theorem nat_case:
∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.∀P:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop.
(n\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 → P \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6) → (∀m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m → P (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m)) → P n.
-#n #P (elim n) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace {}\ 5/span\ 6\ 5/span\ 6/ qed.
+#n #P (elim n) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/ qed.
theorem nat_elim2 :
∀R:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop.
→ (∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. R (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n) \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6)
→ (∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. R n m → R (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n) (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m))
→ ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. R n m.
-#R #ROn #RSO #RSS #n (elim n) // #n0 #Rn0m #m (cases m) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace {}\ 5/span\ 6\ 5/span\ 6/ qed.
+#R #ROn #RSO #RSS #n (elim n) // #n0 #Rn0m #m (cases m) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/ qed.
theorem decidable_eq_nat : ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a href="cic:/matita/basics/logic/decidable.def(1)"\ 6decidable\ 5/a\ 6 (n\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6m).
@\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6 #n [ (cases n) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ | /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/sym_not_eq.def(4)"\ 6sym_not_eq\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ | #m #Hind (cases Hind) /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/not_eq_S.def(4)"\ 6not_eq_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/]
theorem monotonic_le_plus_r:
∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a href="cic:/matita/basics/relations/monotonic.def(1)"\ 6monotonic\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/le.ind(1,0,1)"\ 6le\ 5/a\ 6 (λm.n \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 m).
#n #a #b (elim n) normalize //
-#m #H #leab @\ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace {}\ 5/span\ 6\ 5/span\ 6/ qed.
+#m #H #leab @\ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/ qed.
(*
theorem le_plus_r: ∀p,n,m:nat. n ≤ m → p + n ≤ p + m
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
theorem le_plus_n_r :∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 n.
-/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace {}\ 5/span\ 6\ 5/span\ 6/ qed.
+/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/ qed.
theorem eq_plus_to_le: ∀n,m,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.n\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6m\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6p → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n.
// qed.
theorem lt_plus: ∀n,m,p,q:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → p \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 q → n \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 p \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 q.
#n #m #p #q #ltnm #ltpq
-@(\ 5a href="cic:/matita/arithmetics/nat/transitive_lt.def(3)"\ 6transitive_lt\ 5/a\ 6 ? (n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6q))/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_lt_plus_l.def(9)"\ 6monotonic_lt_plus_l\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_plus_r.def(3)"\ 6monotonic_le_plus_r\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
+@(\ 5a href="cic:/matita/arithmetics/nat/transitive_lt.def(3)"\ 6transitive_lt\ 5/a\ 6 ? (n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6q))/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_plus_r.def(3)"\ 6monotonic_le_plus_r\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/monotonic_lt_plus_l.def(9)"\ 6monotonic_lt_plus_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
theorem lt_plus_to_lt_l :∀n,p,q:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. p\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 q\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6n → p\ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6q.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_plus_to_le.def(5)"\ 6le_plus_to_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
qed.
theorem lt_times:∀n,m,p,q:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n\ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6m → p\ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6q → n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6p \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6q.
-#n #m #p #q #ltnm #ltpq @\ 5a href="cic:/matita/arithmetics/nat/lt_to_le_to_lt_times.def(12)"\ 6lt_to_le_to_lt_times\ 5/a\ 6\ 5span style="text-decoration: underline;"\ 6 \ 5/span\ 6/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_plus_b.def(8)"\ 6le_plus_b\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/ltn_to_ltO.def(5)"\ 6ltn_to_ltO\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+#n #m #p #q #ltnm #ltpq @\ 5a href="cic:/matita/arithmetics/nat/lt_to_le_to_lt_times.def(12)"\ 6lt_to_le_to_lt_times\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_plus_b.def(8)"\ 6le_plus_b\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/ltn_to_ltO.def(5)"\ 6ltn_to_ltO\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
theorem lt_times_n_to_lt_l:
qed.
theorem minus_plus_m_m: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m)\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m.
-/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/plus_minus.def(5)"\ 6plus_minus\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
+/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/plus_minus.def(5)"\ 6plus_minus\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
theorem plus_minus_m_m: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.
m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n → n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m)\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m.
qed.
theorem le_minus_to_plus_r: ∀a,b,c. c \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 b → a \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 b \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 c → a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 c \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 b.
-#a #b #c #Hlecb #H >(\ 5a href="cic:/matita/arithmetics/nat/plus_minus_m_m.def(7)"\ 6plus_minus_m_m\ 5/a\ 6 … Hlecb) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_plus_l.def(6)"\ 6monotonic_le_plus_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+#a #b #c #Hlecb #H >(\ 5a href="cic:/matita/arithmetics/nat/plus_minus_m_m.def(7)"\ 6plus_minus_m_m\ 5/a\ 6 … Hlecb) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_minus_to_plus.def(10)"\ 6le_minus_to_plus\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
theorem le_plus_to_minus: ∀n,m,p. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 p\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m → n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 p.
-#n #m #p #lep /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_minus_l.def(10)"\ 6monotonic_le_minus_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
+#n #m #p #lep /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_minus_l.def(9)"\ 6monotonic_le_minus_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
theorem le_plus_to_minus_r: ∀a,b,c. a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 b \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 c → a \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 c \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6b.
#a #b #c #H @(\ 5a href="cic:/matita/arithmetics/nat/le_plus_to_le_r.def(6)"\ 6le_plus_to_le_r\ 5/a\ 6 … b) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
theorem lt_minus_to_plus_r: ∀a,b,c. a \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 b \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 c → a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 c \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 b.
-#a #b #c #H @\ 5a href="cic:/matita/arithmetics/nat/not_le_to_lt.def(5)"\ 6not_le_to_lt\ 5/a\ 6 @(\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 … (\ 5a href="cic:/matita/arithmetics/nat/le_plus_to_minus.def(11)"\ 6le_plus_to_minus\ 5/a\ 6 …))
+#a #b #c #H @\ 5a href="cic:/matita/arithmetics/nat/not_le_to_lt.def(5)"\ 6not_le_to_lt\ 5/a\ 6 @(\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 … (\ 5a href="cic:/matita/arithmetics/nat/le_plus_to_minus.def(10)"\ 6le_plus_to_minus\ 5/a\ 6 …))
@\ 5a href="cic:/matita/arithmetics/nat/lt_to_not_le.def(7)"\ 6lt_to_not_le\ 5/a\ 6 //
qed.
theorem lt_plus_to_minus: ∀n,m,p. m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n → n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 p\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m → n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 p.
