Section ConstantSubstitution.
Implicit Types (X Y Z : Const).
Fixpoint subst_consts {X Y} (kappa: X -> exp Y) s :=
match s with
| var x => var x
| const c => kappa c
| lambda s => lambda (subst_consts (kappa >> ren shift) s)
| app s t => (subst_consts kappa s) (subst_consts kappa t)
end.
Global Instance step_subst_consts X Y:
Proper (Logic.eq ++> step ++> step) (@subst_consts X Y).
Global Instance steps_subst_consts X Y:
Proper (Logic.eq ++> star step ++> star step) (@subst_consts X Y).
Global Instance equiv_subst_consts X Y:
Proper (Logic.eq ++> equiv step ++> equiv step) (@subst_consts X Y).

Lemma consts_in_subst_consts X Y (kappa: X -> exp Y) c s:
c ∈ consts (subst_consts kappa s) -> exists d, d ∈ consts s /\ c ∈ consts (kappa d).