Is there any support in mathcomp/ssreflect for classical logic

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I don't think there is anything along those lines in the basic mathcomp libraries. The closest might be the classically operator in ssrbool, which mimics classical reasoning in intuitionistic logic. However, the experimental mathcomp analysis library assumes classical axioms and has results like the ones you mentioned (https://github.com/math-comp/analysis/blob/master/theories/boolp.v).

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