-#n #m #p #lenm #H normalize <\ 5a href="cic:/matita/arithmetics/nat/minus_Sn_m.def(5)"\ 6minus_Sn_m\ 5/a\ 6 // @\ 5a href="cic:/matita/arithmetics/nat/le_plus_to_minus.def(11)"\ 6le_plus_to_minus\ 5/a\ 6 //
+#n #m #p #lenm #H normalize <\ 5a href="cic:/matita/arithmetics/nat/minus_Sn_m.def(5)"\ 6minus_Sn_m\ 5/a\ 6 // @\ 5a href="cic:/matita/arithmetics/nat/le_plus_to_minus.def(10)"\ 6le_plus_to_minus\ 5/a\ 6 //
qed.
theorem lt_plus_to_minus_r: ∀a,b,c. a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 b \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 c → a \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 c \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 b.
theorem monotonic_le_minus_r:
∀p,q,n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. q \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 p → n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6p \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6q.
-#p #q #n #lepq @\ 5a href="cic:/matita/arithmetics/nat/le_plus_to_minus.def(11)"\ 6le_plus_to_minus\ 5/a\ 6
+#p #q #n #lepq @\ 5a href="cic:/matita/arithmetics/nat/le_plus_to_minus.def(10)"\ 6le_plus_to_minus\ 5/a\ 6 \ 5span class="error" title="Parse error: illegal begin of statement"\ 6\ 5/span\ 6
@(\ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6 … (\ 5a href="cic:/matita/arithmetics/nat/le_plus_minus_m_m.def(9)"\ 6le_plus_minus_m_m\ 5/a\ 6 ? q)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_plus_r.def(3)"\ 6monotonic_le_plus_r\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
theorem eq_minus_O: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.
n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6.
-#n #m #lenm @(\ 5a href="cic:/matita/arithmetics/nat/le_n_O_elim.def(4)"\ 6le_n_O_elim\ 5/a\ 6 (n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_minus_r.def(12)"\ 6monotonic_le_minus_r\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+#n #m #lenm @(\ 5a href="cic:/matita/arithmetics/nat/le_n_O_elim.def(4)"\ 6le_n_O_elim\ 5/a\ 6 (n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_minus_r.def(11)"\ 6monotonic_le_minus_r\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
theorem distributive_times_minus: \ 5a href="cic:/matita/basics/relations/distributive.def(1)"\ 6distributive\ 5/a\ 6 ? \ 5a href="cic:/matita/arithmetics/nat/times.fix(0,0,2)"\ 6times\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/minus.fix(0,0,1)"\ 6minus\ 5/a\ 6.
#a #b #c
(cases (\ 5a href="cic:/matita/arithmetics/nat/decidable_lt.def(7)"\ 6decidable_lt\ 5/a\ 6 b c)) #Hbc
- [> \ 5a href="cic:/matita/arithmetics/nat/eq_minus_O.def(13)"\ 6eq_minus_O\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_plus_b.def(8)"\ 6le_plus_b\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ >\ 5a href="cic:/matita/arithmetics/nat/eq_minus_O.def(13)"\ 6eq_minus_O\ 5/a\ 6 //
+ [> \ 5a href="cic:/matita/arithmetics/nat/eq_minus_O.def(12)"\ 6eq_minus_O\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_plus_b.def(8)"\ 6le_plus_b\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ >\ 5a href="cic:/matita/arithmetics/nat/eq_minus_O.def(12)"\ 6eq_minus_O\ 5/a\ 6 //
@\ 5a href="cic:/matita/arithmetics/nat/monotonic_le_times_r.def(8)"\ 6monotonic_le_times_r\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_plus_b.def(8)"\ 6le_plus_b\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
|@\ 5a href="cic:/matita/basics/logic/sym_eq.def(2)"\ 6sym_eq\ 5/a\ 6 (applyS \ 5a href="cic:/matita/arithmetics/nat/plus_to_minus.def(7)"\ 6plus_to_minus\ 5/a\ 6) <\ 5a href="cic:/matita/arithmetics/nat/distributive_times_plus.def(7)"\ 6distributive_times_plus\ 5/a\ 6
@\ 5a href="cic:/matita/basics/logic/eq_f.def(3)"\ 6eq_f\ 5/a\ 6 (applyS \ 5a href="cic:/matita/arithmetics/nat/plus_minus_m_m.def(7)"\ 6plus_minus_m_m\ 5/a\ 6) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/not_lt_to_le.def(6)"\ 6not_lt_to_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
[@\ 5a href="cic:/matita/arithmetics/nat/plus_to_minus.def(7)"\ 6plus_to_minus\ 5/a\ 6 @\ 5a href="cic:/matita/arithmetics/nat/plus_to_minus.def(7)"\ 6plus_to_minus\ 5/a\ 6 <\ 5a href="cic:/matita/arithmetics/nat/associative_plus.def(4)"\ 6associative_plus\ 5/a\ 6
@\ 5a href="cic:/matita/arithmetics/nat/minus_to_plus.def(8)"\ 6minus_to_plus\ 5/a\ 6 //
|cut (n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6p) [@(\ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6 … (\ 5a href="cic:/matita/arithmetics/nat/le_n_Sn.def(1)"\ 6le_n_Sn\ 5/a\ 6 …)) @\ 5a href="cic:/matita/arithmetics/nat/not_le_to_lt.def(5)"\ 6not_le_to_lt\ 5/a\ 6 //]
- #H >\ 5a href="cic:/matita/arithmetics/nat/eq_minus_O.def(13)"\ 6eq_minus_O\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/eq_minus_O.def(13)"\ 6eq_minus_O\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_minus_l.def(10)"\ 6monotonic_le_minus_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+ #H >\ 5a href="cic:/matita/arithmetics/nat/eq_minus_O.def(12)"\ 6eq_minus_O\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/eq_minus_O.def(12)"\ 6eq_minus_O\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_minus_l.def(9)"\ 6monotonic_le_minus_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
]
qed.
theorem eqb_elim : ∀ n,m:nat.∀ P:bool → Prop.
(n=m → (P true)) → (n ≠ m → (P false)) → (P (eqb n m)).
@nat_elim2
- [#n (cases n) normalize /3/
- |normalize /3/
- |normalize /4/
+ [#n (cases n) normalize /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/
+ |normalize /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/sym_not_eq.def(4)"\ 6sym_not_eq\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+ |normalize /\ 5span class="autotactic"\ 64\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/not_eq_S.def(4)"\ 6not_eq_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
]
qed.
theorem eqb_n_n: ∀n. eqb n n = true.
#n (elim n) normalize // qed.
-theorem eqb_true_to_eq: ∀n,m:nat. eqb n m = true → n = m.
-#n #m @(eqb_elim n m) // #_ #abs @False_ind /2/ qed.
+theorem eqb_true_to_eq: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"\ 6eqb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 m.
+#n #m @(\ 5a href="cic:/matita/arithmetics/nat/eqb_elim.def(5)"\ 6eqb_elim\ 5/a\ 6 n m) // #_ #abs @\ 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.
-theorem eqb_false_to_not_eq: ∀n,m:nat. eqb n m = false → n ≠ m.
