From b743e4f63f52be49ceb01646c8f1574d89817118 Mon Sep 17 00:00:00 2001
From: matitaweb <claudio.sacerdoticoen@unibo.it>
Date: Mon, 12 Sep 2011 11:36:38 +0000
Subject: [PATCH] commit by user utente1

---
 weblib/arithmetics/nat.ma | 573 +++++++++++++++++++-------------------
 1 file changed, 286 insertions(+), 287 deletions(-)

diff --git a/weblib/arithmetics/nat.ma b/weblib/arithmetics/nat.ma
index 6fa6d57a8..f14392e93 100644
--- a/weblib/arithmetics/nat.ma
+++ b/weblib/arithmetics/nat.ma
@@ -20,55 +20,55 @@ interpretation "Natural numbers" 'N = nat.
 alias num (instance 0) = "natural number".
 
 definition pred ≝
- λn. match n with [ O ⇒ O | S p ⇒ p].
+ λn. match n with [ O ⇒ a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"O/a | S p ⇒ p].
 
-theorem pred_Sn : ∀n.n = pred (S n).
+theorem pred_Sn : ∀n.n a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/arithmetics/nat/pred.def(1)"pred/a (a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a n).
 // qed.
 
-theorem injective_S : injective nat nat S.
+theorem injective_S : 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 a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a.
 // qed.
 
 (*
 theorem inj_S : \forall n,m:nat.(S n)=(S m) \to n=m.
 //. qed. *)
 
-theorem not_eq_S: ∀n,m:nat. n ≠ m → S n ≠ S m.
+theorem not_eq_S: ∀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/nat.con(0,2,0)"S/a n a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"≠/a a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a m.
 /2/ qed.
 
-definition not_zero: nat → Prop ≝
- λn: nat. match n with [ O ⇒ False | (S p) ⇒ True ].
+definition not_zero: a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a → Prop ≝
+ λn: a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. match n with [ O ⇒ a href="cic:/matita/basics/logic/False.ind(1,0,0)"False/a | (S p) ⇒ a href="cic:/matita/basics/logic/True.ind(1,0,0)"True/a ].
 
-theorem not_eq_O_S : ∀n:nat. O ≠ S n.
-#n @nmk #eqOS (change with (not_zero O)) >eqOS // qed.
+theorem not_eq_O_S : ∀n: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="leibnitz's non-equality" href="cic:/fakeuri.def(1)"≠/a a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a n.
+#n @a href="cic:/matita/basics/logic/Not.con(0,1,1)"nmk/a #eqOS (change with (a href="cic:/matita/arithmetics/nat/not_zero.def(1)"not_zero/a a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"O/a)) >eqOS // qed.
 
-theorem not_eq_n_Sn: ∀n:nat. n ≠ S n.
+theorem not_eq_n_Sn: ∀n: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 a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a n.
 #n (elim n) /2/ qed.
 
 theorem nat_case:
- ∀n:nat.∀P:nat → Prop. 
-  (n=O → P O) → (∀m:nat. n= S m → P (S m)) → P n.
+ ∀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) /2/ qed.
 
 theorem nat_elim2 :
- ∀R:nat → nat → Prop.
-  (∀n:nat. R O n) 
-  → (∀n:nat. R (S n) O)
-  → (∀n,m:nat. R n m → R (S n) (S m))
-  → ∀n,m:nat. R n m.
+ ∀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.
+  (∀n:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. R a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"O/a n) 
+  → (∀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) /2/ qed.
 
-theorem decidable_eq_nat : ∀n,m:nat.decidable (n=m).
-@nat_elim2 #n [ (cases n) /2/ | /3/ | #m #Hind (cases Hind) /3/]
+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) /2/ | /3/ | #m #Hind (cases Hind) /3/]
 qed. 
 
 (*************************** plus ******************************)
 
 let rec plus n m ≝ 
- match n with [ O ⇒ m | S p ⇒ S (plus p m) ].
+ match n with [ O ⇒ m | S p ⇒ a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a (plus p m) ].
 
 interpretation "natural plus" 'plus x y = (plus x y).
 
-theorem plus_O_n: ∀n:nat. n = O+n.
+theorem plus_O_n: ∀n: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 a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"O/aa title="natural plus" href="cic:/fakeuri.def(1)"+/an.
 // qed.
 
 (*
@@ -76,10 +76,10 @@ theorem plus_Sn_m: ∀n,m:nat. S (n + m) = S n + m.
 // qed.
 *)
 
-theorem plus_n_O: ∀n:nat. n = n+0.
+theorem plus_n_O: ∀n: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)"+/aa title="natural number" href="cic:/fakeuri.def(1)"0/a.
 #n (elim n) normalize // qed.
 
-theorem plus_n_Sm : ∀n,m:nat. S (n+m) = n + S m.
+theorem plus_n_Sm : ∀n,m:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a (na title="natural plus" href="cic:/fakeuri.def(1)"+/am) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a n a title="natural plus" href="cic:/fakeuri.def(1)"+/a a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a m.
 #n (elim n) normalize // qed.
 
 (*
@@ -92,16 +92,16 @@ theorem plus_n_1 : ∀n:nat. S n = n+1.
 // qed.
 *)
 
-theorem commutative_plus: commutative ? plus.
+theorem commutative_plus: a href="cic:/matita/basics/relations/commutative.def(1)"commutative/a ? a href="cic:/matita/arithmetics/nat/plus.fix(0,0,1)"plus/a.
 #n (elim n) normalize // qed. 
 
-theorem associative_plus : associative nat plus.
+theorem associative_plus : a href="cic:/matita/basics/relations/associative.def(1)"associative/a a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a a href="cic:/matita/arithmetics/nat/plus.fix(0,0,1)"plus/a.
 #n (elim n) normalize // qed.
 
-theorem assoc_plus1: ∀a,b,c. c + (b + a) = b + c + a.
+theorem assoc_plus1: ∀a,b,c. c a title="natural plus" href="cic:/fakeuri.def(1)"+/a (b a title="natural plus" href="cic:/fakeuri.def(1)"+/a a) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a b a title="natural plus" href="cic:/fakeuri.def(1)"+/a c a title="natural plus" href="cic:/fakeuri.def(1)"+/a a.
 // qed. 
 
