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src/HOL/Library/Sublist.thy

changeset 49083 | 01081bca31b6 |

parent 45236 | ac4a2a66707d |

1.1 --- a/src/HOL/Library/Sublist.thy Wed Aug 29 11:05:44 2012 +0900 1.2 +++ b/src/HOL/Library/Sublist.thy Wed Aug 29 12:23:14 2012 +0900 1.3 @@ -10,99 +10,94 @@ 1.4 1.5 subsection {* Prefix order on lists *} 1.6 1.7 -instantiation list :: (type) "{order, bot}" 1.8 -begin 1.9 +definition prefixeq :: "'a list => 'a list => bool" where 1.10 + "prefixeq xs ys \<longleftrightarrow> (\<exists>zs. ys = xs @ zs)" 1.11 1.12 -definition 1.13 - prefixeq_def: "xs \<le> ys \<longleftrightarrow> (\<exists>zs. ys = xs @ zs)" 1.14 - 1.15 -definition 1.16 - prefix_def: "xs < ys \<longleftrightarrow> xs \<le> ys \<and> xs \<noteq> (ys::'a list)" 1.17 +definition prefix :: "'a list => 'a list => bool" where 1.18 + "prefix xs ys \<longleftrightarrow> prefixeq xs ys \<and> xs \<noteq> ys" 1.19 1.20 -definition 1.21 - "bot = []" 1.22 +interpretation prefix_order: order prefixeq prefix 1.23 + by default (auto simp: prefixeq_def prefix_def) 1.24 1.25 -instance proof 1.26 -qed (auto simp add: prefixeq_def prefix_def bot_list_def) 1.27 +interpretation prefix_bot: bot prefixeq prefix Nil 1.28 + by default (simp add: prefixeq_def) 1.29 1.30 -end 1.31 - 1.32 -lemma prefixeqI [intro?]: "ys = xs @ zs ==> xs \<le> ys" 1.33 +lemma prefixeqI [intro?]: "ys = xs @ zs ==> prefixeq xs ys" 1.34 unfolding prefixeq_def by blast 1.35 1.36 lemma prefixeqE [elim?]: 1.37 - assumes "xs \<le> ys" 1.38 + assumes "prefixeq xs ys" 1.39 obtains zs where "ys = xs @ zs" 1.40 using assms unfolding prefixeq_def by blast 1.41 1.42 -lemma prefixI' [intro?]: "ys = xs @ z # zs ==> xs < ys" 1.43 +lemma prefixI' [intro?]: "ys = xs @ z # zs ==> prefix xs ys" 1.44 unfolding prefix_def prefixeq_def by blast 1.45 1.46 lemma prefixE' [elim?]: 1.47 - assumes "xs < ys" 1.48 + assumes "prefix xs ys" 1.49 obtains z zs where "ys = xs @ z # zs" 1.50 proof - 1.51 - from `xs < ys` obtain us where "ys = xs @ us" and "xs \<noteq> ys" 1.52 + from `prefix xs ys` obtain us where "ys = xs @ us" and "xs \<noteq> ys" 1.53 unfolding prefix_def prefixeq_def by blast 1.54 with that show ?thesis by (auto simp add: neq_Nil_conv) 1.55 qed 1.56 1.57 -lemma prefixI [intro?]: "xs \<le> ys ==> xs \<noteq> ys ==> xs < (ys::'a list)" 1.58 +lemma prefixI [intro?]: "prefixeq xs ys ==> xs \<noteq> ys ==> prefix xs ys" 1.59 unfolding prefix_def by blast 1.60 1.61 lemma prefixE [elim?]: 1.62 fixes xs ys :: "'a list" 1.63 - assumes "xs < ys" 1.64 - obtains "xs \<le> ys" and "xs \<noteq> ys" 1.65 + assumes "prefix xs ys" 1.66 + obtains "prefixeq xs ys" and "xs \<noteq> ys" 1.67 using assms unfolding prefix_def by blast 1.68 1.69 1.70 subsection {* Basic properties of prefixes *} 1.71 1.72 -theorem Nil_prefixeq [iff]: "[] \<le> xs" 1.73 +theorem