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I am trying to write some symbolic inequality proofs in Mathematica. What would be a clean way to make the inference that a >= b and b > c implies a > c? Or alternatively, given an inequality b > c, how I could apply a function to the left hand side only performing the transformation f[b] = a? I would be able to code it, but it feels kind of clumsy, I would like to do something simple as 'ApplySides', but for one side of the inequality only. Is there a simple way?

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Reduce[a >= b && b > c, b, Reals]

a > c && c < b <= a

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So far I have come up with this solution:

ApplyLeft[f_, rel_] := ReplacePart[rel, 1 -> f[rel[[1]]]]
ApplyRight[f_, rel_] := ReplacePart[rel, 2 -> f[rel[[2]]]]
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