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 "ground_2/notation/functions/append_2.ma".
16 include "basic_2/notation/functions/snbind2_3.ma".
17 include "basic_2/notation/functions/snabbr_2.ma".
18 include "basic_2/notation/functions/snabst_2.ma".
19 include "basic_2/grammar/lenv_length.ma".
21 (* LOCAL ENVIRONMENTS *******************************************************)
23 let rec append L K on K ≝ match K with
25 | LPair K I V ⇒ (append L K). ⓑ{I} V
28 interpretation "append (local environment)" 'Append L1 L2 = (append L1 L2).
30 interpretation "local environment tail binding construction (binary)"
31 'SnBind2 I T L = (append (LPair LAtom I T) L).
33 interpretation "tail abbreviation (local environment)"
34 'SnAbbr T L = (append (LPair LAtom Abbr T) L).
36 interpretation "tail abstraction (local environment)"
37 'SnAbst L T = (append (LPair LAtom Abst T) L).
39 definition d_appendable_sn: predicate (lenv→relation term) ≝ λR.
40 ∀K,T1,T2. R K T1 T2 → ∀L. R (L @@ K) T1 T2.
42 (* Basic properties *********************************************************)
44 lemma append_atom: ∀L. L @@ ⋆ = L.
47 lemma append_pair: ∀I,L,K,V. L @@ (K.ⓑ{I}V) = (L @@ K).ⓑ{I}V.
50 lemma append_atom_sn: ∀L. ⋆ @@ L = L.
52 #L #I #V >append_pair //
55 lemma append_assoc: associative … append.
56 #L1 #L2 #L3 elim L3 -L3 //
59 lemma append_length: ∀L1,L2. |L1 @@ L2| = |L1| + |L2|.
60 #L1 #L2 elim L2 -L2 //
61 #L2 #I #V2 >append_pair >length_pair >length_pair //
64 lemma ltail_length: ∀I,L,V. |ⓑ{I}V.L| = ⫯|L|.
65 #I #L #V >append_length //
68 (* Basic_1: was just: chead_ctail *)
69 lemma lpair_ltail: ∀L,I,V. ∃∃J,K,W. L.ⓑ{I}V = ⓑ{J}W.K & |L| = |K|.
70 #L elim L -L /2 width=5 by ex2_3_intro/
71 #L #Z #X #IHL #I #V elim (IHL Z X) -IHL
72 #J #K #W #H #_ >H -H >ltail_length
73 @(ex2_3_intro … J (K.ⓑ{I}V) W) /2 width=1 by length_pair/
76 (* Basic inversion lemmas ***************************************************)
78 lemma append_inj_sn: ∀K1,K2,L1,L2. L1 @@ K1 = L2 @@ K2 → |K1| = |K2| →
81 [ * /2 width=1 by conj/
82 #K2 #I2 #V2 #L1 #L2 #_ >length_atom >length_pair
85 [ #L1 #L2 #_ >length_atom >length_pair
87 | #K2 #I2 #V2 #L1 #L2 #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/
95 lemma append_inj_dx: ∀K1,K2,L1,L2. L1 @@ K1 = L2 @@ K2 → |L1| = |L2| →
98 [ * /2 width=1 by conj/
99 #K2 #I2 #V2 #L1 #L2 >append_atom >append_pair #H destruct
100 >length_pair >append_length >plus_n_Sm
101 #H elim (plus_xSy_x_false … H)
103 [ #L1 #L2 >append_pair >append_atom #H destruct
104 >length_pair >append_length >plus_n_Sm #H
105 lapply (discr_plus_x_xy … H) -H #H destruct
106 | #K2 #I2 #V2 #L1 #L2 >append_pair >append_pair #H1 #H2
107 elim (destruct_lpair_lpair_aux … H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *)
108 elim (IH … H1) -IH -H1 /2 width=1 by conj/
113 lemma append_inv_refl_dx: ∀L,K. L @@ K = L → K = ⋆.
114 #L #K #H elim (append_inj_dx … (⋆) … H) //
117 lemma append_inv_pair_dx: ∀I,L,K,V. L @@ K = L.ⓑ{I}V → K = ⋆.ⓑ{I}V.
118 #I #L #K #V #H elim (append_inj_dx … (⋆.ⓑ{I}V) … H) //
121 lemma length_inv_pos_dx_ltail: ∀L,l. |L| = ⫯l →
122 ∃∃I,K,V. |K| = l & L = ⓑ{I}V.K.
123 #Y #l #H elim (length_inv_succ_dx … H) -H #I #L #V #Hl #HLK destruct
124 elim (lpair_ltail L I V) /2 width=5 by ex2_3_intro/
127 lemma length_inv_pos_sn_ltail: ∀L,l. ⫯l = |L| →
128 ∃∃I,K,V. l = |K| & L = ⓑ{I}V.K.
129 #Y #l #H elim (length_inv_succ_sn … H) -H #I #L #V #Hl #HLK destruct
130 elim (lpair_ltail L I V) /2 width=5 by ex2_3_intro/
133 (* Basic eliminators ********************************************************)
135 (* Basic_1: was: c_tail_ind *)
136 lemma lenv_ind_alt: ∀R:predicate lenv.
137 R (⋆) → (∀I,L,T. R L → R (ⓑ{I}T.L)) →
139 #R #IH1 #IH2 #L @(f_ind … length … L) -L #x #IHx * // -IH1
140 #L #I #V #H destruct elim (lpair_ltail L I V) /4 width=1 by/