src/HOL/Real/RealDef.thy
 author wenzelm Fri, 09 Nov 2001 00:09:47 +0100 changeset 12114 a8e860c86252 parent 12018 ec054019c910 child 13487 1291c6375c29 permissions -rw-r--r--
eliminated old "symbols" syntax, use "xsymbols" instead;
```
(*  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"  (Equiv.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

real_of_nat_def   "real n == real_of_posnat n + (- 1)"

"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
```