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I said everything in Lean has a type. What is the type of a proof of a formula X : Prop? Let’s call the proof h.

Well, its type is X itself. In addition to being a term in the (big) type of Prop, X is itself a type whose terms are proofs of X.

We can of course declare we have a proof (without providing one) by saying

variable (A : Prop) 
variable (h : A)

But far more important for us is producing proofs!

Before we look at some basic examples, let’s introduce two new keywords for Lean: example and theorem. Both of these tell Lean that we want to produce a proof for a particular proposition. Here are some examples:

variable (A : Prop) 
example : A := sorry 
theorem bigOne : A := sorry

The main difference between example and theorem is that theorem expects a name whereas example does not.

Each of these is telling Lean that I am going to provide a proof of A. This is why we end the statement with : A. It is informing Lean the type of the coming proof.

The proof goes after :=. sorry is a special command that decreases the volume of the complaints from Lean that we did not actually provide a proof. If we remove sorry, we notice that the message is in red now: unexpected end of input.

Messages like these from the infoview pane are meant to be helpful in constructing our proofs but they can be cryptic. A good rule is to step back a little and meditate on the error message in the context of the question.

In general, we cannot produce a proof out of nothing except for situations. Even so, they are instructive to investigate.

Suppose I want to establish $A \vdash A$. This is trivial in propositional logic: I have $A$ so I have $A$. How does this look in Lean?

For examples/theorems, we can put the assumptions on the left-hand side of the semi-colon.

example (h : A) : A := sorry 

This reads as: “assume I have a proof h of A and I want to prove A”. A completed proof is:

example (h : A) : A := h 

You are providing the assumed proof of A as the desired proof of A.

A formula we can always prove is $\top$. In Lean, this proposition is called True and $\bot$ goes by False.

The introduction rule for True comes in the form of a canonical proof called true of type True.

Implication elimination and introduction

Our rules of inference allowed us to build more complicated proofs from some basic steps. Each rule is encoded in Lean. Let’s start by looking at the introduction and elimination rules for implication.

Say we have A B : Prop and we want to prove B from A → B and A. This is done as

variable (A B : Prop) 
example (f : A  B) (h : A) : B := f h

What is going on here? h is a proof of A as before and f is a proof of A → B. What is a proof of A → B? Well, one way to understand it is as a way to turn a proof of A into a proof B. We generally call such things functions.

By placing f h, we are saying “feed h into f and use the output”. A more common way to say it is that we applied f to h.

Application of f of h is how $\to$-elimination is implemented in Lean.

Next we look at $\to$-introduction. For example, how would we finish the following?

variable (A : Prop)
example : A  A := sorry

This difference between this example and

example (h : A) : A := h 

is the exactly the elimination rule for implication.

We want to make a function f : A → A which mimics our intitution that given a proof h : A that we can output that given proof to get a proof of A.

The syntax for doing so is seen below in the completed example.

variable (A : Prop)
example : A  A := fun (h:A) => h

fun tells Lean that a function is coming (h : A) specifies the input (and its type). The arrow => separates the input from the output.

Strictly speaking this is not the implication introduction rule but it plays an important part.

variable {A B : Prop} 
theorem superProof (h : A) : B := sorry 
example : A  B := fun (h : A) => superProof h

We can treat the (nonexistent) proof of our theorem as a function.

But you may have noticed the braces { A B : Prop }. When we declare (A : Prop), Lean adds it as an assumption to all examples/theorems in our file. So really superProof has the form

theorem {A B : Prop} superProof (h : A) : B := sorry 

Using the parentheses (A B : Prop) tells Lean that I want A and B to also be (explicit) inputs to superProof. (When we saying variable to Lean, it really goes all out.)

Using the braces {A B : Prop} tells Lean “you figure out A and B” from the other information. Which is can do in the case of our example from the desired type is A → B.