PROBLEM $3(5$ marks$)$ Give a formal proof of

p $->$ r $`$ p $$ q $->$ r$.$

Use only the inference rules IR$01$IR$12.$

IR$01$ Modus ponens: $\backslash$alpha $->\backslash$beta $,\backslash$alpha $`\backslash$beta

IR$02$ introduction: $\backslash$alpha $->\backslash$beta $,\backslash$beta $->\backslash$alpha $`\backslash$alpha $\backslash$beta

IR$03$ elimination: $\backslash$alpha $\backslash$beta $,\backslash$beta $`\backslash$alpha and $\backslash$alpha $\backslash$beta $,\backslash$alpha $`\backslash$beta

IR$04$ Case analysis: $\backslash$alpha $\backslash$beta $,\backslash$alpha $->\backslash$gamma $,\backslash$beta $->\backslash$gamma $`\backslash$gamma

IR$05$ introduction: $\backslash$alpha $,\backslash$beta $`\backslash$alpha $\backslash$beta

IR$06$ elimination: $\backslash$alpha $\backslash$beta $`\backslash$alpha and $\backslash$alpha $\backslash$beta $`\backslash$beta

IR$07$ introduction: $\backslash$alpha $`\backslash$alpha $\backslash$beta and $\backslash$alpha $`\backslash$beta $\backslash$alpha

IR$08$ Contradiction: $\backslash$alpha $,\backslash$alpha $`$ F

IR$09$ introduction: $\backslash$alpha $->$ F $`\backslash$alpha

IR$10$ elimination: $\backslash$alpha $->$ F $`\backslash$alpha

IR$11$ Tautology: $`$ T

IR$12$ Deduction rule: $`\backslash$alpha $->\backslash$beta $,$ applicable if there is already a proof of $$A$,\backslash$alpha $`\backslash$beta $.$PROBLEM $3(5$ marks$)$ Give a formal proof of

$pr|--p^{{?}^{?}}qr.$

Use only the inference rules IR$01-$IR$12.$PROBLEM $3(5$ marks$)$ Give a formal proof of

$pr|--p^{{?}^{?}}qr.$

Use only the inference rules IR$01-$IR$12.$

IR$01$ Modus ponens: $,|--$

IR$02$ harr introduction: $,|--harr$

IR$03$ harr elimination: $harr,|--$ and $harr,|--$

IR$04$ Case analysis: $vv,,|--$

IR$05{?}^{{?}^{?}}$ introduction: $,|--{}^{{?}^{?}}$

IR$06{?}^{{?}^{?}}$ elimination: ${}^{{?}^{?}}|--$ and ${}^{{?}^{?}}|--$

IR$07vv$ introduction: $|--vv$ and $|--vv$

IR$08$ Contradiction: $,$not$|--F$

IR$09$ not introduction: $F|--$not$$

IR $10$not elimination: not$F|--$

IR$11$ Tautology: $|--T$

IR$12$ Deduction rule: $|--,$ applicable if there is already a proof of $"A,".$