snapshot

[helm.git] / matita / dama / ordered_groups.ma

diff --git a/matita/dama/ordered_groups.ma b/matita/dama/ordered_groups.ma
index a9912d43f55ccb92bd22ecec4c9e3320e68146ef..11ac6a216ab7e12a187b4f08295ed7d4ca7bf6ad 100644 (file)
--- a/matita/dama/ordered_groups.ma
+++ b/matita/dama/ordered_groups.ma
@@ -14,34 +14,23 @@
 
 set "baseuri" "cic:/matita/ordered_groups/".
 
+include "ordered_sets.ma".
 include "groups.ma".
-include "ordered_sets2.ma".
 
 record pre_ordered_abelian_group : Type ≝
- { og_abelian_group:> abelian_group;
-   og_ordered_set_: ordered_set;
-   og_with: os_carrier og_ordered_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_ordered_set: pre_ordered_abelian_group → ordered_set.
- intro G;
- apply mk_ordered_set;
-  [ apply (carrier (og_abelian_group G))
-  | apply (eq_rect ? ? (λC:Type.C→C→Prop) ? ? (og_with G));
-    apply os_le
-  | apply
-     (eq_rect' ? ?
-       (λa:Type.λH:os_carrier (og_ordered_set_ G) = a.
-        is_order_relation a
-         (eq_rect Type (og_ordered_set_ G) (λC:Type.C→C→Prop)
-          (os_le (og_ordered_set_ G)) a H))
-       ? ? (og_with G));
-    simplify;
-    apply (os_order_relation_properties (og_ordered_set_ G))
-  ]
+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_ordered_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.
@@ -52,20 +41,53 @@ record ordered_abelian_group : Type ≝
     is_ordered_abelian_group og_pre_ordered_abelian_group
  }.
 
-lemma le_zero_x_to_le_opp_x_zero: ∀G:ordered_abelian_group.∀x:G.0 ≤ x → -x ≤ 0.
- intros;
- generalize in match (og_ordered_abelian_group_properties ? ? ? (-x) H); intro;
- rewrite > zero_neutral in H1;
- rewrite > plus_comm in H1;
- rewrite > opp_inverse in H1;
- assumption.
+lemma le_le_eq: ∀E:excedence.∀x,y:E. x ≤ y → y ≤ x → x ≈ y.
+intros 6 (E x y L1 L2 H); cases H; [apply (L1 H1)|apply (L2 H1)]
 qed.
 
-lemma le_x_zero_to_le_zero_opp_x: ∀G:ordered_abelian_group.∀x:G. x ≤ 0 → 0 ≤ -x.
- intros;
- generalize in match (og_ordered_abelian_group_properties ? ? ? (-x) H); intro;
- rewrite > zero_neutral in H1;
- rewrite > plus_comm in H1;
- rewrite > opp_inverse in H1;
- assumption.
+lemma unfold_apart: ∀E:excedence. ∀x,y:E. x ≰ y ∨ y ≰ x → x # y.
+unfold apart_of_excedence; unfold apart; simplify; intros; assumption;
+qed.
+
+lemma le_rewl: ∀E:excedence.∀z,y,x:E. x ≈ y → x ≤ z → y ≤ z.
+intros (E z y x Exy Lxz); apply (le_transitive ???? ? Lxz);
+intro Xyz; apply Exy; apply unfold_apart; right; assumption;
+qed.
+
+lemma le_rewr: ∀E:excedence.∀z,y,x:E. x ≈ y → z ≤ x → z ≤ y.
+intros (E z y x Exy Lxz); apply (le_transitive ???? Lxz);
+intro Xyz; apply Exy; apply unfold_apart; left; assumption;
+qed.
+
+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 (le_rewl ??? (0+x) (zero_neutral ??));
+apply (le_rewl ??? (x+0) (plus_comm ???));
+apply (le_rewl ??? (x+(-z+z))); [apply feq_plusl;apply opp_inverse;]
+apply (le_rewl ??? (x+(z+ -z))); [apply feq_plusl;apply plus_comm;]
+apply (le_rewl ??? (x+z+ -z)); [apply eq_symmetric; apply plus_assoc;]
+apply (le_rewr ??? (0+y) (zero_neutral ??));
+apply (le_rewr ??? (y+0) (plus_comm ???));
+apply (le_rewr ??? (y+(-z+z))); [apply feq_plusl;apply opp_inverse;]
+apply (le_rewr ??? (y+(z+ -z))); [apply feq_plusl;apply plus_comm;]
+apply (le_rewr ??? (y+z+ -z)); [apply eq_symmetric; apply plus_assoc;]
+apply (og_ordered_abelian_group_properties ??? (-z));
+assumption;
+qed.
+
+lemma le_zero_x_to_le_opp_x_zero: 
+  ∀G:ordered_abelian_group.∀x:G.0 ≤ x → -x ≤ 0.
+intros (G x Px); apply (plus_cancr_le ??? x);
+apply (le_rewl ??? 0 (eq_symmetric ??? (opp_inverse ??)));
+apply (le_rewr ??? x (eq_symmetric ??? (zero_neutral ??)));
+assumption;
+qed.
+
+lemma le_x_zero_to_le_zero_opp_x: 
+  ∀G:ordered_abelian_group.∀x:G. x ≤ 0 → 0 ≤ -x.
+intros (G x Lx0); apply (plus_cancr_le ??? x);
+apply (le_rewr ??? 0 (eq_symmetric ??? (opp_inverse ??)));
+apply (le_rewl ??? x (eq_symmetric ??? (zero_neutral ??)));
+assumption; 
 qed.