Questions tagged [proof-assistants]

A proof assistant is software used for creating and checking formal proofs; examples include Coq and HOL. This tag is not to be used for requesting assistance on finding proofs. General questions about proof assistants can also be asked on the Proof Assistants Stack Exchange site.

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Are there any recent advances in formalizing the undecidability of $\mathit{CH}$?

The website Formalizing 100 Theorems by Freek Wiedijk contains a list of some theorems that were chosen at some point as good candidates for formalization (because of their complexity, their ...
14 votes
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559 views

Explicit and complete list of Lean's Axioms

I'm a big fan of the idea of fully formalizing mathematics. So the Lean proof checker appeals to me. Relating to this, one of the biggest appeals of mathematics to me is that there is a (largely) ...
54 votes
2 answers
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Automatically solving olympiad geometry problems

Warning: I am only an amateur in the foundations of mathematics. My understanding of this Wikipedia page about Tarski's axiomatization of plane geometry (and especially the discussion about ...
17 votes
4 answers
2k views

Does formalizing math require search and creativity, or is it near-mechanical?

I remember reading somewhere that it takes about a week to convert a page of math into something a proof-assistant like Isabelle or HOL Light would accept. Is this type of conversion something that ...
0 votes
1 answer
375 views

How bad is Coq proving both $T$ and $\lnot T$? [closed]

Question: How bad is Coq proving both $T$ and $\lnot T$? Can it be abused? Back in 2011 on the coq-club mailing list there was a thread: Is the Daniel Schepler's inconsistency real?. In the thread ...
58 votes
8 answers
12k views

How true are theorems proved by Coq?

Less tongue in cheek, is it known what the relative consistency is for theorems proved with an automatic theorem prover? Of course this depends somewhat on what assumptions one makes with respect to ...
7 votes
1 answer
224 views

Formulas that are valid simultaneously in a power set Boolean algebra $B$ and the 2-element Boolean algebra $\mathbf2$ [duplicate]

Note 1. Early I posted a related question Set-theoretic tautologies. But the answer did not contain any concrete references to the literature. So I posted this, more precisely formulated question, ...
7 votes
1 answer
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Set-theoretic tautologies

Let us consider unquantified formulas of a set theory (for example, NBG), more precisely, the formulas, constructed from variables and the constants $\emptyset, V$ (the empty set and the class of all ...
42 votes
36 answers
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Formalizations of category theory in proof assistants

What are the existing formalizations of category theory in proof assistants? I'm primarily interested in public-domain code implementing category theory in a proof assistant (Coq, Agda, Isabelle/HOL, ...
6 votes
3 answers
1k views

Proof formalization

I read some time ago some papers about proof formalization. Typically, I began whith this one, from Lamport. Are there more recent works in this field ?
4 votes
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Formal and informal proofs: Is there any "bilingual corpus"?

There are extensive libraries of formalized mathematics like those of Lean, Coq or Isabelle/HOL. What I am interested in is documents of formalized mathematics that closely follow certain informal ...
72 votes
13 answers
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The use of computers leading to major mathematical advances II

I would like to ask about recent examples, mainly after 2015, where experimentation by computers or other use of computers has led to major mathematical advances. This is a continuation of a question ...
10 votes
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Feferman's universes for proof assistants?

This question was prompted by a discussion from another MO question about the consistency of ZFC. There are some mathematicians who are comfortable with ZFC but uneasy with large cardinals. For them,...
11 votes
3 answers
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Can computers find zeros of order $2$?

We assume we are given an entire function $f: \mathbb C \to \mathbb C$ with $f(0)=1$ and $f'(0)=0$ and $f$ is real on the real axis. We assume (as a fact about $f$, that we want to demonstrate ...
23 votes
3 answers
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At which level is it currently possible to write formal proofs?

I am wondering whether I should try to have some fun using proof systems. I have never used such a system, but I have some experiences in logic and programming. My question is: At which level of ...
4 votes
2 answers
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Checking elementary proofs with proof checkers

I am not sure if this is the right place to post this, but I have seen discussions related to proof checking here, so let me post it. If there is better place for it, please give me a hint as to where ...
58 votes
9 answers
8k views

How do they verify a verifier of formalized proofs?

