resources

Lean 4 for Systems and Verification Engineers: A Technical Guide

https://gemini.google.com/app/f5fea363d37c331b

https://docs.google.com/document/d/1wDNs8f_RWc0KP_U8Yp9QRO5SW9cBCkpM8H3hbWd2KPI/edit?tab=t.0

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Lean 4 is a self-hosted, dependently typed functional programming language and interactive theorem prover1. Unlike traditional general-purpose functional languages or pure proof checkers, Lean 4 is designed as a unified system where executable algorithms, logical specifications, and mathematical proofs co-exist and compile to optimized machine code via intermediate C1. The execution runtime utilizes reference counting with a destructive update mechanism called Functional-but-in-Place (FBIP) to run pure functional structures at native speed3. Soundness is guaranteed by a compact trusted kernel1.

Theoretical Foundation and Compilation Architecture

The power of Lean 4 lies in its mathematical rigor and its highly performance-optimized compiler. The compilation and verification pipelines run in parallel to decouple logical soundness from optimization and usability1.

Compilation and Verification Pipelines

Lean 4 processes source code through distinct structural phases, distinguishing between proof checking and executable generation:

1. Elaboration: User-facing syntax containing implicit arguments, coercions, and macro structures is translated into fully explicit, type-checked terms of the Calculus of Inductive Constructions (CIC)1. This translation solves higher-order unification and typeclass synthesis problems1.
2. Kernel Verification: The compiled proof terms are passed to a minimal, trusted logical kernel1. The kernel verifies type-correctness and term equivalence, maintaining absolute logical consistency1.
3. Compilation: Computationally relevant definitions are lowered to an intermediate representation, optimized, and compiled to C1. The C code is then compiled to machine code using standard compilers such as GCC or Clang1.

Architectural Division of Labor

The system is constructed with strict separation between verified logical soundness and untrusted proof generation:

• The Trusted Kernel: Implemented in approximately 6,000 lines of C++ code, the kernel is the sole arbiter of logical truth1. It implements type-checking rules for universe polymorphism, inductive types, and proof irrelevance1. The kernel does not understand tactics, editor actions, or code compilation1.
• The Elaborator: Written in Lean 4, this component translates intuitive human declarations into explicit kernel terms1. It is highly complex and untrusted; any bug in the elaborator that produces structurally invalid logic will be rejected immediately by the trusted kernel during type-checking1.
• The Tactic Engine: A metaprogramming framework that constructs proof terms1. Tactics function like imperative code generators for proofs; they manipulate an incomplete proof state consisting of local contexts and target goals, generating raw lambda expressions that are verified by the kernel1.
• The Compiler and Runtime: Lower-level code generation that operates completely outside the logical kernel1. It leverages the FBIP paradigm to perform destructive in-place updates on uniquely referenced heap objects, reducing memory allocation and copy overheads3.

The Propositions-as-Types Paradigm

Lean 4 is mathematically grounded in the Curry-Howard isomorphism, which maps mathematical logic to type theory1. Under this paradigm, a proposition is represented as a type, and a proof of that proposition is represented as a program that inhabits that type1. Proving a theorem is identical to writing a program that type-checks against the specified logical type2.

Lean
-- A logical implication P → Q is literally a function type.
-- A proof of the implication is a function that maps a proof of P to a proof of Q.
theorem imply_transitivity {P Q R : Prop} (hpq : P → Q) (hqr : Q → R) : P → R :=
fun hp => hqr (hpq hp)

During compilation, terms whose types live in Prop (propositions) are erased, leaving only the computationally relevant Type parameters for runtime execution7.

Lean as a Functional Programming Language

Lean 4 supports algebraic data types, pattern matching, structures, recursive functions, type classes, and a robust module system4.

Syntax and Core Declarations

In Lean, variables, algorithms, and computational parameters are defined using the def keyword, whereas types are introduced using inductive and structure13.

