src/HOL/Real/RealDef.thy
author paulson
Thu Oct 01 18:18:01 1998 +0200 (1998-10-01)
changeset 5588 a3ab526bb891
child 5787 4e5c74b7cd9e
permissions -rw-r--r--
Revised version with Abelian group simprocs
     1 (*  Title       : Real/RealDef.thy
     2     Author      : Jacques D. Fleuriot
     3     Copyright   : 1998  University of Cambridge
     4     Description : The reals
     5 *) 
     6 
     7 RealDef = PReal +
     8 
     9 constdefs
    10   realrel   ::  "((preal * preal) * (preal * preal)) set"
    11   "realrel == {p. ? x1 y1 x2 y2. p = ((x1,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, minus}
    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),@#($#1p))})"
    28 
    29   real_minus_def
    30     "- R ==  Abs_real(UN p:Rep_real(R). split (%x y. realrel^^{(y,x)}) p)"
    31 
    32   real_diff_def "x - y == x + -(y::real)"
    33 
    34 constdefs
    35 
    36   real_preal :: preal => real              ("%#_" [80] 80)
    37   "%# m     == Abs_real(realrel^^{(m+@#($#1p),@#($#1p))})"
    38 
    39   rinv       :: real => real
    40   "rinv(R)   == (@S. R ~= 0r & S*R = 1r)"
    41 
    42   real_nat :: nat => real                  ("%%# _" [80] 80) 
    43   "%%# n      == %#(@#($#(*# n)))"
    44 
    45 defs
    46 
    47   real_add_def  
    48   "P + Q == Abs_real(UN p1:Rep_real(P). UN p2:Rep_real(Q).
    49                 split(%x1 y1. split(%x2 y2. realrel^^{(x1+x2, y1+y2)}) p2) p1)"
    50   
    51   real_mult_def  
    52   "P * Q == Abs_real(UN p1:Rep_real(P). UN p2:Rep_real(Q).
    53                 split(%x1 y1. split(%x2 y2. realrel^^{(x1*x2+y1*y2,x1*y2+x2*y1)}) p2) p1)"
    54 
    55   real_less_def
    56   "P < Q == EX x1 y1 x2 y2. x1 + y2 < x2 + y1 & 
    57                                    (x1,y1):Rep_real(P) &
    58                                    (x2,y2):Rep_real(Q)" 
    59   real_le_def
    60   "P <= (Q::real) == ~(Q < P)"
    61 
    62 end