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A discussion on ELU stackexchange has led to the question of whether there is a name for the style of proof in which you start with the proposition to be proven and then proceed via a chain of logically equivalent paraphrases to—in the example below—a definitional criterion (though it could equally be to something obviously true or previously proven).

For instance, you can prove that, if $f$ and $g$ are isomorphisms (with the domain of $g$ subsuming the range of $f$), then $g ∘ f$ too is an isomorphism as follows:

$g ∘ f(x) = g ∘ f(y)$

$⇔ g(f(x)) = g(f(y))$

$⇔ f(x) = f(y)$ — given that $g$ an isomorphism

$⇔ x = y$ — given that $f$ is an isomorphism

This is such a basic method of proof that I can easily imagine that it doesn’t have a name. But does it?

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Daniel Harbour
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It's usually called something like "proof by inspection" or "proof by direct calculation".

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  • Yes, I would call that a "direct proof" – MPW Mar 01 '14 at 15:29
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    My memory of proof by inspection was more along the lines of The solutions to this equation are 1 and 3 and you'd substitute those values and check that the equation equated. But I followed up on direct proof (on wikipedia) and that seems spot on. – Daniel Harbour Mar 01 '14 at 15:45