X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fnlibrary%2Farithmetics%2Fnat.ma;h=25745e370c19d3ed3072e525630234461c6a5725;hb=edccb29109d07b54b48230a280f4351ed042dd9f;hp=d1e401ae5f87de195cfdc4edbdd3f30d532b9470;hpb=71690ba2c49a618d36e5eafcc16726e1ba80f038;p=helm.git

diff --git a/helm/software/matita/nlibrary/arithmetics/nat.ma b/helm/software/matita/nlibrary/arithmetics/nat.ma
index d1e401ae5..25745e370 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/;
  ##]
@@ -112,7 +112,7 @@ ntheorem plus_Sn_m1: ∀n,m:nat. S m + n = n + S m.
 #n; nelim n; nnormalize; //; nqed.
 *)
 
-(*
+(* deleterio 
 ntheorem plus_n_SO : ∀n:nat. S n = n+S O.
 //; nqed. *)
 
@@ -220,6 +220,9 @@ interpretation "natural 'less than'" 'lt x y = (lt x y).
 
 interpretation "natural 'not less than'" 'nless x y = (Not (lt x y)).
 
+(* nlemma eq_lt: ∀n,m. (n < m) = (S n ≤ m).
+//; nqed. *)
+
 ndefinition ge: nat \to nat \to Prop \def
 \lambda n,m:nat.m \leq n.
 
@@ -240,8 +243,9 @@ nqed.
 ntheorem trans_le: \forall n,m,p:nat. n \leq m \to m \leq p \to n \leq p
 \def transitive_le. *)
 
-ntheorem transitive_lt: transitive nat lt.
-#a; #b; #c; #ltab; #ltbc;nelim ltbc;/2/;nqed.
+
+naxiom transitive_lt: transitive nat lt.
+(* #a; #b; #c; #ltab; #ltbc;nelim ltbc;/2/;nqed.*)
 
 (*
 theorem trans_lt: \forall n,m,p:nat. lt n m \to lt m p \to lt n p
@@ -259,13 +263,18 @@ ntheorem le_n_Sn : ∀n:nat. n ≤ S n.
 ntheorem le_pred_n : ∀n:nat. pred n ≤ n.
 #n; nelim n; //; nqed.
 
+(* XXX global problem *)
+nlemma my_trans_le : ∀x,y,z:nat.x ≤ y → y ≤ z → x ≤ z. 
+napply transitive_le.
+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.
+(* XXX *) nletin hint ≝ monotonic. /2/; nqed.
 
-ntheorem lt_S_S_to_lt: ∀n,m. S n < S m \to n < m.
+ntheorem lt_S_S_to_lt: ∀n,m. S n < S m → n < m.
 /2/; nqed. 
 
 ntheorem lt_to_lt_S_S: ∀n,m. n < m → S n < S m.
@@ -284,15 +293,17 @@ ntheorem not_le_to_not_le_S_S: ∀ n,m:nat. n ≰ m → S n ≰ S m.
 ntheorem not_le_S_S_to_not_le: ∀ n,m:nat. S n ≰ S m → n ≰ m.
 /3/; nqed.
 
+naxiom decidable_le: ∀n,m. decidable (n≤m).
+(*
 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 +312,7 @@ 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.
 
@@ -549,7 +560,7 @@ ntheorem le_plus_l: \forall p,n,m:nat. n \le m \to n + p \le m + p
 
 ntheorem le_plus: ∀n1,n2,m1,m2:nat. n1 ≤ n2  \to m1 ≤ m2 
 → n1 + m1 ≤ n2 + m2.
-#n1; #n2; #m1; #m2; #len; #lem; napply transitive_le;
+#n1; #n2; #m1; #m2; #len; #lem; napply (transitive_le ? (n1+m2));
 /2/; nqed.
 
 ntheorem le_plus_n :∀n,m:nat. m ≤ n + m.
@@ -562,15 +573,16 @@ 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. 
+#a; nelim a; nnormalize; /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. 
+/2/; nqed.
 
 (*
 variant lt_plus_r: \forall n,p,q:nat. p < q \to n + p < n + q \def
@@ -629,7 +641,7 @@ napply transitive_le; (* /2/ slow *)
 nqed.
 
 ntheorem lt_times_n: ∀n,m:nat. O < n → m ≤ n*m.
-(* bello *)
+#n; #m; #H; napplyS monotonic_le_times_l;
 /2/; nqed.
 
 ntheorem le_times_to_le: 
@@ -645,9 +657,9 @@ ntheorem le_times_to_le:
   ##]
 nqed.
 
-ntheorem le_S_times_2: ∀n,m.O < m → n ≤ m → n < 2*m.
+ntheorem le_S_times_2: ∀n,m.O < m → n ≤ m → S n ≤ 2*m.
 #n; #m; #posm; #lenm; (* interessante *)
-nnormalize; napplyS (le_plus n); //; nqed.
+napplyS (le_plus n); //; nqed.
 
