Algebraic transformation for dependent type One can make the following type algebraic transformation (where $=$ means isomorphic and $\equiv$ means syntactically equal):
$$
X * (X \to X) \;\; \equiv \;\; X * X^X = \\
X^{(1+X)} \;\; \equiv \;\; (1+X) \to X
$$
Is there an analogous transformation starting with the following dependent type:
$$
P 0 * (\forall n, P n \to P (S n))  \;\; \equiv \;\; ???
$$
 A: (For definiteness, I’ll use Martin-Löf type theory, with the notation of $\sum$ and $\prod$ for dependent sums/products, so my $\prod$ is your $\forall$. If you had a substantially different system or interpretation of the $\forall$ in mind, then my apologies for misunderstanding.)
The following seems a fairly analogous transformation:
$$ \textstyle  \newcommand{\N}{\mathbb{N}} P(0) \times (\prod_{n : \N} P(n) \to P(Sn)) \cong \prod_{z : \left(1 + \sum_{n:\N}P(n)\right)} P(\text{case $z$ of $\operatorname{inl}(x)\mapsto 0,\, \operatorname{inr}(y) \mapsto S(\pi_1(y))$}).$$
This follows from the distributivity properties between dependent function types and binary sums / dependent sums
$$\textstyle \prod_{x:A} (B(x) \to C(x)) \cong \prod_{y : \sum_{x:A} \! B(x)} C(\pi_1(y)) \qquad \qquad A \cong \prod_{x:1} A $$
$$ \textstyle \prod_{x:A} C(x) \times \prod_{y:B} D(y) \cong \prod_{z : A + B} \left(\text{case $z$ of $\operatorname{inl}(x)\mapsto C(x),\, \operatorname{inr}(y) \mapsto D(y)$}\right)$$
which are the dependent versions of the properties involved in your original isomorphism.
