many sed to use notation for rewriting

[helm.git] / helm / software / matita / dama / ordered_group.ma

diff --git a/helm/software/matita/dama/ordered_group.ma b/helm/software/matita/dama/ordered_group.ma
index 9a066a80e9a4d62c345515aa0ff948591547b1c3..8677e755ba6902a649cf990ffcaaceb7332caea4 100644 (file)
--- a/helm/software/matita/dama/ordered_group.ma
+++ b/helm/software/matita/dama/ordered_group.ma
@@ -55,19 +55,19 @@ coercion cic:/matita/ordered_group/fexc_plusr.con nocomposites.
 
 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);
+apply (Ex≪ (z+x) (plus_comm ???));
+apply (Ex≫ (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 (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 ??));
-apply (exc_rewr ??? (0+y) (opp_inverse ??));
-apply (exc_rewl ???? (zero_neutral ??));
-apply (exc_rewr ???? (zero_neutral ??) L);
+apply (Ex≪? (plus_assoc ??z x));
+apply (Ex≫? (plus_assoc ??z y));
+apply (Ex≪ (0+x) (opp_inverse ??));
+apply (Ex≫ (0+y) (opp_inverse ??));
+apply (Ex≪? (zero_neutral ??));
+apply (Ex≫? (zero_neutral ??) L);
 qed.
 
 coercion cic:/matita/ordered_group/fexc_plusl.con nocomposites.
@@ -75,50 +75,50 @@ coercion cic:/matita/ordered_group/fexc_plusl.con nocomposites.
 lemma plus_cancr_le: 
   ∀G:pogroup.∀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)) (opp_inverse ??));
-apply (le_rewl ??? (x+(z+ -z)) (plus_comm ??z));
-apply (le_rewl ??? (x+z+ -z) (plus_assoc ????));
-apply (le_rewr ??? (0+y) (zero_neutral ??));
-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 ????));
+apply (Le≪ (0+x) (zero_neutral ??));
+apply (Le≪ (x+0) (plus_comm ???));
+apply (Le≪ (x+(-z+z)) (opp_inverse ??));
+apply (Le≪ (x+(z+ -z)) (plus_comm ??z));
+apply (Le≪ (x+z+ -z) (plus_assoc ????));
+apply (Le≫ (0+y) (zero_neutral ??));
+apply (Le≫ (y+0) (plus_comm ???));
+apply (Le≫ (y+(-z+z)) (opp_inverse ??));
+apply (Le≫ (y+(z+ -z)) (plus_comm ??z));
+apply (Le≫ (y+z+ -z) (plus_assoc ????));
 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.
 intros (G f g h);
 apply (plus_cancr_le ??? (-h));
-apply (le_rewl ??? (f+h+ -h) (plus_comm ? f h));
-apply (le_rewl ??? (f+(h+ -h)) (plus_assoc ????));
-apply (le_rewl ??? (f+(-h+h)) (plus_comm ? h (-h)));
-apply (le_rewl ??? (f+0) (opp_inverse ??));
-apply (le_rewl ??? (0+f) (plus_comm ???));
-apply (le_rewl ??? (f) (zero_neutral ??));
-apply (le_rewr ??? (g+h+ -h) (plus_comm ? h ?));
-apply (le_rewr ??? (g+(h+ -h)) (plus_assoc ????));
-apply (le_rewr ??? (g+(-h+h)) (plus_comm ??h));
-apply (le_rewr ??? (g+0) (opp_inverse ??));
-apply (le_rewr ??? (0+g) (plus_comm ???));
-apply (le_rewr ??? (g) (zero_neutral ??) H);
+apply (Le≪ (f+h+ -h) (plus_comm ? f h));
+apply (Le≪ (f+(h+ -h)) (plus_assoc ????));
+apply (Le≪ (f+(-h+h)) (plus_comm ? h (-h)));
+apply (Le≪ (f+0) (opp_inverse ??));
+apply (Le≪ (0+f) (plus_comm ???));
+apply (Le≪ (f) (zero_neutral ??));
+apply (Le≫ (g+h+ -h) (plus_comm ? h ?));
+apply (Le≫ (g+(h+ -h)) (plus_assoc ????));
+apply (Le≫ (g+(-h+h)) (plus_comm ??h));
+apply (Le≫ (g+0) (opp_inverse ??));
+apply (Le≫ (0+g) (plus_comm ???));
+apply (Le≫ (g) (zero_neutral ??) H);
 qed.
 
 lemma fle_plusr: ∀G:pogroup. ∀f,g,h:G. f≤g → f+h≤g+h.
-intros (G f g h H); apply (le_rewl ???? (plus_comm ???)); 
-apply (le_rewr ???? (plus_comm ???)); apply fle_plusl; assumption;
+intros (G f g h H); apply (Le≪? (plus_comm ???)); 
+apply (Le≫? (plus_comm ???)); apply fle_plusl; assumption;
 qed.
 
