问题
I am a newbie in using Coq. I want to ask if I want to define a set like
A = {x | f(x) = 0},
how could I do that?
I write something like:
Definition f0 := nat->nat.
Definition A : Set :=
forall x, f0 x -> 0.
They are not working as expected.
Thanks a lot.
回答1:
More or less like you wrote. First, you have to have some function f0 : nat -> nat that you want to apply this definition to. What you did here
Definition f0 := nat -> nat.
was to name the type nat -> nat of functions from naturals to naturals f0. You probably had in mind something like this:
Variable f0 : nat -> nat.
This declares a variable f0 that belongs to the type nat -> nat. Now we can adapt your original description to Coq code:
Definition A : Set := {x : nat | f0 x = 0}.
There are two things to be aware of here. First, you might want to apply this definition later to a particular function f0 : nat -> nat, such as the predecessor function pred : nat -> nat. In this case, you should enclose your code in a section:
Section Test.
Variable f0 : nat -> nat.
Definition A : Set := {x : nat | f0 x = 0}.
End Test.
Outside of the section, A is actually a function (nat -> nat) -> Set, that takes a function f0 : nat -> nat to the type {x : nat | f0 x = 0}. You can use A as you would use any other function, e.g.
Check (A pred).
(* A pred : set *)
The second thing you must keep in mind is that a Set in Coq is not the same thing as a set in conventional mathematics. In math, an element of the set {x | f(x) = 0} is also an element of the set of natural numbers. But not in Coq. In Coq, you need to apply an explicit projection function proj1_sig to convert an element of {x : nat | f0 x = 0} to a nat.
来源：https://stackoverflow.com/questions/46045763/in-coq-how-to-define-a-set-like-a-x-fx-0