Can we say for sure that Black's king has not moved? The following position appears as the 7th puzzle of Raymond Smullyan's Arabian Knights, a book about retrograde analysis.
[FEN &quot;3qk2r/p2pp1bp/bpp2n2/8/3N3Q/1PN1P1P1/P1PPP2P/3K1n2 w - - 0 50&quot;]

The conditions given are that &quot;both [of White's] castles [rooks] captured on the same row, Black['s] queen's castle [rook was] captured by a pawn, and king's bishop given as odds.&quot; The stipulation is that you must find where White's rooks were captured.
I puzzled over this for quite some time, then gave up. Very early in the solution, it is stated that &quot;Since the Black king has not moved...&quot; However, I do not see how this can be inferred from the position. Is it necessarily true? Couldn't the Black king and queen have moved out of the way of the Black rook (which was captured on e3), and then moved back? I do not see &quot;non-movement of the Black king&quot; among the conditions or anything about Black being able to castle.
The conditions you gave are incomplete. The conditions include that black (to move) can castle; it's hidden in Smullyan's original problem text.

From the English book, last paragraph of the problem statement: 


  Just then, however, news came that Black was about to castle. [...]


A clearer problem statement from the Retro Corner:


  Schwarz (am Zuge) darf noch rochieren. [...]


Black (to move) can still castle.