From: Enrico Tassi <enrico.tassi@inria.fr>
Date: Tue, 14 Oct 2008 13:08:54 +0000 (+0000)
Subject: firs step for dualization
X-Git-Tag: make_still_working~4687
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=bf7f52019b3f65b6d635a8b49a63f0d95080f189;p=helm.git

firs step for dualization
---

diff --git a/helm/software/matita/contribs/dama/dama/bishop_set.ma b/helm/software/matita/contribs/dama/dama/bishop_set.ma
index 64ae4495d..68d213bd4 100644
--- a/helm/software/matita/contribs/dama/dama/bishop_set.ma
+++ b/helm/software/matita/contribs/dama/dama/bishop_set.ma
@@ -27,12 +27,12 @@ interpretation "bishop set apartness" 'apart x y = (bs_apart _ 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]
+[1: intro x; simplify; intro H; cases H; clear H;
+    apply (exc_coreflexive x H1);
+|2: intros 3 (x y H); simplify in H ⊢ %; cases H; [right|left]assumption; 
+|3: intros 4 (x y z H);  simplify in H ⊢ %; cases H; clear H;
+    [ cases (exc_cotransitive x y z H1); [left;left|right;left] assumption;
+    | cases (exc_cotransitive y x z H1); [right;right|left;right] assumption;]]
 qed.
 
 (* Definition 2.2 (2) *)
@@ -55,10 +55,10 @@ 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.
+coercion bishop_set_of_ordered_set.
 
 lemma le_antisymmetric: 
-  ∀E:ordered_set.antisymmetric E (le E) (eq E).
+  ∀E:ordered_set.antisymmetric E (le (os_l E)) (eq E).
 intros 5 (E x y Lxy Lyx); intro H; 
 cases H; [apply Lxy;|apply Lyx] assumption;
 qed.
@@ -67,7 +67,8 @@ lemma le_le_eq: ∀E:ordered_set.∀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.
 
-definition lt ≝ λE:ordered_set.λa,b:E. a ≤ b ∧ a # b.
+(*
+definition lt ≝ λE:half_ordered_set.λa,b:E. a ≤ b ∧ a # b.
 
 interpretation "ordered sets less than" 'lt a b = (lt _ a b).
 
@@ -78,16 +79,17 @@ 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;
+split; [apply (le_transitive E ??? 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 (os_cotransitive ??? y H1) (X X); [cases (Lxy X)|cases (os_coreflexive ?? X)]
-|2: cases (os_cotransitive ??? x H2) (X X); [right;assumption|cases (Lxy X)]]
+[1: cases (hos_cotransitive E ?? y H1) (X X); [cases (Lxy X)|cases (hos_coreflexive E ? X)]
+|2: cases (hos_cotransitive E ?? x H2) (X X); [right;assumption|cases (Lxy X)]]
 qed.
 
 theorem lt_to_excess: ∀E:ordered_set.∀a,b:E. (a < b) → (b ≰ a).
 intros (E a b Lab); cases Lab (LEab Aab); cases Aab (H H);[cases (LEab H)]
 assumption;
 qed.
+*)
 
 definition bs_subset ≝ λO:bishop_set.λP,Q:O→Prop.∀x:O.P x → Q x.
 
diff --git a/helm/software/matita/contribs/dama/dama/bishop_set_rewrite.ma b/helm/software/matita/contribs/dama/dama/bishop_set_rewrite.ma
index 7ead93370..27bb10f5a 100644
--- a/helm/software/matita/contribs/dama/dama/bishop_set_rewrite.ma
+++ b/helm/software/matita/contribs/dama/dama/bishop_set_rewrite.ma
@@ -55,12 +55,12 @@ notation > "'Ap'≫" non associative with precedence 50 for @{'aprewriter}.
 interpretation "ap_rewr" 'aprewriter = (ap_rewr _ _ _).
 
