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Is there a Proof assistant which also is an esoteric programming language? If so, then how many there are, approximately?	Yes, there are proof assistants that also function as esoteric programming languages. An example of this is Coq, which is primarily a proof assistant but has been used in various esoteric programming language contexts. Another example is Agda, which is similar in that it can be used for both proof development and progr...	[ { "from": "human", "value": "Is there a Proof assistant which also is an esoteric programming language? If so, then how many there are, approximately?\n" }, { "from": "gpt", "value": "Yes, there are proof assistants that also function as esoteric programming languages. An example of this is Coq,...
In proof mode, if I know an expression e of type T, I can write pose foo : T := e to add foo := e : T as a hypothesis. Often, I don't care about that foo is the same as e, I only care that foo has type T, so in the hypotheses list I would like to see foo : T instead of foo := e : T to reduce mental clutter. How can I d...	In proof mode, if you want to add a hypothesis that only states the type of a term without tying it to an explicit definition, you can use the `assert` tactic instead of `pose`. The `assert` tactic allows you to introduce a hypothesis that has a specified type without implying that it is equal to a particular expressio...	[ { "from": "human", "value": "In proof mode, if I know an expression e of type T, I can write pose foo : T := e to add foo := e : T as a hypothesis. Often, I don't care about that foo is the same as e, I only care that foo has type T, so in the hypotheses list I would like to see foo : T instead of foo := e ...
Is there a Proof assistant which also is an esoteric programming language? If so, then how many there are, approximately?	Yes, there are a few proof assistants that also qualify as esoteric programming languages. The most notable examples include: 1. Coq: While primarily a proof assistant, it has been used in a playful manner, and some implementations have elements that can be considered esoteric. 2. Agda: Similar to Coq, Agda i...	[ { "from": "human", "value": "Is there a Proof assistant which also is an esoteric programming language? If so, then how many there are, approximately?\n" }, { "from": "gpt", "value": "Yes, there are a few proof assistants that also qualify as esoteric programming languages. The most notable exam...
If I had a module declared as follows in file A.v: Section A. Context { : Set}. Inductive abt := Abt_leaf (x : ) \| Abt_node. End A. And in another file, B.v, I had: Require Import A. Definition := nat. Inductive typ := Tpe_num \| Typ_str. Definition typ_to_abt (t : typ) : abt (* @abt *) := Abt_node. The implicit...	To let Coq know that you always want to use the implicit argument of the type `abt` in file `B.v`, you can set the implicit arguments in your current context. Coq allows you to manage implicit arguments using the `Arguments` command, which can specify the default behavior for arguments in a given context. In your `B.v...	[ { "from": "human", "value": "If I had a module declared as follows in file A.v:\nSection A.\n Context { : Set}.\n Inductive abt := Abt_leaf (x : ) \| Abt_node.\nEnd A.\n\nAnd in another file, B.v, I had:\nRequire Import A.\nDefinition := nat.\nInductive typ := Tpe_num \| Typ_str.\nDefinition typ_to_abt (t ...
There is a function which is usually called range in other languages: it takes a natural n, and returns a list of naturals [0,1,2,...,n-1]. Does the Coq standard library have this function, and if so, what is it called?	Yes, the Coq standard library includes a function that behaves like the `range` function found in other programming languages. This function is called `seq`. It is part of the `Coq.Init.Nat` module and can be used to generate a list of natural numbers from `0` to `n-1`. The syntax for using the `seq` function is as f...	[ { "from": "human", "value": "There is a function which is usually called range in other languages: it takes a natural n, and returns a list of naturals [0,1,2,...,n-1]. Does the Coq standard library have this function, and if so, what is it called?\n" }, { "from": "gpt", "value": "Yes, the Coq s...
As mentioned in another question, the way that VSCode and other editors implement unicode characters that cannot be typed directly from most keyboards is an editor feature. As such, different editors could do it differently, so that for instance, $\varphi$ and $\phi$ could be entered as \varphi, \phi, the other way aro...	The handling of Unicode characters in text editors is not governed by a single official standard or recommendation; rather, it is typically determined by the design choices made by the developers of each editor. Different text editors, IDEs, and programming environments adopt their own conventions and methods for enter...	[ { "from": "human", "value": "As mentioned in another question, the way that VSCode and other editors implement unicode characters that cannot be typed directly from most keyboards is an editor feature.\nAs such, different editors could do it differently, so that for instance, $\\varphi$ and $\\phi$ could be...
