(* Title : Real/RealDef.thy
ID : $Id$
Author : Jacques D. Fleuriot
Copyright : 1998 University of Cambridge
Description : The reals
*)
RealDef = PReal +
instance preal :: order (preal_le_refl,preal_le_trans,preal_le_anti_sym,
preal_less_le)
constdefs
realrel :: "((preal * preal) * (preal * preal)) set"
"realrel == {p. ? x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}"
typedef real = "UNIV//realrel" (Equiv.quotient_def)
instance
real :: {ord, zero, plus, times, minus, inverse}
consts
"1r" :: real ("1r")
(*Overloaded constants denoting the Nat and Real subsets of enclosing
types such as hypreal and complex*)
SNat, SReal :: "'a set"
defs
real_zero_def
"0 == Abs_real(realrel^^{(preal_of_prat(prat_of_pnat 1p),
preal_of_prat(prat_of_pnat 1p))})"
real_one_def
"1r == Abs_real(realrel^^{(preal_of_prat(prat_of_pnat 1p) +
preal_of_prat(prat_of_pnat 1p),preal_of_prat(prat_of_pnat 1p))})"
real_minus_def
"- R == Abs_real(UN (x,y):Rep_real(R). realrel^^{(y,x)})"
real_diff_def
"R - (S::real) == R + - S"
real_inverse_def
"inverse (R::real) == (SOME S. (R = 0 & S = 0) | S * R = 1r)"
real_divide_def
"R / (S::real) == R * inverse S"
constdefs
real_of_preal :: preal => real
"real_of_preal m ==
Abs_real(realrel^^{(m+preal_of_prat(prat_of_pnat 1p),
preal_of_prat(prat_of_pnat 1p))})"
real_of_posnat :: nat => real
"real_of_posnat n == real_of_preal(preal_of_prat(prat_of_pnat(pnat_of_nat n)))"
real_of_nat :: nat => real
"real_of_nat n == real_of_posnat n + (-1r)"
defs
real_add_def
"P+Q == Abs_real(UN p1:Rep_real(P). UN p2:Rep_real(Q).
(%(x1,y1). (%(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).
(%(x1,y1). (%(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)"
syntax (symbols)
SReal :: "'a set" ("\\<real>")
SNat :: "'a set" ("\\<nat>")
end