 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]
+intros (A x z y Exy Ayz); cases (hos_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]
+intros (A x z y Exy Azy); cases (hos_cotransitive ???x Azy); [assumption]
 cases (Exy); left; assumption;
 qed.
 
diff --git a/helm/software/matita/contribs/dama/dama/depends b/helm/software/matita/contribs/dama/dama/depends
index 2a59fb1aa..1d54328f5 100644
--- a/helm/software/matita/contribs/dama/dama/depends
+++ b/helm/software/matita/contribs/dama/dama/depends
@@ -18,11 +18,10 @@ russell_support.ma logic/cprop_connectives.ma nat/nat.ma
 models/q_copy.ma models/q_bars.ma
 models/nat_dedekind_sigma_complete.ma models/nat_uniform.ma nat/le_arith.ma russell_support.ma supremum.ma
 models/nat_lebesgue.ma lebesgue.ma models/nat_order_continuous.ma
-models/q_function.ma models/q_copy.ma russell_support.ma
 models/nat_uniform.ma models/discrete_uniformity.ma nat_ordered_set.ma
 uniform.ma supremum.ma
 ordered_uniform.ma uniform.ma
-models/q_rebase.ma Q/q/qtimes.ma models/q_function.ma
+models/q_rebase.ma Q/q/qtimes.ma models/q_copy.ma russell_support.ma
 Q/q/qplus.ma 
 Q/q/qtimes.ma 
 datatypes/constructors.ma 
diff --git a/helm/software/matita/contribs/dama/dama/depends.png b/helm/software/matita/contribs/dama/dama/depends.png
index 87c0beacb..8cbedcae0 100644
Binary files a/helm/software/matita/contribs/dama/dama/depends.png and b/helm/software/matita/contribs/dama/dama/depends.png differ
diff --git a/helm/software/matita/contribs/dama/dama/nat_ordered_set.ma b/helm/software/matita/contribs/dama/dama/nat_ordered_set.ma
index 231cdf941..d4aa1d57d 100644
--- a/helm/software/matita/contribs/dama/dama/nat_ordered_set.ma
+++ b/helm/software/matita/contribs/dama/dama/nat_ordered_set.ma
@@ -44,9 +44,10 @@ cases (nat_discriminable x z); [2: left; assumption] cases H1; clear H1;
 qed.
   
 lemma nat_ordered_set : ordered_set.
-apply (mk_ordered_set ? nat_excess);
-[1: intro x; intro; apply (not_le_Sn_n ? H);
-|2: apply nat_excess_cotransitive]
+letin hos ≝ (mk_half_ordered_set nat nat_excess ? nat_excess_cotransitive);[
+  intro x; intro H; apply (not_le_Sn_n ? H);]
+constructor 1;  
+[ apply hos; | apply (dual_hos hos); | reflexivity]
 qed.
 
 interpretation "ordered set N" 'N = nat_ordered_set.
@@ -54,7 +55,7 @@ interpretation "ordered set N" 'N = nat_ordered_set.
 alias id "le" = "cic:/matita/nat/orders/le.ind#xpointer(1/1)".
 lemma os_le_to_nat_le:
   ∀a,b:nat_ordered_set.a ≤ b → le a b.
-intros; normalize in H; apply (not_lt_to_le ?? H);
+intros; normalize in H; apply (not_lt_to_le b a H);
 qed.
  
 lemma nat_le_to_os_le:
@@ -62,6 +63,7 @@ lemma nat_le_to_os_le:
 intros 3; apply (le_to_not_lt a b);assumption;
 qed.
 
+(*
 lemma nat_lt_to_os_lt:
   ∀a,b:nat_ordered_set.a < b → lt nat_ordered_set a b.
 intros 3; split;
@@ -73,4 +75,5 @@ lemma os_lt_to_nat_lt:
   ∀a,b:nat_ordered_set. lt nat_ordered_set a b → a < b.
 intros; cases H; clear H; cases H2;
 [2: apply H;| cases (H1 H)]
-qed.
\ No newline at end of file
+qed.
+*)
\ No newline at end of file
diff --git a/helm/software/matita/contribs/dama/dama/ordered_set.ma b/helm/software/matita/contribs/dama/dama/ordered_set.ma
index 06c222830..8a1191442 100644
--- a/helm/software/matita/contribs/dama/dama/ordered_set.ma
+++ b/helm/software/matita/contribs/dama/dama/ordered_set.ma
@@ -15,51 +15,141 @@
 include "datatypes/constructors.ma".
 include "logic/cprop_connectives.ma".
 
