From 80ce2b7d10578bf19408c9d98dd287a5b9d9c982 Mon Sep 17 00:00:00 2001
From: Enrico Tassi <enrico.tassi@inria.fr>
Date: Mon, 12 Nov 2007 15:22:43 +0000
Subject: [PATCH] ordered set is over, much new stuff coming from a
 coreflexivee/cotransitive execedence relation.

---
 matita/dama/constructive_connectives.ma |  18 ++--
 matita/dama/ordered_sets.ma             | 137 +++++++++++++++++++++---
 2 files changed, 133 insertions(+), 22 deletions(-)

diff --git a/matita/dama/constructive_connectives.ma b/matita/dama/constructive_connectives.ma
index b7f15e500..2cf0d8d58 100644
--- a/matita/dama/constructive_connectives.ma
+++ b/matita/dama/constructive_connectives.ma
@@ -14,14 +14,20 @@
 
 set "baseuri" "cic:/matita/constructive_connectives/".
 
-inductive or (A,B:Type) : Type \def
-   Left : A → or A B
- | Right : B → or A B.
+inductive Or (A,B:Type) : Type ≝
+   Left : A → Or A B
+ | Right : B → Or A B.
 
 interpretation "constructive or" 'or x y =
-  (cic:/matita/constructive_connectives/or.ind#xpointer(1/1) x y).
+  (cic:/matita/constructive_connectives/Or.ind#xpointer(1/1) x y).
 
-inductive ex (A:Type) (P:A→Prop) : Type \def
+inductive And (A,B:Type) : Type ≝
+ | Conj : A → B → And A B.
+ 
+interpretation "constructive and" 'and x y =
+  (cic:/matita/constructive_connectives/And.ind#xpointer(1/1) x y).
+
+inductive ex (A:Type) (P:A→Prop) : Type ≝
   ex_intro: ∀w:A. P w → ex A P.
 
 notation < "hvbox(Σ ident i opt (: ty) break . p)"
@@ -34,7 +40,7 @@ interpretation "constructive exists" 'sigma \eta.x =
   (cic:/matita/constructive_connectives/ex.ind#xpointer(1/1) _ x).
 
 alias id "False" = "cic:/matita/logic/connectives/False.ind#xpointer(1/1)".
-definition Not ≝ λx:Type.False.
+definition Not ≝ λx:Type.x → False.
 
 interpretation "constructive not" 'not x = 
   (cic:/matita/constructive_connectives/Not.con x).
\ No newline at end of file
diff --git a/matita/dama/ordered_sets.ma b/matita/dama/ordered_sets.ma
index dcae29e18..b8c6952c7 100644
--- a/matita/dama/ordered_sets.ma
+++ b/matita/dama/ordered_sets.ma
@@ -17,27 +17,132 @@ set "baseuri" "cic:/matita/ordered_sets/".
 include "higher_order_defs/relations.ma".
 include "nat/plus.ma".
 include "constructive_connectives.ma".
+include "constructive_higher_order_relations.ma".
 
-definition cotransitive ≝
- λC:Type.λle:C→C→Prop.∀x,y,z:C. le x y → le x z ∨ le z y. 
+record excedence : Type ≝ {
+  exc_carr:> Type;
+  exc_relation: exc_carr → exc_carr → Prop;
+  exc_coreflexive: coreflexive ? exc_relation;
+  exc_cotransitive: cotransitive ? exc_relation 
+}.
 
-definition antisimmetric ≝
- λC:Type.λle:C→C→Prop.∀x,y:C. le x y → le y x → x=y.
+interpretation "excedence" 'nleq a b =
+ (cic:/matita/ordered_sets/exc_relation.con _ a b). 
 
-record is_order_relation (C:Type) (le:C→C→Prop) : Type ≝
- { or_reflexive: reflexive ? le;
-   or_transitive: transitive ? le;
-   or_antisimmetric: antisimmetric ? le
- }.
+definition le ≝ λE:excedence.λa,b:E. ¬ (a ≰ b).
 
