--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *)
+(* *)
+(**************************************************************************)
+
+include "ordered_set.ma".
+
+(* Definition 2.2, setoid *)
+record bishop_set: Type ≝ {
+ bs_carr:> Type;
+ bs_apart: bs_carr → bs_carr → CProp;
+ bs_coreflexive: coreflexive ? bs_apart;
+ bs_symmetric: symmetric ? bs_apart;
+ bs_cotransitive: cotransitive ? bs_apart
+}.
+
+notation "hvbox(a break # b)" non associative with precedence 50
+ for @{ 'apart $a $b}.
+
+interpretation "bishop_setapartness" 'apart x y =
+ (cic:/matita/dama/bishop_set/bs_apart.con _ x y).
+
+definition bishop_set_of_ordered_set: ordered_set → bishop_set.
+intros (E); apply (mk_bishop_set E (λa,b:E. a ≰ b ∨ b ≰ a));
+[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.
+
+(* Definition 2.2 (2) *)
+definition eq ≝ λA:bishop_set.λa,b:A. ¬ (a # b).
+
+notation "hvbox(a break ≈ b)" non associative with precedence 50
+ for @{ 'napart $a $b}.
+
+interpretation "Bishop set alikeness" 'napart a b =
+ (cic:/matita/dama/bishop_set/eq.con _ a b).
+
+lemma eq_reflexive:∀E:bishop_set. reflexive ? (eq E).
+intros (E); unfold; intros (x); apply bs_coreflexive;
+qed.
+
+lemma eq_sym_:∀E:bishop_set.symmetric ? (eq E).
+intros 4 (E x y H); intro T; cases (H (bs_symmetric ??? T));
+qed.
+
+lemma eq_sym:∀E:bishop_set.∀x,y:E.x ≈ y → y ≈ x ≝ eq_sym_.
+
+lemma eq_trans_: ∀E:bishop_set.transitive ? (eq E).
+(* bug. intros k deve fare whd quanto basta *)
+intros 6 (E x y z Exy Eyz); intro Axy; cases (bs_cotransitive ???y Axy);
+[apply Exy|apply Eyz] assumption.
+qed.
+
+coercion cic:/matita/dama/bishop_set/bishop_set_of_ordered_set.con.
+
+lemma le_antisymmetric:
+ ∀E:ordered_set.antisymmetric E (le E) (eq E).
+intros 5 (E x y Lxy Lyx); intro H;
+cases H; [apply Lxy;|apply Lyx] assumption;
+qed.
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *)
+(* *)
+(**************************************************************************)
+
+include "bishop_set.ma".
+
+coercion cic:/matita/dama/bishop_set/eq_sym.con.
+
+lemma eq_trans:∀E:bishop_set.∀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/dama/bishop_set/eq_trans.con _ _ _).
+
+lemma le_rewl: ∀E:ordered_set.∀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; right; assumption;
+qed.
+
+lemma le_rewr: ∀E:ordered_set.∀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; left; assumption;
+qed.
+
+notation > "'Le'≪" non associative with precedence 50 for @{'lerewritel}.
+interpretation "le_rewl" 'lerewritel = (cic:/matita/dama/bishop_set_rewrite/le_rewl.con _ _ _).
+notation > "'Le'≫" non associative with precedence 50 for @{'lerewriter}.
+interpretation "le_rewr" 'lerewriter = (cic:/matita/dama/bishop_set_rewrite/le_rewr.con _ _ _).
+
+lemma ap_rewl: ∀A:bishop_set.∀x,z,y:A. x ≈ y → y # z → x # z.
+intros (A x z y Exy Ayz); cases (bs_cotransitive ???x Ayz); [2:assumption]
+cases (eq_sym ??? Exy H);
+qed.
+
+lemma ap_rewr: ∀A:bishop_set.∀x,z,y:A. x ≈ y → z # y → z # x.
+intros (A x z y Exy Azy); apply bs_symmetric; apply (ap_rewl ???? Exy);
+apply bs_symmetric; assumption;
+qed.
