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Rook ending: The lonely pawn

From a tournament game I had this weekend. Both players had about 2050 ELO, so don't expect brilliances, but it's nevertheless nice for the eyes of a patzer like me. White just took a pawn on a6: Black to move and win ... checked with Lomonosov tablebase; engines may not see it quickly ... maybe you like it
de.lichess.org/study/zdKm18vf
That final position after Kh5 doesn't look like a win, the black king is cut off by the rook and as long as the king can defend the pawn it looks entirely drawn.
Stockfish at depth 102 can't solve it, neither can finalgen. You sure there's a solution?
Btw, the variation with Rd5 is weird, it's draw most of the time, then suddenly winning, then draw again.
EDIT: @Linnemann You don't need shredderchess for that, lichess has 6 men tb too. ;)
I believe Lichess assesses the position based on material until it reaches a depth where a repetition is the best move or the 50 move rule occurs
The position after 7.g5: Its only draw when the black king is on a- or b-file. All other positions are won, because white does not manage to promote his g-pawn in time.

When the position of rook and pawn gets exchanged, so that the rook is behind the pawn, then the black king needs to be in front of the pawn or on h3, f3 or e3, where he wins a tempo by attacking the rook g5. Otherwise the pawn is too fast.

If the pawn is on g7, the rook on g8 and the white king on f6, then the black king needs to be on g-file or e-file or on the seventh rank in order to win. Black then brings his queen to g6, attacking the pawn - he can not bring his queen to this square by giving checks, he needs one quiet move. The rook then can not move away with check - on g-file, because the pawn is in the way, on e-file, because the white king e7 is in the way and on seventh rank because the black king attacks the check square. White then is in zugzwang, losing his rook or pawn. Addon: The white king can also be on a square where he needs just one move to enter g5 or f5, because he then has Qf6+, driving the white king in front of his pawn, followed by Kg6. Addon 2: it should also be noted that the black Queen needs to be on a square where she can give check on d6 or c6, because then the white king needs to go to f7 (followed by Qd5+ and the King needs to go to f8). If he can go to g6 then white draws because black is not able to bring his queen to g6.

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