Questions tagged [coq]

Coq is a formal proof management system. It is often referred to as a proof assistant.

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2 answers
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What are the main differences between Coq and Lean?

Coq and Lean are two of the most common proof assistants out there (but the question of course applies to other proof assistants too). What are the main differences between Coq and Lean? Ideally it ...
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30 votes
2 answers
738 views

What exactly is setoid hell?

One of the only arguments I've heard about why Lean is better than Coq is that you can construct quotients of built-in structures by default. (In Coq, you apparently have to use Setoids instead of ...
24 votes
4 answers
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What are the differences between MLTT and CIC?

In the theory and design of proof assistants based upon dependent types, I feel like there’s a somewhat cultural divide between the "MLTT" world (with Agda as the main representative proof ...
19 votes
1 answer
322 views

What is the state of coinductive types and reasoning in Coq?

Ever since the work by Gimenez for his PhD thesis, Coq has supported positive coinductive types. For example, the type of always-infinite streams containing elements of type ...
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18 votes
1 answer
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When to use coinductive types?

One could define streams in the following manner ...
16 votes
3 answers
851 views

Well-foundedness: classical equivalence of no infinite descent and accessibility

I have often seen the claim that in a classical setting, well-foundedness of a relation > defined as the absence of an infinite descent ...
15 votes
1 answer
213 views

How do Coq's bidirectionality hints (`&`) affect type checking?

I have used Coq's bidirectionality hints (placement of & in a call to Arguments) to some effect, mostly by trial and error. ...
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14 votes
1 answer
812 views

Why not have `Prop : Set` in Coq?

My understanding of Coq is that Prop : Type_1, Set : Type_1, and then Type_1 : Type_2, ...
14 votes
1 answer
186 views

Converting between formulations of reals in Coq

While trying to answer more concretely a question on floating points, I tried proving a simple statement using the Flocq library of Coq. However, I got stuck before really exercising it because I am ...
13 votes
2 answers
232 views

Is eta-equality provable in Coq?

Trying to answer another question, I can't help second guessing myself about whether eta equality is provable in Coq. In particular, can one prove ...
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13 votes
3 answers
715 views

How to extract the witness from exists in Coq in function notation/without destructing?

Assuming I have some definition with a forall and an exists like so: Definition fooable A B P := forall a : A, exists b : B, P a b. Then on an intuitive level, I ...
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3 answers
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Calculus of (inductive) Constructions: Do inductive definitions increase proof strength?

Question Is CiC stronger than CoC, in terms of proof strength? Context To illustrate the kind of confusion I am in, and what I'd like to learn from the answer, here is part of my inner monologue: If I ...
13 votes
2 answers
204 views

What are the upsides and downsides of typed vs untyped conversion?

What are the tradeoffs between untyped and type-directed conversion in dependent type theory, and is there any consensus on what's "better"? Background Generally speaking, in dependent type ...
13 votes
1 answer
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How do I turn off the aggressive auto-indent in Proof General/Coq

I have installed Proof General via Doom Emacs' coq module, keeping most settings as whatever default that module sets. Sometimes the automatic indentation this ...
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11 votes
2 answers
393 views

How does the formal proof of the four color theorem work?

Over a decade ago, Georges Gonthier, gave a formal proof of the four color theorem. I have a mental picture of how the proof works, and I'd like to see if it is correct. The original proof of the ...
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11 votes
1 answer
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What are the differences between jsCoq and the versions of Coq that can be downloaded?

I found a site where I can use Coq in my browser: jsCoq Interactive Online System. Are there any major differences between the experience I'm getting on this site and the downloadable version(s)?
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11 votes
2 answers
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Proving uniqueness of an instance of an indexed inductive type

Consider the simple indexed inductive type Inductive Single : nat -> Set := | single_O : Single O | single_S {n} : Single n -> Single (S n). Intuitively, I ...
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11 votes
1 answer
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What did Coq 8.15 change about divmod?

I noticed that Coq 8.15 (possibly 8.14) made some significant changes to divmod. In particular, Nat.divmod_0q0 seems to have been removed and the ...
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11 votes
1 answer
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Naming conventions (letter case, underscores, &c) for Coq

Does Coq have an established convention/style for constructors, variables, terms, &c? An established convention that isn't exceptionless is totally fine. For example, terms should be lowercase ...
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10 votes
1 answer
514 views

What is the role of impredicativity in program extraction?

Is impredicativity useful for program extraction in Coq? For example is there some kind of realizability argument that depends on impredicativity? Of course it doesn't seem to be necessary for program ...
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10 votes
1 answer
178 views

Is there a Mizar-like sublanguage for Coq?

Isabelle has the frontend Isar which mimics some features of the Mizar system. I'm curious if Coq has anything similar, i.e. an alternative to tactic scripts that's designed to be readable or similar ...
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9 votes
1 answer
545 views

What axioms do I need to search the naturals?

Theorem search {P : nat -> Prop} (dec : forall n, {P n} + {~P n}) : ~~(exists n, P n) -> {n | P n}. Admitted. I don't think this is provable in Coq without ...
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2 answers
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Defining coercion for proof irrelevant equality

Say I would like to define coercion for proof irrelevant equality between types. In Coq I try ...
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9 votes
2 answers
531 views

Is coercion to a higher universe injective?

