src/HOL/Library/List_Prefix.thy
 author huffman Thu Jun 11 09:03:24 2009 -0700 (2009-06-11) changeset 31563 ded2364d14d4 parent 30663 0b6aff7451b2 child 37474 ce943f9edf5e permissions -rw-r--r--
cleaned up some proofs
1 (*  Title:      HOL/Library/List_Prefix.thy
2     Author:     Tobias Nipkow and Markus Wenzel, TU Muenchen
3 *)
5 header {* List prefixes and postfixes *}
7 theory List_Prefix
8 imports List Main
9 begin
11 subsection {* Prefix order on lists *}
13 instantiation list :: (type) order
14 begin
16 definition
17   prefix_def [code del]: "xs \<le> ys = (\<exists>zs. ys = xs @ zs)"
19 definition
20   strict_prefix_def [code del]: "xs < ys = (xs \<le> ys \<and> xs \<noteq> (ys::'a list))"
22 instance
23   by intro_classes (auto simp add: prefix_def strict_prefix_def)
25 end
27 lemma prefixI [intro?]: "ys = xs @ zs ==> xs \<le> ys"
28   unfolding prefix_def by blast
30 lemma prefixE [elim?]:
31   assumes "xs \<le> ys"
32   obtains zs where "ys = xs @ zs"
33   using assms unfolding prefix_def by blast
35 lemma strict_prefixI' [intro?]: "ys = xs @ z # zs ==> xs < ys"
36   unfolding strict_prefix_def prefix_def by blast
38 lemma strict_prefixE' [elim?]:
39   assumes "xs < ys"
40   obtains z zs where "ys = xs @ z # zs"
41 proof -
42   from `xs < ys` obtain us where "ys = xs @ us" and "xs \<noteq> ys"
43     unfolding strict_prefix_def prefix_def by blast
44   with that show ?thesis by (auto simp add: neq_Nil_conv)
45 qed
47 lemma strict_prefixI [intro?]: "xs \<le> ys ==> xs \<noteq> ys ==> xs < (ys::'a list)"
48   unfolding strict_prefix_def by blast
50 lemma strict_prefixE [elim?]:
51   fixes xs ys :: "'a list"
52   assumes "xs < ys"
53   obtains "xs \<le> ys" and "xs \<noteq> ys"
54   using assms unfolding strict_prefix_def by blast
57 subsection {* Basic properties of prefixes *}
59 theorem Nil_prefix [iff]: "[] \<le> xs"
62 theorem prefix_Nil [simp]: "(xs \<le> []) = (xs = [])"
63   by (induct xs) (simp_all add: prefix_def)
65 lemma prefix_snoc [simp]: "(xs \<le> ys @ [y]) = (xs = ys @ [y] \<or> xs \<le> ys)"
66 proof
67   assume "xs \<le> ys @ [y]"
68   then obtain zs where zs: "ys @ [y] = xs @ zs" ..
69   show "xs = ys @ [y] \<or> xs \<le> ys"
70     by (metis append_Nil2 butlast_append butlast_snoc prefixI zs)
71 next
72   assume "xs = ys @ [y] \<or> xs \<le> ys"
73   then show "xs \<le> ys @ [y]"
74     by (metis order_eq_iff strict_prefixE strict_prefixI' xt1(7))
75 qed
77 lemma Cons_prefix_Cons [simp]: "(x # xs \<le> y # ys) = (x = y \<and> xs \<le> ys)"
78   by (auto simp add: prefix_def)
80 lemma same_prefix_prefix [simp]: "(xs @ ys \<le> xs @ zs) = (ys \<le> zs)"
81   by (induct xs) simp_all
83 lemma same_prefix_nil [iff]: "(xs @ ys \<le> xs) = (ys = [])"
84   by (metis append_Nil2 append_self_conv order_eq_iff prefixI)
86 lemma prefix_prefix [simp]: "xs \<le> ys ==> xs \<le> ys @ zs"
87   by (metis order_le_less_trans prefixI strict_prefixE strict_prefixI)
89 lemma append_prefixD: "xs @ ys \<le> zs \<Longrightarrow> xs \<le> zs"
90   by (auto simp add: prefix_def)
92 theorem prefix_Cons: "(xs \<le> y # ys) = (xs = [] \<or> (\<exists>zs. xs = y # zs \<and> zs \<le> ys))"
93   by (cases xs) (auto simp add: prefix_def)
95 theorem prefix_append:
96   "(xs \<le> ys @ zs) = (xs \<le> ys \<or> (\<exists>us. xs = ys @ us \<and> us \<le> zs))"
97   apply (induct zs rule: rev_induct)
98    apply force
99   apply (simp del: append_assoc add: append_assoc [symmetric])
100   apply (metis append_eq_appendI)
101   done
103 lemma append_one_prefix:
104   "xs \<le> ys ==> length xs < length ys ==> xs @ [ys ! length xs] \<le> ys"
105   unfolding prefix_def
106   by (metis Cons_eq_appendI append_eq_appendI append_eq_conv_conj
107     eq_Nil_appendI nth_drop')
