getting-started.md

Getting started

This document assumes you already installed Lean. See https://lean-lang.org/lean4/doc/quickstart.html if you need help doing that.

In order to use Verbose Lean in your teaching, you need to create a Lean project. You can do that from the VSCode Lean menu or type in a terminal lake new teaching lib to create a teaching folder with a Lean project layout (do not use any fancy character in the name of your folder).

You need to make sure your project uses the same lean version as Verbose Lean. For this you can copy the content of the Verbose Lean lean-toolchain file to the lean-toolchain file of your project (this file is automatically created by lake init in the root folder of your project).

Then you need to require the library in your lake file. This means adding at the end of lakefile.toml in your project:

[[require]]
name = "verbose"
git = "https://github.com/PatrickMassot/verbose-lean4.git"
rev = "master"

Important note in case you had an existing project: Verbose Lean will require Mathlib for you, so your lakefile.lean should not contain a require mathlib line, otherwise you could be creating version conflicts.

After adding the above require, you need to run lake update verbose from your project folder. This will update your lake-manifest.json file and download Verbose Lean with all its dependencies. This is large download since it includes a compiled Mathlib.

Assuming you use the teaching name, your teaching folder now contains a Teaching folder where all your Lean files go. It also contains a Teaching.lean file which should import all files from the Teaching folder that you want lake build to build (this typically means all files except for throw away test files). Beware that Lean file whose name start with a number only create pain.

You now need to create at least one teacher file that imports Verbose, configures it and contains definitions you want to work with. Then you can create student files with explanations and exercises.

For instance you can create in the Teaching folder a file Math101.lean containing:

import Mathlib.Topology.Instances.Real.Lemmas
import Verbose.English.All

open Verbose English

-- Let’s define mathematical notions here

def continuous_function_at (f : ℝ → ℝ) (x₀ : ℝ) :=
∀ ε > 0, ∃ δ > 0, ∀ x, |x - x₀| ≤ δ → |f x - f x₀| ≤ ε

def sequence_tendsto (u : ℕ → ℝ) (l : ℝ) :=
∀ ε > 0, ∃ N, ∀ n ≥ N, |u n - l| ≤ ε

-- and some nice notation

notation3:50 f:80 " is continuous at " x₀ => continuous_function_at f x₀
notation3:50 u:80 " converges to " l => sequence_tendsto u l

-- Now configure Verbose Lean 
-- (those configuration commands are explained elsewhere)

configureUnfoldableDefs continuous_function_at sequence_tendsto 

configureAnonymousFactSplittingLemmas le_le_of_abs_le le_le_of_max_le

configureAnonymousGoalSplittingLemmas LogicIntros AbsIntros 

useDefaultDataProviders

useDefaultSuggestionProviders

and a HomeWork1.lean file containing:

import Teaching.Math101

Example "Continuity implies sequential continuity"
  Given: (f : ℝ → ℝ) (u : ℕ → ℝ) (x₀ : ℝ)
  Assume: (hu : u converges to x₀) (hf : f is continuous at x₀)
  Conclusion: (f ∘ u) converges to f x₀
Proof:
  Let's prove that ∀ ε > 0, ∃ N, ∀ n ≥ N, |f (u n) - f x₀| ≤ ε
  Fix ε > 0
  Since f is continuous at x₀ and ε > 0 we get δ such that
    (δ_pos : δ > 0) and (Hf : ∀ x, |x - x₀| ≤ δ ⇒ |f x - f x₀| ≤ ε)
  Since u converges to x₀ and δ > 0 we get N such that Hu : ∀ n ≥ N, |u n - x₀| ≤ δ
  Let's prove that N works : ∀ n ≥ N, |f (u n) - f x₀| ≤ ε
  Fix n ≥ N
  Since ∀ n ≥ N, |u n - x₀| ≤ δ and n ≥ N we get h : |u n - x₀| ≤ δ
  Since ∀ x, |x - x₀| ≤ δ → |f x - f x₀| ≤ ε and |u n - x₀| ≤ δ we conclude that |f (u n) - f x₀| ≤ ε 
QED

Assuming the installation process succeeded, Lean should be happy to process those files. You can then learn more about this library.