--- /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 *)
+(* *)
+(**************************************************************************)
+
+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).
+simplify.
+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.
+
+theorem le_to_lt_to_plus_lt: \forall a,b,c,d:nat.
+a \le c \to b \lt d \to (a + b) \lt (c+d).
+intros.
+cut (a \lt c \lor a = c)
+[ elim Hcut
+ [ apply (lt_plus );
+ assumption
+ | rewrite > H2.
+ apply (lt_plus_r c b d).
+ assumption
+ ]
+| apply le_to_or_lt_eq.
+ 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.
+
+theorem lt_times_eq_O: \forall a,b:nat.
+O \lt a \to (a * b) = O \to b = O.
+intros.
+apply (nat_case1 b)
+[ intros.
+ reflexivity
+| intros.
+ rewrite > H2 in H1.
+ rewrite > (S_pred a) in H1
+ [ apply False_ind.
+ apply (eq_to_not_lt O ((S (pred a))*(S m)))
+ [ apply sym_eq.
+ assumption
+ | apply lt_O_times_S_S
+ ]
+ | assumption
+ ]
+]
+qed.
+
+theorem O_lt_times_to_O_lt: \forall a,c:nat.
+O \lt (a * c) \to O \lt a.
+intros.
+apply (nat_case1 a)
+[ intros.
+ rewrite > H1 in H.
+ simplify in H.
+ assumption
+| intros.
+ apply lt_O_S
+]
+qed.
+
+lemma lt_times_to_lt_O: \forall i,n,m:nat. i < n*m \to O < m.
+intros.
+elim (le_to_or_lt_eq O ? (le_O_n m))
+ [assumption
+ |apply False_ind.
+ rewrite < H1 in H.
+ rewrite < times_n_O in H.
+ apply (not_le_Sn_O ? H)
+ ]
+qed.
+
+(* times *)
+theorem monotonic_lt_times_r:
+\forall n:nat.monotonic nat lt (\lambda m.(S n)*m).
+simplify.
+intros.elim n.
+simplify.rewrite < plus_n_O.rewrite < plus_n_O.assumption.
+apply lt_plus.assumption.assumption.
+qed.
+
+(* a simple variant of the previus monotionic_lt_times *)
+theorem monotonic_lt_times_variant: \forall c:nat.
+O \lt c \to monotonic nat lt (\lambda t.(t*c)).
+intros.
+apply (increasing_to_monotonic).
+unfold increasing.
+intros.
+simplify.
+rewrite > sym_plus.
+rewrite > plus_n_O in \vdash (? % ?).
+apply lt_plus_r.
+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)).
+simplify.
+intros.
+rewrite < sym_times.rewrite < (sym_times (S m)).
+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_r1:
+\forall n,m,p. O < n \to m < p \to n*m < n*p.
+intros.
+elim H;apply lt_times_r;assumption.
+qed.
+
+theorem lt_times_l1:
+\forall n,m,p. O < n \to m < p \to m*n < p*n.
+intros.
+elim H;apply lt_times_l;assumption.
+qed.
+
+theorem lt_to_le_to_lt_times :
+\forall n,n1,m,m1. n < n1 \to m \le m1 \to O < m1 \to n*m < n1*m1.
+intros.
+apply (le_to_lt_to_lt ? (n*m1))
+ [apply le_times_r.assumption
+ |apply lt_times_l1
+ [assumption|assumption]
+ ]
+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_n_to_lt:
+\forall n,p,q:nat. O < n \to p*n < q*n \to p < q.
+intro.
+apply (nat_case n)
+ [intros.apply False_ind.apply (not_le_Sn_n ? H)
+ |intros 4.apply lt_times_to_lt_l
+ ]
+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 lt_times_n_to_lt_r:
+\forall n,p,q:nat. O < n \to n*p < n*q \to lt p q.
+intro.
+apply (nat_case n)
+ [intros.apply False_ind.apply (not_le_Sn_n ? H)
+ |intros 4.apply lt_times_to_lt_r
+ ]
+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.
+
+(* times and plus *)
+theorem lt_times_plus_times: \forall a,b,n,m:nat.
+a < n \to b < m \to a*m + b < n*m.
+intros 3.
+apply (nat_case n)
+ [intros.apply False_ind.apply (not_le_Sn_O ? H)
+ |intros.simplify.
+ rewrite < sym_plus.
