# Questions tagged [usage]

Questions asking how to properly and efficiently use Proof Assistants.

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### Are some proof assistants better suited for given areas of math than others?

There are many different proof assistants out there, and I think it is reasonable to expect that more or less all results we prove in everyday mathematics can be proven in any of them. One nice way to ...
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### What was the 1st Proof Assistant?

What was the first proof assistant? What was it used for? When and by who was it created? Is it still used today? And what was its purpose?
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### Examples of new mathematics discovered through formalization?

In his essay Why formalize mathematics?, Patrick Massot discusses several reasons behind why a working mathematician might be interested in proof formalization. One of the the reasons he discusses is, ...
496 views

### Usage and Importance of Proof Assistants

How would you explain so a beginner in simple terms why Proof Assistants are important, and how / why they are used?
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### What is a Proof Assistant?

How would you explain to a beginner in simple terms, what a Proof Assistant is?
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### Proof Assistants for Vim Users

I remember the last time I was trying to get into proof assistants, I was discouraged by the lack of functionality for vim users. At the time, it seemed like a lot of powerful features (holes, etc) ...
378 views

### Why should you "never resort to polymorphism when initiality would do"?

In the concluding statement of "universe hierarchies", Conor McBride calls it [...] that key lesson which I learned from James McKinna: never resort to polymorphism when initiality will do. ...
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### Incorporating Markov's principle in various proof assistants

The Markov's principle states that if a Turing machine does not run forever, then it halts. Equivalently, if I have a function $f : \mathbb N \to \mathrm{Bool}$, such that I have proved that \$\neg\neg ...
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### What are some good (bad) examples of "green slime"?

I know what this roughly is. But when I'm explaining to a friend, I have trouble pulling out examples from my head. Are there any minimal working examples (maybe "minimal non-working examples&...
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243 views

### How much duplication does universe polymorphism actually save us?

From my rough impression, in (formalizing) everyday mathematics we almost never use universe polymorphism in a way that stretches the proof-theoretic strength. It merely saves us some duplication. My ...
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### What are the practical differences between intensional and extensional type theories?

It is already proved that MLTT with equality reflection is equivalent to MLTT with an intensional equality, plus UIP and function extensionality. So theoretically the differences between intensional ...
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### Could proof assistants be used to prove that some piece of code is free of bugs?

Formal verification is the act of proving or disproving the correctness of intended algorithms underlying a system with respect to a certain formal specification or property, using formal methods of ...
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### How to run Agda?

I want to run Agda, but I do not know how to run functions. I need to run "Hello, World!"; how can I do this?
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### Universe polymorphism and Coq standard library

When developing in Coq with the Universe Polymorphism flag on, the standard library introduces unwelcome universe constraints because it is universe monomorphic. Is there an alternative standard ...
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### General Guidelines and Tips for using Induction

There are many kinds of induction (induction over the natural numbers ā structural induction ā Noetherian induction), when do I use which flavour? And what should I keep in mind when doing proofs by ...
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### Why is it hard to formalize informal proofs?

Say I have some informal but rigorous argument in line with eg real analysis. Currently, it is a massive PITA to do algebraic manipulations in proof assistants like Coq or Isabelle. However, in ...
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### Universe polymorphism and modules in Coq

The following code (without universe polymorphism) is accepted by Coq (8.16.0) : ...
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### .CoqMakefile.d required by CoqMakefile but not generated

I am trying to use CoqMakefile to automatically build my Coq project in Coq 8.15.2. When I did this the compilation failed because a file ".CoqMakefile.d" was expected by make but did not ...
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427 views

### How to install dependencies correctly? [Cannot find a physical path bound to logical path]

There's this library I'd like to use. The idea is to be able to use this library with Require Import easily in any other *.v ...
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### Defining Lists and Prove Associativity of Append [closed]

When I saw this question asking what is the "Hello, World!" for proof assistants I immediately thought of that exercise. Not a long time after this answer by Couchy was proposed. Therefore, ...
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### Packaging Mathematical Structures in Coq: Help Understanding a Definition

Context I am a relatively new user to Coq with a decent understanding of the basics of dependent type theory and am midway through chapter 2 of the Software Foundations Series of books. I want to ...
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### Debug autorewrite in Coq

I often meet proofs using autorewrite which Coq takes a while to process for some reason. (Setoid rewriting) I then manually figure out which rewrite rules were ...
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1 vote
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### Proving that applicative functors compose

For simplicity, here an applicative functor means (in a proof assistant based on dependent type theory) the Haskellian applicative functor, bundled with its equational laws. This I can of course brute ...
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