-lemma drop_split: ∀L1,L2,l,m2,s. ⬇[s, l, m2] L1 ≡ L2 → ∀m1. m1 ≤ m2 →
- ∃∃L. ⬇[s, l, m2 - m1] L1 ≡ L & ⬇[s, l, m1] L ≡ L2.
-#L1 #L2 #l #m2 #s #H elim H -L1 -L2 -l -m2
-[ #l #m2 #Hs #m1 #Hm12 @(ex2_intro … (⋆))
- @drop_atom #H lapply (Hs H) -s #H destruct /2 width=1 by le_n_O_to_eq/
-| #I #L1 #V #m1 #Hm1 lapply (le_n_O_to_eq … Hm1) -Hm1
- #H destruct /2 width=3 by ex2_intro/
-| #I #L1 #L2 #V #m2 #HL12 #IHL12 #m1 @(nat_ind_plus … m1) -m1
- [ /3 width=3 by drop_drop, ex2_intro/
- | -HL12 #m1 #_ #Hm12 lapply (le_plus_to_le_r … Hm12) -Hm12
- #Hm12 elim (IHL12 … Hm12) -IHL12 >minus_plus_plus_l
- #L #HL1 #HL2 elim (lt_or_ge (|L1|) (m2-m1)) #H0
- [ elim (drop_inv_O1_gt … HL1 H0) -HL1 #H1 #H2 destruct
- elim (drop_inv_atom1 … HL2) -HL2 #H #_ destruct
- @(ex2_intro … (⋆)) [ @drop_O1_ge normalize // ]
- @drop_atom #H destruct
- | elim (drop_O1_pair … HL1 H0 I V) -HL1 -H0 /3 width=5 by drop_drop, ex2_intro/
- ]
- ]
-| #I #L1 #L2 #V1 #V2 #l #m2 #_ #HV21 #IHL12 #m1 #Hm12 elim (IHL12 … Hm12) -IHL12
- #L #HL1 #HL2 elim (lift_split … HV21 l m1) -HV21 /3 width=5 by drop_skip, ex2_intro/
-]
-qed-.
-
-(* Basic_2A1: includes: drop_split *)
-lemma drops_split_trans: ∀L1,L2,f,c. ⬇*[c, f] L1 ≡ L2 → ∀f1,f2. f1 ⊚ f2 ≡ f →
- ∃∃L. ⬇*[c, f1] L1 ≡ L & ⬇*[c, f2] L ≡ L2.
-#L1 #L2 #f #c #H elim H -L1 -L2 -f
-[ #f #Hc #f1 #f2 #Hf @(ex2_intro … (⋆)) @drops_atom
- #H lapply (Hc H) -c
- #H elim (after_inv_isid3 … Hf H) -f //
-| #I #L1 #L2 #V #f #HL12 #IHL12 #f1 #f2 #Hf elim (after_inv_xxS … Hf) -Hf *
- [ #g1 #g2 #Hf #H1 #H2 destruct elim (IHL12 … Hf) -f
- #L #HL1 #HL2 @(ex2_intro … (L.ⓑ{I}V)) /2 width=1 by drops_drop/
- @drops_skip //
- | #g1 #Hf #H destruct elim (IHL12 … Hf) -f
- /3 width=5 by ex2_intro, drops_drop/
- ]
-| #I #L1 #L2 #V1 #V2 #f #_ #HV21 #IHL12 #f1 #f2 #Hf elim (after_inv_xxO … Hf) -Hf
- #g1 #g2 #Hf #H1 #H2 destruct elim (lifts_split_trans … HV21 … Hf) -HV21
- elim (IHL12 … Hf) -f /3 width=5 by ex2_intro, drops_skip/
-]
-qed-.
-