 |
assumes neq: "xs \<noteq> ys" |
|
25356 | 549 |
and len: "length xs = length ys" |
550 |
shows "xs \<parallel> ys" |
|
25299 | 551 |
using len neq |
25355 | 552 |
proof (induct rule: list_induct2) |
26445 | 553 |
case Nil |
25356 | 554 |
then show ?case by simp |
25299 | 555 |
next |
26445 | 556 |
case (Cons a as b bs) |
25355 | 557 |
have ih: "as \<noteq> bs \<Longrightarrow> as \<parallel> bs" by fact |
25299 | 558 |
show ?case |
559 |
proof (cases "a = b") |
|
25355 | 560 |
case True |
26445 | 561 |
then have "as \<noteq> bs" using Cons by simp |
25355 | 562 |
then show ?thesis by (rule Cons_parallelI2 [OF True ih]) |
25299 | 563 |
next |
564 |
case False |
|
25355 | 565 |
then show ?thesis by (rule Cons_parallelI1) |
25299 | 566 |
qed |
567 |
qed |
|
22178 | 568 |
|
49087 | 569 |
|
60500 | 570 |
subsection \<open>Homeomorphic embedding on lists\<close> |
49087 | 571 |
|
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
572 |
inductive list_emb :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" |
49087 | 573 |
for P :: "('a \<Rightarrow> 'a \<Rightarrow> bool)" |
574 |
where |
|
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
575 |
list_emb_Nil [intro, simp]: "list_emb P [] ys" |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
576 |
| 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
|
577 |
| list_emb_Cons2 [intro]: "P x y \<Longrightarrow> list_emb P xs ys \<Longrightarrow> list_emb P (x#xs) (y#ys)" |
50516 | 578 |
|
57499
7e22776f2d32
added monotonicity lemma for list embedding
Christian Sternagel
parents:
57498
diff
changeset
|
579 |
lemma list_emb_mono: |
7e22776f2d32
added monotonicity lemma for list embedding
Christian Sternagel
parents:
57498
diff
changeset
|
580 |
assumes "\<And>x y. P x y \<longrightarrow> Q x y" |
7e22776f2d32
added monotonicity lemma for list embedding
Christian Sternagel
parents:
57498
diff
changeset
|
581 |
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
|
582 |
proof |
7e22776f2d32
added monotonicity lemma for list embedding
Christian Sternagel
parents:
57498
diff
changeset
|
583 |
assume "list_emb P xs ys" |
7e22776f2d32
added monotonicity lemma for list embedding
Christian Sternagel
parents:
57498
diff
changeset
|
584 |
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
|
585 |
qed |
7e22776f2d32
added monotonicity lemma for list embedding
Christian Sternagel
parents:
57498
diff
changeset
|
586 |
|
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
587 |
lemma list_emb_Nil2 [simp]: |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
588 |
assumes "list_emb P xs []" shows "xs = []" |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
589 |
using assms by (cases rule: list_emb.cases) auto |
49087 | 590 |
|
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
|
591 |
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
|
592 |
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
|
593 |
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
|
594 |
using assms by (induct xs) auto |
49087 | 595 |
|
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
596 |
lemma list_emb_Cons_Nil [simp]: "list_emb P (x#xs) [] = False" |
49087 | 597 |
proof - |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
598 |
{ assume "list_emb P (x#xs) []" |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
599 |
from list_emb_Nil2 [OF this] have False by simp |
49087 | 600 |
} moreover { |
601 |
assume False |
|
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
602 |
then have "list_emb P (x#xs) []" by simp |
49087 | 603 |
} ultimately show ?thesis by blast |
604 |
qed |
|
605 |
||
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
606 |
lemma list_emb_append2 [intro]: "list_emb P xs ys \<Longrightarrow> list_emb P xs (zs @ ys)" |
49087 | 607 |
by (induct zs) auto |
608 |
||
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
609 |
lemma list_emb_prefix [intro]: |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
610 |
assumes "list_emb P xs ys" shows "list_emb P xs (ys @ zs)" |
49087 | 611 |
using assms |
612 |
