X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=helm%2Fsoftware%2Fmatita%2Fnlibrary%2Farithmetics%2Fnat.ma;h=3152b0dbe3503aeee43bbd0f872b847f87979239;hb=eb4144a401147a44a9620169eb6dafeb8f5a2c17;hp=b6683b267801bac94d220ad5ca573ac644eaf1a5;hpb=8899a3f240f62633f4df58b2ee358fa285a82d1d;p=helm.git

diff --git a/helm/software/matita/nlibrary/arithmetics/nat.ma b/helm/software/matita/nlibrary/arithmetics/nat.ma
index b6683b267..3152b0dbe 100644
--- a/helm/software/matita/nlibrary/arithmetics/nat.ma
+++ b/helm/software/matita/nlibrary/arithmetics/nat.ma
@@ -310,7 +310,8 @@ ntheorem lt_to_not_le: ∀n,m. n < m → m ≰ n.
 
 ntheorem not_lt_to_le: ∀n,m:nat. n ≮ m → m ≤ n.
 #n; #m; #Hnlt; napply lt_S_to_le;
-(* something strange here: /2/ fails *)
+(* something strange here: /2/ fails: 
+   we need an extra depths for unfolding not *)
 napply not_le_to_lt; napply Hnlt; nqed. 
 
 ntheorem le_to_not_lt: ∀n,m:nat. n ≤ m → m ≮ n.
@@ -528,21 +529,23 @@ apply le_to_or_lt_eq.apply H6.
 qed.
 *)
 
-(******************* monotonicity ******************************)
+(*********************** monotonicity ***************************)
 ntheorem monotonic_le_plus_r: 
 ∀n:nat.monotonic nat le (λm.n + m).
 #n; #a; #b; nelim n; nnormalize; //;
 #m; #H; #leab;napply le_S_S; /2/; nqed.
 
+(*
 ntheorem le_plus_r: ∀p,n,m:nat. n ≤ m → p + n ≤ p + m
-≝ monotonic_le_plus_r.
+≝ monotonic_le_plus_r. *)
 
 ntheorem monotonic_le_plus_l: 
 ∀m:nat.monotonic nat le (λn.n + m).
 /2/; nqed.
 
+(*
 ntheorem le_plus_l: \forall p,n,m:nat. n \le m \to n + p \le m + p
-\def monotonic_le_plus_l. 
+\def monotonic_le_plus_l. *)
 
 ntheorem le_plus: ∀n1,n2,m1,m2:nat. n1 ≤ n2  \to m1 ≤ m2 
 → n1 + m1 ≤ n2 + m2.
@@ -559,88 +562,234 @@ ntheorem eq_plus_to_le: ∀n,m,p:nat.n=m+p → m ≤ n.
 //; nqed.
 
 ntheorem le_plus_to_le: ∀a,n,m. a + n ≤ a + m → n ≤ m.
-#a; nelim a; /3/; nqed.
-
-(* times 
-theorem monotonic_le_times_r: 
-\forall n:nat.monotonic nat le (\lambda m. n * m).
-simplify.intros.elim n.
-simplify.apply le_O_n.
-simplify.apply le_plus.
-assumption.
-assumption.
-qed.
+#a; nelim a; /3/; nqed. 
 
-theorem le_times_r: \forall p,n,m:nat. n \le m \to p*n \le p*m
-\def monotonic_le_times_r.
+ntheorem le_plus_to_le_r: ∀a,n,m. n + a ≤ m +a → n ≤ m.
+/2/; nqed. 
 
-theorem monotonic_le_times_l: 
-\forall m:nat.monotonic nat le (\lambda n.n*m).
-simplify.intros.
-rewrite < sym_times.rewrite < (sym_times m).
-apply le_times_r.assumption.
-qed.
+(* times *)
+ntheorem monotonic_le_times_r: 
+∀n:nat.monotonic nat le (λm. n * m).
+#n; #x; #y; #lexy; nelim n; nnormalize;//;(* lento /2/;*)
+#a; #lea; napply le_plus; //;
+nqed.
+
+(*
+ntheorem le_times_r: \forall p,n,m:nat. n \le m \to p*n \le p*m
+\def monotonic_le_times_r. *)
+
+ntheorem monotonic_le_times_l: 
+∀m:nat.monotonic nat le (λn.n*m).
+/2/; nqed.
 
