X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fdama%2Fsequence.ma;h=9990f8c7d6728caf9f54980beec358af4d49e357;hb=892992b24f5476c2b4eed13f64e04854ef919020;hp=9edc871e5ce25ba2f2373d07b01aa6c82f74d043;hpb=a2fe87da00fb5b9a39e9a1c7d796c61d4c7346af;p=helm.git

diff --git a/helm/software/matita/dama/sequence.ma b/helm/software/matita/dama/sequence.ma
index 9edc871e5..9990f8c7d 100644
--- a/helm/software/matita/dama/sequence.ma
+++ b/helm/software/matita/dama/sequence.ma
@@ -14,214 +14,221 @@
 
 set "baseuri" "cic:/matita/sequence/".
 
-include "ordered_set.ma".
+include "excess.ma".
 
-definition sequence := λO:pordered_set.nat → O.
+definition sequence := λO:excess.nat → O.
 
-definition fun_of_sequence: ∀O:pordered_set.sequence O → nat → O.
+definition fun_of_sequence: ∀O:excess.sequence O → nat → O.
 intros; apply s; assumption;
 qed.
 
 coercion cic:/matita/sequence/fun_of_sequence.con 1.
 
 definition upper_bound ≝ 
-  λO:pordered_set.λa:sequence O.λu:O.∀n:nat.a n ≤ u.
+  λO:excess.λa:sequence O.λu:O.∀n:nat.a n ≤ u.
   
 definition lower_bound ≝ 
-  λO:pordered_set.λa:sequence O.λu:O.∀n:nat.u ≤ a n.
+  λO:excess.λa:sequence O.λu:O.∀n:nat.u ≤ a n.
 
 definition strong_sup ≝
-  λO:pordered_set.λs:sequence O.λx.
+  λO:excess.λs:sequence O.λx.
     upper_bound ? s x ∧ (∀y:O.x ≰ y → ∃n.s n ≰ y).
   
 definition strong_inf ≝
-  λO:pordered_set.λs:sequence O.λx.
+  λO:excess.λs:sequence O.λx.
     lower_bound ? s x ∧ (∀y:O.y ≰ x → ∃n.y ≰ s n).
 
 definition weak_sup ≝
-  λO:pordered_set.λs:sequence O.λx.
+  λO:excess.λs:sequence O.λx.
     upper_bound ? s x ∧ (∀y:O.upper_bound ? s y → x ≤ y).
   
 definition weak_inf ≝
-  λO:pordered_set.λs:sequence O.λx.
+  λO:excess.λs:sequence O.λx.
     lower_bound ? s x ∧ (∀y:O.lower_bound ? s y → y ≤ x).
 
 lemma strong_sup_is_weak: 
-  ∀O:pordered_set.∀s:sequence O.∀x:O.strong_sup ? s x → weak_sup ? s x.
+  ∀O:excess.∀s:sequence O.∀x:O.strong_sup ? s x → weak_sup ? s x.
 intros (O s x Ssup); elim Ssup (Ubx M); clear Ssup; split; [assumption]
 intros 3 (y Uby E); cases (M ? E) (n En); unfold in Uby; cases (Uby ? En);
 qed.
  
 lemma strong_inf_is_weak: 
-  ∀O:pordered_set.∀s:sequence O.∀x:O.strong_inf ? s x → weak_inf ? s x.
+  ∀O:excess.∀s:sequence O.∀x:O.strong_inf ? s x → weak_inf ? s x.
 intros (O s x Ssup); elim Ssup (Ubx M); clear Ssup; split; [assumption]
 intros 3 (y Uby E); cases (M ? E) (n En); unfold in Uby; cases (Uby ? En);
 qed.
 
+include "ordered_group.ma".
+include "nat/orders.ma".
 
