Search

(from Software Foundations)

Coq’s Search command is quite helpful with this. Let’s say you’ve forgotten the name of a theorem about rev. The command Search rev will cause Coq to display a list of all theorems involving rev.

Search rev.

Or say you’ve forgotten the name of the theorem showing that plus is commutative. You can use a pattern to search for all theorems involving the equality of two additions.

Search (_ + _ = _ + _).

You’ll see a lot of results there, nearly all of them from the standard library. To restrict the results, you can search inside a particular module:

Search (_ + _ = _ + _) inside Induction.

You can also make the search more precise by using variables in the search pattern instead of wildcards:

Search (?x + ?y = ?y + ?x).

The question mark in front of the variable is needed to indicate that it is a variable in the search pattern, rather than a variable that is expected to be in scope currently.

Search only lemmas(works for Theorem, Example, Definition, etc.):

Search is:Lemma (_ + _ = _ + _)

Check x

Displays the type of term. When called in proof mode, the term is checked in the local context of the selected goal (possibly by using single numbered goal selectors).

Use @def to force implicit arguments of def be passed explicitly. For example,

Definition injective {A B} (f : A -> B) :=
  forall x y : A, f x = f y → x = y.

Note that the braces {} indicate that the types of the function, A, B should be inferred. Checking without @ yields:

> Check injective.
injective
: (?A -> ?B) -> Prop
where
?A : [ |- Type]
?B : [ |- Type]

The first argument automatically expects some function with type A -> B with both being inferred.

Let’s now check @injective with applying A = nat :

Check @injective nat.
@injective nat
: forall B : Type, (nat -> B) -> Prop

Now it expects some type B and a function with type nat -> B.

Locate "x"

Search the definition of a notation. For example,

> Locate "=?"
Notation "x =? y" := (eqb x y) : nat_scope (default interpretation)
Notation "x =? y" := (String.eqb x y) : string_scope