(* Main properties **********************************************************)
-theorem gget_mono: ∀G,G1,e. ⬇[e] G ≡ G1 → ∀G2. ⬇[e] G ≡ G2 → G1 = G2.
-#G #G1 #e #H elim H -G -G1
-[ #G #He #G2 #H
- >(gget_inv_gt … H He) -H -He //
-| #G #He #G2 #H
- >(gget_inv_eq … H He) -H -He //
-| #I #G #G1 #V #He #_ #IHG1 #G2 #H
- lapply (gget_inv_lt … H He) -H -He /2 width=1/
+theorem gget_mono: ∀G,G1,m. ⬇[m] G ≡ G1 → ∀G2. ⬇[m] G ≡ G2 → G1 = G2.
+#G #G1 #m #H elim H -G -G1
+[ #G #Hm #G2 #H
+ >(gget_inv_gt … H Hm) -H -Hm //
+| #G #Hm #G2 #H
+ >(gget_inv_eq … H Hm) -H -Hm //
+| #I #G #G1 #V #Hm #_ #IHG1 #G2 #H
+ lapply (gget_inv_lt … H Hm) -H -Hm /2 width=1/
]
qed-.
-lemma gget_dec: ∀G1,G2,e. Decidable (⬇[e] G1 ≡ G2).
-#G1 #G2 #e
-elim (gget_total e G1) #G #HG1
+lemma gget_dec: ∀G1,G2,m. Decidable (⬇[m] G1 ≡ G2).
+#G1 #G2 #m
+elim (gget_total m G1) #G #HG1
elim (eq_genv_dec G G2) #HG2
[ destruct /2 width=1/
| @or_intror #HG12