lemma drop_inv_O1_gt: ∀L,K,m,s. ⬇[s, 0, m] L ≡ K → |L| < m →
                      s = Ⓣ ∧ K = ⋆.
#L elim L -L [| #L #Z #X #IHL ] #K #m #s #H normalize in ⊢ (?%?→?); #H1m
[ elim (drop_inv_atom1 … H) -H elim s -s /2 width=1 by conj/
  #_ #Hs lapply (Hs ?) // -Hs #H destruct elim (ylt_yle_false … H1m) -H1m //
| elim (drop_inv_O1_pair1 … H) -H * #H2m #HLK destruct
  [ elim (ylt_yle_false … H1m) -H1m //
  | elim (IHL … HLK) -IHL -HLK /2 width=1 by ylt_pred, conj/
  ]
]
qed-.

lemma drop_O1_le: ∀s,m,L. m ≤ |L| → ∃K. ⬇[s, 0, m] L ≡ K.
#s #m @(ynat_ind … m) -m /2 width=2 by ex_intro/
[ #m #IHm *
  [ #H elim (ylt_yle_false … H) -H //
  | #L #I #V #H elim (IHm L) -IHm /3 width=2 by drop_drop, yle_inv_succ, ex_intro/
  ]
| #L #H elim (ylt_yle_false … H) -H //
]
qed-.

lemma drop_O1_lt: ∀s,L,m. m < |L| → ∃∃I,K,V. ⬇[s, 0, m] L ≡ K.ⓑ{I}V.
#s #L elim L -L
[ #m #H elim (ylt_yle_false … H) -H //
| #L #I #V #IHL #m @(ynat_ind … m) -m /2 width=4 by drop_pair, ex1_3_intro/
  [ #m #_#H elim (IHL m) -IHL /3 width=4 by drop_drop, ylt_inv_succ, ex1_3_intro/
  | #H elim (ylt_yle_false … H) -H //
  ]
]
qed-.

lemma drop_O1_pair: ∀L,K,m,s. ⬇[s, 0, m] L ≡ K → m ≤ |L| → ∀I,V.
                    ∃∃J,W. ⬇[s, 0, m] L.ⓑ{I}V ≡ K.ⓑ{J}W.
#L elim L -L [| #L #Z #X #IHL ] #K #m #s #H #Hm #I #V
[ elim (drop_inv_atom1 … H) -H #H >(yle_inv_O2 … Hm) -m
  #Hs destruct /2 width=3 by ex1_2_intro/
| elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK destruct /2 width=3 by ex1_2_intro/
  elim (IHL … HLK … Z X) -IHL -HLK
  /3 width=3 by yle_pred, drop_drop_lt, ex1_2_intro/
]
qed-.

lemma drop_O1_ge: ∀L,m. |L| ≤ m → ⬇[Ⓣ, 0, m] L ≡ ⋆.
#L elim L -L [ #m #_ @drop_atom #H destruct ]
#L #I #V #IHL #m @(ynat_ind … m) -m //
[ #H elim (ylt_yle_false … H) -H /2 width=1 by ylt_inj/
| /4 width=1 by drop_drop, yle_inv_succ/
]
qed.

lemma drop_O1_eq: ∀L,s. ⬇[s, 0, |L|] L ≡ ⋆.
#L elim L -L /2 width=1 by drop_drop/
qed.

lemma drop_fwd_length_ge: ∀L1,L2,l,m,s. ⬇[s, l, m] L1 ≡ L2 → |L1| ≤ l → |L2| = |L1|.
#L1 #L2 #l #m #s #H elim H -L1 -L2 -l -m //
[ #I #L1 #L2 #V #m #_ #_ #H elim (ylt_yle_false … H) -H //
| #I #L1 #L2 #V1 #V2 #l #m #_ #_ #IH #H
  lapply (yle_inv_succ … H) -H #H 
  >length_pair >length_pair /3 width=1 by eq_f/
]
qed-.

lemma drop_fwd_length_le_le: ∀L1,L2,l,m,s. ⬇[s, l, m] L1 ≡ L2 → 
                             ∀l0. l + m + l0 = |L1| → |L2| = l + l0.
#L1 #L2 #l #m #s #H elim H -L1 -L2 -l -m //
[ #l #m #Hm #l0 #H elim (yplus_inv_O … H) -H
  #H #H0 elim (yplus_inv_O … H) -H
  #H1 #_ destruct //
| #I #L1 #L2 #V #m #_ >yplus_O1 >yplus_O1 #IH #l0
  /3 width=1 by ysucc_inv_inj/
| #I #L1 #L2 #V1 #V2 #l #m #_ #_ #IHL12 #l0 >yplus_succ1 >yplus_succ1 #H
  lapply (ysucc_inv_inj … H) -H #Hl1
  >yplus_succ1 /3 width=1 by eq_f/
]
qed-.

lemma drop_fwd_length_le_ge: ∀L1,L2,l,m,s. ⬇[s, l, m] L1 ≡ L2 → l ≤ |L1| → |L1| ≤ l + m → |L2| = l.
#L1 #L2 #l #m #s #H elim H -L1 -L2 -l -m
[ #l #m #_ #H #_ /2 width=1 by yle_inv_O2/
| #I #L #V #_ #H elim (ylt_yle_false … H) -H //
| #I #L1 #L2 #V #m #_ >yplus_O1 >yplus_O1
  /3 width=1 by yle_inv_succ/
| #I #L1 #L2 #V1 #v2 #l #m #_ #_ #IH
  >yplus_SO2 >yplus_SO2
  /4 width=1 by yle_inv_succ, eq_f/
]
qed-.

lemma drop_fwd_be: ∀L,K,s,l,m,i. ⬇[s, l, m] L ≡ K → |K| ≤ i → i < l → |L| ≤ i.
#L #K #s #l #m #i #HLK #HK #Hl elim (ylt_split i (|L|)) //
#HL elim (drop_O1_lt (Ⓕ) … HL) #I #K0 #V #HLK0 -HL
elim (ylt_inv_plus_sn … Hl) -Hl #l0 #H0
elim (drop_conf_lt … HLK … HLK0 … H0) -HLK -HLK0 -H0
#K1 #V1 #HK1 #_ #_ lapply (drop_fwd_length_lt2 … HK1) -I -K1 -V1
#H elim (ylt_yle_false … H) -H //
qed-.

lemma drop_O1_ex: ∀K2,i,L1. |L1| = |K2| + i →
                  ∃∃L2. L1 ⩬[0, i] L2 & ⬇[i] L2 ≡ K2.
#K2 #i @(ynat_ind … i) -i
[ /3 width=3 by lreq_O2, ex2_intro/
| #i #IHi #Y >yplus_succ2 #Hi
  elim (drop_O1_lt (Ⓕ) Y 0) [2: >Hi // ]
  #I #L1 #V #H lapply (drop_inv_O2 … H) -H #H destruct
  >length_pair in Hi; #H lapply (ysucc_inv_inj … H) -H
  #HL1K2 elim (IHi L1) -IHi // -HL1K2
  /3 width=5 by lreq_pair, drop_drop, ex2_intro/
| #L1 >yplus_Y2 #H elim (ylt_yle_false (|L1|) (∞)) //
]
qed-.