In an unrelated thread Sam Nead intrigued me by mentioning a formalized proof of the Jordan curve theorem. I then found that there are at least two, made on two different systems. This is quite an ...
9 votes
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Locally small categories in ZFC

This question is primarily a reference request. It arose from a personal coding/formalization project. I am using a particular form of a definition of a category in ZFC. According to this definition, ...
47 votes
5 answers
8k views

Proof assistants for mathematics

This question is related to (maybe even the same in intent as) Intro to automatic theorem proving / logical foundations?, but none of the answers seem to address what I'm looking for. There are a lot ...
152 votes
6 answers
16k views

Proofs shown to be wrong after formalization with proof assistant

Are there examples of originally widely accepted proofs that were later discovered to be wrong by attempting to formalize them using a proof assistant (e.g. Coq, Agda, Lean, Isabelle, HOL, Metamath, ...
3 votes
0 answers
357 views

Conversion of proofs between HoTT and ZFC

HoTT provides a foundation of math that remains mysterious for many mathematicians including me. Hence this question. There are several implementations of math based on ZFC, an example being MetaMath. ...
7 votes
0 answers
584 views

Theorem proving in Lean, for areas that aren't quite ready for it

While taking a break from being stuck, I have found myself addicted to playing with the Lean Theorem Prover. (Beware, if you visit this link might you might find yourself hooked...) Playing with this ...
0 votes
0 answers
279 views

Can ∞-category be defined in proof assistants?

Can ∞-category be defined in proof assistants? For example, we can directly consider a function such as ...
18 votes
1 answer
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Available frameworks for homotopy type theory

I am thinking about trying to formalise some parts of classical unstable homotopy theory in homotopy type theory, especially the EHP and Toda fibrations, and some related work of Gray, Anick and Cohen-...
38 votes
4 answers
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Is there research on human-oriented theorem proving?

I know there is already a research community that is working on automatic theorem proving mostly using logic (and things like Coq and ACL2). However, I came across a lecture from a fields medalist W.T....
6 votes
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Proof of Tennenbaum's Theorem by McCarty

Tennenbaum's Theorem in its usual form states that for any countable non-standard model $M$ of PA there is no way to code the elements of $M$ as natural numbers such that either the addition or ...
3 votes
1 answer
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Can Tychonoff's theorem be applied to topological spaces generated by program output in ZFC?

I am confused about an issue in set theory. Tychonoff's theorem says that "an arbitrary product of compact topological spaces is compact". We often talk of an index set $I$ and then for each ...
82 votes
4 answers
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Wanted: a "Coq for the working mathematician"

Sorry for a possibly off-topic question -- there are four StackExchange subs each of which could be construed as the proper place for this question, and I've just picked the one I'm most familiar with....
33 votes
4 answers
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What is the endgoal of formalising mathematics?

Recently, I've become interested in proof assistants such as Lean, Coq, Isabelle, and the drive from many mathematicians (Kevin Buzzard, Tom Hales, Metamath, etc) to formalise all of mathematics in ...
152 votes
5 answers
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What makes dependent type theory more suitable than set theory for proof assistants?

In his talk, The Future of Mathematics, Dr. Kevin Buzzard states that Lean is the only existing proof assistant suitable for formalizing all of math. In the Q&A part of the talk (at 1:00:00) he ...
5 votes
1 answer
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Information density of proofs?

I am a CS person so please excuse the hand-waving. Given a set of machine-represented proofs, each different (but not necessarily proving a different thing), what sort of information-theoretic ...
10 votes
2 answers
840 views

Gödel's ontological proof & Benzmüller's work

For a decade or so, Christoph Benzmüller from Berlin has explored Gödel's ontological proof (and variants) of existence of God. He uses the proof assistant Isabelle/HOL. He recently posted a preprint, ...
3 votes
1 answer
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Data abstraction in set theory via Urelements

I am working in a setting of set theory where set theory is embedded in simply-typed higher-order logic, basically as described for example in Chad E. Brown and Cezary Kaliszyk and Karol Pak (2019) ...
137 votes
28 answers
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Which mathematical definitions should be formalised in Lean?

