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I recommend reading Pollack's How to believe a machine-checked proof. It explains how proof assistants are designed to minimize the amount of critical code. There are many levels of formal verification (that's the phrase you're looking for in place of "proven") of a proof assistant: Verify that the algorithms used by the proof assistant are correct. Verify ...

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Most of your statements are elementary algebra, so these can be proved automatically by a computer algebra system (CAS), such as Maple or Mathematica. (In case you're interested in the mathematics behind CAS, I can recommend the book Modern Computer Algebra by Joachim von zur Gathen and Jürgen Gerhard, a beautiful book, considered the 'bible' of the field) ...

12

We have to be carful here. I think you mean that a program given a formula could not, in general, produce a proof of it or its negation. I can write a program that, given a theorem, will produce it's proof as long as the set of proofs is recursively enumerable. This is not possible in general because of Godel's incompleteness theorem (well that was the first ...

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What is second order logic in contrast to first order logic? What is monadic vs non monadic logic? Monadic second-order logic is first-order logic plus quantification over sets. So, as well as being able to say that there exists a domain element with some property ($\exists x\dots$), you can also say that there exists a set of domain elements with some ...

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I don't know why people didn't develop Hoare logics for lambda-calculi earlier. The first work to get this right was Honda et al's A Compositional Program Logic for Polymorphic Higher-Order Functions There were some earlier attempts before this, but they didn't quite nail the problem, for example: how do you denote the value of a functional program? What ...

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There are many related ways you can mechanise your logic. Deep embedding into one of the well-developed provers such as Isabelle/HOL, Coq or Agda. This is (almost) always possible, but makes using the embedded logic awkward. Shallow embedding into one of the well-developed provers such as Isabelle/HOL, Coq or Agda. This is only possible when the embedded ...

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A little google-fu (and my own memory) tells me it was apparently named by Larry Paulson after Gerard Huet's daughter. Gerard Huet happens to be one of the people behind the less poetically named Coq theorem prover. Small world!

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What you need is the idea of "the trusted core". Quoting "A verified runtime for a verified theorem prover": In many theorem provers, the trusted core—the code that must be right to ensure faithfulness—is quite small. As examples, HOL Light is an LCF-style system whose trusted core is 400 lines of Objective Caml, and Milawa is a Boyer-Moore style prover ...

8

To simplify a bit, a system of inference rules generally defines a set of sequents, which are simply pairs of a list (or set) of propositions $\Gamma$ and a proposition $\phi$ written $$\Gamma \vdash \varphi$$ called the derivable sequents. The rules describe which sequents are derivable, and for any reasonable system, given $\Gamma\vdash\varphi$, and ...

8

The existential form of the axioms of set theory is convenient for the meta-theoretic explorations of set theory, such as forcing etc., where it is important to have a minimal language to worry about (only $\in$ relation). But even just stating the axioms of set theory without any constants and operation symbols is pretty haunting, see this gist of mine. If ...

7

Coq is a bit more cruel than paper proofs: when you write "and we are done" or "clearly" in a paper proof, there is often much more to do to convince Coq. Now I did a little clean up of your code, while trying to keep it in the same spirit. You can find it here. Several remarks: I used built in datatypes and definitions where I thought it wouldn't hurt ...

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While this may trend close to self-advertisement, this is essentially the topic of my recent paper Metamath Zero: The Cartesian Theorem Prover (video), and the analogy with bootstrapping compilers is spot on. The introduction of the paper lays out what is needed to make this happen, and it's only a problem of engineering. As Andrej says, there are several ...

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How would you prove inside the pure CoC that the induction principle holds of the Church numerals? See Thomas Streicher's, Independence of the induction principle and the axiom of choice in the pure calculus of constructions.

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This is late for you, but probably might be of help to others. I myself had asked this question and I wrote the summary of my findings to the acl2-help mailing list on 8 and 9 September 2013: https://utlists.utexas.edu/sympa/arc/acl2-help/2013-09/msg00005.html https://utlists.utexas.edu/sympa/arc/acl2-help/2013-09/msg00006.html Disclaimer: What follows is ...

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Yes, there is a generalization of the construction you mentioned. However, it's utility depends on the function F. The guarantees you get are meaningful only if F is robust, in the sense that you have to change a large fraction of the dataset to make any (meaningful) change to the output of the computation. Suppose we have an untrusted agent Agatha who is ...

