author | nipkow |
Fri, 05 Jan 2001 18:48:18 +0100 | |
changeset 10797 | 028d22926a41 |
parent 10778 | 2c6605049646 |
child 10834 | a7897aebbffc |
permissions | -rw-r--r-- |
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(* Title : Real/RealDef.thy |
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ID : $Id$ |
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Author : Jacques D. Fleuriot |
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Copyright : 1998 University of Cambridge |
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Description : The reals |
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*) |
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RealDef = PReal + |
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instance preal :: order (preal_le_refl,preal_le_trans,preal_le_anti_sym, |
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preal_less_le) |
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constdefs |
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realrel :: "((preal * preal) * (preal * preal)) set" |
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"realrel == {p. ? x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}" |
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typedef real = "UNIV//realrel" (Equiv.quotient_def) |
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instance |
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real :: {ord, zero, plus, times, minus, inverse} |
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consts |
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"1r" :: real ("1r") |
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(*Overloaded constants denoting the Nat and Real subsets of enclosing |
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types such as hypreal and complex*) |
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SNat, SReal :: "'a set" |
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defs |
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real_zero_def |
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"0 == Abs_real(realrel```{(preal_of_prat(prat_of_pnat 1p), |
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preal_of_prat(prat_of_pnat 1p))})" |
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real_one_def |
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"1r == Abs_real(realrel```{(preal_of_prat(prat_of_pnat 1p) + |
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preal_of_prat(prat_of_pnat 1p),preal_of_prat(prat_of_pnat 1p))})" |
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real_minus_def |
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"- R == Abs_real(UN (x,y):Rep_real(R). realrel```{(y,x)})" |
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real_diff_def |
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"R - (S::real) == R + - S" |
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real_inverse_def |
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"inverse (R::real) == (SOME S. (R = 0 & S = 0) | S * R = 1r)" |
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real_divide_def |
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"R / (S::real) == R * inverse S" |
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constdefs |
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real_of_preal :: preal => real |
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"real_of_preal m == |
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Abs_real(realrel```{(m+preal_of_prat(prat_of_pnat 1p), |
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preal_of_prat(prat_of_pnat 1p))})" |
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real_of_posnat :: nat => real |
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"real_of_posnat n == real_of_preal(preal_of_prat(prat_of_pnat(pnat_of_nat n)))" |
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real_of_nat :: nat => real |
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"real_of_nat n == real_of_posnat n + (-1r)" |
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defs |
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real_add_def |
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"P+Q == Abs_real(UN p1:Rep_real(P). UN p2:Rep_real(Q). |
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(%(x1,y1). (%(x2,y2). realrel```{(x1+x2, y1+y2)}) p2) p1)" |
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real_mult_def |
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"P*Q == Abs_real(UN p1:Rep_real(P). UN p2:Rep_real(Q). |
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(%(x1,y1). (%(x2,y2). realrel```{(x1*x2+y1*y2,x1*y2+x2*y1)}) |
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p2) p1)" |
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real_less_def |
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"P<Q == EX x1 y1 x2 y2. x1 + y2 < x2 + y1 & |
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(x1,y1):Rep_real(P) & (x2,y2):Rep_real(Q)" |
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real_le_def |
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"P <= (Q::real) == ~(Q < P)" |
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syntax (symbols) |
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SReal :: "'a set" ("\\<real>") |
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SNat :: "'a set" ("\\<nat>") |
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end |