Version 1.0 of linear nat arithmetic.
(* Title : PRat.thy
Author : Jacques D. Fleuriot
Copyright : 1998 University of Cambridge
Description : The positive rationals
*)
PRat = PNat + Equiv +
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 = "{x::(pnat*pnat).True}/ratrel" (Equiv.quotient_def)
instance
prat :: {ord,plus,times}
constdefs
prat_pnat :: pnat => prat ("$#_" [80] 80)
"$# m == Abs_prat(ratrel^^{(m,Abs_pnat 1)})"
qinv :: prat => prat
"qinv(Q) == Abs_prat(UN p:Rep_prat(Q). split (%x y. ratrel^^{(y,x)}) p)"
defs
prat_add_def
"P + Q == Abs_prat(UN p1:Rep_prat(P). UN p2:Rep_prat(Q).
split(%x1 y1. split(%x2 y2. ratrel^^{(x1*y2 + x2*y1, y1*y2)}) p2) p1)"
prat_mult_def
"P * Q == Abs_prat(UN p1:Rep_prat(P). UN p2:Rep_prat(Q).
split(%x1 y1. split(%x2 y2. ratrel^^{(x1*x2, y1*y2)}) p2) p1)"
(*** Gleason p. 119 ***)
prat_less_def
"P < (Q::prat) == ? T. P + T = Q"
prat_le_def
"P <= (Q::prat) == ~(Q < P)"
end