by (induct arbitrary: zs) auto |
|
613 |
||
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
614 |
lemma list_emb_ConsD: |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
615 |
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
|
616 |
shows "\<exists>us v vs. ys = us @ v # vs \<and> P x v \<and> list_emb P xs vs" |
49087 | 617 |
using assms |
49107 | 618 |
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
|
619 |
case list_emb_Cons |
49107 | 620 |
then show ?case by (metis append_Cons) |
49087 | 621 |
next |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
622 |
case (list_emb_Cons2 x y xs ys) |
54483 | 623 |
then show ?case by blast |
49087 | 624 |
qed |
625 |
||
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
626 |
lemma list_emb_appendD: |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
627 |
assumes "list_emb P (xs @ ys) zs" |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
628 |
shows "\<exists>us vs. zs = us @ vs \<and> list_emb P xs us \<and> list_emb P ys vs" |
49087 | 629 |
using assms |
630 |
proof (induction xs arbitrary: ys zs) |
|
49107 | 631 |
case Nil then show ?case by auto |
49087 | 632 |
next |
633 |
case (Cons x xs) |
|
54483 | 634 |
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
|
635 |
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
|
636 |
by (auto dest: list_emb_ConsD) |
54483 | 637 |
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
|
638 |
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 | 639 |
using Cons(1) by (metis (no_types)) |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
640 |
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 | 641 |
thus ?case using lh zs sk by (metis (no_types) append_Cons append_assoc) |
49087 | 642 |
qed |
643 |
||
63149 | 644 |
lemma list_emb_strict_suffix: |
645 |
assumes "list_emb P xs ys" and "strict_suffix ys zs" |
|
646 |
shows "list_emb P xs zs" |
|
647 |
using assms(2) and list_emb_append2 [OF assms(1)] by (auto simp: strict_suffix_def) |
|
648 |
||
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
649 |
lemma list_emb_suffix: |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
650 |
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
|
651 |
shows "list_emb P xs zs" |
63149 | 652 |
using assms and list_emb_strict_suffix |
653 |
unfolding strict_suffix_reflclp_conv[symmetric] by auto |
|
49087 | 654 |
|
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
655 |
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
|
656 |
by (induct rule: list_emb.induct) auto |
49087 | 657 |
|
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
658 |
lemma list_emb_trans: |
57500
5a8b3e9d82a4
weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents:
57499
diff
changeset
|
659 |
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
|
660 |
shows "\<lbrakk>list_emb P xs ys; list_emb P ys zs\<rbrakk> \<Longrightarrow> list_emb P xs zs" |
50516 | 661 |
proof - |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
662 |
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
|
663 |
then show "list_emb P xs zs" using assms |
49087 | 664 |
proof (induction arbitrary: zs) |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
665 |
case list_emb_Nil show ?case by blast |
49087 | 666 |
next |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
667 |
case (list_emb_Cons xs ys y) |
60500 | 668 |
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
|
669 |
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
|
670 |
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
|
671 |
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
|
672 |
from list_emb_Cons.IH [OF this] and list_emb_Cons.prems show ?case by auto |
49087 | 673 |
next |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
674 |
case (list_emb_Cons2 x y xs ys) |
60500 | 675 |
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
|
676 |
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
|
677 |
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
|
678 |
moreover have "P x v" |
49087 | 679 |
proof - |
57500
5a8b3e9d82a4
