fact drop_O1_append_sn_le_aux: ∀L1,L2,s,l,m. ⬇[s, l, m] L1 ≡ L2 →
l = 0 → m ≤ |L1| →
- ∀L. ⬇[s, 0, m] L ;; L1 ≡ L ;; L2.
+ ∀L. ⬇[s, 0, m] L ● L1 ≡ L ● L2.
#L1 #L2 #s #l #m #H elim H -L1 -L2 -l -m normalize
[2,3,4: /4 width=1 by drop_skip_lt, drop_drop, arith_b1, lt_minus_to_plus_r, monotonic_pred/ ]
#l #m #_ #_ #H <(le_n_O_to_eq … H) -H //
qed-.
lemma drop_O1_append_sn_le: ∀L1,L2,s,m. ⬇[s, 0, m] L1 ≡ L2 → m ≤ |L1| →
- ∀L. ⬇[s, 0, m] L ;; L1 ≡ L ;; L2.
+ ∀L. ⬇[s, 0, m] L ● L1 ≡ L ● L2.
/2 width=3 by drop_O1_append_sn_le_aux/ qed.
(* Inversion lemmas on append for local environments ************************)
-lemma drop_O1_inv_append1_ge: ∀K,L1,L2,s,m. ⬇[s, 0, m] L1 ;; L2 ≡ K →
+lemma drop_O1_inv_append1_ge: ∀K,L1,L2,s,m. ⬇[s, 0, m] L1 ● L2 ≡ K →
|L2| ≤ m → ⬇[s, 0, m - |L2|] L1 ≡ K.
#K #L1 #L2 elim L2 -L2 normalize //
#L2 #I #V #IHL2 #s #m #H #H1m
]
qed-.
-lemma drop_O1_inv_append1_le: ∀K,L1,L2,s,m. ⬇[s, 0, m] L1 ;; L2 ≡ K → m ≤ |L2| →
- ∀K2. ⬇[s, 0, m] L2 ≡ K2 → K = L1 ;; K2.
+lemma drop_O1_inv_append1_le: ∀K,L1,L2,s,m. ⬇[s, 0, m] L1 ● L2 ≡ K → m ≤ |L2| →
+ ∀K2. ⬇[s, 0, m] L2 ≡ K2 → K = L1 ● K2.
#K #L1 #L2 elim L2 -L2 normalize
[ #s #m #H1 #H2 #K2 #H3 lapply (le_n_O_to_eq … H2) -H2
#H2 elim (drop_inv_atom1 … H3) -H3 #H3 #_ destruct