(* 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> zs == \<exists>ys. zs = xs @ ys"
strict_prefix_def: "xs < zs == xs \<le> zs \<and> xs \<noteq> (zs::'a list)"
instance list :: ("term") order
proof
fix xs ys zs :: "'a list"
show "xs \<le> xs" by (simp add: prefix_def)
{ assume "xs \<le> ys" and "ys \<le> zs" thus "xs \<le> zs" by (auto simp add: prefix_def) }
{ assume "xs \<le> ys" and "ys \<le> xs" thus "xs = ys" by (auto simp add: prefix_def) }
show "(xs < zs) = (xs \<le> zs \<and> xs \<noteq> zs)" by (simp only: strict_prefix_def)
qed
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 parellelE [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
subsection {* Recursion equations *}
theorem Nil_prefix [iff]: "[] \<le> xs"
apply (simp add: prefix_def)
done
theorem prefix_Nil [simp]: "(xs \<le> []) = (xs = [])"
apply (induct_tac xs)
apply simp
apply (simp add: prefix_def)
done
lemma prefix_snoc [simp]: "(xs \<le> ys @ [y]) = (xs = ys @ [y] \<or> xs \<le> ys)"
apply (unfold prefix_def)
apply (rule iffI)
apply (erule exE)
apply (rename_tac zs)
apply (rule_tac xs = zs in rev_exhaust)
apply simp
apply hypsubst
apply (simp del: append_assoc add: append_assoc [symmetric])
apply force
done
lemma Cons_prefix_Cons [simp]: "(x # xs \<le> y # ys) = (x = y \<and> xs \<le> ys)"
apply (auto simp add: prefix_def)
done
lemma same_prefix_prefix [simp]: "(xs @ ys \<le> xs @ zs) = (ys \<le> zs)"
apply (induct_tac xs)
apply simp_all
done
lemma [iff]: "(xs @ ys \<le> xs) = (ys = [])"
apply (insert same_prefix_prefix [where ?zs = "[]"])
apply simp
apply blast
done
lemma prefix_prefix [simp]: "xs \<le> ys ==> xs \<le> ys @ zs"
apply (unfold prefix_def)
apply clarify
apply simp
done
theorem prefix_Cons: "(xs \<le> y # ys) = (xs = [] \<or> (\<exists>zs. xs = y # zs \<and> zs \<le> ys))"
apply (unfold prefix_def)
apply (case_tac xs)
apply auto
done
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 ys)
apply auto
done
theorem prefix_length_le: "xs \<le> ys ==> length xs \<le> length ys"
apply (auto simp add: prefix_def)
done
subsection {* Prefix cases *}
lemma prefix_Nil_cases [case_names Nil]:
"xs \<le> [] ==>
(xs = [] ==> C) ==> C"
by simp
lemma prefix_Cons_cases [case_names Nil Cons]:
"xs \<le> y # ys ==>
(xs = [] ==> C) ==>
(!!zs. xs = y # zs ==> zs \<le> ys ==> C) ==> C"
by (simp only: prefix_Cons) blast
lemma prefix_snoc_cases [case_names prefix snoc]:
"xs \<le> ys @ [y] ==>
(xs \<le> ys ==> C) ==>
(xs = ys @ [y] ==> C) ==> C"
by (simp only: prefix_snoc) blast
lemma prefix_append_cases [case_names prefix append]:
"xs \<le> ys @ zs ==>
(xs \<le> ys ==> C) ==>
(!!us. xs = ys @ us ==> us \<le> zs ==> C) ==> C"
by (simp only: prefix_append) blast
lemmas prefix_any_cases [cases set: prefix] = (*dummy set name*)
prefix_Nil_cases prefix_Cons_cases
prefix_snoc_cases prefix_append_cases
end