+ unfold.
+ change with (S b+a*m1 \leq m1+m*m1).
+ apply le_plus
+ [assumption
+ |apply le_times
+ [apply le_S_S_to_le.assumption
+ |apply le_n
+ ]
+ ]
+ ]
+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.
+
+theorem eq_div_div_div_times: \forall n,m,q. O < n \to O < m \to
+q/n/m = q/(n*m).
+intros.
+apply (div_mod_spec_to_eq q (n*m) ? (q\mod n+n*(q/n\mod m)) ? (mod q (n*m)))
+ [apply div_mod_spec_intro
+ [apply (lt_to_le_to_lt ? (n*(S (q/n\mod m))))
+ [rewrite < times_n_Sm.
+ apply lt_plus_l.
+ apply lt_mod_m_m.
+ assumption
+ |apply le_times_r.
+ apply lt_mod_m_m.
+ assumption
+ ]
+ |rewrite > sym_times in ⊢ (? ? ? (? (? ? %) ?)).
+ rewrite < assoc_times.
+ rewrite > (eq_times_div_minus_mod ? ? H1).
+ rewrite > sym_times.
+ rewrite > distributive_times_minus.
+ rewrite > sym_times.
+ rewrite > (eq_times_div_minus_mod ? ? H).
+ rewrite < sym_plus in ⊢ (? ? ? (? ? %)).
+ rewrite < assoc_plus.
+ rewrite < plus_minus_m_m
+ [rewrite < plus_minus_m_m
+ [reflexivity
+ |apply (eq_plus_to_le ? ? ((q/n)*n)).
+ rewrite < sym_plus.
+ apply div_mod.
+ assumption
+ ]
+ |apply (trans_le ? (n*(q/n)))
+ [apply le_times_r.
+ apply (eq_plus_to_le ? ? ((q/n)/m*m)).
+ rewrite < sym_plus.
+ apply div_mod.
+ assumption
+ |rewrite > sym_times.
+ rewrite > eq_times_div_minus_mod
+ [apply le_n
+ |assumption
+ ]
+ ]
+ ]
+ ]
+ |apply div_mod_spec_div_mod.
+ rewrite > (times_n_O O).
+ apply lt_times;assumption
+ ]
+qed.
+
+theorem eq_div_div_div_div: \forall n,m,q. O < n \to O < m \to
+q/n/m = q/m/n.
+intros.
+apply (trans_eq ? ? (q/(n*m)))
+ [apply eq_div_div_div_times;assumption
+ |rewrite > sym_times.
+ apply sym_eq.
+ apply eq_div_div_div_times;assumption
+ ]
+qed.
+
+theorem SSO_mod: \forall n,m. O < m \to (S(S O))*n/m = (n/m)*(S(S O)) + mod ((S(S O))*n/m) (S(S O)).
+intros.
+rewrite < (lt_O_to_div_times n (S(S O))) in ⊢ (? ? ? (? (? (? % ?) ?) ?))
+ [rewrite > eq_div_div_div_div
+ [rewrite > sym_times in ⊢ (? ? ? (? (? (? (? % ?) ?) ?) ?)).
+ apply div_mod.
+ apply lt_O_S
+ |apply lt_O_S
+ |assumption
+ ]
+ |apply lt_O_S
+ ]
+qed.
+(* Forall a,b : N. 0 < b \to b * (a/b) <= a < b * (a/b +1) *)
+(* The theorem is shown in two different parts: *)
+
+theorem lt_to_div_to_and_le_times_lt_S: \forall a,b,c:nat.
+O \lt b \to a/b = c \to (b*c \le a \land a \lt b*(S c)).
+intros.
+split
+[ rewrite < H1.
+ rewrite > sym_times.
+ rewrite > eq_times_div_minus_mod
+ [ apply (le_minus_m a (a \mod b))
+ | assumption
+ ]
+| rewrite < (times_n_Sm b c).
+ rewrite < H1.
+ rewrite > sym_times.
+ rewrite > (div_mod a b) in \vdash (? % ?)
+ [ rewrite > (sym_plus b ((a/b)*b)).
+ apply lt_plus_r.
+ apply lt_mod_m_m.
+ assumption
+ | assumption
+ ]
+]
+qed.
+
+theorem lt_to_le_times_to_lt_S_to_div: \forall a,c,b:nat.
