Go forward to Composing Patterns in Rewrite Rules.
Go backward to Algebraic Properties of Rewrite Rules.
Go up to Rewrite Rules.
Other Features of Rewrite Rules
-------------------------------
Certain "function names" serve as markers in rewrite rules. Here is a
complete list of these markers. First are listed the markers that
work inside a pattern; then come the markers that work in the
righthand side of a rule.
One kind of marker, `import(x)', takes the place of a whole
rule. Here `x' is the name of a variable containing another
rule set; those rules are "spliced into" the rule set that
imports them. For example, if `[f(a+b) := f(a) + f(b),
f(a b) := a f(b) :: real(a)]' is stored in variable `linearF',
then the rule set `[f(0) := 0, import(linearF)]' will apply
all three rules. It is possible to modify the imported rules
slightly: `import(x, v1, x1, v2, x2, ...)' imports
the rule set `x' with all occurrences of `v1', as either
a variable name or a function name, replaced with `x1' and
so on. (If `v1' is used as a function name, then `x1'
must be either a function name itself or a `< >' nameless
function; See Specifying Operators.) For example, `[g(0) := 0,
import(linearF, f, g)]' applies the linearity rules to the function
`g' instead of `f'. Imports can be nested, but the
import-with-renaming feature may fail to rename sub-imports properly.
The special functions allowed in patterns are:
`quote(x)'
This pattern matches exactly `x'; variable names in `x' are not
interpreted as meta-variables. The only flexibility is that
numbers are compared for numeric equality, so that the pattern
`f(quote(12))' will match both `f(12)' and `f(12.0)'. (Numbers
are always treated this way by the rewrite mechanism: The rule
`f(x,x) := g(x)' will match `f(12, 12.0)'. The rewrite may
produce either `g(12)' or `g(12.0)' as a result in this case.)
`plain(x)'
Here `x' must be a function call `f(x1,x2,...)'. This
pattern matches a call to function `f' with the specified
argument patterns. No special knowledge of the properties of the
function `f' is used in this case; `+' is not commutative or
associative. Unlike `quote', the arguments `x1,x2,...'
are treated as patterns. If you wish them to be treated "plainly"
as well, you must enclose them with more `plain' markers:
`plain(plain(-a) + plain(b c))'.
`opt(x,def)'
Here `x' must be a variable name. This must appear as an
argument to a function or an element of a vector; it specifies
that the argument or element is optional. As an argument to `+',
`-', `*', `&&', or `||', or as the second argument to `/' or `^',
the value DEF may be omitted. The pattern `x + opt(y)' matches a
sum by binding one summand to `x' and the other to `y', and it
matches anything else by binding the whole expression to `x' and
zero to `y'. The other operators above work similarly.
For general miscellanous functions, the default value `def' must
be specified. Optional arguments are dropped starting with the
rightmost one during matching. For example, the pattern
`f(opt(a,0), b, opt(c,b))' will match `f(b)', `f(a,b)', or
`f(a,b,c)'. Default values of zero and `b' are supplied in this
example for the omitted arguments. Note that the literal
variable `b' will be the default in the latter case, *not* the
value that matched the meta-variable `b'. In other words, the
default DEF is effectively quoted.
`condition(x,c)'
This matches the pattern `x', with the attached condition `c'.
It is the same as `x :: c'.
`pand(x,y)'
This matches anything that matches both pattern `x' and
pattern `y'. It is the same as `x &&& y'.
See Composing Patterns in Rewrite Rules.
`por(x,y)'
This matches anything that matches either pattern `x' or
pattern `y'. It is the same as `x ||| y'.
`pnot(x)'
This matches anything that does not match pattern `x'. It is the
same as `!!! x'.
`cons(h,t)'
This matches any vector of one or more elements. The first
element is matched to `h'; a vector of the remaining elements is
matched to `t'. Note that vectors of fixed length can also be
matched as actual vectors: The rule `cons(a,cons(b,[])) :=
cons(a+b,[])' is equivalent to the rule `[a,b] := [a+b]'.
`rcons(t,h)'
This is like `cons', except that the *last* element is matched to
`h', with the remaining elements matched to `t'.
