snapshot

[helm.git] / helm / software / matita / dama / excess.ma

diff --git a/helm/software/matita/dama/excess.ma b/helm/software/matita/dama/excess.ma
index 959e866fd6bd66bf436b287c5fd023652da20628..9068d297b215aea05faa417f2f3e365146a994c8 100644 (file)
--- a/helm/software/matita/dama/excess.ma
+++ b/helm/software/matita/dama/excess.ma
@@ -26,7 +26,32 @@ record excess_base : Type ≝ {
   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;
@@ -50,19 +75,22 @@ intros (E); apply (mk_apartness E (λa,b:E. a ≰ b ∨ b ≰ a));
 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_;
@@ -70,12 +98,21 @@ record excess : Type ≝ {
   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);
@@ -89,14 +126,15 @@ 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 "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.
 
@@ -105,7 +143,7 @@ lemma eq_sym:∀E:apartness.∀x,y:E.x ≈ y → y ≈ x ≝ eq_sym_.
 (* 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.
@@ -118,7 +156,8 @@ notation > "'Eq'≈" non associative with precedence 50 for @{'eqrewrite}.
 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.
@@ -136,8 +175,8 @@ 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;
 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.
@@ -177,6 +216,7 @@ interpretation "ap_rewl" 'aprewritel = (cic:/matita/excess/ap_rewl.con _ _ _).
 notation > "'Ap'≫" non associative with precedence 50 for @{'aprewriter}.
 interpretation "ap_rewr" 'aprewriter = (cic:/matita/excess/ap_rewr.con _ _ _).
 
+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;
@@ -237,18 +277,3 @@ qed.
 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.