-record ordered_set: Type ≝
- { os_carrier:> Type;
-   os_le: os_carrier → os_carrier → Prop;
-   os_order_relation_properties:> is_order_relation ? os_le
- }.
+interpretation "ordered sets less or equal than" 'leq a b = 
+ (cic:/matita/ordered_sets/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.
+
+definition apart ≝ λE:excedence.λa,b:E. a ≰ b ∨ b ≰ a.
+
+notation "a # b" non associative with precedence 50 for @{ 'apart $a $b}.
+interpretation "apartness" 'apart a b = (cic:/matita/ordered_sets/apart.con _ a b). 
+
+lemma apart_coreflexive: ∀E.coreflexive ? (apart E).
+intros (E); unfold; cases E; simplify; clear E; intros (x); unfold;
+intros (H1); apply (H x); cases H1; assumption;
+qed.
+
+lemma apart_symmetric: ∀E.symmetric ? (apart E).
+intros (E); unfold; intros(x y H); cases H; clear H; [right|left] assumption; 
+qed.
+
+lemma apart_cotrans: ∀E. cotransitive ? (apart E).
+intros (E); unfold; cases E (T f _ cTf); simplify; intros (x y z Axy);
+cases Axy (H); lapply (cTf ? ? z H) as H1; cases H1; clear Axy H1;
+[left; left|right; left|right; right|left; right] assumption.
+qed.
+
+definition eq ≝ λE:excedence.λa,b:E. ¬ (a # b).
+
+notation "a ≈ b" non associative with precedence 50 for @{ 'napart $a $b}.    
+interpretation "alikeness" 'napart a b =
+  (cic:/matita/ordered_sets/eq.con _ a b). 
+
+lemma eq_reflexive:∀E. reflexive ? (eq E).
+intros (E); unfold; cases E (T f cRf _); simplify; unfold Not; intros (x H);
+apply (cRf x); cases H; assumption;
+qed.
+
+lemma eq_symmetric:∀E.symmetric ? (eq E).
+intros (E); unfold; unfold eq; unfold Not;
+intros (x y H1 H2); apply H1; cases H2; [right|left] assumption; 
+qed.
+
+lemma eq_transitive: ∀E.transitive ? (eq E).
+intros (E); unfold; cases E (T f _ cTf); simplify; unfold Not; 
+intros (x y z H1 H2 H3); cases H3 (H4 H4); clear E H3; lapply (cTf ? ? y H4) as H5;
+cases H5; clear H5 H4 cTf; [1,4: apply H1|*:apply H2] clear H1 H2;
+[1,3:left|*:right] assumption;
+qed.
+
+lemma le_antisymmetric: ∀E.antisymmetric ? (le E) (eq E).
+intros (E); unfold; intros (x y Lxy Lyx); unfold; unfold; intros (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/ordered_sets/lt.con _ a b).
+
+lemma lt_coreflexive: ∀E.coreflexive ? (lt E).
+intros (E); unfold; unfold Not; intros (x H); cases H (_ ABS); 
+apply (apart_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 mah: ∀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.
+
+-- altro file
+opposto TH è assioma per ordine totale.
+
+-- 
+
+
+
+
+
+
+
+record is_order_relation (C:Type) (le:C→C→Prop) (eq:C→C→Prop) : Type ≝ { 
+  or_reflexive: reflexive ? le;
+  or_transitive: transitive ? le;
+  or_antisimmetric: antisymmetric ? le eq
+}.
+
+record ordered_set: Type ≝ { 
+  os_carr:> excedence;
+  os_order_relation_properties:> is_order_relation ? (le os_carr) (apart os_carr)
+}.
+
+ordered_set.
+
+E
 
-interpretation "Ordered Sets le" 'leq a b =
- (cic:/matita/ordered_sets/os_le.con _ a b).
+E
 
 theorem antisimmetric_to_cotransitive_to_transitive:
  ∀C.∀le:relation C. antisimmetric ? le → cotransitive ? le →
-- 
2.39.2