+
+(* TEMPLATES
+notation "''" non associative with precedence 90 for @{'}.
+notation "''" non associative with precedence 90 for @{'}.
+
+interpretation "" ' = ( (os_l _)).
+interpretation "" ' = ( (os_r _)).
+*)
+
 (* 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 
+record half_ordered_set: Type ≝ {
+  hos_carr:> Type;
+  hos_excess: hos_carr → hos_carr → CProp;
+  hos_coreflexive: coreflexive ? hos_excess;
+  hos_cotransitive: cotransitive ? hos_excess 
+}.
+
+definition dual_hos : half_ordered_set → half_ordered_set.
+intro; constructor 1;
+[ apply (hos_carr h);
+| apply (λx,y.hos_excess h y x);
+| apply (hos_coreflexive h);
+| intros 4 (x y z H); simplify in H ⊢ %; cases (hos_cotransitive h ?? z H);
+  [right|left] assumption;]
+qed.
+
+record ordered_set : Type ≝ {
+  os_l : half_ordered_set;
+  os_r_ : half_ordered_set;
+  os_with : os_r_ = dual_hos os_l
 }.
 
-interpretation "Ordered set excess" 'nleq a b = (os_excess _ a b). 
+definition os_r : ordered_set → half_ordered_set.
+intro o; apply (dual_hos (os_l o)); qed.
+  
+definition Type_of_ordered_set : ordered_set → Type.
+intro o; apply (hos_carr (os_l o)); qed.
+
+definition Type_of_ordered_set_dual : ordered_set → Type.
+intro o; apply (hos_carr (os_r o)); qed.
+
+coercion Type_of_ordered_set_dual.
+coercion Type_of_ordered_set.
+
+notation "a ≰≰ b" non associative with precedence 45 for @{'nleq_low $a $b}.
+interpretation "Ordered half set excess" 'nleq_low a b = (hos_excess _ a b).
+
+interpretation "Ordered set excess (dual)" 'ngeq a b = (hos_excess (os_r _) a b).
+interpretation "Ordered set excess" 'nleq a b = (hos_excess (os_l _) a b).
+
+notation "'exc_coreflexive'" non associative with precedence 90 for @{'exc_coreflexive}.
+notation "'cxe_coreflexive'" non associative with precedence 90 for @{'cxe_coreflexive}.
+
+interpretation "exc_coreflexive" 'exc_coreflexive = (hos_coreflexive (os_l _)).
+interpretation "cxe_coreflexive" 'cxe_coreflexive = (hos_coreflexive (os_r _)).
+
+notation "'exc_cotransitive'" non associative with precedence 90 for @{'exc_cotransitive}.
+notation "'cxe_cotransitive'" non associative with precedence 90 for @{'cxe_cotransitive}.
+
+interpretation "exc_cotransitive" 'exc_cotransitive = (hos_cotransitive (os_l _)).
+interpretation "cxe_cotransitive" 'cxe_cotransitive = (hos_cotransitive (os_r _)).
 
 (* Definition 2.2 (3) *)
-definition le ≝ λE:ordered_set.λa,b:E. ¬ (a ≰ b).
+definition le ≝ λE:half_ordered_set.λa,b:E. ¬ (a ≰≰ b).
 
-interpretation "Ordered set greater or equal than" 'geq a b = (le _ b a).
+notation "hvbox(a break ≤≤ b)" non associative with precedence 45 for @{ 'leq_low $a $b }.
+interpretation "Ordered half set less or equal than" 'leq_low a b = (le _ a b).
 
