--- a/Sum.thy Fri Nov 25 13:35:32 1994 +0100
+++ b/Sum.thy Fri Nov 25 14:21:14 1994 +0100
@@ -1,55 +1,51 @@
-(* Title: HOL/sum
+(* Title: HOL/Sum.thy
ID: $Id$
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
-The disjoint sum of two types
+The disjoint sum of two types.
*)
Sum = Prod +
-types
- ('a,'b) "+" (infixr 10)
-
-arities
- "+" :: (term,term)term
+(* type definition *)
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"
+ Inl_Rep :: "['a, 'a, 'b, bool] => bool"
+ Inr_Rep :: "['b, 'a, 'b, bool] => bool"
+
+defs
+ Inl_Rep_def "Inl_Rep == (%a. %x y p. x=a & p)"
+ Inr_Rep_def "Inr_Rep == (%b. %x y p. y=b & ~p)"
+
+subtype (Sum)
+ ('a, 'b) "+" (infixr 10)
+ = "{f. (? a. f = Inl_Rep(a::'a)) | (? b. f = Inr_Rep(b::'b))}"
+
+
+(* abstract constants and syntax *)
+
+consts
+ 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, 'b=>'a] => 'a set"
+ "plus" :: "['a set, 'b set] => ('a + 'b) set" (infixr 65)
+ Part :: "['a set, 'b => '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))"
+defs
+ 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)}"
+ Part_def "Part(A, h) == A Int {x. ? z. x = h(z)}"
end