set "baseuri" "cic:/matita/ordered_groups/".
-include "groups.ma".
include "ordered_sets.ma".
+include "groups.ma".
record pre_ordered_abelian_group : Type ≝
- { og_abelian_group:> abelian_group;
- og_tordered_set_: tordered_set;
- og_with: exc_carr og_tordered_set_ = og_abelian_group
+ { og_abelian_group_: abelian_group;
+ og_tordered_set:> tordered_set;
+ og_with: carr og_abelian_group_ = og_tordered_set
}.
-lemma og_tordered_set: pre_ordered_abelian_group → tordered_set.
-intro G; apply mk_tordered_set;
-[1: apply mk_pordered_set;
- [1: apply (mk_excedence G);
- [1: cases G; clear G; simplify; rewrite < H; clear H;
- cases og_tordered_set_; clear og_tordered_set_; simplify;
- cases tos_poset; simplify; cases pos_carr; simplify; assumption;
- |2: cases G; simplify; cases H; simplify; clear H;
- cases og_tordered_set_; simplify; clear og_tordered_set_;
- cases tos_poset; simplify; cases pos_carr; simplify;
- intros; apply H;
- |3: cases G; simplify; cases H; simplify; cases og_tordered_set_; simplify;
- cases tos_poset; simplify; cases pos_carr; simplify;
- intros; apply c; assumption]
- |2: cases G; simplify;
- cases H; simplify; clear H; cases og_tordered_set_; simplify;
- cases tos_poset; simplify; assumption;]
-|2: simplify; (* SLOW, senza la simplify il widget muore *)
- cases G; simplify;
- generalize in match (tos_totality og_tordered_set_);
- unfold total_order_property;
- cases H; simplify; cases og_tordered_set_; simplify;
- cases tos_poset; simplify; cases pos_carr; simplify;
- intros; apply f; assumption;]
+lemma og_abelian_group: pre_ordered_abelian_group → abelian_group.
+intro G; apply (mk_abelian_group G); [1,2,3: rewrite < (og_with G)]
+[apply (plus (og_abelian_group_ G));|apply zero;|apply opp]
+unfold apartness_OF_pre_ordered_abelian_group; cases (og_with G); simplify;
+[apply plus_assoc|apply plus_comm|apply zero_neutral|apply opp_inverse|apply plus_strong_ext]
qed.
-coercion cic:/matita/ordered_groups/og_tordered_set.con.
+coercion cic:/matita/ordered_groups/og_abelian_group.con.
definition is_ordered_abelian_group ≝
λG:pre_ordered_abelian_group. ∀f,g,h:G. f≤g → f+h≤g+h.
is_ordered_abelian_group og_pre_ordered_abelian_group
}.
+lemma le_rewl: ∀E:excedence.∀x,z,y:E. x ≈ y → x ≤ z → y ≤ z.
+intros (E x z y); apply (le_transitive ???? ? H1);
+clear H1 z; unfold in H; unfold; intro H1; apply H; clear H;
+lapply ap_cotransitive;
+intros (G x z y); intro Eyz;
+
+
+lemma plus_cancr_le:
+ ∀G:ordered_abelian_group.∀x,y,z:G.x+z ≤ y + z → x ≤ y.
+intros 5 (G x y z L);
+
+ apply L; clear L; elim (exc_cotransitive ???z Exy);
+
lemma le_zero_x_to_le_opp_x_zero:
∀G:ordered_abelian_group.∀x:G.0 ≤ x → -x ≤ 0.
intros (G x Px);
generalize in match (og_ordered_abelian_group_properties ? ? ? (-x) Px); intro;
(* ma cazzo, qui bisogna rifare anche i gruppi con ≈ ? *)
- rewrite > zero_neutral in H1;
- rewrite > plus_comm in H1;
- rewrite > opp_inverse in H1;
+ rewrite > zero_neutral in H;
+ rewrite > plus_comm in H;
+ rewrite > opp_inverse in H;
assumption.
qed.