From: matitaweb <claudio.sacerdoticoen@unibo.it>
Date: Fri, 30 Sep 2011 16:06:36 +0000 (+0000)
Subject: Matitaweb: test commit.
X-Git-Tag: make_still_working~2257
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=17e998b244d1edf335f6a5162d5e97c90484a3f7;p=helm.git

Matitaweb: test commit.
---

diff --git a/weblib/flagtest/prova.ma b/weblib/flagtest/prova.ma
index 64490b74c..f14392e93 100644
--- a/weblib/flagtest/prova.ma
+++ b/weblib/flagtest/prova.ma
@@ -1 +1,1118 @@
-(* script *)
\ No newline at end of file
+(*
+    ||M||  This file is part of HELM, an Hypertextual, Electronic        
+    ||A||  Library of Mathematics, developed at the Computer Science     
+    ||T||  Department of the University of Bologna, Italy.                     
+    ||I||                                                                 
+    ||T||  
+    ||A||  This file is distributed under the terms of the 
+    \   /  GNU General Public License Version 2        
+     \ /      
+      V_______________________________________________________________ *)
+
+include "basics/relations.ma".
+
+inductive nat : Type[0] ≝
+  | O : nat
+  | S : nat → nat.
+  
+interpretation "Natural numbers" 'N = nat.
+
+alias num (instance 0) = "natural number".
+
+definition pred ≝
+ 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 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 : 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: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: 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: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: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: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: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: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 ⇒ 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: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.
+
+(*
+theorem plus_Sn_m: ∀n,m:nat. S (n + m) = S n + m.
+// qed.
+*)
+
+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: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.
+
+(*
+theorem plus_Sn_m1: ∀n,m:nat. S m + n = n + S m.
+#n (elim n) normalize // qed.
+*)
+
+(* deleterio?
+theorem plus_n_1 : ∀n:nat. S n = n+1.
+// qed.
+*)
+
+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 : 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 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: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
+\def injective_plus_r. 
+
+theorem injective_plus_l: ∀m:nat.injective nat nat (λn.n+m).
+/2/ qed. *)
+
+(* theorem inj_plus_l: \forall p,n,m:nat. n+p = m+p \to n=m
+\def injective_plus_l. *)
+
+(*************************** times *****************************)
+
+let rec times n 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: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: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: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: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 : 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 : 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: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: 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. 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: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? 
+theorem times_O_to_O: ∀n,m:nat.n*m=O → n=O ∨ m=O.
+napply nat_elim2 /2/ 
+#n #m #H normalize #H1 napply False_ind napply not_eq_O_S
+// qed.
+  
+theorem times_n_SO : ∀n:nat. n = n * S O.
+#n // qed.
+
+theorem times_SSO_n : ∀n:nat. n + n = (S(S O)) * n.
+normalize // qed.
+
+nlemma times_SSO: \forall n.(S(S O))*(S n) = S(S((S(S O))*n)).
+// qed.
+
+theorem or_eq_eq_S: \forall n.\exists m. 
+n = (S(S O))*m \lor n = S ((S(S O))*m).
+#n (elim n)
+  ##[@ /2/
+  ##|#a #H nelim H #b#ornelim or#aeq
+    ##[@ b @ 2 //
+    ##|@ (S b) @ 1 /2/
+    ##]
+qed.
+*)
+
+(******************** ordering relations ************************)
+
+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: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: 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)).
+
+(* lemma eq_lt: ∀n,m. (n < m) = (S n ≤ m).
+// qed. *)
+
+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: 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 : 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.
+
+(*
+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: 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: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: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: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: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: 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: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.
+
+(* this are instances of the le versions 
+theorem lt_S_S_to_lt: ∀n,m. S n < S m → n < m.
+/2/ qed. 
+
+theorem lt_to_lt_S_S: ∀n,m. n < m → S n < S m.
+/2/ qed. *)
+
+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: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: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: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. 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. 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: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
+theorem lt_S_to_le: ∀n,m:nat. n < S m → n ≤ m.
+/2/ qed.
+*)
+
+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 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 @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 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: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: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: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: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. 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: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.
+
+(*
+theorem lt_SO_n_to_lt_O_pred_n: \forall n:nat.
+(S O) \lt n \to O \lt (pred n).
+intros.
+apply (ltn_to_ltO (pred (S O)) (pred n) ?).
+ apply (lt_pred (S O) n)
+ [ apply (lt_O_S O) 
+ | assumption
+ ]
+qed. *)
+
+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. 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.
+
+(*
+theorem lt_pred: \forall n,m. 
