- new definition of lazy equivalence for local environments,

[helm.git] / weblib / arithmetics / nat.ma

(*
    ||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".

\ 5img class="anchor" src="icons/tick.png" id="nat"\ 6inductive nat : Type[0] ≝
  | O : nat
  | S : nat → nat.
  
interpretation "Natural numbers" 'N = nat.

alias num (instance 0) = "natural number".

\ 5img class="anchor" src="icons/tick.png" id="pred"\ 6definition pred ≝
 λn. match n with [ O ⇒ \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 | S p ⇒ p].

\ 5img class="anchor" src="icons/tick.png" id="pred_Sn"\ 6theorem pred_Sn : ∀n.n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/pred.def(1)"\ 6pred\ 5/a\ 6 (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n).
// qed.

\ 5img class="anchor" src="icons/tick.png" id="injective_S"\ 6theorem injective_S : \ 5a href="cic:/matita/basics/relations/injective.def(1)"\ 6injective\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6.
// qed.

(*
theorem inj_S : \forall n,m:nat.(S n)=(S m) \to n=m.
//. qed. *)

\ 5img class="anchor" src="icons/tick.png" id="not_eq_S"\ 6theorem not_eq_S: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 m → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

\ 5img class="anchor" src="icons/tick.png" id="not_zero"\ 6definition not_zero: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop ≝
 λn: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. match n with [ O ⇒ \ 5a href="cic:/matita/basics/logic/False.ind(1,0,0)"\ 6False\ 5/a\ 6 | (S p) ⇒ \ 5a href="cic:/matita/basics/logic/True.ind(1,0,0)"\ 6True\ 5/a\ 6 ].

\ 5img class="anchor" src="icons/tick.png" id="not_eq_O_S"\ 6theorem not_eq_O_S : ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n.
#n @\ 5a href="cic:/matita/basics/logic/Not.con(0,1,1)"\ 6nmk\ 5/a\ 6 #eqOS (change with (\ 5a href="cic:/matita/arithmetics/nat/not_zero.def(1)"\ 6not_zero\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6)) >eqOS // qed.

\ 5img class="anchor" src="icons/tick.png" id="not_eq_n_Sn"\ 6theorem not_eq_n_Sn: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n.
#n (elim n) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/not_eq_S.def(4)"\ 6not_eq_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

\ 5img class="anchor" src="icons/tick.png" id="nat_case"\ 6theorem nat_case:
 ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.∀P:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop. 
  (n\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 → P \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6) → (∀m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m → P (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m)) → P n.
#n #P (elim n) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/ qed.

\ 5img class="anchor" src="icons/tick.png" id="nat_elim2"\ 6theorem nat_elim2 :
 ∀R:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop.
  (∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. R \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 n) 
  → (∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. R (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n) \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6)
  → (∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. R n m → R (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n) (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m))
  → ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. R n m.
#R #ROn #RSO #RSS #n (elim n) // #n0 #Rn0m #m (cases m) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/ qed.

\ 5img class="anchor" src="icons/tick.png" id="decidable_eq_nat"\ 6theorem decidable_eq_nat : ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a href="cic:/matita/basics/logic/decidable.def(1)"\ 6decidable\ 5/a\ 6 (n\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6m).
@\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6 #n [ (cases n) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ | /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/sym_not_eq.def(4)"\ 6sym_not_eq\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ | #m #Hind (cases Hind) /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/not_eq_S.def(4)"\ 6not_eq_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/]
qed. 

(*************************** plus ******************************)

\ 5img class="anchor" src="icons/tick.png" id="plus"\ 6let rec plus n m ≝ 
 match n with [ O ⇒ m | S p ⇒ \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (plus p m) ].

interpretation "natural plus" 'plus x y = (plus x y).

\ 5img class="anchor" src="icons/tick.png" id="plus_O_n"\ 6theorem plus_O_n: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6n.
// qed.

(*
theorem plus_Sn_m: ∀n,m:nat. S (n + m) = S n + m.
// qed.
*)

\ 5img class="anchor" src="icons/tick.png" id="plus_n_O"\ 6theorem plus_n_O: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6\ 5a title="natural number" href="cic:/fakeuri.def(1)"\ 60\ 5/a\ 6.
#n (elim n) normalize // qed.

\ 5img class="anchor" src="icons/tick.png" id="plus_n_Sm"\ 6theorem plus_n_Sm : ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m.
#n (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.
*)

\ 5img class="anchor" src="icons/tick.png" id="commutative_plus"\ 6theorem commutative_plus: \ 5a href="cic:/matita/basics/relations/commutative.def(1)"\ 6commutative\ 5/a\ 6 ? \ 5a href="cic:/matita/arithmetics/nat/plus.fix(0,0,1)"\ 6plus\ 5/a\ 6.
#n (elim n) normalize // qed. 

\ 5img class="anchor" src="icons/tick.png" id="associative_plus"\ 6theorem associative_plus : \ 5a href="cic:/matita/basics/relations/associative.def(1)"\ 6associative\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/plus.fix(0,0,1)"\ 6plus\ 5/a\ 6.
#n (elim n) normalize // qed.

\ 5img class="anchor" src="icons/tick.png" id="assoc_plus1"\ 6theorem assoc_plus1: ∀a,b,c. c \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 (b \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 a) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 c \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 a.
// qed. 

\ 5img class="anchor" src="icons/tick.png" id="injective_plus_r"\ 6theorem injective_plus_r: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a href="cic:/matita/basics/relations/injective.def(1)"\ 6injective\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 (λm.n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m).
#n (elim n) normalize /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/injective_S.def(4)"\ 6injective_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

(* theorem inj_plus_r: \forall p,n,m:nat. p+n = p+m \to n=m
\def injective_plus_r. 

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 *****************************)

\ 5img class="anchor" src="icons/tick.png" id="times"\ 6let rec times n m ≝ 
 match n with [ O ⇒ \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 | S p ⇒ m\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6(times p m) ].

interpretation "natural times" 'times x y = (times x y).

\ 5img class="anchor" src="icons/tick.png" id="times_Sn_m"\ 6theorem times_Sn_m: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. m\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m.
// qed.