-theorem injective_plus_r: ∀n:nat.injective nat nat (λm.n+m).
+theorem injective_plus_r: ∀n:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a.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 (λm.na title="natural plus" href="cic:/fakeuri.def(1)"+/am).
 #n (elim n) normalize /3/ qed.
 
 (* theorem inj_plus_r: \forall p,n,m:nat. p+n = p+m \to n=m
@@ -116,42 +116,42 @@ theorem injective_plus_l: ∀m:nat.injective nat nat (λn.n+m).
 (*************************** times *****************************)
 
 let rec times n m ≝ 
- match n with [ O ⇒ O | S p ⇒ m+(times p m) ].
+ match n with [ O ⇒ a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"O/a | S p ⇒ ma title="natural plus" href="cic:/fakeuri.def(1)"+/a(times p m) ].
 
 interpretation "natural times" 'times x y = (times x y).
 
-theorem times_Sn_m: ∀n,m:nat. m+n*m = S n*m.
+theorem times_Sn_m: ∀n,m:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. ma title="natural plus" href="cic:/fakeuri.def(1)"+/ana title="natural times" 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(0,2,0)"S/a na title="natural times" href="cic:/fakeuri.def(1)"*/am.
 // qed.
 
-theorem times_O_n: ∀n:nat. O = O*n.
+theorem times_O_n: ∀n: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="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"O/aa title="natural times" href="cic:/fakeuri.def(1)"*/an.
 // qed.
 
-theorem times_n_O: ∀n:nat. O = n*O.
+theorem times_n_O: ∀n: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="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a na title="natural times" href="cic:/fakeuri.def(1)"*/aa href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"O/a.
 #n (elim n) // qed.
 
-theorem times_n_Sm : ∀n,m:nat. n+(n*m) = n*(S m).
+theorem times_n_Sm : ∀n,m:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. na title="natural plus" href="cic:/fakeuri.def(1)"+/a(na title="natural times" href="cic:/fakeuri.def(1)"*/am) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a na title="natural times" href="cic:/fakeuri.def(1)"*/a(a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a m).
 #n (elim n) normalize // qed.
 
-theorem commutative_times : commutative nat times. 
+theorem commutative_times : a href="cic:/matita/basics/relations/commutative.def(1)"commutative/a a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a a href="cic:/matita/arithmetics/nat/times.fix(0,0,2)"times/a. 
 #n (elim n) normalize // qed. 
 
 (* variant sym_times : \forall n,m:nat. n*m = m*n \def
 symmetric_times. *)
 
-theorem distributive_times_plus : distributive nat times plus.
+theorem distributive_times_plus : a href="cic:/matita/basics/relations/distributive.def(1)"distributive/a a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a a href="cic:/matita/arithmetics/nat/times.fix(0,0,2)"times/a a href="cic:/matita/arithmetics/nat/plus.fix(0,0,1)"plus/a.
 #n (elim n) normalize // qed.
 
 theorem distributive_times_plus_r :
-  ∀a,b,c:nat. (b+c)*a = b*a + c*a.
+  ∀a,b,c:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. (ba title="natural plus" href="cic:/fakeuri.def(1)"+/ac)a title="natural times" href="cic:/fakeuri.def(1)"*/aa a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a ba title="natural times" href="cic:/fakeuri.def(1)"*/aa a title="natural plus" href="cic:/fakeuri.def(1)"+/a ca title="natural times" href="cic:/fakeuri.def(1)"*/aa.
 // qed. 
 
-theorem associative_times: associative nat times.
+theorem associative_times: a href="cic:/matita/basics/relations/associative.def(1)"associative/a a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a a href="cic:/matita/arithmetics/nat/times.fix(0,0,2)"times/a.
 #n (elim n) normalize // qed.
 
-lemma times_times: ∀x,y,z. x*(y*z) = y*(x*z).
+lemma times_times: ∀x,y,z. xa title="natural times" href="cic:/fakeuri.def(1)"*/a(ya title="natural times" href="cic:/fakeuri.def(1)"*/az) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a ya title="natural times" href="cic:/fakeuri.def(1)"*/a(xa title="natural times" href="cic:/fakeuri.def(1)"*/az).
 // qed. 
 
-theorem times_n_1 : ∀n:nat. n = n * 1.
+theorem times_n_1 : ∀n: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 n a title="natural times" href="cic:/fakeuri.def(1)"*/a a title="natural number" href="cic:/fakeuri.def(1)"1/a.
 #n // qed.
 
 (* ci servono questi risultati? 
@@ -182,15 +182,15 @@ qed.
 
 (******************** ordering relations ************************)
 
-inductive le (n:nat) : nat → Prop ≝
+inductive le (n: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 ≝
   | le_n : le n n
-  | le_S : ∀ m:nat. le n m → le n (S m).
+  | le_S : ∀ m:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. le n m → le n (a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a m).
 
 interpretation "natural 'less or equal to'" 'leq x y = (le x y).
 
 interpretation "natural 'neither less nor equal to'" 'nleq x y = (Not (le x y)).
 
-definition lt: nat → nat → Prop ≝ λn,m. S n ≤ m.
+definition lt: 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 ≝ λn,m. a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a n a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a m.
 
 interpretation "natural 'less than'" 'lt x y = (lt x y).
 interpretation "natural 'not less than'" 'nless x y = (Not (lt x y)).
@@ -198,16 +198,16 @@ interpretation "natural 'not less than'" 'nless x y = (Not (lt x y)).
 (* lemma eq_lt: ∀n,m. (n < m) = (S n ≤ m).
 // qed. *)
 
-definition ge: nat → nat → Prop ≝ λn,m.m ≤ n.
+definition ge: 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 ≝ λn,m.m a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a n.
 
 interpretation "natural 'greater or equal to'" 'geq x y = (ge x y).
 
-definition gt: nat → nat → Prop ≝ λn,m.m<n.
+definition gt: 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 ≝ λn,m.ma title="natural 'less than'" href="cic:/fakeuri.def(1)"</an.
 
