X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fnlibrary%2Farithmetics%2Fnat.ma;h=0418444e376e0ecd47ca07a83b8915b1aa8cbc57;hb=221472ea1597505d12677f5742e388125a15e2b9;hp=28764884064208238dd7966591bdfbd5ca53e56f;hpb=a581a54f6118908a21ed3f960c70a0a2c863ca89;p=helm.git

diff --git a/helm/software/matita/nlibrary/arithmetics/nat.ma b/helm/software/matita/nlibrary/arithmetics/nat.ma
index 287648840..0418444e3 100644
--- a/helm/software/matita/nlibrary/arithmetics/nat.ma
+++ b/helm/software/matita/nlibrary/arithmetics/nat.ma
@@ -76,7 +76,7 @@ ntheorem nat_elim2 :
 #R; #ROn; #RSO; #RSS; #n; nelim n;//;
 #n0; #Rn0m; #m; ncases m;/2/; nqed.
 
-ntheorem decidable_eq_nat : \forall n,m:nat.decidable (n=m).
+ntheorem decidable_eq_nat : ∀n,m:nat.decidable (n=m).
 napply nat_elim2; #n;
  ##[ ncases n; /2/;
  ##| /3/;
@@ -260,7 +260,7 @@ ntheorem le_pred_n : ∀n:nat. pred n ≤ n.
 #n; nelim n; //; nqed.
 
 ntheorem monotonic_pred: monotonic ? le pred.
-#n; #m; #lenm; nelim lenm; /2/; nqed.
+#n; #m; #lenm; nelim lenm; //; /2/; nqed.
 
 ntheorem le_S_S_to_le: ∀n,m:nat. S n ≤ S m → n ≤ m.
 /2/; nqed.
@@ -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.
@@ -561,11 +564,47 @@ ntheorem eq_plus_to_le: ∀n,m,p:nat.n=m+p → m ≤ n.
 ntheorem le_plus_to_le: ∀a,n,m. a + n ≤ a + m → n ≤ m.
 #a; nelim a; /3/; nqed. 
 
+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. 
+
+(*
+variant lt_plus_r: \forall n,p,q:nat. p < q \to n + p < n + q \def
+monotonic_lt_plus_r. *)
+
+ntheorem monotonic_lt_plus_l: 
+∀n:nat.monotonic nat lt (λm.m+n).
+/2/;nqed.
+
+(*
+variant lt_plus_l: \forall n,p,q:nat. p < q \to p + n < q + n \def
+monotonic_lt_plus_l. *)
+
+ntheorem lt_plus: ∀n,m,p,q:nat. n < m → p < q → n + p < m + q.
+#n; #m; #p; #q; #ltnm; #ltpq;
+napply (transitive_lt ? (n+q));/2/; nqed.
+
+ntheorem lt_plus_to_lt_l :∀n,p,q:nat. p+n < q+n → p<q.
+/2/; nqed.
+
+ntheorem lt_plus_to_lt_r :∀n,p,q:nat. n+p < n+q → p<q.
+/2/; nqed.
+
+ntheorem le_to_lt_to_plus_lt: ∀a,b,c,d:nat.
+a ≤ c → b < d → a + b < c+d.
+(* bello /2/ un po' lento *)
+#a; #b; #c; #d; #leac; #lebd; 
+nnormalize; napplyS le_plus; //; nqed.
+
 (* times *)
 ntheorem monotonic_le_times_r: 
 ∀n:nat.monotonic nat le (λm. n * m).
-#n; #x; #y; #lexy; nelim n; nnormalize;//;
-#a; #lea; napply le_plus;//; (* lentissimo /2/ *)
+#n; #x; #y; #lexy; nelim n; nnormalize;//;(* lento /2/;*)
+#a; #lea; napply le_plus; //;
 nqed.
 
 (*
@@ -583,7 +622,7 @@ theorem le_times_l: \forall p,n,m:nat. n \le m \to n*p \le m*p
 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; (* too slow *)
+napply transitive_le; (* /2/ slow *)
  ##[ ##| napply monotonic_le_times_l;//; 
      ##| napply monotonic_le_times_r;//;
  ##]
@@ -609,3 +648,336 @@ 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.
+
+(* times & lt *)
+(*
+ntheorem 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. *)
+
+(*
+ntheorem 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. *)
+
+(*
+ntheorem 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. *)
+
+ntheorem monotonic_lt_times_l: 
+  ∀c:nat. O < c → monotonic nat lt (λt.(t*c)).
+#c; #posc; #n; #m; #ltnm;
+nelim ltnm; nnormalize;
+  ##[napplyS monotonic_lt_plus_l;//;
+  ##|#a; #_; #lt1; napply (transitive_le ??? lt1);//;
+  ##]
+nqed.
+
+ntheorem monotonic_lt_times_r: 
+  ∀c:nat. O < c → monotonic nat lt (λt.(c*t)).
+(* /2/ lentissimo *)
+#c; #posc; #n; #m; #ltnm;
+(* why?? napplyS (monotonic_lt_times_l c posc n m ltnm); *)
+nrewrite > (symmetric_times c n);
+nrewrite > (symmetric_times c m);
+napply monotonic_lt_times_l;//;
+nqed.
+
+ntheorem lt_to_le_to_lt_times: 
+∀n,m,p,q:nat. n < m → p ≤ q → O < q → n*p < m*q.
+#n; #m; #p; #q; #ltnm; #lepq; #posq;
+napply (le_to_lt_to_lt ? (n*q));
+  ##[napply monotonic_le_times_r;//;
+  ##|napply monotonic_lt_times_l;//;
+  ##]
+nqed.
+
+ntheorem lt_times:∀n,m,p,q:nat. n<m → p<q → n*p < m*q.
+#n; #m; #p; #q; #ltnm; #ltpq;
+napply lt_to_le_to_lt_times;/2/;
+nqed.
+
+ntheorem lt_times_n_to_lt_l: 
+∀n,p,q:nat. O < n → p*n < q*n → p < q.
+#n; #p; #q; #posn; #Hlt;
+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; //;
+nqed.
+
+ntheorem lt_times_n_to_lt_r: 
+∀n,p,q:nat. O < n → n*p < n*q → p < q.
+#n; #p; #q; #posn; #Hlt;
+napply (lt_times_n_to_lt_l ??? posn);//;
+nqed.
+
+(*
+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 ******************************)
+
+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.
+
+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;//;  
+     nnormalize; #n;napplyS le_S_S;//  
+  ##]
+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.
+simplify.apply H.apply le_S_S_to_le. apply H1.
+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 *)
+
+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.