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 see related question: When installing Agda does one have to be attentive to a version?	Yes, Agda, like many programming languages and systems, can introduce breaking changes between major versions. While it is not as common as in some other languages, there have been instances where updates to Agda included changes that might affect existing codebases. ### Significant Breaking Changes: 1. **Syntax Chang...	[ { "from": "human", "value": "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?\n\nAlso see related question:\nWhen installing Agda does one have to be attentive to a ...
As mentioned in another question, the way that VSCode and other editors implement unicode characters that cannot be typed directly from most keyboards is an editor feature. As such, different editors could do it differently, so that for instance, $\varphi$ and $\phi$ could be entered as \varphi, \phi, the other way aro...	There is no official, universally accepted standard for how text editors should implement the input of Unicode characters that cannot be easily typed from most keyboards. Different editors and development environments have their own conventions and features, often tailored to specific use cases or user needs. However,...	[ { "from": "human", "value": "As mentioned in another question, the way that VSCode and other editors implement unicode characters that cannot be typed directly from most keyboards is an editor feature.\nAs such, different editors could do it differently, so that for instance, $\\varphi$ and $\\phi$ could be...
There was already a question here of this type but I would like to ask more generally: If I see some strange unicode symbols being used in Lean, is there a recommended way to find out how to type them, and finding out what they mean, other than asking here at Proof Assistants or at the Zulip chat? Is the best source th...	If you encounter strange Unicode symbols in Lean and want to understand how to type them and their meanings, here are some recommended approaches beyond asking in forums: 1. Official Documentation: The most reliable source is the official Lean documentation. Lean 3 and Lean 4 have their own documentation that ofte...	[ { "from": "human", "value": "There was already a question here of this type but I would like to ask more generally:\nIf I see some strange unicode symbols being used in Lean, is there a recommended way to find out how to type them, and finding out what they mean, other than asking here at Proof Assistants o...
Let's say I have a simple structure structure foo : Type := (a : ℕ) (b : list (fin a)) when defining the field b I can reference a. If I try to define a function to construct new foo's like so def bar (p : foo) : foo := { a := p.a + 1, b := list (fin a) } where I tried to reference the a I constructed on the fi...	In Lean, when you are constructing a new instance of a structure, you need to refer to the fields of the existing structure through the instance variable itself. In your case, when you want to access the field `a` of the structure `foo`, you should use `p.a` instead of just `a`. Additionally, when you are constructin...	[ { "from": "human", "value": "Let's say I have a simple structure\nstructure foo : Type :=\n (a : ℕ)\n (b : list (fin a))\n\nwhen defining the field b I can reference a. If I try to define a function to construct new foo's like so\ndef bar (p : foo) : foo :=\n{ a := p.a + 1,\n b := list (fin a)\n}\n\nwher...
If I had a module declared as follows in file A.v: Section A. Context { : Set}. Inductive abt := Abt_leaf (x : ) \| Abt_node. End A. And in another file, B.v, I had: Require Import A. Definition := nat. Inductive typ := Tpe_num \| Typ_str. Definition typ_to_abt (t : typ) : abt (* @abt *) := Abt_node. The implicit...	To make Coq infer the implicit arguments for the `abt` type in your file `B.v`, you can use the `Set` as an implicit argument and provide an explicit type for `abt` by making `abt` a global definition with respect to some type in `B.v`. Here's how you can achieve that: 1. First, modify the context in `A.v` so that it ...	[ { "from": "human", "value": "If I had a module declared as follows in file A.v:\nSection A.\n Context { : Set}.\n Inductive abt := Abt_leaf (x : ) \| Abt_node.\nEnd A.\n\nAnd in another file, B.v, I had:\nRequire Import A.\nDefinition := nat.\nInductive typ := Tpe_num \| Typ_str.\nDefinition typ_to_abt (t ...
Andrea Vezzosi said Cubical Agda made cubical subtyping coercion-based (inS and outS) because it's a bad idea to introduce more subtyping relations which complicates the system. Also Agda did not have cumulative universes (it is added now). So does Agda support any subtyping relations? Without using rewriting and cumul...	In Agda, subtyping is not natively supported in the same way as in some other languages that have explicit subtyping relations. Instead, Agda uses a form of type inference and implicit coercions primarily based on dependent types. However, while Agda doesn't have a traditional subtyping system, there are ways to simu...	[ { "from": "human", "value": "Andrea Vezzosi said Cubical Agda made cubical subtyping coercion-based (inS and outS) because it's a bad idea to introduce more subtyping relations which complicates the system. Also Agda did not have cumulative universes (it is added now).\nSo does Agda support any subtyping re...
