Section Reflection.
Variables φ' ψ0' ψ1' : valuation.
Variable Hψ0' : substitute ψ0' (substitute φ' s') = substitute φ' t0'.
Variable Hψ1' : substitute ψ1' (substitute φ' s') = substitute φ' t1'.

Lemma substitute_embed_var {ξ r} :
substitute (fun x => ξ (to_nat (x, 0))) r = substitute ξ (substitute embed_var r).