+
+notation > "'Ap'≪" non associative with precedence 50 for @{'aprewritel}.
+interpretation "ap_rewl" 'aprewritel = (cic:/matita/dama/bishop_set_rewrite/ap_rewl.con _ _ _).
+notation > "'Ap'≫" non associative with precedence 50 for @{'aprewriter}.
+interpretation "ap_rewr" 'aprewriter = (cic:/matita/dama/bishop_set_rewrite/ap_rewr.con _ _ _).
+
+lemma exc_rewl: ∀A:ordered_set.∀x,z,y:A. x ≈ y → y ≰ z → x ≰ z.
+intros (A x z y Exy Ayz); cases (os_cotransitive ??? x Ayz); [2:assumption]
+cases Exy; right; assumption;
+qed.
+
+lemma exc_rewr: ∀A:ordered_set.∀x,z,y:A. x ≈ y → z ≰ y → z ≰ x.
+intros (A x z y Exy Azy); cases (os_cotransitive ???x Azy); [assumption]
+cases (Exy); left; assumption;
+qed.
+
+notation > "'Ex'≪" non associative with precedence 50 for @{'ordered_setrewritel}.
+interpretation "exc_rewl" 'ordered_setrewritel = (cic:/matita/dama/bishop_set_rewrite/exc_rewl.con _ _ _).
+notation > "'Ex'≫" non associative with precedence 50 for @{'ordered_setrewriter}.
+interpretation "exc_rewr" 'ordered_setrewriter = (cic:/matita/dama/bishop_set_rewrite/exc_rewr.con _ _ _).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *)
+(* *)
+(**************************************************************************)
+
+include "logic/equality.ma".
+
+inductive Or (A,B:CProp) : CProp ≝
+ Left : A → Or A B
+ | Right : B → Or A B.
+
+interpretation "constructive or" 'or x y =
+ (cic:/matita/dama/cprop_connectives/Or.ind#xpointer(1/1) x y).
+
+inductive And (A,B:CProp) : CProp ≝
+ | Conj : A → B → And A B.
+
+interpretation "constructive and" 'and x y =
+ (cic:/matita/dama/cprop_connectives/And.ind#xpointer(1/1) x y).
+
+inductive exT (A:Type) (P:A→CProp) : CProp ≝
+ ex_introT: ∀w:A. P w → exT A P.
+
+inductive ex (A:Type) (P:A→Prop) : Type ≝ (* ??? *)
+ ex_intro: ∀w:A. P w → ex A P.
+
+interpretation "constructive exists" 'exists \eta.x =
+ (cic:/matita/dama/cprop_connectives/ex.ind#xpointer(1/1) _ x).
+
+interpretation "constructive exists (Type)" 'exists \eta.x =
+ (cic:/matita/dama/cprop_connectives/exT.ind#xpointer(1/1) _ x).
+
+inductive False : CProp ≝ .
+
+definition Not ≝ λx:CProp.x → False.
+
+interpretation "constructive not" 'not x =
+ (cic:/matita/dama/cprop_connectives/Not.con x).
+
+definition cotransitive ≝
+ λC:Type.λlt:C→C→CProp.∀x,y,z:C. lt x y → lt x z ∨ lt z y.
+
+definition coreflexive ≝ λC:Type.λlt:C→C→CProp. ∀x:C. ¬ (lt x x).
+
+definition symmetric ≝ λC:Type.λlt:C→C→CProp. ∀x,y:C.lt x y → lt y x.
+
+definition antisymmetric ≝ λA:Type.λR:A→A→CProp.λeq:A→A→CProp.∀x:A.∀y:A.R x y→R y x→eq x y.
+
+definition reflexive ≝ λA:Type.λR:A→A→CProp.∀x:A.R x x.
+
+definition transitive ≝ λA:Type.λR:A→A→CProp.∀x,y,z:A.R x y → R y z → R x z.