+  O < n \to n < m \to pred n < pred m. 
+apply nat_elim2
+  [intros.apply False_ind.apply (not_le_Sn_O ? H)
+  |intros.apply False_ind.apply (not_le_Sn_O ? H1)
+  |intros.simplify.unfold.apply le_S_S_to_le.assumption
+  ]
+qed.
+
+theorem le_pred_to_le:
+ ∀n,m. O < m → pred n ≤ pred m → n ≤ m.
+intros 2
+elim n
+[ apply le_O_n
+| simplify in H2
+  rewrite > (S_pred m)
+  [ apply le_S_S
+    assumption
+  | assumption
+  ]
+].
+qed.
+
+*)
+
+(* le to lt or eq *)
+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: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.
+intros.
+unfold Not.
+intros.
+rewrite > H in H1.
+apply (lt_to_not_eq b b)
+[ assumption
+| reflexivity
+]
+qed. 
+
+theorem lt_n_m_to_not_lt_m_Sn: ∀n,m. n < m → m ≮ S n.
+intros
+unfold Not
+intro
+unfold lt in H
+unfold lt in H1
+generalize in match (le_S_S ? ? H)
+intro
+generalize in match (transitive_le ? ? ? H2 H1)
+intro
+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) //
+#Heq /3/ qed.
+(*
+nelim (Hneq Heq) qed. *)
+
+(* le elimination *)
+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: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: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 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: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.
+
+(*
+(* other abstract properties *)
+theorem antisymmetric_le : antisymmetric nat le.
+unfold antisymmetric.intros 2.
+apply (nat_elim2 (\lambda n,m.(n \leq m \to m \leq n \to n=m))).
+intros.apply le_n_O_to_eq.assumption.
+intros.apply False_ind.apply (not_le_Sn_O ? H).
+intros.apply eq_f.apply H.
+apply le_S_S_to_le.assumption.
+apply le_S_S_to_le.assumption.
+qed.
+
+theorem antisym_le: \forall n,m:nat. n \leq m \to m \leq n \to n=m
+\def antisymmetric_le.
+
+theorem le_n_m_to_lt_m_Sn_to_eq_n_m: ∀n,m. n ≤ m → m < S n → n=m.
+intros
+unfold lt in H1
+generalize in match (le_S_S_to_le ? ? H1)
+intro
+apply antisym_le
+assumption.
+qed.
+*)
+
+(* well founded induction principles *)
+
+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: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 *) 
+    @(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 @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: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: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: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: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
+@(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: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)
+  [@(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: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:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a.a href="cic:/matita/basics/relations/monotonic.def(1)"monotonic/a a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a a href="cic:/matita/arithmetics/nat/le.ind(1,0,1)"le/a (λm.n a title="natural plus" href="cic:/fakeuri.def(1)"+/a m).
+#n #a #b (elim n) normalize //
+#m #H #leab @a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"le_S_S/a /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: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: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: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 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 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: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: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 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 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 &amp; lt *)
+
+theorem monotonic_lt_plus_r: 
+∀n:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a.a href="cic:/matita/basics/relations/monotonic.def(1)"monotonic/a a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a a href="cic:/matita/arithmetics/nat/lt.def(1)"lt/a (λm.na title="natural plus" href="cic:/fakeuri.def(1)"+/am).
+/2/ qed.
+
+(*
+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: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:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. n a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a m → p a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a q → n a title="natural plus" href="cic:/fakeuri.def(1)"+/a p a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a m a title="natural plus" href="cic:/fakeuri.def(1)"+/a q.
+#n #m #p #q #ltnm #ltpq
+@(a href="cic:/matita/arithmetics/nat/transitive_lt.def(3)"transitive_lt/a ? (na title="natural plus" href="cic:/fakeuri.def(1)"+/aq))/2/ qed.
+
+theorem lt_plus_to_lt_l :∀n,p,q:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a. pa title="natural plus" href="cic:/fakeuri.def(1)"+/an a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a qa title="natural plus" href="cic:/fakeuri.def(1)"+/an → pa title="natural 'less than'" href="cic:/fakeuri.def(1)"</aq.
+/2/ qed.
+
+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.
+
+(*
+theorem le_to_lt_to_lt_plus: ∀a,b,c,d:nat.
+a ≤ c → b < d → a + b < c+d.
+(* bello /2/ un po' lento *)
+#a #b #c #d #leac #lebd 
+normalize napplyS le_plus // qed.
+*)
+
+(* times *)
+theorem monotonic_le_times_r: 
+∀n:a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a.a href="cic:/matita/basics/relations/monotonic.def(1)"monotonic/a a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a a href="cic:/matita/arithmetics/nat/le.ind(1,0,1)"le/a (λm. n a title="natural times" href="cic:/fakeuri.def(1)"*/a m).
