From 68710656f43bcc362a14df381cc03dc92feab73d Mon Sep 17 00:00:00 2001
From: Andrea Asperti <andrea.asperti@unibo.it>
Date: Wed, 10 Feb 2010 09:23:29 +0000
Subject: [PATCH 1/1] addenda

---
 .../matita/nlibrary/arithmetics/nat.ma        | 66 ++++++++-----------
 1 file changed, 29 insertions(+), 37 deletions(-)

diff --git a/helm/software/matita/nlibrary/arithmetics/nat.ma b/helm/software/matita/nlibrary/arithmetics/nat.ma
index d1e401ae5..d8ac55049 100644
--- a/helm/software/matita/nlibrary/arithmetics/nat.ma
+++ b/helm/software/matita/nlibrary/arithmetics/nat.ma
@@ -78,7 +78,7 @@ ntheorem nat_elim2 :
 
 ntheorem decidable_eq_nat : ∀n,m:nat.decidable (n=m).
 napply nat_elim2; #n;
- ##[ ncases n; /2/;
+ ##[ ncases n; /3/;
  ##| /3/;
  ##| #m; #Hind; ncases Hind; /3/;
  ##]
@@ -285,14 +285,14 @@ ntheorem not_le_S_S_to_not_le: ∀ n,m:nat. S n ≰ S m → n ≰ m.
 /3/; nqed.
 
 ntheorem decidable_le: ∀n,m. decidable (n≤m).
-napply nat_elim2; #n; /2/;
-#m; #dec; ncases dec;/3/; nqed.
+napply nat_elim2; #n; /3/;
+#m; #dec; ncases dec;/4/; nqed.
 
 ntheorem decidable_lt: ∀n,m. decidable (n < m).
 #n; #m; napply decidable_le ; nqed.
 
 ntheorem not_le_Sn_n: ∀n:nat. S n ≰ n.
-#n; nelim n; /2/; nqed.
+#n; nelim n; /3/; nqed.
 
 ntheorem lt_S_to_le: ∀n,m:nat. n < S m → n ≤ m.
 /2/; nqed.
@@ -301,7 +301,7 @@ ntheorem not_le_to_lt: ∀n,m. n ≰ m → m < n.
 napply nat_elim2; #n;
  ##[#abs; napply False_ind;/2/;
  ##|/2/;
- ##|#m;#Hind;#HnotleSS; napply lt_to_lt_S_S;/3/;
+ ##|#m;#Hind;#HnotleSS; napply lt_to_lt_S_S;/4/;
  ##]
 nqed.
 
@@ -568,9 +568,10 @@ ntheorem le_plus_to_le_r: ∀a,n,m. n + a ≤ m +a → n ≤ m.
 /2/; nqed. 
 
 (* plus & lt *)
+
 ntheorem monotonic_lt_plus_r: 
 ∀n:nat.monotonic nat lt (λm.n+m).
-/2/; nqed. 
+/2/; nqed.
 
 (*
 variant lt_plus_r: \forall n,p,q:nat. p < q \to n + p < n + q \def
@@ -749,8 +750,7 @@ ntheorem lt_times_n_to_lt_l:
 nelim (decidable_lt p q);//;
 #nltpq;napply False_ind; 
 napply (lt_to_not_le ? ? Hlt);
-napply monotonic_le_times_l.
-napply not_lt_to_le; //;
+napply monotonic_le_times_l;/3/;
 nqed.
 
 ntheorem lt_times_n_to_lt_r: 
@@ -840,6 +840,13 @@ napply nat_elim2;
   ##]
 nqed.
 
+ntheorem not_eq_to_le_to_le_minus: 
+  ∀n,m.n ≠ m → n ≤ m → n ≤ m - 1.
+#n; #m; ncases m;//; #m; nnormalize;
+#H; #H1; napply le_S_S_to_le;
+napplyS (not_eq_to_le_to_lt n (S m) H H1);
+nqed.
+
 ntheorem eq_minus_S_pred: ∀n,m. n - (S m) = pred(n -m).
 napply nat_elim2; //; nqed.
 
@@ -1013,43 +1020,28 @@ elim ((eqb n1 m1)).
 simplify.apply eq_f.apply H1.
 simplify.unfold Not.intro.apply H1.apply inj_S.assumption.
 qed.
-
-theorem eqb_elim : \forall n,m:nat.\forall P:bool \to Prop.
-(n=m \to (P true)) \to (n \neq m \to (P false)) \to (P (eqb n m)). 
-intros.
-cut 
-(match (eqb n m) with
-[ true  \Rightarrow n = m
-| false \Rightarrow n \neq m] \to (P (eqb n m))).
-apply Hcut.apply eqb_to_Prop.
-elim (eqb n m).
-apply ((H H2)).
-apply ((H1 H2)).
-qed. 
-
 *)
 
+ntheorem eqb_elim : ∀ n,m:nat.∀ P:bool → Prop.
+(n=m → (P true)) → (n ≠ m → (P false)) → (P (eqb n m)). 
+napply nat_elim2; 
+  ##[#n; ncases n; nnormalize; /3/; 
+  ##|nnormalize; /3/;
+  ##|nnormalize; /4/;
+  ##]
+nqed.
+
 ntheorem eqb_n_n: ∀n. eqb n n = true.
 #n; nelim n; nnormalize; //.
 nqed. 
 
 ntheorem eqb_true_to_eq: ∀n,m:nat. eqb n m = true → n = m.
-napply nat_elim2; 
-  ##[#n; ncases n; nnormalize; //; 
-     #m; #abs; napply False_ind;/2/;
-  ##|nnormalize; #m; #abs; napply False_ind;/2/;
-  ##|nnormalize; 
-     #n; #m; #Hind; #eqnm; napply eq_f; napply Hind; //;
-  ##]
+#n; #m; napply (eqb_elim n m);//;
+#_; #abs; napply False_ind;/2/;
 nqed.
 
-ntheorem eqb_false_to_not_eq: ∀n,m:nat.
-  eqb n m = false → n ≠ m.
-napply nat_elim2; 
-  ##[#n; ncases n; nnormalize; /2/; 
-  ##|/2/;
-  ##|nnormalize;/2/; 
-  ##]
+ntheorem eqb_false_to_not_eq: ∀n,m:nat. eqb n m = false → n ≠ m.
+#n; #m; napply (eqb_elim n m);/2/;
 nqed.
 
 ntheorem eq_to_eqb_true: ∀n,m:nat.
@@ -1094,7 +1086,7 @@ ntheorem leb_false_to_not_le:∀n,m.
   leb n m = false → n ≰ m.
 #n; #m; napply leb_elim;
   ##[#_; #abs; napply False_ind; /2/;
-  ##|//;
+  ##|/2/;
   ##]
 nqed.
 
-- 
2.39.2