#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 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.
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 *)
(*
#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.
∀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.
(n=m → (P true)) → (n ≠ m → (P false)) → (P (eqb n m)).
napply nat_elim2;
##[#n; ncases n; nnormalize; /3/;
- ##|nnormalize; /3/;
- ##|nnormalize; /4/;
- ##]
+ ##|nnormalize; /3/;
+ ##|nnormalize; /4/;
+ ##]
nqed.
ntheorem eqb_n_n: ∀n. eqb n n = true.
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;/2/;
+#_; #abs; napply False_ind; /2/;
nqed.
ntheorem eqb_false_to_not_eq: ∀n,m:nat. eqb n m = false → n ≠ m.
(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/;