logic - 如何在 Coq 中编写 ∀x ( P(x) 和 Q(x) )？

Question

我正在尝试 Coq，但我不完全确定我在做什么。是：

Theorem new_theorem : forall x, P:Prop /\ Q:Prop

相当于：

∀x ( P(x) and Q(x) )

编辑：我认为他们是。

Answer 1

你有语法问题吗？

$ coqtop
Welcome to Coq 8.1pl3 (Dec. 2007)

Coq < Section Test.

Coq < Variable X:Set.
X is assumed

Coq < Variables P Q:X -> Prop.
P is assumed
Q is assumed

Coq < Theorem forall_test: forall x:X, P(x) /\ Q(x).
1 subgoal

  X : Set
  P : X -> Prop
  Q : X -> Prop
  ============================
   forall x : X, P x /\ Q x

forall_test < 

Answer 2

好吧，回答你的问题：

Section test.

  Variable A : Type.           (* assume some universe A *)
  Variable P Q : A -> Prop.    (* and two predicates over A, P and Q *)

  Goal forall x, P x /\ Q x.   (* Ax, ( P(x) and Q(x) ) *)

End test.