-interpretation "Ordered set less or equal than" 'leq a b = (le _ a b).
+interpretation "Ordered set greater or equal than" 'geq a b = (le (os_r _) a b).
+interpretation "Ordered set less or equal than" 'leq a b = (le (os_l _) a b).
 
-lemma le_reflexive: ∀E.reflexive ? (le E).
-unfold reflexive; intros 3 (E x H); apply (os_coreflexive ?? H);
+lemma hle_reflexive: ∀E.reflexive ? (le E).
+unfold reflexive; intros 3 (E x H); apply (hos_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);
+notation "'le_reflexive'" non associative with precedence 90 for @{'le_reflexive}.
+notation "'ge_reflexive'" non associative with precedence 90 for @{'ge_reflexive}.
+
+interpretation "le reflexive" 'le_reflexive = (hle_reflexive (os_l _)).
+interpretation "ge reflexive" 'ge_reflexive = (hle_reflexive (os_r _)).
+
+(* DUALITY TESTS 
+lemma test_le_ge_convertible :∀o:ordered_set.∀x,y:o. x ≤ y → y ≥ x.
+intros; assumption; qed.
+
+lemma test_ge_reflexive :∀o:ordered_set.∀x:o. x ≥ x.
+intros; apply ge_reflexive. qed.
+
+lemma test_le_reflexive :∀o:ordered_set.∀x:o. x ≤ x.
+intros; apply le_reflexive. qed.
+*)
+
+lemma hle_transitive: ∀E.transitive ? (le E).
+unfold transitive; intros 7 (E x y z H1 H2 H3); cases (hos_cotransitive ??? y H3) (H4 H4);
 [cases (H1 H4)|cases (H2 H4)]
 qed.
 
+notation "'le_transitive'" non associative with precedence 90 for @{'le_transitive}.
+notation "'ge_transitive'" non associative with precedence 90 for @{'ge_transitive}.
+
+interpretation "le transitive" 'le_transitive = (hle_transitive (os_l _)).
+interpretation "ge transitive" 'ge_transitive = (hle_transitive (os_r _)).
+
 (* Lemma 2.3 *)
-lemma exc_le_variance: 
-  ∀O:ordered_set.∀a,b,a',b':O.a ≰ b → a ≤ a' → b' ≤ b → a' ≰ b'.
+lemma exc_hle_variance: 
+  ∀O:half_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 (hos_cotransitive ??? a1 Eab) (H H); [cases (Laa1 H)]
+cases (hos_cotransitive ??? b1 H) (H1 H1); [assumption]
 cases (Lb1b H1);
 qed.
 
-lemma square_ordered_set: ordered_set → ordered_set.
+notation "'exc_le_variance'" non associative with precedence 90 for @{'exc_le_variance}.
+notation "'exc_ge_variance'" non associative with precedence 90 for @{'exc_ge_variance}.
+
+interpretation "exc_le_variance" 'exc_le_variance = (exc_hle_variance (os_l _)).
+interpretation "exc_ge_variance" 'exc_ge_variance = (exc_hle_variance (os_r _)).
+
+lemma square_half_ordered_set: half_ordered_set → half_ordered_set.
 intro O;
-apply (mk_ordered_set (O × O));
-[1: intros (x y); apply (\fst x ≰ \fst y ∨ \snd x ≰ \snd y);
+apply (mk_half_ordered_set (O × O));
+[1: intros (x y); apply (\fst x ≰≰ \fst y ∨ \snd x ≰≰ \snd y);
 |2: intro x0; cases x0 (x y); clear x0; simplify; intro H;
-    cases H (X X); apply (os_coreflexive ?? X);
+    cases H (X X); apply (hos_coreflexive ?? X);
 |3: intros 3 (x0 y0 z0); cases x0 (x1 x2); cases y0 (y1 y2) ; cases z0 (z1 z2); 
     clear x0 y0 z0; simplify; intro H; cases H (H1 H1); clear H;
-    [1: cases (os_cotransitive ??? z1 H1); [left; left|right;left]assumption;
-    |2: cases (os_cotransitive ??? z2 H1); [left;right|right;right]assumption]]
+    [1: cases (hos_cotransitive ??? z1 H1); [left; left|right;left]assumption;
+    |2: cases (hos_cotransitive ??? z2 H1); [left;right|right;right]assumption]]
+qed.
+
+lemma square_ordered_set: ordered_set → ordered_set.
+intro O; constructor 1;
+[ apply (square_half_ordered_set (os_l O));
+| apply (dual_hos (square_half_ordered_set (os_l O)));
+| reflexivity]
 qed.
 
