Provide proofs for the following statements

The full resolution inference rule $($on Slide $37)$ holds for any $k,n1.$ During the lecture, we show the proof for the simple resolution inference rule. You can use a similar approach.

Horn clauses are closed under resolution: if you resolve two Horn clauses with complementary literals in the two clauses, you get back a Horn clause.

$2.$ Applying Resolution Inference Rule

$$ Unit resolution inference rule:

$$ PVQ$.$

Q

$-($Modus Ponens, Latin for "mode that affirms by affirming"$),$ i$.$e$.,$ both PV Q and $-$P are given, Q can be inferred.

$$ Full resolution inference rule:

PiV. VPk: Q$1$V::VQn

P$,$V$...$VP$-,$ VP$|+$aV$...$VPKVQ$,$ v$...$VQ$-1$VQ$,+1$ V$...$ven where

P$=-$Q;