 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 : transitive nat le.
+theorem transitive_le : a href="cic:/matita/basics/relations/transitive.def(2)"transitive/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.
 #a #b #c #leab #lebc (elim lebc) /2/
 qed.
 
@@ -215,29 +215,29 @@ qed.
 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: transitive nat lt.
+theorem transitive_lt: a href="cic:/matita/basics/relations/transitive.def(2)"transitive/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.
 #a #b #c #ltab #ltbc (elim ltbc) /2/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:nat. n ≤ m → S n ≤ S m.
+theorem le_S_S: ∀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 → a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a n a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a m.
 #n #m #lenm (elim lenm) /2/ qed.
 
-theorem le_O_n : ∀n:nat. O ≤ n.
+theorem le_O_n : ∀n: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 or equal to'" href="cic:/fakeuri.def(1)"≤/a n.
 #n (elim n) /2/ qed.
 
-theorem le_n_Sn : ∀n:nat. n ≤ S n.
+theorem le_n_Sn : ∀n: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 a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a n.
 /2/ qed.
 
-theorem le_pred_n : ∀n:nat. pred n ≤ n.
+theorem le_pred_n : ∀n:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. a href="cic:/matita/arithmetics/nat/pred.def(1)"pred/a n a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a n.
 #n (elim n) // qed.
 
-theorem monotonic_pred: monotonic ? le pred.
+theorem monotonic_pred: a href="cic:/matita/basics/relations/monotonic.def(1)"monotonic/a ? a href="cic:/matita/arithmetics/nat/le.ind(1,0,1)"le/a a href="cic:/matita/arithmetics/nat/pred.def(1)"pred/a.
 #n #m #lenm (elim lenm) /2/ qed.
 
-theorem le_S_S_to_le: ∀n,m:nat. S n ≤ S m → n ≤ m.
+theorem le_S_S_to_le: ∀n,m:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a n a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a m → n a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a m.
 (* demo *)
 /2/ qed.
 
@@ -248,26 +248,26 @@ theorem lt_S_S_to_lt: ∀n,m. S n < S m → n < m.
 theorem lt_to_lt_S_S: ∀n,m. n < m → S n < S m.
 /2/ qed. *)
 
-theorem lt_to_not_zero : ∀n,m:nat. n < m → not_zero m.
+theorem lt_to_not_zero : ∀n,m: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 → a href="cic:/matita/arithmetics/nat/not_zero.def(1)"not_zero/a m.
 #n #m #Hlt (elim Hlt) // qed.
 
 (* lt vs. le *)
-theorem not_le_Sn_O: ∀ n:nat. S n ≰ O.
-#n @nmk #Hlen0 @(lt_to_not_zero ?? Hlen0) qed.
+theorem not_le_Sn_O: ∀ n:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a n a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"≰/a a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"O/a.
+#n @a href="cic:/matita/basics/logic/Not.con(0,1,1)"nmk/a #Hlen0 @(a href="cic:/matita/arithmetics/nat/lt_to_not_zero.def(2)"lt_to_not_zero/a ?? Hlen0) qed.
 
-theorem not_le_to_not_le_S_S: ∀ n,m:nat. n ≰ m → S n ≰ S m.
+theorem not_le_to_not_le_S_S: ∀ n,m:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. n a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"≰/a m → a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a n a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"≰/a a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a m.
 /3/ qed.
 
-theorem not_le_S_S_to_not_le: ∀ n,m:nat. S n ≰ S m → n ≰ m.
+theorem not_le_S_S_to_not_le: ∀ n,m:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a n a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"≰/a a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a m → n a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"≰/a m.
 /3/ qed.
 
-theorem decidable_le: ∀n,m. decidable (n≤m).
-@nat_elim2 #n /2/ #m * /3/ qed.
+theorem decidable_le: ∀n,m. a href="cic:/matita/basics/logic/decidable.def(1)"decidable/a (na title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/am).
+@a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"nat_elim2/a #n /2/ #m * /3/ qed.
 
-theorem decidable_lt: ∀n,m. decidable (n < m).
-#n #m @decidable_le  qed.
+theorem decidable_lt: ∀n,m. a href="cic:/matita/basics/logic/decidable.def(1)"decidable/a (n a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a m).
+#n #m @a href="cic:/matita/arithmetics/nat/decidable_le.def(6)"decidable_le/a  qed.
 
-theorem not_le_Sn_n: ∀n:nat. S n ≰ n.
+theorem not_le_Sn_n: ∀n:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a n a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"≰/a n.
 #n (elim n) /2/ qed.
 
 (* this is le_S_S_to_le
@@ -275,38 +275,38 @@ theorem lt_S_to_le: ∀n,m:nat. n < S m → n ≤ m.
 /2/ qed.
 *)
 
-lemma le_gen: ∀P:nat → Prop.∀n.(∀i. i ≤ n → P i) → P n.
+lemma le_gen: ∀P:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a → Prop.∀n.(∀i. i a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a n → P i) → P n.
 /2/ qed.
 
-theorem not_le_to_lt: ∀n,m. n ≰ m → m < n.
-@nat_elim2 #n
- [#abs @False_ind /2/
+theorem not_le_to_lt: ∀n,m. n a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"≰/a m → m a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a n.
+@a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"nat_elim2/a #n
+ [#abs @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a /2/
  |/2/
- |#m #Hind #HnotleSS @le_S_S /3/
+ |#m #Hind #HnotleSS @a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"le_S_S/a /3/
  ]
 qed.
 
-theorem lt_to_not_le: ∀n,m. n < m → m ≰ n.
+theorem lt_to_not_le: ∀n,m. n a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a m → m a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"≰/a n.
 #n #m #Hltnm (elim Hltnm) /3/ qed.
 
-theorem not_lt_to_le: ∀n,m:nat. n ≮ m → m ≤ n.
+theorem not_lt_to_le: ∀n,m:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. n a title="natural 'not less than'" href="cic:/fakeuri.def(1)"≮/a m → m a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a n.
 /4/ qed.
 
