This is Physics from today:

Hence, for instance, if you input the following expression, the computer just echoes your input:

There is however some structure behind (1.3) and, in most situations, it is convenient to have these structures
apparent, in part because they frequently provide hints on how to proceed ahead, but also because a more
compact expression is, roughly speaking, simpler to understand. To see this automaticsimplification in action,
turn it ON:

Recall this same expression (you could input it with the equation label (1.3) as well) 

What happened: this output, as everything else after you set  and with no
exceptions, is now further processed with simplify/size before being returned. And enjoy computing with frankly
shorter expressions all around! And no need anymore for "simplify(%, size)" every three or four input lines.

Another  example, typical in computer algebra where expressions become uncomfortably large and difficult to
read: convert the following input to 2D math input mode first, in order to compare what is being entered with the
automatically simplified output on the screen

You can turn automaticsimplification OFF the same way

Consider a generic variable, x. Nothing is known about it

Each variable has associated a number that depends on the session, and the computer (internally) uses this
number to refer to the variable.

When using the assume  command to place assumptions on a variable, this number, associated to it, changes,
for example:

Indeed, the variable x got redefined and renamed, it is not anymore the variable x referenced in (1.8).

The semantics may seem confusing but that is what happened, you enter x and the computer thinks x~, not x 
anymore.This means two things:

1) all the equations/expressions, entered before placing the assumptions on x using assume, involve a variable x 
that is different than the one that exists after placing the assumptions, and so these previous expressions cannot
be reused. They involve a different variable.

2) Also, because, after placing the assumptions using assume, x refers to a different object, programs that depend
on the x that existed before placing the assumptions will not recognize the new x redefined by assume .

For example, if x was part of a coordinate system and the spacetime metric depends on it, the new variable x
redefined within assume, being a different symbol, will not be recognized as part of the dependency of  This
posed constant obstacles to working with curved spacetimes that depend on parameters or on coordinates that
have a restricted range. These problems are resolved entirely with this new Library:-Assume, because it does not
redefine the variables. It only places assumptions on them, and in this sense it works like assuming , not assume .
As another example, all the Physics:-Vectors commands look for the cartesian, cylindrical or spherical coordinates
sets  in order to determine how to proceed, but these variables disappear if you use
assume to place assumptions on them. For that reason, only assuming  was fully compatible with Physics, not assume.

To undo assumptions placed using the assume command one reassigns the variable x to itself:

Check the numerical address: it is again equal to (1.8) 

·All these issues get resolved with the new Library:-Assume, that uses all the implementation of the existing 
assume command but with a different approach: the variables being assumed do not get redefined, and hence:
a) you can reuse expressions/equations entered before placing the assumptions, you can also undo the
assumptions and reuse results obtained with assumptions. This is the concept of an extended assuming. Also,
commands that depend on these assumed variables will all continue to work normally, before, during or after
placing the assumption, because the variables do not get redefined.

Example:

So this simplification attempt accomplishes nothing

Let's assume now that

The new command echoes the internal format representing the assumption placed.

a) The address is still the same as (1.8)

So the variable did not get redefined. The system however knows about the assumption - all the machinery of the
assume command is being used

Note that the renaming is to the variable itself - i.e. no renaming.

Hence, expressions entered before placing assumptions can be reused. For example, for (1.12), we now have

To clear the assumptions on x, you can use either of Library:-Assume(x=x) or Library:-Assume(clear = {x, ...}) in
the case of many variables being cleared in one go, or in the case of a single variable being cleared:

The implementation includes the additionally functionality, for that purpose add the keyword additionally 
anywhere in the calling sequence. For example:

In summary, the new Library:-Assume command implements the concept of an extended assuming, that can be
turned ON and OFF at will at any moment without changing the variables involved.