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# 34. Rules and Patterns

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## 34.1 Introduction to Rules and Patterns

This section describes user-defined pattern matching and simplification rules. There are two groups of functions which implement somewhat different pattern matching schemes. In one group are `tellsimp`, `tellsimpafter`, `defmatch`, `defrule`, `apply1`, `applyb1`, and `apply2`. In the other group are `let` and `letsimp`. Both schemes define patterns in terms of pattern variables declared by `matchdeclare`.

Pattern-matching rules defined by `tellsimp` and `tellsimpafter` are applied automatically by the Maxima simplifier. Rules defined by `defmatch`, `defrule`, and `let` are applied by an explicit function call.

There are additional mechanisms for rules applied to polynomials by `tellrat`, and for commutative and noncommutative algebra in `affine` package.

Categories: Simplification · Rules and patterns

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## 34.2 Functions and Variables for Rules and Patterns

Function: apply1 (expr, rule_1, …, rule_n)

  Repeatedly applies rule_1 to expr until it fails, then repeatedly applies the same rule to all subexpressions of expr, left to right, until rule_1 has failed on all subexpressions. Call the result of transforming expr in this manner expr_2. Then rule_2 is applied in the same fashion starting at the top of expr_2. When rule_n fails on the final subexpression, the result is returned.

  `maxapplydepth` is the depth of the deepest subexpressions processed by `apply1` and `apply2`.

  See also `applyb1`, `apply2` and `let`.

  Categories: Rules and patterns

Function: apply2 (expr, rule_1, …, rule_n)

  If rule_1 fails on a given subexpression, then rule_2 is repeatedly applied, etc. Only if all rules fail on a given subexpression is the whole set of rules repeatedly applied to the next subexpression. If one of the rules succeeds, then the same subexpression is reprocessed, starting with the first rule.

  `maxapplydepth` is the depth of the deepest subexpressions processed by `apply1` and `apply2`.

  See also `apply1` and `let`.

  Categories: Rules and patterns

Function: applyb1 (expr, rule_1, …, rule_n)

  Repeatedly applies rule_1 to the deepest subexpression of expr until it fails, then repeatedly applies the same rule one level higher (i.e., larger subexpressions), until rule_1 has failed on the top-level expression. Then rule_2 is applied in the same fashion to the result of rule_1. After rule_n has been applied to the top-level expression, the result is returned.

  `applyb1` is similar to `apply1` but works from the bottom up instead of from the top down.

  `maxapplyheight` is the maximum height which `applyb1` reaches before giving up.

  See also `apply1`, `apply2` and `let`.

  Categories: Rules and patterns

Option variable: current_let_rule_package

  Default value: `default_let_rule_package`

  `current_let_rule_package` is the name of the rule package that is used by functions in the `let` package (`letsimp`, etc.) if no other rule package is specified. This variable may be assigned the name of any rule package defined via the `let` command.

  If a call such as `letsimp (expr, rule_pkg_name)` is made, the rule package `rule_pkg_name` is used for that function call only, and the value of `current_let_rule_package` is not changed.

  Categories: Rules and patterns

Option variable: default_let_rule_package

  Default value: `default_let_rule_package`

  `default_let_rule_package` is the name of the rule package used when one is not explicitly set by the user with `let` or by changing the value of `current_let_rule_package`.

  Categories: Rules and patterns

Function: defmatch
`defmatch` (progname, pattern, x_1, …, x_n)
`defmatch` (progname, pattern)

  Defines a function `progname(expr, x_1, ..., x_n)` which tests expr to see if it matches pattern.

  pattern is an expression containing the pattern arguments x_1, …, x_n (if any) and some pattern variables (if any). The pattern arguments are given explicitly as arguments to `defmatch` while the pattern variables are declared by the `matchdeclare` function. Any variable not declared as a pattern variable in `matchdeclare` or as a pattern argument in `defmatch` matches only itself.

  The first argument to the created function progname is an expression to be matched against the pattern and the other arguments are the actual arguments which correspond to the dummy variables x_1, …, x_n in the pattern.

  If the match is successful, progname returns a list of equations whose left sides are the pattern arguments and pattern variables, and whose right sides are the subexpressions which the pattern arguments and variables matched. The pattern variables, but not the pattern arguments, are assigned the subexpressions they match. If the match fails, progname returns `false`.

  A literal pattern (that is, a pattern which contains neither pattern arguments nor pattern variables) returns `true` if the match succeeds.

  See also `matchdeclare`, `defrule`, `tellsimp` and `tellsimpafter`.