-
+definition tends0 ≝ 
+  λO:pogroup.λs:sequence O.
+    ∀e:O.0 < e → ∃N.∀n.N < n → -e < s n ∧ s n < e.
+    
 definition increasing ≝ 
-  λO:pordered_set.λa:sequence O.∀n:nat.a n ≤ a (S n).
+  λO:excess.λa:sequence O.∀n:nat.a n ≤ a (S n).
 
 definition decreasing ≝ 
-  λO:pordered_set.λa:sequence O.∀n:nat.a (S n) ≤ a n.
+  λO:excess.λa:sequence O.∀n:nat.a (S n) ≤ a n.
+
+
 
 
 (*
 
-definition is_upper_bound ≝ λO:pordered_set.λa:sequence O.λu:O.∀n:nat.a n ≤ u.
-definition is_lower_bound ≝ λO:pordered_set.λa:sequence O.λu:O.∀n:nat.u ≤ a n.
+definition is_upper_bound ≝ λO:excess.λa:sequence O.λu:O.∀n:nat.a n ≤ u.
+definition is_lower_bound ≝ λO:excess.λa:sequence O.λu:O.∀n:nat.u ≤ a n.
 
-record is_sup (O:pordered_set) (a:sequence O) (u:O) : Prop ≝
+record is_sup (O:excess) (a:sequence O) (u:O) : Prop ≝
  { sup_upper_bound: is_upper_bound O a u; 
    sup_least_upper_bound: ∀v:O. is_upper_bound O a v → u≤v
  }.
 
-record is_inf (O:pordered_set) (a:sequence O) (u:O) : Prop ≝
+record is_inf (O:excess) (a:sequence O) (u:O) : Prop ≝
  { inf_lower_bound: is_lower_bound O a u; 
    inf_greatest_lower_bound: ∀v:O. is_lower_bound O a v → v≤u
  }.
 
-record is_bounded_below (O:pordered_set) (a:sequence O) : Type ≝
+record is_bounded_below (O:excess) (a:sequence O) : Type ≝
  { ib_lower_bound: O;
    ib_lower_bound_is_lower_bound: is_lower_bound ? a ib_lower_bound
  }.
 
-record is_bounded_above (O:pordered_set) (a:sequence O) : Type ≝
+record is_bounded_above (O:excess) (a:sequence O) : Type ≝
  { ib_upper_bound: O;
    ib_upper_bound_is_upper_bound: is_upper_bound ? a ib_upper_bound
  }.
 
-record is_bounded (O:pordered_set) (a:sequence O) : Type ≝
+record is_bounded (O:excess) (a:sequence O) : Type ≝
  { ib_bounded_below:> is_bounded_below ? a;
    ib_bounded_above:> is_bounded_above ? a
  }.
 
-record bounded_below_sequence (O:pordered_set) : Type ≝
+record bounded_below_sequence (O:excess) : Type ≝
  { bbs_seq:> sequence O;
    bbs_is_bounded_below:> is_bounded_below ? bbs_seq
  }.
 
-record bounded_above_sequence (O:pordered_set) : Type ≝
+record bounded_above_sequence (O:excess) : Type ≝
  { bas_seq:> sequence O;
    bas_is_bounded_above:> is_bounded_above ? bas_seq
  }.
 
-record bounded_sequence (O:pordered_set) : Type ≝
+record bounded_sequence (O:excess) : Type ≝
  { bs_seq:> sequence O;
    bs_is_bounded_below: is_bounded_below ? bs_seq;
    bs_is_bounded_above: is_bounded_above ? bs_seq
  }.
 
 definition bounded_below_sequence_of_bounded_sequence ≝
- λO:pordered_set.λb:bounded_sequence O.
+ λO:excess.λb:bounded_sequence O.
   mk_bounded_below_sequence ? b (bs_is_bounded_below ? b).
 
 coercion cic:/matita/sequence/bounded_below_sequence_of_bounded_sequence.con.
 
 definition bounded_above_sequence_of_bounded_sequence ≝
- λO:pordered_set.λb:bounded_sequence O.
+ λO:excess.λb:bounded_sequence O.
   mk_bounded_above_sequence ? b (bs_is_bounded_above ? b).
 
 coercion cic:/matita/sequence/bounded_above_sequence_of_bounded_sequence.con.
 
 definition lower_bound ≝
- λO:pordered_set.λb:bounded_below_sequence O.
+ λO:excess.λb:bounded_below_sequence O.
   ib_lower_bound ? b (bbs_is_bounded_below ? b).
 
