Notation lexfg := (e_lexfg _) (only parsing).

(* Specifications: *)
Theorem lexfg_systemSL : forall (b : wsort) (M N : TS b), e_systemSL _ M N -> e_lexfg _ (e_P1 b) (e_P2 b) M N.
Theorem lexfg_app_l : forall a a' b : terms, e_lexfg _ (e_P1 wt) (e_P2 wt) a a' -> e_lexfg _ (e_P1 wt) (e_P2 wt) (app a b) (app a' b).
Theorem lexfg_app_r : forall a b b' : terms, e_lexfg _ (e_P1 wt) (e_P2 wt) b b' -> e_lexfg _ (e_P1 wt) (e_P2 wt) (app a b) (app a b').
Theorem lexfg_lambda : forall a a' : terms, e_lexfg _ (e_P1 wt) (e_P2 wt) a a' -> e_lexfg _ (e_P1 wt) (e_P2 wt) (lambda a) (lambda a').
Theorem lexfg_env_t : forall (a a' : terms) (s : sub_explicits), e_lexfg _ (e_P1 wt) (e_P2 wt) a a' -> e_lexfg _ (e_P1 wt) (e_P2 wt) (env a s) (env a' s).
Theorem lexfg_env_s : forall (a : terms) (s s' : sub_explicits), e_lexfg _ (e_P1 ws) (e_P2 ws) s s' -> e_lexfg _ (e_P1 wt) (e_P2 wt) (env a s) (env a s').
Theorem lexfg_cons_t : forall (a a' : terms) (s : sub_explicits), e_lexfg _ (e_P1 wt) (e_P2 wt) a a' -> e_lexfg _ (e_P1 ws) (e_P2 ws) (cons a s) (cons a' s).
Theorem lexfg_cons_s : forall (a : terms) (s s' : sub_explicits), e_lexfg _ (e_P1 ws) (e_P2 ws) s s' -> e_lexfg _ (e_P1 ws) (e_P2 ws) (cons a s) (cons a s').
Theorem lexfg_comp_l : forall s s' t : sub_explicits, e_lexfg _ (e_P1 ws) (e_P2 ws) s s' -> e_lexfg _ (e_P1 ws) (e_P2 ws) (comp s t) (comp s' t).
Theorem lexfg_comp_r : forall s t t' : sub_explicits, e_lexfg _ (e_P1 ws) (e_P2 ws) t t' -> e_lexfg _ (e_P1 ws) (e_P2 ws) (comp s t) (comp s t').
Theorem lexfg_lift : forall s s' : sub_explicits, e_lexfg _ (e_P1 ws) (e_P2 ws) s s' -> e_lexfg _ (e_P1 ws) (e_P2 ws) (lift s) (lift s').
Theorem lexfg_relSL : forall (b : wsort) (M N : TS b), e_relSL _ M N -> e_lexfg _ (e_P1 b) (e_P2 b) M N.
Theorem relSL_noetherian : forall b : wsort, explicit_noetherian _ (e_relSL b).