Programming Concept	Python	Scala	Lean 4
Variable Binding	x = value	val x = value	let x := value [cite: 17, 18]
Constant Definition	Module-level variables	val / object	def [cite: 13, 19]
Tagged Unions (ADTs)	Union[T1, T2] / Class hierarchies	sealed trait / Case classes	inductive [cite: 14, 15]
Product Records	dataclass / NamedTuple	case class	structure [cite: 14, 15]
Ad-hoc Polymorphism	Protocol / Abstract classes	Implicits / Typeclasses	class and instance [cite: 4, 16]
Logical Invariants	Runtime assert / Unit tests	Types / Design by Contract	Prop theorems verified by kernel2

Structural Representation and Type Classes

Structures are product types with named fields that automatically generate projection functions14. Type classes allow for extensible ad-hoc polymorphism4.

Lean
-- Defining namespaces to avoid global identifier collision
namespace SystemsModel

-- A custom inductive option representation
inductive CustomOption (α : Type) where
| none : CustomOption α
| some : α → CustomOption α

-- A structure representing system state
structure NodeState where
nodeId : Nat
isActive : Bool
payload : String

-- Type class definition for converting types to JSON strings
class ToJson (α : Type) where
toJson : α → String

-- Instantiating the typeclass for NodeState
instance : ToJson NodeState where
toJson state :=
s!”\{ \“id\”: {state.nodeId}, \“active\”: {state.isActive} \}”

end SystemsModel

The standard library includes Nat (which compiles to optimized bignums via arbitrary-precision arithmetic libraries but logically behaves as inductive Peano numbers), List α, and Option α15.

Lean as an Interactive Theorem Prover

Theorem proving in Lean occurs primarily inside tactic-mode blocks, denoted by the by keyword, where commands iteratively manipulate the logical environment10.

Proof State Mechanics

In tactic mode, the compiler provides interactive feedback on the Proof State at the cursor position9. The state is divided into context assumptions and target goals10:

α : Type
xs : List α
h_empty : xs = []
-——————-
⊢ xs.reverse = []

The context (above the turnstile ⊢) defines the current local assumptions10. The goal (below the turnstile ⊢) represents the type that must be inhabited to complete the proof9.

Tactic Mode versus Term-Style Proofs

• Term-Style: Proofs are constructed as explicit functional lambda terms11. This style is concise for simple logical implications but rapidly becomes unreadable for complex derivations26.
• Tactic Mode: Proofs are constructed step-by-step using a sequence of goal-directed commands9. This mode provides real-time compiler feedback10.

Core Proof Tactics

• intro: Binds universal quantifiers or moves the antecedent of an implication into the context10.
• exact: Solves the current goal if the provided term matches the target type exactly10.
• apply: Applies a function or implication backwards, creating subgoals for any missing parameters10.
• rw: Performs syntactic find-and-replace using an equality identity10.
• simp: Automates rewriting using a global simplifier database and local hypotheses10.
• rfl: Proves reflexive equalities (such as a = a) by simplifying both sides computationally10.
• cases: Splits an inductive type into its constructor variants to perform case analysis24.
• induction: Generates proof obligations for the constructors of inductive types, providing an induction hypothesis for recursive cases10.
• calc: Groups transitive equality or inequality rewrites into structured, readable chains17.

Lean
-- A minimal proof illustrating multiple tactics
theorem equality_and_implication (a b : Nat) (h : a = b) : a + 0 = b := by
rw [Nat.add_zero] -- Rewrites “a + 0” to “a” in the goal
exact h -- Goal is now “a = b”, which matches hypothesis h

Type System Essentials

Lean’s type system is based on a cumulative hierarchy of universes, dependent function types (-types), and dependent pair types (-types)1.

Universes and Proof Irrelevance

To prevent logical contradictions, Lean uses a predicative hierarchy of universes19. The base level contains two distinct universes:

• Prop (Sort 0): The universe of mathematical propositions1.
• Type (Sort 1): The universe of computational data structures and standard types19.

The fundamental difference between these two universes is Proof Irrelevance1. If a type P has type Prop, any two elements p1 : P and p2 : P are definitionally equal within the kernel1. This enables aggressive proof erasure during compilation7.

Dependent Functions and Dependent Pairs

A dependent function type (denoted (x : α) → β x or as a -type) represents a function where the output type depends on the input value6. A dependent pair (denoted (x : α) × β x or as a -type) represents a structure where the type of the second element depends on the value of the first10.

Lean
-- A dependent function where the output type “Fin n” depends on the input value “n”
def getFirstElementOfSize (n : Nat) (h : n > 0) : Fin n :=
⟨0, h⟩

Extensible Coercions and Implicit Parameters

Lean uses type classes to manage implicit conversions (coercions)32. Coercions are resolved automatically and shown in proof states via the up-arrow operator ↑33.

