$(**$

Create an inductive relation that holds if$,$ and only if$,$ element $\text{'}$x$\text{'}$

appears before element $\text{'}$y$\text{'}$ in the given list.

We can define $`$succ$`$ inductively as follows:

$($x$,$ y$)$ succ l

$-----------------------$R$1------------------$R$2$

$($x$,$ y$)$ succ x :: y :: l $($x$,$ y$)$ succ z :: l

Rule R$1$ says that x succeeds y in the list that starts with $[$x$,$ y$].$

Rule R$2$ says that if x succeeds y in list l then x succeeds y in a list

the list that results from adding z to list l$.$

$*)$

Inductive succ $\{$X : Type$\}($x:X$)($y:X$)$: list X $->$ Prop :$=$

$|$ succ$\_$here : forall l$,$ succ x y $($x :: y :: l$)$

$|$ succ$\_$later : forall z l$,$ succ x y l $->$ succ x y $($z :: l$).$

Theorem succ$3$:

$(*$ Only one of the following propositions is provable.

Replace 'False' by the only provable proposition and then prove it:

$1)$ succ $24[1$;$2$;$3$;$4]$

$2)$ ~ succ $24[1$;$2$;$3$;$4]$

$*)$

~ succ $24[1$;$2$;$3$;$4].$

Proof.

Qed.

please prove this.