 (* times & lt *)
 (*
@@ -749,8 +761,7 @@ 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: 
@@ -835,11 +846,18 @@ ntheorem minus_Sn_m: ∀m,n:nat. m ≤ n → S n -m = S (n-m).
 #n; #m; #lenm; nelim lenm; napplyS refl_eq. *)
 napply nat_elim2; 
   ##[//
-  ##|#n; #abs; napply False_ind;/2/;
+  ##|#n; #abs; napply False_ind; /2/.
   ##|#n; #m; #Hind; #c; napplyS Hind; /2/;
   ##]
 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.
 
@@ -847,7 +865,7 @@ 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/;
+  ##|#n; #p; #abs; napply False_ind; /2/;
   ##|nnormalize;/3/;
   ##]
 nqed.
@@ -1013,43 +1031,28 @@ 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.
@@ -1075,7 +1078,7 @@ ntheorem leb_elim: ∀n,m:nat. ∀P:bool → Prop.
 (n ≤ m → P true) → (n ≰ m → P false) → P (leb n m).
 napply nat_elim2; nnormalize;
   ##[/2/
-  ##|/3/;
+  ##| /3/;
   ##|#n; #m; #Hind; #P; #Pt; #Pf; napply Hind;
     ##[#lenm; napply Pt; napply le_S_S;//;
     ##|#nlenm; napply Pf; #leSS; /3/;
@@ -1094,7 +1097,7 @@ 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.
 
@@ -1132,3 +1135,99 @@ nqed.
 ntheorem le_to_ltb_false: ∀n,m. m \le n → ltb n m = false.
 #n; #m; #Hltb; napply lt_to_leb_false; /2/;
 nqed. *)
+
+ninductive compare : Type[0] ≝
+| LT : compare
+| EQ : compare
+| GT : compare.
+
+ndefinition compare_invert: compare → compare ≝
+  λc.match c with
+      [ LT ⇒ GT
+      | EQ ⇒ EQ
+      | GT ⇒ LT ].
+
+nlet rec nat_compare n m: compare ≝
+match n with
+[ O ⇒ match m with 
+      [ O ⇒ EQ
+      | (S q) ⇒ LT ]
+| S p ⇒ match m with 
+      [ O ⇒ GT
+      | S q ⇒ nat_compare p q]].
+
+ntheorem nat_compare_n_n: ∀n. nat_compare n n = EQ.
+#n;nelim n
+##[//
+##|#m;#IH;nnormalize;//]
+nqed.
+
+ntheorem nat_compare_S_S: ∀n,m:nat.nat_compare n m = nat_compare (S n) (S m).
+//;
+nqed.
+
+ntheorem nat_compare_pred_pred: 
+  ∀n,m.O < n → O < m → nat_compare n m = nat_compare (pred n) (pred m).
+#n;#m;#Hn;#Hm;
+napply (lt_O_n_elim n Hn);
+napply (lt_O_n_elim m Hm);
+#p;#q;//;
+nqed.
+
+ntheorem nat_compare_to_Prop: 
+  ∀n,m.match (nat_compare n m) with
+    [ LT ⇒ n < m
+    | EQ ⇒ n = m
+    | GT ⇒ m < n ].
+#n;#m;
+napply (nat_elim2 (λn,m.match (nat_compare n m) with
+  [ LT ⇒ n < m
+  | EQ ⇒ n = m
+  | GT ⇒ m < n ]) ?????) (* FIXME: don't want to put all these ?, especially when … does not work! *)
+##[##1,2:#n1;ncases n1;//;
+##|#n1;#m1;nnormalize;ncases (nat_compare n1 m1);
+   ##[##1,3:nnormalize;#IH;napply le_S_S;//;
+   ##|nnormalize;#IH;nrewrite > IH;//]
+nqed.
+
+ntheorem nat_compare_n_m_m_n: 
+  ∀n,m:nat.nat_compare n m = compare_invert (nat_compare m n).
+#n;#m;
+napply (nat_elim2 (λn,m. nat_compare n m = compare_invert (nat_compare m n)))
+##[##1,2:#n1;ncases n1;//;
+##|#n1;#m1;#IH;nnormalize;napply IH]
+nqed.
+     
+ntheorem nat_compare_elim : 
+  ∀n,m. ∀P:compare → Prop.
+    (n < m → P LT) → (n=m → P EQ) → (m < n → P GT) → P (nat_compare n m).
+#n;#m;#P;#Hlt;#Heq;#Hgt;
+ncut (match (nat_compare n m) with
+    [ LT ⇒ n < m
+    | EQ ⇒ n=m
+    | GT ⇒ m < n] →
+  P (nat_compare n m))
+##[ncases (nat_compare n m);
+   ##[napply Hlt
+   ##|napply Heq
+   ##|napply Hgt]
+##|#Hcut;napply Hcut;//;
+nqed.
+
+ninductive cmp_cases (n,m:nat) : CProp[0] ≝
+  | cmp_le : n ≤ m → cmp_cases n m
+  | cmp_gt : m < n → cmp_cases n m.
+
+ntheorem lt_to_le : ∀n,m:nat. n < m → n ≤ m.
+#n;#m;#H;nelim H
+##[//
+##|/2/]
+nqed.
+
+nlemma cmp_nat: ∀n,m.cmp_cases n m.
+#n;#m; nlapply (nat_compare_to_Prop n m);
+ncases (nat_compare n m);#H
+##[@;napply lt_to_le;//
+##|@;//
+##|@2;//]
+nqed.