From: Enrico Tassi <enrico.tassi@inria.fr>
Date: Thu, 6 Dec 2007 10:16:17 +0000 (+0000)
Subject: excedence -> excess
X-Git-Tag: make_still_working~5723
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=87ed0c3e2ccd74f21f81c2cc9ed2945109bf0a9a;p=helm.git

excedence -> excess
---

diff --git a/helm/software/matita/dama/classical_pointfree/ordered_sets.ma b/helm/software/matita/dama/classical_pointfree/ordered_sets.ma
index b42a8c44b..5d9c2da67 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 "excedence.ma".
+include "excess.ma".
 
-record is_dedekind_sigma_complete (O:excedence) : Type ≝
+record is_dedekind_sigma_complete (O:excess) : 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:excedence) : Type ≝
  }.
 
 record dedekind_sigma_complete_ordered_set : Type ≝
- { dscos_ordered_set:> excedence;
+ { dscos_ordered_set:> excess;
    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:excedence.λf:O→O.
+ λO:excess.λf:O→O.
   ∀a:nat→O.∀p:is_increasing ? a.
    is_increasing ? (λi.f (a i)).
 
diff --git a/helm/software/matita/dama/excedence.ma b/helm/software/matita/dama/excedence.ma
deleted file mode 100644
index fa6848e12..000000000
--- a/helm/software/matita/dama/excedence.ma
+++ /dev/null
@@ -1,225 +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/excedence/".
-
-include "higher_order_defs/relations.ma".
-include "nat/plus.ma".
-include "constructive_connectives.ma".
-include "constructive_higher_order_relations.ma".
-
-record excedence : Type ≝ {
-  exc_carr:> Type;
-  exc_relation: exc_carr → exc_carr → Type;
-  exc_coreflexive: coreflexive ? exc_relation;
-  exc_cotransitive: cotransitive ? exc_relation 
-}.
-
-interpretation "excedence" 'nleq a b =
- (cic:/matita/excedence/exc_relation.con _ a b). 
-
-definition le ≝ λE:excedence.λa,b:E. ¬ (a ≰ b).
-
-interpretation "ordered sets less or equal than" 'leq a b = 
- (cic:/matita/excedence/le.con _ a b).
-
-lemma le_reflexive: ∀E.reflexive ? (le E).
-intros (E); unfold; cases E; simplify; intros (x); apply (H x);
-qed.
-
-lemma le_transitive: ∀E.transitive ? (le E).
-intros (E); unfold; cases E; simplify; unfold Not; intros (x y z Rxy Ryz H2); 
-cases (c x z y H2) (H4 H5); clear H2; [exact (Rxy H4)|exact (Ryz H5)] 
-qed.
-
-record apartness : Type ≝ {
-  ap_carr:> Type;
-  ap_apart: ap_carr → ap_carr → Type;
-  ap_coreflexive: coreflexive ? ap_apart;
-  ap_symmetric: symmetric ? ap_apart;
-  ap_cotransitive: cotransitive ? ap_apart
-}.
-
-notation "a break # b" non associative with precedence 50 for @{ 'apart $a $b}.
-interpretation "apartness" 'apart x y = 
-  (cic:/matita/excedence/ap_apart.con _ x y).
-
-definition strong_ext ≝ λA:apartness.λop:A→A.∀x,y. op x # op y → x # y.
-
-definition apart ≝ λE:excedence.λa,b:E. a ≰ b ∨ b ≰ a.
-
-definition apart_of_excedence: excedence → apartness.
-intros (E); apply (mk_apartness E (apart E));  
-[1: unfold; cases E; simplify; clear E; intros (x); unfold;
-    intros (H1); apply (H x); cases H1; assumption;
