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