Cumulative Incidence Function given no event before time $t_d$

Suppose we have a simple competing risks model and have a calculated $$CIF_K$$ for two events $$\{k_1, k_2\} \in K$$. By definition $$CIF_{k_1}(t_i)$$ gives us the probability that event $$k_1$$ has occurred on or before time $$t_i$$. Is it possible for us to recalculate $$CIF_{K}$$ given a known minimum survival time?

That is, if we consider a patient that has already lived to time $$t_d$$ without experiencing either event, is it possible to derive an updated probability $$CIF'_{K}(t_i - t_d)$$ given the additional information that no event $$K$$ occurred before time $$t_d$$ where $$t_0$$ < $$t_d$$ $$\le$$ $$t_i$$?

We would expect $$CIF'_K(t_d)$$ = $$h_K(t_d)$$, the instantaneous hazard rate at time $$t_d$$. But how does this lead us to $$CIF'_K(t_d + 1)$$?

This is done using the rules of conditional probability. Letting $$T$$ denote the time until the event $$K$$ you can write the conditional distribution function of interest as:
\begin{align} CIF_K(t|t_d) &\equiv \mathbb{P}(T \leqslant t | T > t_d) \\[12pt] &= \frac{\mathbb{P}(t_d < T \leqslant t)}{\mathbb{P}(T > t_d)} \\[6pt] &= \frac{CIF_K(t) - CIF_K(t_d)}{1 - CIF_K(t_d)} \cdot \mathbb{I}(t > t_d \geqslant 0). \\[6pt] \end{align}
\begin{align} f_K(t|t_d) &= \frac{d}{dt} CIF_K(t|t_d) \\[6pt] &= \frac{d}{dt} \frac{CIF_K(t) - CIF_K(t_d)}{1 - CIF_K(t_d)} \cdot \mathbb{I}(t > t_d \geqslant 0) \\[6pt] &= \frac{f_K(t)}{1 - CIF_K(t_d)} \cdot \mathbb{I}(t > t_d \geqslant 0), \\[12pt] h_K(t|t_d) &= \frac{f_K(t|t_d)}{1 - CIF_K(t|t_d)} \cdot \mathbb{I}(t > t_d \geqslant 0) \\[6pt] &= \bigg[ \frac{f_K(t)}{1 - CIF_K(t_d)} \bigg/ \frac{1 - CIF_K(t)}{1 - CIF_K(t_d)} \bigg] \cdot \mathbb{I}(t > t_d \geqslant 0) \\[6pt] &= \frac{f_K(t)}{1 - CIF_K(t)} \cdot \mathbb{I}(t > t_d \geqslant 0) \\[6pt] &= \frac{f_K(t)}{1 - CIF_K(t)} \cdot \mathbb{I}(t > t_d \geqslant 0) \\[12pt] &= h_K(t) \cdot \mathbb{I}(t > t_d \geqslant 0). \\[6pt] \end{align}