-|2: unfold; intros(x y H); cases H; clear H; [right|left] assumption;
-|3: intros (E); unfold; cases E (T f _ cTf); simplify; intros (x y z Axy);
-    cases Axy (H H); lapply (cTf ? ? z H) as H1; cases H1; clear Axy H1;
-    [left; left|right; left|right; right|left; right] assumption]
-qed.
-
-coercion cic:/matita/excedence/apart_of_excedence.con.
-
-definition eq ≝ λA:apartness.λa,b:A. ¬ (a # b).
-
-notation "a break ≈ b" non associative with precedence 50 for @{ 'napart $a $b}.    
-interpretation "alikeness" 'napart a b =
-  (cic:/matita/excedence/eq.con _ a b). 
-
-lemma eq_reflexive:∀E. reflexive ? (eq E).
-intros (E); unfold; intros (x); apply ap_coreflexive; 
-qed.
-
-lemma eq_sym_:∀E.symmetric ? (eq E).
-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_.
-
-coercion cic:/matita/excedence/eq_sym.con.
-
-lemma eq_trans_: ∀E.transitive ? (eq E).
-(* bug. intros k deve fare whd quanto basta *)
-intros 6 (E x y z Exy Eyz); intro Axy; cases (ap_cotransitive ???y Axy); 
-[apply Exy|apply Eyz] assumption.
-qed.
-
-lemma eq_trans:∀E:apartness.∀x,z,y:E.x ≈ y → y ≈ z → x ≈ z ≝ 
-  λE,x,y,z.eq_trans_ E x z y.
-
-notation > "'Eq'≈" non associative with precedence 50 for
- @{'eqrewrite}.
- 
-interpretation "eq_rew" 'eqrewrite = 
- (cic:/matita/excedence/eq_trans.con _ _ _).
-
-(* BUG: vedere se ricapita *)
-alias id "antisymmetric" = "cic:/matita/constructive_higher_order_relations/antisymmetric.con".
-lemma le_antisymmetric: ∀E.antisymmetric ? (le E) (eq ?).
-intros 5 (E x y Lxy Lyx); intro H;
-cases H; [apply Lxy;|apply Lyx] assumption;
-qed.
-
-definition lt ≝ λE:excedence.λa,b:E. a ≤ b ∧ a # b.
-
-interpretation "ordered sets less than" 'lt a b =
- (cic:/matita/excedence/lt.con _ a b).
-
-lemma lt_coreflexive: ∀E.coreflexive ? (lt E).
-intros 2 (E x); intro H; cases H (_ ABS); 
-apply (ap_coreflexive ? x ABS);
-qed.
- 
-lemma lt_transitive: ∀E.transitive ? (lt E).
-intros (E); unfold; intros (x y z H1 H2); cases H1 (Lxy Axy); cases H2 (Lyz Ayz); 
-split; [apply (le_transitive ???? Lxy Lyz)] clear H1 H2;
-cases Axy (H1 H1); cases Ayz (H2 H2); [1:cases (Lxy H1)|3:cases (Lyz H2)]
-clear Axy Ayz;lapply (exc_cotransitive E) as c; unfold cotransitive in c;
-lapply (exc_coreflexive E) as r; unfold coreflexive in r;
-[1: lapply (c ?? y H1) as H3; cases H3 (H4 H4); [cases (Lxy H4)|cases (r ? H4)]
-|2: lapply (c ?? x H2) as H3; cases H3 (H4 H4); [right; assumption|cases (Lxy H4)]]
-qed.
-
-theorem lt_to_excede: ∀E:excedence.∀a,b:E. (a < b) → (b ≰ a).
-intros (E a b Lab); cases Lab (LEab Aab);
-cases Aab (H H); [cases (LEab H)] fold normalize (b ≰ a); assumption; (* BUG *)  
-qed.
-
-lemma unfold_apart: ∀E:excedence. ∀x,y:E. x ≰ y ∨ y ≰ x → x # y.
-intros; assumption;
-qed.
-
-lemma le_rewl: ∀E:excedence.∀z,y,x:E. x ≈ y → x ≤ z → y ≤ z.
-intros (E z y x Exy Lxz); apply (le_transitive ???? ? Lxz);
-intro Xyz; apply Exy; apply unfold_apart; right; assumption;
-qed.
-
-notation > "'Ex'≪" non associative with precedence 50 for
- @{'excedencerewritel}.
- 
