set "baseuri" "cic:/matita/nat/lt_arith".
-include "nat/exp.ma".
include "nat/div_and_mod.ma".
(* plus *)
theorem monotonic_lt_plus_r:
-\forall n:nat.monotonic nat lt (\lambda m.plus n m).
+\forall n:nat.monotonic nat lt (\lambda m.n+m).
simplify.intros.
elim n.simplify.assumption.
-simplify.
+simplify.unfold lt.
apply le_S_S.assumption.
qed.
-variant lt_plus_r: \forall n,p,q:nat. lt p q \to lt (plus n p) (plus n q) \def
+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.plus m n).
-change with \forall n,p,q:nat. lt p q \to lt (plus p n) (plus q n).
+\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.
+rewrite < sym_plus. rewrite < (sym_plus n).
apply lt_plus_r.assumption.
qed.
-variant lt_plus_l: \forall n,p,q:nat. lt p q \to lt (plus p n) (plus q n) \def
+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. lt n m \to lt p q \to lt (plus n p) (plus m q).
+theorem lt_plus: \forall n,m,p,q:nat. n < m \to p < q \to n + p < m + q.
intros.
-apply trans_lt ? (plus n q).
+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. lt (plus p n) (plus q n) \to lt p q.
+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.
+rewrite > (plus_n_O q).assumption.
apply H.
-simplify.apply le_S_S_to_le.
+unfold lt.apply le_S_S_to_le.
rewrite > plus_n_Sm.
-rewrite > plus_n_Sm q.
+rewrite > (plus_n_Sm q).
exact H1.
qed.
-theorem lt_plus_to_lt_r :\forall n,p,q:nat. lt (plus n p) (plus n q) \to lt p q.
-intros.apply lt_plus_to_lt_l n.
+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.
+rewrite > (sym_plus q).assumption.
qed.
(* times and zero *)
-theorem lt_O_times_S_S: \forall n,m:nat.lt O (times (S n) (S m)).
-intros.simplify.apply le_S_S.apply le_O_n.
+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.times (S n) m).
-change with \forall n,p,q:nat. lt p q \to lt (times (S n) p) (times (S n) q).
+\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 lt (plus p (times (S n1) p)) (plus q (times (S n1) q)).
+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. lt p q \to lt (times (S n) p) (times (S n) q)
+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.times n (S m)).
+\forall m:nat.monotonic nat lt (\lambda n.n * (S m)).
change with
-\forall n,p,q:nat. lt p q \to lt (times p (S n)) (times q (S n)).
+(\forall n,p,q:nat. p < q \to p*(S n) < q*(S n)).
intros.
-rewrite < sym_times.rewrite < sym_times (S n).
+rewrite < sym_times.rewrite < (sym_times (S n)).
apply lt_times_r.assumption.
qed.
-variant lt_times_l: \forall n,p,q:nat. lt p q \to lt (times p (S n)) (times q (S n))
+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. lt n m \to lt p q \to lt (times n p) (times m q).
+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.
+apply (lt_O_n_elim m H).
intro.
-cut lt O q.
-apply lt_O_n_elim q Hcut.
-intro.change with lt O (times (S m1) (S m2)).
+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 ? (times (S n1) q).
+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.
+cut (lt O q).
+apply (lt_O_n_elim q Hcut).
intro.
apply lt_times_l.
assumption.
-apply ltn_to_ltO p q H2.
+apply (ltn_to_ltO p q H2).
qed.
theorem lt_times_to_lt_l:
-\forall n,p,q:nat. lt (times p (S n)) (times q (S n)) \to lt p q.
+\forall n,p,q:nat. p*(S n) < q*(S n) \to p < q.
intros.
-(* convertibility problem here *)
-cut Or (lt p q) (Not (lt p q)).
+cut (p < q \lor p \nlt q).
elim Hcut.
assumption.
-absurd lt (times p (S n)) (times q (S n)).
+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.
+exact (decidable_lt p q).
qed.
theorem lt_times_to_lt_r:
-\forall n,p,q:nat. lt (times (S n) p) (times(S n) q) \to lt p q.
+\forall n,p,q:nat. (S n)*p < (S n)*q \to lt p q.
intros.
-apply lt_times_to_lt_l n.
+apply (lt_times_to_lt_l n).
rewrite < sym_times.
-rewrite < sym_times (S n).
+rewrite < (sym_times (S n)).
assumption.
qed.
theorem nat_compare_times_l : \forall n,p,q:nat.
-eq compare (nat_compare p q) (nat_compare (times (S n) p) (times (S n) q)).
+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 eq nat p q.
-apply inj_times_r n.assumption.
+intro.absurd (p=q).
+apply (inj_times_r n).assumption.
apply lt_to_not_eq. assumption.
-intro.absurd lt q p.
-apply lt_times_to_lt_r n.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 (lt p q).
-apply lt_times_to_lt_r n.assumption.
+absurd (p<q).
+apply (lt_times_to_lt_r n).assumption.
apply le_to_not_lt.apply lt_to_le.assumption.
-intro.absurd eq nat q p.
+intro.absurd (q=p).
symmetry.
-apply inj_times_r n.assumption.
+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 mod n m = O \to O < div n m.
-intros 4.apply lt_O_n_elim m H.intros.
-apply lt_times_to_lt_r m1.
+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)*(div n (S m1))).
+rewrite > (plus_n_O ((S m1)*(n / (S m1)))).
rewrite < H2.
rewrite < sym_times.
rewrite < div_mod.
rewrite > H2.
assumption.
-simplify.apply le_S_S.apply le_O_n.
+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 div n m \lt n.
+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 (div n m).intro.
+apply (nat_case1 (n / m)).intro.
assumption.intros.rewrite < H2.
rewrite > (div_mod n m) in \vdash (? ? %).
-apply lt_to_le_to_lt ? ((div n m)*m).
-apply lt_to_le_to_lt ? ((div n m)*(S (S O))).
+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.
+simplify.unfold lt.
rewrite < plus_n_O.
rewrite < plus_n_Sm.
apply le_S_S.
assumption.
rewrite < sym_plus.
apply le_plus_n.
-apply trans_lt ? (S O).
-simplify. apply le_n.assumption.
+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.
-simplify.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.
+unfold injective.intros.
+apply (nat_compare_elim x y).
+intro.apply False_ind.apply (not_le_Sn_n (f x)).
+rewrite > H1 in \vdash (? ? %).
+change with (f x < f y).
+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.
+intro.apply False_ind.apply (not_le_Sn_n (f y)).
+rewrite < H1 in \vdash (? ? %).
+change with (f y < f x).
+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.
\ No newline at end of file
+qed.