Questions tagged [foundations]
Use this tag for questions about mathematical or logical foundations of proof assistants. Questions should be related in some way to proof assistants. Possible topics might include mathematical modelling, consistency and computability, universes, etc.
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What is predicativity?
Type systems, and the proof assistants based on them, are frequently divided into predicative and impredicative.
What exactly does this mean? I've heard the slogan "impredicativity means you can'...
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What are the bases for different Proof Assistants?
From the Wikipedia article on Proof Assistant it shows some Proof Assistants are based on Higher Order Logic, (HOL Light) and some are based on Dependent Types, (Coq).
Are there any other means upon ...
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Proof-theoretic comparison table?
I read this CSTheory SE post, which suggests that it is often not clear what variant of MLTT or CIC is being referred to. But I would like to know the proof-theoretic strengths of the various ...
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What is the trade-off to accepting impredicative propositions?
Impredicativity greatly increases the logical strength of a formal system, and impredicative propositions are also a consequence of various axioms including LEM and Zorn's Lemma. An impredicative sort ...
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Do you need a Hilbert style Epsilon operator for definitions in set theory?
I've started to play with mechanizing some set theory stuff. I'm not sure if I want a constructive flavor or not yet.
Anyhow you can do stuff like axiomize the empty set
$$ \top \vdash \exists P. \...
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Are Logics Based on Dependent Types Stronger Than Ones Without?
There have been several times during I came across statements like Isabelle/HOL's logic is not rich enough to formalize X on various places online and in during personal discussions. Or similar ...
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Tools for checking the consistency of a type theory
My question is twofold:
How do you define consistency (analogously to the concept in first-order logic) in the context of a type theory?
Are there any tools that can check consistency?
I have seen a ...
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Attempts to accommodate theories of different consistency strength in single assistant
TL;DR: How / where to formalize results concerning the logical strength of systems? Are proof assistants having a weak base theory but also enough infrastructure to make it feasible?
I'll start by ...
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Integration of proof assistants and Wikipedia-like websites?
I noticed that some of the Wikipedia.org entries that describe proofs of theorems also contain links to proofs of these theorems formalized in the Mizar proof assistant (example). There was one ...
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Unintentionally proven false theorem with type-in-type outside logic and foundations?
We are all familiar with Russell's paradox, and it is known that Per Martin-Löf proved that type-in-type is normalizing and consistent (which is false), by accidentally using an assumption in his meta-...
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What is the role of impredicativity in program extraction?
Is impredicativity useful for program extraction in Coq? For example is there some kind of realizability argument that depends on impredicativity?
Of course it doesn't seem to be necessary for program ...
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Construction of inductive types "the hard way"
Most theorem provers simply axiomize inductive types (or equivalently W types) in the abstract which is fine.
But I'm curious about explicit constructions of inductive types within the theory.
I ...
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Generic proof assistants/modularity of the proof assistants?
One notable feature of Isabelle is that it allows for the definition and in-built integrated support of alternative object logics via the interface of Isabelle/Pure in combination with Isabelle/Isar. ...
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Generating valid statements without a proof goal
Given some initial list of assumptions, I'd like to generate some true statements which follow from them without seeking a specific proof goal. I only need a small number of consequences from those ...
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Can the development of proof assistants make mathematicians switch their framework?
The Stack Exchange bot reminded me that I had committed myself to asking some questions, but please allow a possibly naive question, possibly of a philosophical nature rather than mathematical/...
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What is the computational complexity of theorem proving?
I heard theorem proving is a hard problem, hard enough that it contributed to an early AI winter. But how hard?
While reading about proof assistants, I have come to realize that there are many types ...
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Feferman's universes for proof assistants?
I asked this question on MathOverflow a few months ago, but received no answers.
There are some mathematicians who are comfortable with ZFC but uneasy with large cardinals. For them, it may be ...
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Why is $\Bbb Z = \Bbb N$ independent of Lean?
In this answer, it is noted
For a silly example, in ZFC with the usual encoding, $\mathbb{Z} \neq \mathbb{N}$, but in Lean it is well-known that this is independent. Of course in both ZFC or Lean, ...
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Lean and inaccessible cardinals
It seems well known that Lean's type theory is equiconsistent with ZFC + the existence of $n$ inaccessible cardinals for every $n>0$.
Suppose I want to ensure that my lean proofs depend only on ZFC,...
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Proof-theoretic strength of HOL compared to predicative dependent types
By HOL I mean something like inference rules of HOL Light with the 3 axioms of infinity, extensionality and choice ($\varepsilon$ operator).
By predicative dependent types, I am thinking of MLTT + W-...
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Can a proof engine be built based on graphs?
One of the more common ways to do proofs is using a deductive system.
Can proofs instead be done using graphs?
I am seeking papers that outline from the ground up how such a system works. If example ...
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Uniform notions of pattern matching at dependent types with impredicative proof strength
I am looking for axioms/inference rules that satisfy the following 3 conditions:
can be added to predicative intensional Martin-Löf type theory, so $\Pi$, $\Sigma$, equality type, with W-types, ...
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Does quantification over functions (STLC) increase strength beyond first order logic?
Does quantification over functions (STLC) increase strength beyond first order logic?
I want to add support for binders in my little constructive first order logic formalism I'm working on but I'm ...
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Does a proof assistant have to be interactive?
I have thought of systems like Coq & Isabelle as programming languages specialised for writing proofs.
A programming language might or might not have a REPL making it interactive but the REPL is ...
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MLTT with first-order reasoning and equality-reasoning information preservation
Terms in Extensional MLTT don't contain equality-reasoning information (implicit transports), effectively meaning data is lost, which is bad. But at the same time, higher-order reasoning (reasoning ...
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