Iterate/Loop algebra

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This is still a work in progress. It may be changed in the future.

$c_{n}$ refers to any value independent of the current state of the block. Visit count $VC$ specifically refers to the inner body, shown with an ellipsis.

Stateless

Structure	Claim	Note
*x< ... >	$VC=x$	By definition of the loop construct.
*c1< *c2< ... > >	$VC=c_{1}\cdot c_{2}$	The inner body is executed $c_{2}$ times per each outer iteration, of which there are $c_{1}$ such.
*c1< ⋱ *cn< ... > ⋰ >	$VC=\prod _{i=1}^{n}{c_{i}}$	Repeated application of the above.
*c< *n< ... > >	$VC=\sum _{i=1}^{c}{i}={\frac {c(c+1)}{2}}$	On the $i$-th iteration of the outer body, the inner body executes $i$ times.
*c< *~n< ... > >	$VC=\sum _{i=1}^{c}{c-i}=\sum _{i=1}^{c-1}{i}={\frac {c(c-1)}{2}}$	On the $i$-th iteration of the outer body, the inner body executes $c-i$ times.
*c< ⎧ *n< k ⎨ ⋱ ⎩ *n< ... > ⋰ > >	$VC={\binom {c+k}{k+1}}$	Repeated summation.
*c< ⎧ *~n< k ⎨ ⋱ ⎩ *~n< ... > ⋰ > >	$VC={\binom {c}{k+1}}$	Repeated summation.
*c< ⎧ *X< k ⎨ ⋱ ⎩ *X< ... > ⋰ > > Where X is either n or ~n.	Where $a$ is the total number of n's, and $b$ is the total number of ~n's. $VC={\binom {c+a}{a+b+1}}$
*c< *~n< ... >  ! >	$VC=c-1$	On the first iteration of the outer loop, VC increments by $c-1$. The break immediately halts further iterations of the outer loop.

Stateful

Structure	Claim	Note
*=L1< (L1*)<> >	$VC_{L_{1}}=VC_{L_{1}}+VC_{L_{1}}=2VC_{L_{1}}$
*c< *=L1< (L1*)<> > >	$VC_{L_{1}}=2^{c}VC_{L_{1}}$	Repetition of above.