-#n #m @(eqb_elim n m) /2/ qed.
+theorem eqb_false_to_not_eq: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"\ 6eqb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 → n \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 m.
+#n #m @(\ 5a href="cic:/matita/arithmetics/nat/eqb_elim.def(5)"\ 6eqb_elim\ 5/a\ 6 n m) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
theorem eq_to_eqb_true: ∀n,m:nat.n = m → eqb n m = true.
// qed.
-theorem not_eq_to_eqb_false: ∀n,m:nat.
- n ≠ m → eqb n m = false.
-#n #m #noteq @eqb_elim// #Heq @False_ind /2/ qed.
+theorem not_eq_to_eqb_false: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.
+ n \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 m → \ 5a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"\ 6eqb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6.
+#n #m #noteq @\ 5a href="cic:/matita/arithmetics/nat/eqb_elim.def(5)"\ 6eqb_elim\ 5/a\ 6// #Heq @\ 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 rec leb n m ≝
match n with
theorem leb_elim: ∀n,m:nat. ∀P:bool → Prop.
(n ≤ m → P true) → (n ≰ m → P false) → P (leb n m).
@nat_elim2 normalize
- [/2/
- |/3/
+ [/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/
+ |/\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/
|#n #m #Hind #P #Pt #Pf @Hind
- [#lenm @Pt @le_S_S // |#nlenm @Pf /2/ ]
+ [#lenm @Pt @\ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6 // |#nlenm @Pf /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/not_le_to_not_le_S_S.def(5)"\ 6not_le_to_not_le_S_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ ]
]
qed.
-theorem leb_true_to_le:∀n,m.leb n m = true → n ≤ m.
-#n #m @leb_elim // #_ #abs @False_ind /2/ qed.
+theorem leb_true_to_le:∀n,m.\ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
+#n #m @\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6 // #_ #abs @\ 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.
theorem leb_false_to_not_le:∀n,m.
- leb n m = false → n ≰ m.
-#n #m @leb_elim // #_ #abs @False_ind /2/ qed.
+ \ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 → n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 m.
+#n #m @\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6 // #_ #abs @\ 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.
-theorem le_to_leb_true: ∀n,m. n ≤ m → leb n m = true.
-#n #m @leb_elim // #H #H1 @False_ind /2/ qed.
+theorem le_to_leb_true: ∀n,m. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → \ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
+#n #m @\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6 // #H #H1 @\ 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.
-theorem not_le_to_leb_false: ∀n,m. n ≰ m → leb n m = false.
-#n #m @leb_elim // #H #H1 @False_ind /2/ qed.
+theorem not_le_to_leb_false: ∀n,m. n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 m → \ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6.
+#n #m @\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6 // #H #H1 @\ 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.
theorem lt_to_leb_false: ∀n,m. m < n → leb n m = false.
-/3/ qed.
+/\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/lt_to_not_le.def(7)"\ 6lt_to_not_le\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/not_le_to_leb_false.def(7)"\ 6not_le_to_leb_false\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
(* serve anche ltb?
ndefinition ltb ≝λn,m. leb (S n) m.
qed. *)
(* min e max *)
-definition min: nat →nat →nat ≝
-λn.λm. if_then_else ? (leb n m) n m.
+definition min: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 →\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 →\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 ≝
+λn.λm. if (\ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m) then n else m.
-definition max: nat →nat →nat ≝
-λn.λm. if_then_else ? (leb n m) m n.
+definition max: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 →\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 →\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 ≝
+λn.λm. if (\ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m) then m else n.
-lemma commutative_min: commutative ? min.
-#n #m normalize @leb_elim
- [@leb_elim normalize /2/
- |#notle >(le_to_leb_true …) // @(transitive_le ? (S m)) /2/
+lemma commutative_min: \ 5a href="cic:/matita/basics/relations/commutative.def(1)"\ 6commutative\ 5/a\ 6 ? \ 5a href="cic:/matita/arithmetics/nat/min.def(2)"\ 6min\ 5/a\ 6.
+#n #m normalize @\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6
+ [@\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6 normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_to_le_to_eq.def(5)"\ 6le_to_le_to_eq\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+ |#notle >(\ 5a href="cic:/matita/arithmetics/nat/le_to_leb_true.def(7)"\ 6le_to_leb_true\ 5/a\ 6 …) // @(\ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6 ? (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/not_le_to_lt.def(5)"\ 6not_le_to_lt\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
] qed.
-lemma le_minr: ∀i,n,m. i ≤ min n m → i ≤ m.
-#i #n #m normalize @leb_elim normalize /2/ qed.
+lemma le_minr: ∀i,n,m. i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/min.def(2)"\ 6min\ 5/a\ 6 n m → i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
+#i #n #m normalize @\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6 normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
lemma le_minl: ∀i,n,m. i ≤ min n m → i ≤ n.
-/2/ qed.
+/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_minr.def(7)"\ 6le_minr\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
lemma to_min: ∀i,n,m. i ≤ n → i ≤ m → i ≤ min n m.
#i #n #m #lein #leim normalize (cases (leb n m))
normalize // qed.
-lemma commutative_max: commutative ? max.
-#n #m normalize @leb_elim
- [@leb_elim normalize /2/
- |#notle >(le_to_leb_true …) // @(transitive_le ? (S m)) /2/
+lemma commutative_max: \ 5a href="cic:/matita/basics/relations/commutative.def(1)"\ 6commutative\ 5/a\ 6 ? \ 5a href="cic:/matita/arithmetics/nat/max.def(2)"\ 6max\ 5/a\ 6.
+#n #m normalize @\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6
+ [@\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6 normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_to_le_to_eq.def(5)"\ 6le_to_le_to_eq\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+ |#notle >(\ 5a href="cic:/matita/arithmetics/nat/le_to_leb_true.def(7)"\ 6le_to_leb_true\ 5/a\ 6 …) // @(\ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6 ? (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/not_le_to_lt.def(5)"\ 6not_le_to_lt\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
] qed.
-lemma le_maxl: ∀i,n,m. max n m ≤ i → n ≤ i.
-#i #n #m normalize @leb_elim normalize /2/ qed.
+lemma le_maxl: ∀i,n,m. \ 5a href="cic:/matita/arithmetics/nat/max.def(2)"\ 6max\ 5/a\ 6 n m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i.
+#i #n #m normalize @\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6 normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
lemma le_maxr: ∀i,n,m. max n m ≤ i → m ≤ i.
-/2/ qed.
+/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_maxl.def(7)"\ 6le_maxl\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-lemma to_max: ∀i,n,m. n ≤ i → m ≤ i → max n m ≤ i.
-#i #n #m #leni #lemi normalize (cases (leb n m))
+lemma to_max: ∀i,n,m. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i → \ 5a href="cic:/matita/arithmetics/nat/max.def(2)"\ 6max\ 5/a\ 6 n m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i.