109 theorem prefix_length_le: "xs \<le> ys ==> length xs \<le> length ys"
110   by (auto simp add: prefix_def)
112 lemma prefix_same_cases:
113   "(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"
114   unfolding prefix_def by (metis append_eq_append_conv2)
116 lemma set_mono_prefix: "xs \<le> ys \<Longrightarrow> set xs \<subseteq> set ys"
117   by (auto simp add: prefix_def)
119 lemma take_is_prefix: "take n xs \<le> xs"
120   unfolding prefix_def by (metis append_take_drop_id)
122 lemma map_prefixI: "xs \<le> ys \<Longrightarrow> map f xs \<le> map f ys"
123   by (auto simp: prefix_def)
125 lemma prefix_length_less: "xs < ys \<Longrightarrow> length xs < length ys"
126   by (auto simp: strict_prefix_def prefix_def)
128 lemma strict_prefix_simps [simp]:
129     "xs < [] = False"
130     "[] < (x # xs) = True"
131     "(x # xs) < (y # ys) = (x = y \<and> xs < ys)"
132   by (simp_all add: strict_prefix_def cong: conj_cong)
134 lemma take_strict_prefix: "xs < ys \<Longrightarrow> take n xs < ys"
135   apply (induct n arbitrary: xs ys)
136    apply (case_tac ys, simp_all)[1]
137   apply (metis order_less_trans strict_prefixI take_is_prefix)
138   done
140 lemma not_prefix_cases:
141   assumes pfx: "\<not> ps \<le> ls"
142   obtains
143     (c1) "ps \<noteq> []" and "ls = []"
144   | (c2) a as x xs where "ps = a#as" and "ls = x#xs" and "x = a" and "\<not> as \<le> xs"
145   | (c3) a as x xs where "ps = a#as" and "ls = x#xs" and "x \<noteq> a"
146 proof (cases ps)
147   case Nil then show ?thesis using pfx by simp
148 next
149   case (Cons a as)
150   note c = `ps = a#as`
151   show ?thesis
152   proof (cases ls)
153     case Nil then show ?thesis by (metis append_Nil2 pfx c1 same_prefix_nil)
154   next
155     case (Cons x xs)
156     show ?thesis
157     proof (cases "x = a")
158       case True
159       have "\<not> as \<le> xs" using pfx c Cons True by simp
160       with c Cons True show ?thesis by (rule c2)
161     next
162       case False
163       with c Cons show ?thesis by (rule c3)
164     qed
165   qed
166 qed
168 lemma not_prefix_induct [consumes 1, case_names Nil Neq Eq]:
169   assumes np: "\<not> ps \<le> ls"
170     and base: "\<And>x xs. P (x#xs) []"
171     and r1: "\<And>x xs y ys. x \<noteq> y \<Longrightarrow> P (x#xs) (y#ys)"
172     and r2: "\<And>x xs y ys. \<lbrakk> x = y; \<not> xs \<le> ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys)"
173   shows "P ps ls" using np
174 proof (induct ls arbitrary: ps)
175   case Nil then show ?case
176     by (auto simp: neq_Nil_conv elim!: not_prefix_cases intro!: base)
177 next
178   case (Cons y ys)
179   then have npfx: "\<not> ps \<le> (y # ys)" by simp
180   then obtain x xs where pv: "ps = x # xs"
181     by (rule not_prefix_cases) auto
182   show ?case by (metis Cons.hyps Cons_prefix_Cons npfx pv r1 r2)
183 qed
186 subsection {* Parallel lists *}
188 definition
189   parallel :: "'a list => 'a list => bool"  (infixl "\<parallel>" 50) where
190   "(xs \<parallel> ys) = (\<not> xs \<le> ys \<and> \<not> ys \<le> xs)"
192 lemma parallelI [intro]: "\<not> xs \<le> ys ==> \<not> ys \<le> xs ==> xs \<parallel> ys"
193   unfolding parallel_def by blast
195 lemma parallelE [elim]:
196   assumes "xs \<parallel> ys"
197   obtains "\<not> xs \<le> ys \<and> \<not> ys \<le> xs"
198   using assms unfolding parallel_def by blast
200 theorem prefix_cases:
201   obtains "xs \<le> ys" | "ys < xs" | "xs \<parallel> ys"
202   unfolding parallel_def strict_prefix_def by blast
204 theorem parallel_decomp:
205   "xs \<parallel> ys ==> \<exists>as b bs c cs. b \<noteq> c \<and> xs = as @ b # bs \<and> ys = as @ c # cs"
206 proof (induct xs rule: rev_induct)
207   case Nil
208   then have False by auto
209   then show ?case ..
