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

--- a/Sum.ML	Fri Aug 19 11:10:56 1994 +0200
+++ b/Sum.ML	Fri Aug 19 11:15:01 1994 +0200
@@ -88,7 +88,33 @@
 by (fast_tac (HOL_cs addSEs [Inr_inject]) 1);
 val Inr_eq = result();
 
-val sum_cs = set_cs addSEs [Inl_neq_Inr, Inr_neq_Inl]
+(*** Rules for the disjoint sum of two SETS ***)
+
+(** Introduction rules for the injections **)
+
+goalw Sum.thy [sum_def] "!!a A B. a : A ==> Inl(a) : A plus B";
+by (REPEAT (ares_tac [UnI1,imageI] 1));
+val InlI = result();
+
+goalw Sum.thy [sum_def] "!!b A B. b : B ==> Inr(b) : A plus B";
+by (REPEAT (ares_tac [UnI2,imageI] 1));
+val InrI = result();
+
+(** Elimination rules **)
+
+val major::prems = goalw Sum.thy [sum_def]
+    "[| u: A plus B;  \
+\       !!x. [| x:A;  u=Inl(x) |] ==> P; \
+\       !!y. [| y:B;  u=Inr(y) |] ==> P \
+\    |] ==> P";
+by (rtac (major RS UnE) 1);
+by (REPEAT (rtac refl 1
+     ORELSE eresolve_tac (prems@[imageE,ssubst]) 1));
+val plusE = result();
+
+
+val sum_cs = set_cs addSIs [InlI, InrI] 
+                    addSEs [plusE, Inl_neq_Inr, Inr_neq_Inl]
                     addSDs [Inl_inject, Inr_inject];
 
 
@@ -140,6 +166,8 @@
   (fn [prem] => [rtac (prem RS arg_cong) 1]);
 
 
+
+
 (** Rules for the Part primitive **)
 
 goalw Sum.thy [Part_def]

--- a/Sum.thy	Fri Aug 19 11:10:56 1994 +0200
+++ b/Sum.thy	Fri Aug 19 11:15:01 1994 +0200
@@ -23,6 +23,9 @@
   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
@@ -32,18 +35,20 @@
   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))}"
+  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)))"
+  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)}"