| author | nipkow |
| Tue, 09 Jan 2001 15:32:27 +0100 | |
| changeset 10834 | a7897aebbffc |
| parent 10797 | 028d22926a41 |
| child 13487 | 1291c6375c29 |
| permissions | -rw-r--r-- |
| 5078 | 1 |
(* Title : PRat.thy |
| 7219 | 2 |
ID : $Id$ |
| 5078 | 3 |
Author : Jacques D. Fleuriot |
4 |
Copyright : 1998 University of Cambridge |
|
5 |
Description : The positive rationals |
|
6 |
*) |
|
7 |
||
| 8856 | 8 |
PRat = PNat + |
| 5078 | 9 |
|
10 |
constdefs |
|
11 |
ratrel :: "((pnat * pnat) * (pnat * pnat)) set" |
|
12 |
"ratrel == {p. ? x1 y1 x2 y2. p=((x1::pnat,y1),(x2,y2)) & x1*y2 = x2*y1}"
|
|
13 |
||
| 9391 | 14 |
typedef prat = "UNIV//ratrel" (Equiv.quotient_def) |
| 5078 | 15 |
|
16 |
instance |
|
17 |
prat :: {ord,plus,times}
|
|
18 |
||
19 |
||
20 |
constdefs |
|
21 |
||
|
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
5078
diff
changeset
|
22 |
prat_of_pnat :: pnat => prat |
| 10834 | 23 |
"prat_of_pnat m == Abs_prat(ratrel``{(m,Abs_pnat 1)})"
|
| 5078 | 24 |
|
25 |
qinv :: prat => prat |
|
| 10834 | 26 |
"qinv(Q) == Abs_prat(UN (x,y):Rep_prat(Q). ratrel``{(y,x)})"
|
| 5078 | 27 |
|
28 |
defs |
|
29 |
||
30 |
prat_add_def |
|
|
7376
46f92a120af9
tidied, allowing pattern-matching in defs of prat_add and prat_mult
paulson
parents:
7219
diff
changeset
|
31 |
"P + Q == Abs_prat(UN (x1,y1):Rep_prat(P). UN (x2,y2):Rep_prat(Q). |
| 10834 | 32 |
ratrel``{(x1*y2 + x2*y1, y1*y2)})"
|
| 5078 | 33 |
|
34 |
prat_mult_def |
|
|
7376
46f92a120af9
tidied, allowing pattern-matching in defs of prat_add and prat_mult
paulson
parents:
7219
diff
changeset
|
35 |
"P * Q == Abs_prat(UN (x1,y1):Rep_prat(P). UN (x2,y2):Rep_prat(Q). |
| 10834 | 36 |
ratrel``{(x1*x2, y1*y2)})"
|
| 5078 | 37 |
|
38 |
(*** Gleason p. 119 ***) |
|
39 |
prat_less_def |
|
| 9391 | 40 |
"P < (Q::prat) == EX T. P + T = Q" |
| 5078 | 41 |
|
42 |
prat_le_def |
|
43 |
"P <= (Q::prat) == ~(Q < P)" |
|
44 |
||
45 |
end |
|
46 |
||
47 |
||
48 |
||
49 |
||
50 |