X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Flibrary%2Fnat%2Flt_arith.ma;h=b8339f374e6019dd054719c9928bf735c63aef4b;hb=4167cea65ca58897d1a3dbb81ff95de5074700cc;hp=5bef2237103f27308663712a78f78b67d0c3b621;hpb=7bbce6bc163892cfd99cfcda65db42001b86789f;p=helm.git

diff --git a/helm/matita/library/nat/lt_arith.ma b/helm/matita/library/nat/lt_arith.ma
index 5bef22371..b8339f374 100644
--- a/helm/matita/library/nat/lt_arith.ma
+++ b/helm/matita/library/nat/lt_arith.ma
@@ -18,145 +18,200 @@ 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 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.