exc_cotransitive: cotransitive ? exc_excess
}.
-interpretation "excess" 'nleq a b = (cic:/matita/excess/exc_excess.con _ a b).
+interpretation "Excess base excess" 'nleq a b = (cic:/matita/excess/exc_excess.con _ a b).
+
+(* E(#,≰) → E(#,sym(≰)) *)
+lemma make_dual_exc: excess_base → excess_base.
+intro E;
+apply (mk_excess_base (exc_carr E));
+ [ apply (λx,y:E.y≰x);|apply exc_coreflexive;
+ | unfold cotransitive; simplify; intros (x y z H);
+ cases (exc_cotransitive E ??z H);[right|left]assumption]
+qed.
+
+record excess_dual : Type ≝ {
+ exc_dual_base:> excess_base;
+ exc_dual_dual_ : excess_base;
+ exc_with: exc_dual_dual_ = make_dual_exc exc_dual_base
+}.
+
+lemma mk_excess_dual_smart: excess_base → excess_dual.
+intro; apply mk_excess_dual; [apply e| apply (make_dual_exc e)|reflexivity]
+qed.
+
+definition exc_dual_dual: excess_dual → excess_base.
+intro E; apply (make_dual_exc E);
+qed.
+
+coercion cic:/matita/excess/exc_dual_dual.con.
record apartness : Type ≝ {
ap_carr:> Type;
qed.
record excess_ : Type ≝ {
- exc_exc:> excess_base;
- exc_ap: apartness;
- exc_with: ap_carr exc_ap = exc_carr exc_exc
+ exc_exc:> excess_dual;
+ exc_ap_: apartness;
+ exc_with1: ap_carr exc_ap_ = exc_carr exc_exc
}.
-definition apart_of_excess_: excess_ → apartness.
+definition exc_ap: excess_ → apartness.
intro E; apply (mk_apartness E); unfold Type_OF_excess_;
-cases (exc_with E); simplify;
-[apply (ap_apart (exc_ap E));
+cases (exc_with1 E); simplify;
+[apply (ap_apart (exc_ap_ E));
|apply ap_coreflexive;|apply ap_symmetric;|apply ap_cotransitive]
qed.
-coercion cic:/matita/excess/apart_of_excess_.con.
+coercion cic:/matita/excess/exc_ap.con.
+
+interpretation "Excess excess_" 'nleq a b =
+ (cic:/matita/excess/exc_excess.con (cic:/matita/excess/excess_base_OF_excess_1.con _) a b).
record excess : Type ≝ {
excess_carr:> excess_;
exc2ap: ∀y,x:excess_carr.y ≰ x ∨ x ≰ y → y # x
}.
+interpretation "Excess excess" 'nleq a b =
+ (cic:/matita/excess/exc_excess.con (cic:/matita/excess/excess_base_OF_excess1.con _) a b).
+
+interpretation "Excess (dual) excess" 'ngeq a b =
+ (cic:/matita/excess/exc_excess.con (cic:/matita/excess/excess_base_OF_excess.con _) a b).
+
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).
+definition le ≝ λE:excess_base.λa,b:E. ¬ (a ≰ b).
-interpretation "ordered sets less or equal than" 'leq a b =
- (cic:/matita/excess/le.con _ a b).
+interpretation "Excess less or equal than" 'leq a b =
+ (cic:/matita/excess/le.con (cic:/matita/excess/excess_base_OF_excess1.con _) a b).
+
+interpretation "Excess less or equal than" 'geq a b =
+ (cic:/matita/excess/le.con (cic:/matita/excess/excess_base_OF_excess.con _) a b).
lemma le_reflexive: ∀E.reflexive ? (le E).
unfold reflexive; intros 3 (E x H); apply (exc_coreflexive ?? H);
definition eq ≝ λA:apartness.λa,b:A. ¬ (a # b).
notation "hvbox(a break ≈ b)" non associative with precedence 50 for @{ 'napart $a $b}.
-interpretation "alikeness" 'napart a b =
- (cic:/matita/excess/eq.con _ a b).
+interpretation "Apartness alikeness" 'napart a b = (cic:/matita/excess/eq.con _ a b).
+interpretation "Excess alikeness" 'napart a b = (cic:/matita/excess/eq.con (cic:/matita/excess/excess_base_OF_excess1.con _) a b).
+interpretation "Excess (dual) alikeness" 'napart a b = (cic:/matita/excess/eq.con (cic:/matita/excess/excess_base_OF_excess.con _) a b).
-lemma eq_reflexive:∀E. reflexive ? (eq E).
+lemma eq_reflexive:∀E:apartness. reflexive ? (eq E).
intros (E); unfold; intros (x); apply ap_coreflexive;
qed.
-lemma eq_sym_:∀E.symmetric ? (eq E).
+lemma eq_sym_:∀E:apartness.symmetric ? (eq E).
unfold symmetric; intros 5 (E x y H H1); cases (H (ap_symmetric ??? H1));
qed.
(* SETOID REWRITE *)
coercion cic:/matita/excess/eq_sym.con.
-lemma eq_trans_: ∀E.transitive ? (eq E).
+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.
interpretation "eq_rew" 'eqrewrite = (cic:/matita/excess/eq_trans.con _ _ _).
alias id "antisymmetric" = "cic:/matita/constructive_higher_order_relations/antisymmetric.con".
-lemma le_antisymmetric: ∀E.antisymmetric ? (le E) (eq ?).
+lemma le_antisymmetric:
+ ∀E:excess.antisymmetric ? (le (excess_base_OF_excess1 E)) (eq E).
intros 5 (E x y Lxy Lyx); intro H;
cases (ap2exc ??? H); [apply Lxy;|apply Lyx] assumption;
qed.
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;
elim (ap2exc ??? Axy) (H1 H1); elim (ap2exc ??? 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;
+clear Axy Ayz;lapply (exc_cotransitive (exc_dual_base E)) as c; unfold cotransitive in c;
+lapply (exc_coreflexive (exc_dual_base E)) as r; unfold coreflexive in r;
[1: lapply (c ?? y H1) as H3; elim H3 (H4 H4); [cases (Lxy H4)|cases (r ? H4)]
|2: lapply (c ?? x H2) as H3; elim H3 (H4 H4); [apply exc2ap; right; assumption|cases (Lxy H4)]]
qed.
notation > "'Ap'≫" non associative with precedence 50 for @{'aprewriter}.
interpretation "ap_rewr" 'aprewriter = (cic:/matita/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); elim (exc_cotransitive ???x Ayz); [2:assumption]
cases Exy; apply exc2ap; right; assumption;
definition total_order_property : ∀E:excess. Type ≝
λE:excess. ∀a,b:E. a ≰ b → b < a.
-(* E(#,≰) → E(#,sym(≰)) *)
-lemma dual_exc: excess→ excess.
-intro E; apply mk_excess;
-[1: apply mk_excess_;
- [1: apply (mk_excess_base (exc_carr (excess_carr E)));
- [ apply (λx,y:E.y≰x);|apply exc_coreflexive;
- | unfold cotransitive; simplify; intros (x y z H);
- cases (exc_cotransitive E ??z H);[right|left]assumption]
- |2: apply (exc_ap E);
- |3: apply (exc_with E);]
-|2: simplify; intros (y x H); fold simplify (y#x) in H;
- apply ap2exc; apply ap_symmetric; apply H;
-|3: simplify; intros; fold simplify (y#x); apply exc2ap;
- cases o; [right|left]assumption]
-qed.