-theorem le_to_not_lt: ∀n,m:nat. n ≤ m → m ≮ n.
-#n #m #H @lt_to_not_le /2/ (* /3/ *) qed.
+theorem le_to_not_lt: ∀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 → m a title="natural 'not less than'" href="cic:/fakeuri.def(1)"≮/a n.
+#n #m #H @a href="cic:/matita/arithmetics/nat/lt_to_not_le.def(7)"lt_to_not_le/a /2/ (* /3/ *) qed.
 
 (* lt and le trans *)
 
-theorem lt_to_le_to_lt: ∀n,m,p:nat. n < m → m ≤ p → n < p.
+theorem lt_to_le_to_lt: ∀n,m,p: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 → m a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a p → n a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a p.
 #n #m #p #H #H1 (elim H1) /2/ qed.
 
-theorem le_to_lt_to_lt: ∀n,m,p:nat. n ≤ m → m < p → n < p.
+theorem le_to_lt_to_lt: ∀n,m,p: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 → m a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a p → n a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a p.
 #n #m #p #H (elim H) /3/ qed.
 
-theorem lt_S_to_lt: ∀n,m. S n < m → n < m.
+theorem lt_S_to_lt: ∀n,m. a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a n a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a m → n a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a m.
 /2/ qed.
 
-theorem ltn_to_ltO: ∀n,m:nat. n < m → O < m.
+theorem ltn_to_ltO: ∀n,m: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 → 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.
 /2/ qed.
 
 (*
@@ -320,12 +320,12 @@ apply (ltn_to_ltO (pred (S O)) (pred n) ?).
  ]
 qed. *)
 
-theorem lt_O_n_elim: ∀n:nat. O < n → 
-  ∀P:nat → Prop.(∀m:nat.P (S m)) → P n.
-#n (elim n) // #abs @False_ind /2/
+theorem lt_O_n_elim: ∀n: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 → 
+  ∀P:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a → Prop.(∀m:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a.P (a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a m)) → P n.
+#n (elim n) // #abs @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a /2/
 qed.
 
-theorem S_pred: ∀n. 0 < n → S(pred n) = n.
+theorem S_pred: ∀n. a title="natural number" href="cic:/fakeuri.def(1)"0/a a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a n → a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a(a href="cic:/matita/arithmetics/nat/pred.def(1)"pred/a n) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a n.
 #n #posn (cases posn) //
 qed.
 
@@ -356,12 +356,12 @@ qed.
 *)
 
 (* le to lt or eq *)
-theorem le_to_or_lt_eq: ∀n,m:nat. n ≤ m → n < m ∨ n = m.
+theorem le_to_or_lt_eq: ∀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 → n a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a m a title="logical or" href="cic:/fakeuri.def(1)"∨/a n a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a m.
 #n #m #lenm (elim lenm) /3/ qed.
 
 (* not eq *)
-theorem lt_to_not_eq : ∀n,m:nat. n < m → n ≠ m.
-#n #m #H @not_to_not /2/ qed.
+theorem lt_to_not_eq : ∀n,m: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 → n a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"≠/a m.
+#n #m #H @a href="cic:/matita/basics/logic/not_to_not.def(3)"not_to_not/a /2/ qed.
 
 (*not lt 
 theorem eq_to_not_lt: ∀a,b:nat. a = b → a ≮ b.
@@ -388,30 +388,30 @@ intro
 apply (not_le_Sn_n ? H3).
 qed. *)
 
-theorem not_eq_to_le_to_lt: ∀n,m. n≠m → n≤m → n<m.
-#n #m #Hneq #Hle cases (le_to_or_lt_eq ?? Hle) //
+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) //
 #Heq /3/ qed.
 (*
 nelim (Hneq Heq) qed. *)
 
 (* le elimination *)
-theorem le_n_O_to_eq : ∀n:nat. n ≤ O → O=n.
-#n (cases n) // #a  #abs @False_ind /2/ qed.
+theorem le_n_O_to_eq : ∀n: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 a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"O/a → a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"O/aa title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/an.
+#n (cases n) // #a  #abs @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a /2/ qed.
 
-theorem le_n_O_elim: ∀n:nat. n ≤ O → ∀P: nat →Prop. P O → P n.
-#n (cases n) // #a #abs @False_ind /2/ qed. 
+theorem le_n_O_elim: ∀n: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 a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"O/a → ∀P: a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a →Prop. P a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"O/a → P n.
+#n (cases n) // #a #abs @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a /2/ qed. 
 
-theorem le_n_Sm_elim : ∀n,m:nat.n ≤ S m → 
-∀P:Prop. (S n ≤ S m → P) → (n=S m → P) → P.
+theorem le_n_Sm_elim : ∀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 a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a m → 
+∀P:Prop. (a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a n a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a m → P) → (na title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/aa href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a m → P) → P.
 #n #m #Hle #P (elim Hle) /3/ qed.
 
 (* le and eq *)
 
-theorem le_to_le_to_eq: ∀n,m. n ≤ m → m ≤ n → n = m.
-@nat_elim2 /4/
+theorem le_to_le_to_eq: ∀n,m. n 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 → n a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a m.
+@a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"nat_elim2/a /4/
 qed. 
 
-theorem lt_O_S : ∀n:nat. O < S n.
+theorem lt_O_S : ∀n: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 a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a n.
 /2/ qed.
 
 (*
@@ -441,109 +441,109 @@ qed.
 
 (* well founded induction principles *)
 
-theorem nat_elim1 : ∀n:nat.∀P:nat → Prop. 
-(∀m.(∀p. p < m → P p) → P m) → P n.
+theorem nat_elim1 : ∀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. 
+(∀m.(∀p. p a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a m → P p) → P m) → P n.
 #n #P #H 
-cut (∀q:nat. q ≤ n → P q) /2/
+cut (∀q: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 n → P q) /2/
 (elim n) 
  [#q #HleO (* applica male *) 
-    @(le_n_O_elim ? HleO)
-    @H #p #ltpO @False_ind /2/ (* 3 *)
+    @(a href="cic:/matita/arithmetics/nat/le_n_O_elim.def(4)"le_n_O_elim/a ? HleO)
+    @H #p #ltpO @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a /2/ (* 3 *)
  |#p #Hind #q #HleS 
-    @H #a #lta @Hind @le_S_S_to_le /2/
+    @H #a #lta @Hind @a href="cic:/matita/arithmetics/nat/le_S_S_to_le.def(5)"le_S_S_to_le/a /2/
  ]
 qed.
 
