X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=helm%2Fsoftware%2Fmatita%2Fdama%2Fordered_group.ma;h=9a066a80e9a4d62c345515aa0ff948591547b1c3;hb=4bb9fdc4df84b9659ef3850f09e53aa0284a3250;hp=22c18cfa1cef51fa93f46f8d56233a887855d48a;hpb=feaabb3c45906fafb4b6eb3fb10add6e6da6069b;p=helm.git

diff --git a/helm/software/matita/dama/ordered_group.ma b/helm/software/matita/dama/ordered_group.ma
index 22c18cfa1..9a066a80e 100644
--- a/helm/software/matita/dama/ordered_group.ma
+++ b/helm/software/matita/dama/ordered_group.ma
@@ -12,14 +12,14 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/ordered_group/".
+
 
 include "group.ma".
 
 record pogroup_ : Type ≝ { 
   og_abelian_group_: abelian_group;
-  og_excedence:> excedence;
-  og_with: carr og_abelian_group_ = apart_of_excedence og_excedence
+  og_excess:> excess;
+  og_with: carr og_abelian_group_ = apart_of_excess og_excess
 }.
 
 lemma og_abelian_group: pogroup_ → abelian_group.
@@ -33,12 +33,12 @@ coercion cic:/matita/ordered_group/og_abelian_group.con.
 
 record pogroup : Type ≝ { 
   og_carr:> pogroup_;
-  canc_plusr_exc: ∀f,g,h:og_carr. f+h ≰ g+h → f ≰ g
+  plus_cancr_exc: ∀f,g,h:og_carr. f+h ≰ g+h → f ≰ g
 }.
 
 lemma fexc_plusr: 
   ∀G:pogroup.∀x,y,z:G. x ≰ y → x+z ≰ y + z.
-intros 5 (G x y z L); apply (canc_plusr_exc ??? (-z));
+intros 5 (G x y z L); apply (plus_cancr_exc ??? (-z));
 apply (Ex≪  (x + (z + -z)) (plus_assoc ????));
 apply (Ex≪  (x + (-z + z)) (plus_comm ??z));
 apply (Ex≪  (x+0) (opp_inverse ??));
@@ -53,15 +53,15 @@ qed.
 
 coercion cic:/matita/ordered_group/fexc_plusr.con nocomposites.
 
-lemma canc_plusl_exc: ∀G:pogroup.∀f,g,h:G. h+f ≰ h+g → f ≰ g.
-intros 5 (G x y z L); apply (canc_plusr_exc ??? z);
+lemma plus_cancl_exc: ∀G:pogroup.∀f,g,h:G. h+f ≰ h+g → f ≰ g.
+intros 5 (G x y z L); apply (plus_cancr_exc ??? z);
 apply (exc_rewl ??? (z+x) (plus_comm ???));
 apply (exc_rewr ??? (z+y) (plus_comm ???) L);
 qed.
 
 lemma fexc_plusl: 
   ∀G:pogroup.∀x,y,z:G. x ≰ y → z+x ≰ z+y.
-intros 5 (G x y z L); apply (canc_plusl_exc ??? (-z));
+intros 5 (G x y z L); apply (plus_cancl_exc ??? (-z));
 apply (exc_rewl ???? (plus_assoc ??z x));
 apply (exc_rewr ???? (plus_assoc ??z y));
 apply (exc_rewl ??? (0+x) (opp_inverse ??));
@@ -85,7 +85,7 @@ apply (le_rewr ??? (y+0) (plus_comm ???));
 apply (le_rewr ??? (y+(-z+z)) (opp_inverse ??));
 apply (le_rewr ??? (y+(z+ -z)) (plus_comm ??z));
 apply (le_rewr ??? (y+z+ -z) (plus_assoc ????));
-intro H; apply L; clear L; apply (canc_plusr_exc ??? (-z) H);
+intro H; apply L; clear L; apply (plus_cancr_exc ??? (-z) H);
 qed.
 
 lemma fle_plusl: ∀G:pogroup. ∀f,g,h:G. f≤g → h+f≤h+g.
@@ -137,7 +137,7 @@ qed.
 
 lemma exc_opp_x_zero_to_exc_zero_x: 
   ∀G:pogroup.∀x:G.-x ≰ 0 → 0 ≰ x.
-intros (G x H); apply (canc_plusr_exc ??? (-x));
+intros (G x H); apply (plus_cancr_exc ??? (-x));
 apply (exc_rewr ???? (plus_comm ???));
 apply (exc_rewr ???? (opp_inverse ??));
 apply (exc_rewl ???? (zero_neutral ??) H);
@@ -159,7 +159,7 @@ qed.
 
 lemma exc_zero_opp_x_to_exc_x_zero: 
   ∀G:pogroup.∀x:G. 0 ≰ -x → x ≰ 0.
-intros (G x H); apply (canc_plusl_exc ??? (-x));
+intros (G x H); apply (plus_cancl_exc ??? (-x));
 apply (exc_rewr ???? (plus_comm ???));
 apply (exc_rewl ???? (opp_inverse ??));
 apply (exc_rewr ???? (zero_neutral ??) H);
@@ -181,6 +181,14 @@ apply (lt_rewl ??? x (zero_neutral ??));
 assumption; 
 qed.
 
