End Intervals.
Definition Evar_env'ntoSn (U : nat) (ve : var_env') : var_env' :=
fun n : nat => if beq_nat n U then true else ve n.
Definition Evar_env''ntoSn (U : nat) (ve : var_env'') : var_env'' :=
MapPut _ ve (N_of_nat U) tt.

(* Specifications: *)
Lemma beq_Eq_true : forall m n : nat, beq_nat m n = true <-> eq_nat m n.
Lemma beq_complete : forall m n : nat, beq_nat m n = true -> m = n.
Lemma beq_correct : forall n : nat, beq_nat n n = true.
Lemma Evar_env'ntoSn_lemma :
forall (L U : nat) (ve : var_env'),
L <= U ->
In _ (Evar_env' L U) ve -> In _ (Evar_env' L (S U)) (Evar_env'ntoSn U ve).
Lemma Evar_env''ntoSn_lemma :
forall (L U : nat) (ve : var_env''),
L <= U ->
In _ (Evar_env'' L U) ve -> In _ (Evar_env'' L (S U)) (Evar_env''ntoSn U ve).