#n; nelim n; nnormalize; //; nqed.
*)
-(*
+(* deleterio
ntheorem plus_n_SO : ∀n:nat. S n = n+S O.
//; nqed. *)
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.
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
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.
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.
//; 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.
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:
##]
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 *)
(*
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.