Section Transport.
Variables φ ψ0 ψ1 : valuation.
Variable Hψ0 : substitute ψ0 (substitute φ s0) = substitute φ t.
Variable Hψ1 : substitute ψ1 (substitute φ s1) = substitute φ t.
(* LU2SemiU valuation φ from RU2SemiU valuation φ' *)
Definition φ' : valuation := fun x =>
match of_nat x with
| (x, 0) => substitute embed_var (φ x)
| (0, 1) => substitute embed_var (substitute ψ0 (substitute φ s1))
| (0, 2) => substitute embed_var (substitute ψ1 (substitute φ s0))
| _ => atom x
end.
(* LU2SemiU valuations ψ from RU2SemiU valuations *)
Definition ψ0' : valuation := fun x =>
match of_nat x with
| (x, 0) => substitute embed_var (ψ0 x)
| _ => atom x
end.
Definition ψ1' : valuation := fun x =>
match of_nat x with
| (x, 0) => substitute embed_var (ψ1 x)
| _ => atom x
end.

Lemma substitute_φ'P {r: term} :
substitute φ' (substitute embed_var r) = substitute embed_var (substitute φ r).