X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=weblib%2Farithmetics%2Fnat.ma;h=2260506329463e418b25706942c942bd3cf1c481;hb=346e1c6dc1fd23139ea51fa826105fc93c454754;hp=66ed6ccfd98b416f37986bce6e44cf3fd692da0b;hpb=39dbed815d5a62c639ce9bb276fd44809053afb3;p=helm.git

diff --git a/weblib/arithmetics/nat.ma b/weblib/arithmetics/nat.ma
index 66ed6ccfd..226050632 100644
--- a/weblib/arithmetics/nat.ma
+++ b/weblib/arithmetics/nat.ma
@@ -47,7 +47,7 @@ theorem not_eq_n_Sn: ∀n:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"n
 theorem nat_case:
  ∀n:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a.∀P:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a → Prop. 
   (na title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/aa href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"O/a → P a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"O/a) → (∀m:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. na title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a m → P (a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a m)) → P n.
-#n #P (elim n) /span class="autotactic"2span class="autotrace" trace {}/span/span/ qed.
+#n #P (elim n) /span class="autotactic"2span class="autotrace" trace /span/span/ qed.
 
 theorem nat_elim2 :
  ∀R:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a → a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a → Prop.
@@ -55,7 +55,7 @@ theorem nat_elim2 :
   → (∀n:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. R (a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a n) a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"O/a)
   → (∀n,m:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. R n m → R (a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a n) (a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a m))
   → ∀n,m:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. R n m.
-#R #ROn #RSO #RSS #n (elim n) // #n0 #Rn0m #m (cases m) /span class="autotactic"2span class="autotrace" trace {}/span/span/ qed.
+#R #ROn #RSO #RSS #n (elim n) // #n0 #Rn0m #m (cases m) /span class="autotactic"2span class="autotrace" trace /span/span/ qed.
 
 theorem decidable_eq_nat : ∀n,m:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a.a href="cic:/matita/basics/logic/decidable.def(1)"decidable/a (na title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/am).
 @a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"nat_elim2/a #n [ (cases n) /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a/span/span/ | /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a, a href="cic:/matita/basics/logic/sym_not_eq.def(4)"sym_not_eq/a/span/span/ | #m #Hind (cases Hind) /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a, a href="cic:/matita/arithmetics/nat/not_eq_S.def(4)"not_eq_S/a/span/span/]
@@ -500,7 +500,7 @@ qed.
 theorem monotonic_le_plus_r: 
 ∀n:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a.a href="cic:/matita/basics/relations/monotonic.def(1)"monotonic/a a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a a href="cic:/matita/arithmetics/nat/le.ind(1,0,1)"le/a (λm.n a title="natural plus" href="cic:/fakeuri.def(1)"+/a m).
 #n #a #b (elim n) normalize //
-#m #H #leab @a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"le_S_S/a /span class="autotactic"2span class="autotrace" trace {}/span/span/ qed.
+#m #H #leab @a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"le_S_S/a /span class="autotactic"2span class="autotrace" trace /span/span/ 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 a title="natural plus" href="cic:/fakeuri.def(1)"
 /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"transitive_le/a/span/span/ qed.
 
 theorem le_plus_n_r :∀n,m:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. m a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a m a title="natural plus" href="cic:/fakeuri.def(1)"+/a n.
-/span class="autotactic"2span class="autotrace" trace {}/span/span/ qed.
+/span class="autotactic"2span class="autotrace" trace /span/span/ qed.
 
 theorem eq_plus_to_le: ∀n,m,p:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a.na title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/ama title="natural plus" href="cic:/fakeuri.def(1)"+/ap → m a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a n.
 // qed.
@@ -552,7 +552,7 @@ monotonic_lt_plus_r. *)
 
 theorem monotonic_lt_plus_l: 
 ∀n:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a.a href="cic:/matita/basics/relations/monotonic.def(1)"monotonic/a a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a a href="cic:/matita/arithmetics/nat/lt.def(1)"lt/a (λm.ma title="natural plus" href="cic:/fakeuri.def(1)"+/an).
-/span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/increasing_to_monotonic.def(4)"increasing_to_monotonic/a/span/span/ qed.
+(* /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/increasing_to_monotonic.def(4)"increasing_to_monotonic/a/span/span/ *) #n @a href="cic:/matita/arithmetics/nat/increasing_to_monotonic.def(4)"increasing_to_monotonic/a // qed.
 