+lemma lt_opp_x_zero_to_lt_zero_x: 
+  ∀G:pogroup.∀x:G. -x < 0 → 0 < x.
+intros (G x Lx0); apply (plus_cancr_lt ??? (-x));
+apply (lt_rewl ??? (-x) (zero_neutral ??));
+apply (lt_rewr ??? (-x+x) (plus_comm ???));
+apply (lt_rewr ??? 0 (opp_inverse ??));
+assumption; 
+qed.
 
 lemma lt0plus_orlt: 
   ∀G:pogroup. ∀x,y:G. 0 ≤ x → 0 ≤ y → 0 < x + y → 0 < x ∨ 0 < y.
@@ -263,7 +271,7 @@ record togroup : Type ≝ {
 
 lemma lexxyy_lexy: ∀G:togroup. ∀x,y:G. x+x ≤ y+y → x ≤ y.
 intros (G x y H); intro H1; lapply (tog_total ??? H1) as H2;
-lapply (ltxy_ltyyxx ??? H2) as H3; lapply (lt_to_excede ??? H3) as H4;
+lapply (ltxy_ltyyxx ??? H2) as H3; lapply (lt_to_excess ??? H3) as H4;
 cases (H H4);
 qed. 
 
@@ -272,41 +280,49 @@ intros (G x y H); cases (eq_le_le ??? H); apply le_le_eq;
 apply lexxyy_lexy; assumption;
 qed.
 
-lemma bar: ∀G:abelian_group. ∀x,y:G. 0 # x + y → 0 #x ∨ 0#y.
+lemma applus_orap: ∀G:abelian_group. ∀x,y:G. 0 # x + y → 0 #x ∨ 0#y.
 intros; cases (ap_cotransitive ??? y a); [right; assumption]
 left; apply (plus_cancr_ap ??? y); apply (ap_rewl ???y (zero_neutral ??));
 assumption;
 qed.
 
-lemma pippo: ∀G:pogroup.∀a,b,c,d:G. a < b → c < d → a+c < b + d.
+lemma ltplus: ∀G:pogroup.∀a,b,c,d:G. a < b → c < d → a+c < b + d.
 intros (G a b c d H1 H2);
 lapply (flt_plusr ??? c H1) as H3;
 apply (lt_transitive ???? H3);
 apply flt_plusl; assumption;
 qed.
 
-lemma pippo2: ∀G:pogroup.∀a,b,c,d:G. a+c ≰ b + d →  a ≰ b ∨ c ≰ d.
+lemma excplus_orexc: ∀G:pogroup.∀a,b,c,d:G. a+c ≰ b + d →  a ≰ b ∨ c ≰ d.
 intros (G a b c d H1 H2);
 cases (exc_cotransitive ??? (a + d) H1); [
-  right; apply (canc_plusl_exc ??? a); assumption]
-left; apply (canc_plusr_exc ??? d); assumption;
+  right; apply (plus_cancl_exc ??? a); assumption]
+left; apply (plus_cancr_exc ??? d); assumption;
 qed.
 
-lemma pippo3: ∀G:pogroup.∀a,b,c,d:G. a ≤ b → c ≤ d → a+c ≤ b + d.
-intros (G a b c d H1 H2); intro H3; cases (pippo2 ????? H3);
+lemma leplus: ∀G:pogroup.∀a,b,c,d:G. a ≤ b → c ≤ d → a+c ≤ b + d.
+intros (G a b c d H1 H2); intro H3; cases (excplus_orexc ????? H3);
 [apply H1|apply H2] assumption;
 qed.  
 
-lemma foo: ∀G:togroup.∀x,y:G. 0 ≤ x + y → x < 0 → 0 ≤ y.
-intros; intro; apply H; lapply (lt_to_excede ??? l);
+lemma leplus_lt_le: ∀G:togroup.∀x,y:G. 0 ≤ x + y → x < 0 → 0 ≤ y.
+intros; intro; apply H; lapply (lt_to_excess ??? l);
 lapply (tog_total ??? e);
 lapply (tog_total ??? Hletin);
-lapply (pippo ????? Hletin2 Hletin1);
+lapply (ltplus ????? Hletin2 Hletin1);
 apply (exc_rewl ??? (0+0)); [apply eq_sym; apply zero_neutral]
-apply lt_to_excede; assumption;
+apply lt_to_excess; assumption;
 qed. 
 
-lemma pippo4: ∀G:togroup.∀a,b,c,d:G. a+c < b + d →  a < b ∨ c < d.
-intros (G a b c d H1 H2); lapply (lt_to_excede ??? H1);
-cases (pippo2 ????? Hletin); [left|right] apply tog_total; assumption;
+lemma ltplus_orlt: ∀G:togroup.∀a,b,c,d:G. a+c < b + d →  a < b ∨ c < d.
+intros (G a b c d H1 H2); lapply (lt_to_excess ??? H1);
+cases (excplus_orexc ????? Hletin); [left|right] apply tog_total; assumption;
+qed.
+
+lemma excplus: ∀G:togroup.∀a,b,c,d:G.a ≰ b → c ≰ d → a + c ≰ b + d.
+intros (G a b c d L1 L2); 
+lapply (fexc_plusr ??? (c) L1) as L3;
+elim (exc_cotransitive ??? (b+d) L3); [assumption]
+lapply (plus_cancl_exc ???? t); lapply (tog_total ??? Hletin);
+cases Hletin1; cases (H L2);
 qed.