+#n #x #y #lexy (elim n) normalize//(* lento /2/*)
+#a #lea @a href="cic:/matita/arithmetics/nat/le_plus.def(7)"le_plus/a //
+qed.
+
+(*
+theorem le_times_r: \forall p,n,m:nat. n \le m \to p*n \le p*m
+\def monotonic_le_times_r. *)
+
+(*
+theorem monotonic_le_times_l: 
+∀m:nat.monotonic nat le (λn.n*m).
+/2/ qed.
+*)
+
+(*
+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: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: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. 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 
+     @(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
+     @a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"le_S_S/a @H /2/
+  ]
+qed.
+
+(*
+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 &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.
+qed. *)
+
+(*
+theorem lt_times_eq_O: \forall a,b:nat.
+O < a → a * b = O → b = O.
+intros.
+apply (nat_case1 b)
+[ intros.
+  reflexivity
+| intros.
+  rewrite > H2 in H1.
+  rewrite > (S_pred a) in H1
+  [ apply False_ind.
+    apply (eq_to_not_lt O ((S (pred a))*(S m)))
+    [ apply sym_eq.
+      assumption
+    | apply lt_O_times_S_S
+    ]
+  | assumption
+  ]
+]
+qed. 
+
+theorem O_lt_times_to_O_lt: \forall a,c:nat.
+O \lt (a * c) \to O \lt a.
+intros.
+apply (nat_case1 a)
+[ intros.
+  rewrite > H1 in H.
+  simplify in H.
+  assumption
+| intros.
+  apply lt_O_S
+]
+qed.
+
+lemma lt_times_to_lt_O: \forall i,n,m:nat. i < n*m \to O < m.
+intros.
+elim (le_to_or_lt_eq O ? (le_O_n m))
+  [assumption
+  |apply False_ind.
+   rewrite < H1 in H.
+   rewrite < times_n_O in H.
+   apply (not_le_Sn_O ? H)
+  ]
+qed. *)
+
+(*
+theorem monotonic_lt_times_r: 
+∀n:nat.monotonic nat lt (λm.(S n)*m).
+/2/ 
+simplify.
+intros.elim n.
+simplify.rewrite < plus_n_O.rewrite < plus_n_O.assumption.
+apply lt_plus.assumption.assumption.
+qed. *)
+
+theorem monotonic_lt_times_r: 
+  ∀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 @(a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"transitive_le/a … lt1) //
+  ]
+qed.
+
+theorem monotonic_lt_times_l: 
+  ∀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: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/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: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: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: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.
+
+(*
+theorem nat_compare_times_l : \forall n,p,q:nat. 
+nat_compare p q = nat_compare ((S n) * p) ((S n) * q).
+intros.apply nat_compare_elim.intro.
+apply nat_compare_elim.
+intro.reflexivity.
+intro.absurd (p=q).
+apply (inj_times_r n).assumption.
+apply lt_to_not_eq. assumption.
+intro.absurd (q<p).
+apply (lt_times_to_lt_r n).assumption.
+apply le_to_not_lt.apply lt_to_le.assumption.
+intro.rewrite < H.rewrite > nat_compare_n_n.reflexivity.
+intro.apply nat_compare_elim.intro.
+absurd (p<q).
+apply (lt_times_to_lt_r n).assumption.
+apply le_to_not_lt.apply lt_to_le.assumption.
+intro.absurd (q=p).
+symmetry.
+apply (inj_times_r n).assumption.
+apply lt_to_not_eq.assumption.
+intro.reflexivity.
+qed. *)
+
+(* times and plus 
+theorem lt_times_plus_times: \forall a,b,n,m:nat. 
+a < n \to b < m \to a*m + b < n*m.
+intros 3.
+apply (nat_case n)
+  [intros.apply False_ind.apply (not_le_Sn_O ? H)
+  |intros.simplify.
+   rewrite < sym_plus.
+   unfold.
+   change with (S b+a*m1 \leq m1+m*m1).
+   apply le_plus
+    [assumption
+    |apply le_times
+      [apply le_S_S_to_le.assumption
+      |apply le_n
+      ]
+    ]
+  ]
+qed. *)
+
+(************************** minus ******************************)
+
+let rec minus n m ≝ 
+ match n with 
+ [ O ⇒ a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"O/a
+ | S p ⇒ 
+	match m with
+	  [ 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: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: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: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: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: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: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. *)
+@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/ 
+  |#n #m #Hind #c applyS Hind /2/
+  ]
+qed.
