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We are working in monadic second-order logic (more specifically S2S). The following is defined:

A formula $P(A_1,\ldots, A_m)$ is called principal if it has the form $$\exists x[xw_1\eta_1A_{i_1}\wedge\ldots\wedge xw_k\eta_kA_{i_k}]$$ In this, $\eta_i$ is either $\in$ or $\not\in$, and $xw_i$ stands for the concatenation of the variable $x$ with the word $w_i\in T_2=\{0,1\}^*$, but the latter is not of great importance, it should just be noted that expressions of the form $xw_1$ are all terms in our language.

Then we take any formula $F(A_1,\ldots,A_n)$, and the goal is to transform this into an equivalent formula $G$ which is of the normal form

$$Q_{n+1}\ldots Q_{m} M(A_1,\ldots,A_n),$$

where $M$ is a boolean combination of principal formulas and every $Q_i$ is either $\forall A_i$ or $\exists A_i$.

Now, in the proof we have already established a formula $F'$ of the form

$$Q_{n+1}\ldots Q_{m} M_1(A_1,\ldots,A_n),$$

where $Q_i$ are as above, and $M_1$ is in prenex normal form with only first-order quantifiers, and all atomic subformulas of $F'$ are of the form $\mathsf{term}\in \mathbf{A}$.

But now the problem starts, as they now claim that "pushing the quantifiers of $M_1$ one by one into $M_1$, $M_1$ can be transformed into an equivalent boolean combination of principal formulas".

I have tried multiple things, among which bringing the interior of $M_1$ into Disjunctive / Conjunctive Normal Form (dependant on the innermost quantifier of $M_1$), and then I can construct a formula $$M = q_1\ldots q_{p-1} (\bigvee P_i)$$ for principal formulas $P_i$, but, this is not a boolean combinations of principal formulas (is it? It seems to me to be multiple quantifications over a boolean combination of principal formulas)

Is there anyone who has any idea on how to proceed? I've read some things about combining first-order quantifiers (that is, transforming a formula $\forall x\forall y \phi(x,y)$ into a formula $Q_i\forall x[\phi(...)]$, where $Q_i$ is a second-order quantifier), but they were not entirely clear to me.

This is from Rabin's 1969 article Decidability of Second-Order Theories and Automata on Infinite Trees.

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This can be done using the disjunctive normal form. Suppose we want to reduce $\exists x_s[N(A_1,\ldots,A_m,x_1,\ldots,x_s]$. When you remark that the existential quantifier distributes over disjunctions, and that every $\mathsf{term}\in V$ only contains one variable, we can split all the conjuncts in the DNF into conjunctions over $x_1,\ldots,x_{s-1}$ and conjunctions over $x_s$. The existential quantifier then only applies to the latter conjunctions, thus obtaining a principal formula.

This looks like: $$\exists x_s[N(A_1,\ldots,A_m,x_1,\ldots,x_s] \Rightarrow \text{ (convert to DNF)}$$ $$\exists x_s\Big[\bigvee\Big(\bigwedge x_i w_j \eta_j A_{k_j}\Big)\Big]\Rightarrow$$ $$\bigvee\exists x_s\Big[\bigwedge x_i w_j \eta_j A_{k_j}\Big]\Rightarrow$$ $$\bigvee\exists x_s\Big[\bigwedge_{i

where $P_{x_s}$ is a principal formula over $x_s$.