+#i #n #m #leni #lemi normalize (cases (\ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m))
normalize // qed.
\ No newline at end of file
\ / GNU General Public License Version 2
V_______________________________________________________________ *)
-include "basics/bool.ma".
-(* include "arithmetics/nat.ma". *)
+include "arithmetics/nat.ma".
inductive list (A:Type[0]) : Type[0] :=
| nil: list A
interpretation "nil" 'nil = (nil ?).
interpretation "cons" 'cons hd tl = (cons ? hd tl).
-definition not_nil: ∀A:Type[0].list A → Prop ≝
- λA.λl.match l with [ nil ⇒ True | cons hd tl ⇒ False ].
+definition not_nil: ∀A:Type[0].\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A → Prop ≝
+ λA.λl.match l with [ nil ⇒ \ 5a href="cic:/matita/basics/logic/True.ind(1,0,0)"\ 6True\ 5/a\ 6 | cons hd tl ⇒ \ 5a href="cic:/matita/basics/logic/False.ind(1,0,0)"\ 6False\ 5/a\ 6 ].
theorem nil_cons:
- ∀A:Type[0].∀l:list A.∀a:A. a::l ≠ [].
- #A #l #a @nmk #Heq (change with (not_nil ? (a::l))) >Heq //
+ ∀A:Type[0].∀l:\ 5a href="cic:/matita/basics/list/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].
+ #A #l #a @\ 5a href="cic:/matita/basics/logic/Not.con(0,1,1)"\ 6nmk\ 5/a\ 6 #Heq (change with (\ 5a href="cic:/matita/basics/list/not_nil.def(1)"\ 6not_nil\ 5/a\ 6 ? (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l))) >Heq //
qed.
(*
[ nil => []
| (cons hd tl) => hd :: id_list A tl ]. *)
-let rec append A (l1: list A) l2 on l1 ≝
+let rec append A (l1: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) l2 on l1 ≝
match l1 with
[ nil ⇒ l2
- | cons hd tl ⇒ hd :: append A tl l2 ].
+ | cons hd tl ⇒ hd \ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6: append A tl l2 ].
-definition hd ≝ λA.λl: list A.λd:A.
+definition hd ≝ λA.λl: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.λd:A.
match l with [ nil ⇒ d | cons a _ ⇒ a].
-definition tail ≝ λA.λl: list A.
- match l with [ nil ⇒ [] | cons hd tl ⇒ tl].
+definition tail ≝ λA.λl: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.
+ match l with [ nil ⇒ \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6] | cons hd tl ⇒ tl].
interpretation "append" 'append l1 l2 = (append ? l1 l2).
-theorem append_nil: ∀A.∀l:list A.l @ [] = l.
+theorem append_nil: ∀A.∀l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.l \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 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 l.
#A #l (elim l) normalize // qed.
theorem associative_append:
- ∀A.associative (list A) (append A).
+ ∀A.\ 5a href="cic:/matita/basics/relations/associative.def(1)"\ 6associative\ 5/a\ 6 (\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) (\ 5a href="cic:/matita/basics/list/append.fix(0,1,1)"\ 6append\ 5/a\ 6 A).
#A #l1 #l2 #l3 (elim l1) normalize // qed.
(* deleterio per auto
a :: (l1 @ l2) = (a :: l1) @ l2.
//; nqed. *)
-theorem append_cons:∀A.∀a:A.∀l,l1.l@(a::l1)=(l@[a])@l1.
-/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace trans_eq\ 5/span\ 6\ 5/span\ 6/ qed.
+theorem append_cons:∀A.∀a:A.∀l,l1.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(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\ 6l1.\ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6\ 5span class="autotactic"\ 6\ 5/span\ 6
+#A #a #l1 #l2 >\ 5a href="cic:/matita/basics/list/associative_append.def(4)"\ 6associative_append\ 5/a\ 6 // qed.
-theorem nil_append_elim: ∀A.∀l1,l2: list A.∀P:?→?→Prop.
- l1@l2=[] → P (nil A) (nil A) → P l1 l2.
+theorem nil_append_elim: ∀A.∀l1,l2: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.∀P:?→?→Prop.
+ 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] → P (\ 5a href="cic:/matita/basics/list/list.con(0,1,1)"\ 6nil\ 5/a\ 6 A) (\ 5a href="cic:/matita/basics/list/list.con(0,1,1)"\ 6nil\ 5/a\ 6 A) → P l1 l2.
#A #l1 #l2 #P (cases l1) normalize //
#a #l3 #heq destruct
qed.
-theorem nil_to_nil: ∀A.∀l1,l2:list A.
- l1@l2 = [] → l1 = [] ∧ l2 = [].
-#A #l1 #l2 #isnil @(nil_append_elim A l1 l2) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace conj\ 5/span\ 6\ 5/span\ 6/
+theorem nil_to_nil: ∀A.∀l1,l2:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.
+ 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/basics/list/nil_append_elim.def(4)"\ 6nil_append_elim\ 5/a\ 6 A l1 l2) /\ 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/
qed.
(* iterators *)
-let rec map (A,B:Type[0]) (f: A → B) (l:list A) on l: list B ≝
- match l with [ nil ⇒ nil ? | cons x tl ⇒ f x :: (map A B f tl)].
+let rec map (A,B:Type[0]) (f: A → B) (l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) on l: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 B ≝
+ match l with [ nil ⇒ \ 5a href="cic:/matita/basics/list/list.con(0,1,1)"\ 6nil\ 5/a\ 6 ? | cons x tl ⇒ f x \ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6: (map A B f tl)].
-let rec foldr (A,B:Type[0]) (f:A → B → B) (b:B) (l:list A) on l :B ≝
+lemma map_append : ∀A,B,f,l1,l2.
+ (\ 5a href="cic:/matita/basics/list/map.fix(0,3,1)"\ 6map\ 5/a\ 6 A B f l1) \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 (\ 5a href="cic:/matita/basics/list/map.fix(0,3,1)"\ 6map\ 5/a\ 6 A B f l2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/list/map.fix(0,3,1)"\ 6map\ 5/a\ 6 A B f (l1\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6l2).
+#A #B #f #l1 elim l1
+[ #l2 @\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6
+| #h #t #IH #l2 normalize //
+] qed.
+
+let rec foldr (A,B:Type[0]) (f:A → B → B) (b:B) (l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) on l :B ≝
match l with [ nil ⇒ b | cons a l ⇒ f a (foldr A B f b l)].
definition filter ≝
- λT.λp:T → bool.
- foldr T (list T) (λx,l0.if_then_else ? (p x) (x::l0) l0) (nil T).
+ λT.λp:T → \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6.