Every community can have inside jokes among it, and April Fool’s jokes are a common variety, but proof assistants are particularly susceptible to April Fool’s jokes. They have strong guarantees of correctness so one could claim both that these claims are wrong or that one found irrefutable proof of something crazy. Mo...	Sure! Here's a lighthearted April Fool's joke idea related to proof assistants: The "Infinite Proof" Feature Announcement Announce the release of a new feature in a popular proof assistant (like Coq, Lean, or Agda) that allows users to write "infinite proofs"—proofs that contain no finite steps but still somehow ...	[ { "from": "human", "value": "Every community can have inside jokes among it, and April Fool’s jokes are a common variety, but proof assistants are particularly susceptible to April Fool’s jokes. They have strong guarantees of correctness so one could claim both that these claims are wrong or that one found ...
How would you explain to a beginner in simple terms, what a Proof Assistant is?	A Proof Assistant is a type of computer software that helps people create and check mathematical proofs. Think of it as a very smart calculator for logic and reasoning. When you make a mathematical argument, you need to show that your conclusions are true based on certain rules. A Proof Assistant helps you write down...	[ { "from": "human", "value": "How would you explain to a beginner in simple terms, what a Proof Assistant is?\n" }, { "from": "gpt", "value": "A Proof Assistant is a type of computer software that helps people create and check mathematical proofs. Think of it as a very smart calculator for logic ...
Every community can have inside jokes among it, and April Fool’s jokes are a common variety, but proof assistants are particularly susceptible to April Fool’s jokes. They have strong guarantees of correctness so one could claim both that these claims are wrong or that one found irrefutable proof of something crazy. Mo...	Here’s a classic April Fool’s joke that could be made about proof assistants: Joke: Announce the release of a new proof assistant that utilizes “AI-driven intuition” to automatically generate proofs without user intervention. The assistant claims to have solved some of the hardest problems in mathematics using an ...	[ { "from": "human", "value": "Every community can have inside jokes among it, and April Fool’s jokes are a common variety, but proof assistants are particularly susceptible to April Fool’s jokes. They have strong guarantees of correctness so one could claim both that these claims are wrong or that one found ...
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 Set (suc i), but also a subtype of it. However, in the implementation of Agda, this is not true. We only have the "instance of" relation, not the subtype rel...	Cumulative universes in type theory allow for a hierarchy of types where each universe can contain all lower universes. This can be beneficial in terms of expressiveness and flexibility, as you noted. However, there are several reasons why the designers of Agda may have opted to implement a simpler universe system init...	[ { "from": "human", "value": "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 Set (suc i), but also a subtype of it.\nHowever, in the implementation of Agda, this is not true. We only have the \...
Every community can have inside jokes among it, and April Fool’s jokes are a common variety, but proof assistants are particularly susceptible to April Fool’s jokes. They have strong guarantees of correctness so one could claim both that these claims are wrong or that one found irrefutable proof of something crazy. Mo...	Here's a classic joke that plays on the common experience of proof assistants: "Introducing 'Proof Assistant 2.0': Now with built-in telepathy!" The joke claims that the latest version of a proof assistant can automatically understand and implement the user's thoughts without needing any formal input or commands....	[ { "from": "human", "value": "Every community can have inside jokes among it, and April Fool’s jokes are a common variety, but proof assistants are particularly susceptible to April Fool’s jokes. They have strong guarantees of correctness so one could claim both that these claims are wrong or that one found ...
Jon Sterling said Conor McBride's universe polymorphism proposal to be: seems to have worked well in Swedish experiments I wonder, what does "Swedish" mean here? What kind of type theory or Proof Assistant practice is considered Swedish? Are there other styles (my guess would be French, but I'm unsure)?	In the context of type theory and proof assistants, the term "Swedish" often refers to a particular approach to type systems and polymorphism that is associated with the work of researchers such as Conor McBride and his contributions to functional programming and dependently typed programming. Specifically, the "Swedi...	[ { "from": "human", "value": "Jon Sterling said Conor McBride's universe polymorphism proposal to be:\n\nseems to have worked well in Swedish experiments\n\nI wonder, what does \"Swedish\" mean here? What kind of type theory or Proof Assistant practice is considered Swedish? Are there other styles (my guess ...
