-(* Basic_1: was: drop1_gen_pcons *)
-lemma drops_inv_cons: ∀L1,L2,s,l,m,cs. ⬇*[s, {l, m} @ cs] L1 ≡ L2 →
- ∃∃L. ⬇*[s, cs] L1 ≡ L & ⬇[s, l, m] L ≡ L2.
-/2 width=3 by drops_inv_cons_aux/ qed-.
-
-lemma drops_inv_skip2: ∀I,s,cs,cs2,i. cs ▭ i ≡ cs2 →
- ∀L1,K2,V2. ⬇*[s, cs2] L1 ≡ K2. ⓑ{I} V2 →
- ∃∃K1,V1,cs1. cs + 1 ▭ i + 1 ≡ cs1 + 1 &
- ⬇*[s, cs1] K1 ≡ K2 &
- ⬆*[cs1] V2 ≡ V1 &
- L1 = K1. ⓑ{I} V1.
-#I #s #cs #cs2 #i #H elim H -cs -cs2 -i
-[ #i #L1 #K2 #V2 #H
- >(drops_inv_nil … H) -L1 /2 width=7 by lifts_nil, minuss_nil, ex4_3_intro, drops_nil/
-| #cs #cs2 #l #m #i #Hil #_ #IHcs2 #L1 #K2 #V2 #H
- elim (drops_inv_cons … H) -H #L #HL1 #H
- elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ #K #V <yminus_succ2 #HK2 #HV2 #H destruct
- elim (IHcs2 … HL1) -IHcs2 -HL1 #K1 #V1 #cs1 #Hcs1 #HK1 #HV1 #X destruct
- @(ex4_3_intro … K1 V1 … ) // [3,4: /2 width=7 by lifts_cons, drops_cons/ | skip ]
- >pluss_SO2 >pluss_SO2
- >yminus_succ2 >ylt_inv_O1 /2 width=1 by ylt_to_minus/ <yminus_succ >commutative_plus (**) (* <yminus_succ1_inj does not work *)
- /3 width=1 by minuss_lt, ylt_succ/
-| #cs #cs2 #l #m #i #Hil #_ #IHcs2 #L1 #K2 #V2 #H
- elim (IHcs2 … H) -IHcs2 -H #K1 #V1 #cs1 #Hcs1 #HK1 #HV1 #X destruct
- /4 width=7 by minuss_ge, yle_succ, ex4_3_intro/
+(* Basic_1: includes: drop_gen_skip_l *)
+(* Basic_2A1: includes: drop_inv_skip1 *)
+lemma drops_inv_skip1: ∀I,K1,V1,Y,s,t. ⬇*[s, Ⓣ@t] K1.ⓑ{I}V1 ≡ Y →
+ ∃∃K2,V2. ⬇*[s, t] K1 ≡ K2 & ⬆*[t] V2 ≡ V1 & Y = K2.ⓑ{I}V2.
+/2 width=5 by drops_inv_skip1_aux/ qed-.
+
+fact drops_inv_skip2_aux: ∀X,Y,s,t. ⬇*[s, t] X ≡ Y → ∀I,K2,V2,tl. Y = K2.ⓑ{I}V2 → t = Ⓣ@tl →
+ ∃∃K1,V1. ⬇*[s, tl] K1 ≡ K2 & ⬆*[tl] V2 ≡ V1 & X = K1.ⓑ{I}V1.
+#X #Y #s #t * -X -Y -t
+[ #t #Ht #J #K2 #W2 #tl #H destruct
+| #I #L1 #L2 #V #t #_ #J #K2 #W2 #tl #_ #H2 destruct
+| #I #L1 #L2 #V1 #V2 #t #HL #HV #J #K2 #W2 #tl #H1 #H2 destruct
+ /2 width=5 by ex3_2_intro/