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Debugging output for tactics
idtac can be used to give output when writing ltac tactics for debugging reasons. However, I cannot work out how to convert Coq terms to strings. For example, I would like to see what the failed match is below:
Ltac test x := match x with | _+_ => idtac "Hello world!" | _ => idtac x (* Doesn't work! *) end.
A: From Coq 8.1,
idtac x works and prints the value
In Coq 8.0,
idtac x doesn't work because
idtac accepts only strings as arguments. To observe the value of
x in the branch, a solution is to use
pose (name := x) in place of
Can commutativity of or be proven?
Q: Can commutativity of
or be proven?
Lemma or_comm : forall (A B:Prop), (A \/ B) = (B \/ A).
A: No, but Coq can prove
Lemma or_comm : forall (A B:Prop), (A \/ B) <-> (B \/ A).
Q: Is it consistent to add as an axiom
Hypothesis extprop : forall (C D:Prop), (C <-> D) -> (C = D).
? (I don't suppose it's useful.)
A: It is consistent with respect to the set-theoretic interpretation of the Calculus of Inductive Constructions (the
Set-predicative version of it, as in Coq 8.x).
The axiom is generally not needed. For instance
setoid rewrite is able to "rewrite" equivalence without relying on this axiom.
Syntax Question: What does ... mean?
Q: What do the triple elipses mean after a tactic? E.g. how is
induction 1... different than just
A: If you start your proof with
Proof with foo.
foo is a tactic, then
is equivalent to
induction 1; foo.
This is explained in the Reference Manual in http://coq.inria.fr/V8.1/refman/Reference-Manual010.html#@command186.
Operational Semantics of Coq
Q: Is there a paper which describes a small-(or big) step operational semantics for core coq?
A: Coq is an equational theory generated from four reduction rules named β, δ, ι and ζ (see Section Conversion rules in Chapter Calculus of Inductive Constructions of the Coq Reference Manual). There is no fixed operational semantics and reduction is not restricted to evaluation of closed terms.
Different strategies of reduction are made available to the user such as lazy and strict evaluation (strict here does not mean call-by-value in Plotkin's sense: it means reducing first the arguments into a weak head normal form before contracting β-redexes). See Section Conversion tactics in Chapter Tactics for the list of available strategies of reduction.
Q: I have a problem with implicit arguments. Really, this is only a matter of inconvenience, but since implicit arguments are for convenience, why not ask?
I believe that the following input shows the issues simply:
Set Implicit Arguments. Definition comp (A B C: Set) (f: A -> B) (g: B -> C): A -> C := fun x => g (f x). Print comp. Section comp. Variables (A B C: Set) (f: A -> B) (g: B -> C) (x: A). Check (comp f g x). End comp. Theorem associativity (A B C D: Set) (f: A -> B) (g: B -> C) (h: C -> D): forall (x: A), comp f (comp g h) x = comp (comp f g) h x. Proof. trivial. Qed. Definition id (A: Set): A -> A := fun x => x. Print id. Section id. Variables (A: Set) (x: A). Check (id x). End id. Theorem unit (A B: Set) (f: A -> B): forall (x: A), comp (id (A := A)) f x = f x. Proof. trivial. Qed. Unset Implicit Arguments. Definition id' (A: Set): A -> A := fun x => x. Print id'. Section id'. Variables (A: Set) (x: A). Check (id' A x). End id'. Theorem unit' (A B: Set) (f: A -> B): forall (x: A), comp (id' A) f x = f x. Proof. trivial. Qed.
As you can see, comp is very convenient to use. Thanks to my setting at the start, I can define comp in a perfectly straightforward way. Now, comp depends on three arguments of type Set, but we ordinarily don't refer to these when discussing function composition in mathematics (the symbol is ‘∘’, not ‘∘_A,B,C’), so it's nice that I don't actually have to refer to them! The example in Section comp shows how convenient this is to use; I write simply comp f g instead of comp A B C f g. The theorem associativity gives a more complicated example; all very nice.
