src/HOL/Integ/IntDef.thy

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Integ/IntDef.thy	Fri Sep 18 16:04:00 1998 +0200
@@ -0,0 +1,53 @@
+(*  Title:      IntDef.thy
+    ID:         $Id$
+    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1996  University of Cambridge
+
+The integers as equivalence classes over nat*nat.
+*)
+
+IntDef = Equiv + Arith +
+constdefs
+  intrel      :: "((nat * nat) * (nat * nat)) set"
+  "intrel == {p. ? x1 y1 x2 y2. p=((x1::nat,y1),(x2,y2)) & x1+y2 = x2+y1}"
+
+typedef (Integ)
+  int = "{x::(nat*nat).True}/intrel"            (Equiv.quotient_def)
+
+instance
+  int :: {ord, plus, times, minus}
+
+defs
+  zminus_def
+    "- Z == Abs_Integ(UN p:Rep_Integ(Z). split (%x y. intrel^^{(y,x)}) p)"
+
+constdefs
+
+  znat        :: nat => int                                  ("$# _" [80] 80)
+  "$# m == Abs_Integ(intrel ^^ {(m,0)})"
+
+  znegative   :: int => bool
+  "znegative(Z) == EX x y. x<y & (x,y::nat):Rep_Integ(Z)"
+
+  (*For simplifying equalities*)
+  iszero :: int => bool
+  "iszero z == z = $# 0"
+  
+defs
+  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)"
+
+  zdiff_def "Z1 - Z2 == Z1 + -(Z2::int)"
+
+  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)"
+
+end