Is there a theorem prover that works in natural language?

Question

I'm interested in a computer program, possibly a web app, that could prove theorems and show its proofs.

I essentially want to type in a theorem like "For every bounded sequence, there exists a subsequence that has a limit" and see a proof for that.

I googled theorem provers, and I found Lean with this tutorial, I found HOL and E, but looking at their websites, it seems very complicated. The tutorial I linked to shows the Lean language, which is a whole thing. I was expecting either human language or mathematical language. (Like ∃ and ∀.)

Does something like that exists?

Are you aware of Coq and/or Agda? They have editors that allow mathematical symbols. I don't think you have any high chances for provers in natural language. — frabala, Mar 28 '19 at 12:40
Not only does something like this not exist, I'd bet it won't exist in my lifetime. Of course, for all you know I'm 102. — Stephen, Apr 06 '19 at 01:10
The metamath proof explorer lets you read statements/proofs in very close to standard math symbols (see Bolzano-Weierstrass here, for instance). And those proofs are checked by a computer. But nearly all of those proofs were conceived of/entered in by a human. — Mark S., Apr 06 '19 at 01:16
Statements in a natural language would need to be translated into formal logic in order to be used in an automated theorem prover. — Anderson Green, Feb 11 '24 at 20:54

score 5 · Accepted Answer · answered Apr 06 '19 at 01:04

There's no perfect solution. Finding a proof of an arbitrary statement is undecidable, finding a proof of bounded length is NP-complete. Perhaps one day we'll have a program that heuristically is as good as a human mathematician, but we're nowhere close.
There's a system called Mizar. Here's how you write irrationality of e http://www.mizar.org/JFM/Vol11/irrat_1.abs.html, and as far as I see this is rendered nicely as a PDF http://www.mizar.org/JFM/pdf/irrat_1.pdf.
Proof assistants such as Coq, Agda. Perhaps you might not see ∃ and ∀ in descriptions of those systems, but let this not deter you: they are based on calculus of constructions or type theory, not on first-order logic. Coq was used to formalize the four color theorem, the odd order theorem, and a formally correct C compiler. Proving a statement becomes an interaction: the proof assistant shows the assumptions and you guide it towards the goal; if it's simple enough, you can use a tactic "this is trivial" and the computer will fill the rest.
Regarding human language, there was a program that solved a limited range of problems in metric spaces and returned a text output that was not easily distinguishable from a human proof https://arxiv.org/pdf/1309.4501.pdf.

I also found an implementation of a natural language theorem prover in Prolog. — Anderson Green, Sep 08 '21 at 22:03
I found two other natural-language theorem provers: LangPro and ACE Reasoner. — Anderson Green, Jan 14 '22 at 02:24

Is there a theorem prover that works in natural language?

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