#A #l1 #l2 #l3 (elim l1) normalize // qed.
(* Problemi con la notazione *)
+lemma a_append: ∀A.∀a.∀l:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A. (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]) \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l.
+// qed.
+
theorem append_cons:
∀A.∀a:A.∀l,l1: \ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.l\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6(a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l1)\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (l \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter3/list.con(0,2,1)"\ 6cons\ 5/a\ 6 ? a \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6])) \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 l1.
-/2/ qed.
+// qed.
(* Other typical functions over lists are those computing the length
of a list, and the function returning the nth element *)
authorized to add P to your hypothesis: *)
lemma neg_aux : ∀P:Prop. (P → \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6P) → \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6P.
-#P #PtonegP % /3/ qed.
+#P #PtonegP % /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
theorem diff_cons_nil:
∀A:Type[0].∀l:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.∀a:A. a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6].
lemma 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 cases l1 normalize /2/ #a #tl #l2 #H @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /2/ qed.
+#A #l1 cases l1 normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ #a #tl #l2 #H @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
(* Let us come to some important, higher order, polymorphic functionals
acting over lists. A typical example is the map function, taking a function