I have a question about the: Record and Definition
I have this definition:
Definition rule := term -> term.
and I write a boolean function for it.
Definition beq_rule a b := beq_term a && beq_term b.
where beq_term : term -> term -> bool.
so my definition of beq_rule actually return exactly type of beq_term which is not what I want here. I want it return for me a type: rule -> rule -> bool.
So I changed a definition of rule by Record:
Record rule := mkRule {lhs : term; rhs : term}.
and
Definition beq_rule (a b : rule) : bool :=
beq_term (lhs a) (lhs b) && beq_term (rhs a) (rhs b).
My question is that:
1) What is the different between my first defined rule used Definition and another used Record?
2) If I want to define rule by Definition can I give an alias lhs and rhs likes in Record definition?
Your two definitions of rule are saying totally different things
Definition rule := term -> term
is defining rule as a type (or Prop) alias of the function type term -> term. Hence
Definition not_what_you_meant : rule := fun t => t.
will happily compile.
As to the relation between Record and Definition. Record is just a macro that converts into an Inductive. So
Record rule := mkRule {lhs : term; rhs : term}.
is the same as
Inductive rule := mkRule : term -> term -> rule.
plus accessor functions
Definition lhs (r : rule) : term :=
match r with
mkRule l _ => l
end.
etc.
You should think of Inductive as being fundamentally different from Definition. Definition defines an alias. Another way f saying this is Definitions are "referentially transparent", you can (up to variable renaming) always substitute the right hand side of a definition for any occurrence of its name.
Inductive on the other hand defines type (elements of Coqs universes) by listing off a set of constructors. In more logical way of thinking, Inductive defines a logical proposition in terms of its elimination/introduction rules in a way that ensures "harmony".
Related
I have the following two definitions that result in two different error messages.
The first definition is declined because of strict positivity and the second one because of a universe inconsistency.
(* non-strictly positive *)
Inductive SwitchNSP (A : Type) : Type :=
| switchNSP : SwitchNSP bool -> SwitchNSP A.
Fail Inductive UseSwitchNSP :=
| useSwitchNSP : SwitchNSP UseSwitchNSP -> UseSwitchNSP.
(* universe inconsistency *)
Inductive SwitchNSPI : Type -> Type :=
| switchNSPI : forall A, SwitchNSPI bool -> SwitchNSPI A.
Fail Inductive UseSwitchNSPI :=
| useSwitchNSPI : SwitchNSPI UseSwitchNSPI -> UseSwitchNSPI.
Chatting on gitter revealed that universe (in)consistencies are checked first, that is, the first definition adheres this check, but then fails because of a strict positivity issue.
As far as I understand the strict positivity restriction, if Coq allows non-strictly positivity data type definitions, I could construct non-terminating functions without using fix (which is pretty bad).
In order to make it even more confusing, the first definition is accepted in Agda and the second one gives a strict positivity error.
data Bool : Set where
True : Bool
False : Bool
data SwitchNSP (A : Set) : Set where
switchNSP : SwitchNSP Bool -> SwitchNSP A
data UseSwitchNSP : Set where
useSwitchNSP : SwitchNSP UseSwitchNSP -> UseSwitchNSP
data SwitchNSPI : Set -> Set where
switchNSPI : forall A -> SwitchNSPI Bool -> SwitchNSPI A
data UseSwitchNSPI : Set where
useSwitchNSP : SwitchNSPI UseSwitchNSPI -> UseSwitchNSPI
Now my question is two-folded: first, what is the "evil example" I could construct with the above definition? Second, which of the rules applies to the above definition?
Some notes:
To clarify, I think that I do understand why the second definition is not allowed type-checking-wise, but still feel that there is nothing "evil" happening here, when the definition is allowed.
I first thought that my example is an instance of this question, but enabling universe polymorphism does not help for the second definition.
Can I use some "trick" do adapt my definition such that it is accepted by Coq?