Lean
-- Coerces a boolean to a Prop (decidable proposition)
instance : Coe Bool Prop where
coe b := b = true

Implicit parameters are declared using curly braces {} and are solved using type unification13. Instance implicit parameters, declared using square brackets [], are synthesized via typeclass resolution4. Arguments can also be passed explicitly by name using the (param := value) syntax21.

Unification and Debugging Diagnostics

Unification errors occur when Lean’s elaborator cannot verify that an inferred type matches the expected type18. Missing typeclass instances generate errors such as failed to synthesize instance.
These resolution failures can be debugged in-line by tracing typeclass synthesis:

Lean
set_option trace.Meta.synthInstance true

Mathematical Verification Workflow

Prover development involves translating mathematical concepts into precise code, interacting with the goal state, and utilizing automation tools1.

Navigating the Goal State

When writing proofs, the developer places the cursor inside tactic blocks to inspect the goal state10. Hypotheses are manipulated using forward reasoning (generating new assertions from existing assumptions using tactics like have) or backward reasoning (modifying the goal using apply and refine)10.

Lemma Discovery and Proof Decomposition

Large proofs are decomposed into smaller helper lemmas using the have keyword17. To discover existing lemmas in Lean’s standard library and the mathematical library (Mathlib), developers use:

• Loogle: A query tool that searches by type signatures and patterns38.
• exact?: A tactic that searches the environment for a single matching lemma to solve the goal.

Automated Simplification and Mathlib Integration

The simp tactic repeatedly applies rewrite rules annotated with the @[simp] attribute10. While rw performs strict syntactic matching39, simp normalizes terms to a canonical form10.
Mathlib provides a highly unified algebraic hierarchy where physical dependencies are tightly integrated32. During proof resolution, Mathlib relies on automatically synthesized coercion paths and typeclass derivations32. Over 74% of dependency edges in Mathlib are synthesized implicitly, meaning proof success depends on correct typeclass resolution32.

Functional Programming Workflow and Totality

Lean 4 establishes a mathematically consistent environment by enforcing Totality by default2.

The Totality and Termination Constraint

If non-terminating (infinite) loops were allowed in the logic space, they could represent proofs of false statements, rendering the logic unsound40. Thus, every standard function must be proven to terminate on all inputs2.

Structural versus Well-Founded Recursion

• Structural Recursion: The recursive step operates on a strict subcomponent of an inductive argument40. The compiler automatically translates this into type recursors, proving termination implicitly40.
• Well-Founded Recursion: For non-structural algorithms (such as binary search or gcd), the developer must provide an explicit measure using the termination_by clause41. Lean then generates proof obligations to show that this measure strictly decreases with each recursive call41.

Lean
-- Euclid’s GCD requires well-founded recursion because arguments are not structurally smaller
def gcd (n k : Nat) : Nat :=
if h : n = 0 then
k
else if k = 0 then
n
else if n > k then
gcd (n - k) k
else
gcd n (k - n)
termination_by n + k -- The sum of both inputs serves as the decreasing measure [cite: 42]

To bypass totality constraints in non-logical contexts (such as network servers or external systems bindings), functions can be marked partial or unsafe41.

Executable Implementations versus Logic Specifications

Functional verification in Lean follows a code-first approach:

1. Implementation: Write a pure functional program focused on performance2.
2. Specification: State theorems that define the logical behavior of the program2.
3. Verification: Prove that the program satisfies those theorems2.

The correctness of functional logic is verified through assertions verified at compile-time by the kernel, or evaluated interactively using #eval13.

End-to-End Verification Worked Path

This section presents a complete, compile-valid path that defines a custom datatype, implements helper functions, states properties, and verifies them using rewriting and induction.

