If Γ[x:τ'] |- e:τ AND Γ |- e:τ' then Γ |- e'/xe:τ

should be

If Γ[x:τ'] |- e:τ AND Γ |- e':τ' then Γ |- {e'/x}e:τ

Update 10/14: There is an omission in the typing rules on page 2 of the homework. The missing rules are for numeric constants, which should have type int in any context Gamma, and boolean constants true and false, which should have type bool in any context Gamma. These rules are crucial to the (rather straightforward) proof in problem 2.