210 next
211   case (snoc x xs)
212   show ?case
213   proof (rule prefix_cases)
214     assume le: "xs \<le> ys"
215     then obtain ys' where ys: "ys = xs @ ys'" ..
216     show ?thesis
217     proof (cases ys')
218       assume "ys' = []"
219       then show ?thesis by (metis append_Nil2 parallelE prefixI snoc.prems ys)
220     next
221       fix c cs assume ys': "ys' = c # cs"
222       then show ?thesis
223         by (metis Cons_eq_appendI eq_Nil_appendI parallelE prefixI
224           same_prefix_prefix snoc.prems ys)
225     qed
226   next
227     assume "ys < xs" then have "ys \<le> xs @ [x]" by (simp add: strict_prefix_def)
228     with snoc have False by blast
229     then show ?thesis ..
230   next
231     assume "xs \<parallel> ys"
232     with snoc obtain as b bs c cs where neq: "(b::'a) \<noteq> c"
233       and xs: "xs = as @ b # bs" and ys: "ys = as @ c # cs"
234       by blast
235     from xs have "xs @ [x] = as @ b # (bs @ [x])" by simp
236     with neq ys show ?thesis by blast
237   qed
238 qed
240 lemma parallel_append: "a \<parallel> b \<Longrightarrow> a @ c \<parallel> b @ d"
241   apply (rule parallelI)
242     apply (erule parallelE, erule conjE,
243       induct rule: not_prefix_induct, simp+)+
244   done
246 lemma parallel_appendI: "xs \<parallel> ys \<Longrightarrow> x = xs @ xs' \<Longrightarrow> y = ys @ ys' \<Longrightarrow> x \<parallel> y"
249 lemma parallel_commute: "a \<parallel> b \<longleftrightarrow> b \<parallel> a"
250   unfolding parallel_def by auto
253 subsection {* Postfix order on lists *}
255 definition
256   postfix :: "'a list => 'a list => bool"  ("(_/ >>= _)" [51, 50] 50) where
257   "(xs >>= ys) = (\<exists>zs. xs = zs @ ys)"
259 lemma postfixI [intro?]: "xs = zs @ ys ==> xs >>= ys"
260   unfolding postfix_def by blast
262 lemma postfixE [elim?]:
263   assumes "xs >>= ys"
264   obtains zs where "xs = zs @ ys"
265   using assms unfolding postfix_def by blast
267 lemma postfix_refl [iff]: "xs >>= xs"
268   by (auto simp add: postfix_def)
269 lemma postfix_trans: "\<lbrakk>xs >>= ys; ys >>= zs\<rbrakk> \<Longrightarrow> xs >>= zs"
270   by (auto simp add: postfix_def)
271 lemma postfix_antisym: "\<lbrakk>xs >>= ys; ys >>= xs\<rbrakk> \<Longrightarrow> xs = ys"
272   by (auto simp add: postfix_def)
274 lemma Nil_postfix [iff]: "xs >>= []"
276 lemma postfix_Nil [simp]: "([] >>= xs) = (xs = [])"
277   by (auto simp add: postfix_def)
279 lemma postfix_ConsI: "xs >>= ys \<Longrightarrow> x#xs >>= ys"
280   by (auto simp add: postfix_def)
281 lemma postfix_ConsD: "xs >>= y#ys \<Longrightarrow> xs >>= ys"
282   by (auto simp add: postfix_def)
284 lemma postfix_appendI: "xs >>= ys \<Longrightarrow> zs @ xs >>= ys"
285   by (auto simp add: postfix_def)
286 lemma postfix_appendD: "xs >>= zs @ ys \<Longrightarrow> xs >>= ys"
287   by (auto simp add: postfix_def)
289 lemma postfix_is_subset: "xs >>= ys ==> set ys \<subseteq> set xs"
290 proof -
291   assume "xs >>= ys"
292   then obtain zs where "xs = zs @ ys" ..
