src/HOL/Library/List_Prefix.thy

author | wenzelm |

Fri, 03 Nov 2000 21:35:59 +0100 | |

changeset 10390 | 1d54567bed24 |

parent 10389 | c7d8901ab269 |

child 10408 | d8b3613158b1 |

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

tuned;

(* Title: HOL/Library/List_Prefix.thy ID: $Id$ Author: Tobias Nipkow and Markus Wenzel, TU Muenchen *) header {* \title{List prefixes} \author{Tobias Nipkow and Markus Wenzel} *} theory List_Prefix = Main: subsection {* Prefix order on lists *} instance list :: ("term") 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 :: ("term") 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?]: "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 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) 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 \<le> xs ==> C) ==> (xs \<parallel> ys ==> C) ==> C" by (unfold parallel_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" (concl is "?E xs") proof - assume "xs \<parallel> ys" have "?this --> ?E xs" (is "?P xs") proof (induct (stripped) xs rule: rev_induct) assume "[] \<parallel> ys" hence False by auto thus "?E []" .. next fix x xs assume hyp: "?P xs" assume asm: "xs @ [x] \<parallel> ys" show "?E (xs @ [x])" 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 asm have "[x] \<parallel> []" by auto hence False by blast thus ?thesis .. next fix c cs assume ys': "ys' = c # cs" with asm 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 \<le> xs" hence "ys \<le> xs @ [x]" by simp with asm have False by blast thus ?thesis .. next assume "xs \<parallel> ys" with hyp 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 thus ?thesis .. qed end