+O \lt b \to (b*c) \le a \to a \lt (b*(S c)) \to a/b = c.
+intros.
+apply (le_to_le_to_eq)
+[ apply (leb_elim (a/b) c);intros
+ [ assumption
+ | cut (c \lt (a/b))
+ [ apply False_ind.
+ apply (lt_to_not_le (a \mod b) O)
+ [ apply (lt_plus_to_lt_l ((a/b)*b)).
+ simplify.
+ rewrite < sym_plus.
+ rewrite < div_mod
+ [ apply (lt_to_le_to_lt ? (b*(S c)) ?)
+ [ assumption
+ | rewrite > (sym_times (a/b) b).
+ apply le_times_r.
+ assumption
+ ]
+ | assumption
+ ]
+ | apply le_O_n
+ ]
+ | apply not_le_to_lt.
+ assumption
+ ]
+ ]
+| apply (leb_elim c (a/b));intros
+ [ assumption
+ | cut((a/b) \lt c)
+ [ apply False_ind.
+ apply (lt_to_not_le (a \mod b) b)
+ [ apply (lt_mod_m_m).
+ assumption
+ | apply (le_plus_to_le ((a/b)*b)).
+ rewrite < (div_mod a b)
+ [ apply (trans_le ? (b*c) ?)
+ [ rewrite > (sym_times (a/b) b).
+ rewrite > (times_n_SO b) in \vdash (? (? ? %) ?).
+ rewrite < distr_times_plus.
+ rewrite > sym_plus.
+ simplify in \vdash (? (? ? %) ?).
+ apply le_times_r.
+ assumption
+ | assumption
+ ]
+ | assumption
+ ]
+ ]
+ | apply not_le_to_lt.
+ assumption
+ ]
+ ]
+]
+qed.
+
+
+theorem lt_to_lt_to_eq_div_div_times_times: \forall a,b,c:nat.
+O \lt c \to O \lt b \to (a/b) = (a*c)/(b*c).
+intros.
+apply sym_eq.
+cut (b*(a/b) \le a \land a \lt b*(S (a/b)))
+[ elim Hcut.
+ apply lt_to_le_times_to_lt_S_to_div
+ [ rewrite > (S_pred b)
+ [ rewrite > (S_pred c)
+ [ apply (lt_O_times_S_S)
+ | assumption
+ ]
+ | assumption
+ ]
+ | rewrite > assoc_times.
+ rewrite > (sym_times c (a/b)).
+ rewrite < assoc_times.
+ rewrite > (sym_times (b*(a/b)) c).
+ rewrite > (sym_times a c).
+ apply (le_times_r c (b*(a/b)) a).
+ assumption
+ | rewrite > (sym_times a c).
+ rewrite > (assoc_times ).
+ rewrite > (sym_times c (S (a/b))).
+ rewrite < (assoc_times).
+ rewrite > (sym_times (b*(S (a/b))) c).
+ apply (lt_times_r1 c a (b*(S (a/b))));
+ assumption
+ ]
+| apply (lt_to_div_to_and_le_times_lt_S)
+ [ assumption
+ | reflexivity
+ ]
+]
+qed.
+
+theorem times_mod: \forall a,b,c:nat.
+O \lt c \to O \lt b \to ((a*c) \mod (b*c)) = c*(a\mod b).
+intros.
+apply (div_mod_spec_to_eq2 (a*c) (b*c) (a/b) ((a*c) \mod (b*c)) (a/b) (c*(a \mod b)))
+[ rewrite > (lt_to_lt_to_eq_div_div_times_times a b c)
+ [ apply div_mod_spec_div_mod.
+ rewrite > (S_pred b)
+ [ rewrite > (S_pred c)
+ [ apply lt_O_times_S_S
+ | assumption
+ ]
+ | assumption
+ ]
+ | assumption
+ | assumption
+ ]
+| apply div_mod_spec_intro
+ [ rewrite > (sym_times b c).
+ apply (lt_times_r1 c)
+ [ assumption
+ | apply (lt_mod_m_m).
+ assumption
+ ]
+ | rewrite < (assoc_times (a/b) b c).
+ rewrite > (sym_times a c).
+ rewrite > (sym_times ((a/b)*b) c).
+ rewrite < (distr_times_plus c ? ?).
+ apply eq_f.
+ apply (div_mod a b).
+ 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 (? ? %).
+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 (? ? %).
+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.
+
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