如何证明 (forall x, P x /\\ Q x) -> (forall x, P x)

Question

How does one prove (forall x, P x /\\ Q x) -> (forall x, P x) in Coq? Been trying for hours and can't figure out how to break down the antecedent to something that Coq can digest. (I'm a newb, obviously :)

Answer 1

You can do it more quickly by just applying H, but this script should be more clear.

Lemma foo : forall (A:Type) (P Q: A-> Prop), (forall x, P x /\ Q x) -> (forall x, P x).
intros.
destruct (H x). 
exact H0.
Qed.

Answer 2

Assume ForAll x: P(x) /\ Q(x)
   var x; 
      P(x) //because you assumed it earlier
   ForAll x: P(x)
(ForAll x: P(x) /\ Q(x)) => (ForAll x: P(x))

Intuitivly, if for all x, P(x) AND Q(x) hold, then for all x, P(x) holds.

Answer 3

尝试

elim (H x).

Answer 4

Actually, I figured this one out when I found this:

Mathematics for Computer Scientists 2

In lesson 5 he solves the exact same problem and uses "cut (P x /\\ Q x)" which re-writes the goal from "P x" to "P x /\\ Q x -> P x". From there you can do some manipulations and when the goal is just "P x /\\ Q x" you can apply "forall x : P x /\\ Q x" and the rest is straightforward.

Answer 5

here is the answer:

Lemma fa_dist_and   (A : Set) (P : A -> Prop) (Q: A -> Prop): 

(forall x, P x) /\ (forall x, Q x) <-> (forall x : A, P x /\ Q x).

Proof.

split.

intro H.

(destruct H).

intro H1.

split.

(apply H).

(apply H0).

intro H.

split.

intro H1.

(apply H).

intro H1.

(apply H).

Qed.

you can find other solved exercises in this file: https://cse.buffalo.edu/~knepley/classes/cse191/ClassNotes.pdf