Ltac destruct_match H :=
match type of H with
| context [ match ?x with _ => _  end] => destruct x eqn:?
end.
Section tbled.
Variable tbl : SymEnv.functions RType_typ.
Let RSym_sym := RSym_sym tbl.
Existing Instance RSym_sym.
Let RSym_sym_ok :  RSymOk RSym_sym := _.
Existing Instance RSym_sym_ok.
Let Expr_expr_fs := Expr_expr_fs tbl.
Existing Instance Expr_expr_fs.
Let Expr_ok_fs := Expr_ok_fs tbl.
Existing Instance Expr_ok_fs.
Existing Instance MA.

(* Specifications: *)
Lemma exprD'_App_R_typ  (e1 e2 : expr typ func) tus tvs ty1 ty2 v:
exprD' tus tvs ty2 (App e1 e2) = Some v ->
typeof_expr tus tvs e2 = Some ty1 ->
(exists v1 , exprD' tus tvs (tyArr ty1 ty2) e1 = Some v1) /\
(exists v2, exprD' tus tvs ty1 e2 = Some v2).
Lemma exprD'_App_L_typ  (e1 e2 : expr typ func) tus tvs ty1 ty2 v:
exprD' tus tvs ty2 (App e1 e2) = Some v ->
typeof_expr tus tvs e1 = Some (tyArr ty1 ty2) ->
(exists v1 , exprD' tus tvs (tyArr ty1 ty2) e1 = Some v1) /\
(exists v2, exprD' tus tvs ty1 e2 = Some v2).
Lemma exprD_typeof_Some : forall tus tvs t (e : expr typ func) val,
exprD' tus tvs t e = Some val -> typeof_expr tus tvs e = Some t.
Lemma typeof_app : forall (e1 e2 : expr typ func) t tvs tus,
typeof_expr tus tvs (App e1 e2) = Some t ->
exists t2, typeof_expr tus tvs e1 = Some (tyArr t2 t) /\
typeof_expr tus tvs e2 = Some (t2).
Lemma exprD_ex_typs : forall tus tvs t (e1 e2 : expr typ func) v,
exprD' tus tvs t (App e1 e2) = Some v ->
exists t2, typeof_expr tus tvs e1 = Some (tyArr t2 t) /\
typeof_expr tus tvs e2 = Some (t2).