 lemma lower_bound_is_lower_bound:
- ∀O:pordered_set.∀b:bounded_below_sequence O.
+ ∀O:excess.∀b:bounded_below_sequence O.
   is_lower_bound ? b (lower_bound ? b).
 intros; unfold lower_bound; apply ib_lower_bound_is_lower_bound.
 qed.
 
 definition upper_bound ≝
- λO:pordered_set.λb:bounded_above_sequence O.
+ λO:excess.λb:bounded_above_sequence O.
   ib_upper_bound ? b (bas_is_bounded_above ? b).
 
 lemma upper_bound_is_upper_bound:
- ∀O:pordered_set.∀b:bounded_above_sequence O.
+ ∀O:excess.∀b:bounded_above_sequence O.
   is_upper_bound ? b (upper_bound ? b).
 intros; unfold upper_bound; apply ib_upper_bound_is_upper_bound.
 qed.
 
-definition reverse_excedence: excedence → excedence.
-intros (E); apply (mk_excedence E); [apply (λx,y.exc_relation E y x)] 
+definition reverse_excess: excess → excess.
+intros (E); apply (mk_excess E); [apply (λx,y.exc_relation E y x)] 
 cases E (T f cRf cTf); simplify; 
 [1: unfold Not; intros (x H); apply (cRf x); assumption
 |2: intros (x y z); apply Or_symmetric; apply cTf; assumption;]
 qed. 
 
-definition reverse_pordered_set: pordered_set → pordered_set.
-intros (p); apply (mk_pordered_set (reverse_excedence p));
-generalize in match (reverse_excedence p); intros (E);
+definition reverse_excess: excess → excess.
+intros (p); apply (mk_excess (reverse_excess p));
+generalize in match (reverse_excess p); intros (E);
 apply mk_is_porder_relation;
 [apply le_reflexive|apply le_transitive|apply le_antisymmetric]
 qed. 
  
 lemma is_lower_bound_reverse_is_upper_bound:
- ∀O:pordered_set.∀a:sequence O.∀l:O.
-  is_lower_bound O a l → is_upper_bound (reverse_pordered_set O) a l.
-intros (O a l H); unfold; intros (n); unfold reverse_pordered_set;
-unfold reverse_excedence; simplify; fold unfold le (le ? l (a n)); apply H;    
+ ∀O:excess.∀a:sequence O.∀l:O.
+  is_lower_bound O a l → is_upper_bound (reverse_excess O) a l.
+intros (O a l H); unfold; intros (n); unfold reverse_excess;
+unfold reverse_excess; simplify; fold unfold le (le ? l (a n)); apply H;    
 qed.
 
 lemma is_upper_bound_reverse_is_lower_bound:
- ∀O:pordered_set.∀a:sequence O.∀l:O.
-  is_upper_bound O a l → is_lower_bound (reverse_pordered_set O) a l.
-intros (O a l H); unfold; intros (n); unfold reverse_pordered_set;
-unfold reverse_excedence; simplify; fold unfold le (le ? (a n) l); apply H;    
+ ∀O:excess.∀a:sequence O.∀l:O.
+  is_upper_bound O a l → is_lower_bound (reverse_excess O) a l.
+intros (O a l H); unfold; intros (n); unfold reverse_excess;
+unfold reverse_excess; simplify; fold unfold le (le ? (a n) l); apply H;    
 qed.
 
 lemma reverse_is_lower_bound_is_upper_bound:
- ∀O:pordered_set.∀a:sequence O.∀l:O.
-  is_lower_bound (reverse_pordered_set O) a l → is_upper_bound O a l.
-intros (O a l H); unfold; intros (n); unfold reverse_pordered_set in H;
-unfold reverse_excedence in H; simplify in H; apply H;    
+ ∀O:excess.∀a:sequence O.∀l:O.
+  is_lower_bound (reverse_excess O) a l → is_upper_bound O a l.
+intros (O a l H); unfold; intros (n); unfold reverse_excess in H;
+unfold reverse_excess in H; simplify in H; apply H;    
 qed.
 
