UNION_CONV

pred_setLib.UNION_CONV : conv -> conv

Reduce {t1;...;tn} UNION s to t1 INSERT (... (tn INSERT s)).

The function UNION_CONV is a parameterized conversion for reducing sets of the form {t1;...;tn} UNION s, where {t1;...;tn} and s are sets of type ty->bool. The first argument to UNION_CONV is expected to be a conversion that decides equality between values of the base type ty. Given an equation e1 = e2, where e1 and e2 are terms of type ty, this conversion should return the theorem |- (e1 = e2) = T or the theorem |- (e1 = e2) = F, as appropriate.

Given such a conversion, the function UNION_CONV returns a conversion that maps a term of the form {t1;...;tn} UNION s to the theorem

   |- {t1;...;tn} UNION s = ti INSERT ... (tj INSERT s)

where {ti;...;tj} is the set of all terms t that occur as elements of {t1;...;tn} for which the conversion IN_CONV conv fails to prove that |- (t IN s) = T (that is, either by proving |- (t IN s) = F instead, or by failing outright).

Example

In the following example, REDUCE_CONV is supplied as a parameter to UNION_CONV and used to test for membership of each element of the first finite set {1;2;3} of the union in the second finite set {SUC 0;3;4}.

   - UNION_CONV REDUCE_CONV (Term`{1;2;3} UNION {SUC 0;3;4}`);
   > val it = |- {1; 2; 3} UNION {SUC 0; 3; 4} = {2; SUC 0; 3; 4} : thm

The result is {2;SUC 0;3;4}, rather than {1;2;SUC 0;3;4}, because UNION_CONV is able by means of a call to

   - IN_CONV REDUCE_CONV (Term`1 IN {SUC 0;3;4}`);

to prove that 1 is already an element of the set {SUC 0;3;4}.

The conversion supplied to UNION_CONV need not actually prove equality of elements, if simplification of the resulting set is not desired. For example:

   - UNION_CONV NO_CONV ``{1;2;3} UNION {SUC 0;3;4}``;
   > val it = |- {1;2;3} UNION {SUC 0;3;4} = {1;2;SUC 0;3;4} : thm

In this case, the resulting set is just left unsimplified. Moreover, the second set argument to UNION need not be a finite set:

   - UNION_CONV NO_CONV ``{1;2;3} UNION s``;
   > val it = |- {1;2;3} UNION s = 1 INSERT (2 INSERT (3 INSERT s)) : thm

And, of course, in this case the conversion argument to UNION_CONV is irrelevant.

Failure

UNION_CONV conv fails if applied to a term not of the form {t1;...;tn} UNION s.

See also

pred_setLib.IN_CONV, numLib.REDUCE_CONV