+ \ 5a href="cic:/matita/basics/list/foldr.fix(0,4,1)"\ 6foldr\ 5/a\ 6 T (\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 T) (λx,l0.if (p x) then (x\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l0) else l0) (\ 5a href="cic:/matita/basics/list/list.con(0,1,1)"\ 6nil\ 5/a\ 6 T).
-lemma filter_true : ∀A,l,a,p. p a = true →
- filter A p (a::l) = a :: filter A p l.
+definition compose ≝ λA,B,C.λf:A→B→C.λl1,l2.
+ \ 5a href="cic:/matita/basics/list/foldr.fix(0,4,1)"\ 6foldr\ 5/a\ 6 ?? (λi,acc.(\ 5a href="cic:/matita/basics/list/map.fix(0,3,1)"\ 6map\ 5/a\ 6 ?? (f i) l2)\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6acc) \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 ] l1.
+
+lemma filter_true : ∀A,l,a,p. p a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
+ \ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 A p (a\ 5a title="cons" 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: \ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 A p l.
#A #l #a #p #pa (elim l) normalize >pa normalize // qed.
-lemma filter_false : ∀A,l,a,p. p a = false →
- filter A p (a::l) = filter A p l.
+lemma filter_false : ∀A,l,a,p. p a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 →
+ \ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 A p (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 A p l.
#A #l #a #p #pa (elim l) normalize >pa normalize // qed.
-theorem eq_map : ∀A,B,f,g,l. (∀x.f x = g x) → map A B f l = map A B g l.
+theorem eq_map : ∀A,B,f,g,l. (∀x.f x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 g x) → \ 5a href="cic:/matita/basics/list/map.fix(0,3,1)"\ 6map\ 5/a\ 6 A B f l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/list/map.fix(0,3,1)"\ 6map\ 5/a\ 6 A B g l.
#A #B #f #g #l #eqfg (elim l) normalize // qed.
(*
| cons a tl ⇒ (map ??(dp ?? a) (g a)) @ dprodl A f tl g
]. *)
-(**************************** length ******************************
+(**************************** length *******************************)
-let rec length (A:Type[0]) (l:list A) on l ≝
+let rec length (A:Type[0]) (l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) on l ≝
match l with
- [ nil ⇒ 0
- | cons a tl ⇒ S (length A tl)].
+ [ nil ⇒ \ 5a title="natural number" href="cic:/fakeuri.def(1)"\ 60\ 5/a\ 6
+ | cons a tl ⇒ \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (length A tl)].
notation "|M|" non associative with precedence 60 for @{'norm $M}.
interpretation "norm" 'norm l = (length ? l).
-let rec nth n (A:Type[0]) (l:list A) (d:A) ≝
+lemma length_append: ∀A.∀l1,l2:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.
+ \ 5a title="norm" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6l1\ 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="norm" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6l1|\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6\ 5a title="norm" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6l2|.
+#A #l1 elim l1 // normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/
+qed.
+
+let rec nth n (A:Type[0]) (l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) (d:A) ≝
match n with
- [O ⇒ hd A l d
- |S m ⇒ nth m A (tail A l) d].
+ [O ⇒ \ 5a href="cic:/matita/basics/list/hd.def(1)"\ 6hd\ 5/a\ 6 A l d
+ |S m ⇒ nth m A (\ 5a href="cic:/matita/basics/list/tail.def(1)"\ 6tail\ 5/a\ 6 A l) d].
-**************************** fold *******************************)
+(***************************** fold *******************************)
-let rec fold (A,B:Type[0]) (op:B → B → B) (b:B) (p:A→bool) (f:A→B) (l:list A) on l :B ≝
+let rec fold (A,B:Type[0]) (op:B → B → B) (b:B) (p:A→\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6) (f:A→B) (l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) on l :B ≝
match l with
[ nil ⇒ b
- | cons a l ⇒ if_then_else ? (p a) (op (f a) (fold A B op b p f l))
- (fold A B op b p f l)].
+ | cons a l ⇒ if (p a) then (op (f a) (fold A B op b p f l))
+ else (fold A B op b p f l)
+ ].
notation "\fold [ op , nil ]_{ ident i ∈ l | p} f"
with precedence 80
interpretation "\fold" 'fold op nil p f l = (fold ? ? op nil p f l).
theorem fold_true:
-∀A,B.∀a:A.∀l.∀p.∀op:B→B→B.∀nil.∀f:A→B. p a = true →
- \fold[op,nil]_{i ∈ a::l| p i} (f i) =
- op (f a) \fold[op,nil]_{i ∈ l| p i} (f i).
+∀A,B.∀a:A.∀l.∀p.∀op:B→B→B.∀nil.∀f:A→B. p a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
+ \ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l| p i} (f i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
+ op (f a) \ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ l| p i} (f i).
#A #B #a #l #p #op #nil #f #pa normalize >pa // qed.
theorem fold_false:
∀A,B.∀a:A.∀l.∀p.∀op:B→B→B.∀nil.∀f.
-p a = false → \fold[op,nil]_{i ∈ a::l| p i} (f i) =
- \fold[op,nil]_{i ∈ l| p i} (f i).
+p a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 → \ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l| p i} (f i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
+ \ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ l| p i} (f i).
#A #B #a #l #p #op #nil #f #pa normalize >pa // qed.
theorem fold_filter:
∀A,B.∀a:A.∀l.∀p.∀op:B→B→B.∀nil.∀f:A →B.
- \fold[op,nil]_{i ∈ l| p i} (f i) =
- \fold[op,nil]_{i ∈ (filter A p l)} (f i).
+ \ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ l| p i} (f i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
+ \ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ (\ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 A p l)} (f i).
#A #B #a #l #p #op #nil #f elim l //
-#a #tl #Hind cases(true_or_false (p a)) #pa
- [ >filter_true // > fold_true // >fold_true //
- | >filter_false // >fold_false // ]
+#a #tl #Hind cases(\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (p a)) #pa
+ [ >\ 5a href="cic:/matita/basics/list/filter_true.def(3)"\ 6filter_true\ 5/a\ 6 // > \ 5a href="cic:/matita/basics/list/fold_true.def(3)"\ 6fold_true\ 5/a\ 6 // >\ 5a href="cic:/matita/basics/list/fold_true.def(3)"\ 6fold_true\ 5/a\ 6 //
+ | >\ 5a href="cic:/matita/basics/list/filter_false.def(3)"\ 6filter_false\ 5/a\ 6 // >\ 5a href="cic:/matita/basics/list/fold_false.def(3)"\ 6fold_false\ 5/a\ 6 // ]
qed.
record Aop (A:Type[0]) (nil:A) : Type[0] ≝
{op :2> A → A → A;
- nill:∀a. op nil a = a;
- nilr:∀a. op a nil = a;
- assoc: ∀a,b,c.op a (op b c) = op (op a b) c
+ nill:∀a. op nil a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 a;
+ nilr:∀a. op a nil \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 a;
+ assoc: ∀a,b,c.op a (op b c) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 op (op a b) c
}.
