--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
+(* ||A|| E.Tassi, S.Zacchiroli *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU Lesser General Public License Version 2.1 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/nat/lt_arith".
+
+include "nat/div_and_mod.ma".
+
+(* plus *)
+theorem monotonic_lt_plus_r:
+\forall n:nat.monotonic nat lt (\lambda m.n+m).
+simplify.intros.
+elim n.simplify.assumption.
+simplify.unfold lt.
+apply le_S_S.assumption.
+qed.
+
+variant lt_plus_r: \forall n,p,q:nat. p < q \to n + p < n + q \def
+monotonic_lt_plus_r.
+
+theorem monotonic_lt_plus_l:
+\forall n:nat.monotonic nat lt (\lambda m.m+n).
+change with (\forall n,p,q:nat. p < q \to p + n < q + n).
+intros.
+rewrite < sym_plus. rewrite < (sym_plus n).
+apply lt_plus_r.assumption.
+qed.
+
+variant lt_plus_l: \forall n,p,q:nat. p < q \to p + n < q + n \def
+monotonic_lt_plus_l.
+
+theorem lt_plus: \forall n,m,p,q:nat. n < m \to p < q \to n + p < m + q.
+intros.
+apply (trans_lt ? (n + q)).
+apply lt_plus_r.assumption.
+apply lt_plus_l.assumption.
+qed.
+
+theorem lt_plus_to_lt_l :\forall n,p,q:nat. p+n < q+n \to p<q.
+intro.elim n.
+rewrite > plus_n_O.
+rewrite > (plus_n_O q).assumption.
+apply H.
+unfold lt.apply le_S_S_to_le.
+rewrite > plus_n_Sm.
+rewrite > (plus_n_Sm q).
+exact H1.
+qed.
+
+theorem lt_plus_to_lt_r :\forall n,p,q:nat. n+p < n+q \to p<q.
+intros.apply (lt_plus_to_lt_l n).
+rewrite > sym_plus.
+rewrite > (sym_plus q).assumption.
+qed.
+
+(* times and zero *)
+theorem lt_O_times_S_S: \forall n,m:nat.O < (S n)*(S m).
+intros.simplify.unfold lt.apply le_S_S.apply le_O_n.
+qed.
+
+(* times *)
+theorem monotonic_lt_times_r:
+\forall n:nat.monotonic nat lt (\lambda m.(S n)*m).
+change with (\forall n,p,q:nat. p < q \to (S n) * p < (S n) * q).
+intros.elim n.
+simplify.rewrite < plus_n_O.rewrite < plus_n_O.assumption.
+change with (p + (S n1) * p < q + (S n1) * q).
+apply lt_plus.assumption.assumption.
+qed.
+
+theorem lt_times_r: \forall n,p,q:nat. p < q \to (S n) * p < (S n) * q
+\def monotonic_lt_times_r.
+
+theorem monotonic_lt_times_l:
+\forall m:nat.monotonic nat lt (\lambda n.n * (S m)).
+change with
+(\forall n,p,q:nat. p < q \to p*(S n) < q*(S n)).
+intros.
+rewrite < sym_times.rewrite < (sym_times (S n)).
+apply lt_times_r.assumption.
+qed.
+
+variant lt_times_l: \forall n,p,q:nat. p<q \to p*(S n) < q*(S n)
+\def monotonic_lt_times_l.
+
+theorem lt_times:\forall n,m,p,q:nat. n<m \to p<q \to n*p < m*q.
+intro.
+elim n.
+apply (lt_O_n_elim m H).
+intro.
+cut (lt O q).
+apply (lt_O_n_elim q Hcut).
+intro.change with (O < (S m1)*(S m2)).
+apply lt_O_times_S_S.
+apply (ltn_to_ltO p q H1).
+apply (trans_lt ? ((S n1)*q)).
+apply lt_times_r.assumption.
+cut (lt O q).
+apply (lt_O_n_elim q Hcut).
+intro.
+apply lt_times_l.
+assumption.
+apply (ltn_to_ltO p q H2).
