From: Enrico Tassi <enrico.tassi@inria.fr>
Date: Mon, 3 Dec 2007 15:21:39 +0000 (+0000)
Subject: 1 lemma left!!!!
X-Git-Tag: make_still_working~5743
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=feaabb3c45906fafb4b6eb3fb10add6e6da6069b;p=helm.git

1 lemma left!!!!
---

diff --git a/helm/software/matita/dama/classical_pointfree/ordered_sets.ma b/helm/software/matita/dama/classical_pointfree/ordered_sets.ma
index 127bbf696..b42a8c44b 100644
--- a/helm/software/matita/dama/classical_pointfree/ordered_sets.ma
+++ b/helm/software/matita/dama/classical_pointfree/ordered_sets.ma
@@ -14,9 +14,9 @@
 
 set "baseuri" "cic:/matita/classical_pointwise/ordered_sets/".
 
-include "ordered_set.ma".
+include "excedence.ma".
 
-record is_dedekind_sigma_complete (O:ordered_set) : Type ≝
+record is_dedekind_sigma_complete (O:excedence) : Type ≝
  { dsc_inf: ∀a:nat→O.∀m:O. is_lower_bound ? a m → ex ? (λs:O.is_inf O a s);
    dsc_inf_proof_irrelevant:
     ∀a:nat→O.∀m,m':O.∀p:is_lower_bound ? a m.∀p':is_lower_bound ? a m'.
@@ -30,7 +30,7 @@ record is_dedekind_sigma_complete (O:ordered_set) : Type ≝
  }.
 
 record dedekind_sigma_complete_ordered_set : Type ≝
- { dscos_ordered_set:> ordered_set;
+ { dscos_ordered_set:> excedence;
    dscos_dedekind_sigma_complete_properties:>
     is_dedekind_sigma_complete dscos_ordered_set
  }.
@@ -155,7 +155,7 @@ lemma inf_respects_le:
 qed. 
 
 definition is_sequentially_monotone ≝
- λO:ordered_set.λf:O→O.
+ λO:excedence.λf:O→O.
   ∀a:nat→O.∀p:is_increasing ? a.
    is_increasing ? (λi.f (a i)).
 
diff --git a/helm/software/matita/dama/divisible_group.ma b/helm/software/matita/dama/divisible_group.ma
new file mode 100644
index 000000000..5c7025a9c
--- /dev/null
+++ b/helm/software/matita/dama/divisible_group.ma
@@ -0,0 +1,55 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/divisible_group/".
+
+include "nat/orders.ma".
+include "group.ma".
+
+let rec pow (G : abelian_group) (x : G) (n : nat) on n : G ≝
+  match n with [ O ⇒ 0 | S m ⇒ x + pow ? x m].
+  
+interpretation "abelian group pow" 'exp x n =
+  (cic:/matita/divisible_group/pow.con _ x n).
+  
+record dgroup : Type ≝ {
+  dg_carr:> abelian_group;
+  dg_prop: ∀x:dg_carr.∀n:nat.∃y.pow ? y (S n) ≈ x
+}.
+
+lemma divide: ∀G:dgroup.G → nat → G.
+intros (G x n); cases (dg_prop G x n); apply w; 
+qed.
+
+interpretation "divisible group divide" 'divide x n =
+  (cic:/matita/divisible_group/divide.con _ x n).
+
+lemma divide_divides: 
+  ∀G:dgroup.∀x:G.∀n. pow ? (x / n) (S n) ≈ x.
+intro G; cases G; unfold divide; intros (x n); simplify;
+cases (f x n); simplify; exact H;
+qed.  
+  
+lemma feq_pow: ∀G:dgroup.∀x,y:G.∀n.x≈y → pow ? x n ≈ pow ? y n.
+intros (G x y n H); elim n; [apply eq_reflexive]
+simplify; apply (Eq≈ (x + (pow ? y n1)) H1);
+apply (Eq≈ (y+pow ? y n1) H (eq_reflexive ??));
+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;
+qed.
diff --git a/helm/software/matita/dama/excedence.ma b/helm/software/matita/dama/excedence.ma
index ab7a53af0..1bddcb16e 100644
--- a/helm/software/matita/dama/excedence.ma
+++ b/helm/software/matita/dama/excedence.ma
@@ -86,7 +86,7 @@ intros (E); unfold; intros (x y Exy); unfold; unfold; intros (Ayx); apply Exy;
 apply ap_symmetric; assumption; 
 qed.
 
-lemma eq_sym:∀E:apartness.∀x,y:E.x ≈ y → y ≈ x := eq_sym_.
+lemma eq_sym:∀E:apartness.∀x,y:E.x ≈ y → y ≈ x ≝ eq_sym_.
 
 coercion cic:/matita/excedence/eq_sym.con.
 
@@ -206,12 +206,20 @@ whd in LE LEx APx; cases APx (EXx EXx); [cases (LEx EXx)]
 cases (exc_cotransitive ??? x EXx) (EXz EXz); [right; assumption]
 cases LE; assumption;
 qed.
-
     
 lemma le_le_eq: ∀E:excedence.∀a,b:E. a ≤ b → b ≤ a → a ≈ b.
 intros (E x y L1 L2); intro H; cases H; [apply L1|apply L2] assumption;
 qed.
 