  Examples:

  Define a function `linearp(expr, x)` which tests `expr` to see if it is of the form `a*x + b` such that `a` and `b` do not contain `x` and `a` is nonzero. This match function matches expressions which are linear in any variable, because the pattern argument `x` is given to `defmatch`.

      (%i1) matchdeclare (a, lambda ([e], e#0 and freeof(x, e)), b,
                          freeof(x));
      (%o1)                         done
      (%i2) defmatch (linearp, a*x + b, x);
      (%o2)                        linearp
      (%i3) linearp (3*z + (y + 1)*z + y^2, z);
                               2
      (%o3)              [b = y , a = y + 4, x = z]
      (%i4) a;
      (%o4)                         y + 4
      (%i5) b;
                                      2
      (%o5)                          y
      (%i6) x;
      (%o6)                           x

  Define a function `linearp(expr)` which tests `expr` to see if it is of the form `a*x + b` such that `a` and `b` do not contain `x` and `a` is nonzero. This match function only matches expressions linear in `x`, not any other variable, because no pattern argument is given to `defmatch`.

      (%i1) matchdeclare (a, lambda ([e], e#0 and freeof(x, e)), b,
                          freeof(x));
      (%o1)                         done
      (%i2) defmatch (linearp, a*x + b);
      (%o2)                        linearp
      (%i3) linearp (3*z + (y + 1)*z + y^2);
      (%o3)                         false
      (%i4) linearp (3*x + (y + 1)*x + y^2);
                                   2
      (%o4)                  [b = y , a = y + 4]

  Define a function `checklimits(expr)` which tests `expr` to see if it is a definite integral.

      (%i1) matchdeclare ([a, f], true);
      (%o1)                         done
      (%i2) constinterval (l, h) := constantp (h - l);
      (%o2)        constinterval(l, h) := constantp(h - l)
      (%i3) matchdeclare (b, constinterval (a));
      (%o3)                         done
      (%i4) matchdeclare (x, atom);
      (%o4)                         done
      (%i5) simp : false;
      (%o5)                         false
      (%i6) defmatch (checklimits, 'integrate (f, x, a, b));
      (%o6)                      checklimits
      (%i7) simp : true;
      (%o7)                         true
      (%i8) 'integrate (sin(t), t, %pi + x, 2*%pi + x);
                             x + 2 %pi
                            /
                            [
      (%o8)                 I          sin(t) dt
                            ]
                            /
                             x + %pi
      (%i9) checklimits (%);
      (%o9)    [b = x + 2 %pi, a = x + %pi, x = t, f = sin(t)]

  Categories: Rules and patterns

Function: defrule (rulename, pattern, replacement)

  Defines and names a replacement rule for the given pattern. If the rule named rulename is applied to an expression (by `apply1`, `applyb1`, or `apply2`), every subexpression matching the pattern will be replaced by the replacement. All variables in the replacement which have been assigned values by the pattern match are assigned those values in the replacement which is then simplified.

  The rules themselves can be treated as functions which transform an expression by one operation of the pattern match and replacement. If the match fails, the rule function returns `false`.

  Categories: Rules and patterns

Function: disprule
`disprule` (rulename_1, …, rulename_2)
`disprule` (all)

  Display rules with the names rulename_1, …, rulename_n, as returned by `defrule`, `tellsimp`, or `tellsimpafter`, or a pattern defined by `defmatch`. Each rule is displayed with an intermediate expression label (`%t`).

  `disprule (all)` displays all rules.

  `disprule` quotes its arguments. `disprule` returns the list of intermediate expression labels corresponding to the displayed rules.

  See also `letrules`, which displays rules defined by `let`.

  Examples:

      (%i1) tellsimpafter (foo (x, y), bar (x) + baz (y));
      (%o1)                   [foorule1, false]
      (%i2) tellsimpafter (x + y, special_add (x, y));
      (%o2)                   [+rule1, simplus]
      (%i3) defmatch (quux, mumble (x));
      (%o3)                         quux
      (%i4) disprule (foorule1, ?\+rule1, quux);
      (%t4)        foorule1 : foo(x, y) -> baz(y) + bar(x)

      (%t5)          +rule1 : y + x -> special_add(x, y)

      (%t6)                quux : mumble(x) -> []

      (%o6)                    [%t4, %t5, %t6]
      (%i7) ev(%);
      (%o7) [foorule1 : foo(x, y) -> baz(y) + bar(x),
           +rule1 : y + x -> special_add(x, y), quux : mumble(x) -> []]

  Categories: Rules and patterns · Display functions

Function: let
`let` (prod, repl, predname, arg_1, …, arg_n)
`let` ([prod, repl, predname, arg_1, …, arg_n], package_name)

  Defines a substitution rule for `letsimp` such that prod is replaced by repl. prod is a product of positive or negative powers of the following terms:

  - Atoms which `letsimp` will search for literally unless previous to calling `letsimp` the `matchdeclare` function is used to associate a predicate with the atom. In this case `letsimp` will match the atom to any term of a product satisfying the predicate.
  - Kernels such as `sin(x)`, `n!`, `f(x,y)`, etc. As with atoms above `letsimp` will look for a literal match unless `matchdeclare` is used to associate a predicate with the argument of the kernel.

