src/HOL/ex/Sorting.thy
author obua
Mon Apr 10 16:00:34 2006 +0200 (2006-04-10)
changeset 19404 9bf2cdc9e8e8
parent 15815 62854cac5410
child 19736 d8d0f8f51d69
permissions -rw-r--r--
Moved stuff from Ring_and_Field to Matrix
     1 (*  Title:      HOL/ex/sorting.thy
     2     ID:         $Id$
     3     Author:     Tobias Nipkow
     4     Copyright   1994 TU Muenchen
     5 *)
     6 
     7 header{*Sorting: Basic Theory*}
     8 
     9 theory Sorting
    10 imports Main Multiset
    11 begin
    12 
    13 consts
    14   sorted1:: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"
    15   sorted :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"
    16 
    17 primrec
    18   "sorted1 le [] = True"
    19   "sorted1 le (x#xs) = ((case xs of [] => True | y#ys => le x y) &
    20                         sorted1 le xs)"
    21 
    22 primrec
    23   "sorted le [] = True"
    24   "sorted le (x#xs) = ((\<forall>y \<in> set xs. le x y) & sorted le xs)"
    25 
    26 
    27 constdefs
    28   total  :: "('a \<Rightarrow> 'a \<Rightarrow> bool) => bool"
    29    "total r == (\<forall>x y. r x y | r y x)"
    30   
    31   transf :: "('a \<Rightarrow> 'a \<Rightarrow> bool) => bool"
    32    "transf f == (\<forall>x y z. f x y & f y z --> f x z)"
    33 
    34 
    35 
    36 (* Equivalence of two definitions of `sorted' *)
    37 
    38 lemma sorted1_is_sorted: "transf(le) ==> sorted1 le xs = sorted le xs";
    39 apply(induct xs)
    40  apply simp
    41 apply(simp split: list.split)
    42 apply(unfold transf_def);
    43 apply(blast)
    44 done
    45 
    46 lemma sorted_append [simp]:
    47  "sorted le (xs@ys) = 
    48   (sorted le xs & sorted le ys & (\<forall>x \<in> set xs. \<forall>y \<in> set ys. le x y))"
    49 by (induct xs, auto)
    50 
    51 end