+lemma eq_le_le: ∀E:excedence.∀a,b:E. a ≈ b → a ≤ b ∧ b ≤ a.
+intros (E x y H); unfold apart_of_excedence in H; unfold apart in H;
+simplify in H; split; intro; apply H; [left|right] assumption.
+qed.
+
 lemma ap_le_to_lt: ∀E:excedence.∀a,c:E.c # a → c ≤ a → c < a.
 intros; split; assumption;
-qed.
\ No newline at end of file
+qed.
+
+definition total_order_property : ∀E:excedence. Type ≝
+  λE:excedence. ∀a,b:E. a ≰ b → b < a.
+
diff --git a/helm/software/matita/dama/metric_lattice.ma b/helm/software/matita/dama/metric_lattice.ma
index 81eab3137..ae55b5551 100644
--- a/helm/software/matita/dama/metric_lattice.ma
+++ b/helm/software/matita/dama/metric_lattice.ma
@@ -17,7 +17,7 @@ set "baseuri" "cic:/matita/metric_lattice/".
 include "metric_space.ma".
 include "lattice.ma".
 
-record mlattice_ (R : ogroup) : Type ≝ {
+record mlattice_ (R : todgroup) : Type ≝ {
   ml_mspace_: metric_space R;
   ml_lattice:> lattice;
   ml_with_: ms_carr ? ml_mspace_ = ap_carr (l_carr ml_lattice)
@@ -33,12 +33,12 @@ qed.
 
 coercion cic:/matita/metric_lattice/ml_mspace.con.
 
-record is_mlattice (R : ogroup) (ml: mlattice_ R) : Type ≝ {
+record is_mlattice (R : todgroup) (ml: mlattice_ R) : Type ≝ {
   ml_prop1: ∀a,b:ml. 0 < δ a b → a # b;
   ml_prop2: ∀a,b,c:ml. δ (a∨b) (a∨c) + δ (a∧b) (a∧c) ≤ δ b c
 }.
 
-record mlattice (R : ogroup) : Type ≝ {
+record mlattice (R : todgroup) : Type ≝ {
   ml_carr :> mlattice_ R;
   ml_props:> is_mlattice R ml_carr
 }.
@@ -103,40 +103,6 @@ qed.
 include "sequence.ma".
 include "nat/plus.ma".
 
-definition d2s ≝ 
-  λR.λml:mlattice R.λs:sequence (pordered_set_of_excedence ml).λk.λn. δ (s n) k.
-(*
-notation "s ⇝ 'Zero'" non associative with precedence 50 for @{'tends0 $s }.
-  
-interpretation "tends to" 'tends s x = 
-  (cic:/matita/sequence/tends0.con _ s).    
-*)
-
-lemma lew_opp : ∀O:ogroup.∀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_zero_x_to_le_opp_x_zero;
-assumption;
-qed.
-
-
-lemma ltw_opp : ∀O:ogroup.∀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_zero_x_to_lt_opp_x_zero;
-assumption;
-qed.
-
 lemma ltwl: ∀a,b,c:nat. b + a < c → a < c.
 intros 3 (x y z); elim y (H z IH H); [apply H]
 apply IH; apply lt_S_to_lt; apply H;
@@ -146,17 +112,31 @@ lemma ltwr: ∀a,b,c:nat. a + b < c → a < c.
 intros 3 (x y z); rewrite > sym_plus; apply ltwl;
 qed. 
 
+
+definition d2s ≝ 
+  λR.λml:mlattice R.λs:sequence ml.λk.λn. δ (s n) k.
+(*
+notation "s ⇝ 'Zero'" non associative with precedence 50 for @{'tends0 $s }.
+  
+interpretation "tends to" 'tends s x = 
+  (cic:/matita/sequence/tends0.con _ s).    
+*)
+
+axiom core1: ∀G:todgroup.∀e:G.0<e → e/3 + e/2 + e/2 < e.
+
 alias symbol "leq" = "ordered sets less or equal than".
 alias symbol "and" = "constructive and".
 alias symbol "exists" = "constructive exists (Type)".
-theorem carabinieri: (* non trova la coercion .... *)
-  ∀R.∀ml:mlattice R.∀an,bn,xn:sequence (pordered_set_of_excedence ml).
+theorem carabinieri:
+  ∀R.∀ml:mlattice R.∀an,bn,xn:sequence ml.
     (∀n. (an n ≤ xn n) ∧ (xn n ≤ bn n)) →
     ∀x:ml. tends0 ? (d2s ? ml an x) → tends0 ? (d2s ? ml bn x) →
     tends0 ? (d2s ? ml xn x).
 intros (R ml an bn xn H x Ha Hb); unfold tends0 in Ha Hb ⊢ %. unfold d2s in Ha Hb ⊢ %.
 intros (e He);
-elim (Ha ? He) (n1 H1); clear Ha; elim (Hb e He) (n2 H2); clear Hb;
+alias num (instance 0) = "natural number".
+elim (Ha (e/2) (divide_preserves_lt ??? He)) (n1 H1); clear Ha; 
+elim (Hb (e/3) (divide_preserves_lt ??? He)) (n2 H2); clear Hb;
 constructor 1; [apply (n1 + n2);] intros (n3 Hn3);
 lapply (ltwr ??? Hn3) as Hn1n3; lapply (ltwl ??? Hn3) as Hn2n3;
 elim (H1 ? Hn1n3) (H3 H4); elim (H2 ? Hn2n3) (H5 H6); clear Hn1n3 Hn2n3;
@@ -195,6 +175,16 @@ split; [
   right;  apply exc_opp_x_zero_to_exc_zero_x;
   apply (Ex≪ ? (eq_opp_opp_x_x ??)); assumption;]
 clear Dxm Dym H7 H8 Hn3 H5 H3 n1 n2;
+apply (le_lt_transitive ???? ? (core1 ?? He));
+apply (le_transitive ???? Hcut);
+apply (le_transitive ??  (e/3+ δ(an n3) x+ δ(an n3) x)); [
+  repeat apply fle_plusr; cases H6; assumption;]
+apply (le_transitive ??  (e/3+ e/2 + δ(an n3) x)); [
+  apply fle_plusr; apply fle_plusl; cases H4; assumption;]
+apply (le_transitive ??  (e/3+ e/2 + e/2)); [
+  apply fle_plusl; cases H4; assumption;]
+apply le_reflexive;
+qed.
 