  A term to a positive power will only match a term having at least that power. A term to a negative power on the other hand will only match a term with a power at least as negative. In the case of negative powers in prod the switch `letrat` must be set to `true`. See also `letrat`.

  If a predicate is included in the `let` function followed by a list of arguments, a tentative match (i.e. one that would be accepted if the predicate were omitted) is accepted only if `predname (arg_1', ..., arg_n')` evaluates to `true` where arg_i' is the value matched to arg_i. The arg_i may be the name of any atom or the argument of any kernel appearing in prod. repl may be any rational expression. If any of the atoms or arguments from prod appear in repl the appropriate substitutions are made.

  The global flag `letrat` controls the simplification of quotients by `letsimp`. When `letrat` is `false`, `letsimp` simplifies the numerator and denominator of expr separately, and does not simplify the quotient. Substitutions such as `n!/n` goes to `(n-1)!` then fail. When `letrat` is `true`, then the numerator, denominator, and the quotient are simplified in that order.

  These substitution functions allow you to work with several rule packages at once. Each rule package can contain any number of `let` rules and is referenced by a user-defined name. The command `let ([prod, repl, predname, arg_1, ..., arg_n], package_name)` adds the rule predname to the rule package package_name. The command `letsimp (expr, package_name)` applies the rules in package_name. `letsimp (expr, package_name1, package_name2, ...)` is equivalent to `letsimp (expr, package_name1)` followed by `letsimp (%, package_name2)`, …

  `current_let_rule_package` is the name of the rule package that is presently being used. This variable may be assigned the name of any rule package defined via the `let` command. Whenever any of the functions comprising the `let` package are called with no package name, the package named by `current_let_rule_package` is used. If a call such as `letsimp (expr, rule_pkg_name)` is made, the rule package rule_pkg_name is used for that `letsimp` command only, and `current_let_rule_package` is not changed. If not otherwise specified, `current_let_rule_package` defaults to `default_let_rule_package`.

      (%i1) matchdeclare ([a, a1, a2], true)$
      (%i2) oneless (x, y) := is (x = y-1)$
      (%i3) let (a1*a2!, a1!, oneless, a2, a1);
      (%o3)         a1 a2! --> a1! where oneless(a2, a1)
      (%i4) letrat: true$
      (%i5) let (a1!/a1, (a1-1)!);
                              a1!
      (%o5)                   --- --> (a1 - 1)!
                              a1
      (%i6) letsimp (n*m!*(n-1)!/m);
      (%o6)                      (m - 1)! n!
      (%i7) let (sin(a)^2, 1 - cos(a)^2);
                              2               2
      (%o7)                sin (a) --> 1 - cos (a)
      (%i8) letsimp (sin(x)^4);
                              4           2
      (%o8)                cos (x) - 2 cos (x) + 1

  Categories: Rules and patterns

Option variable: letrat

  Default value: `false`

  When `letrat` is `false`, `letsimp` simplifies the numerator and denominator of a ratio separately, and does not simplify the quotient.

  When `letrat` is `true`, the numerator, denominator, and their quotient are simplified in that order.

      (%i1) matchdeclare (n, true)$
      (%i2) let (n!/n, (n-1)!);
                               n!
      (%o2)                    -- --> (n - 1)!
                               n
      (%i3) letrat: false$
      (%i4) letsimp (a!/a);
                                     a!
      (%o4)                          --
                                     a
      (%i5) letrat: true$
      (%i6) letsimp (a!/a);
      (%o6)                       (a - 1)!

  Categories: Rules and patterns

Function: letrules
`letrules` ()
`letrules` (package_name)

  Displays the rules in a rule package. `letrules ()` displays the rules in the current rule package. `letrules (package_name)` displays the rules in package_name.

  The current rule package is named by `current_let_rule_package`. If not otherwise specified, `current_let_rule_package` defaults to `default_let_rule_package`.

  See also `disprule`, which displays rules defined by `tellsimp` and `tellsimpafter`.

