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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 *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "basic_2/syntax/lenv_length.ma".
16 include "basic_2/syntax/append.ma".
18 (* APPEND FOR LOCAL ENVIRONMENTS ********************************************)
20 (* Properties with length for local environments ****************************)
22 lemma append_length: ∀L1,L2. |L1 @@ L2| = |L1| + |L2|.
23 #L1 #L2 elim L2 -L2 //
24 #L2 #I #V2 >append_pair >length_pair >length_pair //
27 lemma ltail_length: ∀I,L,V. |ⓑ{I}V.L| = ⫯|L|.
28 #I #L #V >append_length //
31 (* Basic_1: was just: chead_ctail *)
32 lemma lpair_ltail: ∀L,I,V. ∃∃J,K,W. L.ⓑ{I}V = ⓑ{J}W.K & |L| = |K|.
33 #L elim L -L /2 width=5 by ex2_3_intro/
34 #L #Z #X #IHL #I #V elim (IHL Z X) -IHL
35 #J #K #W #H #_ >H -H >ltail_length
36 @(ex2_3_intro … J (K.ⓑ{I}V) W) /2 width=1 by length_pair/
39 (* Advanced inversion lemmas on length for local environments ***************)
41 (* Basic_2A1: was: length_inv_pos_dx_ltail *)
42 lemma length_inv_succ_dx_ltail: ∀L,l. |L| = ⫯l →
43 ∃∃I,K,V. |K| = l & L = ⓑ{I}V.K.
44 #Y #l #H elim (length_inv_succ_dx … H) -H #I #L #V #Hl #HLK destruct
45 elim (lpair_ltail L I V) /2 width=5 by ex2_3_intro/
48 (* Basic_2A1: was: length_inv_pos_sn_ltail *)
49 lemma length_inv_succ_sn_ltail: ∀L,l. ⫯l = |L| →
50 ∃∃I,K,V. l = |K| & L = ⓑ{I}V.K.
51 #Y #l #H elim (length_inv_succ_sn … H) -H #I #L #V #Hl #HLK destruct
52 elim (lpair_ltail L I V) /2 width=5 by ex2_3_intro/
55 (* Inversion lemmas with length for local environments **********************)
57 (* Basic_2A1: was: append_inj_sn *)
58 lemma append_inj_length_sn: ∀K1,K2,L1,L2. L1 @@ K1 = L2 @@ K2 → |K1| = |K2| →
61 [ * /2 width=1 by conj/
62 #K2 #I2 #V2 #L1 #L2 #_ >length_atom >length_pair
65 [ #L1 #L2 #_ >length_atom >length_pair
67 | #K2 #I2 #V2 #L1 #L2 #H1 >length_pair >length_pair #H2
68 elim (destruct_lpair_lpair_aux … H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *)
69 elim (IH … H1) -IH -H1 /3 width=4 by conj/
75 (* Basic_2A1: was: append_inj_dx *)
76 lemma append_inj_length_dx: ∀K1,K2,L1,L2. L1 @@ K1 = L2 @@ K2 → |L1| = |L2| →
79 [ * /2 width=1 by conj/
80 #K2 #I2 #V2 #L1 #L2 >append_atom >append_pair #H destruct
81 >length_pair >append_length >plus_n_Sm
82 #H elim (plus_xSy_x_false … H)
84 [ #L1 #L2 >append_pair >append_atom #H destruct
85 >length_pair >append_length >plus_n_Sm #H
86 lapply (discr_plus_x_xy … H) -H #H destruct
87 | #K2 #I2 #V2 #L1 #L2 >append_pair >append_pair #H1 #H2
88 elim (destruct_lpair_lpair_aux … H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *)
89 elim (IH … H1) -IH -H1 /2 width=1 by conj/
94 (* Advanced inversion lemmas ************************************************)
96 lemma append_inj_dx: ∀L,K1,K2. L@@K1 = L@@K2 → K1 = K2.
97 #L #K1 #K2 #H elim (append_inj_length_dx … H) -H //
100 lemma append_inv_refl_dx: ∀L,K. L@@K = L → K = ⋆.
101 #L #K #H elim (append_inj_dx … (⋆) … H) //
104 lemma append_inv_pair_dx: ∀I,L,K,V. L@@K = L.ⓑ{I}V → K = ⋆.ⓑ{I}V.
105 #I #L #K #V #H elim (append_inj_dx … (⋆.ⓑ{I}V) … H) //
108 (* Basic eliminators ********************************************************)
110 (* Basic_1: was: c_tail_ind *)
111 lemma lenv_ind_alt: ∀R:predicate lenv.
112 R (⋆) → (∀I,L,T. R L → R (ⓑ{I}T.L)) →
114 #R #IH1 #IH2 #L @(f_ind … length … L) -L #x #IHx * // -IH1
115 #L #I #V #H destruct elim (lpair_ltail L I V) /4 width=1 by/