hrmacbeth / math2001

Lecture notes for a course on writing proofs, on paper and in the Lean proof assistant

The Mechanics of Proof

This repository contains the Lean code for the book The Mechanics of Proof, by Heather Macbeth, developed for the course Math 2001 at Fordham University.

The Lean files corresponding to each chapter of the book are in the folder Math2001.

You can run the code on the web using Gitpod, by clicking this link:

(You will be prompted to make an account, if you do not yet have one -- the free tier offers 50 hours of use per month.) This is the easiest way to get started if you have never used Lean before. You can accept the default "New Workspace" options and hit the "continue" button, then it will take about five minutes to load up a new workspace and auto-install the Lean mathematical library (wait for all the "cloning", "building", "compiling", "downloading" and "decompressing" steps to finish). The workspace you just created will be accessible later from your Gitpod user page.

You can also install Lean to your own computer, then clone this repository to your computer by typing git clone https://github.com/hrmacbeth/math2001.git at the command line. This will let you run the code on your computer.

math2001's Issues

exercises 1.3.11 (10) and (15) not possible using given tactics?

I am working through the course and have completed all the exercises in 1.3.11 except (10) and (15).

I don't believe these are possible using the two tactics that have been given to the reader at this point in the course (rw and ring).

None of the given examples or hint/tips/tricks seem to apply.

It is entirely possible I am missing the solution, but after 2 days of trying I suspect the solutions require tactics not yet given to the reader.

---

Let's consider (10)

example {p : ℝ} (h1 : 5 * p - 3 = 3 * p + 1) : p = 2 :=
    sorry

No amount of substitution (rw) seems to isolate p.

---

Let's consider (15)

example {x y : ℝ} (h1 : x + 3 = 5) (h2 : 2 * x - y * x = 0) : y = 2 :=
  sorry

The pen and paper solution requires a recognition that (2-y)*x = 0 means either x = 0 or 2-y = 0, or both. But the other hypothesis tells us that x is not 0, and so y must equal 2.

However none of the given tactics, "rw" and "ring" enable this structure of proof at this point in the course.

Am I wrong?

Unable to run locally

$ lake build
error: 'Math2001': no such file or directory (error code: 2)
  file: ./././Math2001.lean

Possible bug in `cancel` tactic?

Given the ways that we see cancel being used in the book, and given some discussions in the Zulip lean community, I think maybe there is a bug in the cancel tactic.

Would the following not have allowed us to arrive at the goal x = y?

example {x y : ℝ} (hx: 0 ≤ x) (hy: 0 ≤ y) (hs: x^2 = y^2) : x = y := by
  cancel 2 at hs

The original context where I ran into this was when trying to do the following:

example {x : ℚ} (h1 : x ^ 2 = 4) (h2 : 1 < x) : x = 2 := by
  have h3 :=
    calc
      x^2 = 4 := by rw [h1]
      _ = 2^2 := by numbers
  cancel 2 at h3

typeclass instance problem is stuck, it is often due to metavariables

Hello

Thank you very much for the awesome book!

When filling out the proof of example 4.4.5, I am getting an error, which I having trouble debugging.
I am trying to apply the theorem abs_lt_of_sq_lt_sq' in order to show
$c < 3 \wedge c > -3$.

The error turns up here:

when I try to use the following theorem:

having these hypotheses:

Using my hypotheses h2 and h3c should allow me to conclude the result of the theorem.
However, it seems as if some of the types are making it throw an error.

Could anyone guide me on how to resolve this?

Thank you very much!

Question: How is Lean automatically graded?

Nice to meet you．I am learning Lean by reading this material. Thank you for creating nice book!

I have a question about Lean's automated grading.

---

I am thinking of using Lean to organise private maths competitions. However, I'm faced with the problem of how to automate grading.

The Note for instructors in this material states that Gradescope is used for automatic grading. Could you tell me how to grade Lean code using Gradescope?

Alternatively, please let me know if you know anything about automatic grading of lean codes, not just Gradescope.

Thank you for reading.

A couple of typos

1. The solution of Example 2.4.2 starts with "We have that p^2 \le 8 = 3^2".
   
2. Problem 3 of Exercises 3.1.10 says "Prove that if the integers m and m are even, then m + n is odd". It should be "if m is odd and n is even ...".
   

Homework 4 problem 2 typo

The problem as stated in the Lean homework file is:

theorem problem2 {t : ℤ} (h : t ≡ 2 [ZMOD 4]) :
    3 * (t ^ 2 + t - 8) ≡ 3 * (2 ^ 2 + 2 - 8) [ZMOD 4] := by

so, 2 substituted for t in all places on the right side.

Homework 7 problem 1 (small) typo

The markup appears to be malformed, minor detail :)