src/HOL/Integ/Relation.ML

 
 val RSLIST = curry (op MRS);
 
 open Relation;
 
-goalw Relation.thy [converse_def] "!!a b r. <a,b>:r ==> <b,a>:converse(r)";
+goalw Relation.thy [converse_def] "!!a b r. (a,b):r ==> (b,a):converse(r)";
 by (simp_tac prod_ss 1);
 by (fast_tac set_cs 1);
 qed "converseI";
 
-goalw Relation.thy [converse_def] "!!a b r. <a,b> : converse(r) ==> <b,a> : r";
+goalw Relation.thy [converse_def] "!!a b r. (a,b) : converse(r) ==> (b,a) : r";
 by (fast_tac comp_cs 1);
 qed "converseD";
 
 qed_goalw "converseE" Relation.thy [converse_def]
     "[| yx : converse(r);  \
-\       !!x y. [| yx=<y,x>;  <x,y>:r |] ==> P \
+\       !!x y. [| yx=(y,x);  (x,y):r |] ==> P \
 \    |] ==> P"
  (fn [major,minor]=>
   [ (rtac (major RS CollectE) 1),
     (REPEAT (eresolve_tac [bexE,exE, conjE, minor] 1)),
     (hyp_subst_tac 1),

 
 val converse_cs = comp_cs addSIs [converseI] 
 			  addSEs [converseD,converseE];
 
 qed_goalw "Domain_iff" Relation.thy [Domain_def]
-    "a: Domain(r) = (EX y. <a,y>: r)"
+    "a: Domain(r) = (EX y. (a,y): r)"
  (fn _=> [ (fast_tac comp_cs 1) ]);
 
-qed_goal "DomainI" Relation.thy "!!a b r. <a,b>: r ==> a: Domain(r)"
+qed_goal "DomainI" Relation.thy "!!a b r. (a,b): r ==> a: Domain(r)"
  (fn _ => [ (etac (exI RS (Domain_iff RS iffD2)) 1) ]);
 
 qed_goal "DomainE" Relation.thy
-    "[| a : Domain(r);  !!y. <a,y>: r ==> P |] ==> P"
+    "[| a : Domain(r);  !!y. (a,y): r ==> P |] ==> P"
  (fn prems=>
   [ (rtac (Domain_iff RS iffD1 RS exE) 1),
     (REPEAT (ares_tac prems 1)) ]);
 
-qed_goalw "RangeI" Relation.thy [Range_def] "!!a b r.<a,b>: r ==> b : Range(r)"
+qed_goalw "RangeI" Relation.thy [Range_def] "!!a b r.(a,b): r ==> b : Range(r)"
  (fn _ => [ (etac (converseI RS DomainI) 1) ]);
 
 qed_goalw "RangeE" Relation.thy [Range_def]
-    "[| b : Range(r);  !!x. <x,b>: r ==> P |] ==> P"
+    "[| b : Range(r);  !!x. (x,b): r ==> P |] ==> P"
  (fn major::prems=>
   [ (rtac (major RS DomainE) 1),
     (resolve_tac prems 1),
     (etac converseD 1) ]);
 
 (*** Image of a set under a function/relation ***)
 
 qed_goalw "Image_iff" Relation.thy [Image_def]
-    "b : r^^A = (? x:A. <x,b>:r)"
+    "b : r^^A = (? x:A. (x,b):r)"
  (fn _ => [ fast_tac (comp_cs addIs [RangeI]) 1 ]);
 
 qed_goal "Image_singleton_iff" Relation.thy
-    "(b : r^^{a}) = (<a,b>:r)"
+    "(b : r^^{a}) = ((a,b):r)"
  (fn _ => [ rtac (Image_iff RS trans) 1,
 	    fast_tac comp_cs 1 ]);
 
 qed_goalw "ImageI" Relation.thy [Image_def]
-    "!!a b r. [| <a,b>: r;  a:A |] ==> b : r^^A"
+    "!!a b r. [| (a,b): r;  a:A |] ==> b : r^^A"
  (fn _ => [ (REPEAT (ares_tac [CollectI,RangeI,bexI] 1)),
             (resolve_tac [conjI ] 1),
             (resolve_tac [RangeI] 1),
             (REPEAT (fast_tac set_cs 1))]);
 
 qed_goalw "ImageE" Relation.thy [Image_def]
-    "[| b: r^^A;  !!x.[| <x,b>: r;  x:A |] ==> P |] ==> P"
+    "[| b: r^^A;  !!x.[| (x,b): r;  x:A |] ==> P |] ==> P"
  (fn major::prems=>
   [ (rtac (major RS CollectE) 1),
     (safe_tac set_cs),
     (etac RangeE 1),
     (rtac (hd prems) 1),