Sum.thy

HOL/Sum: added disjoint sum of two sets as the plus operator, since + is
overloaded with an incompatible type

(*  Title: 	HOL/sum
    ID:         $Id$
    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1992  University of Cambridge

The disjoint sum of two types
*)

Sum = Prod +

types
  ('a,'b) "+"		      (infixl 10)

arities
  "+"      :: (term,term)term

consts
  Inl_Rep  :: "['a,'a,'b,bool] => bool"
  Inr_Rep  :: "['b,'a,'b,bool] => bool"
  Sum      :: "(['a,'b,bool] => bool)set"
  Rep_Sum  :: "'a + 'b => (['a,'b,bool] => bool)"
  Abs_Sum  :: "(['a,'b,bool] => bool) => 'a+'b"
  Inl	   :: "'a => 'a+'b"
  Inr	   :: "'b => 'a+'b"
  sum_case :: "['a=>'c,'b=>'c, 'a+'b] =>'c"

  (*disjoint sum for sets; the operator + is overloaded with wrong type!*)
  "plus"   :: "['a set, 'b set] => ('a+'b)set"      	(infixr 65)
  Part     :: "['a set, 'a=>'a] => 'a set"

translations
  "case p of Inl(x) => a | Inr(y) => b" == "sum_case(%x.a, %y.b, p)"

rules
  Inl_Rep_def	"Inl_Rep == (%a. %x y p. x=a & p)"
  Inr_Rep_def	"Inr_Rep == (%b. %x y p. y=b & ~p)"

  Sum_def 	"Sum == {f. (? a. f = Inl_Rep(a)) | (? b. f = Inr_Rep(b))}"
    (*faking a type definition...*)
  Rep_Sum 		"Rep_Sum(s): Sum"
  Rep_Sum_inverse 	"Abs_Sum(Rep_Sum(s)) = s"
  Abs_Sum_inverse 	"f: Sum ==> Rep_Sum(Abs_Sum(f)) = f"

    (*defining the abstract constants*)
  Inl_def  	"Inl == (%a. Abs_Sum(Inl_Rep(a)))"
  Inr_def 	"Inr == (%b. Abs_Sum(Inr_Rep(b)))"
  sum_case_def	"sum_case(f,g,p) == @z.  (!x. p=Inl(x) --> z=f(x))	\
\                                      & (!y. p=Inr(y) --> z=g(y))"

  sum_def       "A plus B == (Inl``A) Un (Inr``B)"

  (*for selecting out the components of a mutually recursive definition*)
  Part_def	"Part(A,h) == A Int {x. ? z. x = h(z)}"

end