\ 5img class="anchor" src="icons/tick.png" id="times_O_n"\ 6theorem times_O_n: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6n.
// qed.

\ 5img class="anchor" src="icons/tick.png" id="times_n_O"\ 6theorem times_n_O: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6.
#n (elim n) // qed.

\ 5img class="anchor" src="icons/tick.png" id="times_n_Sm"\ 6theorem times_n_Sm : ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6(n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6(\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m).
#n (elim n) normalize // qed.

\ 5img class="anchor" src="icons/tick.png" id="commutative_times"\ 6theorem commutative_times : \ 5a href="cic:/matita/basics/relations/commutative.def(1)"\ 6commutative\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/times.fix(0,0,2)"\ 6times\ 5/a\ 6. 
#n (elim n) normalize // qed. 

(* variant sym_times : \forall n,m:nat. n*m = m*n \def
symmetric_times. *)

\ 5img class="anchor" src="icons/tick.png" id="distributive_times_plus"\ 6theorem distributive_times_plus : \ 5a href="cic:/matita/basics/relations/distributive.def(1)"\ 6distributive\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/times.fix(0,0,2)"\ 6times\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/plus.fix(0,0,1)"\ 6plus\ 5/a\ 6.
#n (elim n) normalize // qed.

\ 5img class="anchor" src="icons/tick.png" id="distributive_times_plus_r"\ 6theorem distributive_times_plus_r :
  ∀a,b,c:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. (b\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6c)\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 c\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6a.
// qed. 

\ 5img class="anchor" src="icons/tick.png" id="associative_times"\ 6theorem associative_times: \ 5a href="cic:/matita/basics/relations/associative.def(1)"\ 6associative\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/times.fix(0,0,2)"\ 6times\ 5/a\ 6.
#n (elim n) normalize // qed.

\ 5img class="anchor" src="icons/tick.png" id="times_times"\ 6lemma times_times: ∀x,y,z. x\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6(y\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6z) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 y\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6(x\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6z).
// qed. 

\ 5img class="anchor" src="icons/tick.png" id="times_n_1"\ 6theorem times_n_1 : ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n \ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 \ 5a title="natural number" href="cic:/fakeuri.def(1)"\ 61\ 5/a\ 6.
#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 ************************)

\ 5img class="anchor" src="icons/tick.png" id="le"\ 6inductive le (n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6) : \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop ≝
  | le_n : le n n
  | le_S : ∀ m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. le n m → le n (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 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)).

\ 5img class="anchor" src="icons/tick.png" id="lt"\ 6definition lt: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop ≝ λn,m. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 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. *)

\ 5img class="anchor" src="icons/tick.png" id="ge"\ 6definition ge: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop ≝ λn,m.m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n.

interpretation "natural 'greater or equal to'" 'geq x y = (ge x y).

\ 5img class="anchor" src="icons/tick.png" id="gt"\ 6definition gt: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop ≝ λn,m.m\ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6n.

interpretation "natural 'greater than'" 'gt x y = (gt x y).
interpretation "natural 'not greater than'" 'ngtr x y = (Not (gt x y)).

\ 5img class="anchor" src="icons/tick.png" id="transitive_le"\ 6theorem transitive_le : \ 5a href="cic:/matita/basics/relations/transitive.def(2)"\ 6transitive\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/le.ind(1,0,1)"\ 6le\ 5/a\ 6.
#a #b #c #leab #lebc (elim lebc) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le.con(0,2,1)"\ 6le_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.

(*
theorem trans_le: \forall n,m,p:nat. n \leq m \to m \leq p \to n \leq p
\def transitive_le. *)

\ 5img class="anchor" src="icons/tick.png" id="transitive_lt"\ 6theorem transitive_lt: \ 5a href="cic:/matita/basics/relations/transitive.def(2)"\ 6transitive\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/lt.def(1)"\ 6lt\ 5/a\ 6.
#a #b #c #ltab #ltbc (elim ltbc) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le.con(0,2,1)"\ 6le_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/qed.

(*
theorem trans_lt: \forall n,m,p:nat. lt n m \to lt m p \to lt n p
\def transitive_lt. *)

\ 5img class="anchor" src="icons/tick.png" id="le_S_S"\ 6theorem le_S_S: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m.
#n #m #lenm (elim lenm) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le.con(0,2,1)"\ 6le_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

\ 5img class="anchor" src="icons/tick.png" id="le_O_n"\ 6theorem le_O_n : ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n.
#n (elim n) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le.con(0,2,1)"\ 6le_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

\ 5img class="anchor" src="icons/tick.png" id="le_n_Sn"\ 6theorem le_n_Sn : ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le.con(0,2,1)"\ 6le_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

\ 5img class="anchor" src="icons/tick.png" id="le_pred_n"\ 6theorem le_pred_n : ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/pred.def(1)"\ 6pred\ 5/a\ 6 n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n.
#n (elim n) // qed.

\ 5img class="anchor" src="icons/tick.png" id="monotonic_pred"\ 6theorem monotonic_pred: \ 5a href="cic:/matita/basics/relations/monotonic.def(1)"\ 6monotonic\ 5/a\ 6 ? \ 5a href="cic:/matita/arithmetics/nat/le.ind(1,0,1)"\ 6le\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/pred.def(1)"\ 6pred\ 5/a\ 6.
#n #m #lenm (elim lenm) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

\ 5img class="anchor" src="icons/tick.png" id="le_S_S_to_le"\ 6theorem le_S_S_to_le: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
(* demo *)
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_pred.def(4)"\ 6monotonic_pred\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

(* this are instances of the le versions 
theorem lt_S_S_to_lt: ∀n,m. S n < S m → n < m.
/2/ qed. 

theorem lt_to_lt_S_S: ∀n,m. n < m → S n < S m.
/2/ qed. *)

\ 5img class="anchor" src="icons/tick.png" id="lt_to_not_zero"\ 6theorem lt_to_not_zero : ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → \ 5a href="cic:/matita/arithmetics/nat/not_zero.def(1)"\ 6not_zero\ 5/a\ 6 m.
#n #m #Hlt (elim Hlt) // qed.

