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_length.ma".
17 (* LOCAL ENVIRONMENTS *******************************************************)
19 let rec append L K on K ≝ match K with
21 | LPair K I V ⇒ (append L K). ⓑ{I} V
24 interpretation "append (local environment)" 'Append L1 L2 = (append L1 L2).
26 (* Basic properties *********************************************************)
28 lemma append_atom_sn: ∀L. ⋆ @@ L = L.
29 #L elim L -L normalize //
32 lemma append_assoc: associative … append.
33 #L1 #L2 #L3 elim L3 -L3 normalize //
36 lemma append_length: ∀L1,L2. |L1 @@ L2| = |L1| + |L2|.
37 #L1 #L2 elim L2 -L2 normalize //
40 (* Basic inversion lemmas ***************************************************)
42 lemma append_inj_sn: ∀K1,K2,L1,L2. L1 @@ K1 = L2 @@ K2 → |K1| = |K2| →
45 [ * normalize /2 width=1/
46 #K2 #I2 #V2 #L1 #L2 #_ <plus_n_Sm #H destruct
47 | #K1 #I1 #V1 #IH * normalize
48 [ #L1 #L2 #_ <plus_n_Sm #H destruct
49 | #K2 #I2 #V2 #L1 #L2 #H1 #H2 destruct (**) (* destruct does not simplify well *)
50 elim (IH … e0 ?) -IH -H1 /2 width=1/ -H2 #H1 #H2 destruct /2 width=1/
56 lemma append_inj_dx: ∀K1,K2,L1,L2. L1 @@ K1 = L2 @@ K2 → |L1| = |L2| →
59 [ * normalize /2 width=1/
60 #K2 #I2 #V2 #L1 #L2 #H1 #H2 destruct
61 normalize in H2; >append_length in H2; #H
62 elim (plus_xySz_x_false … H)
63 | #K1 #I1 #V1 #IH * normalize
64 [ #L1 #L2 #H1 #H2 destruct
65 normalize in H2; >append_length in H2; #H
66 elim (plus_xySz_x_false … (sym_eq … H))
67 | #K2 #I2 #V2 #L1 #L2 #H1 #H2 destruct (**) (* destruct does not simplify well *)
68 elim (IH … e0 ?) -IH -H1 /2 width=1/ -H2 #H1 #H2 destruct /2 width=1/
73 lemma length_inv_pos_dx_append: ∀d,L. |L| = d + 1 →
74 ∃∃I,K,V. |K| = d & L = ⋆.ⓑ{I}V @@ K.
75 #d @(nat_ind_plus … d) -d
77 elim (length_inv_pos_dx … H) -H #I #K #V #H
78 >(length_inv_zero_dx … H) -H #H destruct
79 @ex2_3_intro [4: /2 width=2/ |5: // |1,2,3: skip ] (* /3/ does not work *)
81 elim (length_inv_pos_dx … H) -H #I #K #V #H
82 elim (IHd … H) -IHd -H #I0 #K0 #V0 #H1 #H2 #H3 destruct
83 @(ex2_3_intro … (K0.ⓑ{I}V)) //
87 (* Basic_eliminators ********************************************************)
89 fact lenv_ind_dx_aux: ∀R:predicate lenv. R ⋆ →
90 (∀I,L,V. R L → R (⋆.ⓑ{I}V @@ L)) →
92 #R #Hatom #Hpair #d @(nat_ind_plus … d) -d
93 [ #L #H >(length_inv_zero_dx … H) -H //
95 elim (length_inv_pos_dx_append … H) -H #I #K #V #H1 #H2 destruct /3 width=1/
99 lemma lenv_ind_dx: ∀R:predicate lenv. R ⋆ →
100 (∀I,L,V. R L → R (⋆.ⓑ{I}V @@ L)) →
102 /3 width=2 by lenv_ind_dx_aux/ qed-.