commit by user utente1

diff --git a/weblib/arithmetics/nat.ma b/weblib/arithmetics/nat.ma
index ef8da2f1921c2d3f0510765e8b62f6306b49e8ab..66ed6ccfd98b416f37986bce6e44cf3fd692da0b 100644 (file)
--- a/weblib/arithmetics/nat.ma
+++ b/weblib/arithmetics/nat.ma
@@ -33,7 +33,7 @@ theorem inj_S : \forall n,m:nat.(S n)=(S m) \to n=m.
 //. qed. *)
 
 theorem not_eq_S: ∀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/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="leibnitz's non-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.
-/2/ qed.
+/\ 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.
 
 definition not_zero: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop ≝
  λn: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. match n with [ O ⇒ \ 5a href="cic:/matita/basics/logic/False.ind(1,0,0)"\ 6False\ 5/a\ 6 | (S p) ⇒ \ 5a href="cic:/matita/basics/logic/True.ind(1,0,0)"\ 6True\ 5/a\ 6 ].
@@ -42,12 +42,12 @@ theorem not_eq_O_S : ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6n
 #n @\ 5a href="cic:/matita/basics/logic/Not.con(0,1,1)"\ 6nmk\ 5/a\ 6 #eqOS (change with (\ 5a href="cic:/matita/arithmetics/nat/not_zero.def(1)"\ 6not_zero\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6)) >eqOS // qed.
 
 theorem not_eq_n_Sn: ∀n:\ 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 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n.
-#n (elim n) /2/ qed.
+#n (elim n) /\ 5span class="autotactic"\ 62\ 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 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) /2/ 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.
@@ -55,10 +55,10 @@ 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) /2/ 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) /2/ | /3/ | #m #Hind (cases Hind) /3/]
+@\ 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/]
 qed. 
 
 (*************************** plus ******************************)
@@ -102,7 +102,7 @@ theorem assoc_plus1: ∀a,b,c. c \ 5a title="natural plus" href="cic:/fakeuri.def(
 // qed. 
 
 theorem injective_plus_r: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a href="cic:/matita/basics/relations/injective.def(1)"\ 6injective\ 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 (λm.n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m).
-#n (elim n) normalize /3/ qed.
+#n (elim n) normalize /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/injective_S.def(4)"\ 6injective_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
 
 (* theorem inj_plus_r: \forall p,n,m:nat. p+n = p+m \to n=m
 \def injective_plus_r. 
@@ -208,7 +208,7 @@ interpretation "natural 'greater than'" 'gt x y = (gt x y).
 interpretation "natural 'not greater than'" 'ngtr x y = (Not (gt x y)).
 
 theorem transitive_le : \ 5a href="cic:/matita/basics/relations/transitive.def(2)"\ 6transitive\ 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.
-#a #b #c #leab #lebc (elim lebc) /2/
+#a #b #c #leab #lebc (elim lebc) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le.con(0,2,1)"\ 6le_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
 qed.
 
 (*
@@ -216,30 +216,30 @@ theorem trans_le: \forall n,m,p:nat. n \leq m \to m \leq p \to n \leq p
 \def transitive_le. *)
 
 theorem transitive_lt: \ 5a href="cic:/matita/basics/relations/transitive.def(2)"\ 6transitive\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/lt.def(1)"\ 6lt\ 5/a\ 6.
-#a #b #c #ltab #ltbc (elim ltbc) /2/qed.
+#a #b #c #ltab #ltbc (elim ltbc) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le.con(0,2,1)"\ 6le_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/qed.
 
 (*
 theorem trans_lt: \forall n,m,p:nat. lt n m \to lt m p \to lt n p
 \def transitive_lt. *)
 
 theorem le_S_S: ∀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 → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'less or equal to'" 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.
-#n #m #lenm (elim lenm) /2/ qed.
+#n #m #lenm (elim lenm) /\ 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/le.con(0,2,1)"\ 6le_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
 
 theorem le_O_n : ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n.
-#n (elim n) /2/ qed.
+#n (elim n) /\ 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/le.con(0,2,1)"\ 6le_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
 
 theorem le_n_Sn : ∀n:\ 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 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n.
-/2/ 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/le.con(0,2,1)"\ 6le_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
 
 theorem le_pred_n : ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/pred.def(1)"\ 6pred\ 5/a\ 6 n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n.
 #n (elim n) // qed.
 
 theorem monotonic_pred: \ 5a href="cic:/matita/basics/relations/monotonic.def(1)"\ 6monotonic\ 5/a\ 6 ? \ 5a href="cic:/matita/arithmetics/nat/le.ind(1,0,1)"\ 6le\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/pred.def(1)"\ 6pred\ 5/a\ 6.
-#n #m #lenm (elim lenm) /2/ qed.
+#n #m #lenm (elim lenm) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 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.
 