   
 (* 3.17 conclusione: δ x y ≈ 0 *)
diff --git a/helm/software/matita/dama/metric_space.ma b/helm/software/matita/dama/metric_space.ma
index 6774662a2..e2112d65b 100644
--- a/helm/software/matita/dama/metric_space.ma
+++ b/helm/software/matita/dama/metric_space.ma
@@ -14,9 +14,9 @@
 
 set "baseuri" "cic:/matita/metric_space/".
 
-include "ordered_group.ma".
+include "ordered_divisible_group.ma".
 
-record metric_space (R : ogroup) : Type ≝ {
+record metric_space (R : todgroup) : Type ≝ {
   ms_carr :> Type;
   metric: ms_carr → ms_carr → R;
   mpositive: ∀a,b:ms_carr. 0 ≤ metric a b; 
@@ -34,7 +34,7 @@ interpretation "metric" 'delta = (cic:/matita/metric_space/metric.con _ _).
 definition apart_metric:=
   λR.λms:metric_space R.λa,b:ms.0 < δ a b.
 
-lemma apart_of_metric_space: ∀R:ogroup.metric_space R → apartness.
+lemma apart_of_metric_space: ∀R:todgroup.metric_space R → apartness.
 intros (R ms); apply (mk_apartness ? (apart_metric ? ms)); unfold apart_metric; unfold;
 [1: intros 2 (x H); cases H (H1 H2); 
     lapply (ap_rewr ???? (eq_sym ??? (mreflexive ??x)) H2);
diff --git a/helm/software/matita/dama/ordered_divisible_group.ma b/helm/software/matita/dama/ordered_divisible_group.ma
new file mode 100644
index 000000000..34bd25a23
--- /dev/null
+++ b/helm/software/matita/dama/ordered_divisible_group.ma
@@ -0,0 +1,70 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/ordered_divisible_group/".
+
+include "nat/orders.ma".
+include "nat/times.ma".
+include "ordered_group.ma".
+include "divisible_group.ma".
+
+record todgroup : Type ≝ {
+  todg_order:> togroup;
+  todg_division_: dgroup;
+  todg_with_: dg_carr todg_division_ = og_abelian_group todg_order
+}.
+
+lemma todg_division: todgroup → dgroup.
+intro G; apply (mk_dgroup G); unfold abelian_group_OF_todgroup; 
+cases (todg_with_ G); exact (dg_prop (todg_division_ G));
+qed.
+
+coercion cic:/matita/ordered_divisible_group/todg_division.con.
+
+lemma pow_lt: ∀G:todgroup.∀x:G.∀n.0 < x → 0 < x + pow ? x n.
+intros (G x n H); elim n; [
+  simplify; apply (lt_rewr ???? (plus_comm ???)); 
+  apply (lt_rewr ???x (zero_neutral ??)); assumption]
+simplify; apply (lt_transitive ?? (x+(x)\sup(n1))); [assumption]
+apply flt_plusl; apply (lt_rewr ???? (plus_comm ???));
+apply (lt_rewl ??? (0 + (x \sup n1)) (eq_sym ??? (zero_neutral ??)));
+apply (lt_rewl ???? (plus_comm ???));
+apply flt_plusl; assumption;
+qed.
+
+lemma pow_ge: ∀G:todgroup.∀x:G.∀n.0 ≤ x → 0 ≤ pow ? x n.
+intros (G x n); elim n; simplify; [apply le_reflexive]
+apply (le_transitive ???? H1); 
+apply (le_rewl ??? (0+(x\sup n1)) (zero_neutral ??));
+apply fle_plusr; assumption;
+qed. 
+
+lemma ge_pow: ∀G:todgroup.∀x:G.∀n.0 < pow ? x n → 0 < x.
+intros 3; elim n; [
+  simplify in l; cases (lt_coreflexive ?? l);]
+simplify in l; 
+cut (0+0<x+(x)\sup(n1));[2:
+  apply (lt_rewl ??? 0 (zero_neutral ??)); assumption].
+cases (pippo4 ????? Hcut); [assumption]
+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]
+unfold divide; elim (dg_prop G e (S n1)); simplify; simplify in f;
+apply (ge_pow ?? (S (S n1))); apply (lt_rewr ???? f); assumption;
+qed.
+
+alias num (instance 0) = "natural number".
+axiom core1: ∀G:todgroup.∀e:G.0<e → e/3 + e/2 + e/2 < e.
diff --git a/helm/software/matita/dama/ordered_group.ma b/helm/software/matita/dama/ordered_group.ma
index 7f66d3b80..22c18cfa1 100644
--- a/helm/software/matita/dama/ordered_group.ma
+++ b/helm/software/matita/dama/ordered_group.ma
@@ -12,34 +12,33 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/ordered_gorup/".
+set "baseuri" "cic:/matita/ordered_group/".
 