 (*
 variant lt_plus_l: \forall n,p,q:nat. p < q \to p + n < q + n \def
@@ -560,7 +560,7 @@ monotonic_lt_plus_l. *)
 
 theorem lt_plus: ∀n,m,p,q:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. n a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a m → p a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a q → n a title="natural plus" href="cic:/fakeuri.def(1)"+/a p a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a m a title="natural plus" href="cic:/fakeuri.def(1)"+/a q.
 #n #m #p #q #ltnm #ltpq
-@(a href="cic:/matita/arithmetics/nat/transitive_lt.def(3)"transitive_lt/a ? (na title="natural plus" href="cic:/fakeuri.def(1)"+/aq))/span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/monotonic_lt_plus_l.def(9)"monotonic_lt_plus_l/a, a href="cic:/matita/arithmetics/nat/monotonic_le_plus_r.def(3)"monotonic_le_plus_r/a/span/span/ qed.
+@(a href="cic:/matita/arithmetics/nat/transitive_lt.def(3)"transitive_lt/a ? (na title="natural plus" href="cic:/fakeuri.def(1)"+/aq))/span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/monotonic_le_plus_r.def(3)"monotonic_le_plus_r/a, a href="cic:/matita/arithmetics/nat/monotonic_lt_plus_l.def(9)"monotonic_lt_plus_l/a/span/span/ qed.
 
 theorem lt_plus_to_lt_l :∀n,p,q:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. pa title="natural plus" href="cic:/fakeuri.def(1)"+/an a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a qa title="natural plus" href="cic:/fakeuri.def(1)"+/an → pa title="natural 'less than'" href="cic:/fakeuri.def(1)"</aq.
 /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/le_plus_to_le.def(5)"le_plus_to_le/a/span/span/ qed.
@@ -707,7 +707,7 @@ theorem lt_to_le_to_lt_times:
 qed.
 
 theorem lt_times:∀n,m,p,q:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. na title="natural 'less than'" href="cic:/fakeuri.def(1)"</am → pa title="natural 'less than'" href="cic:/fakeuri.def(1)"</aq → na title="natural times" href="cic:/fakeuri.def(1)"*/ap a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a ma title="natural times" href="cic:/fakeuri.def(1)"*/aq.
-#n #m #p #q #ltnm #ltpq @a href="cic:/matita/arithmetics/nat/lt_to_le_to_lt_times.def(12)"lt_to_le_to_lt_times/aspan style="text-decoration: underline;" /span/span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/le_plus_b.def(8)"le_plus_b/a, a href="cic:/matita/arithmetics/nat/ltn_to_ltO.def(5)"ltn_to_ltO/a/span/span/
+#n #m #p #q #ltnm #ltpq @a href="cic:/matita/arithmetics/nat/lt_to_le_to_lt_times.def(12)"lt_to_le_to_lt_times/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/le_plus_b.def(8)"le_plus_b/a, a href="cic:/matita/arithmetics/nat/ltn_to_ltO.def(5)"ltn_to_ltO/a/span/span/ 
 qed.
 
 theorem lt_times_n_to_lt_l: 
@@ -824,7 +824,7 @@ theorem plus_minus:
 qed.
 