 theorem le_S_S_to_le: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'less or equal to'" 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 → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
 (* demo *)
-/2/ qed.
+/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_pred.def(4)"\ 6monotonic_pred\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
 
 (* this are instances of the le versions 
 theorem lt_S_S_to_lt: ∀n,m. S n < S m → n < m.
@@ -256,19 +256,19 @@ theorem not_le_Sn_O: ∀ n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6
 #n @\ 5a href="cic:/matita/basics/logic/Not.con(0,1,1)"\ 6nmk\ 5/a\ 6 #Hlen0 @(\ 5a href="cic:/matita/arithmetics/nat/lt_to_not_zero.def(2)"\ 6lt_to_not_zero\ 5/a\ 6 ?? Hlen0) qed.
 
 theorem not_le_to_not_le_S_S: ∀ n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. 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/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'neither less nor equal to'" 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.
-/3/ qed.
+/\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/monotonic_pred.def(4)"\ 6monotonic_pred\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
 
 theorem not_le_S_S_to_not_le: ∀ n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'neither less nor equal to'" 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 → n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 m.
-/3/ qed.
+/\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
 
 theorem decidable_le: ∀n,m. \ 5a href="cic:/matita/basics/logic/decidable.def(1)"\ 6decidable\ 5/a\ 6 (n\ 5a title="natural 'less or equal to'" 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 /2/ #m * /3/ qed.
+@\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6 #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/ #m * /\ 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_le_to_not_le_S_S.def(5)"\ 6not_le_to_not_le_S_S\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
 
 theorem decidable_lt: ∀n,m. \ 5a href="cic:/matita/basics/logic/decidable.def(1)"\ 6decidable\ 5/a\ 6 (n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m).
 #n #m @\ 5a href="cic:/matita/arithmetics/nat/decidable_le.def(6)"\ 6decidable_le\ 5/a\ 6  qed.
 
 theorem not_le_Sn_n: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 n.
-#n (elim n) /2/ qed.
+#n (elim n) /\ 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.
 
 (* this is le_S_S_to_le
 theorem lt_S_to_le: ∀n,m:nat. n < S m → n ≤ m.
@@ -276,38 +276,38 @@ theorem lt_S_to_le: ∀n,m:nat. n < S m → n ≤ m.
 *)
 
 lemma le_gen: ∀P:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop.∀n.(∀i. i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n → P i) → P n.
-/2/ 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\ 5/span\ 6\ 5/span\ 6/ qed.
 
 theorem not_le_to_lt: ∀n,m. n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 m → m \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n.
 @\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6 #n
- [#abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /2/
- |/2/
- |#m #Hind #HnotleSS @\ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6 /3/
+ [#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/
+ |/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+ |#m #Hind #HnotleSS @\ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6 /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/not_le_S_S_to_not_le.def(4)"\ 6not_le_S_S_to_not_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
  ]
 qed.
 
 theorem lt_to_not_le: ∀n,m. n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → m \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 n.
-#n #m #Hltnm (elim Hltnm) /3/ qed.
+#n #m #Hltnm (elim Hltnm) /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
 
 theorem not_lt_to_le: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'not less than'" href="cic:/fakeuri.def(1)"\ 6≮\ 5/a\ 6 m → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n.
-/4/ qed.
+/\ 5span class="autotactic"\ 64\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/not_le_to_lt.def(5)"\ 6not_le_to_lt\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/monotonic_pred.def(4)"\ 6monotonic_pred\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
 
 theorem le_to_not_lt: ∀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 → m \ 5a title="natural 'not less than'" href="cic:/fakeuri.def(1)"\ 6≮\ 5/a\ 6 n.
-#n #m #H @\ 5a href="cic:/matita/arithmetics/nat/lt_to_not_le.def(7)"\ 6lt_to_not_le\ 5/a\ 6 /2/ (* /3/ *) qed.
+#n #m #H @\ 5a href="cic:/matita/arithmetics/nat/lt_to_not_le.def(7)"\ 6lt_to_not_le\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ (* /3/ *) qed.
 
