include "ground/notation/functions/oplusleft_3.ma".
include "ground/lib/list_append.ma".
+include "ground/generated/pull_2.ma".
(* RIGHT CONS FOR LISTS *****************************************************)
// qed.
lemma list_append_rcons_dx (A):
- ∀l1,l2,a. l1 ⨁ l2 ⨭ a = l1 ⨁{A} (l2 ⨭ a).
+ ∀l1,l2,a. (l1 ⨁ l2) ⨭ a = l1 ⨁{A} (l2 ⨭ a).
// qed.
(* Basic inversions *********************************************************)
lemma eq_inv_list_empty_rcons (A):
∀l,a. ⓔ = l⨭{A}a → ⊥.
-#A #l #a #H
-elim (eq_inv_list_empty_append … H) -H #_ #H destruct
+#A #l #a #H0
+elim (eq_inv_list_empty_append … H0) -H0 #_ #H0 destruct
+qed-.
+
+lemma eq_inv_list_rcons_empty (A):
+ ∀l,a. l⨭{A}a = ⓔ → ⊥.
+#A #l #a #H0
+elim (eq_inv_list_append_empty … H0) -H0 #_ #H0 destruct
+qed-.
+
+(* Advanced inversions ******************************************************)
+
+lemma eq_inv_list_rcons_bi (A):
+ ∀a1,a2,l1,l2. l1 ⨭{A} a1 = l2 ⨭ a2 →
+ ∧∧ l1 = l2 & a1 = a2.
+#A #a1 #a2 #l1 elim l1 -l1 [| #b1 #l1 #IH ] *
+[ <list_append_empty_sn <list_append_empty_sn #H destruct
+ /2 width=1 by conj/
+| #b2 #l2 <list_append_empty_sn <list_append_lcons_sn #H destruct -H
+ elim (eq_inv_list_empty_rcons ??? e0)
+| <list_append_lcons_sn <list_append_empty_sn #H destruct -H
+ elim (eq_inv_list_empty_rcons ??? (sym_eq … e0))
+| #b2 #l2 <list_append_lcons_sn <list_append_lcons_sn #H destruct -H
+ elim (IH … e0) -IH -e0 #H1 #H2 destruct
+ /2 width=1 by conj/
+]
+qed-.
+
+(* Advanced eliminations ****************************************************)
+
+lemma list_ind_rcons (A) (Q:predicate …):
+ Q (ⓔ{A}) →
+ (∀l,a. Q l -> Q (l⨭a)) →
+ ∀l. Q l.
+#A #Q #IH1 #IH2 #l
+@(list_ind_append_dx … l) -l //
+@pull_2 #l2 elim l2 -l2 //
+#a2 #l2 #IH0 #l1 #IH /3 width=1 by/
qed-.