-include "ordered_set.ma".
 include "group.ma".
 
-record pre_ogroup : Type ≝ { 
+record pogroup_ : Type ≝ { 
   og_abelian_group_: abelian_group;
-  og_tordered_set:> tordered_set;
-  og_with: carr og_abelian_group_ = og_tordered_set
+  og_excedence:> excedence;
+  og_with: carr og_abelian_group_ = apart_of_excedence og_excedence
 }.
 
-lemma og_abelian_group: pre_ogroup → abelian_group.
+lemma og_abelian_group: pogroup_ → 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;
+unfold apartness_OF_pogroup_; 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_gorup/og_abelian_group.con.
+coercion cic:/matita/ordered_group/og_abelian_group.con.
 
-record ogroup : Type ≝ { 
-  og_carr:> pre_ogroup;
-  exc_canc_plusr: ∀f,g,h:og_carr. f+h ≰ g+h → f ≰ g
+record pogroup : Type ≝ { 
+  og_carr:> pogroup_;
+  canc_plusr_exc: ∀f,g,h:og_carr. f+h ≰ g+h → f ≰ g
 }.
 
 lemma fexc_plusr: 
-  ∀G:ogroup.∀x,y,z:G. x ≰ y → x+z ≰ y + z.
-intros 5 (G x y z L); apply (exc_canc_plusr ??? (-z));
+  ∀G:pogroup.∀x,y,z:G. x ≰ y → x+z ≰ y + z.
+intros 5 (G x y z L); apply (canc_plusr_exc ??? (-z));
 apply (Ex≪  (x + (z + -z)) (plus_assoc ????));
 apply (Ex≪  (x + (-z + z)) (plus_comm ??z));
 apply (Ex≪  (x+0) (opp_inverse ??));
@@ -52,17 +51,17 @@ apply (Ex≫  (0+y) (plus_comm ???));
 apply (Ex≫  y (zero_neutral ??) L);
 qed.
 
-coercion cic:/matita/ordered_gorup/fexc_plusr.con nocomposites.
+coercion cic:/matita/ordered_group/fexc_plusr.con nocomposites.
 
-lemma exc_canc_plusl: ∀G:ogroup.∀f,g,h:G. h+f ≰ h+g → f ≰ g.
-intros 5 (G x y z L); apply (exc_canc_plusr ??? z);
+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);
 apply (exc_rewl ??? (z+x) (plus_comm ???));
 apply (exc_rewr ??? (z+y) (plus_comm ???) L);
 qed.
 
 lemma fexc_plusl: 
-  ∀G:ogroup.∀x,y,z:G. x ≰ y → z+x ≰ z+y.
-intros 5 (G x y z L); apply (exc_canc_plusl ??? (-z));
+  ∀G:pogroup.∀x,y,z:G. x ≰ y → z+x ≰ z+y.
+intros 5 (G x y z L); apply (canc_plusl_exc ??? (-z));
 apply (exc_rewl ???? (plus_assoc ??z x));
 apply (exc_rewr ???? (plus_assoc ??z y));
 apply (exc_rewl ??? (0+x) (opp_inverse ??));
@@ -71,10 +70,10 @@ apply (exc_rewl ???? (zero_neutral ??));
 apply (exc_rewr ???? (zero_neutral ??) L);
 qed.
 
-coercion cic:/matita/ordered_gorup/fexc_plusl.con nocomposites.
+coercion cic:/matita/ordered_group/fexc_plusl.con nocomposites.
 
 lemma plus_cancr_le: 
-  ∀G:ogroup.∀x,y,z:G.x+z ≤ y + z → x ≤ y.
+  ∀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 ???));
@@ -86,10 +85,10 @@ 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 (exc_canc_plusr ??? (-z) H);
+intro H; apply L; clear L; apply (canc_plusr_exc ??? (-z) H);
 qed.
 
-lemma fle_plusl: ∀G:ogroup. ∀f,g,h:G. f≤g → h+f≤h+g.
+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));
@@ -106,13 +105,13 @@ apply (le_rewr ??? (0+g) (plus_comm ???));
 apply (le_rewr ??? (g) (zero_neutral ??) H);
 qed.
 
-lemma fle_plusr: ∀G:ogroup. ∀f,g,h:G. f≤g → f+h≤g+h.
+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;
 qed.
 
 lemma plus_cancl_le: 
-  ∀G:ogroup.∀x,y,z:G.z+x ≤ z+y → x ≤ y.
+  ∀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 ??));
@@ -124,50 +123,50 @@ apply (fle_plusl ??? (-z) L);
 qed.
 
 lemma plus_cancl_lt: 
-  ∀G:ogroup.∀x,y,z:G.z+x < z+y → x < y.
+  ∀G:pogroup.∀x,y,z:G.z+x < z+y → x < y.
 intros 5 (G x y z L); elim L (A LE); split; [apply plus_cancl_le; assumption]
 apply (plus_cancl_ap ???? LE);
 qed.
 
 lemma plus_cancr_lt: 
-  ∀G:ogroup.∀x,y,z:G.x+z < y+z → x < y.
+  ∀G:pogroup.∀x,y,z:G.x+z < y+z → x < y.
 intros 5 (G x y z L); elim L (A LE); split; [apply plus_cancr_le; assumption]
 apply (plus_cancr_ap ???? LE);
 qed.
 