 (* lt and le trans *)
 
 theorem lt_to_le_to_lt: ∀n,m,p:\ 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 → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 p → n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 p.
-#n #m #p #H #H1 (elim H1) /2/ qed.
+#n #m #p #H #H1 (elim H1) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/transitive_lt.def(3)"\ 6transitive_lt\ 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.
 
 theorem le_to_lt_to_lt: ∀n,m,p:\ 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 → m \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 p → n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 p.
-#n #m #p #H (elim H) /3/ qed.
+#n #m #p #H (elim H) /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/transitive_lt.def(3)"\ 6transitive_lt\ 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.
 
 theorem lt_S_to_lt: ∀n,m. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m.
-/2/ qed.
+/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/transitive_lt.def(3)"\ 6transitive_lt\ 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.
 
 theorem ltn_to_ltO: ∀n,m:\ 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 → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m.
-/2/ qed.
+/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_to_lt_to_lt.def(4)"\ 6le_to_lt_to_lt\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
 
 (*
 theorem lt_SO_n_to_lt_O_pred_n: \forall n:nat.
@@ -322,7 +322,7 @@ qed. *)
 
 theorem lt_O_n_elim: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n → 
   ∀P:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop.(∀m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.P (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m)) → P n.
-#n (elim n) // #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /2/
+#n (elim n) // #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 S_pred: ∀n. \ 5a title="natural number" href="cic:/fakeuri.def(1)"\ 60\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6(\ 5a href="cic:/matita/arithmetics/nat/pred.def(1)"\ 6pred\ 5/a\ 6 n) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n.
@@ -357,11 +357,11 @@ qed.
 
 (* le to lt or eq *)
 theorem le_to_or_lt_eq: ∀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 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 m.
-#n #m #lenm (elim lenm) /3/ qed.
+#n #m #lenm (elim lenm) /\ 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/le_to_lt_to_lt.def(4)"\ 6le_to_lt_to_lt\ 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.
 
 (* not eq *)
 theorem lt_to_not_eq : ∀n,m:\ 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 → n \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 m.
-#n #m #H @\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 /2/ qed.
+#n #m #H @\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 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, \ 5a href="cic:/matita/basics/logic/Not.con(0,1,1)"\ 6nmk\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
 
 (*not lt 
 theorem eq_to_not_lt: ∀a,b:nat. a = b → a ≮ b.
@@ -389,30 +389,30 @@ apply (not_le_Sn_n ? H3).
 qed. *)
 
 theorem not_eq_to_le_to_lt: ∀n,m. n\ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6m → n\ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6m → n\ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6m.
-#n #m #Hneq #Hle cases (\ 5a href="cic:/matita/arithmetics/nat/le_to_or_lt_eq.def(6)"\ 6le_to_or_lt_eq\ 5/a\ 6 ?? Hle) //
-#Heq /3/ qed.
+#n #m #Hneq #Hle cases (\ 5a href="cic:/matita/arithmetics/nat/le_to_or_lt_eq.def(5)"\ 6le_to_or_lt_eq\ 5/a\ 6 ?? Hle) //
+#Heq /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6, \ 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.
 (*
 nelim (Hneq Heq) qed. *)
 
 (* le elimination *)
 theorem le_n_O_to_eq : ∀n:\ 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 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6n.
-#n (cases n) // #a  #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /2/ qed.
+#n (cases n) // #a  #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_n_O_elim: ∀n:\ 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 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 → ∀P: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 →Prop. P \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 → P n.
-#n (cases n) // #a #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /2/ qed. 
+#n (cases n) // #a #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_n_Sm_elim : ∀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 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m → 
 ∀P:Prop. (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'less or equal to'" 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) → (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) → P.
-#n #m #Hle #P (elim Hle) /3/ qed.
+#n #m #Hle #P (elim Hle) /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
 
 (* le and eq *)
 
 theorem le_to_le_to_eq: ∀n,m. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → 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 m.
-@\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6 /4/
+@\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6 /\ 5span class="autotactic"\ 64\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/eq_f.def(3)"\ 6eq_f\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le_n_O_to_eq.def(4)"\ 6le_n_O_to_eq\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/monotonic_pred.def(4)"\ 6monotonic_pred\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
 qed. 
 
 theorem lt_O_S : ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n.
-/2/ qed.
+/\ 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.
 
 (*
 (* other abstract properties *)
@@ -444,13 +444,13 @@ qed.
 theorem nat_elim1 : ∀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. 
 (∀m.(∀p. p \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → P p) → P m) → P n.
 #n #P #H 
-cut (∀q:\ 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 n → P q) /2/
+cut (∀q:\ 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 n → P q) /\ 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\ 5/span\ 6\ 5/span\ 6/
 (elim n) 
  [#q #HleO (* applica male *) 
     @(\ 5a href="cic:/matita/arithmetics/nat/le_n_O_elim.def(4)"\ 6le_n_O_elim\ 5/a\ 6 ? HleO)
-    @H #p #ltpO @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /2/ (* 3 *)
+    @H #p #ltpO @\ 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/ (* 3 *)
  |#p #Hind #q #HleS 
-    @H #a #lta @Hind @\ 5a href="cic:/matita/arithmetics/nat/le_S_S_to_le.def(5)"\ 6le_S_S_to_le\ 5/a\ 6 /2/
+    @H #a #lta @Hind @\ 5a href="cic:/matita/arithmetics/nat/le_S_S_to_le.def(5)"\ 6le_S_S_to_le\ 5/a\ 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.
 