+
-excedence.ma logic/equality.ma
+bishop_set.ma ordered_set.ma
+extra.ma bishop_set_rewrite.ma
+ordered_set.ma cprop_connectives.ma
+cprop_connectives.ma logic/equality.ma
+bishop_set_rewrite.ma bishop_set.ma
logic/equality.ma
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 *)
-(* *)
-(**************************************************************************)
-
-include "logic/equality.ma".
-
-inductive Or (A,B:CProp) : CProp ≝
- Left : A → Or A B
- | Right : B → Or A B.
-
-interpretation "constructive or" 'or x y =
- (cic:/matita/dama/excess/Or.ind#xpointer(1/1) x y).
-
-inductive And (A,B:CProp) : CProp ≝
- | Conj : A → B → And A B.
-
-interpretation "constructive and" 'and x y =
- (cic:/matita/dama/excess/And.ind#xpointer(1/1) x y).
-
-inductive exT (A:Type) (P:A→CProp) : CProp ≝
- ex_introT: ∀w:A. P w → exT A P.
-
-inductive ex (A:Type) (P:A→Prop) : Type ≝ (* ??? *)
- ex_intro: ∀w:A. P w → ex A P.
-
-interpretation "constructive exists" 'exists \eta.x =
- (cic:/matita/dama/excess/ex.ind#xpointer(1/1) _ x).
-interpretation "constructive exists (Type)" 'exists \eta.x =
- (cic:/matita/dama/excess/exT.ind#xpointer(1/1) _ x).
-
-inductive False : CProp ≝ .
-
-definition Not ≝ λx:CProp.x → False.
-
-interpretation "constructive not" 'not x =
- (cic:/matita/dama/excess/Not.con x).
-
-definition cotransitive ≝
- λC:Type.λlt:C→C→CProp.∀x,y,z:C. lt x y → lt x z ∨ lt z y.
-
-definition coreflexive ≝ λC:Type.λlt:C→C→CProp. ∀x:C. ¬ (lt x x).
-
-definition symmetric ≝ λC:Type.λlt:C→C→CProp. ∀x,y:C.lt x y → lt y x.
-
-definition antisymmetric ≝ λA:Type.λR:A→A→CProp.λeq:A→A→CProp.∀x:A.∀y:A.R x y→R y x→eq x y.
-
-definition reflexive ≝ λA:Type.λR:A→A→CProp.∀x:A.R x x.
-
-definition transitive ≝ λA:Type.λR:A→A→CProp.∀x,y,z:A.R x y → R y z → R x z.
-
-record excess : Type ≝ {
- exc_carr:> Type;
- exc_excess: exc_carr → exc_carr → CProp;
- exc_coreflexive: coreflexive ? exc_excess;
- exc_cotransitive: cotransitive ? exc_excess
-}.
-
-interpretation "Excess" 'nleq a b = (cic:/matita/dama/excess/exc_excess.con _ a b).
-
-record apartness : Type ≝ {
- ap_carr:> Type;
- ap_apart: ap_carr → ap_carr → CProp;
- ap_coreflexive: coreflexive ? ap_apart;
- ap_symmetric: symmetric ? ap_apart;
- ap_cotransitive: cotransitive ? ap_apart
-}.
-
-notation "hvbox(a break # b)" non associative with precedence 50 for @{ 'apart $a $b}.
-interpretation "apartness" 'apart x y = (cic:/matita/dama/excess/ap_apart.con _ x y).
-
-definition apartness_of_excess: excess → apartness.
-intros (E); apply (mk_apartness E (λa,b:E. a ≰ b ∨ b ≰ a));
-[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.
-
-definition strong_ext ≝ λA:apartness.λop:A→A.∀x,y. op x # op y → x # y.
-
-definition le ≝ λE:excess.λa,b:E. ¬ (a ≰ b).
-
-interpretation "Excess less or equal than" 'leq a b =
- (cic:/matita/dama/excess/le.con _ a b).
-
-interpretation "Excess less or equal than" 'geq a b =
- (cic:/matita/excess/le.con _ b a).
