(* ------------------------------------------------------------ *)
(* tactics for substitution proofs. *)
(* Most of the substitution cases below are about showing that the term is locally closed after the substiution. This tactic takes care of that argument. *)
Ltac lc_subst_case x0 b0  :=
let x:= fresh in
lc_inversion x; subst;
try (rewrite tm_subst_tm_tm_open_tm_wrt_tm; eauto);
try (rewrite tm_subst_tm_tm_open_tm_wrt_co; eauto);
econstructor; eauto using Value_lc,
tm_subst_tm_tm_lc_tm, tm_subst_tm_co_lc_co,
tm_subst_tm_constraint_lc_constraint;
apply_lc_exists x;
eauto using tm_subst_tm_tm_lc_tm, tm_subst_tm_co_lc_co,
Value_lc, tm_subst_tm_constraint_lc_constraint;
subst_helper x x0 b0.
(* ------------------------------------------------- *)

Lemma subst_reduction_in_one : forall a a',
reduction_in_one a a' -> forall b x, lc_tm b ->
reduction_in_one (tm_subst_tm_tm b x a)
(tm_subst_tm_tm b x a').