set "baseuri" "cic:/matita/ordered_fields_ch0/".
include "fields.ma".
-include "ordered_sets.ma".
+include "ordered_groups.ma".
(*CSC: non capisco questi alias! Una volta non servivano*)
alias id "plus" = "cic:/matita/groups/plus.con".
alias symbol "plus" = "Abelian group plus".
-record is_ordered_field_ch0 (F:field) (le:F→F→Prop) : Type \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 *)
- (*CSC: qua c'era un ? al posto di F *)
- of_char0: ∀n. n > O → sum F (plus F) 0 1 n ≠ 0
- }.
-
-record ordered_field_ch0 : Type \def
+
+record pre_ordered_field_ch0: Type ≝
{ of_field:> field;
- of_ordered_set:> ordered_set of_field;
- of_reflexive: reflexive ? (os_le ? of_ordered_set);
- of_antisimmetric: antisimmetric ? (os_le ? of_ordered_set);
- of_cotransitive: cotransitive ? (os_le ? of_ordered_set);
- (*CSC: qui c'era un ? al posto di of_field *)
- of_ordered_field_properties:> is_ordered_field_ch0 of_field (os_le ? of_ordered_set)
+ 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
}.
-(*theorem ordered_set_of_ordered_field_ch0:
- ∀F:ordered_field_ch0.ordered_set F.
- intros;
- apply mk_ordered_set;
- [ apply (mk_pre_ordered_set ? (of_le F))
- | apply mk_is_order_relation;
- [ apply (of_reflexive F)
- | apply antisimmetric_to_cotransitive_to_transitive;
- [ apply (of_antisimmetric F)
- | apply (of_cotransitive F)
- ]
- | apply (of_antisimmetric F)
+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.
-coercion cic:/matita/ordered_fields_ch0/ordered_set_of_ordered_field_ch0.con.
-*)
-
-(*interpretation "Ordered field le" 'leq a b =
- (cic:/matita/ordered_fields_ch0/of_le.con _ a b).
- *)
-
-(*CSC: qua c'era uno zero*)
-lemma le_zero_x_to_le_opp_x_zero: ∀F:ordered_field_ch0.∀x:F.(zero F) ≤ 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*)apply H1.
+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.
-(*CSC: qua c'era uno zero*)
-lemma le_x_zero_to_le_zero_opp_x: ∀F:ordered_field_ch0.∀x:F. x ≤ 0 → (zero F) ≤ -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.*) apply H1.
-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.
*)
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).
+ ∀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 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. os_le ? F 0 (inv ? (sum_field ? n) p).
+ ∀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 →
+ ∀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;
- auto new.
+ autobatch.
qed.
(*
apply
(λF:ordered_field_ch0.λf:nat→F.
∀m:nat.
- ∃n:nat.∀N. le n N →
+ ∃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;
- auto.
+ autobatch.
qed.
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