src/HOL/Library/List_Prefix.thy
 author wenzelm Thu Jun 14 23:04:39 2007 +0200 (2007-06-14) changeset 23394 474ff28210c0 parent 23254 99644a53f16d child 25299 c3542f70b0fd permissions -rw-r--r--
tuned proofs;
1 (*  Title:      HOL/Library/List_Prefix.thy
2     ID:         \$Id\$
3     Author:     Tobias Nipkow and Markus Wenzel, TU Muenchen
4 *)
6 header {* List prefixes and postfixes *}
8 theory List_Prefix
9 imports Main
10 begin
12 subsection {* Prefix order on lists *}
14 instance list :: (type) ord ..
17   prefix_def: "xs \<le> ys == \<exists>zs. ys = xs @ zs"
18   strict_prefix_def: "xs < ys == xs \<le> ys \<and> xs \<noteq> (ys::'a list)"
20 instance list :: (type) order
21   by intro_classes (auto simp add: prefix_def strict_prefix_def)
23 lemma prefixI [intro?]: "ys = xs @ zs ==> xs \<le> ys"
24   unfolding prefix_def by blast
26 lemma prefixE [elim?]:
27   assumes "xs \<le> ys"
28   obtains zs where "ys = xs @ zs"
29   using assms unfolding prefix_def by blast
31 lemma strict_prefixI' [intro?]: "ys = xs @ z # zs ==> xs < ys"
32   unfolding strict_prefix_def prefix_def by blast
34 lemma strict_prefixE' [elim?]:
35   assumes "xs < ys"
36   obtains z zs where "ys = xs @ z # zs"
37 proof -
38   from `xs < ys` obtain us where "ys = xs @ us" and "xs \<noteq> ys"
39     unfolding strict_prefix_def prefix_def by blast
40   with that show ?thesis by (auto simp add: neq_Nil_conv)
41 qed
43 lemma strict_prefixI [intro?]: "xs \<le> ys ==> xs \<noteq> ys ==> xs < (ys::'a list)"
44   unfolding strict_prefix_def by blast
46 lemma strict_prefixE [elim?]:
47   fixes xs ys :: "'a list"
48   assumes "xs < ys"
49   obtains "xs \<le> ys" and "xs \<noteq> ys"
50   using assms unfolding strict_prefix_def by blast
53 subsection {* Basic properties of prefixes *}
55 theorem Nil_prefix [iff]: "[] \<le> xs"
58 theorem prefix_Nil [simp]: "(xs \<le> []) = (xs = [])"
59   by (induct xs) (simp_all add: prefix_def)
61 lemma prefix_snoc [simp]: "(xs \<le> ys @ [y]) = (xs = ys @ [y] \<or> xs \<le> ys)"
62 proof
63   assume "xs \<le> ys @ [y]"
64   then obtain zs where zs: "ys @ [y] = xs @ zs" ..
65   show "xs = ys @ [y] \<or> xs \<le> ys"
66   proof (cases zs rule: rev_cases)
67     assume "zs = []"
68     with zs have "xs = ys @ [y]" by simp
69     then show ?thesis ..
70   next
71     fix z zs' assume "zs = zs' @ [z]"
72     with zs have "ys = xs @ zs'" by simp
73     then have "xs \<le> ys" ..
74     then show ?thesis ..
75   qed
76 next
77   assume "xs = ys @ [y] \<or> xs \<le> ys"
78   then show "xs \<le> ys @ [y]"
79   proof
80     assume "xs = ys @ [y]"
81     then show ?thesis by simp
82   next
83     assume "xs \<le> ys"
84     then obtain zs where "ys = xs @ zs" ..
85     then have "ys @ [y] = xs @ (zs @ [y])" by simp
86     then show ?thesis ..
87   qed
88 qed
90 lemma Cons_prefix_Cons [simp]: "(x # xs \<le> y # ys) = (x = y \<and> xs \<le> ys)"
91   by (auto simp add: prefix_def)
93 lemma same_prefix_prefix [simp]: "(xs @ ys \<le> xs @ zs) = (ys \<le> zs)"
94   by (induct xs) simp_all
96 lemma same_prefix_nil [iff]: "(xs @ ys \<le> xs) = (ys = [])"
97 proof -
98   have "(xs @ ys \<le> xs @ []) = (ys \<le> [])" by (rule same_prefix_prefix)
99   then show ?thesis by simp
100 qed
102 lemma prefix_prefix [simp]: "xs \<le> ys ==> xs \<le> ys @ zs"
103 proof -
104   assume "xs \<le> ys"
105   then obtain us where "ys = xs @ us" ..
