X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fgrammar%2Flenv_append.ma;h=aa2e587d80cc08e1a959b1bef6577de34ef07fb0;hb=50997cb3042073d58c2a16885ef0c82217367e63;hp=3fdb1ad3787123b8b99f94e206ad4f9507a88050;hpb=98c91e19a9cc31c77a0151f5df7f7690813cbd07;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/lenv_append.ma b/matita/matita/contribs/lambdadelta/basic_2/grammar/lenv_append.ma
index 3fdb1ad37..aa2e587d8 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/grammar/lenv_append.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/grammar/lenv_append.ma
@@ -36,24 +36,32 @@ interpretation "tail abbreviation (local environment)"
 interpretation "tail abstraction (local environment)"
    'SnAbst L T = (append (LPair LAtom Abst T) L).
 
-definition l_appendable_sn: predicate (lenv→relation term) ≝ λR.
+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.
+// qed.
+
+lemma append_pair: ∀I,L,K,V. L @@ (K.ⓑ{I}V) = (L @@ K).ⓑ{I}V.
+// qed.
+
 lemma append_atom_sn: ∀L. ⋆ @@ L = L.
-#L elim L -L normalize //
+#L elim L -L //
+#L #I #V >append_pair //
 qed.
 
 lemma append_assoc: associative … append.
-#L1 #L2 #L3 elim L3 -L3 normalize //
+#L1 #L2 #L3 elim L3 -L3 //
 qed.
 
 lemma append_length: ∀L1,L2. |L1 @@ L2| = |L1| + |L2|.
-#L1 #L2 elim L2 -L2 normalize //
+#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| + 1.
+lemma ltail_length: ∀I,L,V. |ⓑ{I}V.L| = ⫯|L|.
 #I #L #V >append_length //
 qed.
 
@@ -62,7 +70,7 @@ 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) //
+@(ex2_3_intro … J (K.ⓑ{I}V) W) /2 width=1 by length_pair/
 qed-.
 
 (* Basic inversion lemmas ***************************************************)
@@ -70,12 +78,14 @@ qed-.
 lemma append_inj_sn: ∀K1,K2,L1,L2. L1 @@ K1 = L2 @@ K2 → |K1| = |K2| →
                      L1 = L2 ∧ K1 = K2.
 #K1 elim K1 -K1
-[ * normalize /2 width=1 by conj/
-  #K2 #I2 #V2 #L1 #L2 #_ <plus_n_Sm #H destruct
-| #K1 #I1 #V1 #IH * normalize
-  [ #L1 #L2 #_ <plus_n_Sm #H destruct
+[ * /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 #H2
-    elim (destruct_lpair_lpair … H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *)
+    elim (destruct_lpair_lpair_aux … H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *)
     elim (IH … H1) -IH -H1 /2 width=1 by conj/
   ]
 ]
@@ -85,16 +95,16 @@ qed-.
 lemma append_inj_dx: ∀K1,K2,L1,L2. L1 @@ K1 = L2 @@ K2 → |L1| = |L2| →
                      L1 = L2 ∧ K1 = K2.
 #K1 elim K1 -K1
-[ * normalize /2 width=1 by conj/
-  #K2 #I2 #V2 #L1 #L2 #H1 #H2 destruct
-  normalize in H2; >append_length in H2; #H
-  elim (plus_xySz_x_false … H)
-| #K1 #I1 #V1 #IH * normalize
-  [ #L1 #L2 #H1 #H2 destruct
-    normalize in H2; >append_length in H2; #H
-    elim (plus_xySz_x_false … (sym_eq … H))
-  | #K2 #I2 #V2 #L1 #L2 #H1 #H2
-    elim (destruct_lpair_lpair … H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *)
+[ * /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/
   ]
 ]
@@ -108,24 +118,24 @@ 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-.
 
-lemma length_inv_pos_dx_ltail: ∀L,d. |L| = d + 1 →
-                               ∃∃I,K,V. |K| = d & L = ⓑ{I}V.K.
-#Y #d #H elim (length_inv_pos_dx … H) -H #I #L #V #Hd #HLK destruct
+lemma length_inv_pos_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-.
 
-lemma length_inv_pos_sn_ltail: ∀L,d. d + 1 = |L| →
-                               ∃∃I,K,V. d = |K| & L = ⓑ{I}V.K.
-#Y #d #H elim (length_inv_pos_sn … H) -H #I #L #V #Hd #HLK destruct
+lemma length_inv_pos_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-.
 
-(* Basic_eliminators ********************************************************)
+(* 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 #n #IHn * // -IH1
-#L #I #V normalize #H destruct elim (lpair_ltail L I V) /3 width=1 by/
+#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-.