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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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11 (* v GNU General Public License Version 2 *)
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15 include "basic_2/substitution/ldrop.ma".
17 (* DROPPING *****************************************************************)
19 (* Properties on append for local environments ******************************)
21 fact ldrop_O1_append_sn_le_aux: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 →
23 ∀L. ⇩[0, e] L @@ L1 ≡ L @@ L2.
24 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize // /4 width=1/
26 lapply (le_n_O_to_eq … H) -H //
29 lemma ldrop_O1_append_sn_le: ∀L1,L2,e. ⇩[0, e] L1 ≡ L2 → e ≤ |L1| →
30 ∀L. ⇩[0, e] L @@ L1 ≡ L @@ L2.
31 /2 width=3 by ldrop_O1_append_sn_le_aux/ qed.
33 (* Inversion lemmas on append for local environments ************************)
35 lemma ldrop_O1_inv_append1_ge: ∀K,L1,L2,e. ⇩[0, e] L1 @@ L2 ≡ K →
36 |L2| ≤ e → ⇩[0, e - |L2|] L1 ≡ K.
37 #K #L1 #L2 elim L2 -L2 normalize //
38 #L2 #I #V #IHL2 #e #H #H1e
39 elim (ldrop_inv_O1 … H) -H * #H2e #HL12 destruct
40 [ lapply (le_n_O_to_eq … H1e) -H1e -IHL2
41 >commutative_plus normalize #H destruct
42 | <minus_plus >minus_minus_comm /3 width=1/
46 lemma ldrop_O1_inv_append1_le: ∀K,L1,L2,e. ⇩[0, e] L1 @@ L2 ≡ K → e ≤ |L2| →
47 ∀K2. ⇩[0, e] L2 ≡ K2 → K = L1 @@ K2.
48 #K #L1 #L2 elim L2 -L2 normalize
50 lapply (le_n_O_to_eq … H2) -H2 #H2
51 lapply (ldrop_inv_atom1 … H3) -H3 #H3 destruct
52 >(ldrop_inv_refl … H1) -H1 //
53 | #L2 #I #V #IHL2 #e @(nat_ind_plus … e) -e [ -IHL2 ]
55 lapply (ldrop_inv_refl … H1) -H1 #H1
56 lapply (ldrop_inv_refl … H2) -H2 #H2 destruct //
57 | #e #_ #H1 #H1e #K2 #H2
58 lapply (ldrop_inv_ldrop1 … H1 ?) -H1 //
59 lapply (ldrop_inv_ldrop1 … H2 ?) -H2 // /3 width=4/