 lemma plus_cancl_le: 
   ∀G:pogroup.∀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) (opp_inverse ??));
-apply (le_rewl ??? (-z+(z+x)) (plus_assoc ????));
-apply (le_rewr ??? (0+y) (zero_neutral ??));
-apply (le_rewr ??? ((-z+z)+y) (opp_inverse ??));
-apply (le_rewr ??? (-z+(z+y)) (plus_assoc ????));
+apply (Le≪ (0+x) (zero_neutral ??));
+apply (Le≪ ((-z+z)+x) (opp_inverse ??));
+apply (Le≪ (-z+(z+x)) (plus_assoc ????));
+apply (Le≫ (0+y) (zero_neutral ??));
+apply (Le≫ ((-z+z)+y) (opp_inverse ??));
+apply (Le≫ (-z+(z+y)) (plus_assoc ????));
 apply (fle_plusl ??? (-z) L);
 qed.
 
@@ -138,55 +138,55 @@ qed.
 lemma exc_opp_x_zero_to_exc_zero_x: 
   ∀G:pogroup.∀x:G.-x ≰ 0 → 0 ≰ 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);
+apply (Ex≫? (plus_comm ???));
+apply (Ex≫? (opp_inverse ??));
+apply (Ex≪? (zero_neutral ??) H);
 qed.
   
 lemma le_zero_x_to_le_opp_x_zero: 
   ∀G:pogroup.∀x:G.0 ≤ x → -x ≤ 0.
 intros (G x Px); apply (plus_cancr_le ??? x);
-apply (le_rewl ??? 0 (opp_inverse ??));
-apply (le_rewr ??? x (zero_neutral ??) Px);
+apply (Le≪ 0 (opp_inverse ??));
+apply (Le≫ x (zero_neutral ??) Px);
 qed.
 
 lemma lt_zero_x_to_lt_opp_x_zero: 
   ∀G:pogroup.∀x:G.0 < x → -x < 0.
 intros (G x Px); apply (plus_cancr_lt ??? x);
-apply (lt_rewl ??? 0 (opp_inverse ??));
-apply (lt_rewr ??? x (zero_neutral ??) Px);
+apply (Lt≪ 0 (opp_inverse ??));
+apply (Lt≫ x (zero_neutral ??) Px);
 qed.
 
 lemma exc_zero_opp_x_to_exc_x_zero: 
   ∀G:pogroup.∀x:G. 0 ≰ -x → x ≰ 0.
 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);
+apply (Ex≫? (plus_comm ???));
+apply (Ex≪? (opp_inverse ??));
+apply (Ex≫? (zero_neutral ??) H);
 qed.
 
 lemma le_x_zero_to_le_zero_opp_x: 
   ∀G:pogroup.∀x:G. x ≤ 0 → 0 ≤ -x.
 intros (G x Lx0); apply (plus_cancr_le ??? x);
-apply (le_rewr ??? 0 (opp_inverse ??));
-apply (le_rewl ??? x (zero_neutral ??));
+apply (Le≫ 0 (opp_inverse ??));
+apply (Le≪ x (zero_neutral ??));
 assumption; 
 qed.
 
 lemma lt_x_zero_to_lt_zero_opp_x: 
   ∀G:pogroup.∀x:G. x < 0 → 0 < -x.
 intros (G x Lx0); apply (plus_cancr_lt ??? x);
-apply (lt_rewr ??? 0 (opp_inverse ??));
-apply (lt_rewl ??? x (zero_neutral ??));
+apply (Lt≫ 0 (opp_inverse ??));
+apply (Lt≪ 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 ??));
+apply (Lt≪ (-x) (zero_neutral ??));
+apply (Lt≫ (-x+x) (plus_comm ???));
+apply (Lt≫ 0 (opp_inverse ??));
 assumption; 
 qed.
 
@@ -194,7 +194,7 @@ lemma lt0plus_orlt:
   ∀G:pogroup. ∀x,y:G. 0 ≤ x → 0 ≤ y → 0 < x + y → 0 < x ∨ 0 < y.
 intros (G x y LEx LEy LT); cases LT (H1 H2); cases (ap_cotransitive ??? y H2);
 [right; split; assumption|left;split;[assumption]]
-apply (plus_cancr_ap ??? y); apply (ap_rewl ???? (zero_neutral ??));
+apply (plus_cancr_ap ??? y); apply (Ap≪? (zero_neutral ??));
 assumption;
 qed.
 
