X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=helm%2Fsoftware%2Fmatita%2Fdama%2Fgroup.ma;h=104dcf274e3943727d090dc8ff8fddd5f7bca59e;hb=10f29fdd78ee089a9a94446207b543d33d6c851c;hp=87cc25855cb4ce1c17c47b254e9d7d2e959e0d2b;hpb=39c55604f8114e2e8f9f9769acd328de8f19c7e4;p=helm.git

diff --git a/helm/software/matita/dama/group.ma b/helm/software/matita/dama/group.ma
index 87cc25855..104dcf274 100644
--- a/helm/software/matita/dama/group.ma
+++ b/helm/software/matita/dama/group.ma
@@ -12,9 +12,9 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/group/".
 
-include "excedence.ma".
+
+include "excess.ma".
 
 definition left_neutral ≝ λC:apartness.λop.λe:C. ∀x:C. op e x ≈ x.
 definition right_neutral ≝ λC:apartness.λop. λe:C. ∀x:C. op x e ≈ x.
@@ -80,9 +80,17 @@ coercion cic:/matita/group/feq_plusl.con nocomposites.
 lemma plus_strong_extr: ∀G:abelian_group.∀z:G.strong_ext ? (λx.x + z).
 intros 5 (G z x y A); simplify in A;
 lapply (plus_comm ? z x) as E1; lapply (plus_comm ? z y) as E2;
-lapply (ap_rewl ???? E1 A) as A1; lapply (ap_rewr ???? E2 A1) as A2;
+lapply (Ap≪ ? E1 A) as A1; lapply (Ap≫ ? E2 A1) as A2;
 apply (plus_strong_ext ???? A2);
 qed.
+
+lemma plus_cancl_ap: ∀G:abelian_group.∀x,y,z:G.z+x # z + y → x # y.
+intros; apply plus_strong_ext; assumption;
+qed.
+
+lemma plus_cancr_ap: ∀G:abelian_group.∀x,y,z:G.x+z # y+z → x # y.
+intros; apply plus_strong_extr; assumption;
+qed.
    
 lemma feq_plusr: ∀G:abelian_group.∀x,y,z:G. y ≈ z →  y+x ≈ z+x.
 intros (G x y z Eyz); apply (strong_ext_to_ext ?? (plus_strong_extr ? x));
@@ -103,33 +111,39 @@ coercion cic:/matita/group/feq_plusl_sym_.con nocomposites.
       
 lemma fap_plusl: ∀G:abelian_group.∀x,y,z:G. y # z →  x+y # x+z. 
 intros (G x y z Ayz); apply (plus_strong_ext ? (-x));
-apply (ap_rewl ??? ((-x + x) + y));
+apply (Ap≪ ((-x + x) + y));
 [1: apply plus_assoc; 
-|2: apply (ap_rewr ??? ((-x +x) +z));
+|2: apply (Ap≫ ((-x +x) +z));
     [1: apply plus_assoc; 
-    |2: apply (ap_rewl ??? (0 + y));
+    |2: apply (Ap≪ (0 + y));
         [1: apply (feq_plusr ???? (opp_inverse ??)); 
-        |2: apply (ap_rewl ???? (zero_neutral ? y)); 
-            apply (ap_rewr ??? (0 + z) (opp_inverse ??)); 
-            apply (ap_rewr ???? (zero_neutral ??)); assumption;]]]
+        |2: apply (Ap≪ ? (zero_neutral ? y)); 
+            apply (Ap≫ (0 + z) (opp_inverse ??)); 
+            apply (Ap≫ ? (zero_neutral ??)); assumption;]]]
 qed.
 