  Categories: Rules and patterns

Function: letsimp
`letsimp` (expr)
`letsimp` (expr, package_name)
`letsimp` (expr, package_name_1, …, package_name_n)

  Repeatedly applies the substitution rules defined by `let` until no further change is made to expr.

  `letsimp (expr)` uses the rules from `current_let_rule_package`.

  `letsimp (expr, package_name)` uses the rules from package_name without changing `current_let_rule_package`.

  `letsimp (expr, package_name_1, ..., package_name_n)` is equivalent to `letsimp (expr, package_name_1)`, followed by `letsimp (%, package_name_2)`, and so on.

  See also `let`. For other ways to do substitutions see also `subst`, `psubst`, `at` and `ratsubst`.

      (%i1) e0:e(k)=-(9*y(k))/(5*z)-u(k-1)/(5*z)+(4*y(k))/(5*z^2)+(3*u(k-1))/(5*z^2)+y(k)-(2*u(k-1))/5;
                      9 y(k)    u(k - 1)   4 y(k)   3 u(k - 1)
      (%o1) e(k) = (- ------) - -------- + ------ + ---------- + y(k)
                       5 z        5 z          2          2
                                            5 z        5 z
                                                             2 u(k - 1)
                                                           - ----------
                                                                 5
      (%i2) matchdeclare(h,any)$
      (%i3) let(u(h)/z,u(h-1));
                              u(h)
      (%o3)                   ---- --> u(h - 1)
                               z
      (%i4) let(y(h)/z, y(h-1));
                              y(h)
      (%o4)                   ---- --> y(h - 1)
                               z
      (%i5) e1:letsimp(e0);
                      2 u(k - 1)           3 u(k - 3)   4 y(k - 2)
      (%o5) e(k) = (- ----------) + y(k) + ---------- + ----------
                          5                    5            5
                                             u(k - 2)       9 y(k - 1)
                                        + (- --------) + (- ----------)
                                                5               5

  Categories: Rules and patterns

Option variable: let_rule_packages

  Default value: `[default_let_rule_package]`

  `let_rule_packages` is a list of all user-defined let rule packages plus the default package `default_let_rule_package`.

  Categories: Rules and patterns

Function: matchdeclare (a_1, pred_1, …, a_n, pred_n)

  Associates a predicate pred_k with a variable or list of variables a_k so that a_k matches expressions for which the predicate returns anything other than `false`.

  A predicate is the name of a function, or a lambda expression, or a function call or lambda call missing the last argument, or `true` or `all`. Any expression matches `true` or `all`. If the predicate is specified as a function call or lambda call, the expression to be tested is appended to the list of arguments; the arguments are evaluated at the time the match is evaluated. Otherwise, the predicate is specified as a function name or lambda expression, and the expression to be tested is the sole argument. A predicate function need not be defined when `matchdeclare` is called; the predicate is not evaluated until a match is attempted.

  A predicate may return a Boolean expression as well as `true` or `false`. Boolean expressions are evaluated by `is` within the constructed rule function, so it is not necessary to call `is` within the predicate.

  If an expression satisfies a match predicate, the match variable is assigned the expression, except for match variables which are operands of addition `+` or multiplication `*`. Only addition and multiplication are handled specially; other n-ary operators (both built-in and user-defined) are treated like ordinary functions.

  In the case of addition and multiplication, the match variable may be assigned a single expression which satisfies the match predicate, or a sum or product (respectively) of such expressions. Such multiple-term matching is greedy: predicates are evaluated in the order in which their associated variables appear in the match pattern, and a term which satisfies more than one predicate is taken by the first predicate which it satisfies. Each predicate is tested against all operands of the sum or product before the next predicate is evaluated. In addition, if 0 or 1 (respectively) satisfies a match predicate, and there are no other terms which satisfy the predicate, 0 or 1 is assigned to the match variable associated with the predicate.

  The algorithm for processing addition and multiplication patterns makes some match results (for example, a pattern in which a "match anything" variable appears) dependent on the ordering of terms in the match pattern and in the expression to be matched. However, if all match predicates are mutually exclusive, the match result is insensitive to ordering, as one match predicate cannot accept terms matched by another.

  Calling `matchdeclare` with a variable a as an argument changes the `matchdeclare` property for a, if one was already declared; only the most recent `matchdeclare` is in effect when a rule is defined. Later changes to the `matchdeclare` property (via `matchdeclare` or `remove`) do not affect existing rules.

  `propvars (matchdeclare)` returns the list of all variables for which there is a `matchdeclare` property. `printprops (a, matchdeclare)` returns the predicate for variable `a`. `printprops (all, matchdeclare)` returns the list of predicates for all `matchdeclare` variables. `remove (a, matchdeclare)` removes the `matchdeclare` property from a.