+(*
 theorem le_times_l: \forall p,n,m:nat. n \le m \to n*p \le m*p
-\def monotonic_le_times_l.
+\def monotonic_le_times_l. *)
+
+ntheorem le_times: ∀n1,n2,m1,m2:nat. 
+n1 ≤ n2  → m1 ≤ m2 → n1*m1 ≤ n2*m2.
+#n1; #n2; #m1; #m2; #len; #lem; 
+napply transitive_le; (* /2/ slow *)
+ ##[ ##| napply monotonic_le_times_l;//; 
+     ##| napply monotonic_le_times_r;//;
+ ##]
+nqed.
+
+ntheorem lt_times_n: ∀n,m:nat. O < n → m ≤ n*m.
+(* bello *)
+/2/; nqed.
+
+ntheorem le_times_to_le: 
+∀a,n,m. O < a → a * n ≤ a * m → n ≤ m.
+#a; napply nat_elim2; nnormalize;
+  ##[//;
+  ##|#n; #H1; #H2; napply False_ind;
+     ngeneralize in match H2;
+     napply lt_to_not_le;
+     napply (transitive_le ? (S n));/2/;
+  ##|#n; #m; #H; #lta; #le;
+     napply le_S_S; napply H; /2/;
+  ##]
+nqed.
+
+ntheorem le_S_times_2: ∀n,m.O < m → n ≤ m → n < 2*m.
+#n; #m; #posm; #lenm; (* interessante *)
+nnormalize; napplyS (le_plus n); //; nqed.
+
+(************************** minus ******************************)
+
+nlet rec minus n m ≝ 
+ match n with 
+ [ O ⇒ O
+ | S p ⇒ 
+	match m with
+	  [ O ⇒ S p
+    | S q ⇒ minus p q ]].
+        
+interpretation "natural minus" 'minus x y = (minus x y).
+
+ntheorem minus_S_S: ∀n,m:nat.S n - S m = n -m.
+//; nqed.
+
+ntheorem minus_O_n: ∀n:nat.O=O-n.
+#n; ncases n; //; nqed.
+
+ntheorem minus_n_O: ∀n:nat.n=n-O.
+#n; ncases n; //; nqed.
+
+ntheorem minus_n_n: ∀n:nat.O=n-n.
+#n; nelim n; //; nqed.
+
+ntheorem minus_Sn_n: ∀n:nat. S O = (S n)-n.
+#n; nelim n; //; nqed.
+
+ntheorem minus_Sn_m: ∀m,n:nat. m ≤ n → S n -m = S (n-m).
+(* qualcosa da capire qui 
+#n; #m; #lenm; nelim lenm; napplyS refl_eq. *)
+napply nat_elim2; 
+  ##[//
+  ##|#n; #abs; napply False_ind;/2/;
+  ##|/3/;
+  ##]
+nqed.
+
+ntheorem eq_minus_S_pred: ∀n,m. n - (S m) = pred(n -m).
+napply nat_elim2; //; nqed.
+
+ntheorem plus_minus:
+∀m,n,p:nat. m ≤ n → (n-m)+p = (n+p)-m.
+napply nat_elim2; 
+  ##[//
+  ##|#n; #p; #abs; napply False_ind;/2/;
+  ##|nnormalize;/3/;
+  ##]
+nqed.
 
