ogroups almost finished

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

diff --git a/matita/dama/ordered_groups.ma b/matita/dama/ordered_groups.ma
index 10e8f189ab0d4aeea549902fa4c2671372188f48..fb4b29f0dc3dd59ff9ae31163ca4d2471a1ab6ad 100644 (file)
--- a/matita/dama/ordered_groups.ma
+++ b/matita/dama/ordered_groups.ma
@@ -14,58 +14,91 @@
 
 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_ordered_set_: ordered_set;
-   og_with: os_carrier og_ordered_set_ = og_abelian_group
- }.
+record pre_ogroup : Type ≝ { 
+  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_ogroup → 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_ogroup; 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.
+record ogroup : Type ≝ { 
+  og_carr:> pre_ogroup;
+  fle_plusr: ∀f,g,h:og_carr. f≤g → f+h≤g+h
+}.
+
+lemma plus_cancr_le: 
+  ∀G:ogroup.∀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 (fle_plusr ??? (-z));
+assumption;
+qed.
+
+lemma fle_plusl: ∀G:ogroup. ∀f,g,h:G. f≤g → h+f≤h+g.
+intros (G f g h);
+apply (plus_cancr_le ??? (-h));
+apply (le_rewl ??? (f+h+ -h)); [apply feq_plusr;apply plus_comm;]
+apply (le_rewl ??? (f+(h+ -h)) (plus_assoc ????));
+apply (le_rewl ??? (f+(-h+h))); [apply feq_plusl;apply plus_comm;]
+apply (le_rewl ??? (f+0)); [apply feq_plusl; apply eq_symmetric; apply opp_inverse]
+apply (le_rewl ??? (0+f) (plus_comm ???));
+apply (le_rewl ??? (f) (eq_symmetric ??? (zero_neutral ??)));
+apply (le_rewr ??? (g+h+ -h)); [apply feq_plusr;apply plus_comm;]
+apply (le_rewr ??? (g+(h+ -h)) (plus_assoc ????));
+apply (le_rewr ??? (g+(-h+h))); [apply feq_plusl;apply plus_comm;]
+apply (le_rewr ??? (g+0)); [apply feq_plusl; apply eq_symmetric; apply opp_inverse]
+apply (le_rewr ??? (0+g) (plus_comm ???));
+apply (le_rewr ??? (g) (eq_symmetric ??? (zero_neutral ??)));
+assumption;
+qed.
+
+lemma plus_cancl_le: 
+  ∀G:ogroup.∀x,y,z:G.z+x ≤ z+y → x ≤ y.
+intros 5 (G x y z L);
+apply (le_rewl ??? (0+x) (zero_neutral ??));
+apply (le_rewl ??? ((-z+z)+x)); [apply feq_plusr;apply opp_inverse;]
+apply (le_rewl ??? (-z+(z+x)) (plus_assoc ????));
+apply (le_rewr ??? (0+y) (zero_neutral ??));
+apply (le_rewr ??? ((-z+z)+y)); [apply feq_plusr;apply opp_inverse;]
+apply (le_rewr ??? (-z+(z+y)) (plus_assoc ????));
+apply (fle_plusl ??? (-z));
+assumption;
+qed.
 
-record ordered_abelian_group : Type ≝
- { og_pre_ordered_abelian_group:> pre_ordered_abelian_group;
-   og_ordered_abelian_group_properties:
-    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_zero_x_to_le_opp_x_zero: 
+  ∀G:ogroup.∀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;
- 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_x_zero_to_le_zero_opp_x: 
+  ∀G:ogroup.∀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.