Okay, so your " * " operator means "implication" [1], because the terms "inverse", "contradiction" and "contrapositive" in your mentioned Wikipedia article refe

Okay, so your " * " operator means "implication" [1], because the terms "inverse", "contradiction" and "contrapositive" in your mentioned Wikipedia article refer only to implications. However, the original statement isn't an implication. Also, neither your proposed part "should work" nor the other part "getting proofs correct" are logical statements. Those are just parts of a sentence. In other words, your decomposition doesn't make any sense. [1] That is, "a * b" = "if a then b" = "from a follows b" = "a implies b"