I was going through the Coq book from the maths perspective . I was trying to define a dependently typed function that returned a length list with n trues depending on the number trues we want. Coq complains that things don't have the right type but when I see it if it were to unfold my definitions when doing the type comparison it should have worked but it doesn't. Why?
Code:
Module playing_with_types2.
Inductive Vector {A: Type} : nat -> Type :=
| vnil: Vector 0
| vcons: forall n : nat, A -> Vector n -> Vector (S n).
Definition t {A: Type} (n : nat) : Type :=
match n with
| 0 => @Vector A 0
| S n' => @Vector A (S n')
end.
Check t. (* nat -> Type *)
Check @t. (* Type -> nat -> Type *)
(* meant to mimic Definition g : forall n: nat, t n. *)
Fixpoint g (n : nat) : t n :=
match n with
| 0 => vnil
| S n' => vcons n' true (g n')
end.
End playing_with_types2.
Coq's error:
In environment
g : forall n : nat, t n
n : nat
The term "vnil" has type "Vector 0" while it is expected to have type
"t ?n@{n1:=0}".
Not in proof mode.
ie t?n@{n1:=0}
is Vector 0
...no?
In this case, it looks like Coq does not manage to infer the return type of the match
expression, so the best thing to do is to give it explicitly:
Fixpoint g (n : nat) : t n :=
match n return t n with
| 0 => vnil
| S n' => vcons n' true (g n')
end.
Note the added return
clause.
Then the real error message appears:
In environment
g : forall n : nat, t n
n : nat
n' : nat
The term "g n'" has type "t n'" while it is expected to have type "Vector n'".
And this time it is true that in general t n'
is not the same as Vector n'
because t n'
is stuck (it does not know yet whether n'
is 0
or some S n''
).
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