Spotting LaTeX math common mistakes

I am looking for a program that can read a LaTeX file and give a warning (+ ideally a fix suggestion) for each common mistake it spots in math formulas. If possible, free and works with Windows 7 SP1 x64 Ultimate.

Examples of common mistake in LaTeX math mode:

1) Differentials: $\int_0^1 f(x) dx$ instead of $\int_0^1 f(x) \, dx$ :

2) Multi-letter function names: $log e^x = x$ instead of $\log e^x = x$:

3) Wrong kinds of dots: "$x,...,y$" instead of $x, \ldots, y$.

It would be even greater if it could give warnings for all the LaTeX writing, not just the math mode, e.g. one common mistake is writing "Yes." instead of Yes.'':

There are quite a few but the one I would recommend is ChkTeX.

From the web page:

• Supports over 40 warnings. Warnings include:
• Commands terminated with space. Ignores \tt\'', etc.
• Space in front of references instead of ~''.
• Forgetting to group parenthesis characters when sub-/superscripting.
• Italic correction (/'') mistakes (double, missing, unnecessary).
• Parenthesis and environment matching.
• Ellipsis detection; also checks whether to use \dots'', \cdots'' or \ldots''.
• Enforcement of normal space after abbreviation. Detects most abbreviations automagically.
• Enforcement of end-of-sentence space when the last sentence ended with capital letter.
• Math-mode on/off detection.
• Quote checking, both wrong types ("'') and wrong direction.
• Recommends splitting three quotes in a row.
• Searching for user patterns.
• Space in front of \label'' and similar commands.
• Use of x'' instead of $\times$'' between numbers.