 lemma fap_plusr: ∀G:abelian_group.∀x,y,z:G. y # z →  y+x # z+x. 
 intros (G x y z Ayz); apply (plus_strong_extr ? (-x));
-apply (ap_rewl ??? (y + (x + -x)));
+apply (Ap≪ (y + (x + -x)));
 [1: apply (eq_sym ??? (plus_assoc ????)); 
-|2: apply (ap_rewr ??? (z + (x + -x)));
+|2: apply (Ap≫ (z + (x + -x)));
     [1: apply (eq_sym ??? (plus_assoc ????)); 
-    |2: apply (ap_rewl ??? (y + (-x+x)) (plus_comm ? x (-x)));
-        apply (ap_rewl ??? (y + 0) (opp_inverse ??));
-        apply (ap_rewl ??? (0 + y) (plus_comm ???));
-        apply (ap_rewl ??? y (zero_neutral ??));
-        apply (ap_rewr ??? (z + (-x+x)) (plus_comm ? x (-x)));
-        apply (ap_rewr ??? (z + 0) (opp_inverse ??));
-        apply (ap_rewr ??? (0 + z) (plus_comm ???));
-        apply (ap_rewr ??? z (zero_neutral ??));
+    |2: apply (Ap≪ (y + (-x+x)) (plus_comm ? x (-x)));
+        apply (Ap≪ (y + 0) (opp_inverse ??));
+        apply (Ap≪ (0 + y) (plus_comm ???));
+        apply (Ap≪ y (zero_neutral ??));
+        apply (Ap≫ (z + (-x+x)) (plus_comm ? x (-x)));
+        apply (Ap≫ (z + 0) (opp_inverse ??));
+        apply (Ap≫ (0 + z) (plus_comm ???));
+        apply (Ap≫ z (zero_neutral ??));
         assumption]]
 qed.
+
+lemma applus: ∀E:abelian_group.∀x,a,y,b:E.x + a # y + b → x # y ∨ a # b.
+intros; cases (ap_cotransitive ??? (y+a) a1); [left|right]
+[apply (plus_cancr_ap ??? a)|apply (plus_cancl_ap ??? y)]
+assumption;
+qed.
     
 lemma plus_cancl: ∀G:abelian_group.∀y,z,x:G. x+y ≈ x+z → y ≈ z. 
 intros 6 (G y z x E Ayz); apply E; apply fap_plusl; assumption;
@@ -169,7 +183,6 @@ qed.
 
 lemma feq_oppr: ∀G:abelian_group.∀x,y,z:G. y ≈ z → x ≈ -y → x ≈ -z.
 intros (G x y z H1 H2); apply (plus_cancr ??? z);
-(* apply (eq_trans ??? 0 ? (opp_inverse ??)); *)
 apply (Eq≈ 0 ? (opp_inverse ??));
 apply (Eq≈ (-y + z) H2);
 apply (Eq≈ (-y + y) H1);
@@ -205,25 +218,3 @@ compose feq_plusl with feq_opp(H); apply H; assumption;
 qed.
 
 coercion cic:/matita/group/eq_opp_plusl.con nocomposites.
-
-lemma plus_cancr_ap: ∀G:abelian_group.∀x,y,z:G. x+z # y+z → x # y.
-intros (G x y z H); lapply (fap_plusr ? (-z) ?? H) as H1; clear H;
-lapply (ap_rewl ? (x + (z + -z)) ?? (plus_assoc ? x z (-z)) H1) as H2; clear H1;
-lapply (ap_rewl ? (x + (-z + z)) ?? (plus_comm ?z (-z)) H2) as H1; clear H2;
-lapply (ap_rewl ? (x + 0) ?? (opp_inverse ?z) H1) as H2; clear H1;
-lapply (ap_rewl ? (0+x) ?? (plus_comm ?x 0) H2) as H1; clear H2;
-lapply (ap_rewl ? x ?? (zero_neutral ?x) H1) as H2; clear H1;
-lapply (ap_rewr ? (y + (z + -z)) ?? (plus_assoc ? y z (-z)) H2) as H3;
-lapply (ap_rewr ? (y + (-z + z)) ?? (plus_comm ?z (-z)) H3) as H4;
-lapply (ap_rewr ? (y + 0) ?? (opp_inverse ?z) H4) as H5;
-lapply (ap_rewr ? (0+y) ?? (plus_comm ?y 0) H5) as H6;
-lapply (ap_rewr ? y ?? (zero_neutral ?y) H6);
-assumption;
-qed.
-
-lemma pluc_cancl_ap: ∀G:abelian_group.∀x,y,z:G. z+x # z+y → x # y.
-intros (G x y z H); apply (plus_cancr_ap ??? z);
-apply (ap_rewl ???? (plus_comm ???)); 
-apply (ap_rewr ???? (plus_comm ???));
-assumption;
-qed.