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src/HOL/Library/List_Prefix.thy

author | nipkow |

Wed, 18 Aug 2004 11:09:40 +0200 | |

changeset 15140 | 322485b816ac |

parent 15131 | c69542757a4d |

child 15355 | 0de05d104060 |

permissions | -rw-r--r-- |

import -> imports

(* Title: HOL/Library/List_Prefix.thy ID: $Id$ Author: Tobias Nipkow and Markus Wenzel, TU Muenchen *) header {* List prefixes and postfixes *} theory List_Prefix imports Main begin subsection {* Prefix order on lists *} instance list :: (type) ord .. defs (overloaded) prefix_def: "xs \<le> ys == \<exists>zs. ys = xs @ zs" strict_prefix_def: "xs < ys == xs \<le> ys \<and> xs \<noteq> (ys::'a list)" instance list :: (type) order by intro_classes (auto simp add: prefix_def strict_prefix_def) lemma prefixI [intro?]: "ys = xs @ zs ==> xs \<le> ys" by (unfold prefix_def) blast lemma prefixE [elim?]: "xs \<le> ys ==> (!!zs. ys = xs @ zs ==> C) ==> C" by (unfold prefix_def) blast lemma strict_prefixI' [intro?]: "ys = xs @ z # zs ==> xs < ys" by (unfold strict_prefix_def prefix_def) blast lemma strict_prefixE' [elim?]: "xs < ys ==> (!!z zs. ys = xs @ z # zs ==> C) ==> C" proof - assume r: "!!z zs. ys = xs @ z # zs ==> C" assume "xs < ys" then obtain us where "ys = xs @ us" and "xs \<noteq> ys" by (unfold strict_prefix_def prefix_def) blast with r show ?thesis by (auto simp add: neq_Nil_conv) qed lemma strict_prefixI [intro?]: "xs \<le> ys ==> xs \<noteq> ys ==> xs < (ys::'a list)" by (unfold strict_prefix_def) blast lemma strict_prefixE [elim?]: "xs < ys ==> (xs \<le> ys ==> xs \<noteq> (ys::'a list) ==> C) ==> C" by (unfold strict_prefix_def) blast subsection {* Basic properties of prefixes *} theorem Nil_prefix [iff]: "[] \<le> xs" by (simp add: prefix_def) theorem prefix_Nil [simp]: "(xs \<le> []) = (xs = [])" by (induct xs) (simp_all add: prefix_def) lemma prefix_snoc [simp]: "(xs \<le> ys @ [y]) = (xs = ys @ [y] \<or> xs \<le> ys)" proof assume "xs \<le> ys @ [y]" then obtain zs where zs: "ys @ [y] = xs @ zs" .. show "xs = ys @ [y] \<or> xs \<le> ys" proof (cases zs rule: rev_cases) assume "zs = []" with zs have "xs = ys @ [y]" by simp thus ?thesis .. next fix z zs' assume "zs = zs' @ [z]" with zs have "ys = xs @ zs'" by simp hence "xs \<le> ys" .. thus ?thesis .. qed next assume "xs = ys @ [y] \<or> xs \<le> ys" thus "xs \<le> ys @ [y]" proof assume "xs = ys @ [y]" thus ?thesis by simp next assume "xs \<le> ys" then obtain zs where "ys = xs @ zs" .. hence "ys @ [y] = xs @ (zs @ [y])" by simp thus ?thesis .. qed qed lemma Cons_prefix_Cons [simp]: "(x # xs \<le> y # ys) = (x = y \<and> xs \<le> ys)" by (auto simp add: prefix_def) lemma same_prefix_prefix [simp]: "(xs @ ys \<le> xs @ zs) = (ys \<le> zs)" by (induct xs) simp_all lemma same_prefix_nil [iff]: "(xs @ ys \<le> xs) = (ys = [])" proof - have "(xs @ ys \<le> xs @ []) = (ys \<le> [])" by (rule same_prefix_prefix) thus ?thesis by simp qed lemma prefix_prefix [simp]: "xs \<le> ys ==> xs \<le> ys @ zs" proof - assume "xs \<le> ys" then obtain us where "ys = xs @ us" .. hence "ys @ zs = xs @ (us @ zs)" by simp thus ?thesis .. qed lemma append_prefixD: "xs @ ys \<le> zs \<Longrightarrow> xs \<le> zs" by(simp add:prefix_def) blast theorem prefix_Cons: "(xs \<le> y # ys) = (xs = [] \<or> (\<exists>zs. xs = y # zs \<and> zs \<le> ys))" by (cases xs) (auto simp add: prefix_def) theorem prefix_append: "(xs \<le> ys @ zs) = (xs \<le> ys \<or> (\<exists>us. xs = ys @ us \<and> us \<le> zs))" apply (induct zs rule: rev_induct) apply force apply (simp del: append_assoc add: append_assoc [symmetric]) apply simp apply blast done lemma append_one_prefix: "xs \<le> ys ==> length xs < length ys ==> xs @ [ys ! length xs] \<le> ys" apply (unfold prefix_def) apply (auto simp add: nth_append) apply (case_tac zs) apply auto done theorem prefix_length_le: "xs \<le> ys ==> length xs \<le> length ys" by (auto simp add: prefix_def) lemma prefix_same_cases: "\<lbrakk> (xs\<^isub>1::'a list) \<le> ys; xs\<^isub>2 \<le> ys \<rbrakk> \<Longrightarrow> xs\<^isub>1 \<le> xs\<^isub>2 \<or> xs\<^isub>2 \<le> xs\<^isub>1" apply(simp