(* 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),@#($#1p))})"
real_minus_def
"- R == Abs_real(UN p:Rep_real(R). split (%x y. realrel^^{(y,x)}) p)"
real_diff_def "x - y == x + -(y::real)"
constdefs
real_preal :: preal => real ("%#_" [100] 100)
"%# m == Abs_real(realrel^^{(m+@#($#1p),@#($#1p))})"
rinv :: real => real
"rinv(R) == (@S. R ~= 0r & S*R = 1r)"
real_nat :: nat => real ("%%# _" [100] 100)
"%%# n == %#(@#($#(*# 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