#### 6.1.2.2 The for-as-arithmetic subclause

In the for-as-arithmetic subclause, the `for` or `as` construct iterates from the value supplied by form1 to the value supplied by form2 in increments or decrements denoted by form3. Each expression is evaluated only once and must evaluate to a number. The variable var is bound to the value of form1 in the first iteration and is stepped_1 by the value of form3 in each succeeding iteration, or by 1 if form3 is not provided. The following loop keywords serve as valid prepositions within this syntax. At least one of the prepositions must be used; and at most one from each line may be used in a single subclause.

from | downfrom | upfrom
to | downto | upto | below | above
by

The prepositional phrases in each subclause may appear in any order. For example, either “`from x by y`” or “`by y from x`” is permitted. However, because left-to-right order of evaluation is preserved, the effects will be different in the case of side effects.

Consider:

    (let ((x 1)) (loop for i from x by (incf x) to 10 collect i))
    ⇒  (1 3 5 7 9)
    (let ((x 1)) (loop for i by (incf x) from x to 10 collect i))
    ⇒  (2 4 6 8 10)

The descriptions of the prepositions follow:

from

  The loop keyword `from` specifies the value from which stepping_1 begins, as supplied by form1. Stepping_1 is incremental by default. If decremental stepping_1 is desired, the preposition `downto` or `above` must be used with form2. For incremental stepping_1, the default `from` value is 0.

downfrom, upfrom

  The loop keyword `downfrom` indicates that the variable var is decreased in decrements supplied by form3; the loop keyword `upfrom` indicates that var is increased in increments supplied by form3.

to

  The loop keyword `to` marks the end value for stepping_1 supplied in form2. Stepping_1 is incremental by default. If decremental stepping_1 is desired, the preposition `downfrom` must be used with form1, or else the preposition `downto` or `above` should be used instead of `to` with form2.

downto, upto

  The loop keyword `downto` specifies decremental stepping; the loop keyword `upto` specifies incremental stepping. In both cases, the amount of change on each step is specified by form3, and the loop terminates when the variable var passes the value of form2. Since there is no default for form1 in decremental stepping_1, a form1 value must be supplied (using `from` or `downfrom`) when `downto` is supplied.

below, above

  The loop keywords `below` and `above` are analogous to `upto` and `downto` respectively. These keywords stop iteration just before the value of the variable var reaches the value supplied by form2; the end value of form2 is not included. Since there is no default for form1 in decremental stepping_1, a form1 value must be supplied (using `from` or `downfrom`) when `above` is supplied.

by

  The loop keyword `by` marks the increment or decrement supplied by form3. The value of form3 can be any positive number. The default value is 1.

In an iteration control clause, the `for` or `as` construct causes termination when the supplied limit is reached. That is, iteration continues until the value var is stepped to the exclusive or inclusive limit supplied by form2. The range is exclusive if form3 increases or decreases var to the value of form2 without reaching that value; the loop keywords `below` and `above` provide exclusive limits. An inclusive limit allows var to attain the value of form2; `to`, `downto`, and `upto` provide inclusive limits.