  The functions `defmatch`, `defrule`, `tellsimp`, `tellsimpafter`, and `let` construct rules which test expressions against patterns.

  `matchdeclare` quotes its arguments. `matchdeclare` always returns `done`.

  Examples:

  A predicate is the name of a function, or a lambda expression, or a function call or lambda call missing the last argument, or `true` or `all`.

      (%i1) matchdeclare (aa, integerp);
      (%o1)                         done
      (%i2) matchdeclare (bb, lambda ([x], x > 0));
      (%o2)                         done
      (%i3) matchdeclare (cc, freeof (%e, %pi, %i));
      (%o3)                         done
      (%i4) matchdeclare (dd, lambda ([x, y], gcd (x, y) = 1) (1728));
      (%o4)                         done
      (%i5) matchdeclare (ee, true);
      (%o5)                         done
      (%i6) matchdeclare (ff, all);
      (%o6)                         done

  If an expression satisfies a match predicate, the match variable is assigned the expression.

      (%i1) matchdeclare (aa, integerp, bb, atom);
      (%o1)                         done
      (%i2) defrule (r1, bb^aa, ["integer" = aa, "atom" = bb]);
                          aa
      (%o2)        r1 : bb   -> [integer = aa, atom = bb]
      (%i3) r1 (%pi^8);
      (%o3)               [integer = 8, atom = %pi]

  In the case of addition and multiplication, the match variable may be assigned a single expression which satisfies the match predicate, or a sum or product (respectively) of such expressions.

      (%i1) matchdeclare (aa, atom, bb, lambda ([x], not atom(x)));
      (%o1)                         done
      (%i2) defrule (r1, aa + bb, ["all atoms" = aa, "all nonatoms" =
                     bb]);
      (%o2)  r1 : bb + aa -> [all atoms = aa, all nonatoms = bb]
      (%i3) r1 (8 + a*b + sin(x));
      (%o3)     [all atoms = 8, all nonatoms = sin(x) + a b]
      (%i4) defrule (r2, aa * bb, ["all atoms" = aa, "all nonatoms" =
                     bb]);
      (%o4)   r2 : aa bb -> [all atoms = aa, all nonatoms = bb]
      (%i5) r2 (8 * (a + b) * sin(x));
      (%o5)    [all atoms = 8, all nonatoms = (b + a) sin(x)]

  When matching arguments of `+` and `*`, if all match predicates are mutually exclusive, the match result is insensitive to ordering, as one match predicate cannot accept terms matched by another.

      (%i1) matchdeclare (aa, atom, bb, lambda ([x], not atom(x)));
      (%o1)                         done
      (%i2) defrule (r1, aa + bb, ["all atoms" = aa, "all nonatoms" =
                     bb]);
      (%o2)  r1 : bb + aa -> [all atoms = aa, all nonatoms = bb]
      (%i3) r1 (8 + a*b + %pi + sin(x) - c + 2^n);
                                                           n
      (%o3) [all atoms = %pi + 8, all nonatoms = sin(x) + 2  - c + a b]
      (%i4) defrule (r2, aa * bb, ["all atoms" = aa, "all nonatoms" =
                     bb]);
      (%o4)   r2 : aa bb -> [all atoms = aa, all nonatoms = bb]
      (%i5) r2 (8 * (a + b) * %pi * sin(x) / c * 2^n);
                                                      n + 3
                                             (b + a) 2      sin(x)
      (%o5) [all atoms = %pi, all nonatoms = ---------------------]
                                                       c

  The functions `propvars` and `printprops` return information about match variables.

      (%i1) matchdeclare ([aa, bb, cc], atom, [dd, ee], integerp);
      (%o1)                         done
      (%i2) matchdeclare (ff, floatnump, gg, lambda ([x], x > 100));
      (%o2)                         done
      (%i3) propvars (matchdeclare);
      (%o3)             [aa, bb, cc, dd, ee, ff, gg]
      (%i4) printprops (ee, matchdeclare);
      (%o4)                    [integerp(ee)]
      (%i5) printprops (gg, matchdeclare);
      (%o5)              [lambda([x], x > 100, gg)]
      (%i6) printprops (all, matchdeclare);
      (%o6) [lambda([x], x > 100, gg), floatnump(ff), integerp(ee),
                            integerp(dd), atom(cc), atom(bb), atom(aa)]

  Categories: Rules and patterns · Declarations and inferences

Option variable: maxapplydepth

  Default value: 10000

  `maxapplydepth` is the maximum depth to which `apply1` and `apply2` will delve.