If I had a module declared as follows in file A.v: Section A. Context { : Set}. Inductive abt := Abt_leaf (x : ) \| Abt_node. End A. And in another file, B.v, I had: Require Import A. Definition := nat. Inductive typ := Tpe_num \| Typ_str. Definition typ_to_abt (t : typ) : abt (* @abt *) := Abt_node. The implicit...	To inform Coq that you want to use the inductive type `abt` with the implicit arguments globally in file `B.v`, you can use the `Set Implicit Arguments` command. This command tells Coq to automatically infer implicit arguments for the definitions and constructions in the module. Here’s how you can modify your file `B....	[ { "from": "human", "value": "If I had a module declared as follows in file A.v:\nSection A.\n Context { : Set}.\n Inductive abt := Abt_leaf (x : ) \| Abt_node.\nEnd A.\n\nAnd in another file, B.v, I had:\nRequire Import A.\nDefinition := nat.\nInductive typ := Tpe_num \| Typ_str.\nDefinition typ_to_abt (t ...
If I had a module declared as follows in file A.v: Section A. Context { : Set}. Inductive abt := Abt_leaf (x : ) \| Abt_node. End A. And in another file, B.v, I had: Require Import A. Definition := nat. Inductive typ := Tpe_num \| Typ_str. Definition typ_to_abt (t : typ) : abt (* @abt *) := Abt_node. The implicit...	To avoid having to explicitly specify the implicit arguments of the `abt` inductive type in file `B.v`, you can use the `Arguments` command to set the implicit arguments globally for the `abt` type. This can be done in file `A.v` where you define `abt`. By declaring the arguments as implicit, you will be able to use th...	[ { "from": "human", "value": "If I had a module declared as follows in file A.v:\nSection A.\n Context { : Set}.\n Inductive abt := Abt_leaf (x : ) \| Abt_node.\nEnd A.\n\nAnd in another file, B.v, I had:\nRequire Import A.\nDefinition := nat.\nInductive typ := Tpe_num \| Typ_str.\nDefinition typ_to_abt (t ...
The following snippet can't pass type checking. From mathcomp Require Import choice. Definition exfn (A:countType) := false. (* Fail *) Check exfn nat. Failed with message: The term "nat" has type "Set" while it is expected to have type "countType". How to instantiate A with nat properly?	In Coq's Mathematical Components library (MathComp), `countType` is a type class that represents types with a countable structure. The natural numbers (`nat`) do not directly have an instance of `countType` in the way you might be expecting. To make `nat` work with the `countType` type class, you can use `nat` as a `c...	[ { "from": "human", "value": "The following snippet can't pass type checking.\nFrom mathcomp Require Import choice.\nDefinition exfn (A:countType) := false.\n(* Fail *) Check exfn nat.\n\nFailed with message: The term \"nat\" has type \"Set\" while it is expected to have type \"countType\".\nHow to instantia...
I am learning Coq with ssreflect. Just to understand things, I've proved forall a b : bool, a == b -> a = b but I can't figure out how to prove forall m n : nat, m == n -> m = n. I've tried using the elim tactic first on m, then on n, but in the end I was left with a subgoal I couldn't prove. I've tried destruct m and ...	To prove the lemma `eqn_impl_eq` that states for all natural numbers `m` and `n`, if `m == n` (using the equality operator from the `mathcomp` library) implies that `m = n` (the standard Coq equality), you can leverage the structural properties of natural numbers in Coq. Here's how you can complete the proof: 1. Star...	[ { "from": "human", "value": "I am learning Coq with ssreflect. Just to understand things, I've proved forall a b : bool, a == b -> a = b but I can't figure out how to prove forall m n : nat, m == n -> m = n. I've tried using the elim tactic first on m, then on n, but in the end I was left with a subgoal I c...
Say I'm formalizing something in group theory, and I'm working with some action $\cdot$ of $(G, +)$ on a set. In my math textbook, the identity of $\cdot$ is explicitly mentioned once (if that), and then left implicit. In my formalization, do I have to constantly keep specifying $(G, \cdot)$, or can I temporarily set $...	In formalizing a concept in group theory, it is common and acceptable to adopt a shorthand notation for a group action, provided that you have previously defined what the action is. This is particularly true in mathematical writing, where once you establish the context—such as specifying that \( G \) is a group acting ...	[ { "from": "human", "value": "Say I'm formalizing something in group theory, and I'm working with some action $\\cdot$ of $(G, +)$ on a set. In my math textbook, the identity of $\\cdot$ is explicitly mentioned once (if that), and then left implicit. In my formalization, do I have to constantly keep specifyi...
