(*  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 constant: denotes the Real subset of enclosing types such as

     hypreal and complex*)

   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>")

end