  Categories: Function application

Option variable: maxapplyheight

  Default value: 10000

  `maxapplyheight` is the maximum height to which `applyb1` will reach before giving up.

  Categories: Function application

Function: remlet
`remlet` (prod, name)
`remlet` ()
`remlet` (all)
`remlet` (all, name)

  Deletes the substitution rule, prod -> repl, most recently defined by the `let` function. If name is supplied the rule is deleted from the rule package name.

  `remlet()` and `remlet(all)` delete all substitution rules from the current rule package. If the name of a rule package is supplied, e.g. `remlet (all, name)`, the rule package name is also deleted.

  If a substitution is to be changed using the same product, `remlet` need not be called, just redefine the substitution using the same product (literally) with the `let` function and the new replacement and/or predicate name. Should `remlet (prod)` now be called the original substitution rule is revived.

  See also `remrule`, which removes a rule defined by `tellsimp` or `tellsimpafter`.

  Categories: Rules and patterns

Function: remrule
`remrule` (op, rulename)
`remrule` (op, all)

  Removes rules defined by `tellsimp` or `tellsimpafter`.

  `remrule (op, rulename)` removes the rule with the name rulename from the operator op. When op is a built-in or user-defined operator (as defined by `infix`, `prefix`, etc.), op and rulename must be enclosed in double quote marks.

  `remrule (op, all)` removes all rules for the operator op.

  See also `remlet`, which removes a rule defined by `let`.

  Examples:

      (%i1) tellsimp (foo (aa, bb), bb - aa);
      (%o1)                   [foorule1, false]
      (%i2) tellsimpafter (aa + bb, special_add (aa, bb));
      (%o2)                   [+rule1, simplus]
      (%i3) infix ("@@");
      (%o3)                          @@
      (%i4) tellsimp (aa @@ bb, bb/aa);
      (%o4)                   [@@rule1, false]
      (%i5) tellsimpafter (quux (%pi, %e), %pi - %e);
      (%o5)                  [quuxrule1, false]
      (%i6) tellsimpafter (quux (%e, %pi), %pi + %e);
      (%o6)             [quuxrule2, quuxrule1, false]
      (%i7) [foo (aa, bb), aa + bb, aa @@ bb, quux (%pi, %e),
             quux (%e, %pi)];
                                           bb
      (%o7) [bb - aa, special_add(aa, bb), --, %pi - %e, %pi + %e]
                                           aa
      (%i8) remrule (foo, foorule1);
      (%o8)                          foo
      (%i9) remrule ("+", ?\+rule1);
      (%o9)                           +
      (%i10) remrule ("@@", ?\@\@rule1);
      (%o10)                         @@
      (%i11) remrule (quux, all);
      (%o11)                        quux
      (%i12) [foo (aa, bb), aa + bb, aa @@ bb, quux (%pi, %e),
              quux (%e, %pi)];
      (%o12) [foo(aa, bb), bb + aa, aa @@ bb, quux(%pi, %e),
                                                         quux(%e, %pi)]

  Categories: Rules and patterns

Function: tellsimp (pattern, replacement)

  is similar to `tellsimpafter` but places new information before old so that it is applied before the built-in simplification rules.

  `tellsimp` is used when it is important to modify the expression before the simplifier works on it, for instance if the simplifier "knows" something about the expression, but what it returns is not to your liking. If the simplifier "knows" something about the main operator of the expression, but is simply not doing enough for you, you probably want to use `tellsimpafter`.

  The pattern may not be a sum, product, single variable, or number.

  The system variable `rules` is the list of rules defined by `defrule`, `defmatch`, `tellsimp`, and `tellsimpafter`.

  Examples:

      (%i1) matchdeclare (x, freeof (%i));
      (%o1)                         done
      (%i2) %iargs: false$
      (%i3) tellsimp (sin(%i*x), %i*sinh(x));
      (%o3)                 [sinrule1, simp-%sin]
      (%i4) trigexpand (sin (%i*y + x));
      (%o4)         sin(x) cos(%i y) + %i cos(x) sinh(y)
      (%i5) %iargs:true$
      (%i6) errcatch(0^0);
       0
      0  has been generated
      (%o6)                          []
      (%i7) ev (tellsimp (0^0, 1), simp: false);
      (%o7)                  [^rule1, simpexpt]
      (%i8) 0^0;
      (%o8)                           1
      (%i9) remrule ("^", %th(2)[1]);
      (%o9)                           ^
      (%i10) tellsimp (sin(x)^2, 1 - cos(x)^2);
      (%o10)                 [^rule2, simpexpt]
      (%i11) (1 + sin(x))^2;
                                            2
      (%o11)                    (sin(x) + 1)
      (%i12) expand (%);
                                         2
      (%o12)               2 sin(x) - cos (x) + 2
      (%i13) sin(x)^2;
                                        2
      (%o13)                     1 - cos (x)
      (%i14) kill (rules);
      (%o14)                        done
      (%i15) matchdeclare (a, true);
      (%o15)                        done
      (%i16) tellsimp (sin(a)^2, 1 - cos(a)^2);
      (%o16)                 [^rule3, simpexpt]
      (%i17) sin(y)^2;
                                        2
      (%o17)                     1 - cos (y)