(* lt vs. le *)

\ 5img class="anchor" src="icons/tick.png" id="lt_to_le"\ 6lemma lt_to_le: ∀n,m. n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
#n #m #H @\ 5a href="cic:/matita/arithmetics/nat/le_S_S_to_le.def(5)"\ 6le_S_S_to_le\ 5/a\ 6 @\ 5a href="cic:/matita/arithmetics/nat/le.con(0,2,1)"\ 6le_S\ 5/a\ 6 @H qed-.

theorem not_le_Sn_O: ∀ n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6.
#n @\ 5a href="cic:/matita/basics/logic/Not.con(0,1,1)"\ 6nmk\ 5/a\ 6 #Hlen0 @(\ 5a href="cic:/matita/arithmetics/nat/lt_to_not_zero.def(2)"\ 6lt_to_not_zero\ 5/a\ 6 ?? Hlen0) qed.

theorem not_le_to_not_le_S_S: ∀ n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 m → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m.
/\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/monotonic_pred.def(4)"\ 6monotonic_pred\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

theorem not_le_S_S_to_not_le: ∀ n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m → n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 m.
/\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

theorem decidable_le: ∀n,m. \ 5a href="cic:/matita/basics/logic/decidable.def(1)"\ 6decidable\ 5/a\ 6 (n\ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6m).
@\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6 #n /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ #m * /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/not_le_to_not_le_S_S.def(5)"\ 6not_le_to_not_le_S_S\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

theorem decidable_lt: ∀n,m. \ 5a href="cic:/matita/basics/logic/decidable.def(1)"\ 6decidable\ 5/a\ 6 (n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m).
#n #m @\ 5a href="cic:/matita/arithmetics/nat/decidable_le.def(6)"\ 6decidable_le\ 5/a\ 6  qed.

theorem not_le_Sn_n: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 n.
#n (elim n) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/not_le_to_not_le_S_S.def(5)"\ 6not_le_to_not_le_S_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

(* this is le_S_S_to_le
theorem lt_S_to_le: ∀n,m:nat. n < S m → n ≤ m.
/2/ qed.
*)

lemma le_gen: ∀P:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop.∀n.(∀i. i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n → P i) → P n.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

theorem not_le_to_lt: ∀n,m. n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 m → m \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n.
@\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6 #n
 [#abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
 |/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
 |#m #Hind #HnotleSS @\ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6 /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/not_le_S_S_to_not_le.def(4)"\ 6not_le_S_S_to_not_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
 ]
qed.

theorem lt_to_not_le: ∀n,m. n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → m \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 n.
#n #m #Hltnm (elim Hltnm) /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

theorem not_lt_to_le: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'not less than'" href="cic:/fakeuri.def(1)"\ 6≮\ 5/a\ 6 m → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n.
/\ 5span class="autotactic"\ 64\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/not_le_to_lt.def(5)"\ 6not_le_to_lt\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/monotonic_pred.def(4)"\ 6monotonic_pred\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

theorem le_to_not_lt: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → m \ 5a title="natural 'not less than'" href="cic:/fakeuri.def(1)"\ 6≮\ 5/a\ 6 n.
#n #m #H @\ 5a href="cic:/matita/arithmetics/nat/lt_to_not_le.def(7)"\ 6lt_to_not_le\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ (* /3/ *) qed.

(* lt and le trans *)

theorem lt_to_le_to_lt: ∀n,m,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 p → n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 p.
#n #m #p #H #H1 (elim H1) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/transitive_lt.def(3)"\ 6transitive_lt\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

theorem le_to_lt_to_lt: ∀n,m,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → m \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 p → n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 p.
#n #m #p #H (elim H) /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/transitive_lt.def(3)"\ 6transitive_lt\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

theorem lt_S_to_lt: ∀n,m. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/transitive_lt.def(3)"\ 6transitive_lt\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

theorem ltn_to_ltO: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_to_lt_to_lt.def(4)"\ 6le_to_lt_to_lt\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

theorem lt_O_n_elim: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n → 
  ∀P:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop.(∀m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.P (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m)) → P n.
#n (elim n) // #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.

theorem S_pred: ∀n. \ 5a title="natural number" href="cic:/fakeuri.def(1)"\ 60\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6(\ 5a href="cic:/matita/arithmetics/nat/pred.def(1)"\ 6pred\ 5/a\ 6 n) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n.
#n #posn (cases posn) //
qed.

(* le to lt or eq *)
theorem le_to_or_lt_eq: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 m.
#n #m #lenm (elim lenm) /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le_to_lt_to_lt.def(4)"\ 6le_to_lt_to_lt\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

(* not eq *)
theorem lt_to_not_eq : ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → n \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 m.
#n #m #H @\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Not.con(0,1,1)"\ 6nmk\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

theorem not_eq_to_le_to_lt: ∀n,m. n\ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6m → n\ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6m → n\ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6m.
#n #m #Hneq #Hle cases (\ 5a href="cic:/matita/arithmetics/nat/le_to_or_lt_eq.def(5)"\ 6le_to_or_lt_eq\ 5/a\ 6 ?? Hle) //
#Heq /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/not_le_to_lt.def(5)"\ 6not_le_to_lt\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
(*
nelim (Hneq Heq) qed. *)

(* le elimination *)
theorem le_n_O_to_eq : ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6n.
#n (cases n) // #a  #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

theorem le_n_O_elim: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 → ∀P: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 →Prop. P \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 → P n.
#n (cases n) // #a #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed. 

theorem le_n_Sm_elim : ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m → 
∀P:Prop. (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m → P) → (n\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m → P) → P.
#n #m #Hle #P (elim Hle) /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

(* le and eq *)

theorem le_to_le_to_eq: ∀n,m. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n → n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 m.
@\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6 /\ 5span class="autotactic"\ 64\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/eq_f.def(3)"\ 6eq_f\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le_n_O_to_eq.def(4)"\ 6le_n_O_to_eq\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/monotonic_pred.def(4)"\ 6monotonic_pred\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed. 

theorem lt_O_S : ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/not_le_to_lt.def(5)"\ 6not_le_to_lt\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

(*
(* other abstract properties *)
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:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.∀P:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop. 
(∀m.(∀p. p \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → P p) → P m) → P n.
#n #P #H 
cut (∀q:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. q \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n → P q) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
(elim n) 
 [#q #HleO (* applica male *) 
    @(\ 5a href="cic:/matita/arithmetics/nat/le_n_O_elim.def(4)"\ 6le_n_O_elim\ 5/a\ 6 ? HleO)
    @H #p #ltpO @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ (* 3 *)
 |#p #Hind #q #HleS 
    @H #a #lta @Hind @\ 5a href="cic:/matita/arithmetics/nat/le_S_S_to_le.def(5)"\ 6le_S_S_to_le\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
 ]
qed.

