Classes and subclasses of entities in the Physics Derivation Graph:

- derivations = an ordered set of two or more steps
- steps = a set of one or more statements related by an inference rule
- inference rule = identifies the relation of a set of one or more statements
- statement = two or more expressions (LHS and RHS) and a relational operator
- expressions = an ordered set of symbols
- symbols = a token
- operator = applies to one or more values (aka operands).
*Property*: number of expected values - relational operator = exists between two expressions that are sides of an equation
- equality.
*Members*: =, \approx - inequality.
*Members*: >, <, >=, <= - inequation.
*Member*: != - value.
*Property*: categorized as "variable" xor "constant" - integer = one or more digits. The set of digits depends on the base
- float
- complex
- unit.
*Examples*: "m" for meter, "kg" for kilogram

Some aspects of expressions and derivations I don't have names for yet:

- binary operators {"where", "for all", "when", "for"} used two relate two expressions, the "primary expression" on the left and one or more "scope"/"definition"/"constraint" (equation/inequality)

Some aspects of expressions and derivations I don't need to label in the PDG:

- terms = parts of the expression that are connected with addition and subtraction
- factors = parts of the expression that are connected by multiplication
- coefficients = a number that is multiplied by a variable in a mathematical expression.
- power, base, exponent
- base (as in decimal vs hexadecimal, etc)
- formula
- function

An equation is two expressions linked with an equal sign.

**What is the superclass above "equation" and "inequality"?**So far I'm settling on "statement".

I am intentionally staying out of the realm of {proofs, theorems, axioms} both because that is outside the scope of the Physics Derivation Graph and because the topic is already addressed by OMDoc.

Suppose we have a statement like

y = x^2 + b where x = {5, 3, 1}

In that statement,

- "y = x^2 + b" is an equation
- "x^2 + b" is an expression and is related to the expression "y" by equality.
- "x^2" is a term in the RHS expression
- "x = {5, 3, 1}" is an equation that provides scope for the primary equation.

What is the "where" relation in the statement? The "where" is a binary operator that relates two equations. There are other "statement operators" to relate equations, like "for all"; see the statement

a + c = 2*g + k for all g \in \Re

In that statement, "g \in \Re" is (an equation?) serving as a scope for the primary equation.

All statements have supplemental scope/definition equations that are usually left as implicit. The reader is expected to deduce the scope of the statement from the surrounding context.

The supplemental scope/definition equations describe both per-variable and inter-variable constraints. For example,

x*y + 3 = 94 where ((x \in \Re) AND (y \in \Re) AND (x<y))

More complicated statement:

f(x) = { 0 for x<0

{ 1 for 0<=x<=1

{ 0 for x>1

Here the LHS is a function and the RHS is an integer, but the value of the integer depends on x.

Note that the "0<=x<=1" can be separated into "0<=x AND x<=1". Expanding this even more,

(f(x) = 0 for x<0) AND (f(x) = 1 for (0<=x AND x<=1)) AND (f(x) = 0 for x>1)

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