set "baseuri" "cic:/matita/ordered_fields_ch0/".
include "fields.ma".
+include "ordered_groups.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;
- (* 0 characteristics *)
- of_char0: ∀n. n > O → sum ? (plus F) 0 1 n ≠ 0
- }.
-
-record ordered_field_ch0 : Type \def
+(*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_le: of_field → of_field → Prop;
- of_ordered_field_properties:> is_ordered_field_ch0 of_field of_le
+ of_ordered_abelian_group_: ordered_abelian_group;
+ of_cotransitively_ordered_set_: cotransitively_ordered_set;
+ of_with1_:
+ cos_ordered_set of_cotransitively_ordered_set_ =
+ og_ordered_set of_ordered_abelian_group_;
+ of_with2:
+ og_abelian_group of_ordered_abelian_group_ = r_abelian_group of_field
}.
-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;
- rewrite > zero_neutral in H1;
- rewrite > plus_comm in H1;
- rewrite > opp_inverse in H1;
- assumption.
+lemma of_ordered_abelian_group: pre_ordered_field_ch0 → ordered_abelian_group.
+ intro F;
+ apply mk_ordered_abelian_group;
+ [ apply mk_pre_ordered_abelian_group;
+ [ apply (r_abelian_group F)
+ | apply (og_ordered_set (of_ordered_abelian_group_ F))
+ | apply (eq_f ? ? carrier);
+ apply (of_with2 F)
+ ]
+ |
+ apply
+ (eq_rect' ? ?
+ (λG:abelian_group.λH:og_abelian_group (of_ordered_abelian_group_ F)=G.
+ is_ordered_abelian_group
+ (mk_pre_ordered_abelian_group G (of_ordered_abelian_group_ F)
+ (eq_f abelian_group Type carrier (of_ordered_abelian_group_ F) G
+ H)))
+ ? ? (of_with2 F));
+ simplify;
+ apply (og_ordered_abelian_group_properties (of_ordered_abelian_group_ F))
+ ]
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;
- rewrite > zero_neutral in H1;
- rewrite > plus_comm in H1;
- rewrite > opp_inverse in H1;
- assumption.
+coercion cic:/matita/ordered_fields_ch0/of_ordered_abelian_group.con.
+
+(*CSC: I am not able to prove this since unfold is undone by coercion composition*)
+axiom of_with1:
+ ∀G:pre_ordered_field_ch0.
+ cos_ordered_set (of_cotransitively_ordered_set_ G) =
+ og_ordered_set (of_ordered_abelian_group G).
+
+lemma of_cotransitively_ordered_set : pre_ordered_field_ch0 → cotransitively_ordered_set.
+ intro F;
+ apply mk_cotransitively_ordered_set;
+ [ apply (og_ordered_set F)
+ | apply
+ (eq_rect ? ? (λa:ordered_set.cotransitive (os_carrier a) (os_le a))
+ ? ? (of_with1 F));
+ 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_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_pre_ordered_field_ch0:> pre_ordered_field_ch0;
+ of_ordered_field_properties:> is_ordered_field_ch0 of_pre_ordered_field_ch0
+ }.
+
(*
lemma eq_opp_x_times_opp_one_x: ∀F:ordered_field_ch0.∀x:F.-x = -1*x.
intros;
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.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;
+ autobatch.
+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.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;
+ autobatch.
+qed.
+definition is_complete ≝
+ λF:ordered_field_ch0.
+ ∀f:nat→F. is_cauchy_seq ? f →
+ ex F (λl:F. tends_to ? f l).
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