Inductive hypothesis

$$\displaystyle\frac{a_1\ J \ \dots \ a_k\ J}{a\ J}$$

we typically need to show:

$$\displaystyle\frac{\mathcal{P}(a_1)\ \dots\ \mathcal{P}(a_k)}{\mathcal{P}(a)}$$

In this case, all the Instance of the judgment forms $\mathcal{P}(a_i)$ are the inductive hypotheses.