-interpretation "exc_rewl" 'excedencerewritel = 
- (cic:/matita/excedence/exc_rewl.con _ _ _).
-
-lemma le_rewr: ∀E:excedence.∀z,y,x:E. x ≈ y → z ≤ x → z ≤ y.
-intros (E z y x Exy Lxz); apply (le_transitive ???? Lxz);
-intro Xyz; apply Exy; apply unfold_apart; left; assumption;
-qed.
-
-notation > "'Ex'≫" non associative with precedence 50 for
- @{'excedencerewriter}.
- 
-interpretation "exc_rewr" 'excedencerewriter = 
- (cic:/matita/excedence/exc_rewr.con _ _ _).
-
-lemma ap_rewl: ∀A:apartness.∀x,z,y:A. x ≈ y → y # z → x # z.
-intros (A x z y Exy Ayz); cases (ap_cotransitive ???x Ayz); [2:assumption]
-cases (Exy (ap_symmetric ??? a));
-qed.
-  
-lemma ap_rewr: ∀A:apartness.∀x,z,y:A. x ≈ y → z # y → z # x.
-intros (A x z y Exy Azy); apply ap_symmetric; apply (ap_rewl ???? Exy);
-apply ap_symmetric; assumption;
-qed.
-
-lemma exc_rewl: ∀A:excedence.∀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;
-qed.
-  
-lemma exc_rewr: ∀A:excedence.∀x,z,y:A. x ≈ y → z ≰ y → z ≰ x.
-intros (A x z y Exy Azy); elim (exc_cotransitive ???x Azy); [assumption]
-elim (Exy); left; assumption;
-qed.
-
-lemma lt_rewr: ∀A:excedence.∀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;
-qed.
-
-lemma lt_rewl: ∀A:excedence.∀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;
-qed.
-
-lemma lt_le_transitive: ∀A:excedence.∀x,y,z:A.x < y → y ≤ z → x < z.
-intros (A x y z LT LE); cases LT (LEx APx); split; [apply (le_transitive ???? LEx LE)]
-whd in LE LEx APx; cases APx (EXx EXx); [cases (LEx EXx)]
-cases (exc_cotransitive ??? z EXx) (EXz EXz); [cases (LE EXz)]
-right; assumption;
-qed.
-
-lemma le_lt_transitive: ∀A:excedence.∀x,y,z:A.x ≤ y → y < z → x < z.
-intros (A x y z LE LT); cases LT (LEx APx); split; [apply (le_transitive ???? LE LEx)]
-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.
-
-definition total_order_property : ∀E:excedence. Type ≝
-  λE:excedence. ∀a,b:E. a ≰ b → b < a.
-
diff --git a/helm/software/matita/dama/excess.ma b/helm/software/matita/dama/excess.ma
new file mode 100644
index 000000000..827aaf605
--- /dev/null
+++ b/helm/software/matita/dama/excess.ma
@@ -0,0 +1,225 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/excess/".
+
+include "higher_order_defs/relations.ma".
+include "nat/plus.ma".
+include "constructive_connectives.ma".
+include "constructive_higher_order_relations.ma".
+
+record excess : Type ≝ {
+  exc_carr:> Type;
+  exc_relation: exc_carr → exc_carr → Type;
+  exc_coreflexive: coreflexive ? exc_relation;
+  exc_cotransitive: cotransitive ? exc_relation 
+}.
+
+interpretation "excess" 'nleq a b =
+ (cic:/matita/excess/exc_relation.con _ a b). 
+
+definition le ≝ λE:excess.λa,b:E. ¬ (a ≰ b).
+
+interpretation "ordered sets less or equal than" 'leq a b = 
+ (cic:/matita/excess/le.con _ a b).
+
+lemma le_reflexive: ∀E.reflexive ? (le E).
+intros (E); unfold; cases E; simplify; intros (x); apply (H x);
+qed.
+
+lemma le_transitive: ∀E.transitive ? (le E).
+intros (E); unfold; cases E; simplify; unfold Not; intros (x y z Rxy Ryz H2); 
