I am currently considering if I can prove the conjunction rule (given below) from the above given Hoare logic rules.
My answer would be "no", because from the proof rules of Hoare logic, I can neither use the
Implied rule to prove it (because from $A$ I cannot infer $A \wedge B$) nor introduce it anywhere from the first four proof rules (the only one which introduces conjunction in the post-condition requires a
while). I am fairly confident that this line of reasoning is correct, but am I missing anything? For example, can I argue on a the propositional logic level that since I have $P$ as a precondition, and I have $Q_1$ and $Q_2$ as postconditions, I can simply introduce conjunction on the propositional logic level (instead of inferring purely on Hoare logic)?