-theorem fold_sum: ∀A,B. ∀I,J:list A.∀nil.∀op:Aop B nil.∀f.
- op (\fold[op,nil]_{i∈I} (f i)) (\fold[op,nil]_{i∈J} (f i)) =
- \fold[op,nil]_{i∈(I@J)} (f i).
+theorem fold_sum: ∀A,B. ∀I,J:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.∀nil.∀op:\ 5a href="cic:/matita/basics/list/Aop.ind(1,0,2)"\ 6Aop\ 5/a\ 6 B nil.∀f.
+ op (\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i∈I} (f i)) (\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i∈J} (f i)) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
+ \ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i∈(I\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6J)} (f i).
#A #B #I #J #nil #op #f (elim I) normalize
- [>nill //|#a #tl #Hind <assoc //]
+ [>\ 5a href="cic:/matita/basics/list/nill.fix(0,2,2)"\ 6nill\ 5/a\ 6 //|#a #tl #Hind <\ 5a href="cic:/matita/basics/list/assoc.fix(0,2,2)"\ 6assoc\ 5/a\ 6 //]
qed.
\ No newline at end of file
-(* boolean functions over lists *)
+(* The fact of being able to decide, via a computable boolean function, the
+equality between elements of a given set is an essential prerequisite for
+effectively searching an element of that set inside a data structure. In this
+section we shall define several boolean functions acting on lists of elements in
+a DeqSet, and prove some of their properties.*)
include "basics/list.ma".
-include "basics/sets.ma".
-include "basics/deqsets.ma".
+include "tutorial/chapter4.ma".
(********* search *********)
-let rec memb (S:DeqSet) (x:S) (l: list S) on l ≝
+let rec memb (S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) (x:S) (l: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 S) on l ≝
match l with
- [ nil ⇒ false
- | cons a tl ⇒ (x == a) ∨ memb S x tl
+ [ nil ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6
+ | cons a tl ⇒ (x \ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6= a) \ 5a title="boolean or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 memb S x tl
].
notation < "\memb x l" non associative with precedence 90 for @{'memb $x $l}.
interpretation "boolean membership" 'memb a l = (memb ? a l).
-lemma memb_hd: ∀S,a,l. memb S a (a::l) = true.
-#S #a #l normalize >(proj2 … (eqb_true S …) (refl S a)) //
+lemma memb_hd: ∀S,a,l. \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
+#S #a #l normalize >(\ 5a href="cic:/matita/basics/logic/proj2.def(2)"\ 6proj2\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter4/eqb_true.fix(0,0,4)"\ 6eqb_true\ 5/a\ 6 S …) (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 S a)) //
qed.
lemma memb_cons: ∀S,a,b,l.
- memb S a l = true → memb S a (b::l) = true.
-#S #a #b #l normalize cases (a==b) normalize //
+ \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a (b\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
+#S #a #b #l normalize cases (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=b) normalize //
qed.
-lemma memb_single: ∀S,a,x. memb S a [x] = true → a = x.
-#S #a #x normalize cases (true_or_false … (a==x)) #H
- [#_ >(\P H) // |>H normalize #abs @False_ind /2/]
+lemma memb_single: ∀S,a,x. \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a \ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6x] \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 x.
+#S #a #x normalize cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 … (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=x)) #H
+ [#_ >(\P H) // |>H normalize #abs @\ 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.
lemma memb_append: ∀S,a,l1,l2.
-memb S a (l1@l2) = true →
- memb S a l1= true ∨ memb S a l2 = true.
+\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a (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/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
+ \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a l1\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a l2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#S #a #l1 elim l1 normalize [#l2 #H %2 //]
-#b #tl #Hind #l2 cases (a==b) normalize /2/
+#b #tl #Hind #l2 cases (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=b) normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/bool/orb_true_l.def(2)"\ 6orb_true_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
lemma memb_append_l1: ∀S,a,l1,l2.
- memb S a l1= true → memb S a (l1@l2) = true.
+ \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a l1\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a (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/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#S #a #l1 elim l1 normalize
- [normalize #le #abs @False_ind /2/
- |#b #tl #Hind #l2 cases (a==b) normalize /2/
+ [normalize #le #abs @\ 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/
+ |#b #tl #Hind #l2 cases (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=b) normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/
]
qed.
lemma memb_append_l2: ∀S,a,l1,l2.
- memb S a l2= true → memb S a (l1@l2) = true.
+ \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a l2\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a (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/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#S #a #l1 elim l1 normalize //
-#b #tl #Hind #l2 cases (a==b) normalize /2/
+#b #tl #Hind #l2 cases (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=b) normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/
qed.
-lemma memb_exists: ∀S,a,l.memb S a l = true →
- ∃l1,l2.l=l1@(a::l2).
-#S #a #l elim l [normalize #abs @False_ind /2/]
-#b #tl #Hind #H cases (orb_true_l … H)
- [#eqba @(ex_intro … (nil S)) @(ex_intro … tl) >(\P eqba) //
+lemma memb_exists: ∀S,a,l.\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
+ \ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6l1,l2.l\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6l1\ 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:l2).
+#S #a #l elim l [normalize #abs @\ 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/]
+#b #tl #Hind #H cases (\ 5a href="cic:/matita/basics/bool/orb_true_l.def(2)"\ 6orb_true_l\ 5/a\ 6 … H)
+ [#eqba @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … (\ 5a href="cic:/matita/basics/list/list.con(0,1,1)"\ 6nil\ 5/a\ 6 S)) @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … tl) >(\P eqba) //
|#mem_tl cases (Hind mem_tl) #l1 * #l2 #eqtl
- @(ex_intro … (b::l1)) @(ex_intro … l2) >eqtl //
+ @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … (b\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l1)) @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … l2) >eqtl //
]
qed.
lemma not_memb_to_not_eq: ∀S,a,b,l.
- memb S a l = false → memb S b l = true → a==b = false.
-#S #a #b #l cases (true_or_false (a==b)) //
-#eqab >(\P eqab) #H >H #abs @False_ind /2/
+ \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S b l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=b \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6.
+#S #a #b #l cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=b)) //
+#eqab >(\P eqab) #H >H #abs @\ 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.
-lemma memb_map: ∀S1,S2,f,a,l. memb S1 a l= true →
- memb S2 (f a) (map … f l) = true.