 (* some properties of functions *)
 
-definition increasing ≝ λf:nat → nat. ∀n:nat. f n < f (S n).
+definition increasing ≝ λf: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. ∀n:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. f n 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 n).
 
-theorem increasing_to_monotonic: ∀f:nat → nat.
-  increasing f → monotonic nat lt f.
+theorem increasing_to_monotonic: ∀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/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 f.
 #f #incr #n #m #ltnm (elim ltnm) /2/
 qed.
 
-theorem le_n_fn: ∀f:nat → nat. 
-  increasing f → ∀n:nat. n ≤ f n.
+theorem le_n_fn: ∀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 → ∀n: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 f n.
 #f #incr #n (elim n) /2/
 qed.
 
-theorem increasing_to_le: ∀f:nat → nat. 
-  increasing f → ∀m:nat.∃i.m ≤ f i.
+theorem increasing_to_le: ∀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 → ∀m:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a.a title="exists" href="cic:/fakeuri.def(1)"∃/ai.m a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a f i.
 #f #incr #m (elim m) /2/#n * #a #lenfa
-@(ex_intro ?? (S 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)) /2/
 qed.
 
-theorem increasing_to_le2: ∀f:nat → nat. increasing f → 
-  ∀m:nat. f 0 ≤ m → ∃i. f i ≤ m ∧ m < f (S i).
+theorem increasing_to_le2: ∀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 → 
+  ∀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)
-  [@(ex_intro ? ? O) /2/
-  |#n #len * #a * #len #ltnr (cases(le_to_or_lt_eq … ltnr)) #H
-    [@(ex_intro ? ? a) % /2/ 
-    |@(ex_intro ? ? (S a)) % //
+  [@(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
+    [@(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)) % //
     ]
   ]
 qed.
 
-theorem increasing_to_injective: ∀f:nat → nat.
-  increasing f → injective nat nat f.
-#f #incr #n #m cases(decidable_le n m)
-  [#lenm cases(le_to_or_lt_eq … lenm) //
-   #lenm #eqf @False_ind @(absurd … eqf) @lt_to_not_eq 
-   @increasing_to_monotonic //
-  |#nlenm #eqf @False_ind @(absurd … eqf) @sym_not_eq 
-   @lt_to_not_eq @increasing_to_monotonic /2/
+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 #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 
+   @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 /2/
   ]
 qed.
 
 (*********************** monotonicity ***************************)
 theorem monotonic_le_plus_r: 
-∀n:nat.monotonic nat le (λm.n + m).
+∀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 @le_S_S /2/ qed.
+#m #H #leab @a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"le_S_S/a /2/ qed.
 
 (*
 theorem le_plus_r: ∀p,n,m:nat. n ≤ m → p + n ≤ p + m
 ≝ monotonic_le_plus_r. *)
 
 theorem monotonic_le_plus_l: 
-∀m:nat.monotonic nat le (λn.n + m).
+∀m: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 (λn.n a title="natural plus" href="cic:/fakeuri.def(1)"+/a m).
 /2/ qed.
 
 (*
 theorem le_plus_l: \forall p,n,m:nat. n \le m \to n + p \le m + p
 \def monotonic_le_plus_l. *)
 
-theorem le_plus: ∀n1,n2,m1,m2:nat. n1 ≤ n2  → m1 ≤ m2 
-→ n1 + m1 ≤ n2 + m2.
-#n1 #n2 #m1 #m2 #len #lem @(transitive_le ? (n1+m2))
+theorem le_plus: ∀n1,n2,m1,m2:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. n1 a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a n2  → m1 a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a m2 
+→ n1 a title="natural plus" href="cic:/fakeuri.def(1)"+/a m1 a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a n2 a title="natural plus" href="cic:/fakeuri.def(1)"+/a m2.
+#n1 #n2 #m1 #m2 #len #lem @(a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"transitive_le/a ? (n1a title="natural plus" href="cic:/fakeuri.def(1)"+/am2))
 /2/ qed.
 
-theorem le_plus_n :∀n,m:nat. m ≤ n + m.
+theorem le_plus_n :∀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 a title="natural plus" href="cic:/fakeuri.def(1)"+/a m.
 /2/ qed. 
 
-lemma le_plus_a: ∀a,n,m. n ≤ m → n ≤ a + m.
+lemma le_plus_a: ∀a,n,m. n a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a m → n a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a a a title="natural plus" href="cic:/fakeuri.def(1)"+/a m.
 /2/ qed.
 
-lemma le_plus_b: ∀b,n,m. n + b ≤ m → n ≤ m.
+lemma le_plus_b: ∀b,n,m. n a title="natural plus" href="cic:/fakeuri.def(1)"+/a b a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a m → n a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a m.
 /2/ qed.
 
-theorem le_plus_n_r :∀n,m:nat. m ≤ m + n.
+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.
 /2/ qed.
 
-theorem eq_plus_to_le: ∀n,m,p:nat.n=m+p → m ≤ n.
+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.
 
-theorem le_plus_to_le: ∀a,n,m. a + n ≤ a + m → n ≤ m.
+theorem le_plus_to_le: ∀a,n,m. a a title="natural plus" href="cic:/fakeuri.def(1)"+/a n a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a a a title="natural plus" href="cic:/fakeuri.def(1)"+/a m → n a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a m.
 #a (elim a) normalize /3/ qed. 
 
-theorem le_plus_to_le_r: ∀a,n,m. n + a ≤ m +a → n ≤ m.
+theorem le_plus_to_le_r: ∀a,n,m. n a title="natural plus" href="cic:/fakeuri.def(1)"+/a a a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a m a title="natural plus" href="cic:/fakeuri.def(1)"+/aa → n a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a m.
 /2/ qed. 
 
-(* plus & lt *)
+(* plus &amp; lt *)
 
 theorem monotonic_lt_plus_r: 
-∀n:nat.monotonic nat lt (λm.n+m).
+∀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.na title="natural plus" href="cic:/fakeuri.def(1)"+/am).
 /2/ qed.
 