 notation "s 2 \atop \nleq" non associative with precedence 90
diff --git a/helm/software/matita/contribs/dama/dama/supremum.ma b/helm/software/matita/contribs/dama/dama/supremum.ma
index 6fa8e35ad..a3a341fea 100644
--- a/helm/software/matita/contribs/dama/dama/supremum.ma
+++ b/helm/software/matita/contribs/dama/dama/supremum.ma
@@ -19,16 +19,15 @@ include "nat_ordered_set.ma".
 include "sequence.ma".
 
 (* Definition 2.4 *)
-definition upper_bound ≝ λO:ordered_set.λa:sequence O.λu:O.∀n:nat.a n ≤ u.
-definition lower_bound ≝ λO:ordered_set.λa:sequence O.λu:O.∀n:nat.u ≤ a n.
+definition upper_bound ≝ 
+  λO:half_ordered_set.λa:sequence O.λu:O.∀n:nat.a n ≤≤ u.
 
 definition supremum ≝
-  λO:ordered_set.λs:sequence O.λx.upper_bound ? s x ∧ (∀y:O.x ≰ y → ∃n.s n ≰ y).
-definition infimum ≝
-  λO:ordered_set.λs:sequence O.λx.lower_bound ? s x ∧ (∀y:O.y ≰ x → ∃n.y ≰ s n).
+  λO:half_ordered_set.λs:sequence O.λx.
+    upper_bound ? s x ∧ (∀y:O.x ≰≰ y → ∃n.s n ≰≰ y).
 
-definition increasing ≝ λO:ordered_set.λa:sequence O.∀n:nat.a n ≤ a (S n).
-definition decreasing ≝ λO:ordered_set.λa:sequence O.∀n:nat.a (S n) ≤ a n.
+definition increasing ≝ 
+  λO:half_ordered_set.λa:sequence O.∀n:nat.a n ≤≤ a (S n).
 
 notation < "x \nbsp 'is_upper_bound' \nbsp s" non associative with precedence 45 
   for @{'upper_bound $s $x}.
@@ -42,7 +41,6 @@ notation < "x \nbsp 'is_supremum' \nbsp s" non associative with precedence 45
   for @{'supremum $s $x}.
 notation < "x \nbsp 'is_infimum' \nbsp s" non associative with precedence 45 
   for @{'infimum $s $x}.
-
 notation > "x 'is_upper_bound' s" non associative with precedence 45 
   for @{'upper_bound $s $x}.
 notation > "x 'is_lower_bound' s" non associative with precedence 45 
@@ -56,13 +54,16 @@ notation > "x 'is_supremum' s"  non associative with precedence 45
 notation > "x 'is_infimum' s"  non associative with precedence 45 
   for @{'infimum $s $x}.
 
-interpretation "Ordered set upper bound" 'upper_bound s x = (upper_bound _ s x).
-interpretation "Ordered set lower bound" 'lower_bound s x = (lower_bound _ s x).
-interpretation "Ordered set increasing"  'increasing s = (increasing _ s).
-interpretation "Ordered set decreasing"  'decreasing s = (decreasing _ s).
-interpretation "Ordered set strong sup"  'supremum s x = (supremum _ s x).
-interpretation "Ordered set strong inf"  'infimum s x = (infimum _ s x).
+interpretation "Ordered set upper bound" 'upper_bound s x = (upper_bound (os_l _) s x).
+interpretation "Ordered set lower bound" 'lower_bound s x = (upper_bound (os_r _) s x).
+
+interpretation "Ordered set increasing"  'increasing s = (increasing (os_l _) s).
+interpretation "Ordered set decreasing"  'decreasing s = (increasing (os_r _) s).
+
+interpretation "Ordered set strong sup"  'supremum s x = (supremum (os_l _) s x).
+interpretation "Ordered set strong inf"  'infimum s x = (supremum (os_r _) s x).
   
