example interaction with ChatGPT 3.5 to produce a Lean proof

In a new conversation I provided the following prompt:

prompt

Answer the question based on the context below. Let's work this out in a step by step way to be sure we have the right answer.

Context: Lean is a proof assistant and programming language. Lean is based on the calculus of constructions with inductive types.

Context: There are multiple versions of Lean. This question is about Lean 4

`a=b` is a proposition. The variable `a` is Real valued and `b` is Real valued. Prove in version 4 of Lean that `(a=b) -> (a+2=b+2)`

ChatGTP 3.5 wrote in response

ChatGPT 3.5 response

Certainly! To prove the statement `(a = b) -> (a + 2 = b + 2)` in version 4 of Lean, you can follow these steps:

1. **Introduction of Variables**: Begin by introducing the variables and assumptions using the `variables` keyword. In this case, you have the variables `a` and `b` both being Real valued.

```lean
variables {a b : ℝ}

```

2. **Assumption**: State the assumption that `(a = b)`.

```lean
example (h : a = b) : a + 2 = b + 2 :=

```

3. **Transformation**: Use the assumption to transform the left-hand side of the conclusion `(a + 2)` to `(b + 2)` using the property of equality.

```lean
calc
   a + 2 = b + 2 : by rw h

```

Here, `calc` is used to chain together a sequence of expressions, and `rw h` applies the assumption `h`.

4. **QED (Quod Erat Demonstrandum)**: Conclude the proof.

```lean
   example (h : a = b) : a + 2 = b + 2 :=
   begin
     calc
       a + 2 = b + 2 : by rw h
   end

```

This step-by-step approach uses the Lean proof style to demonstrate the validity of the given proposition. The `calc` tactic is particularly useful for chaining together equalities.

On my computer I created a file "second_chatGPT_attempt.lean" and wrote

variables {a b : ℝ}

example (h : a = b) : a + 2 = b + 2 :=
begin
  calc
    a + 2 = b + 2 : by rw h
end

and then ran

docker run -it --rm -v `pwd`:/scratch --workdir /scratch lean3onubuntu lean --run second_chatGPT_attempt.lean
second_chatGPT_attempt.lean:2:0: error: unexpected identifier; expected command
second_chatGPT_attempt.lean:5:0: error: unknown identifier 'begin'
second_chatGPT_attempt.lean:6:2: error: unexpected token 'calc'; expected command
second_chatGPT_attempt.lean:8:0: error: invalid 'end', insufficient scopes

I then told ChatGPT

prompt

When I run the script

```lean
variables {a b : ℝ}

example (h : a = b) : a + 2 = b + 2 :=
begin
  calc
    a + 2 = b + 2 : by rw h
end

```

Lean 4 returns an error message: `error: unexpected identifier; expected command`

What should I do to fix the error?

ChatGPT 3.5 response

Apologies for the oversight. It seems I made an error in my previous responses. In Lean, you should use `theorem` or `example` as part of the proof declaration. Here's the corrected version:

```lean
variables {a b : ℝ}

theorem add_two_eq_add_two (h : a = b) : a + 2 = b + 2 :=
begin
  calc
    a + 2 = b + 2 : by rw h
end

```

In this corrected version, I replaced `example` with `theorem`. You can use `example` as well, but `theorem` is more appropriate when proving statements. This should resolve the error you encountered.

I then created a second Lean script, third_chatGPT_attempt.lean, and wrote

variables {a b : ℝ}

theorem add_two_eq_add_two (h : a = b) : a + 2 = b + 2 :=
begin
  calc
    a + 2 = b + 2 : by rw h
end

which got the same error as before when I attempted to run the script using Lean 4:

docker run -it --rm -v `pwd`:/scratch --workdir /scratch lean3onubuntu lean --run third_chatGPT_attempt.lean 
third_chatGPT_attempt.lean:1:0: error: unexpected identifier; expected command
third_chatGPT_attempt.lean:4:0: error: unknown identifier 'begin'
third_chatGPT_attempt.lean:5:2: error: unexpected token 'calc'; expected command
third_chatGPT_attempt.lean:7:0: error: invalid 'end', insufficient scopes