Nil_prefixeq [iff]: "prefixeq [] xs" 1.74 by (simp add: prefixeq_def) 1.75 1.76 -theorem prefixeq_Nil [simp]: "(xs \<le> []) = (xs = [])" 1.77 +theorem prefixeq_Nil [simp]: "(prefixeq xs []) = (xs = [])" 1.78 by (induct xs) (simp_all add: prefixeq_def) 1.79 1.80 -lemma prefixeq_snoc [simp]: "(xs \<le> ys @ [y]) = (xs = ys @ [y] \<or> xs \<le> ys)" 1.81 +lemma prefixeq_snoc [simp]: "prefixeq xs (ys @ [y]) \<longleftrightarrow> xs = ys @ [y] \<or> prefixeq xs ys" 1.82 proof 1.83 - assume "xs \<le> ys @ [y]" 1.84 + assume "prefixeq xs (ys @ [y])" 1.85 then obtain zs where zs: "ys @ [y] = xs @ zs" .. 1.86 - show "xs = ys @ [y] \<or> xs \<le> ys" 1.87 + show "xs = ys @ [y] \<or> prefixeq xs ys" 1.88 by (metis append_Nil2 butlast_append butlast_snoc prefixeqI zs) 1.89 next 1.90 - assume "xs = ys @ [y] \<or> xs \<le> ys" 1.91 - then show "xs \<le> ys @ [y]" 1.92 - by (metis order_eq_iff order_trans prefixeqI) 1.93 + assume "xs = ys @ [y] \<or> prefixeq xs ys" 1.94 + then show "prefixeq xs (ys @ [y])" 1.95 + by (metis prefix_order.eq_iff prefix_order.order_trans prefixeqI) 1.96 qed 1.97 1.98 -lemma Cons_prefixeq_Cons [simp]: "(x # xs \<le> y # ys) = (x = y \<and> xs \<le> ys)" 1.99 +lemma Cons_prefixeq_Cons [simp]: "prefixeq (x # xs) (y # ys) = (x = y \<and> prefixeq xs ys)" 1.100 by (auto simp add: prefixeq_def) 1.101 1.102 -lemma less_eq_list_code [code]: 1.103 - "([]\<Colon>'a\<Colon>{equal, ord} list) \<le> xs \<longleftrightarrow> True" 1.104 - "(x\<Colon>'a\<Colon>{equal, ord}) # xs \<le> [] \<longleftrightarrow> False" 1.105 - "(x\<Colon>'a\<Colon>{equal, ord}) # xs \<le> y # ys \<longleftrightarrow> x = y \<and> xs \<le> ys" 1.106 +lemma prefixeq_code [code]: 1.107 + "prefixeq [] xs \<longleftrightarrow> True" 1.108 + "prefixeq (x # xs) [] \<longleftrightarrow> False" 1.109 + "prefixeq (x # xs) (y # ys) \<longleftrightarrow> x = y \<and> prefixeq xs ys" 1.110 by simp_all 1.111 1.112 -lemma same_prefixeq_prefixeq [simp]: "(xs @ ys \<le> xs @ zs) = (ys \<le> zs)" 1.113 +lemma same_prefixeq_prefixeq [simp]: "prefixeq (xs @ ys) (xs @ zs) = prefixeq ys zs" 1.114 by (induct xs) simp_all 1.115 1.116 -lemma same_prefixeq_nil [iff]: "(xs @ ys \<le> xs) = (ys = [])" 1.117 - by (metis append_Nil2 append_self_conv order_eq_iff prefixeqI) 1.118 +lemma same_prefixeq_nil [iff]: "prefixeq (xs @ ys) xs = (ys = [])" 1.119 + by (metis append_Nil2 append_self_conv prefix_order.eq_iff prefixeqI) 1.120 1.121 -lemma prefixeq_prefixeq [simp]: "xs \<le> ys ==> xs \<le> ys @ zs" 1.122 - by (metis order_le_less_trans prefixeqI prefixE prefixI) 1.123 +lemma prefixeq_prefixeq [simp]: "prefixeq xs ys ==> prefixeq xs (ys @ zs)" 1.124 + by (metis prefix_order.le_less_trans prefixeqI prefixE prefixI) 1.125 1.126 -lemma append_prefixeqD: "xs @ ys \<le> zs \<Longrightarrow> xs \<le> zs" 1.127 +lemma append_prefixeqD: "prefixeq (xs @ ys) zs \<Longrightarrow> prefixeq xs zs" 1.128 by (auto simp add: prefixeq_def) 1.129 1.130 -theorem prefixeq_Cons: "(xs \<le> y # ys) = (xs = [] \<or> (\<exists>zs. xs = y # zs \<and> zs \<le> ys))" 1.131 +theorem prefixeq_Cons: "prefixeq xs (y # ys) = (xs = [] \<or> (\<exists>zs. xs = y # zs \<and> prefixeq zs ys))" 1.132 by (cases xs) (auto simp add: prefixeq_def) 1.133 1.134 theorem prefixeq_append: 1.135 - "(xs \<le> ys @ zs) = (xs \<le> ys \<or> (\<exists>us. xs = ys @ us \<and> us \<le> zs))" 1.136 + "prefixeq xs (ys @ zs) = (prefixeq xs ys \<or> (\<exists>us. xs = ys @ us \<and> prefixeq us zs))" 1.137 apply (induct zs rule: rev_induct) 1.138 apply force 1.139 apply (simp del: append_assoc add: append_assoc [symmetric]) 1.140 @@ -110,47 +105,47 @@ 1.141 done 1.142 1.143 lemma append_one_prefixeq: 1.144 - "xs \<le> ys ==> length xs < length ys ==> xs @ [ys ! length xs] \<le> ys" 1.145 + "prefixeq xs ys ==> length xs < length ys ==> prefixeq (xs @ [ys ! length xs]) ys" 1.146 unfolding prefixeq_def 1.147 by (metis Cons_eq_appendI append_eq_appendI append_eq_conv_conj 1.148 eq_Nil_appendI nth_drop') 1.149 1.150 -theorem prefixeq_length_le: "xs \<le> ys ==> length xs \<le> length ys" 1.151 +theorem prefixeq_length_le: "prefixeq xs ys ==> length xs \<le> length ys" 1.152 by (auto simp add: prefixeq_def) 1.153 1.154 lemma prefixeq_same_cases: 1.155 - "(xs\<^isub>1::'a list) \<le> ys \<Longrightarrow> xs\<^isub>2 \<le> ys \<Longrightarrow> xs\<^isub>1 \<le> xs\<^isub>2 \<or> xs\<^isub>2 \<le> xs\<^isub>1" 1.156 + "prefixeq (xs\<^isub>1::'a list) ys \<Longrightarrow> prefixeq xs\<^isub>2 ys \<Longrightarrow> prefixeq xs\<^isub>1 xs\<^isub>2 \<or> prefixeq xs\<^isub>2 xs\<^isub>1" 1.157 unfolding prefixeq_def by (metis append_eq_append_conv2) 1.158 1.159 -lemma set_mono_prefixeq: "xs \<le> ys \<Longrightarrow> set xs \<subseteq> set ys" 1.160 +lemma set_mono_prefixeq: "prefixeq xs ys \<Longrightarrow> set xs \<subseteq> set ys" 1.161 by (auto simp add: prefixeq_def) 1.162 1.163 -lemma take_is_prefixeq: "take n xs \<le> xs" 1.164 +lemma take_is_prefixeq: "prefixeq (take n xs) xs" 1.165 unfolding prefixeq_def by (metis append_take_drop_id) 1.166 1.167 -lemma map_prefixeqI: "xs \<le> ys \<Longrightarrow> map f xs \<le> map f ys" 1.168 +lemma map_prefixeqI: "prefixeq xs ys \<Longrightarrow> prefixeq (map f xs) (map f ys)" 1.169 by (auto simp: prefixeq_def) 1.170 1.171 -lemma prefixeq_length_less: "xs < ys \<Longrightarrow> length xs < length ys" 1.172 +lemma prefixeq_length_less: "prefix xs ys \<Longrightarrow> length xs < length ys" 1.173 by (auto simp: prefix_def prefixeq_def) 1.174 1.175 lemma prefix_simps [simp, code]: 1.176 - "xs < [] \<longleftrightarrow> False" 1.177 - "[] < x # xs \<longleftrightarrow> True" 1.178 - "x # xs < y # ys \<longleftrightarrow> x = y \<and> xs < ys" 1.179 + "prefix