Questions tagged [agda]

For questions regarding Agda: the programming language / proof assistant.

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2 votes
1 answer
42 views

How to write if_then_else function that accepts a type argument explicitly?

While this function example works by declaring type A implicitly between brackets, ...
10 votes
1 answer
428 views

What are the complex induction patterns supported by Agda?

A question was recently asked on the Coq-club mailing list on Coq rejecting a nastily nested inductive type. We encountered a similar difficulty while trying to port code from Agda to Coq: Agda ...
7 votes
0 answers
137 views

What was the first proof assistant with eta equality for records?

The Agda language supports eta equality for (non-(co)inductive) record types: ...
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6 votes
1 answer
160 views

Termination and confluence -- which goes first?

I'm implementing a version of cubical type theory where the well-definedness of pattern matching functions is implied by: the well-typedness of the clauses (type check) the coverage of the patterns (...
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22 votes
0 answers
443 views

Categorical semantics of Agda

I would like to know the state of the art regarding the categorical semantics of the type theory implemented by Agda — or at least some approximation of that type theory that is amenable to ...
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1 vote
1 answer
90 views

Agda - Help with importing Haskell function via FFI

I'm trying out Agda's Foreign Function Interface with a simple correctness proof of quicksort (which I have defined in Haskell), but I'm having trouble importing the Haskell function. I've been ...
6 votes
1 answer
122 views

What are the differences and similarities between records, coinductive records, and codatatypes?

Record types are a common feature in most paradigms. Agda also allows defining records with the coinductive keyword. Lastly there are the seemingly more exotic co(...
5 votes
0 answers
105 views

What are the conditions for Agda to detect that induction-recursion has a least fixed point?

This is a 3rd in a series of questions, spurred by my attempts to encode an argument by Danielsson [1] [2] regarding existence of syntactically non-strictly positive datatype. The idea (rephrased): ...
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5 votes
1 answer
160 views

Why can termination checker affect strict Prop in Agda?

In https://dl.acm.org/doi/10.1145/3290316 section 4 (which I believe is a standard reference), the authors claimed that Voevodsky's propositional resizing rule justifies the consistency of an ...
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5 votes
1 answer
140 views

How to convince Agda that definition is terminating in the presence of unapplied recursion?

In the following, termination check fails for Df: ...
1 vote
0 answers
159 views

Descriptions of heterogenous datatypes

When attempting to describe the datatype as appearing in my previous question, using indexed descriptions in style of The Gentle Art of Levitation to describe this datatype (using Agda for examples): <...
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7 votes
2 answers
235 views

Proving strict positivity in Agda

In Agda we can prove termination of functions by using well-founded relations, is there a guideline for proving datatype declarations strictly positive, possibly via use of some container techniques ...
  • 497
1 vote
1 answer
148 views

Does Agda's --injective-type-constructors flag have canonicity?

Since 2010/01/07, when the Anti-classicality of Agda was proved by Chung-Kil Hur, Agda's --injective-type-constructors is separated from the main branch of Agda (...
3 votes
1 answer
79 views

Can I specify `refl`'s parameter explicitly in Agda?

I'm working on some proofs in Agda that, for educational purposes, explicitly use the path induction principle (which I've defined myself) rather than pattern matching. In the theoretical mathematical ...
5 votes
1 answer
146 views

Application of `SSet` in Agda

Is there any presentable work that makes essential use of SSet in Agda? I kind of get the theory, but I'm not sure about the practical value of this. The ...
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3 votes
1 answer
86 views

Using induction to define Indexed family of HITs in agda

I am looking to define a family of HITs parametrized by $\mathbb{N}$. I want $(-)$-glob : $\mathbb{N} \to Type$, so that $n$-glob is the n-dimensional glob. I know how to construct the n-$glob$ by ...
5 votes
1 answer
149 views

Strictly-monotone "max" operation for constructive Brouwer-trees?

The Setting I'm trying to use Agda's well-founded ordering to prove that something is terminating using Brouwer Trees i.e. ...
3 votes
1 answer
93 views

Agda Error after reload when successfully filling a goal

So I'm not sure what this error is and I'm not even sure how I would find out. I'm trying to code up a proof that refl is a right identity. I've gotten to this stage: ...
5 votes
2 answers
157 views

Agda: Cannot Instantiate Metavariable

I'm running into a certain error while trying to code something up in cubical Agda, but I can't understand the error. Here is my code: ...
9 votes
2 answers
222 views

How to access local definitions in Agda

I'm working in cubical agda. I am wondering how to access local definitions once outside of the local environment. For example, suppose I have the following code: ...
4 votes
2 answers
155 views

Turning off some sProp checks

In Definitional Proof Irrelevance Without K, inductives in sProp need to satisfy three conditions to allow large elimination: (1) Every non-forced argument must be in sProp. (2) The return types of ...
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8 votes
2 answers
485 views

Curly Braces and Lambdas in Agda

The docs on lambdas in Agda provide two forms of lambda: a curly brace based version, and the where syntax. But while writing some programs, I stumbled across a third version: one pattern, no braces, ...
9 votes
2 answers
412 views

Defining coercion for proof irrelevant equality

Say I would like to define coercion for proof irrelevant equality between types. In Coq I try ...
  • 2,051
6 votes
1 answer
198 views

Proof of symmetry of universe-polymorphic Leibniz equality in Agda

Consider the following definition of universe-polymorphic Leibniz equality: ...
  • 5,424
4 votes
1 answer
117 views

Literate Agda: How do I change the size of the displayed code?

