src/HOL/Real/RealDef.thy
changeset 5588 a3ab526bb891
child 5787 4e5c74b7cd9e
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Real/RealDef.thy	Thu Oct 01 18:18:01 1998 +0200
@@ -0,0 +1,62 @@
+(*  Title       : Real/RealDef.thy
+    Author      : Jacques D. Fleuriot
+    Copyright   : 1998  University of Cambridge
+    Description : The reals
+*) 
+
+RealDef = PReal +
+
+constdefs
+  realrel   ::  "((preal * preal) * (preal * preal)) set"
+  "realrel == {p. ? x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}" 
+
+typedef real = "{x::(preal*preal).True}/realrel"          (Equiv.quotient_def)
+
+
+instance
+   real  :: {ord, plus, times, minus}
+
+consts 
+
+  "0r"       :: real               ("0r")   
+  "1r"       :: real               ("1r")  
+
+defs
+
+  real_zero_def  "0r == Abs_real(realrel^^{(@#($#1p),@#($#1p))})"
+  real_one_def   "1r == Abs_real(realrel^^{(@#($#1p) + @#($#1p),@#($#1p))})"
+
+  real_minus_def
+    "- R ==  Abs_real(UN p:Rep_real(R). split (%x y. realrel^^{(y,x)}) p)"
+
+  real_diff_def "x - y == x + -(y::real)"
+
+constdefs
+
+  real_preal :: preal => real              ("%#_" [80] 80)
+  "%# m     == Abs_real(realrel^^{(m+@#($#1p),@#($#1p))})"
+
+  rinv       :: real => real
+  "rinv(R)   == (@S. R ~= 0r & S*R = 1r)"
+
+  real_nat :: nat => real                  ("%%# _" [80] 80) 
+  "%%# n      == %#(@#($#(*# n)))"
+
+defs
+
+  real_add_def  
+  "P + Q == Abs_real(UN p1:Rep_real(P). UN p2:Rep_real(Q).
+                split(%x1 y1. split(%x2 y2. realrel^^{(x1+x2, y1+y2)}) p2) p1)"
+  
+  real_mult_def  
+  "P * Q == Abs_real(UN p1:Rep_real(P). UN p2:Rep_real(Q).
+                split(%x1 y1. split(%x2 y2. realrel^^{(x1*x2+y1*y2,x1*y2+x2*y1)}) p2) p1)"
+
+  real_less_def
+  "P < Q == EX x1 y1 x2 y2. x1 + y2 < x2 + y1 & 
+                                   (x1,y1):Rep_real(P) &
+                                   (x2,y2):Rep_real(Q)" 
+  real_le_def
+  "P <= (Q::real) == ~(Q < P)"
+
+end