Most programmers spend years learning how to make software work.
Very few spend time learning how to make software provably correct.
That difference is exactly why systems like compilers, cryptography libraries, kernels, and formal mathematics increasingly rely on proof assistants such as Lean 4.
Lean 4 is not just another programming language. It sits at the intersection of programming, mathematics, theorem proving, and language design. And once you understand the core concepts, you begin to see software differently: not merely as instructions for machines, but as logical structures that can be verified.
So if you're starting with Lean 4, what are the most important things to actually learn?
This guide focuses on the 10 concepts that give you the biggest practical leverage.
1. Defining Functions
Everything in Lean starts with functions.
Unlike Python or JavaScript, Lean encourages you to think very explicitly about inputs, outputs, and types.
A simple function:
def add (a b : Nat) : Nat :=
a + b
Things to notice:
-
Natmeans natural numbers - Function signatures are explicit
- The return type is part of the definition
This may initially feel verbose, but it creates enormous clarity.
Once you get comfortable defining functions, the rest of Lean becomes much easier.
2. Understanding Types Properly
In Lean, types are central to everything.
A type is not just “integer” or “string.”
Types describe what kinds of values are allowed and what operations are valid.
For example:
#check Nat
#check String
#check List Nat
The #check command is one of the most useful beginner tools because it teaches you how Lean “sees” expressions.
One of the biggest mindset shifts in Lean is realizing:
Types are a form of reasoning.
The compiler is constantly checking logical consistency.
3. Pattern Matching
Pattern matching is one of Lean’s most elegant features.
Example:
def isZero : Nat → Bool
| 0 => true
| _ => false
Instead of writing conditional chains, you directly describe the structure of the input.
Pattern matching becomes especially powerful with:
- recursive data structures
- inductive types
- proofs
This style is heavily used in functional programming languages like Haskell and OCaml.
4. Recursion
Lean prefers recursion over loops.
Example:
def factorial : Nat → Nat
| 0 => 1
| n + 1 => (n + 1) * factorial n
This is important because Lean cares deeply about termination and correctness.
Unlike many languages, Lean wants proof that your recursive functions eventually stop.
That may sound restrictive, but it prevents entire classes of bugs.
5. Working with Theorems
This is where Lean becomes fundamentally different from ordinary languages.
You can state mathematical truths directly:
theorem add_zero (n : Nat) : n + 0 = n := by
rfl
You are not merely “testing” correctness.
You are proving it.
This changes how you think about software reliability.
6. Learning Tactics
Tactics are commands that help construct proofs step by step.
Example:
theorem add_comm (a b : Nat) : a + b = b + a := by
induction a with
| zero =>
simp
| succ a ih =>
simp [ih]
Important beginner tactics include:
simprwexactapplycasesinduction
At first tactics can feel mysterious, but over time they become an interactive problem-solving workflow.
7. Inductive Types
Inductive types are one of the deepest ideas in Lean.
They allow you to define structures from basic building blocks.
Example:
inductive Color where
| red
| green
| blue
Or more complex recursive structures:
inductive Tree where
| leaf
| node (left right : Tree)
This concept underlies:
- algebraic data types
- recursive reasoning
- proofs by induction
Understanding inductive types unlocks much of Lean’s power.
8. Using simp Effectively
Beginners often underestimate simp.
It is one of the most productive tools in Lean.
Example:
example (n : Nat) : n + 0 = n := by
simp
simp automatically simplifies goals using known rules.
As your projects grow larger, mastering simplification becomes essential for keeping proofs manageable.
A surprising amount of Lean expertise is actually expertise in controlling simplification.
9. Reading Error Messages
Lean’s error messages are often dense, but learning to interpret them is a major skill.
Many beginners quit because they treat errors as obstacles instead of guidance.
Usually Lean is telling you one of three things:
- a type mismatch exists
- an argument is missing
- a proof step is invalid
Over time you develop an intuition for debugging proofs the same way programmers debug software.
This is one of the most transferable skills Lean teaches.
10. Understanding the Curry–Howard Correspondence
This sounds intimidating, but it is the core philosophical insight behind Lean.
The idea:
Programs are proofs.
Types are logical propositions.
That means writing a correct program and constructing a proof are deeply related activities.
This is why Lean feels different from ordinary programming languages.
You are not just executing instructions.
You are building logically verified structures.
Once this idea “clicks,” Lean becomes far more intuitive.
Why Lean 4 Matters
Lean 4 represents something larger than theorem proving.
It points toward a future where:
- software can be formally verified
- mathematics becomes machine-checkable
- programming and reasoning merge together
That future is still early, but it is becoming increasingly important in:
- compilers
- security systems
- cryptography
- theorem proving
- AI verification
- formal mathematics
And importantly: Lean 4 is practical enough that ordinary developers can begin exploring these ideas today.
Final Thoughts
The mistake many beginners make is trying to learn Lean 4 like another programming language.
It is better understood as:
- a programming language
- a proof system
- a logical framework
- and a thinking tool
all at once.
The learning curve is real, but the intellectual payoff is unusually high.
Once you begin understanding proofs as executable structures, many parts of programming start looking different forever.
What was the first Lean 4 concept that genuinely changed how you think about programming or mathematics?
