src/HOL/Real/Real.thy
author paulson
Thu Jun 25 13:57:34 1998 +0200 (1998-06-25)
changeset 5078 7b5ea59c0275
child 5588 a3ab526bb891
permissions -rw-r--r--
Installation of target HOL-Real
     1 (*  Title       : Real.thy
     2     Author      : Jacques D. Fleuriot
     3     Copyright   : 1998  University of Cambridge
     4     Description : The reals
     5 *) 
     6 
     7 Real = PReal +
     8 
     9 constdefs
    10     realrel   ::  "((preal * preal) * (preal * preal)) set"
    11     "realrel  ==  {p. ? x1 y1 x2 y2. p=((x1::preal,y1),(x2,y2)) & x1+y2 = x2+y1}" 
    12 
    13 typedef real = "{x::(preal*preal).True}/realrel"          (Equiv.quotient_def)
    14 
    15 
    16 instance
    17    real  :: {ord,plus,times}
    18 
    19 consts 
    20 
    21   "0r"       :: real               ("0r")   
    22   "1r"       :: real               ("1r")  
    23 
    24 defs
    25 
    26   real_zero_def      "0r == Abs_real(realrel^^{(@#($#1p),@#($#1p))})"
    27   real_one_def       "1r == Abs_real(realrel^^{(@#($#1p) + @#($#1p),@#($#1p))})"
    28 
    29 constdefs
    30 
    31   real_preal :: preal => real              ("%#_" [80] 80)
    32   "%# m     == Abs_real(realrel^^{(m+@#($#1p),@#($#1p))})"
    33 
    34   real_minus :: real => real               ("%~ _" [80] 80) 
    35   "%~ R     ==  Abs_real(UN p:Rep_real(R). split (%x y. realrel^^{(y,x)}) p)"
    36 
    37   rinv       :: real => real
    38   "rinv(R)   == (@S. R ~= 0r & S*R = 1r)"
    39 
    40   real_nat :: nat => real                  ("%%# _" [80] 80) 
    41   "%%# n      == %#(@#($#(*# n)))"
    42 
    43 defs
    44 
    45   real_add_def  
    46   "P + Q == Abs_real(UN p1:Rep_real(P). UN p2:Rep_real(Q).
    47                 split(%x1 y1. split(%x2 y2. realrel^^{(x1+x2, y1+y2)}) p2) p1)"
    48   
    49   real_mult_def  
    50   "P * Q == Abs_real(UN p1:Rep_real(P). UN p2:Rep_real(Q).
    51                 split(%x1 y1. split(%x2 y2. realrel^^{(x1*x2+y1*y2,x1*y2+x2*y1)}) p2) p1)"
    52 
    53   real_less_def
    54   "P < (Q::real) == EX x1 y1 x2 y2. x1 + y2 < x2 + y1 & 
    55                                    (x1,y1::preal):Rep_real(P) &
    56                                    (x2,y2):Rep_real(Q)" 
    57 
    58   real_le_def
    59   "P <= (Q::real) == ~(Q < P)"
    60 
    61 end