(* 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.
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.
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.
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.
+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.lt O (times (S n) (S m)).
+theorem lt_O_times_S_S: \forall n,m:nat.O < (S n)*(S m).
intros.simplify.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).
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.
intro.
cut lt O q.
apply lt_O_n_elim q Hcut.
-intro.change with lt O (times (S m1) (S m2)).
+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 trans_lt ? ((S n1)*q).
apply lt_times_r.assumption.
cut lt O q.
apply lt_O_n_elim q Hcut.
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.
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.
rewrite < sym_times.
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.
+intro.absurd p=q.
apply inj_times_r n.assumption.
apply lt_to_not_eq. assumption.
-intro.absurd lt q p.
+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).
+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 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.
+simplify.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.
+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).
+simplify. 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.
+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.