 
 lemma exc_opp_x_zero_to_exc_zero_x: 
-  ∀G:ogroup.∀x:G.-x ≰ 0 → 0 ≰ x.
-intros (G x H); apply (exc_canc_plusr ??? (-x));
+  ∀G:pogroup.∀x:G.-x ≰ 0 → 0 ≰ x.
+intros (G x H); apply (canc_plusr_exc ??? (-x));
 apply (exc_rewr ???? (plus_comm ???));
 apply (exc_rewr ???? (opp_inverse ??));
 apply (exc_rewl ???? (zero_neutral ??) H);
 qed.
   
 lemma le_zero_x_to_le_opp_x_zero: 
-  ∀G:ogroup.∀x:G.0 ≤ x → -x ≤ 0.
+  ∀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);
 qed.
 
 lemma lt_zero_x_to_lt_opp_x_zero: 
-  ∀G:ogroup.∀x:G.0 < x → -x < 0.
+  ∀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);
 qed.
 
 lemma exc_zero_opp_x_to_exc_x_zero: 
-  ∀G:ogroup.∀x:G. 0 ≰ -x → x ≰ 0.
-intros (G x H); apply (exc_canc_plusl ??? (-x));
+  ∀G:pogroup.∀x:G. 0 ≰ -x → x ≰ 0.
+intros (G x H); apply (canc_plusl_exc ??? (-x));
 apply (exc_rewr ???? (plus_comm ???));
 apply (exc_rewl ???? (opp_inverse ??));
 apply (exc_rewr ???? (zero_neutral ??) H);
 qed.
 
 lemma le_x_zero_to_le_zero_opp_x: 
-  ∀G:ogroup.∀x:G. x ≤ 0 → 0 ≤ -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 ??));
@@ -175,7 +174,7 @@ assumption;
 qed.
 
 lemma lt_x_zero_to_lt_zero_opp_x: 
-  ∀G:ogroup.∀x:G. x < 0 → 0 < -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 ??));
@@ -184,7 +183,7 @@ qed.
 
 
 lemma lt0plus_orlt: 
-  ∀G:ogroup. ∀x,y:G. 0 ≤ x → 0 ≤ y → 0 < x + y → 0 < x ∨ 0 < y.
+  ∀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 ??));
@@ -192,7 +191,7 @@ assumption;
 qed.
 
 lemma le0plus_le: 
-  ∀G:ogroup.∀a,b,c:G. 0 ≤ b →  a + b ≤ c → a ≤ c.
+  ∀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 ??));
@@ -203,7 +202,7 @@ assumption;
 qed.
 
 lemma le_le0plus: 
-  ∀G:ogroup.∀a,b:G. 0 ≤ a → 0 ≤ b → 0 ≤ a + b.
+  ∀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 ??));
@@ -213,5 +212,101 @@ apply (le_rewr ??? b (zero_neutral ??));
 assumption;
 qed.
 
-  
-  
\ No newline at end of file
+lemma flt_plusl: 
+  ∀G:pogroup.∀x,y,z:G.x < y → z + x < z + y.
+intros (G x y z H); cases H; split; [apply fle_plusl; assumption]
+apply fap_plusl; assumption;
+qed.
+
+lemma flt_plusr: 
+  ∀G:pogroup.∀x,y,z:G.x < y → x + z < y + z.
+intros (G x y z H); cases H; split; [apply fle_plusr; assumption]
+apply fap_plusr; assumption;
+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 flt_plusl;assumption;
+qed.  
+
+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_zero_x_to_le_opp_x_zero;
+assumption;
+qed.
+
+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_zero_x_to_lt_opp_x_zero;
+assumption;
+qed.
+
+record togroup : Type ≝ {
+  tog_carr:> pogroup;
+  tog_total: ∀x,y:tog_carr.x≰y → y < x
+}.
+
+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;
+cases (H H4);
+qed. 
+
+lemma eqxxyy_eqxy: ∀G:togroup.∀x,y:G. x + x ≈ y + y → x ≈ y.
+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.
+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.
+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.
+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;
+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);
+[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);
+lapply (tog_total ??? e);
+lapply (tog_total ??? Hletin);
+lapply (pippo ????? Hletin2 Hletin1);
+apply (exc_rewl ??? (0+0)); [apply eq_sym; apply zero_neutral]
+apply lt_to_excede; 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;
+qed.
diff --git a/helm/software/matita/dama/ordered_set.ma b/helm/software/matita/dama/ordered_set.ma
deleted file mode 100644
index 7178bd9a7..000000000
--- a/helm/software/matita/dama/ordered_set.ma
+++ /dev/null
@@ -1,47 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-set "baseuri" "cic:/matita/ordered_set/".
-
-include "excedence.ma".
-
-record is_porder_relation (C:Type) (le:C→C→Prop) (eq:C→C→Prop) : Type ≝ { 
-  por_reflexive: reflexive ? le;
-  por_transitive: transitive ? le;
-  por_antisimmetric: antisymmetric ? le eq
-}.
-
-record pordered_set: Type ≝ { 
-  pos_carr:> excedence;
-  pos_order_relation_properties:> is_porder_relation ? (le pos_carr) (eq pos_carr)
-}.
-
-lemma pordered_set_of_excedence: excedence → pordered_set.
-intros (E); apply (mk_pordered_set E); apply (mk_is_porder_relation);
-[apply le_reflexive|apply le_transitive|apply le_antisymmetric]
-qed. 
-
-coercion cic:/matita/ordered_set/pordered_set_of_excedence.con.
-
-alias id "transitive" = "cic:/matita/higher_order_defs/relations/transitive.con".
-alias id "cotransitive" = "cic:/matita/higher_order_defs/relations/cotransitive.con".
-alias id "antisymmetric" = "cic:/matita/higher_order_defs/relations/antisymmetric.con".
-
-definition total_order_property : ∀E:excedence. Type ≝
-  λE:excedence. ∀a,b:E. a ≰ b → b < a.
-
-record tordered_set : Type ≝ {
- tos_poset:> pordered_set;
- tos_totality: total_order_property tos_poset
-}.
diff --git a/helm/software/matita/dama/sequence.ma b/helm/software/matita/dama/sequence.ma
index 40e2f477a..c26ae5721 100644
--- a/helm/software/matita/dama/sequence.ma
+++ b/helm/software/matita/dama/sequence.ma
@@ -14,46 +14,46 @@
 