@@ -460,26 +460,26 @@ definition increasing ≝ λf:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0
 
 theorem increasing_to_monotonic: ∀f:\ 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/increasing.def(2)"\ 6increasing\ 5/a\ 6 f → \ 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/lt.def(1)"\ 6lt\ 5/a\ 6 f.
-#f #incr #n #m #ltnm (elim ltnm) /2/
+#f #incr #n #m #ltnm (elim ltnm) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/transitive_lt.def(3)"\ 6transitive_lt\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
 qed.
 
 theorem le_n_fn: ∀f:\ 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/increasing.def(2)"\ 6increasing\ 5/a\ 6 f → ∀n:\ 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 f n.
-#f #incr #n (elim n) /2/
+#f #incr #n (elim n) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_to_lt_to_lt.def(4)"\ 6le_to_lt_to_lt\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
 qed.
 
 theorem increasing_to_le: ∀f:\ 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/increasing.def(2)"\ 6increasing\ 5/a\ 6 f → ∀m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6i.m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 f i.
-#f #incr #m (elim m) /2/#n * #a #lenfa
-@(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 ?? (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 a)) /2/
+#f #incr #m (elim m) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/#n * #a #lenfa
+@(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 ?? (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 a)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_to_lt_to_lt.def(4)"\ 6le_to_lt_to_lt\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
 qed.
 
 theorem increasing_to_le2: ∀f:\ 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/increasing.def(2)"\ 6increasing\ 5/a\ 6 f → 
   ∀m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. f \ 5a title="natural number" href="cic:/fakeuri.def(1)"\ 60\ 5/a\ 6 \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → \ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6i. f i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 m \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 f (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 i).
 #f #incr #m #lem (elim lem)
-  [@(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 ? ? \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6) /2/
-  |#n #len * #a * #len #ltnr (cases(\ 5a href="cic:/matita/arithmetics/nat/le_to_or_lt_eq.def(6)"\ 6le_to_or_lt_eq\ 5/a\ 6 … ltnr)) #H
-    [@(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 ? ? a) % /2/ 
+  [@(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 ? ? \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 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/
+  |#n #len * #a * #len #ltnr (cases(\ 5a href="cic:/matita/arithmetics/nat/le_to_or_lt_eq.def(5)"\ 6le_to_or_lt_eq\ 5/a\ 6 … ltnr)) #H
+    [@(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 ? ? a) % /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le.con(0,2,1)"\ 6le_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ 
     |@(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 ? ? (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 a)) % //
     ]
   ]
@@ -488,11 +488,11 @@ qed.
 theorem increasing_to_injective: ∀f:\ 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/increasing.def(2)"\ 6increasing\ 5/a\ 6 f → \ 5a href="cic:/matita/basics/relations/injective.def(1)"\ 6injective\ 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 f.
 #f #incr #n #m cases(\ 5a href="cic:/matita/arithmetics/nat/decidable_le.def(6)"\ 6decidable_le\ 5/a\ 6 n m)
-  [#lenm cases(\ 5a href="cic:/matita/arithmetics/nat/le_to_or_lt_eq.def(6)"\ 6le_to_or_lt_eq\ 5/a\ 6 … lenm) //
+  [#lenm cases(\ 5a href="cic:/matita/arithmetics/nat/le_to_or_lt_eq.def(5)"\ 6le_to_or_lt_eq\ 5/a\ 6 … lenm) //
    #lenm #eqf @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 @(\ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6 … eqf) @\ 5a href="cic:/matita/arithmetics/nat/lt_to_not_eq.def(7)"\ 6lt_to_not_eq\ 5/a\ 6 
    @\ 5a href="cic:/matita/arithmetics/nat/increasing_to_monotonic.def(4)"\ 6increasing_to_monotonic\ 5/a\ 6 //
   |#nlenm #eqf @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 @(\ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6 … eqf) @\ 5a href="cic:/matita/basics/logic/sym_not_eq.def(4)"\ 6sym_not_eq\ 5/a\ 6 
-   @\ 5a href="cic:/matita/arithmetics/nat/lt_to_not_eq.def(7)"\ 6lt_to_not_eq\ 5/a\ 6 @\ 5a href="cic:/matita/arithmetics/nat/increasing_to_monotonic.def(4)"\ 6increasing_to_monotonic\ 5/a\ 6 /2/
+   @\ 5a href="cic:/matita/arithmetics/nat/lt_to_not_eq.def(7)"\ 6lt_to_not_eq\ 5/a\ 6 @\ 5a href="cic:/matita/arithmetics/nat/increasing_to_monotonic.def(4)"\ 6increasing_to_monotonic\ 5/a\ 6 /\ 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.
 
