Some integrations to the ng library.

[helm.git] / helm / software / matita / nlibrary / arithmetics / nat.ma

diff --git a/helm/software/matita/nlibrary/arithmetics/nat.ma b/helm/software/matita/nlibrary/arithmetics/nat.ma
index 393f16189a9001d8dacab469c43f285ce8123533..25745e370c19d3ed3072e525630234461c6a5725 100644 (file)
--- a/helm/software/matita/nlibrary/arithmetics/nat.ma
+++ b/helm/software/matita/nlibrary/arithmetics/nat.ma
@@ -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
@@ -289,9 +293,11 @@ 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; /3/;
-#m; #dec; ncases dec;/4/; nqed.
+#m; #dec; ncases dec;/4/; nqed. *)
 
 ntheorem decidable_lt: ∀n,m. decidable (n < m).
 #n; #m; napply decidable_le ; nqed.
@@ -554,7 +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.
@@ -567,7 +573,7 @@ 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. 
@@ -635,7 +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: 
@@ -651,9 +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 *)
 (*
@@ -840,7 +846,7 @@ 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; (* XXX *) napply not_le_Sn_O; /2/.
+  ##|#n; #abs; napply False_ind; /2/.
   ##|#n; #m; #Hind; #c; napplyS Hind; /2/;
   ##]
 nqed.
@@ -859,7 +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; (* XXX *) napply not_le_Sn_O; /2/;
+  ##|#n; #p; #abs; napply False_ind; /2/;
   ##|nnormalize;/3/;
   ##]
 nqed.
@@ -1031,9 +1037,9 @@ 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; (* XXX *) nletin hint ≝ not_eq_O_S; /3/; 
+  ##|nnormalize; /3/; 
   ##|nnormalize; /4/; 
-  ##] (* XXX rimane aperto *) #m; #P; #_; #H; napply H; napply not_eq_O_S.
+  ##] 
 nqed.
 
 ntheorem eqb_n_n: ∀n. eqb n n = true.
@@ -1042,7 +1048,7 @@ nqed.
 
 ntheorem eqb_true_to_eq: ∀n,m:nat. eqb n m = true → n = m.
 #n; #m; napply (eqb_elim n m);//;
-#_; #abs; napply False_ind; (* XXX *) nletin hint ≝ not_eq_true_false; /2/;
+#_; #abs; napply False_ind; /2/;
 nqed.
 
 ntheorem eqb_false_to_not_eq: ∀n,m:nat. eqb n m = false → n ≠ m.
@@ -1072,7 +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/
-  ##| (* XXX *) nletin hint ≝ not_le_Sn_O; /3/;
+  ##| /3/;
   ##|#n; #m; #Hind; #P; #Pt; #Pf; napply Hind;
     ##[#lenm; napply Pt; napply le_S_S;//;
     ##|#nlenm; napply Pf; #leSS; /3/;
@@ -1083,14 +1089,14 @@ nqed.
 ntheorem leb_true_to_le:∀n,m.leb n m = true → n ≤ m.
 #n; #m; napply leb_elim;
   ##[//;
-  ##|#_; #abs; napply False_ind; (* XXX *) nletin hint ≝ not_eq_true_false; /2/;
+  ##|#_; #abs; napply False_ind; /2/;
   ##]
 nqed.
 
 ntheorem leb_false_to_not_le:∀n,m.
   leb n m = false → n ≰ m.
 #n; #m; napply leb_elim;
-  ##[#_; #abs; napply False_ind; (* XXX *) nletin hint ≝ not_eq_true_false; /2/;
+  ##[#_; #abs; napply False_ind; /2/;
   ##|/2/;
   ##]
 nqed.
@@ -1129,3 +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.