Printings removed.

[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 120a0bb16199c888d16866c495c1fbc7a2f1579e..25745e370c19d3ed3072e525630234461c6a5725 100644 (file)
--- a/helm/software/matita/nlibrary/arithmetics/nat.ma
+++ b/helm/software/matita/nlibrary/arithmetics/nat.ma
@@ -1135,3 +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.