weaker assumption for "list_emb_trans"; added lemma
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57499
diff
changeset
|
680 |
from zs have "v \<in> set zs" by auto |
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weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents:
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diff
changeset
|
681 |
moreover have "x \<in> set (x#xs)" and "y \<in> set (y#ys)" by simp_all |
50516 | 682 |
ultimately show ?thesis |
60500 | 683 |
using \<open>P x y\<close> and \<open>P y v\<close> and list_emb_Cons2 |
50516 | 684 |
by blast |
49087 | 685 |
qed |
57497
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renamed "list_hembeq" into slightly shorter "list_emb"
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55579
diff
changeset
|
686 |
ultimately have "list_emb P (x#xs) (v#vs)" by blast |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
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parents:
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diff
changeset
|
687 |
then show ?case unfolding zs by (rule list_emb_append2) |
49087 | 688 |
qed |
689 |
qed |
|
690 |
||
57500
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weaker assumption for "list_emb_trans"; added lemma
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diff
changeset
|
691 |
lemma list_emb_set: |
5a8b3e9d82a4
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Christian Sternagel
parents:
57499
diff
changeset
|
692 |
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
|
693 |
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
|
694 |
using assms by (induct) auto |
5a8b3e9d82a4
weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents:
57499
diff
changeset
|
695 |
|
49087 | 696 |
|
60500 | 697 |
subsection \<open>Sublists (special case of homeomorphic embedding)\<close> |
49087 | 698 |
|
50516 | 699 |
abbreviation sublisteq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" |
57497
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parents:
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diff
changeset
|
700 |
where "sublisteq xs ys \<equiv> list_emb (op =) xs ys" |
49087 | 701 |
|
50516 | 702 |
lemma sublisteq_Cons2: "sublisteq xs ys \<Longrightarrow> sublisteq (x#xs) (x#ys)" by auto |
49087 | 703 |
|
50516 | 704 |
lemma sublisteq_same_length: |
705 |
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
|
706 |
using assms by (induct) (auto dest: list_emb_length) |
49087 | 707 |
|
50516 | 708 |
lemma not_sublisteq_length [simp]: "length ys < length xs \<Longrightarrow> \<not> sublisteq xs ys" |
57497
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Christian Sternagel
parents:
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diff
changeset
|
709 |
by (metis list_emb_length linorder_not_less) |
49087 | 710 |
|
711 |
lemma [code]: |
|
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
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parents:
55579
diff
changeset
|
712 |
"list_emb P [] ys \<longleftrightarrow> True" |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
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parents:
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diff
changeset
|
713 |
"list_emb P (x#xs) [] \<longleftrightarrow> False" |
49087 | 714 |
by (simp_all) |
715 |
||
50516 | 716 |
lemma sublisteq_Cons': "sublisteq (x#xs) ys \<Longrightarrow> sublisteq xs ys" |
57497
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renamed "list_hembeq" into slightly shorter "list_emb"
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parents:
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diff
changeset
|
717 |
by (induct xs, simp, blast dest: list_emb_ConsD) |
49087 | 718 |
|
50516 | 719 |
lemma sublisteq_Cons2': |
720 |
assumes "sublisteq (x#xs) (x#ys)" shows "sublisteq xs ys" |
|
721 |
using assms by (cases) (rule sublisteq_Cons') |
|
49087 | 722 |
|
50516 | 723 |
lemma sublisteq_Cons2_neq: |
724 |
assumes "sublisteq (x#xs) (y#ys)" |
|
725 |
shows "x \<noteq> y \<Longrightarrow> sublisteq (x#xs) ys" |
|