In contrast, id does not work perfectly well. The operator id depends on one argument of type Set, and we often do refer to this in mathematics (writing ‘id_A’ rather than just ‘id’), although sometimes we do still suppress it. As set up here, Coq suppresses it. This works just great in the example in Section id, just like comp worked. However, in unit, I need to put in the very annoying (A := A) or it will complain. (The reason is that simply id A expects A to be an element of a set rather than a set itself, since the set is suppressed; while id alone is not a function at all, of course.)
To be sure, I can change the settings and then define id'. This handles both the example in Section id', as well as the theorem unit', but I do (inconveniently) have to mention A explicitly in both of them. Note that Coq should have no trouble inferring the suppressed argument in any of my uses of id, so the only reason why I must either use id' or the (A := A) construction is to tell Coq (in the latter) that I don't want to refer to the entire operator id by itself.
In my intended applications, I will never want to refer to the entire operator id, but I will often write things involving id like the theorem unit (as well as things like the example in Section id). So my choice is either to use id as above, requiring me to write (A := ...) from time to time (and it must always be A on the left, so I have to remember how the definition of id was phrased); or forgo implicit arguments in this case and use id'. Right now, I'm leaning towards the latter, because I don't want to remember all the local variables that appear in definitions; but I worry that I may eventually want to use even comp in a way analogous to how id is used in unit, making implicit arguments ultimately useless.
So in the end, here's what I'd like to do, and maybe there's some way to do it that I just don't realise: I want to use settings that allow me (preferably automatically, but manually after each definition if necessary) to define comp and id with implicit arguments as above, but also tell Coq that (when they appear in a later type construction) both comp and id should always appear with inferred arguments, even if there are no explicit arguments following. Is this possible?
A: The feature you are looking for is implemented in the development version of Coq (to be released as Coq version 8.2 in early 2008). It is called maximal insertion of implicit arguments and it works as follows:
Set Implicit Arguments. Set Maximal Implicit Insertion. Definition comp (A B C: Set) (f: A -> B) (g: B -> C): A -> C := fun x => g (f x). Definition id (A: Set): A -> A := fun x => x. About id. (* tells you that A is "maximally inserted", i.e. inserted even if no arguments follow *) Theorem unit (A B: Set) (f: A -> B): forall (x: A), comp id f x = f x. (* works *)
Q: Here's another example, which doesn't show all of the issues above, but does demonstrate quickly that I really don't understand what's going on:
Set Implicit Arguments. Set Contextual Implicit. Inductive sum_dep (A: Set) (B: A -> Set): Set := |pair_dep (a: A) (b: B a). Print pair_dep. Definition sum_indep (A B: Set): Set := sum_dep (fun a: A => B). Definition pair_indep (A B: Set) (a: A) (b: B): sum_indep A B := pair_dep (B := fun _ => B) a b. Print pair_indep. Definition pair_indep' (A B: Set) (a: A) (b: B): sum_indep A B := pair_dep a b.
(Note that the final line causes an error.)
I've used an extra setting, just to get Coq to automatically make B implicit, but I could have made it implicit by hand as well. Now, I can understand Coq's reluctance to make the inference called for in pair_indep'; although there's really only one possible answer, it's a strange one, and a good compiler should at least issue a warning in such a circumstance. But what I really don't understand is why that it expects me to make precisely that inference in its printout for pair_indep! I must define it with (B := ...), but Coq doesn't have to print it that way???
A: Coq unfortunately does not implement inference of such kind of implicit :-(. Then, if B is printed in pair_indep, it is to ensure the reversibility of parsing. This is a defensive strategy of printing. From Coq version 8.2, such defensive strategy of printing can be deactivated using the command
Unset Printing Implicit Defensive.
From Coq proof scripts to declarative proofs?
Q: Given a Coq proof object as generated by a proof script, is there any way of generating a declarative proof similar to Coq's DPL syntax?