Skip to main content
Share Your Experience: Take the 2024 Developer Survey

Questions tagged [coinduction]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
0 votes
0 answers
25 views

Can you make the following "coinductive" proof precise?

Question: I have a proof which overwhelmingly feels like it's a proof "by coinduction." Unfortunately I can't coinduct my way out of a paper bag, so I don't know if this actually works. I ...
9 votes
3 answers
1k views

Can a total programming language be Turing-complete?

I've seen two answers to this: Wikipedia says no: These restrictions mean that total functional programming is not Turing-complete. And the Wikipedia article cites D.A. Turner as the coiner of "...
3 votes
1 answer
330 views

Definition of M-type in type theory

According to nLab, M-types are the dual of W-types. What are the introduction and elimination rules for M-types? Edit: For example, the formation/introduction/elimination rules for W-types are: $$\...
1 vote
1 answer
47 views

(Co)-induction, fixpoints and inference systems

I'm learning about induction and co-induction. From what I know, given a set of judgments $U$ and an inference system $\Phi \subseteq \wp(U) \times U$, where $(\left\{ h_1,\dots,h_n \right\}, c) \in \...
2 votes
0 answers
86 views

How to prove by coinduction?

Previously, I've asked a question about coinduction. That gave me a lot of useful high-level insights on what coinduction provides, and what are the usefulness of coinductive proofs. This question is ...
1 vote
2 answers
72 views

"Largest set" in coinductive definitions

In several explanations of coinductive definitions (for example, in the answers to What is coinduction?), we're told that while an inductive definition gives us the smallest set with a specified set ...
3 votes
1 answer
275 views

When can the coinduction hypothesis be used?

We can use the induction hypothesis when we are proving a property for a structure that is well-ordered. I am aware that there is a proof for this. When it comes to coinduction, I'm confused. One of ...
12 votes
0 answers
282 views

Is extensionality for coinductive datatypes consistent with Coq's logic?

Given a coinductive datatype, one can usually (always?) define a bisimulation as the largest equivalence relation over it. I would like to add an axiom stating that if two members of the type are ...
3 votes
1 answer
178 views

Coinduction in mathematical analysis?

Coinduction is a logical principle that is somehow dual to induction. I'm struggling to understand it. Are there any interesting examples of coinduction in analysis? A few examples seem like they ...
2 votes
0 answers
85 views

"Practical coinduction" over $\mathbb N_\infty$?

I've just finished reading the paper Practical Coinduction by Kozen and Silva. What is the difference between induction over $\mathbb N$ and coinduction over $\mathbb N_\infty$? From the paper, it ...
2 votes
0 answers
72 views

Understanding Isabelle's implementation of coinduction

I'm studying how coinduction was encoded in Isabelle. At page 7 of the attached document, the author describes how some datatypes can be encoded as initial algebras. Here is one example: Finite lists ...
5 votes
1 answer
370 views

How to write a coterminating, effectful program?

[Using Idris for code examples and terminology, but the question is not about Idris per se] In a post titled A Neighborhood of Infinity, @sigfpe argues that "the kind of open-ended loop we see in ...
5 votes
1 answer
189 views

Proving with co-induction principles

I'm going through Adam Chlipala's "Certified Programming with Dependent Types" (available here for convenience), and I'm a bit stuck at internalizing the introduction of co-induction principle for the ...
2 votes
0 answers
46 views

How can one flip a stream using corecursion

Following is the definition of codata stream: codata Stream where hd : Stream −> A tl : Stream −> Stream For simplicity I assume I have just a ...
2 votes
1 answer
149 views

Does co-inductive and co-recursive types also have their recursors?

I'm new to type theory, and recently read introductory materials where dependent type are discussed. One of my friend asked me, "Those dependent types are having recursors & 'inductors'(dependent ...
0 votes
1 answer
149 views

Bisimulations: Proof that the following LTS are not bisimilar

I have the two LTS (labeled transition system) as seen in the following picture: And the book is telling me that between those two LTS, their $1$ and $1'$ are non-bisimilar. So I tried to get a ...
5 votes
0 answers
102 views

Is the set finite words over an alphabet a final coalgebra*?

I am studying what coinduction is. In particular, I am reading that coinductive datatypes can be defined as elements of a final coalgebra for a given polynomial endofunctor on $\tt Set$. I've seen ...
1 vote
0 answers
116 views

Typing rules of coinductive types?

Are there typing rules for specific coinductive types such as conat or stream, or even in general the M-types?
80 votes
2 answers
13k views

What is coinduction?

I've heard of (structural) induction. It allows you to build up finite structures from smaller ones and gives you proof principles for reasoning about such structures. The idea is clear enough. But ...