+(* se non faccio il bs_of_hos perdo dualità qui *)  
 lemma uniq_supremum: 
   ∀O:ordered_set.∀s:sequence O.∀t1,t2:O.
     t1 is_supremum s → t2 is_supremum s → t1 ≈ t2.
@@ -73,108 +74,107 @@ apply le_le_eq; intro X;
 qed.
 
 (* Fact 2.5 *)
-lemma supremum_is_upper_bound: 
-  ∀C:ordered_set.∀a:sequence C.∀u:C.
-   u is_supremum a → ∀v.v is_upper_bound a → u ≤ v.
+lemma h_supremum_is_upper_bound: 
+  ∀C:half_ordered_set.∀a:sequence C.∀u:C.
+   supremum ? a u → ∀v.upper_bound ? a v → u ≤≤ v.
 intros 7 (C s u Hu v Hv H); cases Hu (_ H1); clear Hu;
-cases (H1 ? H) (w Hw); apply Hv; assumption;
+cases (H1 ? H) (w Hw); apply Hv; [apply w] assumption;
 qed.
 
-lemma infimum_is_lower_bound: 
+notation "'supremum_is_upper_bound'" non associative with precedence 90 for @{'supremum_is_upper_bound}.
+notation "'infimum_is_lower_bound'" non associative with precedence 90 for @{'infimum_is_lower_bound}.
+
+interpretation "supremum_is_upper_bound" 'supremum_is_upper_bound = (h_supremum_is_upper_bound (os_l _)).
+interpretation "infimum_is_lower_bound" 'infimum_is_lower_bound = (h_supremum_is_upper_bound (os_r _)).
+
+(* TEST DUALITY
+lemma test_infimum_is_lower_bound_duality: 
   ∀C:ordered_set.∀a:sequence C.∀u:C.
-   u is_infimum a → ∀v.v is_lower_bound a → v ≤ u.
-intros 7 (C s u Hu v Hv H); cases Hu (_ H1); clear Hu;
-cases (H1 ? H) (w Hw); apply Hv; assumption;
+   u is_infimum a → ∀v.v is_lower_bound a → u ≥ v. 
+intros; lapply (infimum_is_lower_bound a u H v H1); assumption;
 qed.
+*)
 
 (* Lemma 2.6 *)
 definition strictly_increasing ≝ 
-  λC:ordered_set.λa:sequence C.∀n:nat.a (S n) ≰ a n.
-definition strictly_decreasing ≝ 
-  λC:ordered_set.λa:sequence C.∀n:nat.a n ≰ a (S n).
+  λC:half_ordered_set.λa:sequence C.∀n:nat.a (S n) ≰≰ a n.
 
 notation < "s \nbsp 'is_strictly_increasing'" non associative with precedence 45 
   for @{'strictly_increasing $s}.
 notation > "s 'is_strictly_increasing'" non associative with precedence 45 
   for @{'strictly_increasing $s}.
 interpretation "Ordered set strict increasing"  'strictly_increasing s    = 
-  (strictly_increasing _ s).
+  (strictly_increasing (os_l _) s).
+  
 notation < "s \nbsp 'is_strictly_decreasing'" non associative with precedence 45 
   for @{'strictly_decreasing $s}.
 notation > "s 'is_strictly_decreasing'" non associative with precedence 45 
   for @{'strictly_decreasing $s}.
 interpretation "Ordered set strict decreasing"  'strictly_decreasing s    = 
-  (strictly_decreasing _ s).
+  (strictly_increasing (os_r _) s).
 
