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The
commute-subst-rename
lemma in the Substitution chapter quantifies over polymorphic renamingρ : ∀{Γᵨ} → Rename Γᵨ (Γᵨ , ★)
(i.e. theρ
itself quantifies over the contextΓᵨ
), but the parameterΓᵨ
is only instantiated withΓᵨ = Γ
andΓᵨ = Δ
. Thus we can replace this with just two renamingsρ₁ : Rename Γ (Γ , ★)
andρ₂ : Rename Δ (Δ , ★)
. By this way, we can also avoid defining some what awkwardρ′
. I think the resulting proof is more clean.I also fixed the corresponding prose below the agda code. It seems that the equational reasoning part was just broken, for example in this line:
which should be
even if we keep using polymorphic renaming.