X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fdama%2Fordered_fields_ch0.ma;h=07f32fadef0d8eef0c5e3bb00101a197a5bb3910;hb=0582a602f0b1d6f5430326893a473d78b0aa7dfd;hp=d95108eb327faa8d6d382b3a4ed5be5971e48750;hpb=85fce74d6a0cd8e83e59895e81f1bc09d21ef4c2;p=helm.git

diff --git a/helm/software/matita/dama/ordered_fields_ch0.ma b/helm/software/matita/dama/ordered_fields_ch0.ma
index d95108eb3..07f32fade 100644
--- a/helm/software/matita/dama/ordered_fields_ch0.ma
+++ b/helm/software/matita/dama/ordered_fields_ch0.ma
@@ -15,32 +15,70 @@
 set "baseuri" "cic:/matita/ordered_fields_ch0/".
 
 include "fields.ma".
+include "ordered_sets2.ma".
 
-record is_ordered_field_ch0 (F:field) (le:F→F→Prop) : Prop \def
- { of_mult_compat: ∀a,b. le 0 a → le 0 b → le 0 (a*b);
-   of_plus_compat: ∀a,b,c. le a b → le (a+c) (b+c);
-   of_weak_tricotomy : ∀a,b. a≠b → le a b ∨ le b a;
+(*CSC: non capisco questi alias! Una volta non servivano*)
+alias id "plus" = "cic:/matita/groups/plus.con".
+alias symbol "plus" = "Abelian group plus".
+
+record pre_ordered_field_ch0: Type ≝
+ { of_field:> field;
+   of_cotransitively_ordered_set_: cotransitively_ordered_set;
+   of_with: os_carrier of_cotransitively_ordered_set_ = carrier of_field
+ }.
+
+(*CSC: the following lemma (that is also a coercion) should be automatically
+  generated *)
+lemma of_cotransitively_ordered_set : pre_ordered_field_ch0 → cotransitively_ordered_set.
+ intro F;
+ apply mk_cotransitively_ordered_set;
+ [ apply mk_ordered_set;
+   [ apply (carrier F)
+   | apply
+      (eq_rect ? ? (λC:Type.C→C→Prop) (os_le (of_cotransitively_ordered_set_ F)) ? (of_with F))
+   | apply
+      (eq_rect' Type (os_carrier (of_cotransitively_ordered_set_ F))
+        (λa:Type.λH:os_carrier (of_cotransitively_ordered_set_ F)=a.
+          is_order_relation a (eq_rect Type (os_carrier (of_cotransitively_ordered_set_ F)) (λC:Type.C→C→Prop) (os_le (of_cotransitively_ordered_set_ F)) a H))
+        ? (Type_OF_pre_ordered_field_ch0 F) (of_with F));
+     simplify; 
+     apply (os_order_relation_properties (of_cotransitively_ordered_set_ F))
+   ]
+ | simplify;
+   apply
+    (eq_rect' ? ?
+      (λa:Type.λH:os_carrier (of_cotransitively_ordered_set_ F)=a.
+        cotransitive a
+         match H in eq
+          return
+           λa2:Type.λH1:os_carrier (of_cotransitively_ordered_set_ F)=a2.
+            a2→a2→Prop
+         with 
+          [refl_eq ⇒ os_le (of_cotransitively_ordered_set_ F)])
+      ? ? (of_with F));
+   simplify;
+   apply cos_cotransitive
+ ].
+qed.
+
+coercion cic:/matita/ordered_fields_ch0/of_cotransitively_ordered_set.con.
+
+record is_ordered_field_ch0 (F:pre_ordered_field_ch0) : Type \def
+ { of_mult_compat: ∀a,b:F. 0≤a → 0≤b → 0≤a*b;
+   of_plus_compat: ∀a,b,c:F. a≤b → a+c≤b+c;
+   of_weak_tricotomy : ∀a,b:F. a≠b → a≤b ∨ b≤a;
    (* 0 characteristics *)
    of_char0: ∀n. n > O → sum ? (plus F) 0 1 n ≠ 0
  }.
  
 record ordered_field_ch0 : Type \def
- { of_field:> field;
-   of_le: of_field → of_field → Prop;
-   of_ordered_field_properties:> is_ordered_field_ch0 of_field of_le
+ { of_pre_ordered_field_ch0:> pre_ordered_field_ch0;
+   of_ordered_field_properties:> is_ordered_field_ch0 of_pre_ordered_field_ch0
  }.
 
