Below there is Hoare triple in which variable $a$ is an array of integers, $len$, $x, i$ are integer-valued variables, and $r$ is a Boolean-valued variable. I have to provide a loop invariant (using predicate logic) suitable for proving partial correctness and explain in words why it is a loop invariant and why it is sufficient to prove partial correctness.
$$\begin{align*} &\{0 ≤ len \} \\ &i = 0; \\ &r = \textbf{false}; \\ &\textbf{while}\;(i < len)\;\{ \\ &\quad\textbf{if}\;(a[i] = x)\;\{ \\ &\quad\quad r = \textbf{true}; \\ &\quad\quad i = len; \\ &\quad \}\;\textbf{else}\;\{ \\ &\quad\quad i = i + 1;\\ &\quad \} \\ &\} \\ &\{(r=\textbf{true})\iff(\exists k \in \mathbb{Z}: (0 \le k \land k < len \land a[k] = x ))\} \end{align*}$$
I tried to use loop rule and get the invariant I by the third premise:
$$\{I \land \lnot b\} \; [] \; S \; \{Q\}$$
What I did: $$\{I \land i < len \} \implies \{(r = \textbf{true}) \iff (\exists k \in \mathbb{Z}:(0 \le k \land k < len \land a[k] = x ))\}$$
I can not go anymore at here, because I do not know how to find an invariant $\land$ $a[ i ] \neq x$ $\implies$ postcondition. And if $a[ i ] \neq x$ that means this it does not find $x$ in array $a$. But the postcondition said it will find $x$ because $r = \textbf{true}$.