Lean
-- 1. Defining a custom algebraic datatype representing a linked list
inductive CustomList (α : Type) where
| nil : CustomList α
| cons : α → CustomList α → CustomList α

-- 2. Implementing a structural recursive function to compute length
def customLength {α : Type} (xs : CustomList α) : Nat :=
match xs with
| CustomList.nil => 0
| CustomList.cons _ ys => customLength ys + 1

-- 3. Implementing list concatenation
def customConcat {α : Type} (xs ys : CustomList α) : CustomList α :=
match xs with
| CustomList.nil => ys
| CustomList.cons z zs => CustomList.cons z (customConcat zs ys)

-- 4. Proving a simple equality definitionally using reflexivity
theorem customLength_nil {α : Type} : customLength (CustomList.nil : CustomList α) = 0 :=
rfl

-- 5. Proving a property of concatenation by definitional reduction
theorem customConcat_nil_left {α : Type} (ys : CustomList α) :
customConcat CustomList.nil ys = ys :=
rfl

-- 6. Proving a property by structural induction
-- We prove that appending nil to the right of a list leaves the list unchanged.
theorem customConcat_nil_right {α : Type} (xs : CustomList α) :
customConcat xs CustomList.nil = xs := by
induction xs with
| nil =>
-- Base case: customConcat nil nil reduces definitionally to nil
rfl
| cons z zs ih =>
-- Inductive step: goal is “customConcat (cons z zs) nil = cons z zs”
-- This reduces definitionally to “cons z (customConcat zs nil) = cons z zs”
simp only [customConcat]
-- The induction hypothesis is “ih : customConcat zs nil = zs”
rw [ih]

Practical Systems Engineering Heuristics

When applying Lean 4 to systems engineering, developers often encounter common conceptual roadblocks.

Resolving Common Beginner Failure Modes

• Syntactic vs. Definitional Equality in Rewrites: The rw tactic operates via strict syntactic matching39. If a goal is computationally identical but syntactically different (e.g., trying to rewrite a + 0 using a lemma for 0 + a), rw will fail39.
  • Heuristic: Use simp or dsimp to normalize the expressions to their common logical forms before rewriting, or use change to manually cast the goal to a definitionally equal syntax.
• Alias Transparency and Typeclass Failures: If a custom type alias is defined using def, Lean’s typeclass engine will often fail to resolve basic traits (such as SizeOf or DecidableEq)35.
  • Heuristic: Use abbrev or the @[reducible] attribute to define aliases35. This signals the elaborator to transparently unfold the definition during unification and typeclass resolution35.
• Well-Founded Loop Resolution: When a recursive function fails to compile because Lean cannot prove termination, the compiler generates complex logical helper goals containing invImage or sizeOf35.
  • Heuristic: Use all_goals simp_wf inside a decreasing_by block to automatically simplify the generated relational obligations42.

Architectural Guidelines for Proof Engineering

• When to use Tactics: Use tactics for inductive proof branches, non-trivial equalities, and complex search goals where automated strategies like omega or aesop can close goals10.
• When to use Terms: Use direct proof terms for simple logical mappings, function compositions, and computational helper lemmas17.
• Proof Decomposition: If a tactic proof becomes slow to compile or is deeply nested, extract intermediate steps into standalone private theorem helpers to optimize compilation time and maintain readability46.

Recommended Paths for Continued Mastery

To transition from basic verification to industrial-scale proof engineering, developers should study:

1. Functional Programming in Lean (FPIL): Focuses on monadic side effects, I/O, and programming with dependent types38.
2. Theorem Proving in Lean 4 (TPIL4): Focuses on the logical foundations of dependent type theory and natural deduction38.
3. Mathematics in Lean (MIL): Introduces the design of Mathlib and advanced tactic usage38.

Syntax and Tactic Reference Cheat Sheet

Core Syntax Keywords

Keyword	Operational Purpose	Compiled Runtime Execution
def	Declares a computable function, type alias, or constant13.	Lowers to an executable C function1.
theorem	Declares a logical proposition and its verified proof term10.	Completely erased during compilation7.
example	Introduces an anonymous theorem to check a proof in-line17.	Erased; does not add a symbol to the namespace.
inductive	Declares a custom sum type or algebraic data type14.	Lowers to a tagged structure14.
structure	Declares a product type with projection fields14.	Compiled as a record structure14.
class	Declares a typeclass to support ad-hoc polymorphism16.	Translated to implicit dictionary passing16.
instance	Registers an implementation for a specified typeclass16.	Added to the global synthesis lookup table4.
match … with	Destructures and pattern-matches expressions14.	Lowers to switch statements with FBIP3.
let	Binds a local variable with a defined value17.	Retained in local execution scope.
by	Enters tactic-mode to write imperative proof scripts13.	Interpreted by elaborator; erased on compilation1.