-theorem le_times: \forall n1,n2,m1,m2:nat. n1 \le n2  \to m1 \le m2 
-\to n1*m1 \le n2*m2.
+ntheorem minus_plus_m_m: ∀n,m:nat.n = (n+m)-m.
+#n; #m; napplyS (plus_minus m m n); //; nqed.
+
+ntheorem plus_minus_m_m: ∀n,m:nat.
+m \leq n \to n = (n-m)+m.
+#n; #m; #lemn; napplyS symmetric_eq; 
+napplyS (plus_minus m n m); //; nqed.
+
+ntheorem le_plus_minus_m_m: ∀n,m:nat. n ≤ (n-m)+m.
+#n; nelim n;
+  ##[//
+  ##|#a; #Hind; #m; ncases m;/2/;  
+  ##]
+nqed.
+
+ntheorem minus_to_plus :∀n,m,p:nat.
+  m ≤ n → n-m = p → n = m+p.
+#n; #m; #p; #lemn; #eqp; napplyS plus_minus_m_m; //;
+nqed.
+
+ntheorem plus_to_minus :∀n,m,p:nat.n = m+p → n-m = p.
+(* /4/ done in 43.5 *)
+#n; #m; #p; #eqp; 
+napply symmetric_eq;
+napplyS (minus_plus_m_m p m);
+nqed.
+
+ntheorem minus_pred_pred : ∀n,m:nat. O < n → O < m → 
+pred n - pred m = n - m.
+#n; #m; #posn; #posm;
+napply (lt_O_n_elim n posn);
+napply (lt_O_n_elim m posm);//.
+nqed.
+
+(*
+theorem eq_minus_n_m_O: \forall n,m:nat.
+n \leq m \to n-m = O.
+intros 2.
+apply (nat_elim2 (\lambda n,m.n \leq m \to n-m = O)).
+intros.simplify.reflexivity.
+intros.apply False_ind.
+apply not_le_Sn_O;
+[2: apply H | skip].
 intros.
-apply (trans_le ? (n2*m1)).
-apply le_times_l.assumption.
-apply le_times_r.assumption.
+simplify.apply H.apply le_S_S_to_le. apply H1.
 qed.
 
-theorem le_times_n: \forall n,m:nat.(S O) \le n \to m \le n*m.
-intros.elim H.simplify.
-elim (plus_n_O ?).apply le_n.
-simplify.rewrite < sym_plus.apply le_plus_n.
+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 le_times_to_le: 
-\forall a,n,m. S O \le a \to a * n \le a * m \to n \le m.
-intro.
-apply nat_elim2;intros
-  [apply le_O_n
-  |apply False_ind.
-   rewrite < times_n_O in H1.
-   generalize in match H1.
-   apply (lt_O_n_elim ? H).
-   intros.
-   simplify in H2.
-   apply (le_to_not_lt ? ? H2).
-   apply lt_O_S
-  |apply le_S_S.
-   apply H
-    [assumption
-    |rewrite < times_n_Sm in H2.
-     rewrite < times_n_Sm in H2.
-     apply (le_plus_to_le a).
-     assumption
-    ]
-  ]
+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 le_S_times_SSO: \forall n,m.O < m \to
-n \le m \to S n \le (S(S O))*m.
-intros.
-simplify.
-rewrite > plus_n_O.
-simplify.rewrite > plus_n_Sm.
-apply le_plus
-  [assumption
-  |rewrite < plus_n_O.
-   assumption
-  ]
+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.
-(*0 and times *)
-theorem O_lt_const_to_le_times_const:  \forall a,c:nat.
-O \lt c \to a \le a*c.
-intros.
-rewrite > (times_n_SO a) in \vdash (? % ?).
-apply le_times
-[ apply le_n
-| assumption
-]
-qed. *)
\ No newline at end of file
+
+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 *)
+
+ntheorem monotonic_le_minus_l: 
+∀p,q,n:nat. q ≤ p → q-n ≤ p-n.
+napply nat_elim2; #p; #q;
+  ##[#lePO; napply (le_n_O_elim ? lePO);//;
+  ##|//;
+  ##|#Hind; #n; ncases n;
+    ##[//;
+    ##|#a; #leSS; napply Hind; /2/;
+    ##]
+  ##]
+nqed.
+
+ntheorem le_minus_to_plus: ∀n,m,p. n-m ≤ p → n≤ p+m.
+#n; #m; #p; #lep;
+napply transitive_le;
+  ##[##|napply le_plus_minus_m_m
+  ##|napply monotonic_le_plus_l;//;
+  ##]
+nqed.
+
+ntheorem le_plus_to_minus: ∀n,m,p. n ≤ p+m → n-m ≤ p.
+#n; #m; #p; #lep;
+(* bello *)
+napplyS monotonic_le_minus_l;//;
+nqed.
+
+ntheorem monotonic_le_minus_r: 
+∀p,q,n:nat. q ≤ p → n-p ≤ n-q.
+#p; #q; #n; #lepq;
+napply le_plus_to_minus;
+napply (transitive_le ??? (le_plus_minus_m_m ? q));/2/;
+nqed.