src/HOL/Real/RealDef.thy
author bauerg
Wed Dec 06 12:34:12 2000 +0100 (2000-12-06)
changeset 10606 e3229a37d53f
parent 9391 a6ab3a442da6
child 10648 a8c647cfa31f
permissions -rw-r--r--
converted rinv to inverse;
     1 (*  Title       : Real/RealDef.thy
     2     ID          : $Id$
     3     Author      : Jacques D. Fleuriot
     4     Copyright   : 1998  University of Cambridge
     5     Description : The reals
     6 *) 
     7 
     8 RealDef = PReal +
     9 
    10 constdefs
    11   realrel   ::  "((preal * preal) * (preal * preal)) set"
    12   "realrel == {p. ? x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}" 
    13 
    14 typedef real = "UNIV//realrel"  (Equiv.quotient_def)
    15 
    16 
    17 instance
    18    real  :: {ord, zero, plus, times, minus, inverse}
    19 
    20 consts 
    21 
    22   "1r"       :: real               ("1r")  
    23 
    24 defs
    25 
    26   real_zero_def  
    27   "0 == Abs_real(realrel^^{(preal_of_prat(prat_of_pnat 1p),
    28                                 preal_of_prat(prat_of_pnat 1p))})"
    29   real_one_def   
    30   "1r == Abs_real(realrel^^{(preal_of_prat(prat_of_pnat 1p) + 
    31             preal_of_prat(prat_of_pnat 1p),preal_of_prat(prat_of_pnat 1p))})"
    32 
    33   real_minus_def
    34   "- R ==  Abs_real(UN (x,y):Rep_real(R). realrel^^{(y,x)})"
    35 
    36   real_diff_def
    37   "R - (S::real) == R + - S"
    38 
    39   real_inverse_def
    40   "inverse (R::real) == (@S. R ~= 0 & S*R = 1r)"
    41 
    42   real_divide_def
    43   "R / (S::real) == R * inverse S"
    44   
    45 constdefs
    46 
    47   real_of_preal :: preal => real            
    48   "real_of_preal m     ==
    49            Abs_real(realrel^^{(m+preal_of_prat(prat_of_pnat 1p),
    50                                preal_of_prat(prat_of_pnat 1p))})"
    51 
    52   real_of_posnat :: nat => real             
    53   "real_of_posnat n == real_of_preal(preal_of_prat(prat_of_pnat(pnat_of_nat n)))"
    54 
    55   real_of_nat :: nat => real          
    56   "real_of_nat n    == real_of_posnat n + (-1r)"
    57 
    58 defs
    59 
    60   real_add_def  
    61   "P + Q == Abs_real(UN p1:Rep_real(P). UN p2:Rep_real(Q).
    62                 (%(x1,y1). (%(x2,y2). realrel^^{(x1+x2, y1+y2)}) p2) p1)"
    63   
    64   real_mult_def  
    65   "P * Q == Abs_real(UN p1:Rep_real(P). UN p2:Rep_real(Q).
    66                 (%(x1,y1). (%(x2,y2). realrel^^{(x1*x2+y1*y2,x1*y2+x2*y1)}) p2) p1)"
    67 
    68   real_less_def
    69   "P < Q == EX x1 y1 x2 y2. x1 + y2 < x2 + y1 & 
    70                                    (x1,y1):Rep_real(P) &
    71                                    (x2,y2):Rep_real(Q)" 
    72   real_le_def
    73   "P <= (Q::real) == ~(Q < P)"
    74 
    75 end