 theorem minus_plus_m_m: ∀n,m:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a.n a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a (na title="natural plus" href="cic:/fakeuri.def(1)"+/am)a title="natural minus" href="cic:/fakeuri.def(1)"-/am.
-/span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/plus_minus.def(5)"plus_minus/a, a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"le_n/a/span/span/ qed.
+/span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"le_n/a, a href="cic:/matita/arithmetics/nat/plus_minus.def(5)"plus_minus/a/span/span/ qed.
 
 theorem plus_minus_m_m: ∀n,m:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a.
   m a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a n → n a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a (na title="natural minus" href="cic:/fakeuri.def(1)"-/am)a title="natural plus" href="cic:/fakeuri.def(1)"+/am.
@@ -915,11 +915,11 @@ theorem le_minus_to_plus: ∀n,m,p. na title="natural minus" href="cic:/fakeuri
 qed.
 
 theorem le_minus_to_plus_r: ∀a,b,c. c a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a b → a a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a b a title="natural minus" href="cic:/fakeuri.def(1)"-/a c → a a title="natural plus" href="cic:/fakeuri.def(1)"+/a c a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a b.
-#a #b #c #Hlecb #H >(a href="cic:/matita/arithmetics/nat/plus_minus_m_m.def(7)"plus_minus_m_m/a … Hlecb) /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/monotonic_le_plus_l.def(6)"monotonic_le_plus_l/a/span/span/
+#a #b #c #Hlecb #H >(a href="cic:/matita/arithmetics/nat/plus_minus_m_m.def(7)"plus_minus_m_m/a … Hlecb) /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/le_minus_to_plus.def(10)"le_minus_to_plus/a/span/span/
 qed.
 
 theorem le_plus_to_minus: ∀n,m,p. n a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a pa title="natural plus" href="cic:/fakeuri.def(1)"+/am → na title="natural minus" href="cic:/fakeuri.def(1)"-/am a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a p.
-#n #m #p #lep /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/monotonic_le_minus_l.def(10)"monotonic_le_minus_l/a/span/span/ qed.
+#n #m #p #lep /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/monotonic_le_minus_l.def(9)"monotonic_le_minus_l/a/span/span/ qed.
 
 theorem le_plus_to_minus_r: ∀a,b,c. a a title="natural plus" href="cic:/fakeuri.def(1)"+/a b a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a c → a a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a c a title="natural minus" href="cic:/fakeuri.def(1)"-/ab.
 #a #b #c #H @(a href="cic:/matita/arithmetics/nat/le_plus_to_le_r.def(6)"le_plus_to_le_r/a … b) /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"transitive_le/a/span/span/
@@ -931,12 +931,12 @@ theorem lt_minus_to_plus: ∀a,b,c. a a title="natural minus" href="cic:/fakeur
 qed.
 
 theorem lt_minus_to_plus_r: ∀a,b,c. a a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a b a title="natural minus" href="cic:/fakeuri.def(1)"-/a c → a a title="natural plus" href="cic:/fakeuri.def(1)"+/a c a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a b.
-#a #b #c #H @a href="cic:/matita/arithmetics/nat/not_le_to_lt.def(5)"not_le_to_lt/a @(a href="cic:/matita/basics/logic/not_to_not.def(3)"not_to_not/a … (a href="cic:/matita/arithmetics/nat/le_plus_to_minus.def(11)"le_plus_to_minus/a …))
+#a #b #c #H @a href="cic:/matita/arithmetics/nat/not_le_to_lt.def(5)"not_le_to_lt/a @(a href="cic:/matita/basics/logic/not_to_not.def(3)"not_to_not/a … (a href="cic:/matita/arithmetics/nat/le_plus_to_minus.def(10)"le_plus_to_minus/a …))
 @a href="cic:/matita/arithmetics/nat/lt_to_not_le.def(7)"lt_to_not_le/a //
 qed.
 