 set "baseuri" "cic:/matita/sequence/".
 
-include "ordered_set.ma".
+include "excedence.ma".
 
-definition sequence := λO:pordered_set.nat → O.
+definition sequence := λO:excedence.nat → O.
 
-definition fun_of_sequence: ∀O:pordered_set.sequence O → nat → O.
+definition fun_of_sequence: ∀O:excedence.sequence O → nat → O.
 intros; apply s; assumption;
 qed.
 
 coercion cic:/matita/sequence/fun_of_sequence.con 1.
 
 definition upper_bound ≝ 
-  λO:pordered_set.λa:sequence O.λu:O.∀n:nat.a n ≤ u.
+  λO:excedence.λa:sequence O.λu:O.∀n:nat.a n ≤ u.
   
 definition lower_bound ≝ 
-  λO:pordered_set.λa:sequence O.λu:O.∀n:nat.u ≤ a n.
+  λO:excedence.λa:sequence O.λu:O.∀n:nat.u ≤ a n.
 
 definition strong_sup ≝
-  λO:pordered_set.λs:sequence O.λx.
+  λO:excedence.λs:sequence O.λx.
     upper_bound ? s x ∧ (∀y:O.x ≰ y → ∃n.s n ≰ y).
   
 definition strong_inf ≝
-  λO:pordered_set.λs:sequence O.λx.
+  λO:excedence.λs:sequence O.λx.
     lower_bound ? s x ∧ (∀y:O.y ≰ x → ∃n.y ≰ s n).
 
 definition weak_sup ≝
-  λO:pordered_set.λs:sequence O.λx.
+  λO:excedence.λs:sequence O.λx.
     upper_bound ? s x ∧ (∀y:O.upper_bound ? s y → x ≤ y).
   
 definition weak_inf ≝
-  λO:pordered_set.λs:sequence O.λx.
+  λO:excedence.λs:sequence O.λx.
     lower_bound ? s x ∧ (∀y:O.lower_bound ? s y → y ≤ x).
 
 lemma strong_sup_is_weak: 
-  ∀O:pordered_set.∀s:sequence O.∀x:O.strong_sup ? s x → weak_sup ? s x.
+  ∀O:excedence.∀s:sequence O.∀x:O.strong_sup ? s x → weak_sup ? s x.
 intros (O s x Ssup); elim Ssup (Ubx M); clear Ssup; split; [assumption]
 intros 3 (y Uby E); cases (M ? E) (n En); unfold in Uby; cases (Uby ? En);
 qed.
  
 lemma strong_inf_is_weak: 
-  ∀O:pordered_set.∀s:sequence O.∀x:O.strong_inf ? s x → weak_inf ? s x.
+  ∀O:excedence.∀s:sequence O.∀x:O.strong_inf ? s x → weak_inf ? s x.
 intros (O s x Ssup); elim Ssup (Ubx M); clear Ssup; split; [assumption]
 intros 3 (y Uby E); cases (M ? E) (n En); unfold in Uby; cases (Uby ? En);
 qed.
@@ -62,92 +62,92 @@ include "ordered_group.ma".
 include "nat/orders.ma".
 
 definition tends0 ≝ 
-  λO:ogroup.λs:sequence O.
+  λO:pogroup.λs:sequence O.
     ∀e:O.0 < e → ∃N.∀n.N < n → -e < s n ∧ s n < e.
     
 definition increasing ≝ 
-  λO:pordered_set.λa:sequence O.∀n:nat.a n ≤ a (S n).
+  λO:excedence.λa:sequence O.∀n:nat.a n ≤ a (S n).
 
 definition decreasing ≝ 
-  λO:pordered_set.λa:sequence O.∀n:nat.a (S n) ≤ a n.
+  λO:excedence.λa:sequence O.∀n:nat.a (S n) ≤ a n.
 
 
 
 
 (*
 
-definition is_upper_bound ≝ λO:pordered_set.λa:sequence O.λu:O.∀n:nat.a n ≤ u.
-definition is_lower_bound ≝ λO:pordered_set.λa:sequence O.λu:O.∀n:nat.u ≤ a n.
+definition is_upper_bound ≝ λO:excedence.λa:sequence O.λu:O.∀n:nat.a n ≤ u.
+definition is_lower_bound ≝ λO:excedence.λa:sequence O.λu:O.∀n:nat.u ≤ a n.
 
-record is_sup (O:pordered_set) (a:sequence O) (u:O) : Prop ≝
+record is_sup (O:excedence) (a:sequence O) (u:O) : Prop ≝
  { sup_upper_bound: is_upper_bound O a u; 
    sup_least_upper_bound: ∀v:O. is_upper_bound O a v → u≤v
  }.
 
