(* Title : PRat.thy
ID : $Id$
Author : Jacques D. Fleuriot
Copyright : 1998 University of Cambridge
Description : The positive rationals
*)
PRat = PNat +
constdefs
ratrel :: "((pnat * pnat) * (pnat * pnat)) set"
"ratrel == {p. ? x1 y1 x2 y2. p=((x1::pnat,y1),(x2,y2)) & x1*y2 = x2*y1}"
typedef prat = "UNIV/ratrel" (Equiv.quotient_def)
instance
prat :: {ord,plus,times}
constdefs
prat_of_pnat :: pnat => prat
"prat_of_pnat m == Abs_prat(ratrel^^{(m,Abs_pnat 1)})"
qinv :: prat => prat
"qinv(Q) == Abs_prat(UN (x,y):Rep_prat(Q). ratrel^^{(y,x)})"
defs
prat_add_def
"P + Q == Abs_prat(UN (x1,y1):Rep_prat(P). UN (x2,y2):Rep_prat(Q).
ratrel^^{(x1*y2 + x2*y1, y1*y2)})"
prat_mult_def
"P * Q == Abs_prat(UN (x1,y1):Rep_prat(P). UN (x2,y2):Rep_prat(Q).
ratrel^^{(x1*x2, y1*y2)})"
(*** Gleason p. 119 ***)
prat_less_def
"P < (Q::prat) == ? T. P + T = Q"
prat_le_def
"P <= (Q::prat) == ~(Q < P)"
end