many sed to use notation for rewriting

diff --git a/helm/software/matita/dama/depends b/helm/software/matita/dama/depends
index a047fc9d110a1b06e173fd892edf6959897bd3fb..1fe102822c2a21e267b97e7eb638dad6f7b17ffb 100644 (file)
--- a/helm/software/matita/dama/depends
+++ b/helm/software/matita/dama/depends
@@ -25,16 +25,10 @@ attic/integration_algebras.ma attic/vector_spaces.ma lattice.ma
 attic/vector_spaces.ma attic/reals.ma
 attic/rings.ma group.ma
 attic/ordered_fields_ch0.ma group.ma attic/fields.ma ordered_group.ma
-nat/nat.ma 
-logic/connectives.ma 
-higher_order_defs/relations.ma 
 Q/q.ma 
-nat/plus.ma 
 higher_order_defs/relations.ma 
 logic/connectives.ma 
-nat/orders.ma 
-nat/times.ma 
-nat/orders.ma 
+nat/nat.ma 
 nat/orders.ma 
 nat/plus.ma 
-nat/orders.ma 
+nat/times.ma 

diff --git a/helm/software/matita/dama/divisible_group.ma b/helm/software/matita/dama/divisible_group.ma
index 4f54545c8fbd74c6817037fdba164b6580f9615f..3a79b11bbd969a884307f4a6a86d08e63cbcfff8 100644 (file)
--- a/helm/software/matita/dama/divisible_group.ma
+++ b/helm/software/matita/dama/divisible_group.ma
@@ -50,8 +50,8 @@ qed.
 lemma div1: ∀G:dgroup.∀x:G.x/O ≈ x.
 intro G; cases G; unfold divide; intros; simplify;
 cases (f x O); simplify; simplify in H; intro; apply H;
-apply (ap_rewl ???? (plus_comm ???));
-apply (ap_rewl ???w (zero_neutral ??)); assumption;
+apply (Ap≪ ? (plus_comm ???));
+apply (Ap≪ w (zero_neutral ??)); assumption;
 qed.
 
 lemma apmul_ap: ∀G:dgroup.∀x,y:G.∀n.S n * x # S n * y → x # y.

diff --git a/helm/software/matita/dama/excess.ma b/helm/software/matita/dama/excess.ma
index 6db7e22e6b9363514e6738c265e4e93a30a6c52c..c7d31c2295da159bbbb720713ce260dacbf27d65 100644 (file)
--- a/helm/software/matita/dama/excess.ma
+++ b/helm/software/matita/dama/excess.ma
@@ -173,6 +173,18 @@ intros (A x z y Exy Azy); apply ap_symmetric; apply (ap_rewl ???? Exy);
 apply ap_symmetric; assumption;
 qed.
 
+notation > "'Ap'≪" non associative with precedence 50 for
+ @{'aprewritel}.
+ 
+interpretation "ap_rewl" 'aprewritel = 
+ (cic:/matita/excess/ap_rewl.con _ _ _).
+
+notation > "'Ap'≫" non associative with precedence 50 for
+ @{'aprewriter}.
+ 
+interpretation "ap_rewr" 'aprewriter = 
+ (cic:/matita/excess/ap_rewr.con _ _ _).
+
 lemma exc_rewl: ∀A:excess.∀x,z,y:A. x ≈ y → y ≰ z → x ≰ z.
 intros (A x z y Exy Ayz); elim (exc_cotransitive ???x Ayz); [2:assumption]
 cases Exy; right; assumption;
@@ -197,12 +209,12 @@ interpretation "exc_rewr" 'excessrewriter =
 
 lemma lt_rewr: ∀A:excess.∀x,z,y:A. x ≈ y → z < y → z < x.
 intros (A x y z E H); split; elim H; 
-[apply (le_rewr ???? (eq_sym ??? E));|apply (ap_rewr ???? E)] assumption;
+[apply (Le≫ ? (eq_sym ??? E));|apply (Ap≫ ? E)] assumption;
 qed.
 
 lemma lt_rewl: ∀A:excess.∀x,z,y:A. x ≈ y → y < z → x < z.
 intros (A x y z E H); split; elim H; 
-[apply (le_rewl ???? (eq_sym ??? E));| apply (ap_rewl ???? E);] assumption;
+[apply (Le≪ ? (eq_sym ??? E));| apply (Ap≪ ? E);] assumption;
 qed.
 
