partial update in 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
index 1945bac9f0a9890126a5b1929b652099e628bb0a..1e0de051b2573daeb4bb66eba04a2dca1761f337 100644 (file)
--- a/matita/matita/contribs/lambdadelta/static_2/syntax/append_length.ma
+++ b/matita/matita/contribs/lambdadelta/static_2/syntax/append_length.ma
@@ -12,6 +12,8 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "ground/arith/nat_plus.ma".
+include "ground/arith/wf1_ind_nlt.ma".
 include "static_2/syntax/lenv_length.ma".
 include "static_2/syntax/append.ma".
 
@@ -24,7 +26,7 @@ lemma append_length: ∀L1,L2. |L1 + L2| = |L1| + |L2|.
 #L2 #I >append_bind >length_bind >length_bind //
 qed.
 
-lemma ltail_length: ∀I,L. |ⓘ{I}.L| = ↑|L|.
+lemma ltail_length: ∀I,L. |ⓘ[I].L| = ↑|L|.
 #I #L >append_length //
 qed.
 
@@ -32,33 +34,38 @@ qed.
 
 (* 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.
+                                ∃∃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/
+elim (lenv_case_tail … L) [| * #K #J ] #H destruct
+/2 width=4 by ex2_2_intro/
+@(ex2_2_intro … (J) (K.ⓘ[I])) // (**) (* auto fails *)
+>ltail_length >length_bind //
 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.
+                                ∃∃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/
+elim (lenv_case_tail … L) [| * #K #J ] #H destruct
+/2 width=4 by ex2_2_intro/
+@(ex2_2_intro … (J) (K.ⓘ[I])) // (**) (* auto fails *)
+>ltail_length >length_bind //
 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.
+                            ∧∧ L1 = L2 & K1 = K2.
 #K1 elim K1 -K1
 [ * /2 width=1 by conj/
   #K2 #I2 #L1 #L2 #_ >length_atom >length_bind
-  #H destruct
+  #H elim (eq_inv_zero_nsucc … H)
 | #K1 #I1 #IH *
   [ #L1 #L2 #_ >length_atom >length_bind
-    #H destruct
+    #H elim (eq_inv_nsucc_zero … H)
   | #K2 #I2 #L1 #L2 #H1 >length_bind >length_bind #H2
+    lapply (eq_inv_nsucc_bi … H2) -H2 #H2  
     elim (destruct_lbind_lbind_aux … H1) -H1 #H1 #H3 destruct (**) (* destruct lemma needed *)
     elim (IH … H1) -IH -H1 /3 width=4 by conj/
   ]
@@ -68,16 +75,16 @@ 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.
+                            ∧∧ 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)
+  >length_bind >append_length #H
+  elim (succ_nplus_refl_sn (|L2|) (|K2|) ?) //
 | #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
+    >length_bind >append_length #H
+    elim (succ_nplus_refl_sn … H)
   | #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/
@@ -95,18 +102,18 @@ 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) //
+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 *) 
+(* 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 * //
+                     Q (⋆) → (∀I,L. Q L → Q (ⓘ[I].L)) → ∀L. Q L.
+#Q #IH1 #IH2 #L @(wf1_ind_nlt … length … L) -L #x #IHx * //
 #L #I -IH1 #H destruct
-elim (lenv_case_tail … L) [2: * #K #J ]
+elim (lenv_case_tail … L) [| * #K #J ]
 #H destruct /3 width=1 by/
 qed-.