Generalized case for atoms. Selection of environment lists is allowed more than once.
(* Title: HOL/ex/sorting.thy
ID: $Id$
Author: Tobias Nipkow
Copyright 1994 TU Muenchen
*)
header{*Sorting: Basic Theory*}
theory Sorting
imports Main Multiset
begin
consts
sorted1:: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"
sorted :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"
primrec
"sorted1 le [] = True"
"sorted1 le (x#xs) = ((case xs of [] => True | y#ys => le x y) &
sorted1 le xs)"
primrec
"sorted le [] = True"
"sorted le (x#xs) = ((\<forall>y \<in> set xs. le x y) & sorted le xs)"
definition
total :: "('a \<Rightarrow> 'a \<Rightarrow> bool) => bool" where
"total r = (\<forall>x y. r x y | r y x)"
definition
transf :: "('a \<Rightarrow> 'a \<Rightarrow> bool) => bool" where
"transf f = (\<forall>x y z. f x y & f y z --> f x z)"
(* Equivalence of two definitions of `sorted' *)
lemma sorted1_is_sorted: "transf(le) ==> sorted1 le xs = sorted le xs";
apply(induct xs)
apply simp
apply(simp split: list.split)
apply(unfold transf_def);
apply(blast)
done
lemma sorted_append [simp]:
"sorted le (xs@ys) =
(sorted le xs & sorted le ys & (\<forall>x \<in> set xs. \<forall>y \<in> set ys. le x y))"
by (induct xs) auto
end