Questions tagged [coinduction]
The coinduction tag has no usage guidance.
21 questions
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Question regarding coinduction
I'm rather stuck with the example of proof by coinduction, as generally known the principle is written as:
$$
P\subseteq F(P)\implies P\subseteq \nu F
$$
in an old problem When can the coinduction ...
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Definition of semantics of coinductive type
It is relatively easy to construct an object in set/class theory which has properties of any of the following: dependent sum, dependent product, W-types.
E.g. Dependent sum of a family F is just the ...
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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 ...
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(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 \...
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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 ...
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"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 ...
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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 "...
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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 ...
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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 ...
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"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 ...
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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
...
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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 ...
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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 ...
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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:
$$\...
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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 ...
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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 ...
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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 ...
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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 ...
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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?
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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 ...
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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 ...
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