(* some properties of functions *)

definition increasing ≝ λf:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. f n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 f (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n).

theorem increasing_to_monotonic: ∀f:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.
  \ 5a href="cic:/matita/arithmetics/nat/increasing.def(2)"\ 6increasing\ 5/a\ 6 f → \ 5a href="cic:/matita/basics/relations/monotonic.def(1)"\ 6monotonic\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/lt.def(1)"\ 6lt\ 5/a\ 6 f.
#f #incr #n #m #ltnm (elim ltnm) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/transitive_lt.def(3)"\ 6transitive_lt\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.

theorem le_n_fn: ∀f:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. 
  \ 5a href="cic:/matita/arithmetics/nat/increasing.def(2)"\ 6increasing\ 5/a\ 6 f → ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 f n.
#f #incr #n (elim n) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_to_lt_to_lt.def(4)"\ 6le_to_lt_to_lt\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.

theorem increasing_to_le: ∀f:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. 
  \ 5a href="cic:/matita/arithmetics/nat/increasing.def(2)"\ 6increasing\ 5/a\ 6 f → ∀m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6i.m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 f i.
#f #incr #m (elim m) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/#n * #a #lenfa
@(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 ?? (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 a)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_to_lt_to_lt.def(4)"\ 6le_to_lt_to_lt\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.

theorem increasing_to_le2: ∀f:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/increasing.def(2)"\ 6increasing\ 5/a\ 6 f → 
  ∀m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. f \ 5a title="natural number" href="cic:/fakeuri.def(1)"\ 60\ 5/a\ 6 \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → \ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6i. f i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 m \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 f (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 i).
#f #incr #m #lem (elim lem)
  [@(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 ? ? \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
  |#n #len * #a * #len #ltnr (cases(\ 5a href="cic:/matita/arithmetics/nat/le_to_or_lt_eq.def(5)"\ 6le_to_or_lt_eq\ 5/a\ 6 … ltnr)) #H
    [@(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 ? ? a) % /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le.con(0,2,1)"\ 6le_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ 
    |@(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 ? ? (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 a)) % //
    ]
  ]
qed.

theorem increasing_to_injective: ∀f:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.
  \ 5a href="cic:/matita/arithmetics/nat/increasing.def(2)"\ 6increasing\ 5/a\ 6 f → \ 5a href="cic:/matita/basics/relations/injective.def(1)"\ 6injective\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 f.
#f #incr #n #m cases(\ 5a href="cic:/matita/arithmetics/nat/decidable_le.def(6)"\ 6decidable_le\ 5/a\ 6 n m)
  [#lenm cases(\ 5a href="cic:/matita/arithmetics/nat/le_to_or_lt_eq.def(5)"\ 6le_to_or_lt_eq\ 5/a\ 6 … lenm) //
   #lenm #eqf @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 @(\ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6 … eqf) @\ 5a href="cic:/matita/arithmetics/nat/lt_to_not_eq.def(7)"\ 6lt_to_not_eq\ 5/a\ 6 
   @\ 5a href="cic:/matita/arithmetics/nat/increasing_to_monotonic.def(4)"\ 6increasing_to_monotonic\ 5/a\ 6 //
  |#nlenm #eqf @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 @(\ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6 … eqf) @\ 5a href="cic:/matita/basics/logic/sym_not_eq.def(4)"\ 6sym_not_eq\ 5/a\ 6 
   @\ 5a href="cic:/matita/arithmetics/nat/lt_to_not_eq.def(7)"\ 6lt_to_not_eq\ 5/a\ 6 @\ 5a href="cic:/matita/arithmetics/nat/increasing_to_monotonic.def(4)"\ 6increasing_to_monotonic\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/not_le_to_lt.def(5)"\ 6not_le_to_lt\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
  ]
qed.

(*********************** monotonicity ***************************)
theorem monotonic_le_plus_r: 
∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a href="cic:/matita/basics/relations/monotonic.def(1)"\ 6monotonic\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/le.ind(1,0,1)"\ 6le\ 5/a\ 6 (λm.n \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 m).
#n #a #b (elim n) normalize //
#m #H #leab @\ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/ qed.

(*
theorem le_plus_r: ∀p,n,m:nat. n ≤ m → p + n ≤ p + m
≝ monotonic_le_plus_r. *)

theorem monotonic_le_plus_l: 
∀m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a href="cic:/matita/basics/relations/monotonic.def(1)"\ 6monotonic\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/le.ind(1,0,1)"\ 6le\ 5/a\ 6 (λn.n \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 m).
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_plus_r.def(3)"\ 6monotonic_le_plus_r\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

(*
theorem le_plus_l: \forall p,n,m:nat. n \le m \to n + p \le m + p
\def monotonic_le_plus_l. *)

theorem le_plus: ∀n1,n2,m1,m2:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n1 \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n2  → m1 \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m2 
→ n1 \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 m1 \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n2 \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 m2.
#n1 #n2 #m1 #m2 #len #lem @(\ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6 ? (n1\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m2))
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_plus_l.def(6)"\ 6monotonic_le_plus_l\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_plus_r.def(3)"\ 6monotonic_le_plus_r\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

theorem le_plus_n :∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 m.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_plus_l.def(6)"\ 6monotonic_le_plus_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed. 

lemma le_plus_a: ∀a,n,m. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 m.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_plus.def(7)"\ 6le_plus\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

lemma le_plus_b: ∀b,n,m. n \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 b \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

theorem le_plus_n_r :∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 n.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/ qed.

theorem eq_plus_to_le: ∀n,m,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.n\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6m\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6p → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n.
// qed.

theorem le_plus_to_le: ∀a,n,m. a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 m → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
#a (elim a) normalize /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_pred.def(4)"\ 6monotonic_pred\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed. 

theorem le_plus_to_le_r: ∀a,n,m. n \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 a \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6a → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_plus_to_le.def(5)"\ 6le_plus_to_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed. 

