src/HOL/Integ/Integ.thy

 consts
   intrel      :: "((nat * nat) * (nat * nat)) set"
 
 defs
   intrel_def
-   "intrel == {p. ? x1 y1 x2 y2. p=<<x1::nat,y1>,<x2,y2>> & x1+y2 = x2+y1}"
+   "intrel == {p. ? x1 y1 x2 y2. p=((x1::nat,y1),(x2,y2)) & x1+y2 = x2+y1}"
 
 subtype (Integ)
   int = "{x::(nat*nat).True}/intrel"		("quotient_def")
 
 arities int :: ord

   zpred,zsuc  :: "int=>int"
 
 defs
   zNat_def    "zNat == {x::nat. True}"
 
-  znat_def    "$# m == Abs_Integ(intrel ^^ {<m,0>})"
+  znat_def    "$# m == Abs_Integ(intrel ^^ {(m,0)})"
 
   zminus_def
-	"$~ Z == Abs_Integ(UN p:Rep_Integ(Z). split (%x y. intrel^^{<y,x>}) p)"
+	"$~ Z == Abs_Integ(UN p:Rep_Integ(Z). split (%x y. intrel^^{(y,x)}) p)"
 
   znegative_def
-      "znegative(Z) == EX x y. x<y & <x,y::nat>:Rep_Integ(Z)"
+      "znegative(Z) == EX x y. x<y & (x,y::nat):Rep_Integ(Z)"
 
   zmagnitude_def
-      "zmagnitude(Z) == Abs_Integ(UN p:Rep_Integ(Z).split (%x y. intrel^^{<(y-x) + (x-y),0>}) p)"
+      "zmagnitude(Z) == Abs_Integ(UN p:Rep_Integ(Z).split (%x y. intrel^^{((y-x) + (x-y),0)}) p)"
 
   zadd_def
    "Z1 + Z2 == \
 \       Abs_Integ(UN p1:Rep_Integ(Z1). UN p2:Rep_Integ(Z2).   \
-\           split (%x1 y1. split (%x2 y2. intrel^^{<x1+x2, y1+y2>}) p2) p1)"
+\           split (%x1 y1. split (%x2 y2. intrel^^{(x1+x2, y1+y2)}) p2) p1)"
 
   zdiff_def "Z1 - Z2 == Z1 + zminus(Z2)"
 
   zless_def "Z1<Z2 == znegative(Z1 - Z2)"
 
   zle_def   "Z1 <= (Z2::int) == ~(Z2 < Z1)"
 
   zmult_def
    "Z1 * Z2 == \
 \       Abs_Integ(UN p1:Rep_Integ(Z1). UN p2:Rep_Integ(Z2). split (%x1 y1.   \
-\           split (%x2 y2. intrel^^{<x1*x2 + y1*y2, x1*y2 + y1*x2>}) p2) p1)"
+\           split (%x2 y2. intrel^^{(x1*x2 + y1*y2, x1*y2 + y1*x2)}) p2) p1)"
 
   zdiv_def
    "Z1 zdiv Z2 ==   \
 \       Abs_Integ(UN p1:Rep_Integ(Z1). UN p2:Rep_Integ(Z2). split (%x1 y1.   \
-\           split (%x2 y2. intrel^^{<(x1-y1)div(x2-y2)+(y1-x1)div(y2-x2),   \
-\           (x1-y1)div(y2-x2)+(y1-x1)div(x2-y2)>}) p2) p1)"
+\           split (%x2 y2. intrel^^{((x1-y1)div(x2-y2)+(y1-x1)div(y2-x2),   \
+\           (x1-y1)div(y2-x2)+(y1-x1)div(x2-y2))}) p2) p1)"
 
   zmod_def
    "Z1 zmod Z2 ==   \
 \       Abs_Integ(UN p1:Rep_Integ(Z1).UN p2:Rep_Integ(Z2).split (%x1 y1.   \
-\           split (%x2 y2. intrel^^{<(x1-y1)mod((x2-y2)+(y2-x2)),   \
-\           (x1-y1)mod((x2-y2)+(x2-y2))>}) p2) p1)"
+\           split (%x2 y2. intrel^^{((x1-y1)mod((x2-y2)+(y2-x2)),   \
+\           (x1-y1)mod((x2-y2)+(x2-y2)))}) p2) p1)"
 
   zsuc_def     "zsuc(Z) == Z + $# Suc(0)"
 
   zpred_def    "zpred(Z) == Z - $# Suc(0)"
 end