(* 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.
#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 ***************************************************)
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_aux … H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *)
elim (IH … H1) -IH -H1 /2 width=1 by conj/
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
+[ * /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/
]
#I #L #K #V #H elim (append_inj_dx … (⋆.ⓑ{I}V) … H) //
qed-.
-lemma length_inv_pos_dx_ltail: ∀L,l. |L| = l + 1 →
+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_pos_dx … H) -H #I #L #V #Hl #HLK destruct
+#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,l. l + 1 = |L| →
+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_pos_sn … H) -H #I #L #V #Hl #HLK destruct
+#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-.
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 normalize #H destruct elim (lpair_ltail L I V) /3 width=1 by/
+#L #I #V #H destruct elim (lpair_ltail L I V) /4 width=1 by/
qed-.