(* 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. EX x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}"
typedef (REAL)
real = "UNIV//realrel" (quotient_def)
instance
real :: {ord, zero, one, plus, times, minus, inverse}
consts
(*Overloaded constants denoting the Nat and Real subsets of enclosing
types such as hypreal and complex*)
Nats, Reals :: "'a set"
(*overloaded constant for injecting other types into "real"*)
real :: 'a => real
defs
real_zero_def
"0 == Abs_REAL(realrel``{(preal_of_prat(prat_of_pnat 1),
preal_of_prat(prat_of_pnat 1))})"
real_one_def
"1 == Abs_REAL(realrel``
{(preal_of_prat(prat_of_pnat 1) + preal_of_prat(prat_of_pnat 1),
preal_of_prat(prat_of_pnat 1))})"
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 = 1)"
real_divide_def
"R / (S::real) == R * inverse S"
constdefs
(** these don't use the overloaded "real" function: users don't see them **)
real_of_preal :: preal => real
"real_of_preal m ==
Abs_REAL(realrel``{(m + preal_of_prat(prat_of_pnat 1),
preal_of_prat(prat_of_pnat 1))})"
real_of_posnat :: nat => real
"real_of_posnat n == real_of_preal(preal_of_prat(prat_of_pnat(pnat_of_nat n)))"
defs
(*overloaded*)
real_of_nat_def "real n == real_of_posnat n + (- 1)"
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 (xsymbols)
Reals :: "'a set" ("\\<real>")
Nats :: "'a set" ("\\<nat>")
end