src/HOL/Real/RealDef.thy
 author paulson Fri, 23 Jul 1999 17:29:12 +0200 changeset 7077 60b098bb8b8a parent 5787 4e5c74b7cd9e child 7127 48e235179ffb permissions -rw-r--r--
heavily revised by Jacques: coercions have alphabetic names; exponentiation is available, etc.
```
(*  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^^{(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 "x - y == x + -(y::real)"

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))})"

rinv       :: real => real
"rinv(R)   == (@S. R ~= 0r & S*R = 1r)"

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

"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
```