X = ⓕ{I}V1.T1.
/2 width=3 by lifts_inv_flat2_aux/ qed-.
+(* Advanced inversion lemmas ************************************************)
+
(* Basic_2A1: includes: lift_inv_pair_xy_x *)
lemma lifts_inv_pair_xy_x: ∀f,I,V,T. ⬆*[f] ②{I}V.T ≡ V → ⊥.
#f #J #V elim V -V
]
qed-.
+lemma lifts_inv_lref1_uni: ∀l,Y,i. ⬆*[l] #i ≡ Y → Y = #(l+i).
+#l #Y #i1 #H elim (lifts_inv_lref1 … H) -H /4 width=4 by at_mono, eq_f/
+qed-.
+
(* Basic forward lemmas *****************************************************)
+lemma lifts_fwd_atom2: ∀f,I,X. ⬆*[f] X ≡ ⓪{I} → ∃J. X = ⓪{J}.
+#f * #n #X #H
+[ lapply (lifts_inv_sort2 … H)
+| elim (lifts_inv_lref2 … H)
+| lapply (lifts_inv_gref2 … H)
+] -H /2 width=2 by ex_intro/
+qed-.
+
(* Basic_2A1: includes: lift_inv_O2 *)
lemma lifts_fwd_isid: ∀f,T1,T2. ⬆*[f] T1 ≡ T2 → 𝐈⦃f⦄ → T1 = T2.
#f #T1 #T2 #H elim H -f -T1 -T2
]
qed-.
+(* Properties with uniform relocation ***************************************)
+
+lemma lifts_uni: ∀n1,n2,T,U. ⬆*[𝐔❴n1❵∘𝐔❴n2❵] T ≡ U → ⬆*[n1+n2] T ≡ U.
+/3 width=4 by lifts_eq_repl_back, after_inv_total/ qed.
+
(* Basic_2A1: removed theorems 14:
lifts_inv_nil lifts_inv_cons
lift_inv_Y1 lift_inv_Y2 lift_inv_lref_Y1 lift_inv_lref_Y2 lift_lref_Y lift_Y1