-interpretation "Ordered field le" 'leq a b =
- (cic:/matita/ordered_fields_ch0/of_le.con _ a b).
- 
-definition lt \def λF:ordered_field_ch0.λa,b:F.a ≤ b ∧ a ≠ b.
-
-interpretation "Ordered field lt" 'lt a b =
- (cic:/matita/ordered_fields_ch0/lt.con _ a b).
-
-lemma le_zero_x_to_le_opp_x_zero: ∀F:ordered_field_ch0.∀x:F. 0 ≤ x → -x ≤ 0.
-intros;
- generalize in match (of_plus_compat ? ? F ? ? (-x) H); intro;
+lemma le_zero_x_to_le_opp_x_zero: ∀F:ordered_field_ch0.∀x:F.0 ≤ x → -x ≤ 0.
+ intros;
+ generalize in match (of_plus_compat ? F ? ? (-x) H); intro;
  rewrite > zero_neutral in H1;
  rewrite > plus_comm in H1;
  rewrite > opp_inverse in H1;
@@ -49,7 +87,7 @@ qed.
 
 lemma le_x_zero_to_le_zero_opp_x: ∀F:ordered_field_ch0.∀x:F. x ≤ 0 → 0 ≤ -x.
  intros;
- generalize in match (of_plus_compat ? ? F ? ? (-x) H); intro;
+ generalize in match (of_plus_compat ? F ? ? (-x) H); intro;
  rewrite > zero_neutral in H1;
  rewrite > plus_comm in H1;
  rewrite > opp_inverse in H1;
@@ -68,7 +106,52 @@ lemma not_eq_x_zero_to_lt_zero_mult_x_x:
     generalize in match (of_mult_compat ? ? ? ? ? ? ? ?  F ? ? H2 H2); intro;
 *)  
 
+axiom lt_zero_to_lt_inv_zero:
+ ∀F:ordered_field_ch0.∀x:F.∀p:x≠0. lt F 0 x → lt F 0 (inv ? x p).
+
+alias symbol "lt" = "natural 'less than'".
+
 (* The ordering is not necessary. *)
 axiom not_eq_sum_field_zero: ∀F:ordered_field_ch0.∀n. O<n → sum_field F n ≠ 0.
+axiom le_zero_sum_field: ∀F:ordered_field_ch0.∀n. O<n → lt F 0 (sum_field F n).
+
+axiom lt_zero_to_le_inv_zero:
+ ∀F:ordered_field_ch0.∀n:nat.∀p:sum_field F n ≠ 0. 0 ≤ inv ? (sum_field ? n) p.
+
+definition tends_to : ∀F:ordered_field_ch0.∀f:nat→F.∀l:F.Prop.
+ apply
+  (λF:ordered_field_ch0.λf:nat → F.λl:F.
+    ∀n:nat.∃m:nat.∀j:nat. le m j →
+     l - (inv F (sum_field ? (S n)) ?) ≤ f j ∧
+     f j ≤ l + (inv F (sum_field ? (S n)) ?));
+ apply not_eq_sum_field_zero;
+ unfold;
+ auto new.
+qed.
 
+(*
+definition is_cauchy_seq ≝
+ λF:ordered_field_ch0.λf:nat→F.
+  ∀eps:F. 0 < eps →
+   ∃n:nat.∀M. n ≤ M →
+    -eps ≤ f M - f n ∧ f M - f n ≤ eps.
+*)
+
+
+
+definition is_cauchy_seq : ∀F:ordered_field_ch0.∀f:nat→F.Prop.
+ apply
+  (λF:ordered_field_ch0.λf:nat→F.
+    ∀m:nat.
+     ∃n:nat.∀N. le n N →
+      -(inv ? (sum_field F (S m)) ?) ≤ f N - f n ∧
+      f N - f n ≤ inv ? (sum_field F (S m)) ?);
+ apply not_eq_sum_field_zero;
+ unfold;
+ auto.
+qed.
 
+definition is_complete ≝
+ λF:ordered_field_ch0.
+  ∀f:nat→F. is_cauchy_seq ? f →
+   ex F (λl:F. tends_to ? f l).