Section CompatibilityProperties.
Global Instance equiv_lam_proper:
Proper (equiv step ++> equiv step) (@lam X).
Global Instance equiv_app_proper:
Proper (equiv step ++> equiv step ++> equiv step) (@app X).
Global Instance ren_equiv_proper:
Proper (eq ++> equiv step ++> equiv step) (@ren X).
Global Instance subst_equiv_proper:
Proper (eq ++> equiv step ++> equiv step) (@subst_exp X).

Lemma subst_pointwise_equiv (s: exp X) sigma tau:
(forall x, x ∈ vars s -> sigma x ≡ tau x) -> sigma • s ≡ tau • s.