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Use a model checker. This is exactly what they're good for. You might look at SPIN and NuSMV as a starting point, but there are many others as well.

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Essentially, you can treat uninterpreted predicates as boolean-valued functions (adding a new boolean sort if necessary) and replace them with boolean variables as you would other functions. For the given example: Getting rid of $F$ and $G$ first: \begin{align*} p(z,f_1)&\wedge f_2=g_1\to p(x_1,y)\wedge x_1=f_1\to f_1=f_2\\ &\wedge x_1=x_2\to ...

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Cody's answer is excellent, and fulfils your question about translating your proof to Coq. As a complement to that, I wanted to add the same results, but proven using a different route, mainly as an illustration of some bits of Coq and to demonstrate what you can prove syntactically with very little additional work. This is not a claim however that this is ...

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If you believe that a proof or disproof has reasonable length, you could come up with one SAT instance which states $x_1,\ldots,x_n$ encodes a proof of the Riemann Hypothesis, and another one stating $x_1,\ldots,x_n$ encodes a refutation of the Riemann hypothesis. The formula will have size polynomial in $n$, though probably not linear; if you're careful, ...

4

This is certainly possible, and the language of higher order functions makes this quite nice. If you're doing a Turing/Cook reduction, you can basically just write a function that takes an oracle as a parameter. For example, to show that Graph Coloring is NP-hard, you could write something like this (pardon my terrible Coq-like pseudocode) Theorem ...

3

My goal is to be able to judge what parts of formal verification I could apply to my job as a software/network engineer. If you plan to use formal verification as black boxes, then I would suggest using some tools to have an idea how they work. Here are some open source tools: Error Prone of Google Nullaway of Uber Facebook Infer Ikos of NASA Java ...

3

Metamath has a HUGE chapter-organized archive of formal theorems: http://us.metamath.org/mpeuni/mmtheorems.html Besides that, Freek Wiedijk's QED manifesto is always a good starter for future formalizers. And his list of top 100 formalized theorem is an ever-encouraging source of formalized ideas: http://www.cs.ru.nl/~freek/100/ At the end of that page, ...

3

Not that I know of. A more typical architecture is to decompose the theorem prover into a prover plus a proof checker. Then only the proof checker needs to be validated. If proofs are expressed in a simple enough language, then the proof checker might be extremely simple. All of the smarts can go into the prover (which tries various strategies to try to ...

3

I can speculate/hypothesize. Each solver has many heuristics and proof tactics. Imagine that corresponding to each heuristic is a class of instances that are easy to solve if you use that heuristic, but hard if you don't (e.g., if you don't have a heuristic for it and you fall back to the generic methods, maybe you're effectively exploring the entire ...

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All of the theorems you want to prove (except uniqueAddNeutral) are of the form $\forall x . \phi(x)$ where $\phi(x)$ is quantifier-free. Herbrandization is typically the first step towards proving those theorems: replace $x$ with an arbitrary constant $c$, with no assumptions made about $x$, and prove $\phi(c)$ holds without making any assumptions about $c$...

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This is a project I've been thinking of venturing into for a few years now, and I was fortunate to have a very useful discussion about it at the time I started planning it with Chris Lattner (at the time, one of the main architects of LLVM, although better known these days perhaps for his work on Swift). It does seem that there is a possibility of producing ...

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In case you are not aware, there is a possibility that a theorem prover is implemented 100% correctly and run on a faultless machine and proves itself arithmetically inconsistent, even though it is not. The other existing answers focus on the issue of verifying that the theorem prover runs exactly as it was designed to run, but do not address this aspect of "...

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You do not have to prove things by induction. For example, you can prove $\forall n : \mathbb{N} \,.\, n = n$ without induction by applying reflexivity. In your proof, we use induction on $a$, but then we do not need to use induction on $b$ and $c$ because we can finish the proof simply using other methods. We could have used induction on $b$ and $c$, and ...

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The notion of automating mathematics is a vague one, and that's accounting for the discrepancy here. One interpretation would be: to automate mathematics would be to produce a machine $M$ which could tell whether or not a given sentence is true (or, more weakly, provable from some agreed-upon set of axioms like $\mathsf{ZFC}$). Even the weaker version is ...

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You are conflating two possible meanings of the phrase "mathematics can be automated": "any theorem can be proved true or false by an algorithm" "the practical activity of proving theorems, as presently performed by humans, can instead be performed by computers in an economically-viable fashion" Due to the halting problem, it is impossible for any ...

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