- degree-based equivalene for terms

[helm.git] / matita / matita / contribs / lambdadelta / basic_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 "basic_2/syntax/lenv_length.ma".
+include "basic_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 #V2 >append_pair >length_pair >length_pair //
+qed.
+
+lemma ltail_length: ∀I,L,V. |ⓑ{I}V.L| = ⫯|L|.
+#I #L #V >append_length //
+qed.
+
+(* Basic_1: was just: chead_ctail *)
+lemma lpair_ltail: ∀L,I,V. ∃∃J,K,W. L.ⓑ{I}V = ⓑ{J}W.K & |L| = |K|.
+#L elim L -L /2 width=5 by ex2_3_intro/
+#L #Z #X #IHL #I #V elim (IHL Z X) -IHL
+#J #K #W #H #_ >H -H >ltail_length
+@(ex2_3_intro … J (K.ⓑ{I}V) W) /2 width=1 by length_pair/
+qed-.
+
+(* Advanced inversion lemmas on length for local environments ***************)
+
+(* Basic_2A1: was: length_inv_pos_dx_ltail *)
+lemma length_inv_succ_dx_ltail: ∀L,l. |L| = ⫯l →
+                                ∃∃I,K,V. |K| = l & L = ⓑ{I}V.K.
+#Y #l #H elim (length_inv_succ_dx … H) -H #I #L #V #Hl #HLK destruct
+elim (lpair_ltail L I V) /2 width=5 by ex2_3_intro/
+qed-.
+
+(* Basic_2A1: was: length_inv_pos_sn_ltail *)
+lemma length_inv_succ_sn_ltail: ∀L,l. ⫯l = |L| →
+                                ∃∃I,K,V. l = |K| & L = ⓑ{I}V.K.
+#Y #l #H elim (length_inv_succ_sn … H) -H #I #L #V #Hl #HLK destruct
+elim (lpair_ltail L I V) /2 width=5 by ex2_3_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 #V2 #L1 #L2 #_ >length_atom >length_pair
+  #H destruct
+| #K1 #I1 #V1 #IH *
+  [ #L1 #L2 #_ >length_atom >length_pair
+    #H destruct
+  | #K2 #I2 #V2 #L1 #L2 #H1 >length_pair >length_pair #H2
+    elim (destruct_lpair_lpair_aux … H1) -H1 #H1 #H3 #H4 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 #V2 #L1 #L2 >append_atom >append_pair #H destruct
+  >length_pair >append_length >plus_n_Sm
+  #H elim (plus_xSy_x_false … H)
+| #K1 #I1 #V1 #IH *
+  [ #L1 #L2 >append_pair >append_atom #H destruct
+    >length_pair >append_length >plus_n_Sm #H
+    lapply (discr_plus_x_xy … H) -H #H destruct
+  | #K2 #I2 #V2 #L1 #L2 >append_pair >append_pair #H1 #H2
+    elim (destruct_lpair_lpair_aux … H1) -H1 #H1 #H3 #H4 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 *)
+lemma lenv_ind_alt: ∀R:predicate lenv.
+                    R (⋆) → (∀I,L,T. R L → R (ⓑ{I}T.L)) →
+                    ∀L. R L.
+#R #IH1 #IH2 #L @(f_ind … length … L) -L #x #IHx * // -IH1
+#L #I #V #H destruct elim (lpair_ltail L I V) /4 width=1 by/
+qed-.