106   then have "ys @ zs = xs @ (us @ zs)" by simp
107   then show ?thesis ..
108 qed
110 lemma append_prefixD: "xs @ ys \<le> zs \<Longrightarrow> xs \<le> zs"
111   by (auto simp add: prefix_def)
113 theorem prefix_Cons: "(xs \<le> y # ys) = (xs = [] \<or> (\<exists>zs. xs = y # zs \<and> zs \<le> ys))"
114   by (cases xs) (auto simp add: prefix_def)
116 theorem prefix_append:
117     "(xs \<le> ys @ zs) = (xs \<le> ys \<or> (\<exists>us. xs = ys @ us \<and> us \<le> zs))"
118   apply (induct zs rule: rev_induct)
119    apply force
120   apply (simp del: append_assoc add: append_assoc [symmetric])
121   apply simp
122   apply blast
123   done
125 lemma append_one_prefix:
126     "xs \<le> ys ==> length xs < length ys ==> xs @ [ys ! length xs] \<le> ys"
127   apply (unfold prefix_def)
128   apply (auto simp add: nth_append)
129   apply (case_tac zs)
130    apply auto
131   done
133 theorem prefix_length_le: "xs \<le> ys ==> length xs \<le> length ys"
134   by (auto simp add: prefix_def)
136 lemma prefix_same_cases:
137     "(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"
139   apply (erule exE)+
140   apply (simp add: append_eq_append_conv_if split: if_splits)
141    apply (rule disjI2)
142    apply (rule_tac x = "drop (size xs\<^isub>2) xs\<^isub>1" in exI)
143    apply clarify
144    apply (drule sym)
145    apply (insert append_take_drop_id [of "length xs\<^isub>2" xs\<^isub>1])
146    apply simp
147   apply (rule disjI1)
148   apply (rule_tac x = "drop (size xs\<^isub>1) xs\<^isub>2" in exI)
149   apply clarify
150   apply (insert append_take_drop_id [of "length xs\<^isub>1" xs\<^isub>2])
151   apply simp
152   done
154 lemma set_mono_prefix:
155     "xs \<le> ys \<Longrightarrow> set xs \<subseteq> set ys"
156   by (auto simp add: prefix_def)
159 subsection {* Parallel lists *}
161 definition
162   parallel :: "'a list => 'a list => bool"  (infixl "\<parallel>" 50) where
163   "(xs \<parallel> ys) = (\<not> xs \<le> ys \<and> \<not> ys \<le> xs)"
165 lemma parallelI [intro]: "\<not> xs \<le> ys ==> \<not> ys \<le> xs ==> xs \<parallel> ys"
166   unfolding parallel_def by blast
168 lemma parallelE [elim]:
169   assumes "xs \<parallel> ys"
170   obtains "\<not> xs \<le> ys \<and> \<not> ys \<le> xs"
171   using assms unfolding parallel_def by blast
173 theorem prefix_cases:
174   obtains "xs \<le> ys" | "ys < xs" | "xs \<parallel> ys"
175   unfolding parallel_def strict_prefix_def by blast
177 theorem parallel_decomp:
178   "xs \<parallel> ys ==> \<exists>as b bs c cs. b \<noteq> c \<and> xs = as @ b # bs \<and> ys = as @ c # cs"
179 proof (induct xs rule: rev_induct)
180   case Nil
181   then have False by auto
182   then show ?case ..
183 next
184   case (snoc x xs)
185   show ?case
186   proof (rule prefix_cases)
187     assume le: "xs \<le> ys"
188     then obtain ys' where ys: "ys = xs @ ys'" ..
189     show ?thesis
190     proof (cases ys')
191       assume "ys' = []" with ys have "xs = ys" by simp
192       with snoc have "[x] \<parallel> []" by auto
193       then have False by blast
194       then show ?thesis ..
195     next
196       fix c cs assume ys': "ys' = c # cs"
197       with snoc ys have "xs @ [x] \<parallel> xs @ c # cs" by (simp only:)
198       then have "x \<noteq> c" by auto
199       moreover have "xs @ [x] = xs @ x # []" by simp
200       moreover from ys ys' have "ys = xs @ c # cs" by (simp only:)
201       ultimately show ?thesis by blast
202     qed
203   next
204     assume "ys < xs" then have "ys \<le> xs @ [x]" by (simp add: strict_prefix_def)
205     with snoc have False by blast
206     then show ?thesis ..
