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 *)
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15 include "ground/notation/functions/append_2.ma".
16 include "basic_2A/notation/functions/snbind2_3.ma".
17 include "basic_2A/notation/functions/snabbr_2.ma".
18 include "basic_2A/notation/functions/snabst_2.ma".
19 include "basic_2A/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_sn: ∀L. ⋆ @@ L = L.
45 #L elim L -L normalize //
48 lemma append_assoc: associative … append.
49 #L1 #L2 #L3 elim L3 -L3 normalize //
52 lemma append_length: ∀L1,L2. |L1 @@ L2| = |L1| + |L2|.
53 #L1 #L2 elim L2 -L2 normalize //
56 lemma ltail_length: ∀I,L,V. |ⓑ{I}V.L| = |L| + 1.
57 #I #L #V >append_length //
60 lemma lpair_ltail: ∀L,I,V. ∃∃J,K,W. L.ⓑ{I}V = ⓑ{J}W.K & |L| = |K|.
61 #L elim L -L /2 width=5 by ex2_3_intro/
62 #L #Z #X #IHL #I #V elim (IHL Z X) -IHL
63 #J #K #W #H #_ >H -H >ltail_length
64 @(ex2_3_intro … J (K.ⓑ{I}V) W) //
67 (* Basic inversion lemmas ***************************************************)
69 lemma append_inj_sn: ∀K1,K2,L1,L2. L1 @@ K1 = L2 @@ K2 → |K1| = |K2| →
72 [ * normalize /2 width=1 by conj/
73 #K2 #I2 #V2 #L1 #L2 #_ <plus_n_Sm #H destruct
74 | #K1 #I1 #V1 #IH * normalize
75 [ #L1 #L2 #_ <plus_n_Sm #H destruct
76 | #K2 #I2 #V2 #L1 #L2 #H1 #H2
77 elim (destruct_lpair_lpair_aux … H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *)
78 elim (IH … H1) -IH -H1 /2 width=1 by conj/
84 lemma append_inj_dx: ∀K1,K2,L1,L2. L1 @@ K1 = L2 @@ K2 → |L1| = |L2| →
87 [ * normalize /2 width=1 by conj/
88 #K2 #I2 #V2 #L1 #L2 #H1 #H2 destruct
89 normalize in H2; >append_length in H2; #H
90 elim (plus_xySz_x_false … H)
91 | #K1 #I1 #V1 #IH * normalize
92 [ #L1 #L2 #H1 #H2 destruct
93 normalize in H2; >append_length in H2; #H
94 elim (plus_xySz_x_false … (sym_eq … H))
95 | #K2 #I2 #V2 #L1 #L2 #H1 #H2
96 elim (destruct_lpair_lpair_aux … H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *)
97 elim (IH … H1) -IH -H1 /2 width=1 by conj/
102 lemma append_inv_refl_dx: ∀L,K. L @@ K = L → K = ⋆.
103 #L #K #H elim (append_inj_dx … (⋆) … H) //
106 lemma append_inv_pair_dx: ∀I,L,K,V. L @@ K = L.ⓑ{I}V → K = ⋆.ⓑ{I}V.
107 #I #L #K #V #H elim (append_inj_dx … (⋆.ⓑ{I}V) … H) //
110 lemma length_inv_pos_dx_ltail: ∀L,l. |L| = l + 1 →
111 ∃∃I,K,V. |K| = l & L = ⓑ{I}V.K.
112 #Y #l #H elim (length_inv_pos_dx … H) -H #I #L #V #Hl #HLK destruct
113 elim (lpair_ltail L I V) /2 width=5 by ex2_3_intro/
116 lemma length_inv_pos_sn_ltail: ∀L,l. l + 1 = |L| →
117 ∃∃I,K,V. l = |K| & L = ⓑ{I}V.K.
118 #Y #l #H elim (length_inv_pos_sn … H) -H #I #L #V #Hl #HLK destruct
119 elim (lpair_ltail L I V) /2 width=5 by ex2_3_intro/
122 (* Basic eliminators ********************************************************)
124 lemma lenv_ind_alt: ∀R:predicate lenv.
125 R (⋆) → (∀I,L,T. R L → R (ⓑ{I}T.L)) →
127 #R #IH1 #IH2 #L @(f_ind … length … L) -L #x #IHx * // -IH1
128 #L #I #V normalize #H destruct elim (lpair_ltail L I V) /3 width=1 by/