-(* le to lt or eq *)
-theorem le_to_or_lt_eq: ∀n,m:nat. n ≤ m → n < m ∨ n = m.
-#n #m #lenm (elim lenm) /3/ qed.
-
-(* not eq *)
-theorem lt_to_not_eq : ∀n,m:nat. n < m → n ≠ m.
-#n #m #H @not_to_not /2/ qed.
-
-(*not lt
-theorem eq_to_not_lt: ∀a,b:nat. a = b → a ≮ b.
-intros.
-unfold Not.
-intros.
-rewrite > H in H1.
-apply (lt_to_not_eq b b)
-[ assumption
-| reflexivity
-]
-qed.
-
-theorem lt_n_m_to_not_lt_m_Sn: ∀n,m. n < m → m ≮ S n.
-intros
-unfold Not
-intro
-unfold lt in H
-unfold lt in H1
-generalize in match (le_S_S ? ? H)
-intro
-generalize in match (transitive_le ? ? ? H2 H1)
-intro
-apply (not_le_Sn_n ? H3).
-qed. *)
-
-theorem not_eq_to_le_to_lt: ∀n,m. n≠m → n≤m → n<m.
-#n #m #Hneq #Hle cases (le_to_or_lt_eq ?? Hle) //
-#Heq /3/ qed.
-(*
-nelim (Hneq Heq) qed. *)
-
-(* le elimination *)
-theorem le_n_O_to_eq : ∀n:nat. n ≤ O → O=n.
-#n (cases n) // #a #abs @False_ind /2/ qed.
-
-theorem le_n_O_elim: ∀n:nat. n ≤ O → ∀P: nat →Prop. P O → P n.
-#n (cases n) // #a #abs @False_ind /2/ qed.
-
-theorem le_n_Sm_elim : ∀n,m:nat.n ≤ S m →
-∀P:Prop. (S n ≤ S m → P) → (n=S m → P) → P.
-#n #m #Hle #P (elim Hle) /3/ qed.
-
-(* le and eq *)
-
-theorem le_to_le_to_eq: ∀n,m. n ≤ m → m ≤ n → n = m.
-@nat_elim2 /4/
-qed.
-
-theorem lt_O_S : ∀n:nat. O < S n.
-/2/ qed.
-
-(*
-(* other abstract properties *)
-theorem antisymmetric_le : antisymmetric nat le.
-unfold antisymmetric.intros 2.
-apply (nat_elim2 (\lambda n,m.(n \leq m \to m \leq n \to n=m))).
-intros.apply le_n_O_to_eq.assumption.
-intros.apply False_ind.apply (not_le_Sn_O ? H).
-intros.apply eq_f.apply H.
-apply le_S_S_to_le.assumption.
-apply le_S_S_to_le.assumption.
-qed.
-
-theorem antisym_le: \forall n,m:nat. n \leq m \to m \leq n \to n=m
-\def antisymmetric_le.
-
-theorem le_n_m_to_lt_m_Sn_to_eq_n_m: ∀n,m. n ≤ m → m < S n → n=m.
-intros
-unfold lt in H1
-generalize in match (le_S_S_to_le ? ? H1)
-intro
-apply antisym_le
-assumption.
-qed.
-*)
-
-(* well founded induction principles *)
-
-theorem nat_elim1 : ∀n:nat.∀P:nat → Prop.
-(∀m.(∀p. p < m → P p) → P m) → P n.
-#n #P #H
-cut (∀q:nat. q ≤ n → P q) /2/
-(elim n)
- [#q #HleO (* applica male *)
- @(le_n_O_elim ? HleO)
- @H #p #ltpO @False_ind /2/ (* 3 *)
- |#p #Hind #q #HleS
- @H #a #lta @Hind @le_S_S_to_le /2/
- ]
-qed.
-
-(* some properties of functions *)
-(*
-definition increasing \def \lambda f:nat \to nat.
-\forall n:nat. f n < f (S n).
-
-theorem increasing_to_monotonic: \forall f:nat \to nat.
-increasing f \to monotonic nat lt f.
-unfold monotonic.unfold lt.unfold increasing.unfold lt.intros.elim H1.apply H.
-apply (trans_le ? (f n1)).
-assumption.apply (trans_le ? (S (f n1))).
-apply le_n_Sn.
-apply H.
-qed.
-
-theorem le_n_fn: \forall f:nat \to nat. (increasing f)
-\to \forall n:nat. n \le (f n).
-intros.elim n.
-apply le_O_n.
-apply (trans_le ? (S (f n1))).
-apply le_S_S.apply H1.
-simplify in H. unfold increasing in H.unfold lt in H.apply H.
-qed.
-
-theorem increasing_to_le: \forall f:nat \to nat. (increasing f)
-\to \forall m:nat. \exists i. m \le (f i).
-intros.elim m.
-apply (ex_intro ? ? O).apply le_O_n.
-elim H1.
-apply (ex_intro ? ? (S a)).
-apply (trans_le ? (S (f a))).
-apply le_S_S.assumption.
-simplify in H.unfold increasing in H.unfold lt in H.
-apply H.
-qed.
-
-theorem increasing_to_le2: \forall f:nat \to nat. (increasing f)
-\to \forall m:nat. (f O) \le m \to
-\exists i. (f i) \le m \land m <(f (S i)).
-intros.elim H1.
-apply (ex_intro ? ? O).
-split.apply le_n.apply H.
-elim H3.elim H4.
-cut ((S n1) < (f (S a)) \lor (S n1) = (f (S a))).
-elim Hcut.
-apply (ex_intro ? ? a).
-split.apply le_S. assumption.assumption.
-apply (ex_intro ? ? (S a)).
-split.rewrite < H7.apply le_n.
-rewrite > H7.
-apply H.
-apply le_to_or_lt_eq.apply H6.
-qed.
-*)
-
-(*********************** monotonicity ***************************)