Iterated club shooting and the stationary-logic constructible model

We investigate iterating the construction of $C(\mathtt{a}\mathtt{a})$, the $L$-like inner model constructed using stationary-logic. We show that it is possible to force over generic extensions of $L$ to obtain a model of $V=C(\mathtt{a}\mathtt{a})$, and to obtain models in which the sequence of iterated $C(\mathtt{a}\mathtt{a})$s is decreasing of arbitrarily large order types. For this we prove distributivity and stationary-set preservation properties for countable iterations of club-shooting forcings using mutually stationary sets, and introduce the notion of mutually fat sets which yields better distributivity results even for uncountable iterations.