X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=inline;f=weblib%2Farithmetics%2Fnat.ma;h=ef8da2f1921c2d3f0510765e8b62f6306b49e8ab;hb=7e2f8faa727b72aeebc6eb5a49ca843dcb50dfe0;hp=f14392e9363b16d8b82eea79a35fcc25eafc7ccc;hpb=b743e4f63f52be49ceb01646c8f1574d89817118;p=helm.git

diff --git a/weblib/arithmetics/nat.ma b/weblib/arithmetics/nat.ma
index f14392e93..ef8da2f19 100644
--- a/weblib/arithmetics/nat.ma
+++ b/weblib/arithmetics/nat.ma
@@ -389,7 +389,7 @@ apply (not_le_Sn_n ? H3).
 qed. *)
 
 theorem not_eq_to_le_to_lt: ∀n,m. na title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"≠/am → na title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/am → na title="natural 'less than'" href="cic:/fakeuri.def(1)"</am.
-#n #m #Hneq #Hle cases (a href="cic:/matita/arithmetics/nat/le_to_or_lt_eq.def(5)"le_to_or_lt_eq/a ?? Hle) //
+#n #m #Hneq #Hle cases (a href="cic:/matita/arithmetics/nat/le_to_or_lt_eq.def(6)"le_to_or_lt_eq/a ?? Hle) //
 #Heq /3/ qed.
 (*
 nelim (Hneq Heq) qed. *)
@@ -478,7 +478,7 @@ theorem increasing_to_le2: ∀f:a href="cic:/matita/arithmetics/nat/nat.ind(1,0
   ∀m:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. f a title="natural number" href="cic:/fakeuri.def(1)"0/a a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a m → a title="exists" href="cic:/fakeuri.def(1)"∃/ai. f i a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a m a title="logical and" href="cic:/fakeuri.def(1)"∧/a m a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a f (a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a i).
 #f #incr #m #lem (elim lem)
   [@(a href="cic:/matita/basics/logic/ex.con(0,1,2)"ex_intro/a ? ? a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"O/a) /2/
-  |#n #len * #a * #len #ltnr (cases(a href="cic:/matita/arithmetics/nat/le_to_or_lt_eq.def(5)"le_to_or_lt_eq/a … ltnr)) #H
+  |#n #len * #a * #len #ltnr (cases(a href="cic:/matita/arithmetics/nat/le_to_or_lt_eq.def(6)"le_to_or_lt_eq/a … ltnr)) #H
     [@(a href="cic:/matita/basics/logic/ex.con(0,1,2)"ex_intro/a ? ? a) % /2/ 
     |@(a href="cic:/matita/basics/logic/ex.con(0,1,2)"ex_intro/a ? ? (a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a a)) % //
     ]
@@ -488,7 +488,7 @@ qed.
 theorem increasing_to_injective: ∀f: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/increasing.def(2)"increasing/a f → a href="cic:/matita/basics/relations/injective.def(1)"injective/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 f.
 #f #incr #n #m cases(a href="cic:/matita/arithmetics/nat/decidable_le.def(6)"decidable_le/a n m)
-  [#lenm cases(a href="cic:/matita/arithmetics/nat/le_to_or_lt_eq.def(5)"le_to_or_lt_eq/a … lenm) //
+  [#lenm cases(a href="cic:/matita/arithmetics/nat/le_to_or_lt_eq.def(6)"le_to_or_lt_eq/a … lenm) //
    #lenm #eqf @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a @(a href="cic:/matita/basics/logic/absurd.def(2)"absurd/a … eqf) @a href="cic:/matita/arithmetics/nat/lt_to_not_eq.def(7)"lt_to_not_eq/a 
    @a href="cic:/matita/arithmetics/nat/increasing_to_monotonic.def(4)"increasing_to_monotonic/a //
   |#nlenm #eqf @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a @(a href="cic:/matita/basics/logic/absurd.def(2)"absurd/a … eqf) @a href="cic:/matita/basics/logic/sym_not_eq.def(4)"sym_not_eq/a 
@@ -700,14 +700,14 @@ qed.
 theorem lt_to_le_to_lt_times: 
 ∀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 or equal to'" href="cic:/fakeuri.def(1)"≤/a q → a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"O/a a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a q → 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 #lepq #posq
-@(a href="cic:/matita/arithmetics/nat/le_to_lt_to_lt.def(4)"le_to_lt_to_lt/a ? (na title="natural times" href="cic:/fakeuri.def(1)"*/aq))
+@(a href="cic:/matita/arithmetics/nat/le_to_lt_to_lt.def(5)"le_to_lt_to_lt/a ? (na title="natural times" href="cic:/fakeuri.def(1)"*/aq))
   [@a href="cic:/matita/arithmetics/nat/monotonic_le_times_r.def(8)"monotonic_le_times_r/a //
-  |@a href="cic:/matita/arithmetics/nat/monotonic_lt_times_l.def(9)"monotonic_lt_times_l/a //
+  |@a href="cic:/matita/arithmetics/nat/monotonic_lt_times_l.def(10)"monotonic_lt_times_l/a //
   ]
 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(10)"lt_to_le_to_lt_times/a/2/
+#n #m #p #q #ltnm #ltpq @a href="cic:/matita/arithmetics/nat/lt_to_le_to_lt_times.def(11)"lt_to_le_to_lt_times/aspan style="text-decoration: underline;" /span/2/
 qed.
 
