Bash exit status of shorthand increment notation
a++ is post-increment: it increments after the statement is evaluated. By contrast, ++a increments before. Thus:
$ a=0 ; ((a++)) ; echo $a $?1 1$ a=0 ; ((++a)) ; echo $a $?1 0In the first case, ((a++)), the arithmetic expression is evaluated first, while a is still zero, yielding a value of zero (and hence a nonzero return status). Then, afterward, a is incremented.
In second case, ((++a)), a is incremented to 1 and then ((...)) is evaluated. Since a is nonzero when the arithmetic expression is evaluated, the return status is zero.
From man bash:
id++ id-- variable post-increment and post-decrement ++id --id variable pre-increment and pre-decrement
The exit status of the (()) notation is zero if the arithmetic expression is nonzero, and vice versa.
A=A+1You assign 1 to A, so the expression evaluates to 1, exit status zero.
A++POST-increment operator. The expression evaluates to zero, exit status 1, and then A is incremented.