 theorem lt_plus_to_minus: ∀n,m,p. m a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a n → n a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a pa title="natural plus" href="cic:/fakeuri.def(1)"+/am → na title="natural minus" href="cic:/fakeuri.def(1)"-/am a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a p.
-#n #m #p #lenm #H normalize <a href="cic:/matita/arithmetics/nat/minus_Sn_m.def(5)"minus_Sn_m/a // @a href="cic:/matita/arithmetics/nat/le_plus_to_minus.def(11)"le_plus_to_minus/a //
+#n #m #p #lenm #H normalize <a href="cic:/matita/arithmetics/nat/minus_Sn_m.def(5)"minus_Sn_m/a // @a href="cic:/matita/arithmetics/nat/le_plus_to_minus.def(10)"le_plus_to_minus/a //
 qed.
 
 theorem lt_plus_to_minus_r: ∀a,b,c. a a title="natural plus" href="cic:/fakeuri.def(1)"+/a b a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a c → a a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a c a title="natural minus" href="cic:/fakeuri.def(1)"-/a b.
@@ -945,19 +945,19 @@ qed.
 
 theorem monotonic_le_minus_r: 
 ∀p,q,n:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. q a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a p → na title="natural minus" href="cic:/fakeuri.def(1)"-/ap a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a na title="natural minus" href="cic:/fakeuri.def(1)"-/aq.
-#p #q #n #lepq @a href="cic:/matita/arithmetics/nat/le_plus_to_minus.def(11)"le_plus_to_minus/a
+#p #q #n #lepq @a href="cic:/matita/arithmetics/nat/le_plus_to_minus.def(10)"le_plus_to_minus/a 
 @(a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"transitive_le/a … (a href="cic:/matita/arithmetics/nat/le_plus_minus_m_m.def(9)"le_plus_minus_m_m/a ? q)) /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/monotonic_le_plus_r.def(3)"monotonic_le_plus_r/a/span/span/
 qed.
 
 theorem eq_minus_O: ∀n,m:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a.
   n a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a m → na title="natural minus" href="cic:/fakeuri.def(1)"-/am a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"O/a.
-#n #m #lenm @(a href="cic:/matita/arithmetics/nat/le_n_O_elim.def(4)"le_n_O_elim/a (na title="natural minus" href="cic:/fakeuri.def(1)"-/am)) /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/monotonic_le_minus_r.def(12)"monotonic_le_minus_r/a/span/span/
+#n #m #lenm @(a href="cic:/matita/arithmetics/nat/le_n_O_elim.def(4)"le_n_O_elim/a (na title="natural minus" href="cic:/fakeuri.def(1)"-/am)) /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/monotonic_le_minus_r.def(11)"monotonic_le_minus_r/a/span/span/
 qed.
 
