Step one is to convert our integer into vector notation.
1: 25129925999 3: 25129925999
. 2: 10
1: [11, 10, 9, ..., 1, 0]
.
25129925999 RET 10 RET 12 RET v x 12 RET -
1: 25129925999 1: [0, 2, 25, 251, 2512, ... ]
2: [100000000000, ... ] .
.
V M ^ s 1 V M \
(Recall, the \ command computes an integer quotient.)
1: [0, 2, 5, 1, 2, 9, 9, 2, 5, 9, 9, 9]
.
10 V M % s 2
Next we must increment this number. This involves adding one to the last digit, plus handling carries. There is a carry to the left out of a digit if that digit is a nine and all the digits to the right of it are nines.
1: [0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1] 1: [1, 1, 1, 0, 0, 1, ... ]
. .
9 V M a = v v
1: [1, 1, 1, 0, 0, 0, ... ] 1: [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1]
. .
V U * v v 1 |
Accumulating * across a vector of ones and zeros will preserve only the initial run of ones. These are the carries into all digits except the rightmost digit. Concatenating a one on the right takes care of aligning the carries properly, and also adding one to the rightmost digit.
2: [0, 0, 0, 0, ... ] 1: [0, 0, 2, 5, 1, 2, 9, 9, 2, 6, 0, 0, 0]
1: [0, 0, 2, 5, ... ] .
.
0 r 2 | V M + 10 V M %
Here we have concatenated 0 to the left of the original number; this takes care of shifting the carries by one with respect to the digits that generated them.
Finally, we must convert this list back into an integer.
3: [0, 0, 2, 5, ... ] 2: [0, 0, 2, 5, ... ]
2: 1000000000000 1: [1000000000000, 100000000000, ... ]
1: [100000000000, ... ] .
.
10 RET 12 ^ r 1 |
1: [0, 0, 20000000000, 5000000000, ... ] 1: 25129926000
. .
V M * V R +
Another way to do this final step would be to reduce the formula `10 $$ + $' across the vector of digits.
1: [0, 0, 2, 5, ... ] 1: 25129926000
. .
V R ' 10 $$ + $ RET