The most well-known tutorials/learning materials, including Book 1, Chapter 2 of Software Foundations for Coq and the Natural Number Game for Lean, use natural numbers to introduce induction. But there are multiple complaints by new learners that having to prove trivial facts of natural numbers via induction feels unin...	Yes, there has been increasing recognition within the community that starting with natural numbers for teaching induction can be intimidating for new learners. As a result, several alternative approaches and tutorials have emerged that use different structures to introduce inductive reasoning and proofs. One such appr...	[ { "from": "human", "value": "The most well-known tutorials/learning materials, including Book 1, Chapter 2 of Software Foundations for Coq and the Natural Number Game for Lean, use natural numbers to introduce induction. But there are multiple complaints by new learners that having to prove trivial facts of...
Is there a library (for any proof assistant) which provides formalized definitions of unsolved problems? To clarify, I mean some collection that correctly defines unsolved problems in the language of a proof assistant, without attempting to provide any solutions. By unsolved problems, I mean one which haven't been solv...	Yes, there are libraries and formalizations in proof assistants that specifically address unsolved problems in mathematics. While these libraries do not attempt to provide solutions, they define the problems formally within the framework of the proof assistant. Here are some notable examples: 1. Lean: The Lean the...	[ { "from": "human", "value": "Is there a library (for any proof assistant) which provides formalized definitions of unsolved problems? To clarify, I mean some collection that correctly defines unsolved problems in the language of a proof assistant, without attempting to provide any solutions.\nBy unsolved pr...
In proof mode, if I know an expression e of type T, I can write pose foo : T := e to add foo := e : T as a hypothesis. Often, I don't care about that foo is the same as e, I only care that foo has type T, so in the hypotheses list I would like to see foo : T instead of foo := e : T to reduce mental clutter. How can I d...	In Coq, if you want to introduce a hypothesis that merely asserts the type of an expression without binding the expression itself to a name in a way that reflects its equality, you can use the `assert` tactic or the `pose proof` tactic without associating the expression directly with the hypothesis. Here’s how you can...	[ { "from": "human", "value": "In proof mode, if I know an expression e of type T, I can write pose foo : T := e to add foo := e : T as a hypothesis. Often, I don't care about that foo is the same as e, I only care that foo has type T, so in the hypotheses list I would like to see foo : T instead of foo := e ...
Some time ago I asked this question on Stack Overflow but got no answer: https://stackoverflow.com/questions/60521384/tracing-tactics-in-isabelle Section 9.4 The Classical Reasoner of the Isar Reference Manual writes: The tactics can be traced, and their components can be called directly; in this manner, any proof can...	To trace the classical reasoner in Isabelle, you can use the `trace_tactic` command available in the interactive environment. Although the specific details about tracing the classical reasoner may not be explicitly documented, you can generally apply a few strategies to achieve this. Here’s how you can start tracing t...	[ { "from": "human", "value": "Some time ago I asked this question on Stack Overflow but got no answer:\nhttps://stackoverflow.com/questions/60521384/tracing-tactics-in-isabelle\nSection 9.4 The Classical Reasoner of the Isar Reference Manual writes:\n\nThe tactics can be traced, and their components can be c...
What was the first proof assistant? What was it used for? When and by who was it created? Is it still used today? And what was its purpose?	The first proof assistant is generally considered to be "Automath," which was developed in the late 1960s and early 1970s by Nicolas Bourbaki (a pseudonymous group of mathematicians) and further developed by Adrian K. (Tony) M. Feigenbaum and others, particularly at the University of Amsterdam by **Jan von ...	[ { "from": "human", "value": "What was the first proof assistant? What was it used for? When and by who was it created? Is it still used today? And what was its purpose?\n" }, { "from": "gpt", "value": "The first proof assistant is generally considered to be \"Automath,\" which was developed ...