@@ -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 /2/ 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
@@ -508,7 +508,7 @@ theorem le_plus_r: ∀p,n,m:nat. n ≤ m → p + n ≤ p + m
 
 theorem monotonic_le_plus_l: 
 ∀m:\ 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 (λn.n \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 m).
-/2/ qed.
+/\ 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 le_plus_l: \forall p,n,m:nat. n \le m \to n + p \le m + p
@@ -517,34 +517,34 @@ theorem le_plus_l: \forall p,n,m:nat. n \le m \to n + p \le m + p
 theorem le_plus: ∀n1,n2,m1,m2:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n1 \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n2  → m1 \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m2 
 → n1 \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 m1 \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n2 \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 m2.
 #n1 #n2 #m1 #m2 #len #lem @(\ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6 ? (n1\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m2))
-/2/ qed.
+/\ 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, \ 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 le_plus_n :∀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 \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 m.
-/2/ qed. 
+/\ 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/ qed. 
 
 lemma le_plus_a: ∀a,n,m. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 m.
-/2/ qed.
+/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_plus.def(7)"\ 6le_plus\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
 
 lemma le_plus_b: ∀b,n,m. n \ 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 m → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
-/2/ qed.
+/\ 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.
-/2/ 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 le_plus_to_le: ∀a,n,m. a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 m → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
-#a (elim a) normalize /3/ qed. 
+#a (elim a) normalize /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_pred.def(4)"\ 6monotonic_pred\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed. 
 
 theorem le_plus_to_le_r: ∀a,n,m. n \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 a \ 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\ 6a → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
-/2/ qed. 
+/\ 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. 
 
 (* plus &amp; lt *)
 
 theorem monotonic_lt_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/lt.def(1)"\ 6lt\ 5/a\ 6 (λm.n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m).
-/2/ qed.
+/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/increasing_to_monotonic.def(4)"\ 6increasing_to_monotonic\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
 
 (*
 variant lt_plus_r: \forall n,p,q:nat. p < q \to n + p < n + q \def
@@ -552,7 +552,7 @@ monotonic_lt_plus_r. *)
 
 theorem monotonic_lt_plus_l: 
 ∀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/lt.def(1)"\ 6lt\ 5/a\ 6 (λm.m\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6n).
-/2/ qed.
+/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/increasing_to_monotonic.def(4)"\ 6increasing_to_monotonic\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
 
 (*
 variant lt_plus_l: \forall n,p,q:nat. p < q \to p + n < q + n \def
@@ -560,13 +560,13 @@ 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))/2/ 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_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.
 
 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.
-/2/ qed.
+/\ 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.
 
 theorem lt_plus_to_lt_r :∀n,p,q:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6p \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6q → p\ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6q.
-/2/ qed.
+/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/lt_plus_to_lt_l.def(6)"\ 6lt_plus_to_lt_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
 
 (*
 theorem le_to_lt_to_lt_plus: ∀a,b,c,d:nat.
@@ -599,21 +599,21 @@ theorem le_times_l: \forall p,n,m:nat. n \le m \to n*p \le m*p
 
 theorem le_times: ∀n1,n2,m1,m2:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. 
 n1 \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n2  → m1 \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m2 → n1\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m1 \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n2\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m2.
-#n1 #n2 #m1 #m2 #len #lem @(\ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6 ? (n1\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m2)) /2/
+#n1 #n2 #m1 #m2 #len #lem @(\ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6 ? (n1\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m2)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_times_r.def(8)"\ 6monotonic_le_times_r\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
 qed.
 
 (* interessante *)
 theorem lt_times_n: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m.
-#n #m #H /2/ qed.
+#n #m #H /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_times_r.def(8)"\ 6monotonic_le_times_r\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
 
 theorem le_times_to_le: 
 ∀a,n,m. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 a → a \ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 a \ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 m → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
 #a @\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6 normalize
   [//
   |#n #H1 #H2 
-     @(\ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6 ? (a\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n)) /2/
+     @(\ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6 ? (a\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_times_r.def(8)"\ 6monotonic_le_times_r\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
   |#n #m #H #lta #le
-     @\ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6 @H /2/
+     @\ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6 @H /\ 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.
 