Tactical Guide

Tactic	Syntactic Format	Operational Effect on Proof State
intro	intro h [cite: 10]	Moves a target binder or implication premise into context as h10.
exact	exact h [cite: 10]	Closes the main goal if term h matches the goal type10.
apply	apply h [cite: 10]	Applies implication h backwards, updating the target goal10.
rw	rw [h] / rw [←h] [cite: 10, 25]	Syntactically replaces LHS with RHS using equality lemma h10.
simp	simp [h1, h2] [cite: 10, 29]	Simplifies the goal using simplification rewrites10.
rfl	rfl [cite: 13, 17]	Solves equality if both sides are computationally equivalent13.
cases	cases x [cite: 22, 28]	Performs case analysis, splitting the goal per constructor24.
induction	induction x [cite: 10, 29]	Generates proof branches for induction over variable x10.
calc	calc … [cite: 17, 25]	Groups sequential equality transformations into readable chains17.

Works cited

1. Lean (proof assistant) - Grokipedia, https://grokipedia.com/page/Lean_(proof_assistant)
2. BRIDGE: Building Representations In Domain Guided Program Synthesis - arXiv, https://arxiv.org/html/2511.21104v2
3. The Lean 4 Theorem Prover and Programming Language - Microsoft Research, https://www.microsoft.com/en-us/research/publication/the-lean-4-theorem-prover-and-programming-language/
4. The Lean 4 Theorem Prover and Programming Language - ResearchGate, https://www.researchgate.net/publication/353088178_The_Lean_4_Theorem_Prover_and_Programming_Language

5. functional programming

   Chalmers Security & Privacy Lab, https://www.cse.chalmers.se/research/group/security/tag/functional-programming/

6. The Lean Theorem Prover (system description) - andrew.cmu.ed, https://www.andrew.cmu.edu/user/avigad/Papers/lean_system.pdf

7. “Why not just use Lean?”

