Past exam of the computer science course of the University of Oxford / 2015 / Lambda calculus and types / 2 / a / Solution

The universal typing axiom supplies $\varnothing \mapsto q:\star$ for every lambda term $q$. Use this as the argument premise of the lambda application rule. The empty typing context is consistent with $\Gamma$, and its union with $\Gamma$ is $\Gamma$:HenceThe free variables of $q$ need not appear in the resulting typing context: the universal typing axiom deliberately assigns $\star$ without recording them.