@@ -687,39 +687,39 @@ theorem monotonic_lt_times_r:
   ∀c:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 c → \ 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/lt.def(1)"\ 6lt\ 5/a\ 6 (λt.(c\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6t)).
 #c #posc #n #m #ltnm
 (elim ltnm) normalize
-  [/2/ 
+  [/\ 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\ 5/span\ 6\ 5/span\ 6/ 
   |#a #_ #lt1 @(\ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6 … lt1) //
   ]
 qed.
 
 theorem monotonic_lt_times_l: 
   ∀c:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 c → \ 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/lt.def(1)"\ 6lt\ 5/a\ 6 (λt.(t\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6c)).
-/2/
+/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_lt_times_r.def(10)"\ 6monotonic_lt_times_r\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
 qed.
 
 theorem lt_to_le_to_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\ 6 m → p \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 q → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 q → 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 #lepq #posq
-@(\ 5a href="cic:/matita/arithmetics/nat/le_to_lt_to_lt.def(5)"\ 6le_to_lt_to_lt\ 5/a\ 6 ? (n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6q))
+@(\ 5a href="cic:/matita/arithmetics/nat/le_to_lt_to_lt.def(4)"\ 6le_to_lt_to_lt\ 5/a\ 6 ? (n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6q))
   [@\ 5a href="cic:/matita/arithmetics/nat/monotonic_le_times_r.def(8)"\ 6monotonic_le_times_r\ 5/a\ 6 //
-  |@\ 5a href="cic:/matita/arithmetics/nat/monotonic_lt_times_l.def(10)"\ 6monotonic_lt_times_l\ 5/a\ 6 //
+  |@\ 5a href="cic:/matita/arithmetics/nat/monotonic_lt_times_l.def(11)"\ 6monotonic_lt_times_l\ 5/a\ 6 //
   ]
 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(11)"\ 6lt_to_le_to_lt_times\ 5/a\ 6\ 5span style="text-decoration: underline;"\ 6 \ 5/span\ 6/2/
+#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/
 qed.
 
 theorem lt_times_n_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 times" 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 times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6n → p \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 q.
 #n #p #q #Hlt (elim (\ 5a href="cic:/matita/arithmetics/nat/decidable_lt.def(7)"\ 6decidable_lt\ 5/a\ 6 p q)) //
 #nltpq @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 @(\ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6 ? ? (\ 5a href="cic:/matita/arithmetics/nat/lt_to_not_le.def(7)"\ 6lt_to_not_le\ 5/a\ 6 ? ? Hlt))
-applyS \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_times_r.def(8)"\ 6monotonic_le_times_r\ 5/a\ 6 /2/
+applyS \ 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/not_lt_to_le.def(6)"\ 6not_lt_to_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
 qed.
 
 theorem lt_times_n_to_lt_r: 
 ∀n,p,q:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. 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 n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6q → p \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 q.
-/2/ qed.
+/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/lt_times_n_to_lt_l.def(9)"\ 6lt_times_n_to_lt_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
 
 (*
 theorem nat_compare_times_l : \forall n,p,q:nat. 
@@ -797,8 +797,8 @@ theorem minus_Sn_m: ∀m,n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6
 #n #m #lenm nelim lenm napplyS refl_eq. *)
 @\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6 
   [//
-  |#n #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /2/ 
-  |#n #m #Hind #c applyS Hind /2/
+  |#n #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/ 
+  |#n #m #Hind #c applyS Hind /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_pred.def(4)"\ 6monotonic_pred\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
   ]
 qed.
 
@@ -818,20 +818,20 @@ theorem plus_minus:
 ∀m,n,p:\ 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="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m)\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6p \ 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\ 6p)\ 5a title="natural minus" 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 #p #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /2/
-  |normalize/3/
+  |#n #p #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/
+  |normalize/\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_pred.def(4)"\ 6monotonic_pred\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
   ]
 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.
-/2/ qed.
+/\ 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.
 
 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.
-#n #m #lemn @\ 5a href="cic:/matita/basics/logic/sym_eq.def(2)"\ 6sym_eq\ 5/a\ 6 /2/ qed.
+#n #m #lemn @\ 5a href="cic:/matita/basics/logic/sym_eq.def(2)"\ 6sym_eq\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/plus_minus.def(5)"\ 6plus_minus\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
 
 theorem le_plus_minus_m_m: ∀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 (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.
-#n (elim n) // #a #Hind #m (cases m) // normalize #n/2/  
+#n (elim n) // #a #Hind #m (cases m) // normalize #n/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/  
 qed.
 
 theorem minus_to_plus :∀n,m,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.
@@ -845,7 +845,7 @@ qed.
 
 theorem minus_pred_pred : ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → 
 \ 5a href="cic:/matita/arithmetics/nat/pred.def(1)"\ 6pred\ 5/a\ 6 n \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/pred.def(1)"\ 6pred\ 5/a\ 6 m \ 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\ 6 m.
-#n #m #posn #posm @(\ 5a href="cic:/matita/arithmetics/nat/lt_O_n_elim.def(7)"\ 6lt_O_n_elim\ 5/a\ 6 n posn) @(\ 5a href="cic:/matita/arithmetics/nat/lt_O_n_elim.def(7)"\ 6lt_O_n_elim\ 5/a\ 6 m posm) //.
+#n #m #posn #posm @(\ 5a href="cic:/matita/arithmetics/nat/lt_O_n_elim.def(4)"\ 6lt_O_n_elim\ 5/a\ 6 n posn) @(\ 5a href="cic:/matita/arithmetics/nat/lt_O_n_elim.def(4)"\ 6lt_O_n_elim\ 5/a\ 6 m posm) //.
 qed.
 
