`INSERT_CONV : conv -> conv`
STRUCTURE
SYNOPSIS
Reduce t INSERT {t1;...;t;...;tn} to {t1;...;t;...;tn}.
LIBRARY
pred_set
DESCRIPTION
The function INSERT_CONV is a parameterized conversion for reducing finite sets of the form t INSERT {t1;...;tn}, where {t1;...;tn} is a set of type ty->bool and t is equal to some element ti of this set. The first argument to INSERT_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 INSERT_CONV returns a conversion that maps a term of the form t INSERT {t1;...;tn} to the theorem

```   |- t INSERT {t1;...;tn} = {t1;...;tn}
```
if t is alpha-equivalent to any ti in the set {t1,...,tn}, or if the supplied conversion proves |- (t = ti) = T for any ti.
EXAMPLE
In the following example, the conversion REDUCE_CONV is supplied as a parameter and used to test equality of the inserted value 2 with the remaining elements of the set.
```   - INSERT_CONV REDUCE_CONV ``2 INSERT {1;SUC 1;3}``;
> val it = |- {2; 1; SUC 1; 3} = {1; SUC 1; 3} : thm
```
In this example, the supplied conversion REDUCE_CONV is able to prove that 2 is equal to SUC 1 and the set is therefore reduced. Note that 2 INSERT {1; SUC 1; 3} is just {2; 1; SUC 1; 3}.

A call to INSERT_CONV fails when the value being inserted is provably not equal to any of the remaining elements:

```   - INSERT_CONV REDUCE_CONV ``1 INSERT {2;3}``;
! Uncaught exception:
! HOL_ERR
```
But this failure can, if desired, be caught using TRY_CONV.

The behaviour of the supplied conversion is irrelevant when the inserted value is alpha-equivalent to one of the remaining elements:

```   - INSERT_CONV NO_CONV ``y INSERT {x;y;z}``;
> val it = |- {y; x; y; z} = {x; y; z} : thm
```
The conversion NO_CONV always fails, but INSERT_CONV is nontheless able in this case to prove the required result.

Note that DEPTH_CONV(INSERT_CONV conv) can be used to remove duplicate elements from a finite set, but the following conversion is faster:

```   - fun SETIFY_CONV conv tm =
(SUB_CONV (SETIFY_CONV conv)
THENC
TRY_CONV (INSERT_CONV conv)) tm;
> val SETIFY_CONV = fn : conv -> conv

- SETIFY_CONV REDUCE_CONV ``{1;2;1;3;2;4;3;5;6}``;
> val it = |- {1; 2; 1; 3; 2; 4; 3; 5; 6} = {1; 2; 4; 3; 5; 6} : thm
```
FAILURE
INSERT_CONV conv fails if applied to a term not of the form t INSERT {t1;...;tn}. A call INSERT_CONV conv ``t INSERT {t1;...;tn} fails unless t is alpha-equivalent to some ti, or conv ``t = ti`` returns |- (t = ti) = T for some ti.
SEEALSO