What do the symbols / and || represent in coq?
I've checked the reference manual and searched the web but couldn't find it.
These symbols (like most operators) can be redefined. It depends on the context.
You'll get the most precise answer by asking Coq.
Use e.g. Locate "||". to display all the currently notations containing the token ||.
If you aren't sure which notation is used in a particular expression, ask Coq to print it back with pretty-printing disabled.
Unset Printing Notations.
Check (fun a b => a || b).
Set Printing Notations.
Related
I'm learning Ltac2 and reading the official documentation of coq 8.13.2.
I do not get what role Control.refine play in the evaluation inside of quotations, as there doesn't seem to be a lot of explanations to this function.
For example, using variables in tactic expressions inside a constr quotation should be done by: constr:(... $x ...), where $x is the syntactic sugar of ltac2:(Control.refine (fun () => x)).
Why does, say simply, ltac2:(x) not work? (and it indeed doesn't, as coq gives an error of Error: Cannot infer an existential variable of type ..., at the position where the constr represented by x should be inserted).
So my questions are:
What does Control.refine do in general?
It seems to be an idiom to sometimes do this: constr:( ... ltac2:(Control.refine (fun () => ...) ...), in what situation where such anti-quotation idiom should (or shouldn't) be used?
Thanks.
The ltac2:(...) in-term quotation expects an Ltac2 term of type unit, not of type constr. Actually, your example should trigger a warning because of this.
Since the content of the quotation has type unit, it means that it can only act through side-effects. The warning tells you that the value returned by the quotation will be discarded. In particular, it cannot be used to fill the hole.
Rather, the semantics of the quotation is to evaluate its content inside a goal Γ ⊢ A where Γ is the current list of hypotheses, and A the inferred type of the hole. It is expected that doing this will solve the goal, and the resulting partial proof will be used as the Coq term filler for that hole. This is precisely the rôle of Control.refine : (unit -> constr) -> unit. It takes a (for technical reasons, thunked) term as an argument and uses it to solve the goal(s) under focus.
What happens in your example is that you provide a term that is just ignored, the goal is left untouched and the quotation rightfully complains that it was not solved.
Regarding your second point, I cannot really tell. It depends on what you want do do. But in general, if you stick to antiquotations $x I would consider this more readable. It is not always possible to do so, though, e.g. if the term being built depends on a context introduced in the constr:(...) quotation.
I'd like to view the definition of a Standard Library theorem which I found through Search. I think seeing the definition will help me complete a similar theorem.
Doing Print Rdiv_lt_0_compat. yields:
Rdiv_lt_0_compat =
fun (a b : R) (H : (0 < a)%R) (H0 : (0 < b)%R) =>
Rmult_lt_0_compat a (/ b) H (Rinv_0_lt_compat b H0)
: forall a b : R, (0 < a)%R -> (0 < b)%R -> (0 < a / b)%R
Argument scopes are [R_scope R_scope _ _]
Setting Set Printing All. doesn't help. There's nothing extra available in the docs page.
The whole Coq system is based on the idea Proofs are programs, logical formulas are types. When you consider a theorem, it is a proof (a program) and its statement is a logical formula (the type of a program). In the very
first years of Coq, there was no tactic language, every proof was defined using the same keywords as when defining a program.
After a few years, it was recognized that writing the programs entirely by hand was long and tiresome, so a tactic language was invented to explain how to construct the proof-programs in a shorter and less difficult way. But what is recorded and eventually checked are still the programs that you see using Print.
When building a proof-program, the tactic intros constructs anonymous function expressions (also known as lambdas, usually written with the keyword fun, and apply constructs an application of a function to a certain number arguments, which apply infers or leaves to the user as goals. The tactics induction and rewrite are similar, but they apply theorems that are not
given by the user. The tactic destruct essentially produces a piece of programs that is a pattern-matching construct.
With Rdiv_lt_0_compat, you are lucky that the proof built by the tactic is quite short. Often, proofs written using tactics produce programs that are much longer.
If instead of the program, you want to see the sequence of tactics that generated it, you need to find it in the sources of the system, because this
is not kept in the memory of the proof assistant. Here are the clues.
Require Import Reals.
