# Tag Info

Accepted

### Curious about computer-assisted NP-completeness proofs

As for question 2, there are at least two examples of $NP$-completeness proofs that involve computer-assistant. Erickson and Ruskey provided a computer-aided proof that Domino Tatami Covering is NP-...
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### Formalizing Homotopy Type theory in Idris

Here is a small, incomplete, and inconsistent formalization of HoTT in Idris. It shows that you can derive a contradiction in Idris just by postulating univalence. There are two barriers to ...

### What should a proof of correctness for a typechecker actually be proving?

That's a good question! It asks what we expect from types in a typed language. First note that we can type any programing language with the unitype: just pick a letter, say ...
• 26.8k
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• 10.4k

### Formally Verified Complexity Theory

In the following paper my colleague Uli Schöpp presents a formal verification (in Coq) of a nontrivial result by Cook and Rackoff on the computational power of graph automata. https://scholar.google....
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### Is Buchberger's algorithm or Wu's method valuable theoretically when we have the Tarski–Seidenberg theorem?

For Buchberger, it depends what you want it for, but generally speaking the answer is no. First, as pointed out on the Wikipedia article, the complexity upper bound given by Tarski-Seidenberg is ...
• 35.9k
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### Formal semantics of tactics

I'm not sure this answers your question, but the first (?) paper on the subject of tactics appears to have been Milner's The Use of Machines to Assist in Rigorous Proof.
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### Formal semantics of tactics

Obviously there is an operational semantics of Ltac by Jedynak et al.
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### Small kernel (i.e. proof-verifier) for Agda?

It is true that Agda currently has a much shakier foundation than say Coq or Lean. It does have an internal term syntax that could be seen as a core language (https://github.com/agda/agda/blob/master/...
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### Structural equality of Pi Types with heterogeneous equality?

I am not aware that J or K exists for heterogeneous equality. It does not need an elimination principle, because it can be simply defined as a sigma type: ...
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### Proving running time upper bounds for algorithms in dependent type theory

As usual, (a) the high-level conceptual approach is basically the same as it is on paper, but (b) mechanization makes new things reasonable to attempt. The way you do things is to define a cost ...
• 31.8k

### Proof assistant usage in complexity theory research?

One very prominent example is of course Gonthiers Coq formalization of the 4 color theorem in Coq which uses a lot of combinatorics. My colleague Uli Schöpp used the ssreflect library developed by ...

### Formally Verified Complexity Theory

A nice example is Hugo Férée, Samuel Hym, Micaela Mayero, Jean-Yves Moyen, David Nowak: Formal proof of polynomial-time complexity with quasi-interpretations. CPP 2018: 146-157 Their abstract (my ...
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### Why REFL rule is primitive in HOL Light?

DEDUCT_ANTISYM_RULE only applies to propositions, while REFL applies to all terms of all types. Your suggestion only shows that ...
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### What should a proof of correctness for a typechecker actually be proving?

There are a few different things you could mean by "prove that my typechecker works". Which, I suppose, is part of what your question is asking ;) One half of this question is proving that your type ...
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### Representations of Planar Graphs in Coq

The obvious resource for planar graphs in Coq would be the (modern port of) the four color theorem in Coq/SSReflect, by Georges Gonthier (and others) which obviously does need to define planar graphs. ...
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### Where is the quote "Informal proofs are algorithms, formal proofs are code" from?

He doesn't cite any references for it and Google doesn't return any results so I don't think he is really quoting from anywhere. The idea that a proof is a "construction" (a term in intuitionistic/...
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### Has the compactness theorem for FOL been formalized in Coq/Isabelle/etc?

Compactness for FOL was done in HOL by John Harrison, and reported at TPHOLs 1998. See Formalizing basic first order model theory.

### How would I go about learning the underlying theory of the Coq proof assistant?

The current Software Foundations book does explain all this later on: https://softwarefoundations.cis.upenn.edu/lf-current/ProofObjects.html So if you're following the book, just read on :)