wneuper/isa: src/HOL/Real/RealDef.thy@ec054019c910

     1 (*  Title       : Real/RealDef.thy

     2     ID          : $Id$

     3     Author      : Jacques D. Fleuriot

     4     Copyright   : 1998  University of Cambridge

     5     Description : The reals

     6 *)

     8 RealDef = PReal +

    10 instance preal :: order (preal_le_refl,preal_le_trans,preal_le_anti_sym,

    11                          preal_less_le)

    13 constdefs

    14   realrel   ::  "((preal * preal) * (preal * preal)) set"

    15   "realrel == {p. EX x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}"

    17 typedef (REAL)

    18   real = "UNIV//realrel"  (Equiv.quotient_def)

    21 instance

    22    real  :: {ord, zero, one, plus, times, minus, inverse}

    24 consts

    25    (*Overloaded constants denoting the Nat and Real subsets of enclosing

    26      types such as hypreal and complex*)

    27    Nats, Reals :: "'a set"

    29    (*overloaded constant for injecting other types into "real"*)

    30    real :: 'a => real

    33 defs

    35   real_zero_def

    36   "0 == Abs_REAL(realrel``{(preal_of_prat(prat_of_pnat 1),

    37 			    preal_of_prat(prat_of_pnat 1))})"

    39   real_one_def

    40   "1 == Abs_REAL(realrel``

    41                {(preal_of_prat(prat_of_pnat 1) + preal_of_prat(prat_of_pnat 1),

    42 		 preal_of_prat(prat_of_pnat 1))})"

    44   real_minus_def

    45   "- R ==  Abs_REAL(UN (x,y):Rep_REAL(R). realrel``{(y,x)})"

    47   real_diff_def

    48   "R - (S::real) == R + - S"

    50   real_inverse_def

    51   "inverse (R::real) == (SOME S. (R = 0 & S = 0) | S * R = 1)"

    53   real_divide_def

    54   "R / (S::real) == R * inverse S"

    56 constdefs

    58   (** these don't use the overloaded "real" function: users don't see them **)

    60   real_of_preal :: preal => real

    61   "real_of_preal m     ==

    62            Abs_REAL(realrel``{(m + preal_of_prat(prat_of_pnat 1),

    63                                preal_of_prat(prat_of_pnat 1))})"

    65   real_of_posnat :: nat => real

    66   "real_of_posnat n == real_of_preal(preal_of_prat(prat_of_pnat(pnat_of_nat n)))"

    69 defs

    71   (*overloaded*)

    72   real_of_nat_def   "real n == real_of_posnat n + (- 1)"

    74   real_add_def

    75   "P+Q == Abs_REAL(UN p1:Rep_REAL(P). UN p2:Rep_REAL(Q).

    76                    (%(x1,y1). (%(x2,y2). realrel``{(x1+x2, y1+y2)}) p2) p1)"

    78   real_mult_def

    79   "P*Q == Abs_REAL(UN p1:Rep_REAL(P). UN p2:Rep_REAL(Q).

    80                    (%(x1,y1). (%(x2,y2). realrel``{(x1*x2+y1*y2,x1*y2+x2*y1)})

    81 		   p2) p1)"

    83   real_less_def

    84   "P<Q == EX x1 y1 x2 y2. x1 + y2 < x2 + y1 &

    85                             (x1,y1):Rep_REAL(P) & (x2,y2):Rep_REAL(Q)"

    86   real_le_def

    87   "P <= (Q::real) == ~(Q < P)"

    89 syntax (symbols)

    90   Reals     :: "'a set"                   ("\\<real>")

    91   Nats      :: "'a set"                   ("\\<nat>")

    93 end

author	paulson
	Fri, 02 Nov 2001 17:55:24 +0100
changeset 12018	ec054019c910
parent 11713	883d559b0b8c
child 12114	a8e860c86252
permissions	-rw-r--r--