`apply(f,args)'
This matches any function call. The name of the function, in the
form of a variable, is matched to `f'. The arguments of the
function, as a vector of zero or more objects, are matched to
`args'. Constants, variables, and vectors do *not* match an
`apply' pattern. For example, `apply(f,x)' matches any function
call, `apply(quote(f),x)' matches any call to the function `f',
`apply(f,[a,b])' matches any function call with exactly two
arguments, and `apply(quote(f), cons(a,cons(b,x)))' matches any
call to the function `f' with two or more arguments. Another way
to implement the latter, if the rest of the rule does not need to
refer to the first two arguments of `f' by name, would be
`apply(quote(f), x :: vlen(x) >= 2)'. Here's a more interesting
sample use of `apply':
apply(f,[x+n]) := n + apply(f,[x])
:: in(f, [floor,ceil,round,trunc]) :: integer(n)
Note, however, that this will be slower to match than a rule set
with four separate rules. The reason is that Calc sorts the
rules of a rule set according to top-level function name; if the
top-level function is `apply', Calc must try the rule for every
single formula and sub-formula. If the top-level function in the
pattern is, say, `floor', then Calc invokes the rule only for
sub-formulas which are calls to `floor'.
Formulas normally written with operators like `+' are still
considered function calls: `apply(f,x)' matches `a+b' with `f =
add', `x = [a,b]'.
You must use `apply' for meta-variables with function names
on both sides of a rewrite rule: `apply(f, [x]) := f(x+1)'
is *not* correct, because it rewrites `spam(6)' into
`f(7)'. The righthand side should be `apply(f, [x+1])'.
Also note that you will have to use no-simplify (`m O')
mode when entering this rule so that the `apply' isn't
evaluated immediately to get the new rule `f(x) := f(x+1)'.
Or, use `s e' to enter the rule without going through the stack,
or enter the rule as `apply(f, [x]) := apply(f, [x+1]) :: 1'.
See Conditional Rewrite Rules.
`select(x)'
This is used for applying rules to formulas with selections;
See Selections with Rewrite Rules.
Special functions for the righthand sides of rules are:
`quote(x)'
The notation `quote(x)' is changed to `x' when the righthand side
is used. As far as the rewrite rule is concerned, `quote' is
invisible. However, `quote' has the special property in Calc
that its argument is not evaluated. Thus, while it will not work
to put the rule `t(a) := typeof(a)' on the stack because
`typeof(a)' is evaluated immediately to produce `t(a) := 100',
you can use `quote' to protect the righthand side: `t(a) :=
quote(typeof(a))'. (See Conditional Rewrite Rules, for
another trick for protecting rules from evaluation.)
`plain(x)'
Special properties of and simplifications for the function call
`x' are not used. One interesting case where `plain' is useful
is the rule, `q(x) := quote(x)', trying to expand a shorthand
notation for the `quote' function. This rule will not work as
shown; instead of replacing `q(foo)' with `quote(foo)', it will
replace it with `foo'! The correct rule would be `q(x) :=
plain(quote(x))'.
`cons(h,t)'
Where `t' is a vector, this is converted into an expanded vector
during rewrite processing. Note that `cons' is a regular Calc
function which normally does this anyway; the only way `cons' is
treated specially by rewrites is that `cons' on the righthand
side of a rule will be evaluated even if default simplifications
have been turned off.
`rcons(t,h)'
Analogous to `cons' except putting `h' at the *end* of the vector
`t'.
`apply(f,args)'
Where `f' is a variable and ARGS is a vector, this is converted
to a function call. Once again, note that `apply' is also a
regular Calc function.
`eval(x)'
The formula `x' is handled in the usual way, then the default
simplifications are applied to it even if they have been turned
off normally. This allows you to treat any function similarly to
the way `cons' and `apply' are always treated. However, there is
a slight difference: `cons(2+3, [])' with default simplifications
off will be converted to `[2+3]', whereas `eval(cons(2+3, []))'
will be converted to `[5]'.
`evalsimp(x)'
The formula `x' has meta-variables substituted in the usual way,
then algebraically simplified as if by the `a s' command.
`evalextsimp(x)'
The formula `x' has meta-variables substituted in the normal way,
then "extendedly" simplified as if by the `a e' command.
`select(x)'
See Selections with Rewrite Rules.
There are also some special functions you can use in conditions.
`let(v := x)'
The expression `x' is evaluated with meta-variables substituted.