(* plus &amp; lt *)

theorem monotonic_lt_plus_r: 
∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a href="cic:/matita/basics/relations/monotonic.def(1)"\ 6monotonic\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/lt.def(1)"\ 6lt\ 5/a\ 6 (λm.n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m).
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/increasing_to_monotonic.def(4)"\ 6increasing_to_monotonic\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

(*
variant lt_plus_r: \forall n,p,q:nat. p < q \to n + p < n + q \def
monotonic_lt_plus_r. *)

theorem monotonic_lt_plus_l: 
∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a href="cic:/matita/basics/relations/monotonic.def(1)"\ 6monotonic\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/lt.def(1)"\ 6lt\ 5/a\ 6 (λm.m\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6n).
(* /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/increasing_to_monotonic.def(4)"\ 6increasing_to_monotonic\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ *) #n @\ 5a href="cic:/matita/arithmetics/nat/increasing_to_monotonic.def(4)"\ 6increasing_to_monotonic\ 5/a\ 6 // 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:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → p \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 q → n \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 p \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 q.
#n #m #p #q #ltnm #ltpq
@(\ 5a href="cic:/matita/arithmetics/nat/transitive_lt.def(3)"\ 6transitive_lt\ 5/a\ 6 ? (n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6q))/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_plus_r.def(3)"\ 6monotonic_le_plus_r\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/monotonic_lt_plus_l.def(9)"\ 6monotonic_lt_plus_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

theorem lt_plus_to_lt_l :∀n,p,q:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. p\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 q\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6n → p\ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6q.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_plus_to_le.def(5)"\ 6le_plus_to_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

theorem lt_plus_to_lt_r :∀n,p,q:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6p \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6q → p\ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6q.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/lt_plus_to_lt_l.def(6)"\ 6lt_plus_to_lt_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

(*
theorem le_to_lt_to_lt_plus: ∀a,b,c,d:nat.
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:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a href="cic:/matita/basics/relations/monotonic.def(1)"\ 6monotonic\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/le.ind(1,0,1)"\ 6le\ 5/a\ 6 (λm. n \ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 m).
#n #x #y #lexy (elim n) normalize//(* lento /2/*)
#a #lea @\ 5a href="cic:/matita/arithmetics/nat/le_plus.def(7)"\ 6le_plus\ 5/a\ 6 //
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:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. 
n1 \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n2  → m1 \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m2 → n1\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m1 \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n2\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m2.
#n1 #n2 #m1 #m2 #len #lem @(\ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6 ? (n1\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m2)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_times_r.def(8)"\ 6monotonic_le_times_r\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.

(* interessante *)
theorem lt_times_n: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m.
#n #m #H /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_times_r.def(8)"\ 6monotonic_le_times_r\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

theorem le_times_to_le: 
∀a,n,m. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 a → a \ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 a \ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 m → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
#a @\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6 normalize
  [//
  |#n #H1 #H2 
     @(\ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6 ? (a\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_times_r.def(8)"\ 6monotonic_le_times_r\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
  |#n #m #H #lta #le
     @\ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6 @H /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_plus_to_le.def(5)"\ 6le_plus_to_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
  ]
qed.

(*
theorem 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:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 c → \ 5a href="cic:/matita/basics/relations/monotonic.def(1)"\ 6monotonic\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/lt.def(1)"\ 6lt\ 5/a\ 6 (λt.(c\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6t)).
#c #posc #n #m #ltnm
(elim ltnm) normalize
  [/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_lt_plus_l.def(9)"\ 6monotonic_lt_plus_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ 
  |#a #_ #lt1 @(\ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6 … lt1) //
  ]
qed.

theorem monotonic_lt_times_l: 
  ∀c:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 c → \ 5a href="cic:/matita/basics/relations/monotonic.def(1)"\ 6monotonic\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/lt.def(1)"\ 6lt\ 5/a\ 6 (λt.(t\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6c)).
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_lt_times_r.def(10)"\ 6monotonic_lt_times_r\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.

theorem lt_to_le_to_lt_times: 
∀n,m,p,q:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → p \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 q → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 q → n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6p \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6q.
#n #m #p #q #ltnm #lepq #posq
@(\ 5a href="cic:/matita/arithmetics/nat/le_to_lt_to_lt.def(4)"\ 6le_to_lt_to_lt\ 5/a\ 6 ? (n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6q))
  [@\ 5a href="cic:/matita/arithmetics/nat/monotonic_le_times_r.def(8)"\ 6monotonic_le_times_r\ 5/a\ 6 //
  |@\ 5a href="cic:/matita/arithmetics/nat/monotonic_lt_times_l.def(11)"\ 6monotonic_lt_times_l\ 5/a\ 6 //
  ]
qed.

theorem lt_times:∀n,m,p,q:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n\ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6m → p\ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6q → n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6p \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6q.
#n #m #p #q #ltnm #ltpq @\ 5a href="cic:/matita/arithmetics/nat/lt_to_le_to_lt_times.def(12)"\ 6lt_to_le_to_lt_times\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_plus_b.def(8)"\ 6le_plus_b\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/ltn_to_ltO.def(5)"\ 6ltn_to_ltO\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ 
qed.

theorem lt_times_n_to_lt_l: 
∀n,p,q:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. p\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 q\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6n → p \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 q.
#n #p #q #Hlt (elim (\ 5a href="cic:/matita/arithmetics/nat/decidable_lt.def(7)"\ 6decidable_lt\ 5/a\ 6 p q)) //
#nltpq @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 @(\ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6 ? ? (\ 5a href="cic:/matita/arithmetics/nat/lt_to_not_le.def(7)"\ 6lt_to_not_le\ 5/a\ 6 ? ? Hlt))
applyS \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_times_r.def(8)"\ 6monotonic_le_times_r\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/not_lt_to_le.def(6)"\ 6not_lt_to_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.

theorem lt_times_n_to_lt_r: 
∀n,p,q:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6p \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6q → p \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 q.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/lt_times_n_to_lt_l.def(9)"\ 6lt_times_n_to_lt_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