A type that is a member of a universe can be coerced into a higher universe. Is that coercion injective? That is, if two elements of U1 are equal after being coerced to U2, does that imply they are ...
8 votes
2 answers
685 views

Explanation of Coq math-comp repositories

How are the Coq math-comp account and repositories related? Details One of my side goals is to try to keep the tags on this site meaningful and useful. Today I ran into this question: How to prove ...
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8 votes
2 answers
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Learning Math Proof via Proof Assistants

I want to learn proof based mathematics and it looks like a proof assistant like Coq and Lean could be a good way to go about verifying my proofs, without needing a PhD on hand to check through all my ...
8 votes
3 answers
188 views

How to prove that addition is commutative for conatural numbers in Coq

I have defined the conatural numbers, bisimulation (extensional equality?) and addition as follows, ...
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8 votes
2 answers
151 views

How to prove `forall m n : nat, m == n -> m = n`?

I am learning Coq with ssreflect. Just to understand things, I've proved forall a b : bool, a == b -> a = b but I can't figure out how to prove ...
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8 votes
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Coq: Recursive Smart Constructors and Sigma types, how to avoid axioms

I am using a recursive smart constructor to return a sigma type, which includes the property that the type was constructed in a smart way. This is very basic compared to the smart constructors and ...
8 votes
3 answers
254 views

Has anyone accidentally "proven" a false theorem using what was later found to be a critical bug?

Critical bugs are periodically found in Coq, and I assume in other proof assistants as well. We are still happy to mostly trust the proof assistants, partly because these critical bugs are relatively ...
7 votes
1 answer
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Tactic unification vs evarconv in Coq

I gather, from practical experience and Zulip hearsay, that Coq has two unification algorithms, known as “tactic unification” and “evarconv”. However, I can't find any documentation on these from a ...
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1 answer
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Why Coq's `Include` is designed to instantiate functor with current interactive defining module?

It is surprising for me to see that Coq can Include a functor and will instantiate it with the current interactive module. Coq Ref Manual: Command Include ...
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1 answer
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How to replace a function by its body

I have this function: Definition bexp x y := bexp_r x y [true]. And I have this goal: value (bexp [] y) = 0 ^ value y I want ...
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6 votes
2 answers
251 views

Problem proving a binary add function

I'm fairly new to the Coq language and I want to prove a function that does an binary add from numbers represented as a list (least significant bit upfront). I have created this badd function that ...
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6 votes
1 answer
98 views

Does Coq's Module and Functor type-check incrementally?

I am trying to search the following questions online but I failed: When applying a functor (parametrized module), will the contents inside the functor be re-type-checked? Will Coq's command ...
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1 answer
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How to set defaults for implicit arguments when they can't be inferred?

If I had a module declared as follows in file A.v: Section A. Context {𝒳 : Set}. Inductive abt := Abt_leaf (x : 𝒳) | Abt_node. End A. And in another file, B....
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In VSCoq, how to issue an arbitrary Gallina query via the prompt?

I’m currently trying out Visual Studio + VSCoq, having previously used Emacs + Proof General for Coq, and there’s one part of my usual workflow that I haven’t managed to replicate yet. Is there a way ...
6 votes
0 answers
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How to write heavily indexed proofs?

I've been playing with hereditary substitution. However, things get very awkward because substitution isn't total unless you index by the environment somehow. In my old approach terms were not indexed ...
5 votes
1 answer
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Formal description of Coq’s termination checker

Coq’s built-in termination checker accepts some rather intricate recursion patterns with functional values in data types, as shown by this example ...
5 votes
2 answers
112 views

Basics of real numbers in mathcomp (Coq)

I'm trying to play around with the Mathematical Components library in Coq but am having trouble writing basic, concrete statements about real numbers. E.g., I'd like to define ...
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3 answers
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Hide the value of a hypothesis introduced by `pose`, show only its type

In proof mode, if I know an expression e of type T, I can write pose foo : T := e to add <...
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5 votes
2 answers
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How to provide a countType when using mathcomp?

The following snippet can't pass type checking. From mathcomp Require Import choice. Definition exfn (A:countType) := false. (* Fail *) Check exfn nat. Failed with ...
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1 answer
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How to implement first-order relational structures in Coq?

I'm trying to define a first-order relational structure in Coq. I have a way to define a pre-first-order-relational-structure, which is not a standard notion, but seems simple enough. I also have a ...
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5 votes
1 answer
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Unexpected eta expansion in constant definition

Start from this example in section 2.1.14: Polymorphic Universes in Coq's reference maunual (slightly modified): ...
5 votes
1 answer
143 views

How to fold simplified goal?

I have written a Program Fixpoint called generic_debugging_algorithm with a measure, and I have one obligation left to prove. To me the obligation reads like: ...
5 votes
1 answer
62 views

How to evaluate proof terms through opaque definitions?

Is there is a way to force computation over opaque terms, for the purposes of debugging/meta-analysis of proof scripts. I understand why Coq doesn’t do this by default, and guess it would probably ...
5 votes
0 answers
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What are Generic Arguments in Coq and how are they structured in their OCaml code?

I was trying to figure out why it seems that in a Coq generic argument there seems to be 3 arguments to the constructor GenArg when according to me there should ...
5 votes
0 answers
119 views

How does one systematically traverse OCaml representations of Coq ASTs terms?

I want to be able to (tree) traverse Coq terms (e.g. in OCaml or using their s-expression representation in any language). My main challenge is to figure out how to do this systematically because I ...
5 votes
0 answers
82 views

Display style proofs using Coq

How to display proofs using in Gentzen tree style and (or) Fitch-style, using CoqIDE or JsCoq? PS: I'm rookie used coq.
5 votes
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Prove equality in a record type

I am trying to prove something about monoids an categories. This results in the following (partial) proof: ...