Let's say I want to formally prove a statement of the form $$p \implies q$$ So I do a bit of work,some re-arranging and eventually I arrive at a statement of the form $$p \implies p$$ which is a tautology, as this is true for all values of $p$, but also an obvious truth. Have I proved $p \implies q$?
Reaching "Obvious Truths"
Generally when doing proofs, reaching something that is obviously true" is not a proof. Like for example if you're proving that $\sqrt{2}$ is irrational, reaching $1 = 1$ has not proved anything. But in this case I've reached a tautology, does that make my original implication true?
To me this seems like a contradiction, because the tautology I've reached is an obvious truth, which is a big "no-no" in proof validation.
Furthermore does the convention that reaching something an "obvious truth" is not a mathematically rigorous way of completing a proof, extend to all proof techniques?
For example does reaching an "obvious truth" still not constitute a proof for the following proof techniques :
- Direct Proofs
- Indirect Proofs (Proofs by Contrapositive)
- Proof by Contradiction
- Proof by Mathematical Induction
- Vacuous Proofs
- Trivial Proofs
- Proof by Cases
Last question, in the example I've given above seems I've started off with trying to Directly prove $p \implies q$, but when I've reached $p \implies p$, it's now a Trivial Proof, which would the above proof example I've given fall under? Direct Proof or Trivial Proof?
Some background context if needed :
I've asked this question in response to a previous questions I've asked, on the $\epsilon - \delta$ definition of a limit, where I try to prove a statement of the form $p \implies q$ by reducing it to the form $p \implies p$, and I argue that it is not mathematically rigorous, as I've reached an obvious truth, however another user argues that I have formally proven what I set out to prove as I reached a tautology, by manipulating the original implication : https://math.stackexchange.com/a/1745657/266135
Another extremely relevant question on reaching "obvious truths" : Is this direct proof of an inequality wrong?