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paperproof

Lean theorem proving interface which feels like pen-and-paper proofs.

Paperproof

A new proof interface for Lean 4.

Paperproof vscode
Try Paperproof online

Paperproof will inspect how the hypotheses and goals were changing throughout the Lean 4 proof, and display this history - making it equivalent to how we think of a mathematical proof on paper.

In the following tables, you can see what tactics such as apply, rw, or cases look like in Paperproof; and how Paperproof renders real proofs from well-known repos.

Common tactics
Lean Paperproof

apply

 
theorem apply (a b: ℝ) : a = b := by
  apply le_antisymm
image

have

theorem have (a b: ℝ)
(h1: a ≤ b) (h2: b ≤ a) : True := by
  have hi := le_antisymm h1 h2
image

intro

theorem intro
: ∀ (N: ℕ), ∃ M, N + N = M := by
  intro n
image

rw

theorem rw (a b: ℕ)
(h1: a = b) : (10 * a = 666) := by
  rw [h1]
image

by_contra

theorem by_contra (m: ℕ)
: 2 ≤ m := by
  by_contra h
image

use

theorem use
: ∃ x: ℕ, x = 5 := by
  use 42
image

induction

theorem induction (n: ℕ)
: Nat.mul 0 n = 0 := by
  induction' n with k ih
image

cases

theorem casesN (n: ℕ)
: Nat.mul 0 n = 0 := by
  cases' n with m
image 
theorem casesAnd (A B C: Prop)
(h: A ∧ B) : C := by
  cases' h with a b
image 
theorem casesOr (A B C: Prop)
(h: A ∨ B) : C := by
  cases' h with a b
image 
inductive Random where
  | hi: ℕ → String → Random
  | hello: (2 + 2 = 4) → Random
  | wow: Random
theorem casesRandom (C: Prop)
(h: Random) : C := by
  cases' h with a b c
image
Full-fledged proofs

Mathematics in Lean (Jeremy Avigad, Patrick Massot)
(mathematics_in_lean/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean:155)
Mathematics in Lean - Paperproof

Mathlib
(mathlib4/Mathlib/Algebra/Field/Power.lean:30)
Mathlib - Paperproof

Hitchhiker's Guide to Logical Verification
(Anne Baanen, Alexander Bentkamp, Jasmin Blanchette, Johannes Hölzl, Jannis Limperg)
(logical_verification_2023/blob/main/lean/LoVe/LoVe05_FunctionalProgramming_Demo.lean:316)
Hitchhiker's Guide to Logical Verification - Paperproof

Installation

  1. Install the "Paperproof" vscode extension (link).

  2. In your lakefile.lean, write:

    -- if you are on a Lean version >= 4.12.0:
require Paperproof from git "https://github.com/Paper-Proof/paperproof.git"@"main"/"lean"
-- if you are on a Lean version <= 4.8.0:
require Paperproof from git "https://github.com/Paper-Proof/paperproof.git"@"21e4ec999c23fc6181d6528cbaecb600a2db6851"/"lean"

    Note: if you are on a Lean version in-between these two versions, one of these lines should work.

  3. Then, in your terminal, run:

    lake update Paperproof

    Note: if you're getting "error: unexpected arguments: Paperproof", it means you're on the older version of Lean, and it doesn't support per-package updates. In that case, just run lake build.

  4. In a Lean file with your theorems, write:

    import Paperproof

  5. You're done!

    Now, click on the paperproof icon (after you installed the Paperproof extension, it should appear in all .lean files), this will open a Paperproof panel within vscode.

    You can click on any theorem now (well, only tactic-based proofs, those starting with by, are supported now) - you should see your proof tree rendered.

[!TIP] If you get a build error when you try to build the paperproof package, please copypaste the error message and create the corresponding issue in Paperproof. Lean updates its metaprogramming api frequently - we try to go along with it, and it's helpful when incompatibilities are discovered on time.
The quickfix for any Paperproof build error is to use the Lean version mentioned in Paperproof's lean-toolchain.

Tutorial

If you worked with formal proofs before, you might find Paperproof most similar to proof trees/Gentzen trees. The resemblance is not spurious, we can easily mimic Semantic Tableaux and Natural Deduction trees with Paperproof. All of these interfaces show "the history of a proof" - the way hypotheses and nodes were changing throughout the proof.

Unlike Gentzen, we can make use of css and javascript - so there are many visual syntax sugars on top of what would be a formal proof tree:

  • hypotheses aren't repeated when used multiple times,
  • goals and hypotheses are visually differentiated,
  • variable scopes are shown as darkening backgrounds,
  • available hypotheses are indicated via node transparencies,
  • and so on.

Below, you will see a table with the main features of Paperproof.

Paperproof walkthrough
Lean Paperproof
Hypotheses are displayed as green nodes,
goals are displayed as red nodes,
tactics are displayed as transparent nodes with dashed borders.
image image
A proof should be read "towards the middle".
So, hypotheses should be read from top to bottom; and goals should be read bottom up.
image image
To zoom in on a particular dark box, you can click on it.
Hint: these boxes represent variable scopes. Don't overthink this however, we'll always highlight the available hypotheses as you're writing the proof, consider these boxes a visual hint that will eventually become second nature.
Screenshot 2024-08-26 at 13 58 30
Nodes becomes transparent when they are not in scope.
So, an opaque red node represents a currently focused goal, and opaque green nodes represent currently available hypotheses.
image image

Updating

To update Paperproof, you only need to rerun lake update Paperproof. This will fetch the newest version of the Paperproof Lean library from this github repo, and build it.

Vscode extensions are automatically updated, however you can check for new updates with
cmd+shift+p => "Extensions: Show Extension Updates".

Paperproof is a package that's usually only used during development, so you might want to remove it from your lakefile.lean when you're pushing to production. In order to do that, just remove the Paperproof require from lakefile.lean, and run lake update Paperproof. This will clean up lake-manifest.json and lake-packages for you.

About Paperproof

Theoretical:

  • Paperproof ♥️ Semantic Tableaux: link
  • Paperproof ♥️ Natural Deduction: link
  • Paperproof ⚔️ Lean's #explode: link
  • Lean ⚔️ Coq ⚔️ Metamath ⚔️ Isabel proof trees: link

Practical:

  • How to add my Lean repo to Github Codespaces: link
  • Can Paperproof render proof terms: link
  • How to parse the InfoTree: link

Videos:

Development

You're welcome to contribute to Paperproof, see the instructions in CONTRIBUTING.md.