I am trying to proof the following algorithm to see if a there exists a path from u to v in a graph G = (V,E).   Reach := {u} LastCardinality := 0 while LastCardinality &lt; |Reach| do   LastCardinality := |Reach|   Next := null   foreach i &lt;- Reach do     foreach j &lt;- V do       if(i,j) exists in E then         Next := Next + {j}   Reach := Reach + Next return Reach   So far I have got that the output needs to be Reach = Goal where Goal = {r | r exists V, there is a path from u to r}. I also have three invariants that I am pretty sure will help prove this.  Reach is always a subset of Goal and Next is always a subset of Goal.  After an arbitrary K times through the while loop, for every t that exsits in V, if t has a path of length k+1 from u, then t exists in Reach  If Reach is a subset of Goal before the while loop then LastCardinality &lt; |Reach|  I know that to finish up the proof, I need to prove termination, the invariants, and correctness but I have no idea how. I think I need to use induction on the while loop but I am not exactly sure how.   How do I proof those three characteristics about an algorithm?
I am trying to proof the following algorithm to see if a there exists a path from u to v in a graph G = (V,E).   Reach := {u} LastCardinality := 0 while LastCardinality &lt; |Reach| do   LastCardinality := |Reach|   Next := null   foreach i &lt;- Reach do     foreach j &lt;- V do       if(i,j) exists in E then         Next := Next + {j}   Reach := Reach + Next return Reach   So far I have got that the output needs to be Reach = Goal where Goal = {r | r exists V, there is a path from u to r}. I also have three invariants that I am pretty sure will help prove this.   Reach is always a subset of Goal and Next is always a subset of Goal. After an arbitrary K times through the while loop, for every t that exsits in V, if t has a path of length k+1 from u, then t exists in Reach If Reach is a subset of Goal before the while loop then LastCardinality &lt; |Reach|   I know that to finish up the proof, I need to prove termination, the invariants, and correctness but I have no idea how. I think I need to use induction on the while loop but I am not exactly sure how.   How do I proof those three characteristics about an algorithm?
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Obviously, the content of the reference might well be in Welsh. (Though it might be in English, too.)  As can be seen, while elements of the document controlled by babel are handled correctly, those under the control of biblatex are not. However, I cannot simply modify the solution offered by lockstep for changing strings to customised English variants. That is, I can't just use a Welsh version of  \DefineBibliographyStrings{english}{%   bibliography = {Works Cited},   references = {Works Cited}, }   because biblatex has never heard of it.  What is the most appropriate way to configure biblatex for use with an unsupported language? What does it mean if a professor does not answer your email in time? I recently joined a research group as a postdoc. I wrote the professor an email to ask quick but important questions including asking him when when he wanted me to start. It's been over a week and I haven't received get an answer. 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Difference In Eigenvectors