Monotonic Boolfuck

This esolang is a brainfuck derivative.

Monotonic Boolfuck is an esoteric programming language invented by User:DigitalDetective47. It is a variant of Boolfuck where the number of set bits is monotonically increasing with respect to time.

Commands

Command	Meaning
1	Set the bit that the current tape cell holds.
,	Input if tape head's cell is 0; consume a bit of input with no effect if tape head's cell is 1.
;	Output.
<	Move tape head left.
>	Move tape head right.
[	Jump past matching ] if current tape cell holds 0.
]	Jump back to matching [.

Examples

Hello, world!

1
>;;;<;>;;<;;;  H
<;>;<;>;;<;;;; e
;;<;;>;<;;;;   l
;;<;;>;<;;;;   l
<;;;;>;<;;;;   o
;;<;;>;<;;;
;;;;;<;;;
<;;;>;<;;;;;   W
<;;;;>;<;;;;   o
;<;>;;<;;;;;   r
;;<;;>;<;;;;   l
;;<;>;;<;;;;   d
<;>;;;;<;;;

Truth-machine

,;>;;;1;;>;;<<[;>>;;;<;;>;;<<]