-record is_inf (O:pordered_set) (a:sequence O) (u:O) : Prop ≝
+record is_inf (O:excedence) (a:sequence O) (u:O) : Prop ≝
  { inf_lower_bound: is_lower_bound O a u; 
    inf_greatest_lower_bound: ∀v:O. is_lower_bound O a v → v≤u
  }.
 
-record is_bounded_below (O:pordered_set) (a:sequence O) : Type ≝
+record is_bounded_below (O:excedence) (a:sequence O) : Type ≝
  { ib_lower_bound: O;
    ib_lower_bound_is_lower_bound: is_lower_bound ? a ib_lower_bound
  }.
 
-record is_bounded_above (O:pordered_set) (a:sequence O) : Type ≝
+record is_bounded_above (O:excedence) (a:sequence O) : Type ≝
  { ib_upper_bound: O;
    ib_upper_bound_is_upper_bound: is_upper_bound ? a ib_upper_bound
  }.
 
-record is_bounded (O:pordered_set) (a:sequence O) : Type ≝
+record is_bounded (O:excedence) (a:sequence O) : Type ≝
  { ib_bounded_below:> is_bounded_below ? a;
    ib_bounded_above:> is_bounded_above ? a
  }.
 
-record bounded_below_sequence (O:pordered_set) : Type ≝
+record bounded_below_sequence (O:excedence) : Type ≝
  { bbs_seq:> sequence O;
    bbs_is_bounded_below:> is_bounded_below ? bbs_seq
  }.
 
-record bounded_above_sequence (O:pordered_set) : Type ≝
+record bounded_above_sequence (O:excedence) : Type ≝
  { bas_seq:> sequence O;
    bas_is_bounded_above:> is_bounded_above ? bas_seq
  }.
 
-record bounded_sequence (O:pordered_set) : Type ≝
+record bounded_sequence (O:excedence) : Type ≝
  { bs_seq:> sequence O;
    bs_is_bounded_below: is_bounded_below ? bs_seq;
    bs_is_bounded_above: is_bounded_above ? bs_seq
  }.
 
 definition bounded_below_sequence_of_bounded_sequence ≝
- λO:pordered_set.λb:bounded_sequence O.
+ λO:excedence.λb:bounded_sequence O.
   mk_bounded_below_sequence ? b (bs_is_bounded_below ? b).
 
 coercion cic:/matita/sequence/bounded_below_sequence_of_bounded_sequence.con.
 
 definition bounded_above_sequence_of_bounded_sequence ≝
- λO:pordered_set.λb:bounded_sequence O.
+ λO:excedence.λb:bounded_sequence O.
   mk_bounded_above_sequence ? b (bs_is_bounded_above ? b).
 
 coercion cic:/matita/sequence/bounded_above_sequence_of_bounded_sequence.con.
 
 definition lower_bound ≝
- λO:pordered_set.λb:bounded_below_sequence O.
+ λO:excedence.λb:bounded_below_sequence O.
   ib_lower_bound ? b (bbs_is_bounded_below ? b).
 
 lemma lower_bound_is_lower_bound:
- ∀O:pordered_set.∀b:bounded_below_sequence O.
+ ∀O:excedence.∀b:bounded_below_sequence O.
   is_lower_bound ? b (lower_bound ? b).
 intros; unfold lower_bound; apply ib_lower_bound_is_lower_bound.
 qed.
 
 definition upper_bound ≝
- λO:pordered_set.λb:bounded_above_sequence O.
+ λO:excedence.λb:bounded_above_sequence O.
   ib_upper_bound ? b (bas_is_bounded_above ? b).
 
 lemma upper_bound_is_upper_bound:
- ∀O:pordered_set.∀b:bounded_above_sequence O.
+ ∀O:excedence.∀b:bounded_above_sequence O.
   is_upper_bound ? b (upper_bound ? b).
 intros; unfold upper_bound; apply ib_upper_bound_is_upper_bound.
 qed.
@@ -159,76 +159,76 @@ cases E (T f cRf cTf); simplify;
 |2: intros (x y z); apply Or_symmetric; apply cTf; assumption;]
 qed. 
 
-definition reverse_pordered_set: pordered_set → pordered_set.
-intros (p); apply (mk_pordered_set (reverse_excedence p));
+definition reverse_excedence: excedence → excedence.
+intros (p); apply (mk_excedence (reverse_excedence p));
 generalize in match (reverse_excedence p); intros (E);
 apply mk_is_porder_relation;
 [apply le_reflexive|apply le_transitive|apply le_antisymmetric]
 qed. 
  
 lemma is_lower_bound_reverse_is_upper_bound:
- ∀O:pordered_set.∀a:sequence O.∀l:O.
-  is_lower_bound O a l → is_upper_bound (reverse_pordered_set O) a l.
-intros (O a l H); unfold; intros (n); unfold reverse_pordered_set;
+ ∀O:excedence.∀a:sequence O.∀l:O.
+  is_lower_bound O a l → is_upper_bound (reverse_excedence O) a l.
+intros (O a l H); unfold; intros (n); unfold reverse_excedence;
 unfold reverse_excedence; simplify; fold unfold le (le ? l (a n)); apply H;    
 qed.
 