 
@@ -905,62 +905,62 @@ theorem monotonic_le_minus_l:
 @\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6 #p #q
   [#lePO @(\ 5a href="cic:/matita/arithmetics/nat/le_n_O_elim.def(4)"\ 6le_n_O_elim\ 5/a\ 6 ? lePO) //
   |//
-  |#Hind #n (cases n) // #a #leSS @Hind /2/
+  |#Hind #n (cases n) // #a #leSS @Hind /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_pred.def(4)"\ 6monotonic_pred\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
   ]
 qed.
 
 theorem le_minus_to_plus: ∀n,m,p. 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\ 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 #m #p #lep @\ 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(8)"\ 6le_plus_minus_m_m\ 5/a\ 6 | @\ 5a href="cic:/matita/arithmetics/nat/monotonic_le_plus_l.def(6)"\ 6monotonic_le_plus_l\ 5/a\ 6 // ]
+  [|@\ 5a href="cic:/matita/arithmetics/nat/le_plus_minus_m_m.def(9)"\ 6le_plus_minus_m_m\ 5/a\ 6 | @\ 5a href="cic:/matita/arithmetics/nat/monotonic_le_plus_l.def(6)"\ 6monotonic_le_plus_l\ 5/a\ 6 // ]
 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) /2/
+#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/
 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 /2/ 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(10)"\ 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) /2/
+#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: ∀a,b,c. a \ 5a title="natural minus" 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 plus" 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/lt_to_not_le.def(7)"\ 6lt_to_not_le\ 5/a\ 6 …H)) /2/
+@(\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 … (\ 5a href="cic:/matita/arithmetics/nat/lt_to_not_le.def(7)"\ 6lt_to_not_le\ 5/a\ 6 …H)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_plus_to_minus_r.def(10)"\ 6le_plus_to_minus_r\ 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(9)"\ 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(11)"\ 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(9)"\ 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(11)"\ 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.
-#a #b #c #H @\ 5a href="cic:/matita/arithmetics/nat/le_plus_to_minus_r.def(9)"\ 6le_plus_to_minus_r\ 5/a\ 6 //
+#a #b #c #H @\ 5a href="cic:/matita/arithmetics/nat/le_plus_to_minus_r.def(10)"\ 6le_plus_to_minus_r\ 5/a\ 6 //
 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(9)"\ 6le_plus_to_minus\ 5/a\ 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(8)"\ 6le_plus_minus_m_m\ 5/a\ 6 ? q)) /2/
+#p #q #n #lepq @\ 5a href="cic:/matita/arithmetics/nat/le_plus_to_minus.def(11)"\ 6le_plus_to_minus\ 5/a\ 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)) /2/
+#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/
 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(11)"\ 6eq_minus_O\ 5/a\ 6 /2/ >\ 5a href="cic:/matita/arithmetics/nat/eq_minus_O.def(11)"\ 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 /2/
+ [> \ 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/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) /2/
+  @\ 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/
 qed.
 
 theorem minus_plus: ∀n,m,p. n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6p \ 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\ 6(m\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6p).
@@ -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(11)"\ 6eq_minus_O\ 5/a\ 6 /2/ >\ 5a href="cic:/matita/arithmetics/nat/eq_minus_O.def(11)"\ 6eq_minus_O\ 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/
   ]
 qed.
 
@@ -979,11 +979,11 @@ theorem plus_minus: ∀n,m,p. p ≤ m → (n+m)-p = n +(m-p).
 >associative_plus <plus_minus_m_m //
 qed.  *)
 
-theorem minus_minus: ∀n,m,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. p \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n →
-  p\ 5a title="natural plus" 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="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\ 6(m\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6p).
+theorem minus_minus: ∀n,m,p:nat. p ≤ m → m ≤ n →
+  p+(n-m) = n-(m-p).
 #n #m #p #lepm #lemn
-@\ 5a href="cic:/matita/basics/logic/sym_eq.def(2)"\ 6sym_eq\ 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/plus_minus_m_m.def(7)"\ 6plus_minus_m_m\ 5/a\ 6 //
-<\ 5a href="cic:/matita/arithmetics/nat/commutative_plus.def(5)"\ 6commutative_plus\ 5/a\ 6 <\ 5a href="cic:/matita/arithmetics/nat/plus_minus_m_m.def(7)"\ 6plus_minus_m_m\ 5/a\ 6 //
+@sym_eq @plus_to_minus <associative_plus <plus_minus_m_m //
+<commutative_plus <plus_minus_m_m //
 qed.
 
