When do proof by contradiction, or by negation, or direct proof, or proof by contrapositive all work equally well? When can any one be chosen as the proof form?
I'm interested in a pragmatic answer and not a relatively theoretical one.
As mooted here, I always want to supplant any proof that's not a direct proof or proof by either such, thus I must firstly be enlightened that this switch is admissible.
Nevertheless, if a reference fails to advise/divulge that any one can be superseded,
how would a student divine/previse of this freedom of choice/interchange, before trying to prove?
In the following instances, I didn't know if direct proofs or by contrapositive had existed. If I hadn't Googled, I would've been benighted about easier proofs, a tribulation which I want to prevent.
In Linear Algebra 3rd ed, David Poole proves by contradiction P308 Theorem 4.20 (cp Theorem 1 in this PDF) and P316 Theorem 4.2.4, but doesn't impart the existence and validity of direct proofs.
I reference related questions: When to use the contrapositive to prove a statment, Proof by contradiction vs Prove the contrapositive.