  Categories: Rules and patterns

Function: tellsimpafter (pattern, replacement)

  Defines a simplification rule which the Maxima simplifier applies after built-in simplification rules. pattern is an expression, comprising pattern variables (declared by `matchdeclare`) and other atoms and operators, considered literals for the purpose of pattern matching. replacement is substituted for an actual expression which matches pattern; pattern variables in replacement are assigned the values matched in the actual expression.

  pattern may be any nonatomic expression in which the main operator is not a pattern variable; the simplification rule is associated with the main operator. The names of functions (with one exception, described below), lists, and arrays may appear in pattern as the main operator only as literals (not pattern variables); this rules out expressions such as `aa(x)` and `bb[y]` as patterns, if `aa` and `bb` are pattern variables. Names of functions, lists, and arrays which are pattern variables may appear as operators other than the main operator in pattern.

  There is one exception to the above rule concerning names of functions. The name of a subscripted function in an expression such as `aa[x](y)` may be a pattern variable, because the main operator is not `aa` but rather the Lisp atom `mqapply`. This is a consequence of the representation of expressions involving subscripted functions.

  Simplification rules are applied after evaluation (if not suppressed through quotation or the flag `noeval`). Rules established by `tellsimpafter` are applied in the order they were defined, and after any built-in rules. Rules are applied bottom-up, that is, applied first to subexpressions before application to the whole expression. It may be necessary to repeatedly simplify a result (for example, via the quote-quote operator `''` or the flag `infeval`) to ensure that all rules are applied.

  Pattern variables are treated as local variables in simplification rules. Once a rule is defined, the value of a pattern variable does not affect the rule, and is not affected by the rule. An assignment to a pattern variable which results from a successful rule match does not affect the current assignment (or lack of it) of the pattern variable. However, as with all atoms in Maxima, the properties of pattern variables (as declared by `put` and related functions) are global.

  The rule constructed by `tellsimpafter` is named after the main operator of pattern. Rules for built-in operators, and user-defined operators defined by `infix`, `prefix`, `postfix`, `matchfix`, and `nofix`, have names which are Lisp identifiers. Rules for other functions have names which are Maxima identifiers.

  The treatment of noun and verb forms is slightly confused. If a rule is defined for a noun (or verb) form and a rule for the corresponding verb (or noun) form already exists, the newly-defined rule applies to both forms (noun and verb). If a rule for the corresponding verb (or noun) form does not exist, the newly-defined rule applies only to the noun (or verb) form.

  The rule constructed by `tellsimpafter` is an ordinary Lisp function. If the name of the rule is `$foorule1`, the construct `:lisp (trace $foorule1)` traces the function, and `:lisp (symbol-function '$foorule1)` displays its definition.

  `tellsimpafter` quotes its arguments. `tellsimpafter` returns the list of rules for the main operator of pattern, including the newly established rule.

  See also `matchdeclare`, `defmatch`, `defrule`, `tellsimp`, `let`, `kill`, `remrule` and `clear_rules`.

  Examples:

  pattern may be any nonatomic expression in which the main operator is not a pattern variable.

      (%i1) matchdeclare (aa, atom, [ll, mm], listp, xx, true)$
      (%i2) tellsimpafter (sin (ll), map (sin, ll));
      (%o2)                 [sinrule1, simp-%sin]
      (%i3) sin ([1/6, 1/4, 1/3, 1/2, 1]*%pi);
                          1     1     sqrt(3)
      (%o3)              [-, -------, -------, 1, 0]
                          2  sqrt(2)     2
      (%i4) tellsimpafter (ll^mm, map ("^", ll, mm));
      (%o4)                  [^rule1, simpexpt]
      (%i5) [a, b, c]^[1, 2, 3];
                                      2   3
      (%o5)                      [a, b , c ]
      (%i6) tellsimpafter (foo (aa (xx)), aa (foo (xx)));
      (%o6)                   [foorule1, false]
      (%i7) foo (bar (u - v));
      (%o7)                    bar(foo(u - v))

  Rules are applied in the order they were defined. If two rules can match an expression, the rule which was defined first is applied.