   Hacker News, https://news.ycombinator.com/item?id=47922079

8. The Lean FRO Year 1 Roadmap - Lean Programming Language, https://lean-lang.org/fro/roadmap/y1/
9. Tactic Proofs - Lean Programming Language, https://lean-lang.org/doc/reference/latest/Tactic-Proofs/
10. The hitchhiker’s guide to reading Lean 4 theorems - LambdaClass Blog, https://blog.lambdaclass.com/the-hitchhikers-guide-to-reading-lean-4-theorems/
11. Propositions and Proofs - Lean Programming Language, https://lean-lang.org/theorem_proving_in_lean4/Propositions-and-Proofs/
12. 11. Axioms and Computation — Theorem Proving in Lean 3 (outdated) 3.23.0 documentation, https://leanprover.github.io/theorem_proving_in_lean/axioms_and_computation.html
13. The Lean 4 Theorem Prover and Programming Language (System Description), https://lean-lang.org/papers/lean4.pdf
14. Datatypes and Patterns - Lean Programming Language, https://lean-lang.org/functional_programming_in_lean/Getting-to-Know-Lean/Datatypes-and-Patterns/
15. 4.4. Inductive Types - Lean Programming Language, https://lean-lang.org/doc/reference/latest/The-Type-System/Inductive-Types/
16. 10. Type Classes - Lean Programming Language, https://lean-lang.org/theorem_proving_in_lean4/Type-Classes/
17. tactic - When is the lean 4 “by” required? - Proof Assistants Stack Exchange, https://proofassistants.stackexchange.com/questions/4029/when-is-the-lean-4-by-required
18. Coercion Insertion - Lean Programming Language, https://lean-lang.org/doc/reference/latest/Coercions/Coercion-Insertion/
19. 2. Dependent Type Theory - Lean Programming Language, https://lean-lang.org/theorem_proving_in_lean4/Dependent-Type-Theory/
20. Learning Lean 4 as a programming language 4 – Proofs - Unreasonable Effectiveness, https://unreasonableeffectiveness.com/learning-lean-4-as-a-programming-language-4-proofs/
21. Lecture 19: Inductive types & proofs, https://course.ccs.neu.edu/cs2800sp23/l19.html
22. lean4/doc/examples/tc.lean at master - GitHub, https://github.com/leanprover/lean4/blob/master/doc/examples/tc.lean
23. 20.1. Natural Numbers - Lean Programming Language, https://lean-lang.org/doc/reference/latest/Basic-Types/Natural-Numbers/
24. Natural numbers - Universitat de València, https://www.uv.es/coslloen/Lean4/Leancap06.html
25. 2. Basics — Mathematics in Lean v4.19.0 documentation, https://leanprover-community.github.io/mathematics_in_lean/C02_Basics.html
26. Topic: Term mode vs tactic mode - Zulip Chat Archive, https://leanprover-community.github.io/archive/stream/187764-Lean-for-teaching/topic/Term.20mode.20vs.20tactic.20mode.html
27. Lean 4: Formal Theorems - Emergent Mind, https://www.emergentmind.com/topics/formal-lean-4-theorems
28. Arrays and Termination - Functional Programming in Lean, https://leanprover.github.io/functional_programming_in_lean/programs-proofs/arrays-termination.html
29. Interlude: Tactics, Induction, and Proofs - Lean Programming Language, https://lean-lang.org/functional_programming_in_lean/Interlude___-Tactics___-Induction___-and-Proofs/
30. Don’t trust, verify: guarantees for the Lean proof assistant - MFoCS Seminar Presentation, https://cs.ru.nl/~freek/courses/mfocs-2024/slides/rutger.pdf
31. Learn Lean 4 in Y Minutes, https://learnxinyminutes.com/lean4/
32. The Network Structure of Mathlib - arXiv, https://arxiv.org/html/2604.24797v1
33. 11.2. Coercing Between Types - Lean Programming Language, https://lean-lang.org/doc/reference/latest/Coercions/Coercing-Between-Types/
34. Coercions - Lean Programming Language, https://lean-lang.org/functional_programming_in_lean/Overloading-and-Type-Classes/Coercions/
35. lean - Termination for Wrapped Fin in Lean4 - Proof Assistants Stack Exchange, https://proofassistants.stackexchange.com/questions/3840/termination-for-wrapped-fin-in-lean4
36. invalid apply tactic, failed to unify - Proof Assistants Stack Exchange, https://proofassistants.stackexchange.com/questions/2210/invalid-apply-tactic-failed-to-unify
37. Lean 4 Tutorial - Brandon Rozek, https://brandonrozek.com/blog/lean4-tutorial/
38. Learn — Lean Lang, https://lean-lang.org/learn/
39. Difference between `Nat.add` and `+` in Lean - Proof Assistants Stack Exchange, https://proofassistants.stackexchange.com/questions/5278/difference-between-nat-add-and-in-lean
40. Termination Checking: Comparing Structural Recursion and Sized Types by Examples - McGill School Of Computer Science, https://www.cs.mcgill.ca/~dthibo1/papers/termination.pdf
41. 7.6. Recursive Definitions - Lean Programming Language, https://lean-lang.org/doc/reference/latest/Definitions/Recursive-Definitions/
42. Lean 4.6.0 - Lean Programming Language, https://lean-lang.org/blog/2024-2-29-lean-460
43. Using `decreasing_by` to prove termination in Lean 4 - Proof Assistants Stack Exchange, https://proofassistants.stackexchange.com/questions/2712/using-decreasing-by-to-prove-termination-in-lean-4
44. Interacting with Lean - Lean Programming Language, https://lean-lang.org/doc/reference/latest/Interacting-with-Lean/
45. Termination for Wrapped `Fin n` in Lean4 - Stack Overflow, https://stackoverflow.com/questions/78183967/termination-for-wrapped-fin-n-in-lean4
46. How can I do induction on a field in Lean 4? - Proof Assistants Stack Exchange, https://proofassistants.stackexchange.com/questions/5127/how-can-i-do-induction-on-a-field-in-lean-4
47. Functional Programming in Lean, https://lean-lang.org/functional_programming_in_lean/
48. Introduction - Lean Programming Language, https://lean-lang.org/theorem_proving_in_lean4/Introduction/
49. Datatypes, Patterns and Recursion - Functional Programming in Lean, https://leanprover.github.io/functional_programming_in_lean/getting-to-know/datatypes-and-patterns.html
50. Lean for Scala programmers - Part 4 - Typista.org, https://typista.org/lean-for-scala-programmers-4/