 theorem distributive_times_minus: a href="cic:/matita/basics/relations/distributive.def(1)"distributive/a ? a href="cic:/matita/arithmetics/nat/times.fix(0,0,2)"times/a a href="cic:/matita/arithmetics/nat/minus.fix(0,0,1)"minus/a.
 #a #b #c
 (cases (a href="cic:/matita/arithmetics/nat/decidable_lt.def(7)"decidable_lt/a b c)) #Hbc
- [> a href="cic:/matita/arithmetics/nat/eq_minus_O.def(13)"eq_minus_O/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/le_plus_b.def(8)"le_plus_b/a/span/span/ >a href="cic:/matita/arithmetics/nat/eq_minus_O.def(13)"eq_minus_O/a // 
+ [> a href="cic:/matita/arithmetics/nat/eq_minus_O.def(12)"eq_minus_O/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/le_plus_b.def(8)"le_plus_b/a/span/span/ >a href="cic:/matita/arithmetics/nat/eq_minus_O.def(12)"eq_minus_O/a // 
   @a href="cic:/matita/arithmetics/nat/monotonic_le_times_r.def(8)"monotonic_le_times_r/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/le_plus_b.def(8)"le_plus_b/a/span/span/
  |@a href="cic:/matita/basics/logic/sym_eq.def(2)"sym_eq/a (applyS a href="cic:/matita/arithmetics/nat/plus_to_minus.def(7)"plus_to_minus/a) <a href="cic:/matita/arithmetics/nat/distributive_times_plus.def(7)"distributive_times_plus/a 
   @a href="cic:/matita/basics/logic/eq_f.def(3)"eq_f/a (applyS a href="cic:/matita/arithmetics/nat/plus_minus_m_m.def(7)"plus_minus_m_m/a) /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/not_lt_to_le.def(6)"not_lt_to_le/a/span/span/
@@ -969,7 +969,7 @@ cases (a href="cic:/matita/arithmetics/nat/decidable_le.def(6)"decidable_le/a
   [@a href="cic:/matita/arithmetics/nat/plus_to_minus.def(7)"plus_to_minus/a @a href="cic:/matita/arithmetics/nat/plus_to_minus.def(7)"plus_to_minus/a <a href="cic:/matita/arithmetics/nat/associative_plus.def(4)"associative_plus/a
    @a href="cic:/matita/arithmetics/nat/minus_to_plus.def(8)"minus_to_plus/a //
   |cut (n a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a ma title="natural plus" href="cic:/fakeuri.def(1)"+/ap) [@(a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"transitive_le/a … (a href="cic:/matita/arithmetics/nat/le_n_Sn.def(1)"le_n_Sn/a …)) @a href="cic:/matita/arithmetics/nat/not_le_to_lt.def(5)"not_le_to_lt/a //]
-   #H >a href="cic:/matita/arithmetics/nat/eq_minus_O.def(13)"eq_minus_O/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/eq_minus_O.def(13)"eq_minus_O/a, a href="cic:/matita/arithmetics/nat/monotonic_le_minus_l.def(10)"monotonic_le_minus_l/a/span/span/
+   #H >a href="cic:/matita/arithmetics/nat/eq_minus_O.def(12)"eq_minus_O/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/eq_minus_O.def(12)"eq_minus_O/a, a href="cic:/matita/arithmetics/nat/monotonic_le_minus_l.def(9)"monotonic_le_minus_l/a/span/span/ 
   ]
 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:nat. p ≤ m → m ≤ n →
-  p+(n-m) = n-(m-p).
+theorem minus_minus: ∀n,m,p:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. p a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a m → m a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a n →
+  pa title="natural plus" href="cic:/fakeuri.def(1)"+/a(na title="natural minus" href="cic:/fakeuri.def(1)"-/am) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a na title="natural minus" href="cic:/fakeuri.def(1)"-/a(ma title="natural minus" href="cic:/fakeuri.def(1)"-/ap).
 #n #m #p #lepm #lemn
-@sym_eq @plus_to_minus <associative_plus <plus_minus_m_m //
-<commutative_plus <plus_minus_m_m //
+@a href="cic:/matita/basics/logic/sym_eq.def(2)"sym_eq/a @a href="cic:/matita/arithmetics/nat/plus_to_minus.def(7)"plus_to_minus/a <a href="cic:/matita/arithmetics/nat/associative_plus.def(4)"associative_plus/a <a href="cic:/matita/arithmetics/nat/plus_minus_m_m.def(7)"plus_minus_m_m/a //
+<a href="cic:/matita/arithmetics/nat/commutative_plus.def(5)"commutative_plus/a <a href="cic:/matita/arithmetics/nat/plus_minus_m_m.def(7)"plus_minus_m_m/a //
 qed.
 
 (*********************** boolean arithmetics ********************) 
@@ -991,68 +991,68 @@ include "basics/bool.ma".
 
 let rec eqb n m ≝ 
 match n with 
-  [ O ⇒ match m with [ O ⇒ true | S q ⇒ false] 
-  | S p ⇒ match m with [ O ⇒ false | S q ⇒ eqb p q]
+  [ O ⇒ match m with [ O ⇒ a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a | S q ⇒ a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a] 
+  | S p ⇒ match m with [ O ⇒ a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a | S q ⇒ eqb p q]
   ].
 