 (*********************** boolean arithmetics ********************) 
@@ -991,67 +991,67 @@ include "basics/bool.ma".
 
 let rec eqb n m ≝ 
 match n with 
-  [ O ⇒ match m with [ O ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 | S q ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6] 
-  | S p ⇒ match m with [ O ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 | S q ⇒ eqb p q]
+  [ O ⇒ match m with [ O ⇒ true | S q ⇒ false] 
+  | S p ⇒ match m with [ O ⇒ false | S q ⇒ eqb p q]
   ].
 
-theorem eqb_elim : ∀ n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.∀ P:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 → Prop.
-(n\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6m → (P \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6)) → (n \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 m → (P \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6)) → (P (\ 5a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"\ 6eqb\ 5/a\ 6 n m)). 
-@\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6 
+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/ 
   ] 
 qed.
 
-theorem eqb_n_n: ∀n. \ 5a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"\ 6eqb\ 5/a\ 6 n n \ 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.
+theorem eqb_n_n: ∀n. eqb n n = true.
 #n (elim n) normalize // 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 /2/ 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_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) /2/ 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 eq_to_eqb_true: ∀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 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(0,1,0)"\ 6true\ 5/a\ 6.
+theorem eq_to_eqb_true: ∀n,m:nat.n = m → eqb n m = true.
 // 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 /2/ 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.
 
 let rec leb n m ≝ 
 match n with 
-    [ O ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6
+    [ O ⇒ true
     | (S p) ⇒
        match m with 
-        [ O ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6
+        [ O ⇒ false
              | (S q) ⇒ leb p q]].
 
-theorem leb_elim: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. ∀P:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 → Prop. 
-(n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → P \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6) → (n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 m → P \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6) → P (\ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m).
-@\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6 normalize
+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/
   |#n #m #Hind #P #Pt #Pf @Hind
-    [#lenm @Pt @\ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6 // |#nlenm @Pf /2/ ]
+    [#lenm @Pt @le_S_S // |#nlenm @Pf /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 /2/ 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_false_to_not_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(0,2,0)"\ 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 /2/ qed.
+  leb n m = false → n ≰ m.
+#n #m @leb_elim // #_ #abs @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 /2/ 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 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 /2/ 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 lt_to_leb_false: ∀n,m. m \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n → \ 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.
+theorem lt_to_leb_false: ∀n,m. m < n → leb n m = false.
 /3/ qed.
 
 (* serve anche ltb? 
@@ -1080,40 +1080,40 @@ theorem le_to_ltb_false: ∀n,m. m \le n → ltb n m = false.
 qed. *)
 
 (* min e max *)
-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. \ 5a href="cic:/matita/basics/bool/if_then_else.def(1)"\ 6if_then_else\ 5/a\ 6 ? (\ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m) n m.
+definition min: nat →nat →nat ≝
+λn.λm. if_then_else ? (leb n m) n m.
 
-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. \ 5a href="cic:/matita/basics/bool/if_then_else.def(1)"\ 6if_then_else\ 5/a\ 6 ? (\ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m) m n.
+definition max: nat →nat →nat ≝
+λn.λm. if_then_else ? (leb n m) m n.
 
-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 /2/
-  |#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)) /2/
+lemma commutative_min: commutative ? min.
+#n #m normalize @leb_elim 
+  [@leb_elim normalize /2/
+  |#notle >(le_to_leb_true …) // @(transitive_le ? (S m)) /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 /2/ qed. 
+lemma le_minr: ∀i,n,m. i ≤ min n m → i ≤ m.
+#i #n #m normalize @leb_elim normalize /2/ qed. 
 
-lemma le_minl: ∀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 n.
+lemma le_minl: ∀i,n,m. i ≤ min n m → i ≤ n.
 /2/ qed.
 
-lemma to_min: ∀i,n,m. i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n → i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 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 #n #m #lein #leim normalize (cases (\ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m)) 
+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: \ 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 /2/
-  |#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)) /2/
+lemma commutative_max: commutative ? max.
+#n #m normalize @leb_elim 
+  [@leb_elim normalize /2/
+  |#notle >(le_to_leb_true …) // @(transitive_le ? (S m)) /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 /2/ qed. 
+lemma le_maxl: ∀i,n,m. max n m ≤ i → n ≤ i.
+#i #n #m normalize @leb_elim normalize /2/ qed. 
 
-lemma le_maxr: ∀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 → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i.
+lemma le_maxr: ∀i,n,m. max n m ≤ i → m ≤ i.
 /2/ qed.
 
-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)) 
+lemma to_max: ∀i,n,m. n ≤ i → m ≤ i → max n m ≤ i.
+#i #n #m #leni #lemi normalize (cases (leb n m)) 
 normalize // qed.
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