commit by user andrea

[helm.git] / weblib / arithmetics / nat.ma

diff --git a/weblib/arithmetics/nat.ma b/weblib/arithmetics/nat.ma
index 66ed6ccfd98b416f37986bce6e44cf3fd692da0b..24cfc5ef66d3e8d62c0eb5a6977134333b403cff 100644 (file)
--- a/weblib/arithmetics/nat.ma
+++ b/weblib/arithmetics/nat.ma
@@ -47,7 +47,7 @@ theorem not_eq_n_Sn: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6n
 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.
@@ -55,7 +55,7 @@ theorem nat_elim2 :
   → (∀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/]
@@ -500,7 +500,7 @@ qed.
 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
@@ -529,7 +529,7 @@ lemma le_plus_b: ∀b,n,m. n \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6
 /\ 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.
@@ -560,7 +560,7 @@ monotonic_lt_plus_l. *)
 
 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.
@@ -707,7 +707,7 @@ theorem lt_to_le_to_lt_times:
 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: 
@@ -824,7 +824,7 @@ theorem plus_minus:
 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.
@@ -915,11 +915,11 @@ theorem le_minus_to_plus: ∀n,m,p. n\ 5a title="natural minus" href="cic:/fakeuri
 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/
@@ -931,12 +931,12 @@ theorem lt_minus_to_plus: ∀a,b,c. a \ 5a title="natural minus" href="cic:/fakeur
 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.
@@ -945,19 +945,19 @@ qed.
 
 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/
@@ -969,7 +969,7 @@ cases (\ 5a href="cic:/matita/arithmetics/nat/decidable_le.def(6)"\ 6decidable_le\ 5/a
   [@\ 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.
 
@@ -998,27 +998,27 @@ match n with
 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 
@@ -1031,28 +1031,28 @@ 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.
@@ -1080,40 +1080,40 @@ theorem le_to_ltb_false: ∀n,m. m \le n → ltb n m = false.
 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.
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