-include "basic_2/grammar/lenv_length.ma".
-
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
]
qed-.
-lemma drop_fwd_length: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → |L1| = |L2| + m.
-#L1 #L2 #l #m #H elim H -L1 -L2 -l -m //
-#l #m #H >H -m //
-qed-.
-
-lemma drop_fwd_length_le2: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → m ≤ |L1|.
-#L1 #L2 #l #m #H lapply (drop_fwd_length … H) -H //
-qed-.
-
-lemma drop_fwd_length_le4: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → |L2| ≤ |L1|.
-#L1 #L2 #l #m #H lapply (drop_fwd_length … H) -H //
-qed-.
-
-lemma drop_fwd_length_lt2: ∀L1,I2,K2,V2,l,m.
- ⬇[Ⓕ, l, m] L1 ≡ K2. ⓑ{I2} V2 → m < |L1|.
-#L1 #I2 #K2 #V2 #l #m #H
-lapply (drop_fwd_Y2 … H) #Hm
-lapply (drop_fwd_length … H) -l #H <(yplus_O2 m) >H -L1
-/2 width=1 by monotonic_ylt_plus_sn/
-qed-.
-
-lemma drop_fwd_length_lt4: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → 0 < m → |L2| < |L1|.
-#L1 #L2 #l #m #H
-lapply (drop_fwd_Y2 … H) #Hm
-lapply (drop_fwd_length … H) -l
-/2 width=1 by monotonic_ylt_plus_sn/
-qed-.
-
-lemma drop_fwd_length_eq1: ∀L1,L2,K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 →
- |L1| = |L2| → |K1| = |K2|.
-#L1 #L2 #K1 #K2 #l #m #HLK1 #HLK2 #HL12
-lapply (drop_fwd_Y2 … HLK1) #Hm
-lapply (drop_fwd_length … HLK1) -HLK1
-lapply (drop_fwd_length … HLK2) -HLK2
-#H #H0 >H in HL12; -H >H0 -H0 #H
-@(yplus_inv_monotonic_dx … H) -H // (**) (* auto fails *)
-qed-.
-
-lemma drop_fwd_length_eq2: ∀L1,L2,K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 →
- |K1| = |K2| → |L1| = |L2|.
-#L1 #L2 #K1 #K2 #l #m #HLK1 #HLK2 #HL12
-lapply (drop_fwd_length … HLK1) -HLK1
-lapply (drop_fwd_length … HLK2) -HLK2 //
-qed-.
-
-lemma drop_inv_length_eq: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 →
- |L1| = |L2| → m = 0.
-#L1 #L2 #l #m #H #HL12 lapply (drop_fwd_length … H) -H
->HL12 -L1 #H elim (discr_yplus_x_xy … H) -H //
-#H elim (ylt_yle_false (|L2|) (∞)) //
-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
#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-.