This is a related question to Is there a consolidated or partial list noting the use of quasiquotation with provers? for which it seems there is no answer. However as there might be an answer some day am leaving that question and asking this question that others are dancing with in the comments of that question. As t...	Here’s a list of theorem provers and proof assistants that utilize systems of quoting or quasiquotation influenced by Lisp: 1. Coq: Coq employs a system of quotations, particularly in its metaprogramming extensions such as Ltac. Ltac allows for quoted tactics, which can be manipulated and executed dynamically, som...	[ { "from": "human", "value": "This is a related question to\n\nIs there a consolidated or partial list noting the use of quasiquotation with provers?\n\nfor which it seems there is no answer. However as there might be an answer some day am leaving that question and asking this question that others are dancin...
In proof mode, if I know an expression e of type T, I can write pose foo : T := e to add foo := e : T as a hypothesis. Often, I don't care about that foo is the same as e, I only care that foo has type T, so in the hypotheses list I would like to see foo : T instead of foo := e : T to reduce mental clutter. How can I d...	In proof mode, you can use the `pose` tactic in a slightly different way to achieve what you're looking for. Instead of using the regular `pose` tactic which binds the expression directly to a name while showing its definition, you can use the `assert` tactic instead. This will allow you to state that something of a ce...	[ { "from": "human", "value": "In proof mode, if I know an expression e of type T, I can write pose foo : T := e to add foo := e : T as a hypothesis. Often, I don't care about that foo is the same as e, I only care that foo has type T, so in the hypotheses list I would like to see foo : T instead of foo := e ...
My understanding of most automated theorem provers (which is possibly an incorrect understanding!) is that they start with some premises stored as unprocessed statements, and repeatedly select one to "process", generating different consequences and substitutions. Because the total number of statements is so big, they r...	Your understanding of automated theorem provers (ATPs) is largely correct; they typically start with a set of axioms and conjectures, systematically exploring the implications of these statements using inference rules. However, there are several factors that contribute to their limitations in comparison to human mathem...	[ { "from": "human", "value": "My understanding of most automated theorem provers (which is possibly an incorrect understanding!) is that they start with some premises stored as unprocessed statements, and repeatedly select one to \"process\", generating different consequences and substitutions. Because the t...
I have this function: Definition bexp x y := bexp_r x y [true]. And I have this goal: value (bexp [] y) = 0 ^ value y I want to replace the call to bexp with the underlying bexp_r. I tried to use the simpl tactic but it did not replace it.	In order to replace the call to the `bexp` function with its underlying definition `bexp_r`, you can use the `rewrite` tactic if you have the definition available in your context. Here’s how you can do it: 1. Ensure that the definition of `bexp` is in scope. If it is defined in a module, you may need to import that mo...	[ { "from": "human", "value": "I have this function:\nDefinition bexp x y := bexp_r x y [true].\n\nAnd I have this goal:\nvalue (bexp [] y) = 0 ^ value y\n\nI want to replace the call to bexp with the underlying bexp_r.\nI tried to use the simpl tactic but it did not replace it.\n" }, { "from": "gpt",...
I want to simplify the expression 0 * 1 * 1 * 1 * 0 using simp only [mul_zero, zero_mul]. I would like mul_zero to fire first, so that the entire expression is simplified to zero in only one step. If mul_zero fires first, I would expect the simplification process to take 4 steps, due to there being one step to eliminat...	In Lean, the simplifier uses a set of rules to determine how to simplify expressions, but it does not allow you to directly specify the order in which specific rules are applied. However, you can control the simplification process by manipulating the structure of the expression or by using specific tactics to force the...	[ { "from": "human", "value": "I want to simplify the expression 0 * 1 * 1 * 1 * 0 using simp only [mul_zero, zero_mul]. I would like mul_zero to fire first, so that the entire expression is simplified to zero in only one step. If mul_zero fires first, I would expect the simplification process to take 4 steps...
I found a site where I can use Coq in my browser: jsCoq Interactive Online System. Are there any major differences between the experience I'm getting on this site and the downloadable version(s)?	Using Coq through the jsCoq Interactive Online System in your browser offers several conveniences and advantages, but there are also some differences compared to using a local installation of Coq. Here are the major points to consider: ### Advantages of jsCoq: 1. No Installation Required: You can start using Coq i...	[ { "from": "human", "value": "I found a site where I can use Coq in my browser: jsCoq Interactive Online System. Are there any major differences between the experience I'm getting on this site and the downloadable version(s)?\n" }, { "from": "gpt", "value": "Using Coq through the jsCoq Interactiv...
