(* This version of the proof is incredibly slow. *)
(* all: pre; basic_solve. all: try oh_c'mon. all: try invert_open_wrt. all: try pick fresh c for L. all: try eapply lc_a_UCAbs_exists; eauto. all: try apply H; eauto. all: eapply (lc_a_UAbs_exists c). rewrite e. eauto. done. Qed.*)
Definition Typing_lc  := first lc_mutual.
Definition PropWff_lc := second lc_mutual.
Definition Iso_lc     := third lc_mutual.
Definition DefEq_lc   := fourth lc_mutual.
Definition Ctx_lc     := fifth lc_mutual.

(* Specifications: *)
Lemma Typing_lc1 : forall G0 a A, Typing G0 a A -> lc_tm a.
Lemma Typing_lc2 : forall G0 a A, Typing G0 a A -> lc_tm A.
Lemma Iso_lc1 : forall G0 D p1 p2, Iso G0 D p1 p2 -> lc_constraint p1.
Lemma Iso_lc2 : forall G0 D p1 p2, Iso G0 D p1 p2 -> lc_constraint p2.
Lemma DefEq_lc1 : forall G0 D A B T,   DefEq G0 D A B T -> lc_tm A.
Lemma DefEq_lc2 : forall G0 D A B T,   DefEq G0 D A B T -> lc_tm B.
Lemma DefEq_lc3 : forall G0 D A B T,   DefEq G0 D A B T -> lc_tm T.
Hint Resolve Typing_lc1 Typing_lc2 Iso_lc1 Iso_lc2 DefEq_lc1 DefEq_lc2 DefEq_lc3 Ctx_lc : lc.
Lemma Toplevel_lc : forall c s, binds c s toplevel -> lc_sig_sort s.