+cases (c x z y H2) (H4 H5); clear H2; [exact (Rxy H4)|exact (Ryz H5)] 
+qed.
+
+record apartness : Type ≝ {
+  ap_carr:> Type;
+  ap_apart: ap_carr → ap_carr → Type;
+  ap_coreflexive: coreflexive ? ap_apart;
+  ap_symmetric: symmetric ? ap_apart;
+  ap_cotransitive: cotransitive ? ap_apart
+}.
+
+notation "a break # b" non associative with precedence 50 for @{ 'apart $a $b}.
+interpretation "apartness" 'apart x y = 
+  (cic:/matita/excess/ap_apart.con _ x y).
+
+definition strong_ext ≝ λA:apartness.λop:A→A.∀x,y. op x # op y → x # y.
+
+definition apart ≝ λE:excess.λa,b:E. a ≰ b ∨ b ≰ a.
+
+definition apart_of_excess: excess → apartness.
+intros (E); apply (mk_apartness E (apart E));  
+[1: unfold; cases E; simplify; clear E; intros (x); unfold;
+    intros (H1); apply (H x); cases H1; assumption;
+|2: unfold; intros(x y H); cases H; clear H; [right|left] assumption;
+|3: intros (E); unfold; cases E (T f _ cTf); simplify; intros (x y z Axy);
+    cases Axy (H H); lapply (cTf ? ? z H) as H1; cases H1; clear Axy H1;
+    [left; left|right; left|right; right|left; right] assumption]
+qed.
+
+coercion cic:/matita/excess/apart_of_excess.con.
+
+definition eq ≝ λA:apartness.λa,b:A. ¬ (a # b).
+
+notation "a break ≈ b" non associative with precedence 50 for @{ 'napart $a $b}.    
+interpretation "alikeness" 'napart a b =
+  (cic:/matita/excess/eq.con _ a b). 
+
+lemma eq_reflexive:∀E. reflexive ? (eq E).
+intros (E); unfold; intros (x); apply ap_coreflexive; 
+qed.
+
+lemma eq_sym_:∀E.symmetric ? (eq E).
+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_.
+
+coercion cic:/matita/excess/eq_sym.con.
+
+lemma eq_trans_: ∀E.transitive ? (eq E).
+(* bug. intros k deve fare whd quanto basta *)
+intros 6 (E x y z Exy Eyz); intro Axy; cases (ap_cotransitive ???y Axy); 
+[apply Exy|apply Eyz] assumption.
+qed.
+
+lemma eq_trans:∀E:apartness.∀x,z,y:E.x ≈ y → y ≈ z → x ≈ z ≝ 
+  λE,x,y,z.eq_trans_ E x z y.
+
+notation > "'Eq'≈" non associative with precedence 50 for
+ @{'eqrewrite}.
+ 
+interpretation "eq_rew" 'eqrewrite = 
+ (cic:/matita/excess/eq_trans.con _ _ _).
+
+(* BUG: vedere se ricapita *)
+alias id "antisymmetric" = "cic:/matita/constructive_higher_order_relations/antisymmetric.con".
+lemma le_antisymmetric: ∀E.antisymmetric ? (le E) (eq ?).
+intros 5 (E x y Lxy Lyx); intro H;
+cases H; [apply Lxy;|apply Lyx] assumption;
+qed.
+
+definition lt ≝ λE:excess.λa,b:E. a ≤ b ∧ a # b.
+
+interpretation "ordered sets less than" 'lt a b =
+ (cic:/matita/excess/lt.con _ a b).
+
+lemma lt_coreflexive: ∀E.coreflexive ? (lt E).
+intros 2 (E x); intro H; cases H (_ ABS); 
+apply (ap_coreflexive ? x ABS);
+qed.
+ 
+lemma lt_transitive: ∀E.transitive ? (lt E).
+intros (E); unfold; intros (x y z H1 H2); cases H1 (Lxy Axy); cases H2 (Lyz Ayz); 
+split; [apply (le_transitive ???? Lxy Lyz)] clear H1 H2;
+cases Axy (H1 H1); cases Ayz (H2 H2); [1:cases (Lxy H1)|3:cases (Lyz H2)]
+clear Axy Ayz;lapply (exc_cotransitive E) as c; unfold cotransitive in c;