 lemma is_upper_bound_reverse_is_lower_bound:
- ∀O:pordered_set.∀a:sequence O.∀l:O.
-  is_upper_bound O a l → is_lower_bound (reverse_pordered_set O) a l.
-intros (O a l H); unfold; intros (n); unfold reverse_pordered_set;
+ ∀O:excedence.∀a:sequence O.∀l:O.
+  is_upper_bound O a l → is_lower_bound (reverse_excedence O) a l.
+intros (O a l H); unfold; intros (n); unfold reverse_excedence;
 unfold reverse_excedence; simplify; fold unfold le (le ? (a n) l); apply H;    
 qed.
 
 lemma reverse_is_lower_bound_is_upper_bound:
- ∀O:pordered_set.∀a:sequence O.∀l:O.
-  is_lower_bound (reverse_pordered_set O) a l → is_upper_bound O a l.
-intros (O a l H); unfold; intros (n); unfold reverse_pordered_set in H;
+ ∀O:excedence.∀a:sequence O.∀l:O.
+  is_lower_bound (reverse_excedence O) a l → is_upper_bound O a l.
+intros (O a l H); unfold; intros (n); unfold reverse_excedence in H;
 unfold reverse_excedence in H; simplify in H; apply H;    
 qed.
 
 lemma reverse_is_upper_bound_is_lower_bound:
- ∀O:pordered_set.∀a:sequence O.∀l:O.
-  is_upper_bound (reverse_pordered_set O) a l → is_lower_bound O a l.
-intros (O a l H); unfold; intros (n); unfold reverse_pordered_set in H;
+ ∀O:excedence.∀a:sequence O.∀l:O.
+  is_upper_bound (reverse_excedence O) a l → is_lower_bound O a l.
+intros (O a l H); unfold; intros (n); unfold reverse_excedence in H;
 unfold reverse_excedence in H; simplify in H; apply H;    
 qed.
 
 lemma is_inf_to_reverse_is_sup:
- ∀O:pordered_set.∀a:bounded_below_sequence O.∀l:O.
-  is_inf O a l → is_sup (reverse_pordered_set O) a l.
-intros (O a l H); apply (mk_is_sup (reverse_pordered_set O));
+ ∀O:excedence.∀a:bounded_below_sequence O.∀l:O.
+  is_inf O a l → is_sup (reverse_excedence O) a l.
+intros (O a l H); apply (mk_is_sup (reverse_excedence O));
 [1: apply is_lower_bound_reverse_is_upper_bound; apply inf_lower_bound; assumption
-|2: unfold reverse_pordered_set; simplify; unfold reverse_excedence; simplify; 
+|2: unfold reverse_excedence; simplify; unfold reverse_excedence; simplify; 
     intros (m H1); apply (inf_greatest_lower_bound ? ? ? H); apply H1;]
 qed.
 
 lemma is_sup_to_reverse_is_inf:
- ∀O:pordered_set.∀a:bounded_above_sequence O.∀l:O.
-  is_sup O a l → is_inf (reverse_pordered_set O) a l.
-intros (O a l H); apply (mk_is_inf (reverse_pordered_set O));
+ ∀O:excedence.∀a:bounded_above_sequence O.∀l:O.
+  is_sup O a l → is_inf (reverse_excedence O) a l.
+intros (O a l H); apply (mk_is_inf (reverse_excedence O));
 [1: apply is_upper_bound_reverse_is_lower_bound; apply sup_upper_bound; assumption
-|2: unfold reverse_pordered_set; simplify; unfold reverse_excedence; simplify; 
+|2: unfold reverse_excedence; simplify; unfold reverse_excedence; simplify; 
     intros (m H1); apply (sup_least_upper_bound ? ? ? H); apply H1;]
 qed.
 
 lemma reverse_is_sup_to_is_inf:
- ∀O:pordered_set.∀a:bounded_above_sequence O.∀l:O.
-  is_sup (reverse_pordered_set O) a l → is_inf O a l.
+ ∀O:excedence.∀a:bounded_above_sequence O.∀l:O.
+  is_sup (reverse_excedence O) a l → is_inf O a l.
 intros (O a l H); apply mk_is_inf;
 [1: apply reverse_is_upper_bound_is_lower_bound; 
-    apply (sup_upper_bound (reverse_pordered_set O)); assumption
-|2: intros (v H1); apply (sup_least_upper_bound (reverse_pordered_set O) a l H v);
+    apply (sup_upper_bound (reverse_excedence O)); assumption
+|2: intros (v H1); apply (sup_least_upper_bound (reverse_excedence O) a l H v);
     apply is_lower_bound_reverse_is_upper_bound; assumption;]
 qed.
 
 lemma reverse_is_inf_to_is_sup:
- ∀O:pordered_set.∀a:bounded_above_sequence O.∀l:O.
-  is_inf (reverse_pordered_set O) a l → is_sup O a l.
+ ∀O:excedence.∀a:bounded_above_sequence O.∀l:O.
+  is_inf (reverse_excedence O) a l → is_sup O a l.
 intros (O a l H); apply mk_is_sup;
 [1: apply reverse_is_lower_bound_is_upper_bound; 
-    apply (inf_lower_bound (reverse_pordered_set O)); assumption
-|2: intros (v H1); apply (inf_greatest_lower_bound (reverse_pordered_set O) a l H v);
+    apply (inf_lower_bound (reverse_excedence O)); assumption
+|2: intros (v H1); apply (inf_greatest_lower_bound (reverse_excedence O) a l H v);
     apply is_upper_bound_reverse_is_lower_bound; assumption;]
 qed.