+lemma memb_map: ∀S1,S2,f,a,l. \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S1 a l\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
+ \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S2 (f a) (\ 5a href="cic:/matita/basics/list/map.fix(0,3,1)"\ 6map\ 5/a\ 6 … f l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#S1 #S2 #f #a #l elim l normalize [//]
-#x #tl #memba cases (true_or_false (a==x))
- [#eqx >eqx >(\P eqx) >(\b (refl … (f x))) normalize //
- |#eqx >eqx cases (f a==f x) normalize /2/
+#x #tl #memba cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=x))
+ [#eqx >eqx >(\P eqx) >(\b (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 … (f x))) normalize //
+ |#eqx >eqx cases (f a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=f x) normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/
]
qed.
lemma memb_compose: ∀S1,S2,S3,op,a1,a2,l1,l2.
- memb S1 a1 l1 = true → memb S2 a2 l2 = true →
- memb S3 (op a1 a2) (compose S1 S2 S3 op l1 l2) = true.
+ \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S1 a1 l1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S2 a2 l2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
+ \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S3 (op a1 a2) (\ 5a href="cic:/matita/basics/list/compose.def(2)"\ 6compose\ 5/a\ 6 S1 S2 S3 op l1 l2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#S1 #S2 #S3 #op #a1 #a2 #l1 elim l1 [normalize //]
-#x #tl #Hind #l2 #memba1 #memba2 cases (orb_true_l … memba1)
- [#eqa1 >(\P eqa1) @memb_append_l1 @memb_map //
- |#membtl @memb_append_l2 @Hind //
+#x #tl #Hind #l2 #memba1 #memba2 cases (\ 5a href="cic:/matita/basics/bool/orb_true_l.def(2)"\ 6orb_true_l\ 5/a\ 6 … memba1)
+ [#eqa1 >(\P eqa1) @\ 5a href="cic:/matita/tutorial/chapter5/memb_append_l1.def(5)"\ 6memb_append_l1\ 5/a\ 6 @\ 5a href="cic:/matita/tutorial/chapter5/memb_map.def(5)"\ 6memb_map\ 5/a\ 6 //
+ |#membtl @\ 5a href="cic:/matita/tutorial/chapter5/memb_append_l2.def(5)"\ 6memb_append_l2\ 5/a\ 6 @Hind //
]
qed.
(**************** unicity test *****************)
-let rec uniqueb (S:DeqSet) l on l : bool ≝
+let rec uniqueb (S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) l on l : \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 ≝
match l with
- [ nil ⇒ true
- | cons a tl ⇒ notb (memb S a tl) ∧ uniqueb S tl
+ [ nil ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6
+ | cons a tl ⇒ \ 5a href="cic:/matita/basics/bool/notb.def(1)"\ 6notb\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a tl) \ 5a title="boolean and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 uniqueb S tl
].
(* unique_append l1 l2 add l1 in fornt of l2, but preserving unicity *)
-let rec unique_append (S:DeqSet) (l1,l2: list S) on l1 ≝
+let rec unique_append (S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) (l1,l2: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 S) on l1 ≝
match l1 with
[ nil ⇒ l2
| cons a tl ⇒
let r ≝ unique_append S tl l2 in
- if memb S a r then r else a::r
+ if \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a r then r else a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:r
].
-axiom unique_append_elim: ∀S:DeqSet.∀P: S → Prop.∀l1,l2.
-(∀x. memb S x l1 = true → P x) → (∀x. memb S x l2 = true → P x) →
-∀x. memb S x (unique_append S l1 l2) = true → P x.
+axiom unique_append_elim: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀P: S → Prop.∀l1,l2.
+(∀x. \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S x l1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → P x) → (∀x. \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S x l2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → P x) →
+∀x. \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S x (\ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 S l1 l2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → P x.
-lemma unique_append_unique: ∀S,l1,l2. uniqueb S l2 = true →
- uniqueb S (unique_append S l1 l2) = true.
+lemma unique_append_unique: ∀S,l1,l2. \ 5a href="cic:/matita/tutorial/chapter5/uniqueb.fix(0,1,5)"\ 6uniqueb\ 5/a\ 6 S l2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
+ \ 5a href="cic:/matita/tutorial/chapter5/uniqueb.fix(0,1,5)"\ 6uniqueb\ 5/a\ 6 S (\ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 S l1 l2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#S #l1 elim l1 normalize // #a #tl #Hind #l2 #uniquel2
-cases (true_or_false … (memb S a (unique_append S tl l2)))
+cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a (\ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 S tl l2)))
#H >H normalize [@Hind //] >H normalize @Hind //
qed.
(******************* sublist *******************)
definition sublist ≝
- λS,l1,l2.∀a. memb S a l1 = true → memb S a l2 = true.
+ λS,l1,l2.∀a. \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a l1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a l2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
lemma sublist_length: ∀S,l1,l2.
- uniqueb S l1 = true → sublist S l1 l2 → |l1| ≤ |l2|.
+ \ 5a href="cic:/matita/tutorial/chapter5/uniqueb.fix(0,1,5)"\ 6uniqueb\ 5/a\ 6 S l1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 S l1 l2 → \ 5a title="norm" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6l1| \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a title="norm" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6l2|.
#S #l1 elim l1 //
#a #tl #Hind #l2 #unique #sub
-cut (∃l3,l4.l2=l3@(a::l4)) [@memb_exists @sub //]
-* #l3 * #l4 #eql2 >eql2 >length_append normalize
-applyS le_S_S <length_append @Hind [@(andb_true_r … unique)]
+cut (\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6l3,l4.l2\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6l3\ 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:l4)) [@\ 5a href="cic:/matita/tutorial/chapter5/memb_exists.def(5)"\ 6memb_exists\ 5/a\ 6 @sub //]
+* #l3 * #l4 #eql2 >eql2 >\ 5a href="cic:/matita/basics/list/length_append.def(2)"\ 6length_append\ 5/a\ 6 normalize
+applyS \ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6 <\ 5a href="cic:/matita/basics/list/length_append.def(2)"\ 6length_append\ 5/a\ 6 @Hind [@(\ 5a href="cic:/matita/basics/bool/andb_true_r.def(4)"\ 6andb_true_r\ 5/a\ 6 … unique)]
>eql2 in sub; #sub #x #membx
-cases (memb_append … (sub x (orb_true_r2 … membx)))
- [#membxl3 @memb_append_l1 //
- |#membxal4 cases (orb_true_l … membxal4)
- [#eqxa @False_ind lapply (andb_true_l … unique)
- <(\P eqxa) >membx normalize /2/ |#membxl4 @memb_append_l2 //
+cases (\ 5a href="cic:/matita/tutorial/chapter5/memb_append.def(5)"\ 6memb_append\ 5/a\ 6 … (sub x (\ 5a href="cic:/matita/basics/bool/orb_true_r2.def(3)"\ 6orb_true_r2\ 5/a\ 6 … membx)))
+ [#membxl3 @\ 5a href="cic:/matita/tutorial/chapter5/memb_append_l1.def(5)"\ 6memb_append_l1\ 5/a\ 6 //
+ |#membxal4 cases (\ 5a href="cic:/matita/basics/bool/orb_true_l.def(2)"\ 6orb_true_l\ 5/a\ 6 … membxal4)
+ [#eqxa @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 lapply (\ 5a href="cic:/matita/basics/bool/andb_true_l.def(4)"\ 6andb_true_l\ 5/a\ 6 … unique)
+ <(\P eqxa) >membx normalize /\ 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/ |#membxl4 @\ 5a href="cic:/matita/tutorial/chapter5/memb_append_l2.def(5)"\ 6memb_append_l2\ 5/a\ 6 //
]
]
qed.
lemma sublist_unique_append_l1:
- ∀S,l1,l2. sublist S l1 (unique_append S l1 l2).