49087 | 726 |
using assms by (cases) auto |
727 |
||
50516 | 728 |
lemma sublisteq_Cons2_iff [simp, code]: |
729 |
"sublisteq (x#xs) (y#ys) = (if x = y then sublisteq xs ys else sublisteq (x#xs) ys)" |
|
57497
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renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
730 |
by (metis list_emb_Cons sublisteq_Cons2 sublisteq_Cons2' sublisteq_Cons2_neq) |
49087 | 731 |
|
50516 | 732 |
lemma sublisteq_append': "sublisteq (zs @ xs) (zs @ ys) \<longleftrightarrow> sublisteq xs ys" |
49087 | 733 |
by (induct zs) simp_all |
734 |
||
50516 | 735 |
lemma sublisteq_refl [simp, intro!]: "sublisteq xs xs" by (induct xs) simp_all |
49087 | 736 |
|
50516 | 737 |
lemma sublisteq_antisym: |
738 |
assumes "sublisteq xs ys" and "sublisteq ys xs" |
|
49087 | 739 |
shows "xs = ys" |
740 |
using assms |
|
741 |
proof (induct) |
|
57497
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renamed "list_hembeq" into slightly shorter "list_emb"
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parents:
55579
diff
changeset
|
742 |
case list_emb_Nil |
4106a2bc066a
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parents:
55579
diff
changeset
|
743 |
from list_emb_Nil2 [OF this] show ?case by simp |
49087 | 744 |
next |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
745 |
case list_emb_Cons2 |
54483 | 746 |
thus ?case by simp |
49087 | 747 |
next |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
748 |
case list_emb_Cons |
54483 | 749 |
hence False using sublisteq_Cons' by fastforce |
750 |
thus ?case .. |
|
49087 | 751 |
qed |
752 |
||
50516 | 753 |
lemma sublisteq_trans: "sublisteq xs ys \<Longrightarrow> sublisteq ys zs \<Longrightarrow> sublisteq xs zs" |
57500
5a8b3e9d82a4
weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents:
57499
diff
changeset
|
754 |
by (rule list_emb_trans [of _ _ _ "op ="]) auto |
49087 | 755 |
|
50516 | 756 |
lemma sublisteq_append_le_same_iff: "sublisteq (xs @ ys) ys \<longleftrightarrow> xs = []" |
57497
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renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
757 |
by (auto dest: list_emb_length) |
49087 | 758 |
|
57497
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Christian Sternagel
parents:
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diff
changeset
|
759 |
lemma list_emb_append_mono: |
4106a2bc066a
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Christian Sternagel
parents:
55579
diff
changeset
|
760 |
"\<lbrakk> list_emb P xs xs'; list_emb P ys ys' \<rbrakk> \<Longrightarrow> list_emb P (xs@ys) (xs'@ys')" |
4106a2bc066a
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Christian Sternagel
parents:
55579
diff
changeset
|
761 |
apply (induct rule: list_emb.induct) |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
762 |
apply (metis eq_Nil_appendI list_emb_append2) |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
763 |
apply (metis append_Cons list_emb_Cons) |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
764 |
apply (metis append_Cons list_emb_Cons2) |
49107 | 765 |
done |
49087 | 766 |
|
767 |
||
60500 | 768 |
subsection \<open>Appending elements\<close> |
49087 | 769 |
|
50516 | 770 |
lemma sublisteq_append [simp]: |
771 |
"sublisteq (xs @ zs) (ys @ zs) \<longleftrightarrow> sublisteq xs ys" (is "?l = ?r") |
|
49087 | 772 |
proof |
50516 | 773 |
{ fix xs' ys' xs ys zs :: "'a list" assume "sublisteq xs' ys'" |
774 |
then have "xs' = xs @ zs & ys' = ys @ zs \<longrightarrow> sublisteq xs ys" |
|
49087 | 775 |
proof (induct arbitrary: xs ys zs) |
57497
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Christian Sternagel
parents:
55579
diff
changeset
|
776 |
case list_emb_Nil show ?case by simp |
49087 | 777 |
next |
57497
4106a2bc066a
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Christian Sternagel
parents:
55579
diff
changeset
|
778 |
case (list_emb_Cons xs' ys' x) |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
779 |
{ assume "ys=[]" then have ?case using list_emb_Cons(1) by auto } |
49087 | 780 |
moreover |