 (*
@@ -551,21 +551,21 @@ variant lt_plus_r: \forall n,p,q:nat. p < q \to n + p < n + q \def
 monotonic_lt_plus_r. *)
 
 theorem monotonic_lt_plus_l: 
-∀n:nat.monotonic nat lt (λm.m+n).
+∀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).
 /2/ qed.
 
 (*
 variant lt_plus_l: \forall n,p,q:nat. p < q \to p + n < q + n \def
 monotonic_lt_plus_l. *)
 
-theorem lt_plus: ∀n,m,p,q:nat. n < m → p < q → n + p < m + q.
+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
-@(transitive_lt ? (n+q))/2/ qed.
+@(a href="cic:/matita/arithmetics/nat/transitive_lt.def(3)"transitive_lt/a ? (na title="natural plus" href="cic:/fakeuri.def(1)"+/aq))/2/ qed.
 
-theorem lt_plus_to_lt_l :∀n,p,q:nat. p+n < q+n → p<q.
+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.
 /2/ qed.
 
-theorem lt_plus_to_lt_r :∀n,p,q:nat. n+p < n+q → p<q.
+theorem lt_plus_to_lt_r :∀n,p,q:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. na title="natural plus" href="cic:/fakeuri.def(1)"+/ap a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a na title="natural plus" href="cic:/fakeuri.def(1)"+/aq → pa title="natural 'less than'" href="cic:/fakeuri.def(1)"</aq.
 /2/ qed.
 
 (*
@@ -578,9 +578,9 @@ normalize napplyS le_plus // qed.
 
 (* times *)
 theorem monotonic_le_times_r: 
-∀n:nat.monotonic nat le (λm. n * m).
+∀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 times" href="cic:/fakeuri.def(1)"*/a m).
 #n #x #y #lexy (elim n) normalize//(* lento /2/*)
-#a #lea @le_plus //
+#a #lea @a href="cic:/matita/arithmetics/nat/le_plus.def(7)"le_plus/a //
 qed.
 
 (*
@@ -597,23 +597,23 @@ theorem monotonic_le_times_l:
 theorem le_times_l: \forall p,n,m:nat. n \le m \to n*p \le m*p
 \def monotonic_le_times_l. *)
 
-theorem le_times: ∀n1,n2,m1,m2:nat. 
-n1 ≤ n2  → m1 ≤ m2 → n1*m1 ≤ n2*m2.
-#n1 #n2 #m1 #m2 #len #lem @(transitive_le ? (n1*m2)) /2/
+theorem le_times: ∀n1,n2,m1,m2:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. 
+n1 a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a n2  → m1 a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a m2 → n1a title="natural times" href="cic:/fakeuri.def(1)"*/am1 a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a n2a title="natural times" href="cic:/fakeuri.def(1)"*/am2.
+#n1 #n2 #m1 #m2 #len #lem @(a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"transitive_le/a ? (n1a title="natural times" href="cic:/fakeuri.def(1)"*/am2)) /2/
 qed.
 
 (* interessante *)
-theorem lt_times_n: ∀n,m:nat. O < n → m ≤ n*m.
+theorem lt_times_n: ∀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 → m a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a na title="natural times" href="cic:/fakeuri.def(1)"*/am.
 #n #m #H /2/ qed.
 
 theorem le_times_to_le: 
-∀a,n,m. O < a → a * n ≤ a * m → n ≤ m.
-#a @nat_elim2 normalize
+∀a,n,m. a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"O/a a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a a → a a title="natural times" href="cic:/fakeuri.def(1)"*/a n a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a a a title="natural times" href="cic:/fakeuri.def(1)"*/a m → n a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a m.
+#a @a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"nat_elim2/a normalize
   [//
   |#n #H1 #H2 
-     @(transitive_le ? (a*S n)) /2/
+     @(a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"transitive_le/a ? (aa title="natural times" href="cic:/fakeuri.def(1)"*/aa href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a n)) /2/
   |#n #m #H #lta #le
-     @le_S_S @H /2/
+     @a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"le_S_S/a @H /2/
   ]
 qed.
 
@@ -622,7 +622,7 @@ theorem le_S_times_2: ∀n,m.O < m → n ≤ m → S n ≤ 2*m.
 #n #m #posm #lenm  (* interessante *)
 applyS (le_plus n m) // qed. *)
 
-(* times & lt *)
+(* times &amp; lt *)
 (*
 theorem lt_O_times_S_S: ∀n,m:nat.O < (S n)*(S m).
 intros.simplify.unfold lt.apply le_S_S.apply le_O_n.
@@ -684,41 +684,41 @@ apply lt_plus.assumption.assumption.
 qed. *)
 
 theorem monotonic_lt_times_r: 
-  ∀c:nat. O < c → monotonic nat lt (λt.(c*t)).
+  ∀c: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 c → 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 (λt.(ca title="natural times" href="cic:/fakeuri.def(1)"*/at)).
 #c #posc #n #m #ltnm
 (elim ltnm) normalize
   [/2/ 
-  |#a #_ #lt1 @(transitive_le … lt1) //
+  |#a #_ #lt1 @(a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"transitive_le/a … lt1) //
   ]
 qed.
 
 theorem monotonic_lt_times_l: 
-  ∀c:nat. O < c → monotonic nat lt (λt.(t*c)).
+  ∀c: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 c → 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 (λt.(ta title="natural times" href="cic:/fakeuri.def(1)"*/ac)).
 /2/
 qed.
 
 theorem lt_to_le_to_lt_times: 
-∀n,m,p,q:nat. n < m → p ≤ q → O < q → n*p < m*q.
+∀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
-@(le_to_lt_to_lt ? (n*q))
-  [@monotonic_le_times_r //
-  |@monotonic_lt_times_l //
+@(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/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 //
   ]
 qed.
 
