Given this assertion in Hoare Logic:
\begin{align} &\mathbf{\{p >= 0\}}\\ &s = 0 ; n = 1 ;\\ &\mathtt{while}\ (n <= p)\ \{\\ &\quad s = s + n ;\\ &\quad n = n + 1\\ &\}\\ &\mathbf{\{s == p * (p + 1) / 2\}} \end{align}
How to prove this using the rules of Hoare Logic. I am not sure where to really begin, I only currently understand (most) of the meaning of the axioms, but not what is required for a proof.
Its a question of chaining/glueing the axioms between them.
Let's start from the end:
$$\mathbf{\{s == p * (p + 1) / 2\}}$$
Use the strengthening rule to change the postcondition to
$$\mathbf{\{s == p * (p + 1) / 2 \wedge p == n \wedge p >= 0\}} $$
which is equivalent to
$$\mathbf{\{s == n * (n + 1) / 2 \wedge p == n \wedge p >= 0\}}$$
By the Loop rule in order for the triple:
$$ \mathbf{\{s == n * (n + 1) / 2 \}} \mathtt{while}\ (n <= p)\ \{s = s + n ; n = n + 1\} \mathbf{\{s == n * (n + 1) / 2 \wedge p == n \}} $$
we need to prove that the triple
$$\mathbf{\{s == n * (n + 1) / 2 \}} \{s = s + n ; n = n + 1\} \mathbf{\{s == n * (n + 1) / 2\}}$$
is valid.
Let then use the composition to reduce this to proving that the two following triples are valid:
$$\mathbf{\{s == n * (n + 1) / 2 \}} \{s = s + n\} \mathbf{\{s == (n+1) * (n + 2) / 2\}}$$
and
$$\mathbf{\{s == (n+1) * (n + 2) / 2 \}} \{ n = n + 1\} \mathbf{\{s == n * (n + 1) / 2\}}$$
The latter one is just an application of the Assignment rule (just replace the occurrences of $n$ by $n+1$ in the postcond to get the precondition).
To prove the first one, we need to apply Strengthening to reduce it to the triple $$\mathbf{\{s == n * (n + 1) / 2 \}} \{s = s + n\} \mathbf{\{s == (n) * (n +1) / 2 + n\}}$$ and then use the Assignment rule trivially.
Okay so now we only need (by Sequence rule) to prove that the triple
$$\mathbf{\{p>=0 \}} s:=0; n:= 1 \mathbf{\{s == n * (n + 1) / 2 \wedge p >= 0\}}$$ which can be done in the same way by splitting with sequence.