Unfortunately, there's nothing super deep about this example. As you noted Agda accepts it, and what trips Coq is the lack of uniformity in the parameters. For example, it accepts this:
Inductive SwitchNSPA (A : Type) : Type :=
| switchNSPA : SwitchNSPA A -> SwitchNSPA A.
Inductive UseSwitchNSPA :=
| useSwitchNSPA : SwitchNSPA UseSwitchNSPA -> UseSwitchNSPA.
Positivity criteria like the one used by Coq are not complete, so they will reject harmless types; the problem with supporting more types is that it often makes the positivity checker more complex, and that's already one of the most complex pieces of the kernel.
As for the concrete details of why it rejects it, well, I'm not 100% sure. Going by the rules in the manual, I think it should be accepted?
EDIT: The manual is being updated.
Specifically, using the following shorter names to simplify the following:
Inductive Inner (A : Type) : Type := inner : Inner bool -> Inner A.
Inductive Outer := outer : Inner Outer -> Outer.
Correctness rules
Positivity condition
Here,
X = Outer
T = forall x: Inner X, X
So we're in the second case with
U = Inner X
V = X
V is easy, so let's do that first:
V = (X) falls in the first case, with no t_i, hence is positive for X
For U: is U = Inner X strictly positive wrt X?
Here,
T = Inner X
Hence we're in the last case: T converts to (I a1) (no t_i) with
I = Inner
a1 = X
and X does not occur in the t_i, since there are no t_i.
Do the instantiated types of the constructors satisfy the nested positivity condition?
There is only one constructor:
inner : Inner bool -> Inner X.
Does this satisfy the nested positivity condition?
Here,
T = forall x: Inner bool, Inner X.
So we're in the second case with
U = Inner bool
V = Inner X
X does not occur in U, so X is strictly positive in U.
Does V satisfy the nested positivity condition for X?
Here,
T = Inner X
Hence we're in the first case: T converts to (I b1) (no u_i) with
I = Inner
b1 = X
There are no u_i, so V satisfies the nested positivity condition.
I have opened a bug report. The manual is being fixed.
Two more small things:
I can't resist pointing that your type is empty:
Theorem Inner_empty: forall A, Inner A -> False.
Proof. induction 1; assumption. Qed.
You wrote:
if Coq allows non-strictly positivity data type definitions, I could construct non-terminating functions without using fix (which is pretty bad).
That's almost correct, but not exactly: if Coq didn't enforce strict positivity, you could construct non-terminating functions period, which is bad. It doesn't matter whether they use fix or not: having non-termination in the logic basically makes it unsound (and hence Coq prevents you from writing fixpoints that do not terminate by lazy reduction).
What does the with keyword without the match do inside a inductive type in Coq?, example:
Inductive Block : Type :=
| EmptyBlk : Block
| Blk : Statement -> Block
with Statement : Type :=
| Assignment : string -> AExp -> Statement
| Seq : Statement -> Statement -> Statement
| IfElse : BExp -> Block -> Block -> Statement
| While : BExp -> Block -> Statement.
I tried checking the type of Statement and it seems its not of type block or something...So what is the point of defining it inside another inductive type rather than by itself. At least checking the type of Statement gives Set the same as for Block...
It is used to specify mutually recursive definitions. For example, consider the following two functions:
Fixpoint even (n : nat) : bool :=
match n with
| O => true
| S n => odd n
end
with odd (n : nat) : bool :=
match n with
| O => false
| S n => even n
end.
Here, you cannot define even first because it needs odd to be defined. You cannot define odd first either because it needs even. You need to be able to define both at the same time - and you do that by using the with keyword.
Your example is similar but defines inductive datatype rather than recursive function - Statement uses Block in its definition and Block uses Statement. Hence, with to define them both at the same time.
Note that this with is completely different keyword than with from the match expressions. In fact, they belong to two different languages: the former one is part of Vernacular whereas the latter is part of Gallina.
Is there a way to define subtype relationship in Coq?