+lapply (exc_coreflexive E) as r; unfold coreflexive in r;
+[1: lapply (c ?? y H1) as H3; cases H3 (H4 H4); [cases (Lxy H4)|cases (r ? H4)]
+|2: lapply (c ?? x H2) as H3; cases H3 (H4 H4); [right; assumption|cases (Lxy H4)]]
+qed.
+
+theorem lt_to_excede: ∀E:excess.∀a,b:E. (a < b) → (b ≰ a).
+intros (E a b Lab); cases Lab (LEab Aab);
+cases Aab (H H); [cases (LEab H)] fold normalize (b ≰ a); assumption; (* BUG *)  
+qed.
+
+lemma unfold_apart: ∀E:excess. ∀x,y:E. x ≰ y ∨ y ≰ x → x # y.
+intros; assumption;
+qed.
+
+lemma le_rewl: ∀E:excess.∀z,y,x:E. x ≈ y → x ≤ z → y ≤ z.
+intros (E z y x Exy Lxz); apply (le_transitive ???? ? Lxz);
+intro Xyz; apply Exy; apply unfold_apart; right; assumption;
+qed.
+
+notation > "'Ex'≪" non associative with precedence 50 for
+ @{'excessrewritel}.
+ 
+interpretation "exc_rewl" 'excessrewritel = 
+ (cic:/matita/excess/exc_rewl.con _ _ _).
+
+lemma le_rewr: ∀E:excess.∀z,y,x:E. x ≈ y → z ≤ x → z ≤ y.
+intros (E z y x Exy Lxz); apply (le_transitive ???? Lxz);
+intro Xyz; apply Exy; apply unfold_apart; left; assumption;
+qed.
+
+notation > "'Ex'≫" non associative with precedence 50 for
+ @{'excessrewriter}.
+ 
+interpretation "exc_rewr" 'excessrewriter = 
+ (cic:/matita/excess/exc_rewr.con _ _ _).
+
+lemma ap_rewl: ∀A:apartness.∀x,z,y:A. x ≈ y → y # z → x # z.
+intros (A x z y Exy Ayz); cases (ap_cotransitive ???x Ayz); [2:assumption]
+cases (Exy (ap_symmetric ??? a));
+qed.
+  
+lemma ap_rewr: ∀A:apartness.∀x,z,y:A. x ≈ y → z # y → z # x.
+intros (A x z y Exy Azy); apply ap_symmetric; apply (ap_rewl ???? Exy);
+apply ap_symmetric; assumption;
+qed.
+
+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;
+qed.
+  
+lemma exc_rewr: ∀A:excess.∀x,z,y:A. x ≈ y → z ≰ y → z ≰ x.
+intros (A x z y Exy Azy); elim (exc_cotransitive ???x Azy); [assumption]
+elim (Exy); left; assumption;
+qed.
+
+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;
+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;
+qed.
+
+lemma lt_le_transitive: ∀A:excess.∀x,y,z:A.x < y → y ≤ z → x < z.
+intros (A x y z LT LE); cases LT (LEx APx); split; [apply (le_transitive ???? LEx LE)]
+whd in LE LEx APx; cases APx (EXx EXx); [cases (LEx EXx)]
+cases (exc_cotransitive ??? z EXx) (EXz EXz); [cases (LE EXz)]
+right; assumption;
+qed.
+
+lemma le_lt_transitive: ∀A:excess.∀x,y,z:A.x ≤ y → y < z → x < z.
+intros (A x y z LE LT); cases LT (LEx APx); split; [apply (le_transitive ???? LE LEx)]
+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:excess.∀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:excess.∀a,b:E. a ≈ b → a ≤ b ∧ b ≤ a.
+intros (E x y H); unfold apart_of_excess in H; unfold apart in H;
+simplify in H; split; intro; apply H; [left|right] assumption.
+qed.
+
+lemma ap_le_to_lt: ∀E:excess.∀a,c:E.c # a → c ≤ a → c < a.
+intros; split; assumption;
+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 b298e82e0..0e2668c2d 100644
--- a/helm/software/matita/dama/group.ma
+++ b/helm/software/matita/dama/group.ma
@@ -14,7 +14,7 @@
 
