Along the same lines, does there exist an "algebra checker" that could, say, take in two successive lines of latex, with perhaps a hint of how to get from one to the other, and confirm that there are no algebra errors?
1) It's designed for describe graphical representation and not semantics.
2) parsing it properly is very difficult, especially considering how macros and other definitions can influence the syntax.
1) assume a standard interpretation of symbols (e.g. this site uses a-g for scalars, h-z for vectors and A-Z for matrices) and make those assumptions explicit and modifiable (e.g. WolframAlpha displays "assuming X is a Y, use as a Z instead").
2) Support only the subset that MathJax can handle, which seems to be enough for most purposes.
I definitely agree that LaTeX is not the optimal input format for that purpose, though.
If you want to start with LaTeX then the parser from this library might be useful: https://github.com/Khan/KAS
Methods are highly heuristic, and some languages (all expressions with,say, \pi, and exp function, or something similar) are in general undecidable.
import sympy
from sympy.abc import x, y, alpha, s
quad = s ** 2 - alpha * s - 2
# Let s1 and s2 be the two solutions to the quadratic equation 'quad == 0'
s1, s2 = sympy.solve(quad, s)
u = (x - s2) / (x - s1) * (y - s1) / (y - s2)
f1 = (s2 - s1 * u) / (1 - u)
f2 = (x * y - alpha * x - 2) / (y - x)
# Claim: f1 is equal to f2
print(sympy.simplify(sympy.Eq(f1, f2)))
# Prints "True" >>> import sympy as sp
>>> x = sp.Symbol('x')
>>> sp.simplify(sp.Implies(sp.Eq(x**2 + 2*x + 1, 0), sp.Eq(x, -1)))
Eq(x, -1) | Ne(x**2 + 2*x + 1, 0)
>>> sp.solve(sp.simplify(sp.Implies(sp.Eq(x**2 + 2*x + 1, 0), sp.Eq(x, -1))))
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "/home/user/.local/lib/python3.5/site-packages/sympy/solvers/solvers.py", line 1065, in solve
solution = _solve(f[0], *symbols, **flags)
File "/home/user/.local/lib/python3.5/site-packages/sympy/solvers/solvers.py", line 1401, in _solve
f_num, sol = solve_linear(f, symbols=symbols)
File "/home/user/.local/lib/python3.5/site-packages/sympy/solvers/solvers.py", line 1971, in solve_linear
eq = lhs - rhs
TypeError: unsupported operand type(s) for -: 'Or' and 'int'
>>> sp.solveset(sp.simplify(sp.Implies(sp.Eq(x**2 + 2*x + 1, 0), sp.Eq(x, -1))))
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "/home/user/.local/lib/python3.5/site-packages/sympy/solvers/solveset.py", line 880, in solveset
raise ValueError("%s is not a valid SymPy expression" % (f))
ValueError: Eq(x, -1) | Ne(x**2 + 2*x + 1, 0) is not a valid SymPy expression
From this I learnt the hard way that tokenisation and representation (in LATEX or MathML) do not belong in the same place as a CAS (Computer-assisted Algebra System).
But maybe your example can convince me otherwise. Could you show the specific LaTeX code in question and describe how your program mishandled it?
I’m not a programmer. I erred in trying to do a programmer’s job. Also, I discovered why Nicholas Nassim Taleb has the reputation for being rather uncharitable (but deep down I feel like I deserved it, because hey, I stated a mathematical untruth).
Internally we check it numerically, i.e., generate some random data for the given variables and check the derivative by comparing it to an approximation via finite differences. We will ship this code (hopefully soon) with one of the next versions. You can then check it yourself. Otherwise, as far as I know there does not exist any other tool that computes matrix derivatives so I understand it is hard to convince anyone of the correctness of the results. But I hope the numerical tests will be helpful.
Implies[x^2 + 2 x + 1 == 0, x == -1] // FullSimplify [In] FullSimplify[expr1 == expr2]
[Out] True