(*
theorem nat_compare_times_l : \forall n,p,q:nat. 
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 ⇒ \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6
 | S p ⇒ 
        match m with
          [ O ⇒ \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 p
    | S q ⇒ minus p q ]].
        
interpretation "natural minus" 'minus x y = (minus x y).

theorem minus_S_S: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m.
// qed.

theorem minus_O_n: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6n.
#n (cases n) // qed.

theorem minus_n_O: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.n\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6.
#n (cases n) // qed.

theorem minus_n_n: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6n.
#n (elim n) // qed.

theorem minus_Sn_n: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n)\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6n.
#n (elim n) normalize // qed.

theorem minus_Sn_m: ∀m,n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m).
(* qualcosa da capire qui 
#n #m #lenm nelim lenm napplyS refl_eq. *)
@\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6 
  [//
  |#n #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ 
  |#n #m #Hind #c applyS Hind /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_pred.def(4)"\ 6monotonic_pred\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
  ]
qed.

(*
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 \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/pred.def(1)"\ 6pred\ 5/a\ 6(n \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m).
@\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6 normalize //
qed.

theorem plus_minus:
∀m,n,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n → (n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m)\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6p \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6p)\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m.
@\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6 
  [//
  |#n #p #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
  |normalize/\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_pred.def(4)"\ 6monotonic_pred\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
  ]
qed.

theorem minus_plus_m_m: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m)\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/plus_minus.def(5)"\ 6plus_minus\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

theorem plus_minus_m_m: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.
  m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n → n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m)\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m.
#n #m #lemn @\ 5a href="cic:/matita/basics/logic/sym_eq.def(2)"\ 6sym_eq\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/plus_minus.def(5)"\ 6plus_minus\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

theorem le_plus_minus_m_m: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 (n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m)\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m.
#n (elim n) // #a #Hind #m (cases m) // normalize #n/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/  
qed.

theorem minus_to_plus :∀n,m,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.
  m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n → n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 p → n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 m\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6p.
#n #m #p #lemn #eqp (applyS \ 5a href="cic:/matita/arithmetics/nat/plus_minus_m_m.def(7)"\ 6plus_minus_m_m\ 5/a\ 6) //
qed.

theorem plus_to_minus :∀n,m,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 m\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6p → n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 p.
#n #m #p #eqp @\ 5a href="cic:/matita/basics/logic/sym_eq.def(2)"\ 6sym_eq\ 5/a\ 6 (applyS (\ 5a href="cic:/matita/arithmetics/nat/minus_plus_m_m.def(6)"\ 6minus_plus_m_m\ 5/a\ 6 p m))
qed.

theorem minus_pred_pred : ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → 
\ 5a href="cic:/matita/arithmetics/nat/pred.def(1)"\ 6pred\ 5/a\ 6 n \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/pred.def(1)"\ 6pred\ 5/a\ 6 m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 m.
#n #m #posn #posm @(\ 5a href="cic:/matita/arithmetics/nat/lt_O_n_elim.def(4)"\ 6lt_O_n_elim\ 5/a\ 6 n posn) @(\ 5a href="cic:/matita/arithmetics/nat/lt_O_n_elim.def(4)"\ 6lt_O_n_elim\ 5/a\ 6 m posm) //.
qed.


(*

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:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. q \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 p → q\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 p\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6n.
@\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6 #p #q
  [#lePO @(\ 5a href="cic:/matita/arithmetics/nat/le_n_O_elim.def(4)"\ 6le_n_O_elim\ 5/a\ 6 ? lePO) //
  |//
  |#Hind #n (cases n) // #a #leSS @Hind /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_pred.def(4)"\ 6monotonic_pred\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
  ]
qed.

theorem le_minus_to_plus: ∀n,m,p. n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 p → n\ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 p\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m.
#n #m #p #lep @\ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6
  [|@\ 5a href="cic:/matita/arithmetics/nat/le_plus_minus_m_m.def(9)"\ 6le_plus_minus_m_m\ 5/a\ 6 | @\ 5a href="cic:/matita/arithmetics/nat/monotonic_le_plus_l.def(6)"\ 6monotonic_le_plus_l\ 5/a\ 6 // ]
qed.

theorem le_minus_to_plus_r: ∀a,b,c. c \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 b → a \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 b \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 c → a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 c \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 b.
#a #b #c #Hlecb #H >(\ 5a href="cic:/matita/arithmetics/nat/plus_minus_m_m.def(7)"\ 6plus_minus_m_m\ 5/a\ 6 … Hlecb) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_minus_to_plus.def(10)"\ 6le_minus_to_plus\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.

theorem le_plus_to_minus: ∀n,m,p. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 p\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m → n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 p.
#n #m #p #lep /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_minus_l.def(9)"\ 6monotonic_le_minus_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

theorem le_plus_to_minus_r: ∀a,b,c. a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 b \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 c → a \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 c \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6b.
#a #b #c #H @(\ 5a href="cic:/matita/arithmetics/nat/le_plus_to_le_r.def(6)"\ 6le_plus_to_le_r\ 5/a\ 6 … b) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.

theorem lt_minus_to_plus: ∀a,b,c. a \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 b \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 c → a \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 c \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 b.
#a #b #c #H @\ 5a href="cic:/matita/arithmetics/nat/not_le_to_lt.def(5)"\ 6not_le_to_lt\ 5/a\ 6 
@(\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 … (\ 5a href="cic:/matita/arithmetics/nat/lt_to_not_le.def(7)"\ 6lt_to_not_le\ 5/a\ 6 …H)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_plus_to_minus_r.def(10)"\ 6le_plus_to_minus_r\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.

theorem lt_minus_to_plus_r: ∀a,b,c. a \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 b \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 c → a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 c \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 b.
#a #b #c #H @\ 5a href="cic:/matita/arithmetics/nat/not_le_to_lt.def(5)"\ 6not_le_to_lt\ 5/a\ 6 @(\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 … (\ 5a href="cic:/matita/arithmetics/nat/le_plus_to_minus.def(10)"\ 6le_plus_to_minus\ 5/a\ 6 …))
@\ 5a href="cic:/matita/arithmetics/nat/lt_to_not_le.def(7)"\ 6lt_to_not_le\ 5/a\ 6 //
qed.