 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.
diff --git a/helm/software/matita/dama/lattice.ma b/helm/software/matita/dama/lattice.ma
index d6b637a0d..768c86df1 100644
--- a/helm/software/matita/dama/lattice.ma
+++ b/helm/software/matita/dama/lattice.ma
@@ -14,7 +14,7 @@
 
 set "baseuri" "cic:/matita/lattice/".
 
-include "excedence.ma".
+include "excess.ma".
 
 record lattice : Type ≝ {
   l_carr:> apartness;
@@ -40,8 +40,8 @@ interpretation "Lattice join" 'or a b =
 
 definition excl ≝ λl:lattice.λa,b:l.a # (a ∧ b).
 
-lemma excedence_of_lattice: lattice → excedence.
-intro l; apply (mk_excedence l (excl l));
+lemma excess_of_lattice: lattice → excess.
+intro l; apply (mk_excess l (excl l));
 [ intro x; unfold; intro H; unfold in H; apply (ap_coreflexive l x);
   apply (ap_rewr ??? (x∧x) (meet_refl l x)); assumption;
 | intros 3 (x y z); unfold excl; intro H;
@@ -55,7 +55,7 @@ intro l; apply (mk_excedence l (excl l));
   assumption]
 qed.    
 
-coercion cic:/matita/lattice/excedence_of_lattice.con.
+coercion cic:/matita/lattice/excess_of_lattice.con.
 
 lemma feq_ml: ∀ml:lattice.∀a,b,c:ml. a ≈ b → (c ∧ a) ≈ (c ∧ b).
 intros (l a b c H); unfold eq in H ⊢ %; unfold Not in H ⊢ %;
@@ -69,7 +69,7 @@ qed.
 
 lemma le_to_eqm: ∀ml:lattice.∀a,b:ml. a ≤ b → a ≈ (a ∧ b).
 intros (l a b H); 
-  unfold le in H; unfold excedence_of_lattice in H;
+  unfold le in H; unfold excess_of_lattice in H;
   unfold excl in H; simplify in H; 
 unfold eq; assumption;
 qed.
diff --git a/helm/software/matita/dama/ordered_group.ma b/helm/software/matita/dama/ordered_group.ma
index e6b6f55ef..a00b6eb4a 100644
--- a/helm/software/matita/dama/ordered_group.ma
+++ b/helm/software/matita/dama/ordered_group.ma
@@ -18,8 +18,8 @@ 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.
diff --git a/helm/software/matita/dama/sequence.ma b/helm/software/matita/dama/sequence.ma
index c26ae5721..9990f8c7d 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 "excedence.ma".
+include "excess.ma".
 
-definition sequence := λO:excedence.nat → O.
+definition sequence := λO:excess.nat → O.
 
-definition fun_of_sequence: ∀O:excedence.sequence O → nat → O.
+definition fun_of_sequence: ∀O:excess.sequence O → nat → O.
 intros; apply s; assumption;
 qed.
 
 coercion cic:/matita/sequence/fun_of_sequence.con 1.
 
 definition upper_bound ≝ 
-  λO:excedence.λa:sequence O.λu:O.∀n:nat.a n ≤ u.
+  λO:excess.λa:sequence O.λu:O.∀n:nat.a n ≤ u.
   
 definition lower_bound ≝ 
-  λO:excedence.λa:sequence O.λu:O.∀n:nat.u ≤ a n.
+  λO:excess.λa:sequence O.λu:O.∀n:nat.u ≤ a n.
 
 definition strong_sup ≝
-  λO:excedence.λs:sequence O.λx.
+  λO:excess.λs:sequence O.λx.
     upper_bound ? s x ∧ (∀y:O.x ≰ y → ∃n.s n ≰ y).
   
 definition strong_inf ≝
-  λO:excedence.λs:sequence O.λx.
+  λO:excess.λs:sequence O.λx.
     lower_bound ? s x ∧ (∀y:O.y ≰ x → ∃n.y ≰ s n).
 
 definition weak_sup ≝
-  λO:excedence.λs:sequence O.λx.
+  λO:excess.λs:sequence O.λx.
     upper_bound ? s x ∧ (∀y:O.upper_bound ? s y → x ≤ y).
   
 definition weak_inf ≝
-  λO:excedence.λs:sequence O.λx.
+  λO:excess.λs:sequence O.λx.
     lower_bound ? s x ∧ (∀y:O.lower_bound ? s y → y ≤ x).
 
