--- a/src/HOL/Real/RealDef.thy Tue Jan 16 00:40:57 2001 +0100
+++ b/src/HOL/Real/RealDef.thy Tue Jan 16 12:20:52 2001 +0100
@@ -12,9 +12,10 @@
constdefs
realrel :: "((preal * preal) * (preal * preal)) set"
- "realrel == {p. ? x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}"
+ "realrel == {p. EX x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}"
-typedef real = "UNIV//realrel" (Equiv.quotient_def)
+typedef (REAL)
+ real = "UNIV//realrel" (Equiv.quotient_def)
instance
@@ -25,20 +26,23 @@
(*Overloaded constants denoting the Nat and Real subsets of enclosing
types such as hypreal and complex*)
- SNat, SReal :: "'a set"
+ Nats, Reals :: "'a set"
+ (*overloaded constant for injecting other types into "real"*)
+ real :: 'a => real
+
defs
real_zero_def
- "0 == Abs_real(realrel``{(preal_of_prat(prat_of_pnat 1p),
+ "0 == Abs_REAL(realrel``{(preal_of_prat(prat_of_pnat 1p),
preal_of_prat(prat_of_pnat 1p))})"
real_one_def
- "1r == Abs_real(realrel``{(preal_of_prat(prat_of_pnat 1p) +
+ "1r == Abs_REAL(realrel``{(preal_of_prat(prat_of_pnat 1p) +
preal_of_prat(prat_of_pnat 1p),preal_of_prat(prat_of_pnat 1p))})"
real_minus_def
- "- R == Abs_real(UN (x,y):Rep_real(R). realrel``{(y,x)})"
+ "- R == Abs_REAL(UN (x,y):Rep_REAL(R). realrel``{(y,x)})"
real_diff_def
"R - (S::real) == R + - S"
@@ -51,36 +55,39 @@
constdefs
+ (** these don't use the overloaded real because users don't see them **)
+
real_of_preal :: preal => real
"real_of_preal m ==
- Abs_real(realrel``{(m+preal_of_prat(prat_of_pnat 1p),
+ Abs_REAL(realrel``{(m+preal_of_prat(prat_of_pnat 1p),
preal_of_prat(prat_of_pnat 1p))})"
real_of_posnat :: nat => real
"real_of_posnat n == real_of_preal(preal_of_prat(prat_of_pnat(pnat_of_nat n)))"
- real_of_nat :: nat => real
- "real_of_nat n == real_of_posnat n + (-1r)"
defs
+ (*overloaded*)
+ real_of_nat_def "real n == real_of_posnat n + (-1r)"
+
real_add_def
- "P+Q == Abs_real(UN p1:Rep_real(P). UN p2:Rep_real(Q).
+ "P+Q == Abs_REAL(UN p1:Rep_REAL(P). UN p2:Rep_REAL(Q).
(%(x1,y1). (%(x2,y2). realrel``{(x1+x2, y1+y2)}) p2) p1)"
real_mult_def
- "P*Q == Abs_real(UN p1:Rep_real(P). UN p2:Rep_real(Q).
+ "P*Q == Abs_REAL(UN p1:Rep_REAL(P). UN p2:Rep_REAL(Q).
(%(x1,y1). (%(x2,y2). realrel``{(x1*x2+y1*y2,x1*y2+x2*y1)})
p2) p1)"
real_less_def
"P<Q == EX x1 y1 x2 y2. x1 + y2 < x2 + y1 &
- (x1,y1):Rep_real(P) & (x2,y2):Rep_real(Q)"
+ (x1,y1):Rep_REAL(P) & (x2,y2):Rep_REAL(Q)"
real_le_def
"P <= (Q::real) == ~(Q < P)"
syntax (symbols)
- SReal :: "'a set" ("\\<real>")
- SNat :: "'a set" ("\\<nat>")
+ Reals :: "'a set" ("\\<real>")
+ Nats :: "'a set" ("\\<nat>")
end