What does Proof. simpl. reflexivity. Qed. mean in Coq?

Link:: https://stackoverflow.com/q/64246592

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Question

I'm reading the book Software Foundations and got stuck at the very beginning.

The author defined a boolean type and common operations:

Inductive bool: Type :=
| true
| false.

Definition orb (b1: bool) (b2: bool) : bool :=
  match b1 with
  | true => true
  | false => b2
  end.

So let's say we want to prove the correctness of the or function. The author wrote a test followed by a proof:

Example test_orb1: (orb true false) = true.orb true false = true
Proof.orb true false = true simpl.true = true reflexivity. Qed.

Could someone explain to me what simpl. reflexivity. mean? Is there any other way we can prove this simple test?

Answer

simpl is a tactic evaluating the goal. In your case, after executing it, the goal will be left to true = true. reflexivity is a tactic discharging goals of the shape x = x (in its simplest incarnation). What it does under the hood is to provide the proof term eq_refl : x = x as a solution to the current proof obligation.

Now, there are many ways to achieve this thing that ultimately will produce the same (rather trivial) proof eq_refl (try doing Print test_orb1.). First, the simpl operation is not needed because Coq will do some computations when applying a term (in particular when calling reflexivity). Second, you could obtain the same effect as reflexivity by calling constructor, apply eq_refl or refine eq_refl. These are tactics with different goals but that happen to coincide here.