Multiple-twin problem

The problem we present has a peculiarity. It is a multiple problem, but not exactly a twin, because the produced problems have different number of moves in their solutions.
The American Joseph Wainwright (1851 – 1921) is the composer, known for his tasks with two-mover problems.

Each problem is producing the next one just after the key-move is played, while the number of moves for the solution is increased by 1. To be exact...
...the initial position is Mate in 2 moves,
after the key is Mate in 3 moves,
after the key is Mate in 4 moves,
after the key is Mate in 5 moves.

	(Problem 388) J. C. J. Wainwright, American Chess Bulletin, 1910, Mate in 2 moves. (a) #2 (10 + 9), (b) after the key of (a) #3, (c) after the key of (b) #4, (d) after the key of (c) #5
[8/2p1p1p1/p1PkP1P1/B1p2K2/2P5/pPP4p/P6p/7B]

In the initial position Black is stalemated. The solutions are simple (with possible exception the five-mover) :

(a) 1.b4! (zugzwang) cxb4 2.Bxb4#

(b) 1.b5! (zz) axb5 2.cxb5 (zz) c4 3.Bb4#

(c) 1.b6! (zz) cxb6 2.Bxb6 a5 3.c7 a4 4.c8=S#

(d) 1.Kg5! cxb6 2.Bxb6 a5 3.c7
3...a4 4.c8=Q/B Ke5 5.Bc7#
3...Kxe6 4.c8=Q+ Kd6/Ke5 5.Bc7#
3...Ke5 4.c8=Q a4/Kd6 5.Bc7#

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27-11-2009 : The friend reader Alotan has posted a comment :
Nice problem. The mate in 5 had many variations and I had to set it on the chessboard. The reason for comment, however, is that it reminded me a nice helpmate problem by Caillaud, with similar twinning mechanism :

	(Problem 389) Michel Caillaud, First prize, Pitlochry TT 2003 (a) h#2 (5+2), (b) Position of (a) before the mating move and h#2, (c) Position of (b) before the mating move and h#2.
[8/4p3/3S4/8/SRBk3K/8/8/8]

It is not exceptional or difficult, but it belongs to those problems that remain carved in the memory of the solver.

Dear readers, send the solution.