Suppose two separate files define decidability instances for a particular predicate in different ways. If I import both of these files, and attempt to run #eval to_bool ... to evaluate the predicate for a particular input, how does Lean decide which instance to use in the evaluation?	In Lean, when you import multiple files (modules) that define the same predicate, the resolution of which instance to use during evaluation depends on several factors, including the order of imports and the specific namespace where the instances are defined. Here are the key points to consider: 1. Import Order: W...	[ { "from": "human", "value": "Suppose two separate files define decidability instances for a particular predicate in different ways. If I import both of these files, and attempt to run #eval to_bool ... to evaluate the predicate for a particular input, how does Lean decide which instance to use in the evalua...
Suppose two separate files define decidability instances for a particular predicate in different ways. If I import both of these files, and attempt to run #eval to_bool ... to evaluate the predicate for a particular input, how does Lean decide which instance to use in the evaluation?	In Lean, when you import multiple files (modules) that define the same instance (in this case, the decidability instance for a predicate), the resolution of which instance to use during evaluation depends on the specific order of imports and Lean's handling of instances. 1. Instance Resolution: Lean uses an implic...	[ { "from": "human", "value": "Suppose two separate files define decidability instances for a particular predicate in different ways. If I import both of these files, and attempt to run #eval to_bool ... to evaluate the predicate for a particular input, how does Lean decide which instance to use in the evalua...
Is there a library (for any proof assistant) which provides formalized definitions of unsolved problems? To clarify, I mean some collection that correctly defines unsolved problems in the language of a proof assistant, without attempting to provide any solutions. By unsolved problems, I mean one which haven't been solv...	Yes, there are some libraries and projects in various proof assistants that focus on formalizing mathematical concepts and problems, including some unsolved problems. However, it's important to note that these libraries often don't aim to solve these problems but rather to provide formal definitions, formulations, and ...	[ { "from": "human", "value": "Is there a library (for any proof assistant) which provides formalized definitions of unsolved problems? To clarify, I mean some collection that correctly defines unsolved problems in the language of a proof assistant, without attempting to provide any solutions.\nBy unsolved pr...
For example, if I want to define multiplication inverse on the rational type, intuitively we would define: ÷ : Q -> <Q \ {0}> -> Q But how would I remove a zero from the second argument? One solution would be to add an extra parameter ÷ : Q -> (q : Q) -> q ≠ 0 -> Q, but this makes the signature ugly as it's no longer ...	In type theory and functional programming, handling operations that involve restrictions on the types of inputs—like division where the divisor cannot be zero—can indeed lead to some complex type signatures. Your suggestion to add an additional parameter representing the non-zero condition is valid, but as you mentione...	[ { "from": "human", "value": "For example, if I want to define multiplication inverse on the rational type, intuitively we would define:\n÷ : Q -> <Q \\ {0}> -> Q\n\nBut how would I remove a zero from the second argument?\nOne solution would be to add an extra parameter ÷ : Q -> (q : Q) -> q ≠ 0 -> Q, but th...
The paper in question: An Implementation of a Dependently Typed Lambda Calculus by Andres Löh, Conor McBride and Wouter Swierstra. I'm wondering whether it's correct in what's it is doing. Precisely on page 7. In type checker. It's got these evaluations with empty environments. Eg. let v = eval↓ t []. Precisely when th...	In the context of the paper "An Implementation of a Dependently Typed Lambda Calculus" by Löh, McBride, and Swierstra, the type checker and evaluator are designed to work with dependently typed terms in a way that ensures that types and terms are checked correctly. On page 7, the evaluation of terms with an empty envi...	[ { "from": "human", "value": "The paper in question: An Implementation of a Dependently Typed Lambda Calculus by Andres Löh, Conor McBride and Wouter Swierstra.\nI'm wondering whether it's correct in what's it is doing. Precisely on page 7. In type checker. It's got these evaluations with empty environments....
How would you explain so a beginner in simple terms why Proof Assistants are important, and how / why they are used?	Proof assistants are like powerful tools that help us check the correctness of mathematical proofs and computer programs. Here’s a simple way to understand why they are important and how they are used: ### Why They Are Important: 1. Accuracy: Just like how a calculator helps avoid mistakes in math, proof assistan...	[ { "from": "human", "value": "How would you explain so a beginner in simple terms why Proof Assistants are important, and how / why they are used?\n" }, { "from": "gpt", "value": "Proof assistants are like powerful tools that help us check the correctness of mathematical proofs and computer progr...