Questions tagged [agda]
For questions regarding Agda: the programming language / proof assistant.
58
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Examples of theories where tactic language is required for simple proofs
I was always under the impression, that separate tactic languages were generally considered to be vital for writing long proofs. I see tactic languages as a kind of interpreted DSL to generate ...
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0
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83
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How to improve/understand typechecking performance in Agda?
I've recently been trying to do some basic formalization of category theory in Agda. As part of this I need to prove a bunch of basic properties around products/monoidal categories. A bunch of these ...
- 171
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2
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299
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Reasoning about CwFs in a proof assistant
I've been chatting with folks on Mastodon about this but the perspective there is markedly Agda-focused, so I thought I'd ask here for some broader opinions.
What tools/libraries are there for ...
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96
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Inspecting proof terms in Agda
In Coq, if I am unsure how a definition foo is defined, I can use the command Print foo. to inspect what ...
- 4,153
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114
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Cumulativity unsafe in Agda 2.6.3
I've been reading and understand the design choices behind Agda not having a cumulative universe hierarchy by default. However, while tinkering with the language, I've noticed that, at least for Agda ...
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When we can use refl?
I am confused with the context related to identity type in HoTT
In particular, for a family $C : (a =_A a) \to U$ dependent on a loop, we cannot apply path induction and consider only the case for $C(...
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764
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86
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6
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208
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Is there a formalism of "coinductive" data types with negative occurrences?
Consider the following program in Haskell:
...
2
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1
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68
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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,
...
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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 ...
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173
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What was the first proof assistant with eta equality for records?
The Agda language supports eta equality for (non-(co)inductive) record types:
...
- 486
8
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1
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199
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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
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0
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513
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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 ...
- 2,912
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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 ...
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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(...
- 299
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135
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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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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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184
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How to convince Agda that definition is terminating in the presence of unapplied recursion?
In the following, termination check fails for Df:
...
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168
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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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2
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256
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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 ...
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1
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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 (...
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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 ...
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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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93
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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 ...
- 305
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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.
...
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1
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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:
...
- 305
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198
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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:
...
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2
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240
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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:
...
- 305
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2
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167
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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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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
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2
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432
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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
...
- 2,171
6
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1
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209
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Proof of symmetry of universe-polymorphic Leibniz equality in Agda
Consider the following definition of universe-polymorphic Leibniz equality:
...
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138
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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
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1
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292
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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 <...
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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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343
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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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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 ...
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238
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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 ...
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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 ...
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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 ...
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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 ...
- 1,015
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2
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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:
...
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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 ...
- 345
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2
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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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356
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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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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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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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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
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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
...
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