Could White have won this double rook endgame? I reached this position as white that led to a draw:

[FEN ""]
[White "Bad_Bishop"]
[Black ""]
[StartPly "61"]
[Result "1/2-1/2"]

1. d4 d5 2. c4 dxc4 3. Nf3 Nf6 4. e3 Bg4 5. Bxc4 e6 6. h3 Bh5 7. Nc3 Nbd7 8. O-O Bd6 9. e4 e5 10. g4 Bg6 11. dxe5 Nxe5 12. Nxe5 Bxe5 13. f4 Qd4+ 14. Qxd4 Bxd4+ 15. Kh2 Bxc3 16. bxc3 Bxe4 17. g5 Bd5 18. Re1+ Kf8 19. Bb5 Ne4 20. c4 Nd6 21. cxd5 Nxb5 22. Rb1 a6 23. a4 Nc3 24. Rxb7 Nxd5 25. Re5 c6 26. Bb2 Rd8 27. Rf5 f6 28. gxf6 Nxf6 29. Bxf6 gxf6 30. Rxf6+ Ke8 31. Re6+ Kf8 32. Rxc6 Rg8 33. Rc2 Rg7 34. Rb6 Ra7 35. Rcc6 a5 36. Ra6 Rxa6 37. Rxa6 Rd5 38. Ra7 h5 39. Kg3 Rc5 40. Ra6 Kf7 41. Kh4 Ke7 42. Rh6 Rc4 43. Kg5 Rxa4 44. Rxh5 Ra1 45. Rh7+ Kd6 46. Ra7 a4 47. f5 a3 48. f6 a2 49. f7 Rg1+ 50. Kh6 Rf1 51. Rxa2 Ke6  1/2-1/2


I was up a pawn and had more active rooks. How could white have won this endgame (if there is a win at all)?
48.Kf6 should be winning. After say 48...a2, white goes h4-h5-h6 and something involving ...Rh1, Rxa2 Rxh6+, Kg5 followed by Re2 will lead to the Lucena position.

Much earlier on, I believe 31.Kg3 is winning, but it's also the best try by elimination: without an active king (which is the most important non-trivial factor in rook endgames), white won't be able to win with just his f- and h- pawns vs. black's h-pawn if he loses his a-pawn, and after Rg8 as happened in the game, white's own king is exposed to a perpetual. Black is doomed to passivity because his rooks are disconnected, and white can slowly capture the rest of black's pawns and/or get his king to e6 and promote the f-pawn.

P/S: 17.Re1 Kf8 18.Ba3+ Kg8 19.Rad1 followed by 20.Re7 looks crushing.