Invertibility of weakening (reverse weakening)

Show that weakening is invertible for definitional equality: if e and e′ do not mention any variables in Δ, then Γ, Δ, Γ′ ⊢ e ≡ e′ if and only if Γ, Γ′ ⊢ e ≡ e′.

Background

Weakening allows adding assumptions to the context; invertibility of weakening asserts that adding irrelevant variables has no effect on definitional equality for terms that do not mention those variables. This property supports modular reasoning about contexts and is used to derive several corollaries about types and well-formed contexts.

The paper frames this as a conjecture and enumerates consequences that would follow. Establishing reverse weakening would strengthen the metatheory and aid in the verification of context manipulations in the kernel.