How do you determine which terms to call intros on in coq - coq

Question

I am a beginner with coq, so this may be a trivial question. Sometimes I can't figure out which terms I need to call intros on, when writing a Theorem. A simple example,
Theorem silly1 : forall (n m o p : nat),
n = m ->
[n;o] = [n;p] ->
[n;o] = [m;p].
Proof.
intros n m o p eq1 eq2.
rewrite <- eq1.
apply eq2. Qed.
I know based on the goal, that I will probably need to call intros on (n m o p), but why do I need to use it on eq1 and eq2.
Also, in some other Theorems, you may need to use intros on the type parameter, the hypothesis, or the inductive hypothesis. Example
Theorem trans_eq : forall (X:Type) (n m o : X),
n = m -> m = o -> n = o.
Proof.
intros X n m o eq1 eq2. rewrite -> eq1. rewrite -> eq2.
reflexivity. Qed.
Theorem silly3' : forall (n : nat),
(beq_nat n 5 = true -> beq_nat (S (S n)) 7 = true) ->
true = beq_nat n 5 ->
true = beq_nat (S (S n)) 7.
Proof.
intros n eq H.
symmetry in H. apply eq in H. symmetry in H.
apply H. Qed.
So I guess what I'm asking is...when I start proving a theorem, how should I go about reasoning through the goals, to determine which terms I need to call intros on?

Answer 1

An example of what gallais is refering to is this.
Theorem example_1 : forall A B, (A -> B) -> A -> B.
Proof. intros ? ? H1. apply H1. Qed.
Theorem example_2 : forall A B, (A -> B) -> A -> B.
Proof. intros ? ? H1 H2. apply H1. apply H2. Qed.
Print example_1.
Print example_2.
Another example of when it can be problematic is using introduction before using induction. This makes the induction hypothesis different.
Fixpoint reverse_helper {A : Type} (l1 l2 : list A) : list A :=
match l1 with
| nil => l2
| cons x l1 => reverse_helper l1 (cons x l2)
end.
Theorem example_3 : forall A (l1 l2 : list A), reverse_helper l1 l2 = app (reverse_helper l1 nil) l2.
Proof. intros. induction l1. simpl. reflexivity. simpl. try rewrite IHl1. Abort.
Theorem example_4 : forall A (l1 l2 : list A), reverse_helper l1 l2 = app (reverse_helper l1 nil) l2.
Proof. induction l1. intros. simpl. reflexivity. intros. simpl. rewrite (IHl1 (cons a l2)). rewrite (IHl1 (cons a nil)). Admitted.
Otherwise, you should use introduction whenever you can. You won't be able to use whatever is being quantified over or the antecedents of an implication until you do.
By the way
H1 : A1
...
Hn : An
___
B
is equivalent to
H1: A1, ..., Hn: An ⊢ B.
When you prove something interactively, you're using a sequent calculus starting from the conclusion and working your way back to the hypotheses.