-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/ 
+theorem eqb_elim : ∀ n,m:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a.∀ P:a href="cic:/matita/basics/bool/bool.ind(1,0,0)"bool/a → Prop.
+(na title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/am → (P a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a)) → (n a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"≠/a m → (P a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a)) → (P (a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"eqb/a n m)). 
+@a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"nat_elim2/a 
+  [#n (cases n) normalize /span class="autotactic"3span class="autotrace" trace /span/span/ 
+  |normalize /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/sym_not_eq.def(4)"sym_not_eq/a/span/span/
+  |normalize /span class="autotactic"4span class="autotrace" trace a href="cic:/matita/arithmetics/nat/not_eq_S.def(4)"not_eq_S/a/span/span/ 
   ] 
 qed.
 
-theorem eqb_n_n: ∀n. eqb n n = true.
+theorem eqb_n_n: ∀n. a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"eqb/a n n a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a.
 #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:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"eqb/a n m a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a → n a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a m.
+#n #m @(a href="cic:/matita/arithmetics/nat/eqb_elim.def(5)"eqb_elim/a n m) // #_ #abs @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/absurd.def(2)"absurd/a/span/span/ 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:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"eqb/a n m a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a → n a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"≠/a m.
+#n #m @(a href="cic:/matita/arithmetics/nat/eqb_elim.def(5)"eqb_elim/a n m) /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/not_to_not.def(3)"not_to_not/a/span/span/ qed.
 
-theorem eq_to_eqb_true: ∀n,m:nat.n = m → eqb n m = true.
+theorem eq_to_eqb_true: ∀n,m:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a.n a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a m → a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"eqb/a n m a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a.
 // 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:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a.
+  n a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"≠/a  m → a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"eqb/a n m a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a.
+#n #m #noteq @a href="cic:/matita/arithmetics/nat/eqb_elim.def(5)"eqb_elim/a// #Heq @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/absurd.def(2)"absurd/a/span/span/ qed.
 
 let rec leb n m ≝ 
 match n with 
-    [ O ⇒ true
+    [ O ⇒ a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a
     | (S p) ⇒
 	match m with 
-        [ O ⇒ false
+        [ O ⇒ a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a
 	      | (S q) ⇒ leb p q]].
 
-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/
+theorem leb_elim: ∀n,m:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. ∀P:a href="cic:/matita/basics/bool/bool.ind(1,0,0)"bool/a → Prop. 
+(n a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a m → P a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a) → (n a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"≰/a m → P a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a) → P (a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"leb/a n m).
+@a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"nat_elim2/a normalize
+  [/span class="autotactic"2span class="autotrace" trace /span/span/
+  |/span class="autotactic"3span class="autotrace" trace /span/span/
   |#n #m #Hind #P #Pt #Pf @Hind
-    [#lenm @Pt @le_S_S // |#nlenm @Pf /2/ ]
+    [#lenm @Pt @a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"le_S_S/a // |#nlenm @Pf /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/not_le_to_not_le_S_S.def(5)"not_le_to_not_le_S_S/a/span/span/ ]
   ]
 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.a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"leb/a n m a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a → n a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a m.
+#n #m @a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"leb_elim/a // #_ #abs @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/absurd.def(2)"absurd/a/span/span/ qed.
 
 theorem leb_false_to_not_le:∀n,m.
-  leb n m = false → n ≰ m.
-#n #m @leb_elim // #_ #abs @False_ind /2/ qed.
+  a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"leb/a n m a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a → n a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"≰/a m.
+#n #m @a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"leb_elim/a // #_ #abs @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/absurd.def(2)"absurd/a/span/span/ 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 a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a m → a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"leb/a n m a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a.
+#n #m @a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"leb_elim/a // #H #H1 @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/absurd.def(2)"absurd/a/span/span/ 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 a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"≰/a m → a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"leb/a n m a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a.
+#n #m @a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"leb_elim/a // #H #H1 @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/absurd.def(2)"absurd/a/span/span/ qed.
 