-theorem lt_times:∀n,m,p,q:nat. n<m → p<q → n*p < m*q.
-#n #m #p #q #ltnm #ltpq @lt_to_le_to_lt_times/2/
+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/
 qed.
 
 theorem lt_times_n_to_lt_l: 
-∀n,p,q:nat. p*n < q*n → p < q.
-#n #p #q #Hlt (elim (decidable_lt p q)) //
-#nltpq @False_ind @(absurd ? ? (lt_to_not_le ? ? Hlt))
-applyS monotonic_le_times_r /2/
+∀n,p,q:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. pa title="natural times" href="cic:/fakeuri.def(1)"*/an a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a qa title="natural times" href="cic:/fakeuri.def(1)"*/an → p a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a q.
+#n #p #q #Hlt (elim (a href="cic:/matita/arithmetics/nat/decidable_lt.def(7)"decidable_lt/a p q)) //
+#nltpq @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 ? ? (a href="cic:/matita/arithmetics/nat/lt_to_not_le.def(7)"lt_to_not_le/a ? ? Hlt))
+applyS a href="cic:/matita/arithmetics/nat/monotonic_le_times_r.def(8)"monotonic_le_times_r/a /2/
 qed.
 
 theorem lt_times_n_to_lt_r: 
-∀n,p,q:nat. n*p < n*q → p < q.
+∀n,p,q:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. na title="natural times" href="cic:/fakeuri.def(1)"*/ap a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a na title="natural times" href="cic:/fakeuri.def(1)"*/aq → p a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a q.
 /2/ qed.
 
 (*
@@ -769,35 +769,35 @@ qed. *)
 
 let rec minus n m ≝ 
  match n with 
- [ O ⇒ O
+ [ O ⇒ a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"O/a
  | S p ⇒ 
 	match m with
-	  [ O ⇒ S p
+	  [ O ⇒ a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a p
     | S q ⇒ minus p q ]].
         
 interpretation "natural minus" 'minus x y = (minus x y).
 
-theorem minus_S_S: ∀n,m:nat.S n - S m = n -m.
+theorem minus_S_S: ∀n,m:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a.a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a 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 n a title="natural minus" href="cic:/fakeuri.def(1)"-/am.
 // qed.
 
-theorem minus_O_n: ∀n:nat.O=O-n.
+theorem minus_O_n: ∀n: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/aa title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/aa href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"O/aa title="natural minus" href="cic:/fakeuri.def(1)"-/an.
 #n (cases n) // qed.
 
-theorem minus_n_O: ∀n:nat.n=n-O.
+theorem minus_n_O: ∀n:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a.na title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/ana title="natural minus" href="cic:/fakeuri.def(1)"-/aa href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"O/a.
 #n (cases n) // qed.
 
-theorem minus_n_n: ∀n:nat.O=n-n.
+theorem minus_n_n: ∀n: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/aa title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/ana title="natural minus" href="cic:/fakeuri.def(1)"-/an.
 #n (elim n) // qed.
 
-theorem minus_Sn_n: ∀n:nat. S O = (S n)-n.
+theorem minus_Sn_n: ∀n:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"O/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a (a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a n)a title="natural minus" href="cic:/fakeuri.def(1)"-/an.
 #n (elim n) normalize // qed.
 
-theorem minus_Sn_m: ∀m,n:nat. m ≤ n → S n -m = S (n-m).
+theorem minus_Sn_m: ∀m,n: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 → a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a n a 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(0,2,0)"S/a (na title="natural minus" href="cic:/fakeuri.def(1)"-/am).
 (* qualcosa da capire qui 
 #n #m #lenm nelim lenm napplyS refl_eq. *)
-@nat_elim2 
+@a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"nat_elim2/a 
   [//
-  |#n #abs @False_ind /2/ 
+  |#n #abs @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a /2/ 
   |#n #m #Hind #c applyS Hind /2/
   ]
 qed.
@@ -810,41 +810,41 @@ theorem not_eq_to_le_to_le_minus:
 napplyS (not_eq_to_le_to_lt n (S m) H H1)
 qed. *)
 
-theorem eq_minus_S_pred: ∀n,m. n - (S m) = pred(n -m).
-@nat_elim2 normalize // 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.
 
 theorem plus_minus:
-∀m,n,p:nat. m ≤ n → (n-m)+p = (n+p)-m.
-@nat_elim2 
+∀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.
+@a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"nat_elim2/a 
   [//
-  |#n #p #abs @False_ind /2/
+  |#n #p #abs @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a /2/
   |normalize/3/
   ]
 qed.
 
-theorem minus_plus_m_m: ∀n,m:nat.n = (n+m)-m.
+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.
 /2/ qed.
 
-theorem plus_minus_m_m: ∀n,m:nat.
-  m ≤ n → n = (n-m)+m.
-#n #m #lemn @sym_eq /2/ 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.
+#n #m #lemn @a href="cic:/matita/basics/logic/sym_eq.def(2)"sym_eq/a /2/ qed.
 
-theorem le_plus_minus_m_m: ∀n,m:nat. n ≤ (n-m)+m.
+theorem le_plus_minus_m_m: ∀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 (na title="natural minus" href="cic:/fakeuri.def(1)"-/am)a title="natural plus" href="cic:/fakeuri.def(1)"+/am.
 #n (elim n) // #a #Hind #m (cases m) // normalize #n/2/  
 qed.
 
-theorem minus_to_plus :∀n,m,p:nat.
-  m ≤ n → n-m = p → n = m+p.
-#n #m #p #lemn #eqp (applyS plus_minus_m_m) //
+theorem minus_to_plus :∀n,m,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="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a p → n a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a ma title="natural plus" href="cic:/fakeuri.def(1)"+/ap.
+#n #m #p #lemn #eqp (applyS a href="cic:/matita/arithmetics/nat/plus_minus_m_m.def(7)"plus_minus_m_m/a) //
 qed.
 
-theorem plus_to_minus :∀n,m,p:nat.n = m+p → n-m = p.
-#n #m #p #eqp @sym_eq (applyS (minus_plus_m_m p m))
+theorem plus_to_minus :∀n,m,p: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 ma title="natural plus" href="cic:/fakeuri.def(1)"+/ap → na title="natural minus" href="cic:/fakeuri.def(1)"-/am a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a p.
+#n #m #p #eqp @a href="cic:/matita/basics/logic/sym_eq.def(2)"sym_eq/a (applyS (a href="cic:/matita/arithmetics/nat/minus_plus_m_m.def(6)"minus_plus_m_m/a p m))
 qed.
 
