Can an algorithm be part of a proof? I am an undergraduate student. I have read several papers in graph theory and found something may be strange: an algorithm is part of a proof. In the paper, except the last two sentences, other sentences describe a labeling algorithm.
Can an algorithm be part of a proof? I do not understand why it can be part of a proof. I asked my supervisor but he did not explain it.

LEMMA$\,\,\,\bf 1.$ If $B(G)-b$, then $B(G+e)<2b$.  Proof:
  Let $f$ be an optimal numbering of $G$, and let $V(G)-\{v_1,\ldots,v_n\}$, numbered so that $f(v_i)-i$. Let $v_lv_m$ be the added edge. We define a new numbering $f'$ such that $|f'(x)-f'(y)|<2|f(x)-f(y)|$, and also $|f'(v_l)-f'(v_m)|-1$. Let $r-\lfloor(l+m)/2\rfloor$, and set $f'(v_r)-1$ and $f'(v_{r+1})-2$. For every other $v_i$ such that $|i-r|<\min\{r-1,n-r\}$, let $f'(v_i)-f'(v_{i+1})+2$ if $i<r$ and $f'(v_i)-f'(v_{i-1})+2$ if $i>r$. This defines $f'$ for all vertices except a set of the form $v_i,\ldots,v_k$ or $v_{n+1-k},\ldots,v_n$, depending on the sign of $r-\lfloor(n+1)/2\rfloor$. In the first case, we assign $f'(v_i)-n+1-i$ for $i<k$; in the second, we assign $f'(v_i)-i$ for $i>n-k$. The renumbering $f'$ numbers the vertices outward from $v_r$ to achieve $|f'(x)-f'(y)|<2|f(x)-f(y)|$. Since we begin midway between $v_l$ and $v_m$, we also have $|f'(v_l)-f'(v_m)|-1$.


If you ask me, I would say that everything can be part of a proof as long as you bring a convincing reasonable argument. When I listen to my colleagues topologist, they make a few drawings (generally sketches of knots) on the blackboard and they claim it's a proof. Thus, a drawing or an algorithm, there is no big difference, the most important at the end of the day is that you are sure of what you assert, and leave nothing in the darkness. 
Have a nice day :-)