293   then show ?thesis by (induct zs) auto
294 qed
296 lemma postfix_ConsD2: "x#xs >>= y#ys ==> xs >>= ys"
297 proof -
298   assume "x#xs >>= y#ys"
299   then obtain zs where "x#xs = zs @ y#ys" ..
300   then show ?thesis
301     by (induct zs) (auto intro!: postfix_appendI postfix_ConsI)
302 qed
304 lemma postfix_to_prefix: "xs >>= ys \<longleftrightarrow> rev ys \<le> rev xs"
305 proof
306   assume "xs >>= ys"
307   then obtain zs where "xs = zs @ ys" ..
308   then have "rev xs = rev ys @ rev zs" by simp
309   then show "rev ys <= rev xs" ..
310 next
311   assume "rev ys <= rev xs"
312   then obtain zs where "rev xs = rev ys @ zs" ..
313   then have "rev (rev xs) = rev zs @ rev (rev ys)" by simp
314   then have "xs = rev zs @ ys" by simp
315   then show "xs >>= ys" ..
316 qed
318 lemma distinct_postfix: "distinct xs \<Longrightarrow> xs >>= ys \<Longrightarrow> distinct ys"
319   by (clarsimp elim!: postfixE)
321 lemma postfix_map: "xs >>= ys \<Longrightarrow> map f xs >>= map f ys"
322   by (auto elim!: postfixE intro: postfixI)
324 lemma postfix_drop: "as >>= drop n as"
325   unfolding postfix_def
326   apply (rule exI [where x = "take n as"])
327   apply simp
328   done
330 lemma postfix_take: "xs >>= ys \<Longrightarrow> xs = take (length xs - length ys) xs @ ys"
331   by (clarsimp elim!: postfixE)
333 lemma parallelD1: "x \<parallel> y \<Longrightarrow> \<not> x \<le> y"
334   by blast
336 lemma parallelD2: "x \<parallel> y \<Longrightarrow> \<not> y \<le> x"
337   by blast
339 lemma parallel_Nil1 [simp]: "\<not> x \<parallel> []"
340   unfolding parallel_def by simp
342 lemma parallel_Nil2 [simp]: "\<not> [] \<parallel> x"
343   unfolding parallel_def by simp
345 lemma Cons_parallelI1: "a \<noteq> b \<Longrightarrow> a # as \<parallel> b # bs"
346   by auto
348 lemma Cons_parallelI2: "\<lbrakk> a = b; as \<parallel> bs \<rbrakk> \<Longrightarrow> a # as \<parallel> b # bs"
349   by (metis Cons_prefix_Cons parallelE parallelI)
351 lemma not_equal_is_parallel:
352   assumes neq: "xs \<noteq> ys"
353     and len: "length xs = length ys"
354   shows "xs \<parallel> ys"
355   using len neq
356 proof (induct rule: list_induct2)
357   case Nil
358   then show ?case by simp
359 next
360   case (Cons a as b bs)
361   have ih: "as \<noteq> bs \<Longrightarrow> as \<parallel> bs" by fact
362   show ?case
363   proof (cases "a = b")
364     case True
365     then have "as \<noteq> bs" using Cons by simp
366     then show ?thesis by (rule Cons_parallelI2 [OF True ih])
367   next
368     case False
369     then show ?thesis by (rule Cons_parallelI1)
370   qed
371 qed
374 subsection {* Executable code *}
376 lemma less_eq_code [code]:
377     "([]\<Colon>'a\<Colon>{eq, ord} list) \<le> xs \<longleftrightarrow> True"
378     "(x\<Colon>'a\<Colon>{eq, ord}) # xs \<le> [] \<longleftrightarrow> False"
379     "(x\<Colon>'a\<Colon>{eq, ord}) # xs \<le> y # ys \<longleftrightarrow> x = y \<and> xs \<le> ys"
380   by simp_all
382 lemma less_code [code]:
383     "xs < ([]\<Colon>'a\<Colon>{eq, ord} list) \<longleftrightarrow> False"
384     "[] < (x\<Colon>'a\<Colon>{eq, ord})# xs \<longleftrightarrow> True"
385     "(x\<Colon>'a\<Colon>{eq, ord}) # xs < y # ys \<longleftrightarrow> x = y \<and> xs < ys"
386   unfolding strict_prefix_def by auto
388 lemmas [code] = postfix_to_prefix
390 end