+qed.
+
+theorem lt_times_to_lt_l:
+\forall n,p,q:nat. p*(S n) < q*(S n) \to p < q.
+intros.
+cut (p < q \lor p \nlt q).
+elim Hcut.
+assumption.
+absurd (p * (S n) < q * (S n)).
+assumption.
+apply le_to_not_lt.
+apply le_times_l.
+apply not_lt_to_le.
+assumption.
+exact (decidable_lt p q).
+qed.
+
+theorem lt_times_to_lt_r:
+\forall n,p,q:nat. (S n)*p < (S n)*q \to lt p q.
+intros.
+apply (lt_times_to_lt_l n).
+rewrite < sym_times.
+rewrite < (sym_times (S n)).
+assumption.
+qed.
+
+theorem nat_compare_times_l : \forall n,p,q:nat.
+nat_compare p q = nat_compare ((S n) * p) ((S n) * q).
+intros.apply nat_compare_elim.intro.
+apply nat_compare_elim.
+intro.reflexivity.
+intro.absurd (p=q).
+apply (inj_times_r n).assumption.
+apply lt_to_not_eq. assumption.
+intro.absurd (q<p).
+apply (lt_times_to_lt_r n).assumption.
+apply le_to_not_lt.apply lt_to_le.assumption.
+intro.rewrite < H.rewrite > nat_compare_n_n.reflexivity.
+intro.apply nat_compare_elim.intro.
+absurd (p<q).
+apply (lt_times_to_lt_r n).assumption.
+apply le_to_not_lt.apply lt_to_le.assumption.
+intro.absurd (q=p).
+symmetry.
+apply (inj_times_r n).assumption.
+apply lt_to_not_eq.assumption.
+intro.reflexivity.
+qed.
+
+(* div *)
+
+theorem eq_mod_O_to_lt_O_div: \forall n,m:nat. O < m \to O < n\to n \mod m = O \to O < n / m.
+intros 4.apply (lt_O_n_elim m H).intros.
+apply (lt_times_to_lt_r m1).
+rewrite < times_n_O.
+rewrite > (plus_n_O ((S m1)*(n / (S m1)))).
+rewrite < H2.
+rewrite < sym_times.
+rewrite < div_mod.
+rewrite > H2.
+assumption.
+unfold lt.apply le_S_S.apply le_O_n.
+qed.
+
+theorem lt_div_n_m_n: \forall n,m:nat. (S O) < m \to O < n \to n / m \lt n.
+intros.
+apply (nat_case1 (n / m)).intro.
+assumption.intros.rewrite < H2.
+rewrite > (div_mod n m) in \vdash (? ? %).
+apply (lt_to_le_to_lt ? ((n / m)*m)).
+apply (lt_to_le_to_lt ? ((n / m)*(S (S O)))).
+rewrite < sym_times.
+rewrite > H2.
+simplify.unfold lt.
+rewrite < plus_n_O.
+rewrite < plus_n_Sm.
+apply le_S_S.
+apply le_S_S.
+apply le_plus_n.
+apply le_times_r.
+assumption.
+rewrite < sym_plus.
+apply le_plus_n.
+apply (trans_lt ? (S O)).
+unfold lt. apply le_n.assumption.
+qed.
+
+(* general properties of functions *)
+theorem monotonic_to_injective: \forall f:nat\to nat.
+monotonic nat lt f \to injective nat nat f.
+unfold injective.intros.
+apply (nat_compare_elim x y).
+intro.apply False_ind.apply (not_le_Sn_n (f x)).
+rewrite > H1 in \vdash (? ? %).apply H.apply H2.
+intros.assumption.
+intro.apply False_ind.apply (not_le_Sn_n (f y)).
+rewrite < H1 in \vdash (? ? %).apply H.apply H2.
+qed.
+
+theorem increasing_to_injective: \forall f:nat\to nat.
+increasing f \to injective nat nat f.
+intros.apply monotonic_to_injective.
+apply increasing_to_monotonic.assumption.
+qed.
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