Lecture 9: The Substitution Lemma

The Substitution Lemma

If Γ, x: τ₁ |- x : τ₂ and e' : τ₁, then Γ |- [e'/x]e : τ₂

Proof by induction on the derivation of Γ, x: τ₁ |- x : τ₂, and the crucial case is the one for λ.

In this case, we have
Γ, x: τ₁, y: τ |- e₁ : τ'
-----------------------------
Γ, x: τ₁ |- λy:τ.e₁ : τ -> τ'

(we have e = λy:τ.e₁ and τ₂ = τ -> τ' in this case, so we need to show Γ |- [e'/x]λy:τ.e₁ : τ -> τ')

Due to the magic of alpha-variance, we can assume, "without loss of generality," that x != y and y !ε FV(e'), so that by the rules of substitution from before, [e'/x]λy:τ.e₁ = λy:τ.[e'/x]e₁. That means that we need to show that Γ |- λy:τ.[e'/x]e₁ : τ -> τ' in order to complete the case.

Because we said we would treat the context Γ like a set, we can rewrite Γ, x: τ₁, y: τ as Γ, y: τ, x: τ₁. We have Γ, y: τ, x: τ₁ |- e₁ : τ' and e' : τ'₁, so by the induction hypothesis we have Γ, y: τ |- [e'/x]e₁ : τ'.

Now, by rule, we have Γ |- λy:τ.[e'/x]e₁ : τ -> τ'. We're done!