🧾 Coq — A Language for Formal Proofs, Verified Logic, and Certified Software

What is Coq?

Coq is a formal proof assistant and dependently typed language used to write mathematical theorems and mechanically verify their correctness. Unlike general-purpose programming languages, Coq focuses on proving statements, constructing proofs, and ensuring absolute logical soundness through type theory.

It has been used to formally verify mathematical theorems, cryptographic systems, programming languages, and even certified compilers (such as CompCert).

Specs

Language Type: Dependently typed proof language

Released: 1989 (actively developed)

Built On: Calculus of Inductive Constructions (CIC)

Primary Use: Formal verification, theorem proving, logic research

Execution Model: Proof checking rather than typical program execution

Typing: Dependent, constructive, rigorous

Example Code (Hello World Equivalent)

Coq does not have a traditional print model; instead, a simple statement might look like:

Theorem hello: True.
Proof.
  exact I.
Qed.

This proves the statement True — the closest conceptual analog to "Hello World".

How It Works

Coq programs consist of:

Definitions (types, functions, logic foundations)
Theorems and lemmas
Proof scripts guiding the proof engine

Proofs are constructed interactively using tactics — commands that manipulate the proof state.

Common tactics include:

Tactic	Purpose
`intros`	Bring variables into context
`simpl`	Reduce expressions
`apply`	Use a known lemma or rule
`rewrite`	Replace equivalent forms
`exact`	Provide a final proof term
`auto`	Automated proof search

Once complete, the proof is machine-verified and guaranteed correct.

Strengths

Extremely high assurance of correctness
Used in academic research, cryptography, and verified compilers
Supports formal reasoning beyond traditional programming
Stable, well-established tooling and community

Weaknesses

Very steep conceptual learning curve
Proof construction can be slow and tedious
Not intended for ordinary application programming
Requires mathematical and logical background to use effectively

Where to Run

Coq can be used through:

CoqIDE
VS Code plugin (VSCoq)
Emacs Proof General
Web-based proof playgrounds
TIO.run (subset support)

Many university courses teach Coq for formal methods.

Should You Learn It?

For normal development: No
For systems requiring mathematical guarantees: Yes
For cryptography, compilers, or formal reasoning: Very useful
For esolang curiosity or theory exploration: Fascinating (but hard)

Summary

Coq isn't about programming in the traditional sense — it's about certainty. It treats code and mathematical logic as interchangeable expressions, enabling verified systems and provable correctness. While demanding to learn, Coq remains one of the most important tools for building software where correctness is non-negotiable.