ex/Sorting.thy

some more sorting algorithms

(*  Title: 	HOL/ex/sorting.thy
    ID:         $Id$
    Author: 	Tobias Nipkow
    Copyright   1994 TU Muenchen

Specifikation of sorting
*)

Sorting = List +
consts
  sorted1:: "[['a,'a] => bool, 'a list] => bool"
  sorted :: "[['a,'a] => bool, 'a list] => bool"
  mset   :: "'a list => ('a => nat)"
  total  :: "(['a,'a] => bool) => bool"
  transf :: "(['a,'a] => bool) => bool"

rules

sorted1_Nil  "sorted1(f,[])"
sorted1_One  "sorted1(f,[x])"
sorted1_Cons "sorted1(f,Cons(x,Cons(y,zs))) = (f(x,y) & sorted1(f,Cons(y,zs)))"

sorted_Nil "sorted(le,[])"
sorted_Cons "sorted(le,Cons(x,xs)) = ((Alls y:xs. le(x,y)) & sorted(le,xs))"

mset_Nil "mset([],y) = 0"
mset_Cons "mset(Cons(x,xs),y) = if(x=y, Suc(mset(xs,y)), mset(xs,y))"

total_def  "total(r) == (!x y. r(x,y) | r(y,x))"
transf_def "transf(f) == (!x y z. f(x,y) & f(y,z) --> f(x,z))"
end