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Trying to prove the following in coq: Prove that the universal quantifier distributes over conjunction ∀x ∈ A, P x ∧ Qx ⇐⇒ (∀x ∈ A, P x) ∧ (∀x ∈ A, Qx)

My Proof so far-

Parameter (A : Type).
Parameter (P Q : A -> Prop).

Lemma II3: (forall x : A, P x /\ Q x) <->
           (forall x : A, P x) /\ (forall x : A, Q x).
split.
intro H.
split.
apply H.
intros H1.
Proof.

I have tried to split, destruct, and introduce a new hypothesis, but I just can't seem to make it past this point. Any advice would be greatly appreciated.

gallais
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maria
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    Just a small tip: it would be simpler if you added a proof script written in a regular `.v` file, as opposed to the transcript of an interaction with the proof assistant. That way, it is easier for us to copy your attempt in Coq to diagnose the problem. – Arthur Azevedo De Amorim Mar 15 '18 at 20:17
  • You can edit her question tho. – ejgallego Mar 16 '18 at 02:43
  • Two strange things in your script: the use of `Proof.` in the middle of the proof instead of at the beginning of it; the fact that you handle similar goals (with conclusions `forall x, P x` and `forall x, Q x`) very differently. – eponier Mar 16 '18 at 10:40

1 Answers1

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this is a pretty easy proof, for example now intuition; apply H will solve your goal.

In your case, you should first figure out how the proof works using pen and paper, and once you've done that, the proof in Coq will be trivial.

ejgallego
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