+
+(*
+theorem not_eq_to_le_to_le_minus: 
+  ∀n,m.n ≠ m → n ≤ m → n ≤ m - 1.
+#n * #m (cases m// #m normalize
+#H #H1 napply le_S_S_to_le
+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.
+
+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.
+@a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"nat_elim2/a 
+  [//
+  |#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: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: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: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: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: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: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.
+
+
+(*
+
+theorem le_SO_minus: \forall n,m:nat.S n \leq m \to S O \leq m-n.
+intros.elim H.elim (minus_Sn_n n).apply le_n.
+rewrite > minus_Sn_m.
+apply le_S.assumption.
+apply lt_to_le.assumption.
+qed.
+
+theorem minus_le_S_minus_S: \forall n,m:nat. m-n \leq S (m-(S n)).
+intros.
+apply (nat_elim2 (\lambda n,m.m-n \leq S (m-(S n)))).
+intro.elim n1.simplify.apply le_n_Sn.
+simplify.rewrite < minus_n_O.apply le_n.
+intros.simplify.apply le_n_Sn.
+intros.simplify.apply H.
+qed.
+
+theorem lt_minus_S_n_to_le_minus_n : \forall n,m,p:nat. m-(S n) < p \to m-n \leq p. 
+intros 3.intro.
+(* autobatch *)
+(* end auto($Revision: 9739 $) proof: TIME=1.33 SIZE=100 DEPTH=100 *)
+apply (trans_le (m-n) (S (m-(S n))) p).
+apply minus_le_S_minus_S.
+assumption.
+qed.
+
+theorem le_minus_m: \forall n,m:nat. n-m \leq n.
+intros.apply (nat_elim2 (\lambda m,n. n-m \leq n)).
+intros.rewrite < minus_n_O.apply le_n.
+intros.simplify.apply le_n.
+intros.simplify.apply le_S.assumption.
+qed.
+
+theorem lt_minus_m: \forall n,m:nat. O < n \to O < m \to n-m \lt n.
+intros.apply (lt_O_n_elim n H).intro.
+apply (lt_O_n_elim m H1).intro.
+simplify.unfold lt.apply le_S_S.apply le_minus_m.
+qed.
+
+theorem minus_le_O_to_le: \forall n,m:nat. n-m \leq O \to n \leq m.
+intros 2.
+apply (nat_elim2 (\lambda n,m:nat.n-m \leq O \to n \leq m)).
+intros.apply le_O_n.
+simplify.intros. assumption.
+simplify.intros.apply le_S_S.apply H.assumption.
+qed.
+*)
+
+(* monotonicity and galois *)
+
+theorem monotonic_le_minus_l: 
+∀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. 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 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 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 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 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 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 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 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: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: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: 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/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. 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 (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.
+
+(*
+theorem plus_minus: ∀n,m,p. p ≤ m → (n+m)-p = n +(m-p).
+#n #m #p #lepm @plus_to_minus >(commutative_plus p)
+>associative_plus <plus_minus_m_m //
+qed.  *)
+
+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
+@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 ********************) 
+include "basics/bool.ma".
+
+let rec eqb n m ≝ 
+match n with 
+  [ 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: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. 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: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: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: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: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 ⇒ a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a
+    | (S p) ⇒
+	match m with 
+        [ O ⇒ a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a
+	      | (S q) ⇒ leb p q]].
+
+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 @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.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.
+  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 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 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 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? 
+ndefinition ltb ≝λn,m. leb (S n) m.
+
+theorem ltb_elim: ∀n,m:nat. ∀P:bool → Prop. 
+(n < m → P true) → (n ≮ m → P false) → P (ltb n m).
+#n #m #P #Hlt #Hnlt
+napply leb_elim /3/ qed.
+
+theorem ltb_true_to_lt:∀n,m.ltb n m = true → n < m.
+#n #m #Hltb napply leb_true_to_le nassumption
+qed.
+
+theorem ltb_false_to_not_lt:∀n,m.
+  ltb n m = false → n ≮ m.
+#n #m #Hltb napply leb_false_to_not_le nassumption
+qed.
+
+theorem lt_to_ltb_true: ∀n,m. n < m → ltb n m = true.
+#n #m #Hltb napply le_to_leb_true nassumption
+qed.
+
+theorem le_to_ltb_false: ∀n,m. m \le n → ltb n m = false.
+#n #m #Hltb napply lt_to_leb_false /2/
+qed. *)
+
+(* min e max *)
+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: 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: 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 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 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 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: 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. 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. 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 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.