Locate Rdiv_lt_0_compat.
the answer is Constant Coq.Reals.RIneq.Rdiv_lt_0_compat
This sequence of names indicates the hierarchy of modules in which the theorem is kept. The first name Coq expresses that this theorem is in the Coq sources, essentially in directory ...theories/, the second name Reals, indicates that you should look in tge sub directory ...theories/Reals.
The fourth name should not be used as a directory name, but as file name. So you should look in the file RIneq.v
So go an look in https://github.com/coq/coq/tree/v8.12/theories/Reals/RIneq.v and you will probably find the script fragment that was used to generate your theorem (for the 8.12 version of Coq). I just checked, the theorem appears at line https://github.com/coq/coq/blob/c95bd4cf015a3084a8bddf6d3640458c9c25b455/theories/Reals/RIneq.v#L2106
The sequence of names provided by Locate is not a sure way to find the file where the script for a theorem is stored. The correspondence between the long name and the file path is broken when the theorem is defined using modules and functor instantiation. In that case, you have to rely on stronger knowledge of how the Coq system works.
I'm extending an existing project (Featherweight Java formalization),
and there are a number of constants, such as:
Notation env := (list (var * typ)).
What would change if I used Definition instead:
Definition env := (list (var * typ)).
Why did the author use Notation here?
Whenever you try to apply or rewrite with a lemma, there's a component in Coq called the unifier that tries to find out how to instantiate your lemma so that it can work with the situation at hand (and checking that it indeed applies there). The behavior of this unifier is a bit different depending on whether you use notations or definitions.
Notations are invisible to Coq's theory: they only affect the parsing and printing behavior of the system. In particular, the unifier doesn't need to explicitly unfold a notation when analyzing a term. Definitions, on the other hand, must be explicitly unfolded by the unifier. The problem is that the unifier works heuristically, and cannot tell with 100% certainty when some definition must be unfolded or not. As a consequence, we often find ourselves with a goal that mentions a definition that the unifier doesn't unfold by itself, preventing us from applying a lemma or rewriting, and having to manually add a call to unfold ourselves to get it to work.
Thus, notations can be used as a hack to help the unifier understand what an abbreviation means without manual unfolding steps.
In studying Coq proofs of other authors, I often encounter a tactic, lets say "inv eq Heq" or "intro_b". I want to understand such tactics.
How can I find if it is a Coq tactic or a Tactic Notation defined somewhere in my current project?
Second, is there a way to find its definition?
I used SearchAbout, Search, Locate and Print but could not find answers to the above questions.
You should be able to use
Print Ltac <tacticname>.
to print the code of a user-defined tactic (according to the documentation).
To find where it is defined... I guess you're going to need grep unfortunately, Locate does not work for tactics names it seems.
As mentioned before, Print Ltac ... prints the code of a user-defined tactic.
To locate a user-defined tactic (i.e. to know where its defined), use Locate Ltac .... It gives you the fully qualified name. Then use Locate Library to find the corresponding file.
Is there any function/command to get/check if a free variable, lets say n:U, exists in a term/expression e, using Coq? Please share.
For example, I want to state this "n does not occur in the free names of e" in Coq.
Thanks,
Wilayat
Let's assume you're talking about free variables in Coq terms:
Dealing with an elementary Coq proof (using nothing exterior), what you manipulate is, outside of the proof context, a closed term, i.e. a term with only bound variables. If in proof mode a term ein your goal appears to have a free variable n(meaning the variable n is somewhere in your proof context), you can simply make the binding explicit (and close the goal term) using the generalize tactic.
In more advanced cases, your less-vanilla proof may involve free variables in the form of assumptions or parameters, in which case you can list them using Print Assumptions.
If on the other hand you are talking about using Coq terms to represent the notion of term in a particular language (e.g. you are formalizing such a language), you simply have to give a special treatment to the free variables of your language.
Provided you give them a specific constructor in the inductive definition of your language's terms, it should be easy to state whether some term has free variables or not. If you're unfamiliar with the (not so trivial) notion of representing substitution & the free variables of a language, you'll find pointers at increasing levels of precision from the TAPL to B.C.Pierce's SF course to the results of the POPLMark Challenge.