 notation > "'Lt'≪" non associative with precedence 50 for
@@ -246,4 +258,3 @@ qed.
 
 definition total_order_property : ∀E:excess. Type ≝
   λE:excess. ∀a,b:E. a ≰ b → b < a.
-

diff --git a/helm/software/matita/dama/group.ma b/helm/software/matita/dama/group.ma
index 9da386ef7bfefbb1797e86bcec43cddfb0c067c7..104dcf274e3943727d090dc8ff8fddd5f7bca59e 100644 (file)
--- a/helm/software/matita/dama/group.ma
+++ b/helm/software/matita/dama/group.ma
@@ -80,7 +80,7 @@ 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.
 
@@ -111,31 +111,31 @@ 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.
 
@@ -183,7 +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);

diff --git a/helm/software/matita/dama/ordered_divisible_group.ma b/helm/software/matita/dama/ordered_divisible_group.ma
index a9671d934b082c1bcc3300601108af6c9fce5aee..15dd52cdb853ba822b7d35e4a1ab959b0bf6e311 100644 (file)
--- a/helm/software/matita/dama/ordered_divisible_group.ma
+++ b/helm/software/matita/dama/ordered_divisible_group.ma
@@ -35,7 +35,7 @@ coercion cic:/matita/ordered_divisible_group/todg_division.con.
 lemma mul_ge: ∀G:todgroup.∀x:G.∀n.0 ≤ x → 0 ≤ n * x.
 intros (G x n); elim n; simplify; [apply le_reflexive]
 apply (le_transitive ???? H1); 
-apply (le_rewl ??? (0+(n1*x)) (zero_neutral ??));
+apply (Le≪ (0+(n1*x)) (zero_neutral ??));
 apply fle_plusr; assumption;
 qed. 
 
@@ -51,25 +51,25 @@ apply f; assumption;
 qed.
 
 lemma divide_preserves_lt: ∀G:todgroup.∀e:G.∀n.0<e → 0 < e/n.
-intros; elim n; [apply (lt_rewr ???? (div1 ??));assumption]
+intros; elim n; [apply (Lt≫ ? (div1 ??));assumption]
 unfold divide; elim (dg_prop G e (S n1)); simplify; simplify in f;
-apply (ltmul_lt ??? (S n1)); simplify; apply (lt_rewr ???? f);
-apply (lt_rewl ???? (zero_neutral ??)); 
-apply (lt_rewl ???? (zero_neutral ??)); 
-apply (lt_rewl ???? (mulzero ?n1));
+apply (ltmul_lt ??? (S n1)); simplify; apply (Lt≫ ? f);
+apply (Lt≪ ? (zero_neutral ??)); (* bug se faccio repeat *)
+apply (Lt≪ ? (zero_neutral ??));  
+apply (Lt≪ ? (mulzero ?n1));
 assumption;
 qed.
 
 lemma muleqplus_lt: ∀G:todgroup.∀x,y:G.∀n,m.
    0<x → 0<y → S n * x ≈ S (n + S m) * y → y < x.
-intros (G x y n m H1 H2 H3); apply (ltmul_lt ??? n); apply (lt_rewr ???? H3);
+intros (G x y n m H1 H2 H3); apply (ltmul_lt ??? n); apply (Lt≫ ? H3);
 clear H3; elim m; [
-  rewrite > sym_plus; simplify; apply (lt_rewl ??? (0+(y+n*y))); [
+  rewrite > sym_plus; simplify; apply (Lt≪ (0+(y+n*y))); [
     apply eq_sym; apply zero_neutral]
   apply flt_plusr; assumption;]
 apply (lt_transitive ???? l); rewrite > sym_plus; simplify;  
 rewrite > (sym_plus n); simplify; repeat apply flt_plusl;
-apply (lt_rewl ???(0+(n1+n)*y)); [apply eq_sym; apply zero_neutral]
+apply (Lt≪ (0+(n1+n)*y)); [apply eq_sym; apply zero_neutral]
 apply flt_plusr; assumption;
 qed.  
 

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.