I am making Beamer slides for a talk about an Agda formalisation of mine. In order to make the code fit onto the slide, I need to scale the font size. One would think that this might not be so hard, ...
9 votes
1 answer
238 views

Case splitting with quotient types in Cubical Agda

I'm getting started with Cubcal Agda and I'm quite confused. I've got a HIT A defined, with a path constructor eq returning <...
9 votes
2 answers
240 views

What's the benefit of having pi and sigma types with an invariant parameter?

Ulf Norell wrote this in his PhD thesis (figure 1.6): This contradicts my stereotype on pi & sigma types, where pi parameter should be contravariant and sigma parameter is covariant. Why is Agda ...
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5 votes
1 answer
290 views

What subtyping rules does Agda support?

Andrea Vezzosi said Cubical Agda made cubical subtyping coercion-based (inS and outS) because it's a bad idea to introduce more ...
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10 votes
3 answers
939 views

Constructive proof of strong normalization for simply typed lambda calculus

I'm reading Girard's Proofs and Types, and in section 4.4 he writes: Lemma: t is strongly normalisable iff there is a number ...
6 votes
1 answer
221 views

Parameterized Datatypes in a Universe à la Tarski?

I'm wondering, is there a way to make a Universe à la Tarski that models all of the types in an open type theory, where there can be user defined parameterized inductive types? For context, I'm trying ...
7 votes
1 answer
101 views

Decidability for an infinite sequence of decidable propositions

I've been trying to prove this in Agda: lemma : (P : ℕ → Set) → ((n : ℕ) → Dec (P n)) → Dec (Σ[ n ∈ ℕ ] ¬ (P n)) In words: We have an infinite sequence of ...
8 votes
1 answer
126 views

Are `P x` and `▸ ((next P) ⊛ (next x))` equivalent in Guarded Cubical Agda?

In Guarded Cubical Agda there's ▹_ : Set i → Set i and ▸_ : ▹ Set i → Set i. If I've got ...
8 votes
1 answer
209 views

Why is `--irrelevant-projections` unsafe in Agda?

In Agda, irrelevance is an annotation which marks a parameter, record field, or definition which "will only be typechecked but never evaluated", with the consequence that irrelevant ...
6 votes
2 answers
90 views

Can I use if_then_else on indexed paths in HITs?

I want to define a function out of an indexed higher inductive type, and am running into some problems. Here is a somewhat contrived minimal example of what I'm doing: ...
6 votes
2 answers
104 views

Data constructors and universe constraints

In the code below: data Foo₁ : Set where foo₁ : Set → Foo₁ data Foo₂ (A : Set) : Set where foo₂ : Foo₂ A type₂ : {A : Set} → Foo₂ A type₂ = foo₂ for the ...
3 votes
2 answers
142 views

How do I make use of an irrelevant equality in a proof?

open import Agda.Primitive import Relation.Binary.PropositionalEquality as Eq open Eq public open Eq.≡-Reasoning Suppose I have a dependent pair whose second ...
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21 votes
2 answers
304 views

In what intensional type theories can absurdity be made definitionally proof irrelevant?

Various type theories have, over the years, explored extending the definitional equality with a variety of eta-laws and various forms of proof irrelevance. Quite a lot of systems manage eta for ...
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7 votes
1 answer
174 views

When should I use `no-eta-equality` in Agda records?

In a recent development, I used the no-eta-equality attribute on a record. The motivation in that case was ad hoc: I noticed that I'd get better metavariable ...
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15 votes
2 answers
294 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 ...
6 votes
0 answers
217 views

Can Agda proofs be trusted and do they conform to a specific, well-defined type theory?

When using Agda proofs, how can you know whether to trust it? Also, do they conform to a specific, well-defined type theory? Credit
  • 1,258
8 votes
1 answer
118 views

Is it possible to define `fib` coinductive stream w/o sized types?

In Agda with --guardedness, we can define productive definitions using copatterns like the following ...
  • 5,424
4 votes
1 answer
104 views

Why does Agda's `--without-K` option also affect the indices of inductive types, where they have to be of lower levels?

In my understanding, --without-K changes the index unification algorithm so that deletion rules are no longer used, as described in Jesper Cockx' "pattern ...
  • 5,424
24 votes
4 answers
770 views

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 ...
13 votes
2 answers
292 views

In Agda, why is universe polymorphism explicit?

I've noticed that the process of coding a definition in Agda that is universe polymorphic usually requires little thought in how the Level parameters are specified. ...
11 votes
1 answer
318 views

Why did Agda give up cumulative universes?

In Ulf Norell's PhD thesis, which is considered the standard reference of the Agda 2 language, the universes are cumulative, say, Set i is not just an instance of <...
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7 votes
1 answer
100 views

How to state and prove the associativity of append of sized vectors with homogeneous equality?

Append (denoted ++) for sized vectors is obviously associative, i.e. a ++ (b ++ c) = (a ++ b) ++ c for ...
  • 5,424
4 votes
1 answer
65 views

When installing Agda does one have to be attentive to a version?

If I want to install Agda do I have to be attentive to versions, breaking changes, etc.? A comment from an earlier question noted Agda version for installed libraries
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6 votes
1 answer
146 views

Does Agda have breaking changes?

Since Agda is software and sometimes software has breaking changes, e.g. Python 2 vs Python 3 (ref), does Agda have any significant breaking changes that one would need to be concerned about? Also ...
  • 2,796
6 votes
2 answers
449 views

How to run Agda?

I want to run Agda, but I do not know how to run functions. I need to run "Hello, World!"; how can I do this?