Consider the pseudocode given for Decreasekey on a min-heap. The two loop invariants below together lead to proving that, after DecreaseKey is performed, the tree satisfies the min-heap property again. Loop invariant 1 for all nodes u in H except u and its root, keylparentfu)] < keyfu). Loop invariant 2 if u has a parent, then for each child c of t, keylparentfu < keyle). Decreasekey(H, v, k) Input a min-heap H containing node v Output a min-heap in which the key of node v has been set of k 1. key(v) +k 2. while v is not the root and keyſparentfu]] > keyfu) do 3. swap u and its parent Prove maintenance of Loop invariant 1. Hint think carerdny about which nodes get a new parent! Also, use Loop invariant 2.

can this be proved mathematically?

Formal approaches are based on the use of logical reasoning. In terms of general popularity, propositional logic and predicate logic are your best bets. An undergraduate student in discrete mathematics wonders whether propositional and predicate logics may be useful in software testing. This student really needs your help in finding what they're looking for. The benefits and drawbacks of logic as a tool for software testing might be outlined as one such approach. How should formal programme verification be performed, and which logic is optimal for doing so?

This is Discrete Math. Please show all work and give explanations for solutions. Thank you!