 lemma strong_sup_is_weak: 
-  ∀O:excedence.∀s:sequence O.∀x:O.strong_sup ? s x → weak_sup ? s x.
+  ∀O:excess.∀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:excedence.∀s:sequence O.∀x:O.strong_inf ? s x → weak_inf ? s x.
+  ∀O:excess.∀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.
@@ -66,169 +66,169 @@ definition tends0 ≝
     ∀e:O.0 < e → ∃N.∀n.N < n → -e < s n ∧ s n < e.
     
 definition increasing ≝ 
-  λO:excedence.λa:sequence O.∀n:nat.a n ≤ a (S n).
+  λO:excess.λa:sequence O.∀n:nat.a n ≤ a (S n).
 
 definition decreasing ≝ 
-  λO:excedence.λa:sequence O.∀n:nat.a (S n) ≤ a n.
+  λO:excess.λa:sequence O.∀n:nat.a (S n) ≤ 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.
+definition is_upper_bound ≝ λO:excess.λa:sequence O.λu:O.∀n:nat.a n ≤ u.
+definition is_lower_bound ≝ λO:excess.λa:sequence O.λu:O.∀n:nat.u ≤ a n.
 
-record is_sup (O:excedence) (a:sequence O) (u:O) : Prop ≝
+record is_sup (O:excess) (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:excedence) (a:sequence O) (u:O) : Prop ≝
+record is_inf (O:excess) (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:excedence) (a:sequence O) : Type ≝
+record is_bounded_below (O:excess) (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:excedence) (a:sequence O) : Type ≝
+record is_bounded_above (O:excess) (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:excedence) (a:sequence O) : Type ≝
+record is_bounded (O:excess) (a:sequence O) : Type ≝
  { ib_bounded_below:> is_bounded_below ? a;
    ib_bounded_above:> is_bounded_above ? a
  }.
 
-record bounded_below_sequence (O:excedence) : Type ≝
+record bounded_below_sequence (O:excess) : Type ≝
  { bbs_seq:> sequence O;
    bbs_is_bounded_below:> is_bounded_below ? bbs_seq
  }.
 
-record bounded_above_sequence (O:excedence) : Type ≝
+record bounded_above_sequence (O:excess) : Type ≝
  { bas_seq:> sequence O;
    bas_is_bounded_above:> is_bounded_above ? bas_seq
  }.
 
-record bounded_sequence (O:excedence) : Type ≝
+record bounded_sequence (O:excess) : 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:excedence.λb:bounded_sequence O.
+ λO:excess.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:excedence.λb:bounded_sequence O.
+ λO:excess.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:excedence.λb:bounded_below_sequence O.
+ λO:excess.λb:bounded_below_sequence O.
   ib_lower_bound ? b (bbs_is_bounded_below ? b).
 
 lemma lower_bound_is_lower_bound:
- ∀O:excedence.∀b:bounded_below_sequence O.
+ ∀O:excess.∀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:excedence.λb:bounded_above_sequence O.
+ λO:excess.λb:bounded_above_sequence O.
   ib_upper_bound ? b (bas_is_bounded_above ? b).
 
 lemma upper_bound_is_upper_bound:
- ∀O:excedence.∀b:bounded_above_sequence O.
+ ∀O:excess.∀b:bounded_above_sequence O.
   is_upper_bound ? b (upper_bound ? b).
 intros; unfold upper_bound; apply ib_upper_bound_is_upper_bound.
 qed.
 
-definition reverse_excedence: excedence → excedence.
-intros (E); apply (mk_excedence E); [apply (λx,y.exc_relation E y x)] 
+definition reverse_excess: excess → excess.
+intros (E); apply (mk_excess E); [apply (λx,y.exc_relation E y x)] 
 cases E (T f cRf cTf); simplify; 
 [1: unfold Not; intros (x H); apply (cRf x); assumption
 |2: intros (x y z); apply Or_symmetric; apply cTf; assumption;]
 qed. 
 
