X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fdama%2Fexcess.ma;h=959e866fd6bd66bf436b287c5fd023652da20628;hb=10f29fdd78ee089a9a94446207b543d33d6c851c;hp=d8cf097668382310eb8eda47d9c382a72c04d0aa;hpb=6067115471521e8b9ea805531cb94a0a80774314;p=helm.git

diff --git a/helm/software/matita/dama/excess.ma b/helm/software/matita/dama/excess.ma
index d8cf09766..959e866fd 100644
--- a/helm/software/matita/dama/excess.ma
+++ b/helm/software/matita/dama/excess.ma
@@ -12,36 +12,21 @@
 (*                                                                        *)
 (**************************************************************************)
 
-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".
+include "constructive_connectives.ma".
 
-record excess : Type ≝ {
+record excess_base : Type ≝ {
   exc_carr:> Type;
-  exc_relation: exc_carr → exc_carr → Type;
-  exc_coreflexive: coreflexive ? exc_relation;
-  exc_cotransitive: cotransitive ? exc_relation 
+  exc_excess: exc_carr → exc_carr → Type;
+  exc_coreflexive: coreflexive ? exc_excess;
+  exc_cotransitive: cotransitive ? exc_excess 
 }.
 
-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.
+interpretation "excess" 'nleq a b = (cic:/matita/excess/exc_excess.con _ a b). 
 
 record apartness : Type ≝ {
   ap_carr:> Type;
@@ -51,16 +36,11 @@ record apartness : Type ≝ {
   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.
+notation "hvbox(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 apart ≝ λE:excess.λa,b:E. a ≰ b ∨ b ≰ a.
-
-definition apart_of_excess: excess → apartness.
-intros (E); apply (mk_apartness E (apart E));  
+definition apartness_of_excess_base: excess_base → 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;
@@ -69,11 +49,46 @@ intros (E); apply (mk_apartness E (apart E));
     [left; left|right; left|right; right|left; right] assumption]
 qed.
 
-coercion cic:/matita/excess/apart_of_excess.con.
+record excess_ : Type ≝ {
+  exc_exc:> excess_base;
+  exc_ap: apartness;
+  exc_with: ap_carr exc_ap = exc_carr exc_exc
+}.
+
+definition apart_of_excess_: excess_ → apartness.
+intro E; apply (mk_apartness E); unfold Type_OF_excess_; 
+cases (exc_with 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.
+
+record excess : Type ≝ {
+  excess_carr:> excess_;
+  ap2exc: ∀y,x:excess_carr. y # x → y ≰ x ∨ x ≰ y;
+  exc2ap: ∀y,x:excess_carr.y ≰ x ∨ x ≰ y →  y # x 
+}.
+
+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 "ordered sets less or equal than" 'leq a b = 
+ (cic:/matita/excess/le.con _ a b).
+
+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 "a break ≈ b" non associative with precedence 50 for @{ 'napart $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). 
 
@@ -82,12 +97,12 @@ 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; 
+unfold symmetric; intros 5 (E x y H H1); cases (H (ap_symmetric ??? H1)); 
 qed.
 
 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).
@@ -99,23 +114,18 @@ 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 _ _ _).
+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;
+intros 5 (E x y Lxy Lyx); intro H; 
+cases (ap2exc ??? 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).
+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); 
@@ -125,43 +135,32 @@ 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)]
+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;
-[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)]]
+[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.
 
 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 unfold_apart: ∀E:excess. ∀x,y:E. x ≰ y ∨ y ≰ x → x # y.
-intros; assumption;
+intros (E a b Lab); elim Lab (LEab Aab);
+elim (ap2exc ??? 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; apply unfold_apart; right; assumption;
+intro Xyz; apply Exy; apply exc2ap; 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; apply unfold_apart; left; assumption;
+intro Xyz; apply Exy; apply exc2ap; left; assumption;
 qed.
 
-notation > "'Le'≪" non associative with precedence 50 for
- @{'lerewritel}.
- 
-interpretation "le_rewl" 'lerewritel = 
- (cic:/matita/excess/le_rewl.con _ _ _).
-
-notation > "'Le'≫" non associative with precedence 50 for
- @{'lerewriter}.
- 
-interpretation "le_rewr" 'lerewriter = 
- (cic:/matita/excess/le_rewr.con _ _ _).
+notation > "'Le'≪" non associative with precedence 50 for @{'lerewritel}.
+interpretation "le_rewl" 'lerewritel = (cic:/matita/excess/le_rewl.con _ _ _).
+notation > "'Le'≫" non associative with precedence 50 for @{'lerewriter}.
+interpretation "le_rewr" 'lerewriter = (cic:/matita/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]
@@ -173,77 +172,83 @@ 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/excess/ap_rewl.con _ _ _).
+notation > "'Ap'≫" non associative with precedence 50 for @{'aprewriter}.
+interpretation "ap_rewr" 'aprewriter = (cic:/matita/excess/ap_rewr.con _ _ _).
+
 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;
+cases Exy; apply exc2ap; 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;
+elim (Exy); apply exc2ap; left; assumption;
 qed.
 
-notation > "'Ex'≪" non associative with precedence 50 for
- @{'excessrewritel}.
- 
-interpretation "exc_rewl" 'excessrewritel = 
- (cic:/matita/excess/exc_rewl.con _ _ _).
-
-notation > "'Ex'≫" non associative with precedence 50 for
- @{'excessrewriter}.
- 
-interpretation "exc_rewr" 'excessrewriter = 
- (cic:/matita/excess/exc_rewr.con _ _ _).
+notation > "'Ex'≪" non associative with precedence 50 for @{'excessrewritel}.
+interpretation "exc_rewl" 'excessrewritel = (cic:/matita/excess/exc_rewl.con _ _ _).
+notation > "'Ex'≫" non associative with precedence 50 for @{'excessrewriter}.
+interpretation "exc_rewr" 'excessrewriter = (cic:/matita/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; elim H; 
-[apply (le_rewr ???? (eq_sym ??? E));|apply (ap_rewr ???? E)] assumption;
+[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; elim H; 
-[apply (le_rewl ???? (eq_sym ??? E));| apply (ap_rewl ???? E);] assumption;
+[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/excess/lt_rewl.con _ _ _).
-
-notation > "'Lt'≫" non associative with precedence 50 for
- @{'ltrewriter}.
- 
-interpretation "lt_rewr" 'ltrewriter = 
- (cic:/matita/excess/lt_rewr.con _ _ _).
+notation > "'Lt'≪" non associative with precedence 50 for @{'ltrewritel}.
+interpretation "lt_rewl" 'ltrewritel = (cic:/matita/excess/lt_rewl.con _ _ _).
+notation > "'Lt'≫" non associative with precedence 50 for @{'ltrewriter}.
+interpretation "lt_rewr" 'ltrewriter = (cic:/matita/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)]
-whd in LE LEx APx; cases APx (EXx EXx); [cases (LEx EXx)]
+apply exc2ap; cases (ap2exc ??? 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]
+elim (ap2exc ??? APx) (EXx EXx); [cases (LEx EXx)]
+elim (exc_cotransitive ??? x EXx) (EXz EXz); [apply exc2ap; 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;
+intros (E x y L1 L2); intro H; cases (ap2exc ??? 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.
+intros (E x y H); whd in H;
+split; intro; apply H; apply exc2ap; [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 ≝
+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.