-theorem minus_pred_pred : ∀n,m:nat. O < n → O < m → 
-pred n - pred m = n - m.
-#n #m #posn #posm @(lt_O_n_elim n posn) @(lt_O_n_elim m posm) //.
+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) //.
 qed.
 
 
@@ -900,75 +900,75 @@ qed.
 (* monotonicity and galois *)
 
 theorem monotonic_le_minus_l: 
-∀p,q,n:nat. q ≤ p → q-n ≤ p-n.
-@nat_elim2 #p #q
-  [#lePO @(le_n_O_elim ? lePO) //
+∀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 → qa title="natural minus" href="cic:/fakeuri.def(1)"-/an a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a pa title="natural minus" href="cic:/fakeuri.def(1)"-/an.
+@a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"nat_elim2/a #p #q
+  [#lePO @(a href="cic:/matita/arithmetics/nat/le_n_O_elim.def(4)"le_n_O_elim/a ? lePO) //
   |//
   |#Hind #n (cases n) // #a #leSS @Hind /2/
   ]
 qed.
 
-theorem le_minus_to_plus: ∀n,m,p. n-m ≤ p → n≤ p+m.
-#n #m #p #lep @transitive_le
-  [|@le_plus_minus_m_m | @monotonic_le_plus_l // ]
+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 // ]
 qed.
 
-theorem le_minus_to_plus_r: ∀a,b,c. c ≤ b → a ≤ b - c → a + c ≤ b.
-#a #b #c #Hlecb #H >(plus_minus_m_m … Hlecb) /2/
+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) /2/
 qed.
 
-theorem le_plus_to_minus: ∀n,m,p. n ≤ p+m → n-m ≤ p.
+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 /2/ qed.
 
-theorem le_plus_to_minus_r: ∀a,b,c. a + b ≤ c → a ≤ c -b.
-#a #b #c #H @(le_plus_to_le_r … b) /2/
+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) /2/
 qed.
 
-theorem lt_minus_to_plus: ∀a,b,c. a - b < c → a < c + b.
-#a #b #c #H @not_le_to_lt 
-@(not_to_not … (lt_to_not_le …H)) /2/
+theorem lt_minus_to_plus: ∀a,b,c. a a title="natural minus" 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 plus" 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/lt_to_not_le.def(7)"lt_to_not_le/a …H)) /2/
 qed.
 
-theorem lt_minus_to_plus_r: ∀a,b,c. a < b - c → a + c < b.
-#a #b #c #H @not_le_to_lt @(not_to_not … (le_plus_to_minus …))
-@lt_to_not_le //
+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 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 ≤ n → n < p+m → n-m < p.
-#n #m #p #lenm #H normalize <minus_Sn_m // @le_plus_to_minus //
+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 //
 qed.
 
-theorem lt_plus_to_minus_r: ∀a,b,c. a + b < c → a < c - b.
-#a #b #c #H @le_plus_to_minus_r //
+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 //
 qed. 
 
 theorem monotonic_le_minus_r: 
-∀p,q,n:nat. q ≤ p → n-p ≤ n-q.
-#p #q #n #lepq @le_plus_to_minus
-@(transitive_le … (le_plus_minus_m_m ? q)) /2/
+∀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/
 qed.
 
-theorem eq_minus_O: ∀n,m:nat.
-  n ≤ m → n-m = O.
-#n #m #lenm @(le_n_O_elim (n-m)) /2/
+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)) /2/
 qed.
 
-theorem distributive_times_minus: distributive ? times minus.
+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 (decidable_lt b c)) #Hbc
- [> eq_minus_O /2/ >eq_minus_O // 
-  @monotonic_le_times_r /2/
- |@sym_eq (applyS plus_to_minus) <distributive_times_plus 
-  @eq_f (applyS plus_minus_m_m) /2/
+(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/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/
 qed.
 
-theorem minus_plus: ∀n,m,p. n-m-p = n -(m+p).
+theorem minus_plus: ∀n,m,p. na title="natural minus" href="cic:/fakeuri.def(1)"-/ama title="natural minus" href="cic:/fakeuri.def(1)"-/ap a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a n a title="natural minus" href="cic:/fakeuri.def(1)"-/a(ma title="natural plus" href="cic:/fakeuri.def(1)"+/ap).
 #n #m #p 
-cases (decidable_le (m+p) n) #Hlt
-  [@plus_to_minus @plus_to_minus <associative_plus
-   @minus_to_plus //
-  |cut (n ≤ m+p) [@(transitive_le … (le_n_Sn …)) @not_le_to_lt //]
-   #H >eq_minus_O /2/ >eq_minus_O // 
+cases (a href="cic:/matita/arithmetics/nat/decidable_le.def(6)"decidable_le/a (ma title="natural plus" href="cic:/fakeuri.def(1)"+/ap) n) #Hlt
+  [@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 // 
   ]
 qed.
 
@@ -978,11 +978,11 @@ 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 ********************) 
@@ -990,67 +990,67 @@ 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 
+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 /3/ 
   |normalize /3/
   |normalize /4/ 
   ] 
 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 /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 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) /2/ 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 /2/ 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
+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
   [/2/
   |/3/
   |#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 /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_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 /2/ 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 /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 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 /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 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 /2/ qed.
 
-theorem lt_to_leb_false: ∀n,m. m < n → leb n m = false.
+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.
 /3/ qed.
 
 (* serve anche ltb? 
@@ -1079,41 +1079,40 @@ 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. a href="cic:/matita/basics/bool/if_then_else.def(1)"if_then_else/a ? (a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"leb/a n m) n 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. a href="cic:/matita/basics/bool/if_then_else.def(1)"if_then_else/a ? (a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"leb/a n m) m 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 /2/
+  |#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)) /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_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 /2/ qed. 
 
-lemma le_minl: ∀i,n,m. i ≤ min n m → i ≤ n.
+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.
 /2/ 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 /2/
+  |#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)) /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_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 /2/ qed. 
 
-lemma le_maxr: ∀i,n,m. max n m ≤ i → m ≤ i.
+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.
 /2/ 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.
-
-- 
2.39.2