Questions tagged [equality]
49 questions
12
votes
0 answers
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 related by the bisimulation, they are equal in the…
Jannis Limperg
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8
votes
2 answers
Decidability of Equality of Radical Expressions
Consider terms built from elements of $\mathbb Q$ and the operations $+,\times,-,/$, and $\sqrt[n]{\,\cdot\,}$ for each natural number $n$. Given the promise that two terms are well-formed -- that is, there is no division by zero, and no even roots…
Mees de Vries
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8
votes
1 answer
Why is `map insertionsort` not to equal to`map mergesort`?
In the type theory podcast ep. 3, Dan Licata claims that the fact that for every input, insertionsort and mergesort give the same result does not imply that the result would be equal when used as higher order functions as arguments to a third…
Drathier
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8
votes
1 answer
Elimination rule for the equality type aka J axiom
I'm implementing a interpreter for lambda calculus,
and now I want to add the equality type.
The introduction rule for it is easy,
but the elimination rule is rather obscure for me.
I found this stackoverflow thread, but it explains the J axiom only…
盛安安
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7
votes
0 answers
Extensional constructs in minimal extensional type theory without eta equality
The extensional version of Intuitionistic Type Theory is usually formulated in a way that makes extensional concepts like functional extensionality derivable. In particular, equality reflection, together with $\xi$- and $\eta$-rules for $\Pi$ types…
fsestini
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7
votes
2 answers
Computability of equality to zero for a simple language
Suppose we have a tree in which leaves are labeled with a set of numbers $L$, and internal nodes with a set of operations $O$.
In particular $L$ can be $\mathbb{N}, \mathbb{Z}$ or $\mathbb{Q}$, and can optionally contain $\pi$ and/or $e$.
$O$ can be…
miniBill
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6
votes
2 answers
Elegant algorithm to semi-decide if two lambda calculus terms are equivalent
Given two lambda terms $t_1$ and $t_2$, it is semi-decidable if they are equivalent (i.e. can be rewritten as each other using alpha, beta, and eta conversions). An algorithm to do this is to try every sequence of rewrites starting with $t_1$, and…
Christopher King
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5
votes
1 answer
Is the validity of some instance of an equational problem decidable?
Is the following FOL-problem (equality is a logical symbol)
effectively decidable?
Given. A finite equation system $E$ and an equation $s = t$.
Question. Is there a substitution $\sigma$, such that $\sigma(E)
\models \sigma(s = t)$?
Some useful…
Steffen Schuler
- 153
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5
votes
1 answer
Definitional equality of two propositions about propositional equality
Martin Hofmann states in Extensional Concepts in Intensional Type Theory (§1.1 p.3) that:
It is important that definition equality is pervasive so if M and N are definitionally equal then P(M) is definitionally equal to P(N) whatever P is. In…
al pal
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4
votes
1 answer
Check if two items are equal after replacing
Let's say that an item is either a natural number or a list of items. Examples of items are:
1
[2]
[4, [3, 1], 3, 4]
A rule states that two items are equal. For example:
1 = 2
3 = [3, 1]
[4, 3] = [1, 5]
When using these example rules, we can…
Paul
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4
votes
2 answers
Unification --- most specific unifier
In unification, given a set of equations, a standard problem is to compute a most general unifier (mgu). I am interested in a somewhat reversed problem. Imagine having a set of equations that do not have an mgu, like this one:
x = a
x = b
x here is…
zpavlinovic
- 1,664
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4
votes
1 answer
Decidability of equality, and soundness of expressions involving elementary arithmetic and exponentials
Let's have expressions that are composed of elements of $\mathbb N$ and a limited set of binary operations {$+,\times,-,/$} and functions {$\exp, \ln$}. The expressions are always well-formed and form finite trees, with numbers as leaf nodes and…
GolDDranks
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4
votes
1 answer
Is unification over regular expression equations doable?
By way of example, suppose I know that $X + a = b + Y$ where $X$ and $Y$ are variables standing for regular expressions, then $(X, Y) = (b, a)$ is a solution to this set of equations.
Generalizing over all such equations, is there a computable…
Jonas Kölker
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4
votes
2 answers
Why values can not be replaced with their extensionally equal values in an intensional system?
Thomas Streicher states in Investigations into Intensional Type Theory(§Introduction p.5) that:
Although in Intensional constructive set theory (Intensional Type Theory) one can do most of the things one wants to do... certain theorems simply do…
al pal
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4
votes
1 answer
Definition of extensional and propositional equality in Martin-Lof extensional type theory
Martin Hofmann states in Extensional Concepts in Intensional Type Theory (§1.1 p.[4-5]) that:
A similar situation occurs in extensional Martin-Lof type theory where propositional and definitional equality are forcefully identified by the equality…
al pal
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