Questions tagged [coq]
Coq is a formal proof management system. It is often referred to as a proof assistant.
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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
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Can a Prop show that all entries of a list equal Type@{U} for any U?
In Coq, is it possible to write a predicate (list Type -> Prop) that is only provable if all entries of the list are of the form ...
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Coq defining a hierarchy of collections of integers with infinitely many "levels"
I'm trying to formalize a small part of higher-order arithmetic in Coq as an exercise (Wikipedia article for second-order arithmetic).
It's straightforward to formalize something resembling second-...
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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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Recursive notations with forall quantifier
How can I implement a notation of the form: ∀ x ≤ y ≤ .. ≤ z ≤ t, φ in Coq?
A similar notation (but without quantifiers) appears here
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SSReflect tuple constructor: why not use phantom?
I was reading the mathcomp book learning about canonical structures and following along with the mathcomp source to compare how things were done in practice. Specifically I was looking at sections 6....
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Unfolding expressions in Coq by one layer
Are there any ways to unfold an expression in Coq by a single layer?
I have only come up with this obvious solution:
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I'm stuck trying to prove ∀x : ℕ, 3 | (x + 5x) with Coq
Specifically, I think what's got me is showing that ∀x y z : ℕ, (z|x and z|y) → z|(x + y), or that ∀x y z : ℕ, (x mod y) = 0 → z mod y = (z + x) mod y, depending on how you want to look at it. I know ...
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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 ...
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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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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.
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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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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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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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How can I prove this theorem with induction in Coq?
Lemma sum_square_p : forall n, 6 * sum_n2 n = n * (n + 1) * (2 * n + 1).
where sum_n2 is defined
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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:
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Found type UU where "?T" was expected
I am trying to solve a couple of exercises in coq. However, with the following code:
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How can I prove has_esp when using mathcomp.analysis?
How can I prove the following goal (which I believe to be true) using mathcomp.analysis?
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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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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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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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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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Cannot discriminate `0 = 1`
I am just practicing a bit with coq, doing some UniMath exercises and am trying to prove (0 = 1) -> empty. However, for some reason, I seem unable to reason ...
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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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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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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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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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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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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 ...
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Why does this trivial proof fail with structuring tacticals?
Given this:
Inductive color := Black | White.
Inductive point_state :=
| Occupied of color
| Empty
.
this works:
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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 ...
JNat♦
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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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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 ...
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For functions that are eq_dep equal, are their applications eq_dep equal without axioms?
Is it possible to prove the following theorem without axioms in Coq? Or is the following theorem equivalent to any well known axioms?
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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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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 ...
CommunityBot
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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 ...
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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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List of general purpose Coq sublanguages for defining custom tactics
I've been tweaking the Coq plugin template recently to try to get a feel for writing custom Coq tactics in OCaml.
It's tricky. You need to define an .mlg file (...
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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 ...
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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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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 ...
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Coq produce instance of a type `{x : T | P x}` inside an explicit definition given an `x'` of type `T`
I'm trying to formalize a simple type system in Coq as an exercise.
I have a type Item and a type {x : Item | IsNormal Item}. If ...
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Form of intros in Coq specifically for `forall` and explicitly for `->`
Are there tactics in Coq that are more limited versions (subtactics?) of intros?
I'm curious if there are any specifically for ...
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Code Review: Proving that a simple propositional logic satisfies Aristotle's Thesis
I'm proving that a simple propositional logic satisfies Aristotle's thesis.
I'm curious how to improve the code in question.
Here are the things I know that are wrong with it:
I'm using ...
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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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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 ...
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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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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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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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