I have a to prove problem $P_{2}$ is NP-Complete. But for that I need to prove another intermediate problem $P_{1}$ NP-complete too. To prove $P_{1}$ I need to state that problem $P_{0}$ is already proven to be NP-complete. Now how should I arrange things while writing ?
Lemma 1: Problem $P_{2}$ is NP-complete (with full definition of $P_{2}$)
Begin Proof:
Lemma 2: Problem $P_{1}$ is NP-complete (with full definition of $P_{1}$)
Begin Proof
Definition $P_{0}$ is known to be NP-complete
...
End Proof
End proof
or should I first define all these problems
State following defines our problem and how Definition 2 and 3 relates
Definition 1: $P_{2}$
Definition 2: $P_{1}$
Definition 3: $P_{0}$
State the sequence of the proof as follows
Lemma 1: Problem $P_{1}$ is NP-complete
Begin Proof
Refer $P_{0}$
$\dots$
End proof
Lemma 2: Problem $P_{2}$ is NP-complete
Begin proof
$\dots$
End Proof
Or I should use Lemma for all my problems
State the sequence of the proof as follows (referred with acronyms)
Lemma 1(Problem Acronym): Problem $P_{1}$ is NP-complete (with full definition of $P_{1}$)
Begin Proof
Definition 3: $P_{0}$
Refer $P_{0}$
$\dots$
End proof
Lemma 2(Problem Acronym): Problem $P_{2}$ is NP-complete (with full definition of $P_{2}$)
Begin proof
$\dots$
End Proof
In my case definitions are not small, it takes 8 lines of one column in 2 column paper. What style of writing is easier to follow ?