theorem lt_plus_to_minus: ∀n,m,p. m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n → n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 p\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m → n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 p.
#n #m #p #lenm #H normalize <\ 5a href="cic:/matita/arithmetics/nat/minus_Sn_m.def(5)"\ 6minus_Sn_m\ 5/a\ 6 // @\ 5a href="cic:/matita/arithmetics/nat/le_plus_to_minus.def(10)"\ 6le_plus_to_minus\ 5/a\ 6 //
qed.

theorem lt_plus_to_minus_r: ∀a,b,c. a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 b \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 c → a \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 c \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 b.
#a #b #c #H @\ 5a href="cic:/matita/arithmetics/nat/le_plus_to_minus_r.def(10)"\ 6le_plus_to_minus_r\ 5/a\ 6 //
qed. 

theorem monotonic_le_minus_r: 
∀p,q,n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. q \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 p → n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6p \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6q.
#p #q #n #lepq @\ 5a href="cic:/matita/arithmetics/nat/le_plus_to_minus.def(10)"\ 6le_plus_to_minus\ 5/a\ 6 
@(\ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6 … (\ 5a href="cic:/matita/arithmetics/nat/le_plus_minus_m_m.def(9)"\ 6le_plus_minus_m_m\ 5/a\ 6 ? q)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_plus_r.def(3)"\ 6monotonic_le_plus_r\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.

theorem monotonic_lt_minus_l: ∀p,q,n. n ≤ q → q < p → q - n < p - n.
#p #q #n #H1 #H2
@lt_plus_to_minus_r <plus_minus_m_m //
qed.

theorem eq_minus_O: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.
  n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6.
#n #m #lenm @(\ 5a href="cic:/matita/arithmetics/nat/le_n_O_elim.def(4)"\ 6le_n_O_elim\ 5/a\ 6 (n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_minus_r.def(11)"\ 6monotonic_le_minus_r\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.

theorem distributive_times_minus: \ 5a href="cic:/matita/basics/relations/distributive.def(1)"\ 6distributive\ 5/a\ 6 ? \ 5a href="cic:/matita/arithmetics/nat/times.fix(0,0,2)"\ 6times\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/minus.fix(0,0,1)"\ 6minus\ 5/a\ 6.
#a #b #c
(cases (\ 5a href="cic:/matita/arithmetics/nat/decidable_lt.def(7)"\ 6decidable_lt\ 5/a\ 6 b c)) #Hbc
 [> \ 5a href="cic:/matita/arithmetics/nat/eq_minus_O.def(12)"\ 6eq_minus_O\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_plus_b.def(8)"\ 6le_plus_b\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ >\ 5a href="cic:/matita/arithmetics/nat/eq_minus_O.def(12)"\ 6eq_minus_O\ 5/a\ 6 // 
  @\ 5a href="cic:/matita/arithmetics/nat/monotonic_le_times_r.def(8)"\ 6monotonic_le_times_r\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_plus_b.def(8)"\ 6le_plus_b\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
 |@\ 5a href="cic:/matita/basics/logic/sym_eq.def(2)"\ 6sym_eq\ 5/a\ 6 (applyS \ 5a href="cic:/matita/arithmetics/nat/plus_to_minus.def(7)"\ 6plus_to_minus\ 5/a\ 6) <\ 5a href="cic:/matita/arithmetics/nat/distributive_times_plus.def(7)"\ 6distributive_times_plus\ 5/a\ 6 
  @\ 5a href="cic:/matita/basics/logic/eq_f.def(3)"\ 6eq_f\ 5/a\ 6 (applyS \ 5a href="cic:/matita/arithmetics/nat/plus_minus_m_m.def(7)"\ 6plus_minus_m_m\ 5/a\ 6) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/not_lt_to_le.def(6)"\ 6not_lt_to_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.

theorem minus_plus: ∀n,m,p. n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6p \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6(m\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6p).
#n #m #p 
cases (\ 5a href="cic:/matita/arithmetics/nat/decidable_le.def(6)"\ 6decidable_le\ 5/a\ 6 (m\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6p) n) #Hlt
  [@\ 5a href="cic:/matita/arithmetics/nat/plus_to_minus.def(7)"\ 6plus_to_minus\ 5/a\ 6 @\ 5a href="cic:/matita/arithmetics/nat/plus_to_minus.def(7)"\ 6plus_to_minus\ 5/a\ 6 <\ 5a href="cic:/matita/arithmetics/nat/associative_plus.def(4)"\ 6associative_plus\ 5/a\ 6
   @\ 5a href="cic:/matita/arithmetics/nat/minus_to_plus.def(8)"\ 6minus_to_plus\ 5/a\ 6 //
  |cut (n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6p) [@(\ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6 … (\ 5a href="cic:/matita/arithmetics/nat/le_n_Sn.def(1)"\ 6le_n_Sn\ 5/a\ 6 …)) @\ 5a href="cic:/matita/arithmetics/nat/not_le_to_lt.def(5)"\ 6not_le_to_lt\ 5/a\ 6 //]
   #H >\ 5a href="cic:/matita/arithmetics/nat/eq_minus_O.def(12)"\ 6eq_minus_O\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/eq_minus_O.def(12)"\ 6eq_minus_O\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_minus_l.def(9)"\ 6monotonic_le_minus_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ 
  ]
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:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. p \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n →
  p\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6(n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6(m\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6p).
#n #m #p #lepm #lemn
@\ 5a href="cic:/matita/basics/logic/sym_eq.def(2)"\ 6sym_eq\ 5/a\ 6 @\ 5a href="cic:/matita/arithmetics/nat/plus_to_minus.def(7)"\ 6plus_to_minus\ 5/a\ 6 <\ 5a href="cic:/matita/arithmetics/nat/associative_plus.def(4)"\ 6associative_plus\ 5/a\ 6 <\ 5a href="cic:/matita/arithmetics/nat/plus_minus_m_m.def(7)"\ 6plus_minus_m_m\ 5/a\ 6 //
<\ 5a href="cic:/matita/arithmetics/nat/commutative_plus.def(5)"\ 6commutative_plus\ 5/a\ 6 <\ 5a href="cic:/matita/arithmetics/nat/plus_minus_m_m.def(7)"\ 6plus_minus_m_m\ 5/a\ 6 //
qed.

