Determining Causality and Time-Invariance of a system Consider the following system:
$$y(t-1)=\int_{-\infty}^\infty x()u(-t) d $$
where $u(t)$ is the unit step function, which is zero for $t&lt;0$ and equals $1$ for $t&gt;0$.
$(1)$ Is the system causal? Why or why not?

I think if $u(t)=0$ for all $t&lt;0$. This means that $u(τ−t)=0$ for all $τ&lt;t$ or, equivalently, for all $t&gt;τ$ and the integrand is zero in range $({-\infty}, t)$.Therefore, we can show that:
$$\int_{-\infty}^\infty x()u(-t) d = \int_{t}^\infty x()u(-t) d =  \int_{t}^\infty x() d$$ So, the system is not causal! Am I right?!

$(2)$ Is the system time-invariant? Why or why not?

Let $ s=t-1$
$$y(s)=\int_{-\infty}^\infty x()u(-s+1) d $$
For invariance, we need to show output resulted from $x(s−s_o)$ = $y(s−s_o)$.
$$y_1(s)=\int_{-\infty}^\infty x(-s_0)u(-s+1) d $$
Now when I change of variable to $z=τ−s_o$, $τ=z+s_0$ , $dz=dτ$ it leads to:
$$y_1(s)=\int_{-\infty}^\infty x(z)u(z+s_0-s+1) dz $$
For $y(s-s_o)$ integral becomes:
$$y_2(s-s_0)=\int_{-\infty}^\infty x()u(-s+1-s_0) d $$
So it seems to time-variant system.
Update:I would appreciate if anyone can answer my question because it's not a homework anymore and yet I didn't find the answer.
Hint
Substitute $s = t-1$. That gets you the equations in a more standard form $y(s) = ...$ Then go through the same excercise.