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quantifier restrictions in natural deduction

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$P_1 \vee P_2 $ , $Q_1 \vee Q_2 $ are semi-decidable predicates and $P_2 = \overline{Q_2}$. What can be said about $P_1 \vee Q_1 $?

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#predicate-logic #computability
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How can I show $ (p \to r) \vee (q \to r) \equiv (p \vee q) \to r $ by Natural Deduction?

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Proof writing involving propositional logic: (x ∨ y) ≡ ( x ∧ y ) → x ≡ y

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#discrete-mathematics #propositional-calculus #predicate-logic
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