-
-lemma le_reflexive: ∀E.reflexive ? (le E).
-unfold reflexive; intros 3 (E x H); apply (exc_coreflexive ?? H);
-qed.
-
-lemma le_transitive: ∀E.transitive ? (le E).
-unfold transitive; intros 7 (E x y z H1 H2 H3); cases (exc_cotransitive ??? y H3) (H4 H4);
-[cases (H1 H4)|cases (H2 H4)]
-qed.
-
-definition eq ≝ λA:apartness.λa,b:A. ¬ (a # b).
-
-notation "hvbox(a break ≈ b)" non associative with precedence 50 for @{ 'napart $a $b}.
-interpretation "Apartness alikeness" 'napart a b = (cic:/matita/dama/excess/eq.con _ a b).
-
-lemma eq_reflexive:∀E:apartness. reflexive ? (eq E).
-intros (E); unfold; intros (x); apply ap_coreflexive;
-qed.
-
-lemma eq_sym_:∀E:apartness.symmetric ? (eq E).
-intros 4 (E x y H); intro T; cases (H (ap_symmetric ??? T));
-qed.
-
-lemma eq_sym:∀E:apartness.∀x,y:E.x ≈ y → y ≈ x ≝ eq_sym_.
-
-(* SETOID REWRITE *)
-coercion cic:/matita/dama/excess/eq_sym.con.
-
-lemma eq_trans_: ∀E:apartness.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/dama/excess/eq_trans.con _ _ _).
-
-coercion cic:/matita/dama/excess/apartness_of_excess.con.
-
-lemma le_antisymmetric:
- ∀E:excess.antisymmetric E (le E) (eq E).
-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/dama/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;
-[1: cases (exc_cotransitive ??? y H1) (X X); [cases (Lxy X)|cases (exc_coreflexive ?? X)]
-|2: cases (exc_cotransitive ??? x H2) (X X); [right;assumption|cases (Lxy X)]]
-qed.
-
-theorem lt_to_excess: ∀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 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; right; assumption;
-qed.
-
-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; left; assumption;
-qed.
-
-notation > "'Le'≪" non associative with precedence 50 for @{'lerewritel}.
-interpretation "le_rewl" 'lerewritel = (cic:/matita/dama/excess/le_rewl.con _ _ _).
-notation > "'Le'≫" non associative with precedence 50 for @{'lerewriter}.
-interpretation "le_rewr" 'lerewriter = (cic:/matita/dama/excess/le_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 (eq_sym ??? Exy H);
-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.
-
-notation > "'Ap'≪" non associative with precedence 50 for @{'aprewritel}.
-interpretation "ap_rewl" 'aprewritel = (cic:/matita/dama/excess/ap_rewl.con _ _ _).
-notation > "'Ap'≫" non associative with precedence 50 for @{'aprewriter}.
-interpretation "ap_rewr" 'aprewriter = (cic:/matita/dama/excess/ap_rewr.con _ _ _).
-
-alias symbol "napart" = "Apartness alikeness".
-lemma exc_rewl: ∀A:excess.∀x,z,y:A. x ≈ y → y ≰ z → x ≰ z.
-intros (A x z y Exy Ayz); cases (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); cases (exc_cotransitive ???x Azy); [assumption]
-cases (Exy); left; assumption;
-qed.
-
-notation > "'Ex'≪" non associative with precedence 50 for @{'excessrewritel}.
-interpretation "exc_rewl" 'excessrewritel = (cic:/matita/dama/excess/exc_rewl.con _ _ _).
-notation > "'Ex'≫" non associative with precedence 50 for @{'excessrewriter}.
-interpretation "exc_rewr" 'excessrewriter = (cic:/matita/dama/excess/exc_rewr.con _ _ _).
-
-lemma lt_rewr: ∀A:excess.∀x,z,y:A. x ≈ y → z < y → z < x.
-intros (A x y z E H); split; cases H;
-[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; cases H;
-[apply (Le≪ ? (eq_sym ??? E));| apply (Ap≪ ? E);] assumption;
-qed.