207   next
208     assume "xs \<parallel> ys"
209     with snoc obtain as b bs c cs where neq: "(b::'a) \<noteq> c"
210       and xs: "xs = as @ b # bs" and ys: "ys = as @ c # cs"
211       by blast
212     from xs have "xs @ [x] = as @ b # (bs @ [x])" by simp
213     with neq ys show ?thesis by blast
214   qed
215 qed
218 subsection {* Postfix order on lists *}
220 definition
221   postfix :: "'a list => 'a list => bool"  ("(_/ >>= _)" [51, 50] 50) where
222   "(xs >>= ys) = (\<exists>zs. xs = zs @ ys)"
224 lemma postfixI [intro?]: "xs = zs @ ys ==> xs >>= ys"
225   unfolding postfix_def by blast
227 lemma postfixE [elim?]:
228   assumes "xs >>= ys"
229   obtains zs where "xs = zs @ ys"
230   using assms unfolding postfix_def by blast
232 lemma postfix_refl [iff]: "xs >>= xs"
233   by (auto simp add: postfix_def)
234 lemma postfix_trans: "\<lbrakk>xs >>= ys; ys >>= zs\<rbrakk> \<Longrightarrow> xs >>= zs"
235   by (auto simp add: postfix_def)
236 lemma postfix_antisym: "\<lbrakk>xs >>= ys; ys >>= xs\<rbrakk> \<Longrightarrow> xs = ys"
237   by (auto simp add: postfix_def)
239 lemma Nil_postfix [iff]: "xs >>= []"
241 lemma postfix_Nil [simp]: "([] >>= xs) = (xs = [])"
242   by (auto simp add: postfix_def)
244 lemma postfix_ConsI: "xs >>= ys \<Longrightarrow> x#xs >>= ys"
245   by (auto simp add: postfix_def)
246 lemma postfix_ConsD: "xs >>= y#ys \<Longrightarrow> xs >>= ys"
247   by (auto simp add: postfix_def)
249 lemma postfix_appendI: "xs >>= ys \<Longrightarrow> zs @ xs >>= ys"
250   by (auto simp add: postfix_def)
251 lemma postfix_appendD: "xs >>= zs @ ys \<Longrightarrow> xs >>= ys"
252   by (auto simp add: postfix_def)
254 lemma postfix_is_subset: "xs >>= ys ==> set ys \<subseteq> set xs"
255 proof -
256   assume "xs >>= ys"
257   then obtain zs where "xs = zs @ ys" ..
258   then show ?thesis by (induct zs) auto
259 qed
261 lemma postfix_ConsD2: "x#xs >>= y#ys ==> xs >>= ys"
262 proof -
263   assume "x#xs >>= y#ys"
264   then obtain zs where "x#xs = zs @ y#ys" ..
265   then show ?thesis
266     by (induct zs) (auto intro!: postfix_appendI postfix_ConsI)
267 qed
269 lemma postfix_to_prefix: "xs >>= ys \<longleftrightarrow> rev ys \<le> rev xs"
270 proof
271   assume "xs >>= ys"
272   then obtain zs where "xs = zs @ ys" ..
273   then have "rev xs = rev ys @ rev zs" by simp
274   then show "rev ys <= rev xs" ..
275 next
276   assume "rev ys <= rev xs"
277   then obtain zs where "rev xs = rev ys @ zs" ..
278   then have "rev (rev xs) = rev zs @ rev (rev ys)" by simp
279   then have "xs = rev zs @ ys" by simp
280   then show "xs >>= ys" ..
281 qed
284 subsection {* Exeuctable code *}
286 lemma less_eq_code [code func]:
287   "([]\<Colon>'a\<Colon>{eq, ord} list) \<le> xs \<longleftrightarrow> True"
288   "(x\<Colon>'a\<Colon>{eq, ord}) # xs \<le> [] \<longleftrightarrow> False"
289   "(x\<Colon>'a\<Colon>{eq, ord}) # xs \<le> y # ys \<longleftrightarrow> x = y \<and> xs \<le> ys"
290   by simp_all
292 lemma less_code [code func]:
293   "xs < ([]\<Colon>'a\<Colon>{eq, ord} list) \<longleftrightarrow> False"
294   "[] < (x\<Colon>'a\<Colon>{eq, ord})# xs \<longleftrightarrow> True"
295   "(x\<Colon>'a\<Colon>{eq, ord}) # xs < y # ys \<longleftrightarrow> x = y \<and> xs < ys"
296   unfolding strict_prefix_def by auto
298 lemmas [code func] = postfix_to_prefix
300 end