From d3e99ab90a47397d9822df58772ff3e2ac5e8a97 Mon Sep 17 00:00:00 2001
From: rocky <rb@dustyfeet.com>
Date: Mon, 16 Sep 2024 10:28:35 -0400
Subject: [PATCH] Add mmatera's information on Rules and application

---
 mathics/core/rules.py | 51 +++++++++++++++++++++++++++++++++++++------
 1 file changed, 44 insertions(+), 7 deletions(-)

diff --git a/mathics/core/rules.py b/mathics/core/rules.py
index 4285b72c6..6af8f5ff5 100644
--- a/mathics/core/rules.py
+++ b/mathics/core/rules.py
@@ -1,15 +1,51 @@
 # -*- coding: utf-8 -*-
-"""Rules are a core part of the way WMA and Mathics3 executes a
+"""Rules are a core part of the way Mathematica and Mathics3 execute a
 program.
 
 Expressions which are transformed by rewrite rules (AKA transformation
-rules) are handed by the Rule class.
+rules) are handed by the `Rule` class.
 
-There are also rules for how to match, assign parameter arguments, and
-apply Mathics3 functions that are implemented in Python code. The
-FunctionApplyRule class handles this.
+There are also rules for how to match, assign function parameter
+arguments, and then apply a Python "evaluation" function to a Mathics3 Expression.
+These kinds of rules are handled by objects in the `FunctionApplyRule` class.
 
 This module contains the classes for these two types of rules.
+
+In a `FunctionApplyRule` rule, the match status of a rule depends on the evaluation return.
+
+For example, suppose that we try to apply rule `F[x_]->x^2` to the expression `F[2]`. The pattern part of the rule,`F[x_]` matches
+the expression, `Blank[x]` (or `x_`) is replaced by `2`, giving the substitution expression `2^2`. Evaluation then stops
+looking for other rules to be applied over `F[2]`.
+
+On the other hand, suppose that we define a `FunctionApplyRule` that associates `F[x_]` with the function:
+
+```
+...
+class MyFunction(Builtin):
+    ...
+    def eval_f(self, x, evaluation) -> Optional[Expression]:
+        "F[x_]"   # pattern part of FunctionApplyRule
+        if x>3:
+            return Expression(SymbolPower, x, Integer2)
+        return None
+```
+
+Then, if we apply the rule to `F[2]`, the function is evaluated returning `None`. Then, in the evaluation loop, we get the same
+effect as if the pattern didn't match with the expression. The loop continues then with the next rule associated with `F`.
+
+Why do things this way?
+
+Sometimes, the cost of deciding if the rule match is similar to the cost of evaluating the function. Suppose for example a rule
+
+   F[x_/;(G[x]>0)]:=G[x]
+
+with G[x] a computationally expensive function. To decide if G[x] is larger than 0, we need to evaluate it,
+and once we have evaluated it, just need to return its value.
+
+Also, this allows us to handle several rules in the same function, without relying on our very slow pattern-matching routines.
+In particular, this is used for for some critical low-level tasks like building lists in iterators, processing arithmetic expressions,
+plotting functions, or evaluating derivatives and integrals.
+
 """
 
 
@@ -224,7 +260,8 @@ def __repr__(self) -> str:
 
 
 class FunctionApplyRule(BaseRule):
-    """A FunctionApplyRule is a rule that has a replacement term that
+    """
+    A FunctionApplyRule is a rule that has a replacement term that
     is associated a Python function rather than a Mathics Expression
     as happens in a transformation Rule.
 
@@ -260,7 +297,7 @@ class FunctionApplyRule(BaseRule):
     sets the attribute ``NumericFunction`` in the  definition of the symbol ``F`` and
     returns Null (``SymbolNull`)`.
 
-    This will cause `Expression.evalate() to perform an additional
+    This will cause `Expression.evaluate() to perform an additional
     ``rewrite_apply_eval()`` step.
 
     """