Re: List of possible algebraic transformations

[Mike Griebling <grieblm@trt.allied.com>]

>   Juergen Zimmermann
> 
> Here the list:
> - integer types
> 
>   a + 0 = a
>   0 + a = a
>   a - 0 = a
>   0 - a = a

I think this should be 0 - a = -a

>   
>   --a = a

> - real types
> 
>   a + 0.0 = a
>   0.0 + a = a
>   a - 0.0 = a
>   0.0 - a = a

Same problem, should be 0.0 - a = -a
   
>   --a = a
>   
>   a * 1.0 = a
>   1.0 * a = a
>   a * (-1.0) = -a
>   (-1.0) * a = -a
>   
>   a * 0.0 = 0
>   0.0 * a = 0
>   
>   a / 1.0 = a
>   a / (-1.0) = -a

Would a / 1 = a where a is a REAL?  ie., are expression type conversions
part of the optimizations?
 
> - boolean types
> 
>   a & a = a
>   a OR a = a

You could add:

    a & ~a = ~a & a = FALSE
    a OR ~a = ~a OR a = TRUE

>   ~~a = a
>   
>   a & TRUE = a
>   TRUE & a = a
>   FALSE & a = FALSE
>   
>   a OR FALSE = a
>   FALSE OR a = a
>   TRUE OR a = TRUE

I assume you would also support the equivalent reductions
for when a is replaced by ~a? i.e., ~a & TRUE = ~a, etc.
 
> - set types
> 
>   a - a = {}
>   a + a = a
>   a * a = a
>   a / a = {}
>   
>   a - {} = a
>   a + {} = a
>   a * {} = a

Should be: a * {} = {}

>   a / {} = ?

I believe: a / {} = a
   
>   {} - a = ?

Should be: {} - a = {}

>   {} + a = a
>   {} * a = {}
>   {} / a = ?

Should be: {} / a = a

You may wish to add the following:

  Let F = ~{}

  F + a = a + F = F
  F - a = ~a
  a - F = {}
  F * a = a * F = a
  F / a = a / F = ~a

As well, consider the replacement of a with ~a.  
   
>   
> -string types
>   
>  str = "" -> str[0] = 0X
>  str # "" -> str[0] # 0X
>  str >= "" -> TRUE
>  str < ""  -> FALSE
>  "" <= str -> TRUE
>  "" > str  -> FALSE
>  
>  str ( < | > | # ) str -> FALSE
>  str ( <= | >= | = ) str -> TRUE

Michael G.