set "baseuri" "cic:/matita/ordered_fields_ch0/".
include "fields.ma".
+include "ordered_sets2.ma".
-record is_total_order_relation (C:Type) (le:C→C→Prop) : Type \def
- { to_cotransitive: ∀x,y,z:C. le x y → le x z ∨ le z y;
- to_antisimmetry: ∀x,y:C. le x y → le y x → x=y
+(*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
}.
-theorem to_transitive:
- ∀C.∀le:C→C→Prop. is_total_order_relation ? le → transitive ? le.
- intros;
- unfold transitive;
- intros;
- elim (to_cotransitive ? ? i ? ? z H);
- [ assumption
- | rewrite < (to_antisimmetry ? ? i ? ? H1 t);
- assumption
- ].
+(*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.
-record is_ordered_field_ch0 (F:field) (le:F→F→Prop) : Type \def
- { of_total_order_relation:> is_total_order_relation ? le;
- 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;
+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_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;
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;
*)
axiom lt_zero_to_lt_inv_zero:
- ∀F:ordered_field_ch0.∀x:F.∀p:x≠0. 0 < x → 0 < inv ? x p.
-
+ ∀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 → 0 < sum_field F n.
+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 →
+ ∀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;
apply
(λF:ordered_field_ch0.λf:nat→F.
∀m:nat.
- ∃n:nat.∀N. n ≤ N →
+ ∃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;
definition is_complete ≝
λF:ordered_field_ch0.
∀f:nat→F. is_cauchy_seq ? f →
- ex F (λl:F. tends_to ? f l).
\ No newline at end of file
+ ex F (λl:F. tends_to ? f l).
projects / helm.git / blobdiff