Questions tagged [agda]
For questions regarding Agda: the programming language / proof assistant.
43
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Parametric adjunctions induce cowedges, a real proof [migrated]
$\require{AMScd}$ Since I am trying to formalise the following theorem in a proof assistant, I need to complete the fine details of it, no handwaving of sorts.
Theorem. Let $L_c\dashv R_c$ be a ...
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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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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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150
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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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211
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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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129
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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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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 ...
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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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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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131
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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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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:
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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, ...
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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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Proof of symmetry of universe-polymorphic Leibniz equality in Agda
Consider the following definition of universe-polymorphic Leibniz equality:
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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, ...
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215
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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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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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211
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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 ...
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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 ...
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133
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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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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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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 ...
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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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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. ...
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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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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 ...
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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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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 ...
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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?
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