I read about subset typing, in which a predicate is used to determine what goes into the subtype, but this is not what I am aiming for. I just want to define a theory in which there is a type (U) and another type (I), which is subtype of (U).
There is no true subtyping in Coq (except for universe subtyping, which is probably not what you want). The closest alternative is to use coercions, which are functions that the Coq type checker inserts automatically whenever it is expecting an element of one type but finds an element of another type instead. For instance, consider the following coercion from booleans to natural numbers:
Definition nat_of_bool (b : bool) : nat :=
if b then 1 else 0.
Coercion nat_of_bool : bool >-> nat.
After running this snippet, Coq uses nat_of_bool to convert bool to nat, as shown here:
Check true + 3.
(* true + 3 : nat *)
Thus, bool starts behaving almost as if it were a subtype of nat.
Though nat_of_bool does not appear here, it is just being hidden by Coq's printer. This term is actually the same thing as nat_of_bool true + 3, as we can see by asking Coq to print all coercions:
Set Printing Coercions.
Check true + 3.
(* nat_of_bool true + 3 : nat *)
The :> symbol you had asked about earlier, when used in a record declaration, is doing the same thing. For instance, the code
Record foo := Foo {
sort :> Type
}.
is equivalent to
Record foo := Foo {
sort : Type
}.
Coercion sort : foo >-> Sortclass.
where Sortclass is a special coercion target for Type, Prop and Set.
The Coq user manual describes coercions in more detail.
Suppose I have a definition f : x -> y -> z where x can be easily inferred.
I therefore choose to make x an implicit argument using Arguments.
Consider the following example:
Definition id : forall (S : Set), S -> S :=
fun S s => s.
Arguments id {_} s.
Check (id 1).
Clearly S = nat can be and is inferred by Coq, and Coq replies:
id 1
: nat
However, at a later time, I want to make the implicit argument explicit, say, for readability.
In other words, I would like something like:
Definition foo :=
id {nat} 1. (* We want to make it clear that the 2nd argument is nat*)
Is this possible at all?
If so, what is the appropriate syntax?
You can prepend the name with # to remove all implicits and provide them explicitly:
Check #id nat 1.
You can also use (a:=v) to pass an implicit argument by name. This can both clarify what argument is being passed and also allows you to pass some implicits without passing _ for the others:
Check id (S:=nat) 1.
Definition third {A B C:Type} (a:A) (b:B) (c:C) := c.
Check third (B:=nat) (A:=unit) tt 1 2.
One possible way is by prepending # to the definition.
For example:
Definition id : forall (S : Set), S -> S :=
fun S s => s.
Arguments id {_} s.
Check #id nat 1.
Which results in:
id 1
: nat
I want to set Coq up, without redefining the : with a notation (and without a plugin, and without replacing the standard library or redefining the constants I'm using---no cheating like that), so that I have something like a partial coercion from option nat to nat, which is defined only on Some _. In particular, I want
Eval compute in Some 0 : nat.
to evaluate to 0, and I want
Check None : nat.
to raise an error.
The closest I've managed is the ability to do this with two :s:
Definition dummy {A} (x : option A) := A.
Definition inverted_option {A} (x : option A)
:= match x with Some _ => A | _ => True end.
Definition invert_Some {A} (x : option A) : inverted_option x
:= match x with Some v => v | None => I end.
Coercion invert_Some : option >-> inverted_option.
Notation nat' := (inverted_option (A:=nat) (Some _)).
Eval compute in (Some 0 : nat') : nat.
Check (None : nat') : nat.
(* The term "None" has type "option ?A" while it is expected to have type
"nat'". *)
However, this only works when nat' is a notation, and I can't define a coercion out of a notation. (And trying to define a coercion from inverted_option (Some _) to nat violated the uniform inheritance condition.) I thought I might be able to get around this issue by using canonical structures, but I haven't managed to figure out how to interleave canonical structure resolution with coercion insertion (see also Can canonical structure resolution be interleaved with coercion insertion?).
(I ran into this issue when attempting to answer Coq: Defining a subtype.)