(*********************** boolean arithmetics ********************) 
include "basics/bool.ma".

let rec eqb n m ≝ 
match n with 
  [ O ⇒ match m with [ O ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 | S q ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6] 
  | S p ⇒ match m with [ O ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 | S q ⇒ eqb p q]
  ].

theorem eqb_elim : ∀ n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.∀ P:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 → Prop.
(n\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6m → (P \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6)) → (n \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 m → (P \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6)) → (P (\ 5a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"\ 6eqb\ 5/a\ 6 n m)). 
@\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6 
  [#n (cases n) normalize /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/ 
  |normalize /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/sym_not_eq.def(4)"\ 6sym_not_eq\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
  |normalize /\ 5span class="autotactic"\ 64\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/not_eq_S.def(4)"\ 6not_eq_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ 
  ] 
qed.

theorem eqb_n_n: ∀n. \ 5a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"\ 6eqb\ 5/a\ 6 n n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#n (elim n) normalize // qed. 

theorem eqb_true_to_eq: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"\ 6eqb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 m.
#n #m @(\ 5a href="cic:/matita/arithmetics/nat/eqb_elim.def(5)"\ 6eqb_elim\ 5/a\ 6 n m) // #_ #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

theorem eqb_false_to_not_eq: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"\ 6eqb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 → n \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 m.
#n #m @(\ 5a href="cic:/matita/arithmetics/nat/eqb_elim.def(5)"\ 6eqb_elim\ 5/a\ 6 n m) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

theorem eq_to_eqb_true: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 m → \ 5a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"\ 6eqb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
// qed.

theorem not_eq_to_eqb_false: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.
  n \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6  m → \ 5a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"\ 6eqb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6.
#n #m #noteq @\ 5a href="cic:/matita/arithmetics/nat/eqb_elim.def(5)"\ 6eqb_elim\ 5/a\ 6// #Heq @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

let rec leb n m ≝ 
match n with 
    [ O ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6
    | (S p) ⇒
        match m with 
        [ O ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6
              | (S q) ⇒ leb p q]].

theorem leb_elim: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. ∀P:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 → Prop. 
(n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → P \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6) → (n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 m → P \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6) → P (\ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m).
@\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6 normalize
  [/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/
  |/\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/
  |#n #m #Hind #P #Pt #Pf @Hind
    [#lenm @Pt @\ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6 // |#nlenm @Pf /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/not_le_to_not_le_S_S.def(5)"\ 6not_le_to_not_le_S_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ ]
  ]
qed.

theorem leb_true_to_le:∀n,m.\ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
#n #m @\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6 // #_ #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

theorem leb_false_to_not_le:∀n,m.
  \ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 → n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 m.
#n #m @\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6 // #_ #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

theorem le_to_leb_true: ∀n,m. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → \ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#n #m @\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6 // #H #H1 @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

theorem not_le_to_leb_false: ∀n,m. n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 m → \ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6.
#n #m @\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6 // #H #H1 @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

theorem lt_to_leb_false: ∀n,m. m \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n → \ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6.
/\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/lt_to_not_le.def(7)"\ 6lt_to_not_le\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/not_le_to_leb_false.def(7)"\ 6not_le_to_leb_false\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ 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: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 →\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 →\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 ≝
λn.λm. if (\ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m) then n else m.

definition max: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 →\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 →\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 ≝
λn.λm. if (\ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m) then m else n.

lemma commutative_min: \ 5a href="cic:/matita/basics/relations/commutative.def(1)"\ 6commutative\ 5/a\ 6 ? \ 5a href="cic:/matita/arithmetics/nat/min.def(2)"\ 6min\ 5/a\ 6.
#n #m normalize @\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6 
  [@\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6 normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_to_le_to_eq.def(5)"\ 6le_to_le_to_eq\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
  |#notle >(\ 5a href="cic:/matita/arithmetics/nat/le_to_leb_true.def(7)"\ 6le_to_leb_true\ 5/a\ 6 …) // @(\ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6 ? (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/not_le_to_lt.def(5)"\ 6not_le_to_lt\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
  ] qed.

lemma le_minr: ∀i,n,m. i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/min.def(2)"\ 6min\ 5/a\ 6 n m → i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
#i #n #m normalize @\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6 normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed. 

lemma le_minl: ∀i,n,m. i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/min.def(2)"\ 6min\ 5/a\ 6 n m → i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_minr.def(7)"\ 6le_minr\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

lemma to_min: ∀i,n,m. i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n → i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/min.def(2)"\ 6min\ 5/a\ 6 n m.
#i #n #m #lein #leim normalize (cases (\ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m)) 
normalize // qed.

lemma commutative_max: \ 5a href="cic:/matita/basics/relations/commutative.def(1)"\ 6commutative\ 5/a\ 6 ? \ 5a href="cic:/matita/arithmetics/nat/max.def(2)"\ 6max\ 5/a\ 6.
#n #m normalize @\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6 
  [@\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6 normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_to_le_to_eq.def(5)"\ 6le_to_le_to_eq\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
  |#notle >(\ 5a href="cic:/matita/arithmetics/nat/le_to_leb_true.def(7)"\ 6le_to_leb_true\ 5/a\ 6 …) // @(\ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6 ? (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/not_le_to_lt.def(5)"\ 6not_le_to_lt\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
  ] qed.

lemma le_maxl: ∀i,n,m. \ 5a href="cic:/matita/arithmetics/nat/max.def(2)"\ 6max\ 5/a\ 6 n m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i.
#i #n #m normalize @\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6 normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed. 

lemma le_maxr: ∀i,n,m. \ 5a href="cic:/matita/arithmetics/nat/max.def(2)"\ 6max\ 5/a\ 6 n m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_maxl.def(7)"\ 6le_maxl\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.

lemma to_max: ∀i,n,m. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i → \ 5a href="cic:/matita/arithmetics/nat/max.def(2)"\ 6max\ 5/a\ 6 n m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i.
#i #n #m #leni #lemi normalize (cases (\ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m)) 
normalize // qed.