Examples of Rewrite Rules
-------------------------

Returning to the example of substituting the pattern `sin(x)^2 +
cos(x)^2' with 1, we saw that the rule `opt(a) sin(x)^2 + opt(a)
cos(x)^2 := a' does a good job of finding suitable cases.  Another
solution would be to use the rule `cos(x)^2 := 1 - sin(x)^2', followed
by algebraic simplification if necessary.  This rule will be the most
effective way to do the job, but at the expense of making some changes
that you might not desire.

Another algebraic rewrite rule is `exp(x+y) := exp(x) exp(y)'.  To
make this work with the `j r' command so that it can be easily
targeted to a particular exponential in a large formula, you might
wish to write the rule as `select(exp(x+y)) := select(exp(x) exp(y))'.
The `select' markers will be ignored by the regular `a r' command
(See Selections with Rewrite Rules).

A surprisingly useful rewrite rule is `a/(b-c) := a*(b+c)/(b^2-c^2)'.
This will simplify the formula whenever `b' and/or `c' can be made
simpler by squaring.  For example, applying this rule to `2 / (sqrt(2)
+ 3)' yields `6:7 - 2:7 sqrt(2)' (assuming Symbolic Mode has been
enabled to keep the square root from being evaulated to a
floating-point approximation).  This rule is also useful when working
with symbolic complex numbers, e.g., `(a + b i) / (c + d i)'.

As another example, we could define our own "triangular numbers" function
with the rules `[tri(0) := 0, tri(n) := n + tri(n-1) :: n>0]'.  Enter
this vector and store it in a variable:  `s t trirules'.  Now, given
a suitable formula like `tri(5)' on the stack, type `a r trirules'
to apply these rules repeatedly.  After six applications, `a r' will
stop with 15 on the stack.  Once these rules are debugged, it would probably
be most useful to add them to `EvalRules' so that Calc will evaluate
the new `tri' function automatically.  We could then use `Z K' on
the keyboard macro `' tri($) RET' to make a command that applies
`tri' to the value on the top of the stack.  See Programming.

The following rule set, contributed by Francois Pinard, implements
"quaternions", a generalization of the concept of complex numbers.
Quaternions have four components, and are here represented by function
calls `quat(W, [X, Y, Z])' with "real part" W and the three
"imaginary" parts collected into a vector.  Various arithmetical
operations on quaternions are supported.  To use these rules, either
add them to `EvalRules', or create a command based on `a r' for
simplifying quaternion formulas.  A convenient way to enter
quaternions would be a command defined by a keyboard macro containing:
`' quat($$$$, [$$$, $$, $]) RET'.

     [ quat(w, x, y, z) := quat(w, [x, y, z]),
       quat(w, [0, 0, 0]) := w,
       abs(quat(w, v)) := hypot(w, v),
       -quat(w, v) := quat(-w, -v),
       r + quat(w, v) := quat(r + w, v) :: real(r),
       r - quat(w, v) := quat(r - w, -v) :: real(r),
       quat(w1, v1) + quat(w2, v2) := quat(w1 + w2, v1 + v2),
       r * quat(w, v) := quat(r * w, r * v) :: real(r),
       plain(quat(w1, v1) * quat(w2, v2))
          := quat(w1 * w2 - v1 * v2, w1 * v2 + w2 * v1 + cross(v1, v2)),
       quat(w1, v1) / r := quat(w1 / r, v1 / r) :: real(r),
       z / quat(w, v) := z * quatinv(quat(w, v)),
       quatinv(quat(w, v)) := quat(w, -v) / (w^2 + v^2),
       quatsqr(quat(w, v)) := quat(w^2 - v^2, 2 * w * v),
       quat(w, v)^k := quatsqr(quat(w, v)^(k / 2))
                    :: integer(k) :: k > 0 :: k % 2 = 0,
       quat(w, v)^k := quatsqr(quat(w, v)^((k - 1) / 2)) * quat(w, v)
                    :: integer(k) :: k > 2,
       quat(w, v)^-k := quatinv(quat(w, v)^k) :: integer(k) :: k > 0 ]

Quaternions, like matrices, have non-commutative multiplication.  In
other words, `q1 * q2 = q2 * q1' is not necessarily true if `q1' and
`q2' are `quat' forms.  The `quat*quat' rule above uses `plain' to
prevent Calc from rearranging the product.  It may also be wise to add
the line `[quat(), matrix]' to the `Decls' matrix, to ensure that
Calc's other algebraic operations will not rearrange a quaternion
product.  See Declarations.

These rules also accept a four-argument `quat' form, converting it to
the preferred form in the first rule.  If you would rather see results
in the four-argument form, just append the two items `phase(2),
quat(w, [x, y, z]) := quat(w, x, y, z)' to the end of the rule set.
(But remember that multi-phase rule sets don't work in `EvalRules'.)