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I noticed that not a lot of authors of math books don't use the $\leadsto$ to denote the next step. As far as I understand it, that is supposed to mean "the next step of the proof is". Instead, I notice that they use $=$ to mean the same thing, or also $\implies$ or $\iff$. Which symbol is the most appropriate for stating the next step? Can we use $\implies$ and $\leadsto$ interchangeably? Is this accepted in formal proofs?

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  • $\begingroup$ @kimchilover partially, yes, but it's not clear if we can interchangeably use $\leadsto$ with $\implies$...it seems to suggest that $\leadsto$ can be used, but also $\implies$ provided that it is used with a condition and somewhere else in the proof that the condition exists. I'd still like to know what would be much clearer to a reader who reads proofs $\endgroup$ – Paco G Mar 5 '20 at 0:03
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Mathematics is about understanding, not loads of symbols. Use words. Most often just placing one equation after the other, perhaps with a few well-placed comments interspersed, is much easier to read.

Don't introduce new notation if you can help it. Yes, it might sometimes be worthwhile to introduce new notation, for a rather complex concept that repeats often enough (e.g. Landau's $O()$, later extended by Knuth to $\Omega()$, $\Theta()$, and so on). "Next step is..." just doesn't cut it.

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