(* Inversion lemmas on equivalence ******************************************)

lemma drop_O1_inj: ∀i,L1,L2,K. ⬇[i] L1 ≡ K → ⬇[i] L2 ≡ K → L1 ⩬[i, ∞] L2.
#i @(ynat_ind … i) -i
[ #L1 #L2 #K #H <(drop_inv_O2 … H) -K #H <(drop_inv_O2 … H) -L1 //
| #i #IHi * [2: #L1 #I1 #V1 ] * [2,4: #L2 #I2 #V2 ] #K #HLK1 #HLK2 //
  lapply (drop_fwd_length … HLK1)
  <(drop_fwd_length … HLK2) [ /4 width=5 by drop_inv_drop1, lreq_succ/ ]
  #H [ elim (ysucc_inv_O_sn … H) | elim (ysucc_inv_O_dx … H) ]
| #L1 #L2 #K #H1 lapply (drop_fwd_Y2 … H1) -H1
  #H elim (ylt_yle_false … H) //
]
qed-.