- lambda: first half of the standardization theorem is proved!

[helm.git] / matita / matita / contribs / lambda_delta / basic_2 / grammar / lenv_append.ma

diff --git a/matita/matita/contribs/lambda_delta/basic_2/grammar/lenv_append.ma b/matita/matita/contribs/lambda_delta/basic_2/grammar/lenv_append.ma
index 336cd7b2f642c62eb55c9f7f5f033ccd5f702bab..ab90ddf298c9d7dee96e8d2d81108ba21f781200 100644 (file)
--- a/matita/matita/contribs/lambda_delta/basic_2/grammar/lenv_append.ma
+++ b/matita/matita/contribs/lambda_delta/basic_2/grammar/lenv_append.ma
@@ -29,32 +29,102 @@ lemma append_atom_sn: ∀L. ⋆ @@ L = L.
 #L elim L -L normalize //
 qed.
 
+lemma append_assoc: associative … append.
+#L1 #L2 #L3 elim L3 -L3 normalize //
+qed.
+
 lemma append_length: ∀L1,L2. |L1 @@ L2| = |L1| + |L2|.
 #L1 #L2 elim L2 -L2 normalize //
 qed.
 
 (* Basic inversion lemmas ***************************************************)
 
-axiom discr_lpair_append_xy_x: ∀I,L,K,V. (L @@ K).ⓑ{I}V = L → ⊥.
-(*
-#I #L #K #V #H
-lapply (refl … (|L|)) <H in ⊢ (? ? % ? → ?); -H
-normalize >append_length -I -V #H
-*)
-lemma append_inv_sn: ∀L1,L2,L. L1 @@ L = L2 @@ L → L1 = L2.
-#L1 #L2 #L elim L -L normalize //
-#L #I #V #IHL #HL12 destruct /2 width=1/ (**) (* destruct does not simplify well *)
-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/
+  #K2 #I2 #V2 #L1 #L2 #_ <plus_n_Sm #H destruct
+| #K1 #I1 #V1 #IH * normalize
+  [ #L1 #L2 #_ <plus_n_Sm #H destruct
+  | #K2 #I2 #V2 #L1 #L2 #H1 #H2
+    elim (destruct_lpair_lpair … H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *)
+    elim (IH … H1 ?) -IH -H1 // -H2 /2 width=1/
+  ]
+]
+qed-.
 
-lemma append_inv_dx: ∀L1,L2,L. L @@ L1 = L @@ L2 → L1 = L2.
-#L1 elim L1 -L1
-[ * normalize //
-  #L2 #I2 #V2 #L #H
-  elim (discr_lpair_append_xy_x I2 L L2 V2 ?) //
-| #L1 #I1 #V1 #IHL1 * normalize
-  [ #L #H -IHL1 elim (discr_lpair_append_xy_x … H)
-  | #L2 #I2 #V2 #L normalize #H destruct (**) (* destruct does not simplify well *)
-    -H >e0 /3 width=2/
+(* Note: lemma 750 *)
+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/
+  #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 *)
+    elim (IH … H1 ?) -IH -H1 // -H2 /2 width=1/
   ]
 ]
-qed.
+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-.
+
+lemma length_inv_pos_dx_append: ∀d,L. |L| = d + 1 →
+                                ∃∃I,K,V. |K| = d & L = ⋆.ⓑ{I}V @@ K.
+#d @(nat_ind_plus … d) -d
+[ #L #H 
+  elim (length_inv_pos_dx … H) -H #I #K #V #H
+  >(length_inv_zero_dx … H) -H #H destruct
+  @ex2_3_intro [4: /2 width=2/ |5: // |1,2,3: skip ] (**) (* /3/ does not work *)
+| #d #IHd #L #H
+  elim (length_inv_pos_dx … H) -H #I #K #V #H
+  elim (IHd … H) -IHd -H #I0 #K0 #V0 #H1 #H2 #H3 destruct
+  @(ex2_3_intro … (K0.ⓑ{I}V)) //
+]
+qed-.
+
+(* Basic_eliminators ********************************************************)
+
+fact lenv_ind_dx_aux: ∀R:predicate lenv. R ⋆ →
+                      (∀I,L,V. R L → R (⋆.ⓑ{I}V @@ L)) →
+                      ∀d,L. |L| = d → R L.
+#R #Hatom #Hpair #d @(nat_ind_plus … d) -d
+[ #L #H >(length_inv_zero_dx … H) -H //
+| #d #IH #L #H
+  elim (length_inv_pos_dx_append … H) -H #I #K #V #H1 #H2 destruct /3 width=1/
+]
+qed-.
+
+lemma lenv_ind_dx: ∀R:predicate lenv. R ⋆ →
+                   (∀I,L,V. R L → R (⋆.ⓑ{I}V @@ L)) →
+                   ∀L. R L.
+/3 width=2 by lenv_ind_dx_aux/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma length_inv_pos_sn_append: ∀d,L. 1 + d = |L| →
+                                ∃∃I,K,V. d = |K| & L = ⋆. ⓑ{I}V @@ K.
+#d >commutative_plus @(nat_ind_plus … d) -d
+[ #L #H elim (length_inv_pos_sn … H) -H #I #K #V #H1 #H2 destruct
+  >(length_inv_zero_sn … H1) -K
+  @(ex2_3_intro … (⋆)) // (**) (* explicit constructor *)
+| #d #IHd #L #H elim (length_inv_pos_sn … H) -H #I #K #V #H1 #H2 destruct
+  >H1 in IHd; -H1 #IHd
+  elim (IHd K ?) -IHd // #J #L #W #H1 #H2 destruct
+  @(ex2_3_intro … (L.ⓑ{I}V)) // (**) (* explicit constructor *)
+  >append_length /2 width=1/
+]
+qed-.