Sympy: working with equalities manually

Question

I am currently doing a math course where my goal is to understand concepts and process as quickly as possible, rather than crunching through a set of tasks. When solving equations, I would like to be sure that I could deflate them myself, and not allow for me.

Say we have a very simple equation z + 1 = 4 - if I solved it myself, I would obviously subtract 1 from both sides, but I can’t figure out if sympy provides an easy way to do this. At the moment, the best the solution I can come up with is:

 from sympy import * z = symbols('z') eq1 = Eq(z + 1, 4) Eq(eq1.lhs - 1, eq1.rhs - 1) # Output: # z == 3 

If the more obvious expression eq1 - 1 is subtracted only on the left side. How can I use sympy to work through equality in stages, like this (i.e., not getting the solve() method, just to give me an answer)? Any pointers to manipulations that are really possible with pretty equalities would be appreciated.

Answer 1

The https://github.com/sympy/sympy/issues/5031#issuecomment-36996878 page has a “do” method and discussion that will allow you to “perform” operations on both sides of the equality. It was not accepted as an add-on to SymPy, but it is a simple add-on that you can use. It is pasted here for convenience:

 def do(self, e, i=None, doit=False): """Return a new Eq using function given or a model model expression in which a variable represents each side of the expression. Examples ======== >>> from sympy import Eq >>> from sympy.abc import i, x, y, z >>> eq = Eq(x, y) When the argument passed is an expression with one free symbol that symbol is used to indicate a "side" in the Eq and an Eq will be returned with the sides from self replaced in that expression. For example, to add 2 to both sides: >>> eq.do(i + 2) Eq(x + 2, y + 2) To add x to both sides: >>> eq.do(i + x) Eq(2*x, x + y) In the preceding it was actually ambiguous whether x or i was to be added but the rule is that any symbol that are already in the expression are not to be interpreted as the dummy variable. If we try to add z to each side, however, an error is raised because now it is unclear whether i or z is being added: >>> eq.do(i + z) Traceback (most recent call last): ... ValueError: not sure what symbol is being used to represent a side The ambiguity must be resolved by indicating with another parameter which is the dummy variable representing a side: >>> eq.do(i + z, i) Eq(x + z, y + z) Alternatively, if only one Dummy symbol appears in the expression then it will be automatically used to represent a side of the Eq. >>> eq.do(2*Dummy() + z) Eq(2*x + z, 2*y + z) It is also possible to do Eq/Eq operations: >>> eq.do(i + Eq(x, 2)) Eq(2*x, y + 2) Operations like differentiation must be passed as a lambda: >>> Eq(x, y).do(lambda i: i.diff(x)) Eq(1, 0) Because doit=False by default, the result is not evaluated. to evaluate it, either use the doit method or pass doit=True. >>> _.doit == Eq(x, y).do(lambda i: i.diff(x), doit=True) True """ if not isinstance(e, (FunctionClass, Lambda, type(lambda:1))): e = S(e) if len(e.args) == 2: a, b = e.args if isinstance(a, Equality) or isinstance(b, Equality): if isinstance(b, Symbol): return self.func(e.func(a.lhs, self.lhs), e.func(a.rhs, self.rhs), evaluate=doit) if isinstance(a, Symbol): return self.func(e.func(self.lhs, b.lhs), e.func(self.rhs, b.rhs), evaluate=doit) raise ValueError('expecting 1 arg to be a symbol') imaybe = e.free_symbols - self.free_symbols if not imaybe: raise ValueError('expecting a symbol') if imaybe and i and i not in imaybe: raise ValueError('indicated i not in given expression') if len(imaybe) != 1 and not i: d = [i for i in i if isinstance(i, Dummy)] if len(d) != 1: raise ValueError( 'not sure what symbol is being used to represent a side') i = set(d) i = i.pop() f = lambda side: e.subs(i, side) else: f = e return self.func(*[f(side) for side in self.args], evaluate=doit) from sympy.core.relational import Equality Equality.do = do