syntactic components detached from basic_2 become static_2

[helm.git] / matita / matita / contribs / lambdadelta / static_2 / syntax / append.ma

diff --git a/matita/matita/contribs/lambdadelta/static_2/syntax/append.ma b/matita/matita/contribs/lambdadelta/static_2/syntax/append.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "static_2/notation/functions/snitem_2.ma".
+include "static_2/notation/functions/snbind1_2.ma".
+include "static_2/notation/functions/snbind2_3.ma".
+include "static_2/notation/functions/snvoid_1.ma".
+include "static_2/notation/functions/snabbr_2.ma".
+include "static_2/notation/functions/snabst_2.ma".
+include "static_2/syntax/lenv.ma".
+
+(* APPEND FOR LOCAL ENVIRONMENTS ********************************************)
+
+rec definition append L K on K ≝ match K with
+[ LAtom     ⇒ L
+| LBind K I ⇒ (append L K).ⓘ{I}
+].
+
+interpretation "append (local environment)" 'plus L1 L2 = (append L1 L2).
+
+interpretation "local environment tail binding construction (generic)"
+   'SnItem I L = (append (LBind LAtom I) L).
+
+interpretation "local environment tail binding construction (unary)"
+   'SnBind1 I L = (append (LBind LAtom (BUnit I)) L).
+
+interpretation "local environment tail binding construction (binary)"
+   'SnBind2 I T L = (append (LBind LAtom (BPair I T)) L).
+
+interpretation "tail exclusion (local environment)"
+   'SnVoid L = (append (LBind LAtom (BUnit Void)) L).
+
+interpretation "tail abbreviation (local environment)"
+   'SnAbbr T L = (append (LBind LAtom (BPair Abbr T)) L).
+
+interpretation "tail abstraction (local environment)"
+   'SnAbst L T = (append (LBind LAtom (BPair Abst T)) L).
+
+definition d_appendable_sn: predicate (lenv→relation term) ≝ λR.
+                            ∀K,T1,T2. R K T1 T2 → ∀L. R (L+K) T1 T2.
+
+(* Basic properties *********************************************************)
+
+lemma append_atom: ∀L. (L + ⋆) = L. (**) (* () should be redundant *)
+// qed.
+
+(* Basic_2A1: uses: append_pair *)
+lemma append_bind: ∀I,L,K. L+(K.ⓘ{I}) = (L+K).ⓘ{I}.
+// qed.
+
+lemma append_atom_sn: ∀L. ⋆ + L = L.
+#L elim L -L //
+#L #I >append_bind //
+qed.
+
+lemma append_assoc: associative … append.
+#L1 #L2 #L3 elim L3 -L3 //
+qed.
+
+lemma append_shift: ∀L,K,I. L+(ⓘ{I}.K) = (L.ⓘ{I})+K.
+#L #K #I <append_assoc //
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma append_inv_atom3_sn: ∀L,K. ⋆ = L + K → ∧∧ ⋆ = L & ⋆ = K.
+#L * /2 width=1 by conj/
+#K #I >append_bind #H destruct
+qed-.
+
+lemma append_inv_bind3_sn: ∀I0,L,L0,K. L0.ⓘ{I0} = L + K →
+                           ∨∨ ∧∧ L0.ⓘ{I0} = L & ⋆ = K
+                            | ∃∃K0. K = K0.ⓘ{I0} & L0 = L + K0.
+#I0 #L #L0 * /3 width=1 by or_introl, conj/
+#K #I >append_bind #H destruct /3 width=3 by ex2_intro, or_intror/
+qed-.
+
+lemma append_inj_sn: ∀K,L1,L2. L1+K = L2+K → L1 = L2.
+#K elim K -K //
+#K #I #IH #L1 #L2 >append_bind #H
+elim (destruct_lbind_lbind_aux … H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
+qed-.
+
+(* Basic_1: uses: chead_ctail *)
+(* Basic_2A1: uses: lpair_ltail *)
+lemma lenv_case_tail: ∀L. L = ⋆ ∨ ∃∃K,I. L = ⓘ{I}.K.
+#L elim L -L /2 width=1 by or_introl/
+#L #I * [2: * ] /3 width=3 by ex1_2_intro, or_intror/
+qed-.