(* The following, trivial lemmas are only meant for rewriting purposes. *)
lemma sem_pre_true : ∀S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
- \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6〉} \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i} \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6}.
+ \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6〉} \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i} \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6}.
// qed.
lemma sem_pre_false : ∀S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
- \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6〉} \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i}.
+ \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6〉} \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i}.
// qed.
lemma sem_cat: ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.