To prove this using inference theory, we can follow the process of logical deduction based on the premises given. Let's break down the information we have and see how it leads to any conclusion.

Premises:

1. $A \rightarrow (B \lor C)$
2. $B \rightarrow \neg A$
3. $D \rightarrow \neg C$

We will use these to try to deduce any valid conclusions using rules of inference, such as modus ponens, modus tollens, and disjunction elimination.

Step-by-Step Proof:

1. Assume $A$ is true:

   • From premise (1), $A \rightarrow (B \lor C)$, using modus ponens, we can conclude that $B \lor C$ is true.

2. Assume $B$ is true:

   • If $B$ is true, from premise (2), $B \rightarrow \neg A$, by modus ponens, $\neg A$ must be true, which contradicts our assumption that $A$ is true. Therefore, $B$ must be false.

3. Since $B$ is false, $C$ must be true:

   • From the disjunction $B \lor C$, if $B$ is false, $C$ must be true to satisfy the disjunction.

4. Now assume $D$ is true:

   • From premise (3), $D \rightarrow \neg C$, by modus ponens, we can deduce $\neg C$.

5. Contradiction arises:

   • From step 3, we concluded that $C$ is true, but from step 4, we derived $\neg C$, which contradicts our earlier conclusion.

Conclusion:

Since assuming $A$ leads to a contradiction, we conclude that $A$ must be false. This is the only conclusion we can draw without further premises.

Thus, by inference theory, $A$ must be false, as any assumption of its truth leads to a contradiction.

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Would you like further clarification or details on specific steps?

Here are five questions to extend this:

1. How does modus ponens work in logical inference?
2. What is the role of disjunction elimination in logical proofs?
3. Can you explain how modus tollens is used in logical deductions?
4. What are the limitations of using inference rules in proofs?
5. How can contradictions be used to prove logical negations?

Tip: Always check for contradictions in logical proofs to validate the truth or falsity of assumptions!