-definition reverse_excedence: excedence → excedence.
-intros (p); apply (mk_excedence (reverse_excedence p));
-generalize in match (reverse_excedence p); intros (E);
+definition reverse_excess: excess → excess.
+intros (p); apply (mk_excess (reverse_excess p));
+generalize in match (reverse_excess 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: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;    
+ ∀O:excess.∀a:sequence O.∀l:O.
+  is_lower_bound O a l → is_upper_bound (reverse_excess O) a l.
+intros (O a l H); unfold; intros (n); unfold reverse_excess;
+unfold reverse_excess; simplify; fold unfold le (le ? l (a n)); apply H;    
 qed.
 
 lemma is_upper_bound_reverse_is_lower_bound:
- ∀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;    
+ ∀O:excess.∀a:sequence O.∀l:O.
+  is_upper_bound O a l → is_lower_bound (reverse_excess O) a l.
+intros (O a l H); unfold; intros (n); unfold reverse_excess;
+unfold reverse_excess; simplify; fold unfold le (le ? (a n) l); apply H;    
 qed.
 
 lemma reverse_is_lower_bound_is_upper_bound:
- ∀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;    
+ ∀O:excess.∀a:sequence O.∀l:O.
+  is_lower_bound (reverse_excess O) a l → is_upper_bound O a l.
+intros (O a l H); unfold; intros (n); unfold reverse_excess in H;
+unfold reverse_excess in H; simplify in H; apply H;    
 qed.
 
 lemma reverse_is_upper_bound_is_lower_bound:
- ∀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;    
+ ∀O:excess.∀a:sequence O.∀l:O.
+  is_upper_bound (reverse_excess O) a l → is_lower_bound O a l.
+intros (O a l H); unfold; intros (n); unfold reverse_excess in H;
+unfold reverse_excess in H; simplify in H; apply H;    
 qed.
 
 lemma is_inf_to_reverse_is_sup:
- ∀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));
+ ∀O:excess.∀a:bounded_below_sequence O.∀l:O.
+  is_inf O a l → is_sup (reverse_excess O) a l.
+intros (O a l H); apply (mk_is_sup (reverse_excess O));
 [1: apply is_lower_bound_reverse_is_upper_bound; apply inf_lower_bound; assumption
-|2: unfold reverse_excedence; simplify; unfold reverse_excedence; simplify; 
+|2: unfold reverse_excess; simplify; unfold reverse_excess; simplify; 
     intros (m H1); apply (inf_greatest_lower_bound ? ? ? H); apply H1;]
 qed.
 
 lemma is_sup_to_reverse_is_inf:
- ∀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));
+ ∀O:excess.∀a:bounded_above_sequence O.∀l:O.
+  is_sup O a l → is_inf (reverse_excess O) a l.
+intros (O a l H); apply (mk_is_inf (reverse_excess O));
 [1: apply is_upper_bound_reverse_is_lower_bound; apply sup_upper_bound; assumption
-|2: unfold reverse_excedence; simplify; unfold reverse_excedence; simplify; 
+|2: unfold reverse_excess; simplify; unfold reverse_excess; simplify; 
     intros (m H1); apply (sup_least_upper_bound ? ? ? H); apply H1;]
 qed.
 
 lemma reverse_is_sup_to_is_inf:
- ∀O:excedence.∀a:bounded_above_sequence O.∀l:O.
-  is_sup (reverse_excedence O) a l → is_inf O a l.
+ ∀O:excess.∀a:bounded_above_sequence O.∀l:O.
+  is_sup (reverse_excess 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_excedence O)); assumption
-|2: intros (v H1); apply (sup_least_upper_bound (reverse_excedence O) a l H v);
+    apply (sup_upper_bound (reverse_excess O)); assumption
+|2: intros (v H1); apply (sup_least_upper_bound (reverse_excess O) a l H v);
     apply is_lower_bound_reverse_is_upper_bound; assumption;]
 qed.
 
 lemma reverse_is_inf_to_is_sup:
- ∀O:excedence.∀a:bounded_above_sequence O.∀l:O.
-  is_inf (reverse_excedence O) a l → is_sup O a l.
+ ∀O:excess.∀a:bounded_above_sequence O.∀l:O.
+  is_inf (reverse_excess 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_excedence O)); assumption
-|2: intros (v H1); apply (inf_greatest_lower_bound (reverse_excedence O) a l H v);
+    apply (inf_lower_bound (reverse_excess O)); assumption
+|2: intros (v H1); apply (inf_greatest_lower_bound (reverse_excess O) a l H v);
     apply is_upper_bound_reverse_is_lower_bound; assumption;]
 qed.