The question. Which mathematical objects would you like to see formally defined in the Lean Theorem Prover? Examples. In the current stable version of the Lean Theorem Prover, topological groups ...
-3 votes
1 answer
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PCP theorem to check hard proofs [closed]

Is it technically possible to check formidable proofs like Mochizuki's using PCP theorem before mathematicians spend time in understanding the mechanics of the proof? If so why have mathematicians not ...
45 votes
2 answers
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On proof-verification using Coq

So i recently learnt that there is now a certain software called ''Coq'' by which one can check the validity of mathematical proofs. My questions are: Are there limitations on the kinds of proofs ...
99 votes
2 answers
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Extent of “unscientific”, and of wrong, papers in research mathematics

This question is cross-posted from academia.stackexchange.com where it got closed with the advice of posting it on MO. Kevin Buzzard's slides (PDF version) at a recent conference have really ...
0 votes
0 answers
425 views

Artificial intelligence simulating mathematicians (what a distopia!)

This is kind of soft and naive question, so feel free to shame on me :) I start from the fact that, in my opinion, what humans are interested in about mathematics are things that we find deep and ...
14 votes
2 answers
1k views

How does proof assistant organize knowledge?

I am reading a paper Ittay Weiss, The QED Manifesto after Two Decades — Version 2.0, Journal of Software, 11 no. 8 (2016) pp. 803–815, doi:10.17706/jsw.11.8.803-815 The paper says Goal 7: ...
37 votes
1 answer
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Real manifolds in a theorem prover?

Which of the formal computer proof verification systems (like Lean, Coq, Agda, Idris, Isabelle-HOL, HOL-Light, Mizar etc) have a basic theory of real manifolds? Up to, say, the definition of a smooth ...
6 votes
3 answers
3k views

The Lucas argument vs the theorem-provers -- who wins and why?

In his paper, "Minds, Machines and Gödel", J.R. Lucas writes the following: Gödel's theorem [First Incompleteness Theorem, that is—my comment] must apply to cybernetic machines, because it is of ...
3 votes
4 answers
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A complete formalization of EGA in Lean

I have been lately thinking about the feasibility of creating a "mediocre algebraic geometer" AI. I thought that to train it, one could feed it some large chunks of algebraic geometry presented in an ...
13 votes
3 answers
2k views

Is there research on Machine Learning techniques to discover conjectures (theorems) in a wide range of mathematics beyond mathematical logic?

Although there already exists active research area, so-called, automated theorem proving, mostly work on logic and elementary geometry. Rather than only logic and elementary geometry, are there ...
8 votes
1 answer
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Creativity and the mechanization of elementary geometry

In plane geometry, it is customary to say that checking proofs is a mechanical process but that finding new theorems is a creative activity. Citing J. Hadamard, "logic only sanctions the conquests of ...
9 votes
1 answer
343 views

Automated geometry theorem provers

What is the state of the art concerning automated geometry theorem provers (AGTP)? I can see that a few computer algebra softwares and dynamic geometry softwares (e.g. geogebra) have embedded provers ...
33 votes
1 answer
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Proof assistant for working in weaker foundations?

In some of my works I need to prove some results within the internal logic of categories with not much structures (like pretoposes or even just categories with finite limits). The kind of things I ...
20 votes
1 answer
802 views

Proof assistant, Cura te ipsum

By a bona fide bug in a proof assistant I mean a software flaw which is serious enough to create a possibility of "proving" something which is actually false. This is not a purely ...
37 votes
1 answer
4k views

How much mathematics has been formally verified?

That's a vague question so allow me to tighten it up a bit. I recently noticed that there is a formal machine verified proof of the Central Limit Theorem (CLT) implemented with Isabelle. This ...
6 votes
0 answers
290 views

formalization of coordinate-free linear algebra in a proof assistant

I am aware of projects that formalize linear algebra in existing proof assistants (i.e. Coq), but it seems like most of them are based on matrices. I was wondering if it's done in a coordinate-free ...
16 votes
1 answer
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Axioms of Choice in constructive mathematics

There is a widely accepted opinion that the Axiom of Countable Choice (further, ACC) $$ \forall n\in \mathbb{N} . \exists x \in X . \varphi [n, x] \implies \exists f: \mathbb{N} \longrightarrow X . \...