syntactic components detached from basic_2 become static_2

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

diff --git a/matita/matita/contribs/lambdadelta/static_2/syntax/append_length.ma b/matita/matita/contribs/lambdadelta/static_2/syntax/append_length.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/syntax/lenv_length.ma".
+include "static_2/syntax/append.ma".
+
+(* APPEND FOR LOCAL ENVIRONMENTS ********************************************)
+
+(* Properties with length for local environments ****************************)
+
+lemma append_length: ∀L1,L2. |L1 + L2| = |L1| + |L2|.
+#L1 #L2 elim L2 -L2 //
+#L2 #I >append_bind >length_bind >length_bind //
+qed.
+
+lemma ltail_length: ∀I,L. |ⓘ{I}.L| = ↑|L|.
+#I #L >append_length //
+qed.
+
+(* Advanced inversion lemmas on length for local environments ***************)
+
+(* Basic_2A1: was: length_inv_pos_dx_ltail *)
+lemma length_inv_succ_dx_ltail: ∀L,n. |L| = ↑n →
+                                ∃∃I,K. |K| = n & L = ⓘ{I}.K.
+#Y #n #H elim (length_inv_succ_dx … H) -H #I #L #Hn #HLK destruct
+elim (lenv_case_tail … L) [2: * #K #J ]
+#H destruct /2 width=4 by ex2_2_intro/
+qed-.
+
+(* Basic_2A1: was: length_inv_pos_sn_ltail *)
+lemma length_inv_succ_sn_ltail: ∀L,n. ↑n = |L| →
+                                ∃∃I,K. n = |K| & L = ⓘ{I}.K.
+#Y #n #H elim (length_inv_succ_sn … H) -H #I #L #Hn #HLK destruct
+elim (lenv_case_tail … L) [2: * #K #J ]
+#H destruct /2 width=4 by ex2_2_intro/
+qed-.
+
+(* Inversion lemmas with length for local environments **********************)
+
+(* Basic_2A1: was: append_inj_sn *)
+lemma append_inj_length_sn: ∀K1,K2,L1,L2. L1 + K1 = L2 + K2 → |K1| = |K2| →
+                            L1 = L2 ∧ K1 = K2.
+#K1 elim K1 -K1
+[ * /2 width=1 by conj/
+  #K2 #I2 #L1 #L2 #_ >length_atom >length_bind
+  #H destruct
+| #K1 #I1 #IH *
+  [ #L1 #L2 #_ >length_atom >length_bind
+    #H destruct
+  | #K2 #I2 #L1 #L2 #H1 >length_bind >length_bind #H2
+    elim (destruct_lbind_lbind_aux … H1) -H1 #H1 #H3 destruct (**) (* destruct lemma needed *)
+    elim (IH … H1) -IH -H1 /3 width=4 by conj/
+  ]
+]
+qed-.
+
+(* Note: lemma 750 *)
+(* Basic_2A1: was: append_inj_dx *)
+lemma append_inj_length_dx: ∀K1,K2,L1,L2. L1 + K1 = L2 + K2 → |L1| = |L2| →
+                            L1 = L2 ∧ K1 = K2.
+#K1 elim K1 -K1
+[ * /2 width=1 by conj/
+  #K2 #I2 #L1 #L2 >append_atom >append_bind #H destruct
+  >length_bind >append_length >plus_n_Sm
+  #H elim (plus_xSy_x_false … H)
+| #K1 #I1 #IH *
+  [ #L1 #L2 >append_bind >append_atom #H destruct
+    >length_bind >append_length >plus_n_Sm #H
+    lapply (discr_plus_x_xy … H) -H #H destruct
+  | #K2 #I2 #L1 #L2 >append_bind >append_bind #H1 #H2
+    elim (destruct_lbind_lbind_aux … H1) -H1 #H1 #H3 destruct (**) (* destruct lemma needed *)
+    elim (IH … H1) -IH -H1 /2 width=1 by conj/
+  ]
+]
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma append_inj_dx: ∀L,K1,K2. L+K1 = L+K2 → K1 = K2.
+#L #K1 #K2 #H elim (append_inj_length_dx … H) -H //
+qed-.
+
+lemma append_inv_refl_dx: ∀L,K. L+K = L → K = ⋆.
+#L #K #H elim (append_inj_dx … (⋆) … H) //
+qed-.
+
+lemma append_inv_pair_dx: ∀I,L,K,V. L+K = L.ⓑ{I}V → K = ⋆.ⓑ{I}V.
+#I #L #K #V #H elim (append_inj_dx … (⋆.ⓑ{I}V) … H) //
+qed-.
+
+(* Basic eliminators ********************************************************)
+
+(* Basic_1: was: c_tail_ind *)
+(* Basic_2A1: was: lenv_ind_alt *) 
+lemma lenv_ind_tail: ∀Q:predicate lenv.
+                     Q (⋆) → (∀I,L. Q L → Q (ⓘ{I}.L)) → ∀L. Q L.
+#Q #IH1 #IH2 #L @(f_ind … length … L) -L #x #IHx * //
+#L #I -IH1 #H destruct
+elim (lenv_case_tail … L) [2: * #K #J ]
+#H destruct /3 width=1 by/
+qed-.