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Assuming you are on Lean 3, the best way to answer these sort of questions is via the mathlib tactic documentation (which also documents the built in tactics in core Lean). … of common tactics) Similarly, import tactic also works (which I think is the same as import tactic.basic but I could be mistaken). …

Further, to prove that two structures are equal, it is enough to use the congr tactic. (Note from a theorem-proving perspective Arrays are just defined as structures wrapping a list data. … I notice that the Array lemmas are often stated as <expression for an array>.data = ... and I think if you state your lemmas in that format then you would find that simp tactic works much of the time for …

Without the by, then one would be giving a term proof instead of a tactic proof. example : True := by -- tactic mode have h0 : True ∨ False -> True := by -- this is a tactic proof inside a tactic … Still in tactic mode. simp How to know if you are in tactic mode or term mode? If you see a goal in the info view, you are likely in tactic mode. …

The lemma proof are short, often mostly relying on the simp tactic and various rewrite lemmas. … For that library, I heard they developed the tidy tactic to do a lot of the tedious follow-your-nose sort of reasoning. …

So I just had to use the contradiction tactic to solve the subgoal. The second case was a bit more tricky, so I did cases on b to prove the hypotheses were inconsistent. …

(Wrote this before you updated your question.) I'm not sure where that goal came up, either in a real situation or just something you want to prove, but here is my best approximation to your situation …