Learn Lean 4 in 40 Minutes — A Visual Crash Course

What does it actually look like to *write a mathematical proof in Lean 4*? Want the prompts, Mathify projects, source code, scripts, and production workflow behind this video? Get permanent access to the Mathify Long-Form Source Vault: https://mathify.dev/digital-products/... It includes the production files behind Mathify’s long-form videos, all organized in a private GitHub repo that I keep expanding as new long-form videos are published. One-time $99 purchase. In this animated crash course, you'll learn Lean by watching real code, proof states, tactics, equations, and mathematical ideas transform on screen step by step. We start with the basic language of Lean — types, terms, functions, and `#check` — then build toward propositions as types, interactive theorem proving, tactics, rewriting, Mathlib, induction, and the trusted kernel that ultimately checks every proof. You'll see actual Lean code throughout the course, including: • `def`, `example`, and `theorem` • `intro`, `exact`, and `apply` • Lean's goal state and local context • `constructor`, existential witnesses, and logical proofs • Equality with `rfl` and `rw` • Simplification with `simp` • Algebra with `ring` • Numerical proofs with `norm_num` • Inequalities with `linarith` • Mathematical induction • Structures, type classes, and Mathlib • How tactics become proof terms that Lean's kernel can check But this isn't just a syntax tutorial. The animations show **what each line of code actually does**. When `intro` runs, you'll see an assumption move into the local context. When `apply` runs, you'll see Lean work backward and create the premises that still need to be proved. When `rw` runs, you'll watch the exact mathematical expression being rewritten. When a tactic closes a goal, you'll see why the proof step was valid. By the end, you should understand the central mental model behind Lean: A theorem is a type. A proof is a term of that type. A goal tells you what proof is still missing. Tactics help construct it. The kernel checks the result. This course uses modern Lean 4 and Mathlib. CHAPTERS 0:00 What Does Lean Actually Do? 1:57 Everything Has a Type 4:20 Functions Are Values Too 6:27 Propositions Are Types 9:16 What the Lean Kernel Actually Checks 11:35 Understanding the Proof State 13:36 intro, exact, and apply 16:23 Logic Becomes Construction 19:47 Equality: rfl, rw, and calc 22:40 How simp Works 24:52 Mathlib, ring, norm_num, and linarith 28:26 Inductive Types 31:35 Induction and Recursion 34:56 Structures and Type Classes 37:45 From Mathematics to a Checked Theorem Lean: https://lean-lang.org/ Lean Language Reference: https://lean-lang.org/doc/reference/l... Theorem Proving in Lean 4: https://lean-lang.org/theorem_proving... Mathematics in Lean: https://leanprover-community.github.i... Mathify: https://mathify.dev/ Want the prompts, Mathify projects, source code, scripts, and production workflow behind Mathify’s long-form videos? Mathify Long-Form Source Vault: https://mathify.dev/digital-products/... #Lean4 #Lean #TheoremProving #Mathematics #FormalVerification #Mathlib #Programming #ComputerScience