# Scott's normal form for $\exists y \forall x R(x,y)$

In this paper Scott's reduction is discussed, which reduces any FO2 formula to Scott's Normal form. As far as I understand the reduction process explained in the paper can get you to formulas with only $$\forall \forall$$ and $$\forall \exists$$ quantification prefixes. Which has three possibilities of prefixes $$\forall x \forall y$$, $$\forall x \exists y$$ and $$\exists y \forall x$$. Now scott's normal form only admits $$\forall x \forall y$$ and $$\forall x \exists y$$ quantification. Then how is the explained process sound for a formula like $$\exists x \forall y R(x,y)$$ ?

In other words, what is the scott's reduction for $$\Phi(x,y) = \exists x \forall y R(x,y)$$.

My attempt on reduction according to the paper:

$$Sc( \Phi(x,y) )= \exists x P(x) \land \forall x \forall y (P(x) \leftrightarrow R(x,y))$$

Your solution is correct. It suffices to see that the formulae $$\exists{y} P(y)$$ and $$\forall{x} \exists{y} P(y)$$ are equi-satisfiable, which allows you to put your formulae in the desired form.