-theorem nil_to_nil: ∀A.∀l1,l2:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 \ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6A.
- l1\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6l2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6] → l1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6] \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 l2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6].
-#A #l1 #l2 #isnil @(\ 5a href="cic:/matita/tutorial/chapter3/nil_append_elim.def(3)"\ 6nil_append_elim\ 5/a\ 6 A l1 l2) /2/
-qed.
+let rec length (A:Type[0]) (l:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) on l ≝
+match l with
+ [ nil ⇒ \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6
+ | cons a tl ⇒ \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (length A tl)].
+
+let rec nth n (A:Type[0]) (l:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) (d:A) ≝
+ match n with
+ [O ⇒ \ 5a href="cic:/matita/tutorial/chapter3/hd.def(1)"\ 6hd\ 5/a\ 6 A l d
+ |S m ⇒ nth m A (\ 5a href="cic:/matita/tutorial/chapter3/tail.def(1)"\ 6tail\ 5/a\ 6 A l) d].
+
+example ex_length: \ 5a href="cic:/matita/tutorial/chapter3/length.fix(0,1,1)"\ 6length\ 5/a\ 6 ? (\ 5a href="cic:/matita/tutorial/chapter3/list.con(0,2,1)"\ 6cons\ 5/a\ 6 ? \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6.
+normalize // qed.
+
+example ex_nth: \ 5a href="cic:/matita/tutorial/chapter3/nth.fix(0,0,2)"\ 6nth\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6) ? (\ 5a href="cic:/matita/tutorial/chapter3/list.con(0,2,1)"\ 6cons\ 5/a\ 6 ? (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6) (\ 5a href="cic:/matita/tutorial/chapter3/list.con(0,2,1)"\ 6cons\ 5/a\ 6 ? \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6])) \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6.
+normalize // qed.
+
+lemma length_add: ∀A.∀l1,l2:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.
+ \ 5a href="cic:/matita/tutorial/chapter3/length.fix(0,1,1)"\ 6length\ 5/a\ 6 ? (l1\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6l2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter3/length.fix(0,1,1)"\ 6length\ 5/a\ 6 ? l1) (\ 5a href="cic:/matita/tutorial/chapter3/length.fix(0,1,1)"\ 6length\ 5/a\ 6 ? l2).
+#A #l1 elim l1 normalize // qed.