@@ -202,10 +202,10 @@ lemma le0plus_le:
   ∀G:pogroup.∀a,b,c:G. 0 ≤ b →  a + b ≤ c → a ≤ c.
 intros (G a b c L H); apply (le_transitive ????? H);
 apply (plus_cancl_le ??? (-a)); 
-apply (le_rewl ??? 0 (opp_inverse ??));
-apply (le_rewr ??? (-a + a + b) (plus_assoc ????));
-apply (le_rewr ??? (0 + b) (opp_inverse ??));
-apply (le_rewr ??? b (zero_neutral ??));
+apply (Le≪ 0 (opp_inverse ??));
+apply (Le≫ (-a + a + b) (plus_assoc ????));
+apply (Le≫ (0 + b) (opp_inverse ??));
+apply (Le≫ b (zero_neutral ??));
 assumption;
 qed.
 
@@ -213,10 +213,10 @@ lemma le_le0plus:
   ∀G:pogroup.∀a,b:G. 0 ≤ a → 0 ≤ b → 0 ≤ a + b.
 intros (G a b L1 L2); apply (le_transitive ???? L1);
 apply (plus_cancl_le ??? (-a));
-apply (le_rewl ??? 0 (opp_inverse ??));
-apply (le_rewr ??? (-a + a + b) (plus_assoc ????));
-apply (le_rewr ??? (0 + b) (opp_inverse ??));
-apply (le_rewr ??? b (zero_neutral ??));
+apply (Le≪ 0 (opp_inverse ??));
+apply (Le≫ (-a + a + b) (plus_assoc ????));
+apply (Le≫ (0 + b) (opp_inverse ??));
+apply (Le≫ b (zero_neutral ??));
 assumption;
 qed.
 
@@ -235,8 +235,8 @@ qed.
 
 lemma ltxy_ltyyxx: ∀G:pogroup.∀x,y:G. y < x → y+y < x+x.
 intros; apply (lt_transitive ?? (y+x));[2: 
-  apply (lt_rewl ???? (plus_comm ???));
-  apply (lt_rewr ???? (plus_comm ???));]
+  apply (Lt≪? (plus_comm ???));
+  apply (Lt≫? (plus_comm ???));]
 apply flt_plusl;assumption;
 qed.  
 
@@ -244,10 +244,10 @@ lemma lew_opp : ∀O:pogroup.∀a,b,c:O.0 ≤ b → a ≤ c → a + -b ≤ c.
 intros (O a b c L0 L);
 apply (le_transitive ????? L);
 apply (plus_cancl_le ??? (-a));
-apply (le_rewr ??? 0 (opp_inverse ??));
-apply (le_rewl ??? (-a+a+-b) (plus_assoc ????));
-apply (le_rewl ??? (0+-b) (opp_inverse ??));
-apply (le_rewl ??? (-b) (zero_neutral ?(-b)));
+apply (Le≫ 0 (opp_inverse ??));
+apply (Le≪ (-a+a+-b) (plus_assoc ????));
+apply (Le≪ (0+-b) (opp_inverse ??));
+apply (Le≪ (-b) (zero_neutral ?(-b)));
 apply le_zero_x_to_le_opp_x_zero;
 assumption;
 qed.
@@ -256,10 +256,10 @@ lemma ltw_opp : ∀O:pogroup.∀a,b,c:O.0 < b → a < c → a + -b < c.
 intros (O a b c P L);
 apply (lt_transitive ????? L);
 apply (plus_cancl_lt ??? (-a));
-apply (lt_rewr ??? 0 (opp_inverse ??));
-apply (lt_rewl ??? (-a+a+-b) (plus_assoc ????));
-apply (lt_rewl ??? (0+-b) (opp_inverse ??));
-apply (lt_rewl ??? ? (zero_neutral ??));
+apply (Lt≫ 0 (opp_inverse ??));
+apply (Lt≪ (-a+a+-b) (plus_assoc ????));
+apply (Lt≪ (0+-b) (opp_inverse ??));
+apply (Lt≪ ? (zero_neutral ??));
 apply lt_zero_x_to_lt_opp_x_zero;
 assumption;
 qed.
@@ -282,7 +282,7 @@ qed.
 
 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 ??));
+left; apply (plus_cancr_ap ??? y); apply (Ap≪y (zero_neutral ??));
 assumption;
 qed.
 
@@ -310,7 +310,7 @@ intros; intro; apply H; lapply (lt_to_excess ??? l);
 lapply (tog_total ??? e);
 lapply (tog_total ??? Hletin);
 lapply (ltplus ????? Hletin2 Hletin1);
-apply (exc_rewl ??? (0+0)); [apply eq_sym; apply zero_neutral]
+apply (Ex≪ (0+0)); [apply eq_sym; apply zero_neutral]
 apply lt_to_excess; assumption;
 qed.