      (%i1) matchdeclare (aa, integerp);
      (%o1)                         done
      (%i2) tellsimpafter (foo (aa), bar_1 (aa));
      (%o2)                   [foorule1, false]
      (%i3) tellsimpafter (foo (aa), bar_2 (aa));
      (%o3)              [foorule2, foorule1, false]
      (%i4) foo (42);
      (%o4)                       bar_1(42)

  Pattern variables are treated as local variables in simplification rules. (Compare to `defmatch`, which treats pattern variables as global variables.)

      (%i1) matchdeclare (aa, integerp, bb, atom);
      (%o1)                         done
      (%i2) tellsimpafter (foo(aa, bb), bar('aa=aa, 'bb=bb));
      (%o2)                   [foorule1, false]
      (%i3) bb: 12345;
      (%o3)                         12345
      (%i4) foo (42, %e);
      (%o4)                 bar(aa = 42, bb = %e)
      (%i5) bb;
      (%o5)                         12345

  As with all atoms, properties of pattern variables are global even though values are local. In this example, an assignment property is declared via `define_variable`. This is a property of the atom `bb` throughout Maxima.

      (%i1) matchdeclare (aa, integerp, bb, atom);
      (%o1)                         done
      (%i2) tellsimpafter (foo(aa, bb), bar('aa=aa, 'bb=bb));
      (%o2)                   [foorule1, false]
      (%i3) foo (42, %e);
      (%o3)                 bar(aa = 42, bb = %e)
      (%i4) define_variable (bb, true, boolean);
      (%o4)                         true
      (%i5) foo (42, %e);
      translator: bb was declared with mode boolean, but it has value:
                                                                     %e
       -- an error. To debug this try: debugmode(true);

  Rules are named after main operators. Names of rules for built-in and user-defined operators are Lisp identifiers, while names for other functions are Maxima identifiers.

      (%i1) tellsimpafter (foo (%pi + %e), 3*%pi);
      (%o1)                   [foorule1, false]
      (%i2) tellsimpafter (foo (%pi * %e), 17*%e);
      (%o2)              [foorule2, foorule1, false]
      (%i3) tellsimpafter (foo (%i ^ %e), -42*%i);
      (%o3)         [foorule3, foorule2, foorule1, false]
      (%i4) tellsimpafter (foo (9) + foo (13), quux (22));
      (%o4)                   [+rule1, simplus]
      (%i5) tellsimpafter (foo (9) * foo (13), blurf (22));
      (%o5)                  [*rule1, simptimes]
      (%i6) tellsimpafter (foo (9) ^ foo (13), mumble (22));
      (%o6)                  [^rule1, simpexpt]
      (%i7) rules;
      (%o7) [foorule1, foorule2, foorule3, +rule1, *rule1, ^rule1]
      (%i8) foorule_name: first (%o1);
      (%o8)                       foorule1
      (%i9) plusrule_name: first (%o4);
      (%o9)                        +rule1
      (%i10) remrule (foo, foorule1);
      (%o10)                         foo
      (%i11) remrule ("^", ?\^rule1);
      (%o11)                          ^
      (%i12) rules;
      (%o12)        [foorule2, foorule3, +rule1, *rule1]

  A worked example: anticommutative multiplication.

      (%i1) gt (i, j) := integerp(j) and i < j;
      (%o1)          gt(i, j) := integerp(j) and (i < j)
      (%i2) matchdeclare (i, integerp, j, gt(i));
      (%o2)                         done
      (%i3) tellsimpafter (s[i]^^2, 1);
      (%o3)                 [^^rule1, simpncexpt]
      (%i4) tellsimpafter (s[i] . s[j], -s[j] . s[i]);
      (%o4)                   [.rule1, simpnct]
      (%i5) s[1] . (s[1] + s[2]);
      (%o5)                    s  . (s  + s )
                                1     2    1
      (%i6) expand (%);
      (%o6)                      1 - s  . s
                                      2    1
      (%i7) factor (expand (sum (s[i], i, 0, 9)^^5));
      (%o7) 100 (s  + s  + s  + s  + s  + s  + s  + s  + s  + s )
                  9    8    7    6    5    4    3    2    1    0

  Categories: Rules and patterns

Function: clear_rules ()

  Executes `kill (rules)` and then resets the next rule number to 1 for addition `+`, multiplication `*`, and exponentiation `^`.

  Categories: Rules and patterns

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