add:prefix_def) apply(erule exE)+ apply(simp add: append_eq_append_conv_if split:if_splits) apply(rule disjI2) apply(rule_tac x = "drop (size xs\<^isub>2) xs\<^isub>1" in exI) apply clarify apply(drule sym) apply(insert append_take_drop_id[of "length xs\<^isub>2" xs\<^isub>1]) apply simp apply(rule disjI1) apply(rule_tac x = "drop (size xs\<^isub>1) xs\<^isub>2" in exI) apply clarify apply(insert append_take_drop_id[of "length xs\<^isub>1" xs\<^isub>2]) apply simp done lemma set_mono_prefix: "xs \<le> ys \<Longrightarrow> set xs \<subseteq> set ys" by(fastsimp simp add:prefix_def) subsection {* Parallel lists *} constdefs parallel :: "'a list => 'a list => bool" (infixl "\<parallel>" 50) "xs \<parallel> ys == \<not> xs \<le> ys \<and> \<not> ys \<le> xs" lemma parallelI [intro]: "\<not> xs \<le> ys ==> \<not> ys \<le> xs ==> xs \<parallel> ys" by (unfold parallel_def) blast lemma parallelE [elim]: "xs \<parallel> ys ==> (\<not> xs \<le> ys ==> \<not> ys \<le> xs ==> C) ==> C" by (unfold parallel_def) blast theorem prefix_cases: "(xs \<le> ys ==> C) ==> (ys < xs ==> C) ==> (xs \<parallel> ys ==> C) ==> C" by (unfold parallel_def strict_prefix_def) blast theorem parallel_decomp: "xs \<parallel> ys ==> \<exists>as b bs c cs. b \<noteq> c \<and> xs = as @ b # bs \<and> ys = as @ c # cs" proof (induct xs rule: rev_induct) case Nil hence False by auto thus ?case .. next case (snoc x xs) show ?case proof (rule prefix_cases) assume le: "xs \<le> ys" then obtain ys' where ys: "ys = xs @ ys'" .. show ?thesis proof (cases ys') assume "ys' = []" with ys have "xs = ys" by simp with snoc have "[x] \<parallel> []" by auto hence False by blast thus ?thesis .. next fix c cs assume ys': "ys' = c # cs" with snoc ys have "xs @ [x] \<parallel> xs @ c # cs" by (simp only:) hence "x \<noteq> c" by auto moreover have "xs @ [x] = xs @ x # []" by simp moreover from ys ys' have "ys = xs @ c # cs" by (simp only:) ultimately show ?thesis by blast qed next assume "ys < xs" hence "ys \<le> xs @ [x]" by (simp add: strict_prefix_def) with snoc have False by blast thus ?thesis .. next assume "xs \<parallel> ys" with snoc obtain as b bs c cs where neq: "(b::'a) \<noteq> c" and xs: "xs = as @ b # bs" and ys: "ys = as @ c # cs" by blast from xs have "xs @ [x] = as @ b # (bs @ [x])" by simp with neq ys show ?thesis by blast qed qed subsection {* Postfix order on lists *} constdefs postfix :: "'a list => 'a list => bool" ("(_/ >= _)" [51, 50] 50) "xs >= ys == \<exists>zs. xs = zs @ ys" lemma postfix_refl [simp, intro!]: "xs >= xs" by (auto simp add: postfix_def) lemma postfix_trans: "\<lbrakk>xs >= ys; ys >= zs\<rbrakk> \<Longrightarrow> xs >= zs" by (auto simp add: postfix_def) lemma postfix_antisym: "\<lbrakk>xs >= ys; ys >= xs\<rbrakk> \<Longrightarrow> xs = ys" by (auto simp add: postfix_def) lemma Nil_postfix [iff]: "xs >= []" by (simp add: postfix_def) lemma postfix_Nil [simp]: "([] >= xs) = (xs = [])" by (auto simp add:postfix_def) lemma postfix_ConsI: "xs >= ys \<Longrightarrow> x#xs >= ys" by (auto simp add: postfix_def) lemma postfix_ConsD: "xs >= y#ys \<Longrightarrow> xs >= ys" by (auto simp add: postfix_def) lemma postfix_appendI: "xs >= ys \<Longrightarrow> zs @ xs >= ys" by (auto simp add: postfix_def) lemma postfix_appendD: "xs >= zs @ ys \<Longrightarrow> xs >= ys" by(auto simp add: postfix_def) lemma postfix_is_subset_lemma: "xs = zs @ ys \<Longrightarrow> set ys \<subseteq> set xs" by (induct zs, auto) lemma postfix_is_subset: "xs >= ys \<Longrightarrow> set ys \<subseteq> set xs" by (unfold postfix_def, erule exE, erule postfix_is_subset_lemma) lemma postfix_ConsD2_lemma [rule_format]: "x#xs = zs @ y#ys \<longrightarrow> xs >= ys" by (induct zs, auto intro!: postfix_appendI postfix_ConsI) lemma postfix_ConsD2: "x#xs >= y#ys \<Longrightarrow> xs >= ys" by (auto simp add: postfix_def dest!: postfix_ConsD2_lemma) lemma postfix2prefix: "(xs >= ys) = (rev ys <= rev xs)" apply (unfold prefix_def postfix_def, safe) apply (rule_tac x = "rev zs" in exI, simp) apply (rule_tac x = "rev zs" in exI) apply (rule rev_is_rev_conv [THEN iffD1], simp) done end