X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Farithmetics%2Fnat.ma;h=edb99b8c0c9dac85517e253e9282d7f34597bd89;hb=8a5391064ec6e84333444453f836ef9cb91fee7b;hp=cc6b19debaf147377cf02e9bfb39c26eea1201a5;hpb=bdc0a7a8c1de693a40f116742f5d2d3f3290c381;p=helm.git

diff --git a/matita/matita/lib/arithmetics/nat.ma b/matita/matita/lib/arithmetics/nat.ma
index cc6b19deb..edb99b8c0 100644
--- a/matita/matita/lib/arithmetics/nat.ma
+++ b/matita/matita/lib/arithmetics/nat.ma
@@ -135,9 +135,6 @@ theorem injective_plus_r: ∀n:nat.injective nat nat (λm.n+m).
 theorem injective_plus_l: ∀n:nat.injective nat nat (λm.m+n). 
 /2/ qed.
 
-theorem not_eq_S: ∀n,m:nat. n ≠ m → S n ≠ S m.
-/2/ qed.
-
 theorem times_Sn_m: ∀n,m:nat. m+n*m = S n*m.
 // qed.
 
@@ -192,6 +189,9 @@ lemma plus_plus_comm_23: ∀x,y,z. x + y + z = x + z + y.
 
 (* Negated equalities *******************************************************)
 
+theorem not_eq_S: ∀n,m:nat. n ≠ m → S n ≠ S m.
+/2/ qed.
+
 theorem not_eq_O_S : ∀n:nat. 0 ≠ S n.
 #n @nmk #eqOS (change with (not_zero O)) >eqOS // qed.
 
@@ -242,7 +242,7 @@ theorem monotonic_le_plus_l:
 theorem le_plus: ∀n1,n2,m1,m2:nat. n1 ≤ n2  → m1 ≤ m2 
 → n1 + m1 ≤ n2 + m2.
 #n1 #n2 #m1 #m2 #len #lem @(transitive_le ? (n1+m2))
-/2/ qed-.
+/2/ qed.
 
 theorem le_plus_n :∀n,m:nat. m ≤ n + m.
 /2/ qed. 
@@ -299,6 +299,12 @@ theorem le_plus_to_minus_r: ∀a,b,c. a + b ≤ c → a ≤ c -b.
 #a #b #c #H @(le_plus_to_le_r … b) /2/
 qed.
 
+lemma lt_to_le: ∀x,y. x < y → x ≤ y.
+/2 width=2/ qed.
+
+lemma inv_eq_minus_O: ∀x,y. x - y = 0 → x ≤ y.
+// qed-.
+
 (* lt *)
 
 theorem transitive_lt: transitive nat lt.
@@ -595,8 +601,13 @@ theorem increasing_to_le: ∀f:nat → nat.
 @(ex_intro ?? (S a)) /2/
 qed.
 
+(* thus is le_plus
 lemma le_plus_compatible: ∀x1,x2,y1,y2. x1 ≤ y1 → x2 ≤ y2 → x1 + x2 ≤ y1 + y2.
-/3 by le_minus_to_plus, monotonic_le_plus_r, transitive_le/ qed.
+#x1 #y1 #x2 #y2 #H1 #H2 /2/ @le_plus // /2/ /3 by le_minus_to_plus, monotonic_le_plus_r, transitive_le/ qed.
+*)
+
+lemma minus_le: ∀x,y. x - y ≤ x.
+/2 width=1/ qed.
 
 (* lt *)
 
@@ -634,6 +645,14 @@ theorem monotonic_lt_minus_l: ∀p,q,n. n ≤ q → q < p → q - n < p - n.
 @lt_plus_to_minus_r <plus_minus_m_m //
 qed.
 
+(* More compound conclusion *************************************************)
+
+lemma discr_minus_x_xy: ∀x,y. x = x - y → x = 0 ∨ y = 0.
+* /2 width=1/ #x * /2 width=1/ #y normalize #H 
+lapply (minus_le x y) <H -H #H
+elim (not_le_Sn_n x) #H0 elim (H0 ?) //
+qed-.
+
 (* Still more equalities ****************************************************)
 
 theorem eq_minus_O: ∀n,m:nat.
@@ -677,6 +696,9 @@ qed.
 lemma minus_minus_m_m: ∀m,n. n ≤ m → m - (m - n) = n.
 /2 width=1/ qed.
 
+lemma minus_plus_plus_l: ∀x,y,h. (x + h) - (y + h) = x - y.
+// qed.
+
 (* Stilll more atomic conclusion ********************************************)
 
 (* le *)
@@ -794,4 +816,3 @@ lemma le_maxr: ∀i,n,m. max n m ≤ i → m ≤ i.
 lemma to_max: ∀i,n,m. n ≤ i → m ≤ i → max n m ≤ i.
 #i #n #m #leni #lemi normalize (cases (leb n m)) 
 normalize // qed.
-