-theorem lt_to_leb_false: ∀n,m. m < n → leb n m = false.
-/3/ qed.
+theorem lt_to_leb_false: ∀n,m. m a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a n → a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"leb/a n m a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a.
+/span class="autotactic"3span class="autotrace" trace a href="cic:/matita/arithmetics/nat/lt_to_not_le.def(7)"lt_to_not_le/a, a href="cic:/matita/arithmetics/nat/not_le_to_leb_false.def(7)"not_le_to_leb_false/a/span/span/ 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: a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a →a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a →a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a ≝
+λn.λm. if (a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"leb/a n m) then n else m.
 
-definition max: nat →nat →nat ≝
-λn.λm. if_then_else ? (leb n m) m n.
+definition max: a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a →a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a →a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a ≝
+λn.λm. if (a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"leb/a 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: a href="cic:/matita/basics/relations/commutative.def(1)"commutative/a ? a href="cic:/matita/arithmetics/nat/min.def(2)"min/a.
+#n #m normalize @a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"leb_elim/a 
+  [@a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"leb_elim/a normalize /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/le_to_le_to_eq.def(5)"le_to_le_to_eq/a/span/span/
+  |#notle >(a href="cic:/matita/arithmetics/nat/le_to_leb_true.def(7)"le_to_leb_true/a …) // @(a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"transitive_le/a ? (a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a m)) /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/not_le_to_lt.def(5)"not_le_to_lt/a/span/span/
   ] 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 a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a a href="cic:/matita/arithmetics/nat/min.def(2)"min/a n m → i a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a m.
+#i #n #m normalize @a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"leb_elim/a normalize /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"transitive_le/a/span/span/ qed. 
 
-lemma le_minl: ∀i,n,m. i ≤ min n m → i ≤ n.
-/2/ qed.
+lemma le_minl: ∀i,n,m. i a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a a href="cic:/matita/arithmetics/nat/min.def(2)"min/a n m → i a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a n.
+/span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/le_minr.def(7)"le_minr/a/span/span/ 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)) 
+lemma to_min: ∀i,n,m. i a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a n → i a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a m → i a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a a href="cic:/matita/arithmetics/nat/min.def(2)"min/a n m.
+#i #n #m #lein #leim normalize (cases (a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"leb/a 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: a href="cic:/matita/basics/relations/commutative.def(1)"commutative/a ? a href="cic:/matita/arithmetics/nat/max.def(2)"max/a.
+#n #m normalize @a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"leb_elim/a 
+  [@a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"leb_elim/a normalize /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/le_to_le_to_eq.def(5)"le_to_le_to_eq/a/span/span/
+  |#notle >(a href="cic:/matita/arithmetics/nat/le_to_leb_true.def(7)"le_to_leb_true/a …) // @(a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"transitive_le/a ? (a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a m)) /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/not_le_to_lt.def(5)"not_le_to_lt/a/span/span/
   ] 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. a href="cic:/matita/arithmetics/nat/max.def(2)"max/a n m a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a i → n a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a i.
+#i #n #m normalize @a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"leb_elim/a normalize /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"transitive_le/a/span/span/ qed. 
 
-lemma le_maxr: ∀i,n,m. max n m ≤ i → m ≤ i.
-/2/ qed.
+lemma le_maxr: ∀i,n,m. a href="cic:/matita/arithmetics/nat/max.def(2)"max/a n m a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a i → m a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a i.
+/span class="autotactic"2span class="autotrace" trace a href="cic:/matita/arithmetics/nat/le_maxl.def(7)"le_maxl/a/span/span/ 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 a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a i → m a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a i → a href="cic:/matita/arithmetics/nat/max.def(2)"max/a n m a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a i.
+#i #n #m #leni #lemi normalize (cases (a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"leb/a n m)) 
 normalize // qed.
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