781 |
{ fix us assume "ys = x#us" |
|
57497
4106a2bc066a
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Christian Sternagel
parents:
55579
diff
changeset
|
782 |
then have ?case using list_emb_Cons(2) by(simp add: list_emb.list_emb_Cons) } |
49087 | 783 |
ultimately show ?case by (auto simp:Cons_eq_append_conv) |
784 |
next |
|
57497
4106a2bc066a
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Christian Sternagel
parents:
55579
diff
changeset
|
785 |
case (list_emb_Cons2 x y xs' ys') |
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
786 |
{ assume "xs=[]" then have ?case using list_emb_Cons2(1) by auto } |
49087 | 787 |
moreover |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
788 |
{ fix us vs assume "xs=x#us" "ys=x#vs" then have ?case using list_emb_Cons2 by auto} |
49087 | 789 |
moreover |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
790 |
{ fix us assume "xs=x#us" "ys=[]" then have ?case using list_emb_Cons2(2) by bestsimp } |
60500 | 791 |
ultimately show ?case using \<open>op = x y\<close> by (auto simp: Cons_eq_append_conv) |
49087 | 792 |
qed } |
793 |
moreover assume ?l |
|
794 |
ultimately show ?r by blast |
|
795 |
next |
|
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
796 |
assume ?r then show ?l by (metis list_emb_append_mono sublisteq_refl) |
49087 | 797 |
qed |
798 |
||
50516 | 799 |
lemma sublisteq_drop_many: "sublisteq xs ys \<Longrightarrow> sublisteq xs (zs @ ys)" |
49087 | 800 |
by (induct zs) auto |
801 |
||
50516 | 802 |
lemma sublisteq_rev_drop_many: "sublisteq xs ys \<Longrightarrow> sublisteq xs (ys @ zs)" |
57497
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Christian Sternagel
parents:
55579
diff
changeset
|
803 |
by (metis append_Nil2 list_emb_Nil list_emb_append_mono) |
49087 | 804 |
|
805 |
||
60500 | 806 |
subsection \<open>Relation to standard list operations\<close> |
49087 | 807 |
|
50516 | 808 |
lemma sublisteq_map: |
809 |
assumes "sublisteq xs ys" shows "sublisteq (map f xs) (map f ys)" |
|
49087 | 810 |
using assms by (induct) auto |
811 |
||
50516 | 812 |
lemma sublisteq_filter_left [simp]: "sublisteq (filter P xs) xs" |
49087 | 813 |
by (induct xs) auto |
814 |
||
50516 | 815 |
lemma sublisteq_filter [simp]: |
816 |
assumes "sublisteq xs ys" shows "sublisteq (filter P xs) (filter P ys)" |
|
54483 | 817 |
using assms by induct auto |
49087 | 818 |
|
50516 | 819 |
lemma "sublisteq xs ys \<longleftrightarrow> (\<exists>N. xs = sublist ys N)" (is "?L = ?R") |
49087 | 820 |
proof |
821 |
assume ?L |
|
49107 | 822 |
then show ?R |
49087 | 823 |
proof (induct) |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
824 |
case list_emb_Nil show ?case by (metis sublist_empty) |
49087 | 825 |
next |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
826 |
case (list_emb_Cons xs ys x) |
49087 | 827 |
then obtain N where "xs = sublist ys N" by blast |
49107 | 828 |
then have "xs = sublist (x#ys) (Suc ` N)" |
49087 | 829 |
by (clarsimp simp add:sublist_Cons inj_image_mem_iff) |
49107 | 830 |
then show ?case by blast |
49087 | 831 |
next |
57497
4106a2bc066a
renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents:
55579
diff
changeset
|
832 |
case (list_emb_Cons2 x y xs ys) |
49087 | 833 |
then obtain N where "xs = sublist ys N" by blast |
49107 | 834 |
then have "x#xs = sublist (x#ys) (insert 0 (Suc ` N))" |
49087 | 835 |
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
|
836 |
moreover from list_emb_Cons2 have "x = y" by simp |
50516 | 837 |
ultimately show ?case by blast |
49087 | 838 |
qed |
839 |
next |
|
840 |
assume ?R |
|
841 |
then obtain N where "xs = sublist ys N" .. |
|
50516 | 842 |
moreover have "sublisteq (sublist ys N) ys" |
49107 | 843 |
proof (induct ys arbitrary: N) |
49087 | 844 |
case Nil show ?case by simp |
845 |
next |
|
49107 | 846 |
case Cons then show ?case by (auto simp: sublist_Cons) |
49087 | 847 |
qed |
848 |
ultimately show ?L by simp |
|
849 |
qed |
|
850 |
||
10330
4362e906b745
"List prefixes" library theory (replaces old Lex/Prefix);
wenzelm
parents:
diff
changeset
|
851 |
end |