-#S #l1 elim l1 normalize [#l2 #S #abs @False_ind /2/]
+ ∀S,l1,l2. \ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 S l1 (\ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 S l1 l2).
+#S #l1 elim l1 normalize [#l2 #S #abs @\ 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/]
#x #tl #Hind #l2 #a
-normalize cases (true_or_false … (a==x)) #eqax >eqax
-[<(\P eqax) cases (true_or_false (memb S a (unique_append S tl l2)))
- [#H >H normalize // | #H >H normalize >(\b (refl … a)) //]
-|cases (memb S x (unique_append S tl l2)) normalize
- [/2/ |>eqax normalize /2/]
+normalize cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 … (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=x)) #eqax >eqax
+[<(\P eqax) cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a (\ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 S tl l2)))
+ [#H >H normalize // | #H >H normalize >(\b (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 … a)) //]
+|cases (\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S x (\ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 S tl l2)) normalize
+ [/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/ |>eqax normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/]
]
qed.
lemma sublist_unique_append_l2:
- ∀S,l1,l2. sublist S l2 (unique_append S l1 l2).
+ ∀S,l1,l2. \ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 S l2 (\ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 S l1 l2).
#S #l1 elim l1 [normalize //] #x #tl #Hind normalize
-#l2 #a cases (memb S x (unique_append S tl l2)) normalize
-[@Hind | cases (a==x) normalize // @Hind]
+#l2 #a cases (\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S x (\ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 S tl l2)) normalize
+[@Hind | cases (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=x) normalize // @Hind]
qed.
lemma decidable_sublist:∀S,l1,l2.
- (sublist S l1 l2) ∨ ¬(sublist S l1 l2).
+ (\ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 S l1 l2) \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 S l1 l2).
#S #l1 #l2 elim l1
- [%1 #a normalize in ⊢ (%→?); #abs @False_ind /2/
+ [%1 #a normalize in ⊢ (%→?); #abs @\ 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/
|#a #tl * #subtl
- [cases (true_or_false (memb S a l2)) #memba
- [%1 whd #x #membx cases (orb_true_l … membx)
+ [cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a l2)) #memba
+ [%1 whd #x #membx cases (\ 5a href="cic:/matita/basics/bool/orb_true_l.def(2)"\ 6orb_true_l\ 5/a\ 6 … membx)
[#eqax >(\P eqax) // |@subtl]
- |%2 @(not_to_not … (eqnot_to_noteq … true memba)) #H1 @H1 @memb_hd
+ |%2 @(\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 … (\ 5a href="cic:/matita/basics/bool/eqnot_to_noteq.def(4)"\ 6eqnot_to_noteq\ 5/a\ 6 … \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 memba)) #H1 @H1 @\ 5a href="cic:/matita/tutorial/chapter5/memb_hd.def(5)"\ 6memb_hd\ 5/a\ 6
]
- |%2 @(not_to_not … subtl) #H1 #x #H2 @H1 @memb_cons //
+ |%2 @(\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 … subtl) #H1 #x #H2 @H1 @\ 5a href="cic:/matita/tutorial/chapter5/memb_cons.def(5)"\ 6memb_cons\ 5/a\ 6 //
]
]
qed.
(********************* filtering *****************)
lemma filter_true: ∀S,f,a,l.
- memb S a (filter S f l) = true → f a = true.
-#S #f #a #l elim l [normalize #H @False_ind /2/]
-#b #tl #Hind cases (true_or_false (f b)) #H
+ \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a (\ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 S f l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → f a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
+#S #f #a #l elim l [normalize #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/]
+#b #tl #Hind cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (f b)) #H
normalize >H normalize [2:@Hind]
-cases (true_or_false (a==b)) #eqab
+cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=b)) #eqab
[#_ >(\P eqab) // | >eqab normalize @Hind]
qed.
lemma memb_filter_memb: ∀S,f,a,l.
- memb S a (filter S f l) = true → memb S a l = true.
+ \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a (\ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 S f l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#S #f #a #l elim l [normalize //]
#b #tl #Hind normalize (cases (f b)) normalize
-cases (a==b) normalize // @Hind
+cases (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=b) normalize // @Hind
qed.
-lemma memb_filter: ∀S,f,l,x. memb S x (filter ? f l) = true →
-memb S x l = true ∧ (f x = true).
-/3/ qed.
+lemma memb_filter: ∀S,f,l,x. \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S x (\ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 ? f l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
+\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S x l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 (f x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6).
+/\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter5/filter_true.def(5)"\ 6filter_true\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter5/memb_filter_memb.def(5)"\ 6memb_filter_memb\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-lemma memb_filter_l: ∀S,f,x,l. (f x = true) → memb S x l = true →
-memb S x (filter ? f l) = true.
+lemma memb_filter_l: ∀S,f,x,l. (f x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6) → \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S x l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
+\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S x (\ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 ? f l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#S #f #x #l #fx elim l normalize //
-#b #tl #Hind cases (true_or_false (x==b)) #eqxb
- [<(\P eqxb) >(\b (refl … x)) >fx normalize >(\b (refl … x)) normalize //
+#b #tl #Hind cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (x\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=b)) #eqxb
+ [<(\P eqxb) >(\b (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 … x)) >fx normalize >(\b (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 … x)) normalize //
|>eqxb cases (f b) normalize [>eqxb normalize @Hind| @Hind]
]
qed.
(********************* exists *****************)
-let rec exists (A:Type[0]) (p:A → bool) (l:list A) on l : bool ≝
+let rec exists (A:Type[0]) (p:A → \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6) (l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) on l : \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 ≝
match l with
-[ nil ⇒ false
-| cons h t ⇒ orb (p h) (exists A p t)
+[ nil ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6
+| cons h t ⇒ \ 5a href="cic:/matita/basics/bool/orb.def(1)"\ 6orb\ 5/a\ 6 (p h) (exists A p t)
].
-
-lemma Exists_exists : ∀A,P,l.
- Exists A P l →
- ∃x. P x.
-#A #P #l elim l [ * | #hd #tl #IH * [ #H %{hd} @H | @IH ]
-qed.
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