1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "basic_2/grammar/lenv_append.ma".
16 include "basic_2/relocation/ldrop.ma".
18 (* DROPPING *****************************************************************)
20 (* Properties on append for local environments ******************************)
22 fact ldrop_O1_append_sn_le_aux: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 →
24 ∀L. ⇩[0, e] L @@ L1 ≡ L @@ L2.
25 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize // /4 width=1/
27 lapply (le_n_O_to_eq … H) -H //
30 lemma ldrop_O1_append_sn_le: ∀L1,L2,e. ⇩[0, e] L1 ≡ L2 → e ≤ |L1| →
31 ∀L. ⇩[0, e] L @@ L1 ≡ L @@ L2.
32 /2 width=3 by ldrop_O1_append_sn_le_aux/ qed.
34 (* Inversion lemmas on append for local environments ************************)
36 lemma ldrop_O1_inv_append1_ge: ∀K,L1,L2,e. ⇩[0, e] L1 @@ L2 ≡ K →
37 |L2| ≤ e → ⇩[0, e - |L2|] L1 ≡ K.
38 #K #L1 #L2 elim L2 -L2 normalize //
39 #L2 #I #V #IHL2 #e #H #H1e
40 elim (ldrop_inv_O1 … H) -H * #H2e #HL12 destruct
41 [ lapply (le_n_O_to_eq … H1e) -H1e -IHL2
42 >commutative_plus normalize #H destruct
43 | <minus_plus >minus_minus_comm /3 width=1/
47 lemma ldrop_O1_inv_append1_le: ∀K,L1,L2,e. ⇩[0, e] L1 @@ L2 ≡ K → e ≤ |L2| →
48 ∀K2. ⇩[0, e] L2 ≡ K2 → K = L1 @@ K2.
49 #K #L1 #L2 elim L2 -L2 normalize
51 lapply (le_n_O_to_eq … H2) -H2 #H2
52 lapply (ldrop_inv_atom1 … H3) -H3 #H3 destruct
53 >(ldrop_inv_refl … H1) -H1 //
54 | #L2 #I #V #IHL2 #e @(nat_ind_plus … e) -e [ -IHL2 ]
56 lapply (ldrop_inv_refl … H1) -H1 #H1
57 lapply (ldrop_inv_refl … H2) -H2 #H2 destruct //
58 | #e #_ #H1 #H #K2 #H2
59 lapply (le_plus_to_le_r … H) -H
60 lapply (ldrop_inv_ldrop1 … H1 ?) -H1 //
61 lapply (ldrop_inv_ldrop1 … H2 ?) -H2 //
62 <minus_plus_m_m /2 width=4/