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22 votes

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 ...
Andrej Bauer's user avatar
  • 29.1k
18 votes
Accepted

Yet another constructive (Coq) proof that `nat -> nat -> nat` is not bijective. How to explain it to myself?

First, let me say that "constructive" does not imply "all maps are Turing computable". It means “no excluded middle and axiom of choice were used“. In constructive mathematics the ...
Andrej Bauer's user avatar
  • 29.1k
14 votes
Accepted

Defining inductive types in intensional type theory purely in terms of type-theoretic data

It turns out that $W$ types plus identity types (eq/= in Coq) allow you to construct pretty much all the general inductive types ...
Jasper Hugunin's user avatar
12 votes

Formal semantics of OCaml in Coq

Have you seen Arthur Charguéraud's PhD thesis, Characteristic Formulae for Mechanized Program Verification? Rather than building the type system and small-step semantics as inductive relations, he ...
Neel Krishnaswami's user avatar
10 votes
Accepted

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

The question can be interpreted in two ways: Whether the implementation does implement a given typing system $T$? Whether the typing system $T$ does prevent the errors you think it should? The ...
Martin Berger's user avatar
10 votes
Accepted

What exactly is "large elimination"?

You are incorrect about the definition of large elimination: it refers to the ability to build values of type $\mathrm{Type}$ by eliminating an inductive value. The canonical example: ...
cody's user avatar
  • 13.9k
8 votes

Formal semantics of OCaml in Coq

You could be interested in Jacques Garrigue's A Certified Implementation of ML with Structural Polymorphism and Recursive Types, which establishes the soundness of static and dynamic semantics and ...
gasche's user avatar
  • 2,040
8 votes
Accepted

Is there a formalization of normalization of impredicative system F?

Coq without Prop is not strong enough, because it's basically Martin-Löf type theory with universes. Coq with Prop is strong enough, because you can encode sets of normalizing terms via predicates $S :...
Neel Krishnaswami's user avatar
6 votes
Accepted

Impredicativity + large eliminations (with no excluded middle) in Coq

No: strong elimination of large inductive types (SELIT) is itself inconsistent because it breaks the layering of universes by trivially allowing you to smuggle a large value in a Prop box and take it ...
Stefan's user avatar
  • 390
6 votes
Accepted

Defining finite sets inductively in a proof assistant?

There are many variants of finite sets in constructive mathematics. One that can be defined using just inductive definitions, and is therefore amenable to formalization in type theory, is the ...
Andrej Bauer's user avatar
  • 29.1k
6 votes
Accepted

How does axiom K contradict univalence?

You will certainly find it natural that most types, like structures, admit different isomorphisms. Just take the type $\textbf{2}$, with inhabitants $0_\textbf{2}$ and $1_\textbf{2}$. It admits 2 ...
L. Garde's user avatar
  • 176
6 votes
Accepted

Formalization of matching logic (logic behind K Framework)

There is a number of formalizations of matching logic in various proof assistants. I am a co-author of the first one in the following list; thus I have more insight into that one. I am not aware of ...
Jan Tušil's user avatar
5 votes

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 ...
wren romano's user avatar
5 votes
Accepted

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. ...
cody's user avatar
  • 13.9k
5 votes
Accepted

Universe polymorphism: the inference of universes and their constraints

It's complicated because universe constraints are simplified during inference (in order to avoid an explosion of constraints). Have a look at: Matthieu Sozeau and Nicolas Tabareau: Universe ...
Andrej Bauer's user avatar
  • 29.1k
5 votes

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.
Michael Norrish's user avatar
4 votes

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 :)
Catalin Hritcu's user avatar
4 votes

Representations of Planar Graphs in Coq

I just wanted to make some additional comments not already covered by Cody's nice answer, and also address question (2). First, Gonthier goes into detail about the representation of planar maps used ...
Noam Zeilberger's user avatar
4 votes

How can you build a coinductive memoization table for recursive functions over binary trees?

It's easy enough to get the recursion pattern to work with sized types. Hopefully the sharing is preserved through compilation![1] ...
Saizan's user avatar
  • 141
4 votes

Dependent eliminator for empty type in intensional Martin-Löf type theory

$\text{absurd} : (A : \text{U}) \to 0 \to A$ and $\text{elim} : (A : 0 \to \text{U}) \to (x : 0) \to A\,x$ are equivalent. To go right, use $\text{absurd}\,(A\,x)\,x$. To go left, use $\text{elim}\,(\...
András Kovács's user avatar
3 votes

Notation problem

You need to add a space after the d, before the ].
Pierre Jouvelot's user avatar
3 votes

What exactly is "large elimination"?

Here is another attempt to illustrate large elimination. Consider the two following two ways to define a type: as an inductive or by induction. For instance, here are two definitions of a vector with ...
drcicero's user avatar
3 votes

Modeling objects (OOP) in dependent type theory

There is a substantially expanded follow paper Andreas Abel, Stephan Adelsberger, Anton Setzer: Interactive Programming in Agda - Objects and Graphical User Interfaces. It contains an Agda library ...
Anton Setzer's user avatar
2 votes

Why does Coq have Prop?

The principle of propositions-as-types (or formulas-as-types), also known as the Curry-Howard correspondence, is the key idea for viewing (intuitionistic) type theories as logical systems and to apply ...
Postmortem's user avatar
2 votes

How does axiom K contradict univalence?

For a quick reference, here's (equation 8) a proof sketched in Agda. But I guess you're asking for the idea, and I think the reference is kinda technical. When you say 'univalence', you not only mean ...
ice1000's user avatar
  • 965
1 vote
Accepted

Can I define nested mutually dependent types in Coq?

You can do mutually defined inductive types in Coq with the «with» syntax You end up with this definition: ...
Mysaa's user avatar
  • 26
1 vote

Yet another constructive (Coq) proof that `nat -> nat -> nat` is not bijective. How to explain it to myself?

To answer very directly: You have a constructive proof in Coq but it is not the case that the enum : nat -> nat is assumed (in Coq) to be computable. In a sense, ...
ezrakilty's user avatar
1 vote
Accepted

A Coq question : How to prove the image of the two same valued variables under a function are same?

You can't do that. You can actually define a function which doesn't respect Q's setoid structure. ...
Kazuhiro Kobayashi's user avatar
1 vote

Defining functions on non-inductive types using LEM in Coq

I’m not sure I get what you are trying to do – or, rather, how your example corresponds to your goal. Indeed, in your code you postulate those three types a, ...
Meven Lennon-Bertrand's user avatar
1 vote
Accepted

Lack of atomic propositions in the Calculus of Constructions from ATTAPL textbook

I have no thoughts on whether this is on-topic for TCS and if need be I can repost my answer elsewhere. But I think the question is a good opportunity to demonstrate how to think about the CoC pending ...
dunnl's user avatar
  • 34

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