convert unary to binary, rules below are buggy and don't work on all inputs

rules

$|[A]->$an|[A]
[A]n|[B]->[A]u[B]
[A]g|[B]->[A]u[B]
[A]u|[B]->[A]|g[B]
$a[A]g$->$0n[A]$
$0[A]g$->$00n[A]$
$1[A]g$->$01n[A]$
$a[A]u$->$0|n[A]$
$0[A]u$->$10n[A]$
$1[A]u$->$11[A]n$
[A]n$->[A]$

input

$||||||$