 lemma reverse_is_upper_bound_is_lower_bound:
- ∀O:pordered_set.∀a:sequence O.∀l:O.
-  is_upper_bound (reverse_pordered_set O) a l → is_lower_bound O a l.
-intros (O a l H); unfold; intros (n); unfold reverse_pordered_set in H;
-unfold reverse_excedence in H; simplify in H; apply H;    
+ ∀O:excess.∀a:sequence O.∀l:O.
+  is_upper_bound (reverse_excess O) a l → is_lower_bound O a l.
+intros (O a l H); unfold; intros (n); unfold reverse_excess in H;
+unfold reverse_excess in H; simplify in H; apply H;    
 qed.
 
 lemma is_inf_to_reverse_is_sup:
- ∀O:pordered_set.∀a:bounded_below_sequence O.∀l:O.
-  is_inf O a l → is_sup (reverse_pordered_set O) a l.
-intros (O a l H); apply (mk_is_sup (reverse_pordered_set O));
+ ∀O:excess.∀a:bounded_below_sequence O.∀l:O.
+  is_inf O a l → is_sup (reverse_excess O) a l.
+intros (O a l H); apply (mk_is_sup (reverse_excess O));
 [1: apply is_lower_bound_reverse_is_upper_bound; apply inf_lower_bound; assumption
-|2: unfold reverse_pordered_set; simplify; unfold reverse_excedence; simplify; 
+|2: unfold reverse_excess; simplify; unfold reverse_excess; simplify; 
     intros (m H1); apply (inf_greatest_lower_bound ? ? ? H); apply H1;]
 qed.
 
 lemma is_sup_to_reverse_is_inf:
- ∀O:pordered_set.∀a:bounded_above_sequence O.∀l:O.
-  is_sup O a l → is_inf (reverse_pordered_set O) a l.
-intros (O a l H); apply (mk_is_inf (reverse_pordered_set O));
+ ∀O:excess.∀a:bounded_above_sequence O.∀l:O.
+  is_sup O a l → is_inf (reverse_excess O) a l.
+intros (O a l H); apply (mk_is_inf (reverse_excess O));
 [1: apply is_upper_bound_reverse_is_lower_bound; apply sup_upper_bound; assumption
-|2: unfold reverse_pordered_set; simplify; unfold reverse_excedence; simplify; 
+|2: unfold reverse_excess; simplify; unfold reverse_excess; simplify; 
     intros (m H1); apply (sup_least_upper_bound ? ? ? H); apply H1;]
 qed.
 
 lemma reverse_is_sup_to_is_inf:
- ∀O:pordered_set.∀a:bounded_above_sequence O.∀l:O.
-  is_sup (reverse_pordered_set O) a l → is_inf O a l.
+ ∀O:excess.∀a:bounded_above_sequence O.∀l:O.
+  is_sup (reverse_excess O) a l → is_inf O a l.
 intros (O a l H); apply mk_is_inf;
 [1: apply reverse_is_upper_bound_is_lower_bound; 
-    apply (sup_upper_bound (reverse_pordered_set O)); assumption
-|2: intros (v H1); apply (sup_least_upper_bound (reverse_pordered_set O) a l H v);
+    apply (sup_upper_bound (reverse_excess O)); assumption
+|2: intros (v H1); apply (sup_least_upper_bound (reverse_excess O) a l H v);
     apply is_lower_bound_reverse_is_upper_bound; assumption;]
 qed.
 
 lemma reverse_is_inf_to_is_sup:
- ∀O:pordered_set.∀a:bounded_above_sequence O.∀l:O.
-  is_inf (reverse_pordered_set O) a l → is_sup O a l.
+ ∀O:excess.∀a:bounded_above_sequence O.∀l:O.
+  is_inf (reverse_excess O) a l → is_sup O a l.
 intros (O a l H); apply mk_is_sup;
 [1: apply reverse_is_lower_bound_is_upper_bound; 
-    apply (inf_lower_bound (reverse_pordered_set O)); assumption
-|2: intros (v H1); apply (inf_greatest_lower_bound (reverse_pordered_set O) a l H v);
+    apply (inf_lower_bound (reverse_excess O)); assumption
+|2: intros (v H1); apply (inf_greatest_lower_bound (reverse_excess O) a l H v);
     apply is_upper_bound_reverse_is_lower_bound; assumption;]
 qed.