Whenever I want to verify some simple construction involving chain complexes of simplicial complexes, I get lost in all the alternating signs and stuff. First verifying that the boundary operator really satisfies $\partial^2 = 0$ requires some work. Then comes the somewhat confusing definition of chain map, given a simplicial map, which I think looks like: If $f:X\to Y$ is a simplicial map, and orientations are chosen for $X$ and $Y$, then: $$ f_* \{x_1,…,x_n\} = \begin{cases} \varepsilon(\sigma)\{f(x_1),…,f(x_n)\} & \text{if $\{f(x_1),…,f(x_n),\}$ has cardinality $n$ and $\sigma$ is the permutation making the orientations match} \\ 0 & \text{otherwise.} \end{cases} $$
I'm guaranteed to get lost verifying commutativity of the squares here.
Another example: You can define another chain complex for $X$ as $C_n$ being formal sums of $n$-tuples of points forming simplices. Thus you allow degenerate simplices in $C_n$ as well as multiple occurences of a given simplex, but oriented as you please. This makes functoriality easier to verify since if $f:X\to Y$ is a simplicial map, then $f_n : C_n(X) \to C_n(Y)$ sends a tuple $(x_1,…,x_n)$ to $(fx_1,…,fx_n)$. But now verifying that the homology you get is the same as the one for the standard construct again involves getting lost in indices.
It seems that the only way to work it all out is to take some A3 sheet of paper, lay down the equality you want to prove, and rewrite as loong as needed, until you get the desired equality.
Question: How do I deal with that? Is there a nice "incremental" construction to do that?