-
-notation > "'Lt'≪" non associative with precedence 50 for @{'ltrewritel}.
-interpretation "lt_rewl" 'ltrewritel = (cic:/matita/dama/excess/lt_rewl.con _ _ _).
-notation > "'Lt'≫" non associative with precedence 50 for @{'ltrewriter}.
-interpretation "lt_rewr" 'ltrewriter = (cic:/matita/dama/excess/lt_rewr.con _ _ _).
-
-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)]
-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)]
-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); whd 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.
-
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *)
+(* *)
+(**************************************************************************)
+
+include "bishop_set_rewrite.ma".
+
+definition strong_ext ≝ λA:bishop_set.λop:A→A.∀x,y. op x # op y → x # y.
+
+
+
+definition lt ≝ λE:excess.λa,b:E. a ≤ b ∧ a # b.
+
+interpretation "ordered sets less than" 'lt a b = (cic:/matita/dama/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;
+[1: cases (exc_cotransitive ??? y H1) (X X); [cases (Lxy X)|cases (exc_coreflexive ?? X)]
+|2: cases (exc_cotransitive ??? x H2) (X X); [right;assumption|cases (Lxy X)]]
+qed.
+
+theorem lt_to_excess: ∀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 lt_rewr: ∀A:excess.∀x,z,y:A. x ≈ y → z < y → z < x.
+intros (A x y z E H); split; cases H;
+[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; cases H;
+[apply (Le≪ ? (eq_sym ??? E));| apply (Ap≪ ? E);] assumption;
+qed.
+
+notation > "'Lt'≪" non associative with precedence 50 for @{'ltrewritel}.
+interpretation "lt_rewl" 'ltrewritel = (cic:/matita/dama/excess/lt_rewl.con _ _ _).
+notation > "'Lt'≫" non associative with precedence 50 for @{'ltrewriter}.
+interpretation "lt_rewr" 'ltrewriter = (cic:/matita/dama/excess/lt_rewr.con _ _ _).
+
+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)]
+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)]
+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); whd 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.
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *)
+(* *)
+(**************************************************************************)
+
+include "cprop_connectives.ma".
+
+(* Definition 2.1 *)
+record ordered_set: Type ≝ {
+ os_carr:> Type;
+ os_excess: os_carr → os_carr → CProp;
+ os_coreflexive: coreflexive ? os_excess;
+ os_cotransitive: cotransitive ? os_excess
+}.
+
+interpretation "Ordered set excess" 'nleq a b =
+ (cic:/matita/dama/ordered_set/os_excess.con _ a b).
+
+(* Definition 2.2 (3) *)
+definition le ≝ λE:ordered_set.λa,b:E. ¬ (a ≰ b).
+
+interpretation "Ordered set greater or equal than" 'geq a b =
+ (cic:/matita/dama/ordered_set/le.con _ b a).
+
+interpretation "Ordered set less or equal than" 'leq a b =
+ (cic:/matita/dama/ordered_set/le.con _ a b).
+
+lemma le_reflexive: ∀E.reflexive ? (le E).
+unfold reflexive; intros 3 (E x H); apply (os_coreflexive ?? H);
+qed.
+
+lemma le_transitive: ∀E.transitive ? (le E).
+unfold transitive; intros 7 (E x y z H1 H2 H3); cases (os_cotransitive ??? y H3) (H4 H4);
+[cases (H1 H4)|cases (H2 H4)]
+qed.
+
+(* Lemma 2.3 *)
+lemma exc_le_variance:
+ ∀O:ordered_set.∀a,b,a',b':O.a ≰ b → a ≤ a' → b' ≤ b → a' ≰ b'.
+intros (O a b a1 b1 Eab Laa1 Lb1b);
+cases (os_cotransitive ??? a1 Eab) (H H); [cases (Laa1 H)]
+cases (os_cotransitive ??? b1 H) (H1 H1); [assumption]
+cases (Lb1b H1);
+qed.
+
+
\ No newline at end of file
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