Equations SymPEP

Equations.md

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+SymPEP for behavior of the Equation class
+=========================================
+Fundamentally this is an object with two SymPy expressions connected by an equals sign.
+Potentially this could be extended to a general relational expression, but that is not
+part of this proposal.
+
+The initial motivation for this class is to facilitate manipulating mathematical expressions
+in a manner as similar to what is done on paper as possible. For operations/methods where
+it makes sense they should be applied to both sides of the equation. As much as possible,
+operations on the equations should be initiated using syntax mimicking standard
+mathematical notation.
+
+__Proposed Behaviors__
+
+_Binary Operations (+,-, *, /, %, **)_
+* If combining an equation with a SymPy expression the expression will be applied to both
+sides of the equation as specified by the binary operator.
+```
+>>> a, b, c = Symbol('a b c')
+>>> eq1 = Equation(a, b/c)
+>>> eq1
+a = b/c
+>>> eq1*c
+a*c = b
+```
+* If combining two equations the lhs will combine with the lhs and the rhs with the rhs.
+```
+>>> eq2 = Equation(b, c**2)
+>>> eq2
+b = c**2
+>>> eq1 + eq2
+a + b = b/c + c**2
+```
+_Functions (sin, cos, exp, etc...)_
+* Functions will apply to both sides.
+```
+>>> cos(eq1)
+cos(a) = cos(b/c)
+```
+* Keywords and additional parameters will be passed through to the function.
+
+_Simplification_
+* Behavior should mimic the behavior for expressions and default to applying to both sides.
+
+_Differentiation_
+
+_Integration_
+
+_Solve_
+* Solve should accept equations and lists of equations and return equations or lists of
+equations as solutions. To avoid breaking past behavior this should only occur when solve
+is used on equations or lists of equations.
+
+_Applications of methods_
+* Support `.method()` calls for any method that applies to expressions. Apply the method
+to both sides of the equation.
+* Support `.apply()`, `.applylhs()`, `.applyrhs()`, `.do.`, `.dolhs.`,`.dorhs.`.
+* Methods that can be applied to a SymPy expression as a Python method should also work
+on an Equation:
+```
+>>> collect(a*x + b*x**2 + c*x, x)
+(a + c)*x + b*x**2
+>>> collect(Equation(a*x + b*x**2 + c*x,a*x**2 + b*x + c*x, x)
+(a + c)*x + b*x**2 = a*x**2 + (b + c)*x
+```
+_Miscellaneous_
+* `.as_Boolean()` returns the equation cast as an `Equality`.
+* `.lhs` returns the expression on the left hand side.
+* `.rhs` returns the expression on the right hand side.
+* `.swap` returns the equation with the lhs and rhs swapped.
+* `.check()` shortcut for `.as_Boolean().simplify()`.