xs [] \<longleftrightarrow> False" 1.180 + "prefix [] (x # xs) \<longleftrightarrow> True" 1.181 + "prefix (x # xs) (y # ys) \<longleftrightarrow> x = y \<and> prefix xs ys" 1.182 by (simp_all add: prefix_def cong: conj_cong) 1.183 1.184 -lemma take_prefix: "xs < ys \<Longrightarrow> take n xs < ys" 1.185 +lemma take_prefix: "prefix xs ys \<Longrightarrow> prefix (take n xs) ys" 1.186 apply (induct n arbitrary: xs ys) 1.187 apply (case_tac ys, simp_all)[1] 1.188 - apply (metis order_less_trans prefixI take_is_prefixeq) 1.189 + apply (metis prefix_order.less_trans prefixI take_is_prefixeq) 1.190 done 1.191 1.192 lemma not_prefixeq_cases: 1.193 - assumes pfx: "\<not> ps \<le> ls" 1.194 + assumes pfx: "\<not> prefixeq ps ls" 1.195 obtains 1.196 (c1) "ps \<noteq> []" and "ls = []" 1.197 - | (c2) a as x xs where "ps = a#as" and "ls = x#xs" and "x = a" and "\<not> as \<le> xs" 1.198 + | (c2) a as x xs where "ps = a#as" and "ls = x#xs" and "x = a" and "\<not> prefixeq as xs" 1.199 | (c3) a as x xs where "ps = a#as" and "ls = x#xs" and "x \<noteq> a" 1.200 proof (cases ps) 1.201 case Nil then show ?thesis using pfx by simp 1.202 @@ -165,7 +160,7 @@ 1.203 show ?thesis 1.204 proof (cases "x = a") 1.205 case True 1.206 - have "\<not> as \<le> xs" using pfx c Cons True by simp 1.207 + have "\<not> prefixeq as xs" using pfx c Cons True by simp 1.208 with c Cons True show ?thesis by (rule c2) 1.209 next 1.210 case False 1.211 @@ -175,17 +170,17 @@ 1.212 qed 1.213 1.214 lemma not_prefixeq_induct [consumes 1, case_names Nil Neq Eq]: 1.215 - assumes np: "\<not> ps \<le> ls" 1.216 + assumes np: "\<not> prefixeq ps ls" 1.217 and base: "\<And>x xs. P (x#xs) []" 1.218 and r1: "\<And>x xs y ys. x \<noteq> y \<Longrightarrow> P (x#xs) (y#ys)" 1.219 - and r2: "\<And>x xs y ys. \<lbrakk> x = y; \<not> xs \<le> ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys)" 1.220 + and r2: "\<And>x xs y ys. \<lbrakk> x = y; \<not> prefixeq xs ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys)" 1.221 shows "P ps ls" using np 1.222 proof (induct ls arbitrary: ps) 1.223 case Nil then show ?case 1.224 by (auto simp: neq_Nil_conv elim!: not_prefixeq_cases intro!: base) 1.225 next 1.226 case (Cons y ys) 1.227 - then have npfx: "\<not> ps \<le> (y # ys)" by simp 1.228 + then have npfx: "\<not> prefixeq ps (y # ys)" by simp 1.229 then obtain x xs where pv: "ps = x # xs" 1.230 by (rule not_prefixeq_cases) auto 1.231 show ?case by (metis Cons.hyps Cons_prefixeq_Cons npfx pv r1 r2) 1.232 @@ -196,18 +191,18 @@ 1.233 1.234 definition 1.235 parallel :: "'a list => 'a list => bool" (infixl "\<parallel>" 50) where 1.236 - "(xs \<parallel> ys) = (\<not> xs \<le> ys \<and> \<not> ys \<le> xs)" 1.237 + "(xs \<parallel> ys) = (\<not> prefixeq xs ys \<and> \<not> prefixeq ys xs)" 1.238 1.239 -lemma parallelI [intro]: "\<not> xs \<le> ys ==> \<not> ys \<le> xs ==> xs \<parallel> ys" 1.240 +lemma parallelI [intro]: "\<not> prefixeq xs ys ==> \<not> prefixeq ys xs ==> xs \<parallel> ys" 1.241 unfolding parallel_def by blast 1.242 1.243 lemma parallelE [elim]: 1.244 assumes "xs \<parallel> ys" 1.245 - obtains "\<not> xs \<le> ys \<and> \<not> ys \<le> xs" 1.246 + obtains "\<not> prefixeq xs ys \<and> \<not> prefixeq ys xs" 1.247 using assms unfolding parallel_def by blast 1.248 1.249 theorem prefixeq_cases: 1.250 - obtains "xs \<le> ys" | "ys < xs" | "xs \<parallel> ys" 1.251 + obtains "prefixeq xs ys" | "prefix ys xs" | "xs \<parallel> ys" 1.252 unfolding parallel_def prefix_def by blast 1.253 1.254 theorem parallel_decomp: 1.255 @@ -220,7 +215,7 @@ 1.256 case (snoc x xs) 1.257 show ?case 1.258 proof (rule prefixeq_cases) 1.259 - assume le: "xs \<le> ys" 1.260 + assume le: "prefixeq xs ys" 1.261 then obtain ys' where ys: "ys = xs @ ys'" .. 1.262 show ?thesis 1.263 proof (cases ys') 1.264 @@ -233,7 +228,7 @@ 1.265 same_prefixeq_prefixeq snoc.prems ys) 1.266 qed 1.267 next 1.268 - assume "ys < xs" then have "ys \<le> xs @ [x]" by (simp add: prefix_def) 1.269 + assume "prefix ys xs" then have "prefixeq ys (xs @ [x])" by (simp add: prefix_def) 1.270 with snoc have False by blast 1.271 then show ?thesis .. 1.272 next 1.273 @@ -325,14 +320,14 @@ 1.274 by (induct zs) (auto intro!: suffixeq_appendI suffixeq_ConsI) 1.275 qed 1.276 1.277 -lemma suffixeq_to_prefixeq [code]: "suffixeq xs ys \<longleftrightarrow> rev xs \<le> rev ys" 1.278 +lemma suffixeq_to_prefixeq [code]: "suffixeq xs ys \<longleftrightarrow> prefixeq (rev xs) (rev ys)" 1.279 proof 1.280 assume "suffixeq xs ys" 1.281 then obtain zs where "ys = zs @ xs" .. 1.282 then have "rev ys = rev xs @ rev zs" by simp 1.283 - then show "rev xs <= rev ys" .. 1.284 + then show "prefixeq (rev xs) (rev ys)" .. 1.285 next 1.286 - assume "rev xs <= rev ys" 1.287 + assume "prefixeq (rev xs) (rev ys)" 1.288 then obtain zs where "rev ys = rev xs @ zs" .. 1.289 then have "rev (rev ys) = rev zs @ rev (rev xs)" by simp 1.290 then have "ys = rev zs @ xs" by simp 1.291 @@ -375,10 +370,10 @@ 1.292 qed 1.293 qed 1.294 1.295 -lemma parallelD1: "x \<parallel> y \<Longrightarrow> \<not> x \<le> y" 1.296 +lemma parallelD1: "x \<parallel> y \<Longrightarrow> \<not> prefixeq x y" 1.297 by blast 1.298 1.299 -lemma parallelD2: "x \<parallel> y \<Longrightarrow> \<not> y \<le> x" 1.300 +lemma parallelD2: "x \<parallel> y \<Longrightarrow> \<not> prefixeq y x" 1.301 by blast 1.302 1.303 lemma parallel_Nil1 [simp]: "\<not> x \<parallel> []" 1.304 @@ -481,4 +476,7 @@ 1.305 shows "emb P xs zs" 1.306 using assms and emb_suffix unfolding suffixeq_suffix_reflclp_conv by auto 1.307 1.308 +lemma emb_length: "emb P xs ys \<Longrightarrow> length xs \<le> length ys" 1.309 + by (induct rule: emb.induct) auto 1.310 + 1.311 end