The `a s' command's simplifications are *not* applied by default,
but `x' can include calls to `evalsimp' or `evalextsimp' as
described above to invoke higher levels of simplification. The
result of `x' is then bound to the meta-variable `v'. As usual,
if this meta-variable has already been matched to something else
the two values must be equal; if the meta-variable is new then it
is bound to the result of the expression. This variable can then
appear in later conditions, and on the righthand side of the
rule. In fact, `v' may be any pattern in which case the result
of evaluating `x' is matched to that pattern, binding any
meta-variables that appear in that pattern. Note that `let' can
only appear by itself as a condition, or as one term of an `&&'
which is a whole condition: It cannot be inside an `||' term or
otherwise buried.
The alternate, equivalent form `let(v, x)' is also recognized.
Note that the use of `:=' by `let', while still being
assignment-like in character, is unrelated to the use of `:=' in
the main part of a rewrite rule.
As an example, `f(a) := g(ia) :: let(ia := 1/a) :: constant(ia)'
replaces `f(a)' with `g' of the inverse of `a', if that inverse
exists and is constant. For example, if `a' is a singular matrix
the operation `1/a' is left unsimplified and `constant(ia)'
fails, but if `a' is an invertible matrix then the rule succeeds.
Without `let' there would be no way to express this rule that
didn't have to invert the matrix twice. Note that, because the
meta-variable `ia' is otherwise unbound in this rule, the `let'
condition itself always "succeeds" because no matter what `1/a'
evaluates to, it can successfully be bound to `ia'.
Here's another example, for integrating cosines of linear terms:
`myint(cos(y),x) := sin(y)/b :: let([a,b,x] := lin(y,x))'. The
`lin' function returns a 3-vector if its argument is linear, or
leaves itself unevaluated if not. But an unevaluated `lin' call
will not match the 3-vector on the lefthand side of the `let', so
this `let' both verifies that `y' is linear, and binds the
coefficients `a' and `b' for use elsewhere in the rule. (It
would have been possible to use `sin(a x + b)/b' for the
righthand side instead, but using `sin(y)/b' avoids gratuitous
rearrangement of the argument of the sine.)
Similarly, here is a rule that implements an inverse-`erf'
function. It uses `root' to search for a solution. If `root'
succeeds, it will return a vector of two numbers where the first
number is the desired solution. If no solution is found, `root'
remains in symbolic form. So we use `let' to check that the
result was indeed a vector.
ierf(x) := y :: let([y,z] := root(erf(a) = x, a, .5))
`matches(v,p)'
The meta-variable V, which must already have been matched to
something elsewhere in the rule, is compared against pattern P.
Since `matches' is a standard Calc function, it can appear
anywhere in a condition. But if it appears alone or as a term of
a top-level `&&', then you get the special extra feature that
meta-variables which are bound to things inside P can be used
elsewhere in the surrounding rewrite rule.
The only real difference between `let(p := v)' and `matches(v,
p)' is that the former evaluates `v' using the default
simplifications, while the latter does not.
`remember'
This is actually a variable, not a function. If `remember'
appears as a condition in a rule, then when that rule succeeds
the original expression and rewritten expression are added to the
front of the rule set that contained the rule. If the rule set
was not stored in a variable, `remember' is ignored. The
lefthand side is enclosed in `quote' in the added rule if it
contains any variables.
For example, the rule `f(n) := n f(n-1) :: remember' applied to
`f(7)' will add the rule `f(7) := 7 f(6)' to the front of the
rule set. The rule set `EvalRules' works slightly differently:
There, the evaluation of `f(6)' will complete before the result
is added to the rule set, in this case as `f(7) := 5040'. Thus
`remember' is most useful inside `EvalRules'.
It is up to you to ensure that the optimization performed by
`remember' is safe. For example, the rule `foo(n) := n ::
evalv(eatfoo) > 0 :: remember' is a bad idea (`evalv' is the
function equivalent of the `=' command); if the variable `eatfoo'
ever contains 1, rules like `foo(7) := 7' will be added to the
rule set and will continue to operate even if `eatfoo' is later
changed to 0.
`remember(c)'
Remember the match as described above, but only if condition `c'
is true. For example, `remember(n % 4 = 0)' in the above
factorial rule remembers only every fourth result. Note that
`remember(1)' is equivalent to `remember', and `remember(0)' has
no effect.