 theorem lt_times_n_to_lt_l: 
@@ -811,7 +811,8 @@ napplyS (not_eq_to_le_to_lt n (S m) H H1)
 qed. *)
 
 theorem eq_minus_S_pred: ∀n,m. n a title="natural minus" href="cic:/fakeuri.def(1)"-/a (a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a m) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/arithmetics/nat/pred.def(1)"pred/a(n a title="natural minus" href="cic:/fakeuri.def(1)"-/am).
-@a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"nat_elim2/a normalize // qed.
+@a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"nat_elim2/a normalize //
+qed.
 
 theorem plus_minus:
 ∀m,n,p: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 → (na title="natural minus" href="cic:/fakeuri.def(1)"-/am)a title="natural plus" href="cic:/fakeuri.def(1)"+/ap a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a (na title="natural plus" href="cic:/fakeuri.def(1)"+/ap)a title="natural minus" href="cic:/fakeuri.def(1)"-/am.
@@ -844,7 +845,7 @@ qed.
 
 theorem minus_pred_pred : ∀n,m:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"O/a a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a n → a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"O/a a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a m → 
 a href="cic:/matita/arithmetics/nat/pred.def(1)"pred/a n a title="natural minus" href="cic:/fakeuri.def(1)"-/a a href="cic:/matita/arithmetics/nat/pred.def(1)"pred/a m a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a n a title="natural minus" href="cic:/fakeuri.def(1)"-/a m.
-#n #m #posn #posm @(a href="cic:/matita/arithmetics/nat/lt_O_n_elim.def(4)"lt_O_n_elim/a n posn) @(a href="cic:/matita/arithmetics/nat/lt_O_n_elim.def(4)"lt_O_n_elim/a m posm) //.
+#n #m #posn #posm @(a href="cic:/matita/arithmetics/nat/lt_O_n_elim.def(7)"lt_O_n_elim/a n posn) @(a href="cic:/matita/arithmetics/nat/lt_O_n_elim.def(7)"lt_O_n_elim/a m posm) //.
 qed.
 
 
@@ -910,7 +911,7 @@ qed.
 
 theorem le_minus_to_plus: ∀n,m,p. na title="natural minus" href="cic:/fakeuri.def(1)"-/am a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a p → na title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a pa title="natural plus" href="cic:/fakeuri.def(1)"+/am.
 #n #m #p #lep @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(6)"le_plus_minus_m_m/a | @a href="cic:/matita/arithmetics/nat/monotonic_le_plus_l.def(6)"monotonic_le_plus_l/a // ]
+  [|@a href="cic:/matita/arithmetics/nat/le_plus_minus_m_m.def(8)"le_plus_minus_m_m/a | @a href="cic:/matita/arithmetics/nat/monotonic_le_plus_l.def(6)"monotonic_le_plus_l/a // ]
 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.
@@ -930,22 +931,22 @@ 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(7)"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(9)"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(7)"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(9)"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.
-#a #b #c #H @a href="cic:/matita/arithmetics/nat/le_plus_to_minus_r.def(7)"le_plus_to_minus_r/a //
+#a #b #c #H @a href="cic:/matita/arithmetics/nat/le_plus_to_minus_r.def(9)"le_plus_to_minus_r/a //
 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(7)"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(6)"le_plus_minus_m_m/a ? q)) /2/
+#p #q #n #lepq @a href="cic:/matita/arithmetics/nat/le_plus_to_minus.def(9)"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(8)"le_plus_minus_m_m/a ? q)) /2/
 qed.
 
 theorem eq_minus_O: ∀n,m:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a.
@@ -956,7 +957,7 @@ 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(9)"eq_minus_O/a /2/ >a href="cic:/matita/arithmetics/nat/eq_minus_O.def(9)"eq_minus_O/a // 
+ [> a href="cic:/matita/arithmetics/nat/eq_minus_O.def(11)"eq_minus_O/a /2/ >a href="cic:/matita/arithmetics/nat/eq_minus_O.def(11)"eq_minus_O/a // 
   @a href="cic:/matita/arithmetics/nat/monotonic_le_times_r.def(8)"monotonic_le_times_r/a /2/
  |@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) /2/
@@ -968,7 +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(9)"eq_minus_O/a /2/ >a href="cic:/matita/arithmetics/nat/eq_minus_O.def(9)"eq_minus_O/a // 
+   #H >a href="cic:/matita/arithmetics/nat/eq_minus_O.def(11)"eq_minus_O/a /2/ >a href="cic:/matita/arithmetics/nat/eq_minus_O.def(11)"eq_minus_O/a // 
   ]
 qed.
 
@@ -1115,4 +1116,4 @@ lemma le_maxr: ∀i,n,m. a href="cic:/matita/arithmetics/nat/max.def(2)"max/a
 
 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.
+normalize // qed.
\ No newline at end of file