 definition uparrow ≝
-  λC:ordered_set.λs:sequence C.λu:C.
-   s is_increasing ∧ u is_supremum s.
-   
-definition downarrow ≝
-  λC:ordered_set.λs:sequence C.λu:C.
-   s is_decreasing ∧ u is_infimum s.
-     
+  λC:half_ordered_set.λs:sequence C.λu:C.
+   increasing ? s ∧ supremum ? s u.
+(*   
 notation < "a \uparrow \nbsp u" non associative with precedence 45 for @{'sup_inc $a $u}.
 notation > "a \uparrow u" non associative with precedence 45 for @{'sup_inc $a $u}.
-interpretation "Ordered set uparrow" 'sup_inc s u = (uparrow _ s u).
+*)
+interpretation "Ordered set uparrow" 'funion s u = (uparrow (os_l _) s u).
 
+(*
 notation < "a \downarrow \nbsp u" non associative with precedence 45 for @{'inf_dec $a $u}.
 notation > "a \downarrow u" non associative with precedence 45 for @{'inf_dec $a $u}.
-interpretation "Ordered set downarrow" 'inf_dec s u = (downarrow _ s u).
+*)
+interpretation "Ordered set downarrow" 'fintersects s u = (uparrow (os_r _) s u).
 
-lemma trans_increasing: 
-  ∀C:ordered_set.∀a:sequence C.a is_increasing → 
-   ∀n,m:nat_ordered_set. n ≤ m → a n ≤ a m.
+lemma h_trans_increasing: 
+  ∀C:half_ordered_set.∀a:sequence C.increasing ? a → 
+   ∀n,m:nat_ordered_set. n ≤ m → a n ≤≤ a m.
 intros 5 (C a Hs n m); elim m; [
   rewrite > (le_n_O_to_eq n (not_lt_to_le O n H));
-  intro X; cases (os_coreflexive ?? X);]
+  intro X; cases (hos_coreflexive ?? X);]
 cases (le_to_or_lt_eq ?? (not_lt_to_le (S n1) n H1)); clear H1;
-[2: rewrite > H2; intro; cases (os_coreflexive ?? H1);
-|1: apply (le_transitive ???? (H ?) (Hs ?));
+[2: rewrite > H2; intro; cases (hos_coreflexive ?? H1);
+|1: apply (hle_transitive ???? (H ?) (Hs ?));
     intro; whd in H1; apply (not_le_Sn_n n); apply (transitive_le ??? H2 H1);]
 qed.
 
-lemma trans_decreasing: 
+notation "'trans_increasing'" non associative with precedence 90 for @{'trans_increasing}.
+notation "'trans_decreasing'" non associative with precedence 90 for @{'trans_decreasing}.
+
+interpretation "trans_increasing" 'trans_increasing = (h_trans_increasing (os_l _)).
+interpretation "trans_decreasing" 'trans_decreasing = (h_trans_increasing (os_r _)).
+
+(* TEST DUALITY
+lemma test_trans_decreasing_duality: 
   ∀C:ordered_set.∀a:sequence C.a is_decreasing → 
    ∀n,m:nat_ordered_set. n ≤ m → a m ≤ a n.
-intros 5 (C a Hs n m); elim m; [
-  rewrite > (le_n_O_to_eq n (not_lt_to_le O n H));
-  intro X; cases (os_coreflexive ?? X);]
-cases (le_to_or_lt_eq ?? (not_lt_to_le (S n1) n H1)); clear H1;
-[2: rewrite > H2; intro; cases (os_coreflexive ?? H1);
-|1: apply (le_transitive ???? (Hs ?) (H ?));
-    intro; whd in H1; apply (not_le_Sn_n n); apply (transitive_le ??? H2 H1);]
-qed.
+intros; apply (trans_decreasing ? H ?? H1); qed.
+*)
 
-lemma trans_increasing_exc: 
-  ∀C:ordered_set.∀a:sequence C.a is_increasing → 
-   ∀n,m:nat_ordered_set. m ≰ n → a n ≤ a m.
+lemma h_trans_increasing_exc: 
+  ∀C:half_ordered_set.∀a:sequence C.increasing ? a → 
+   ∀n,m:nat_ordered_set. m ≰ n → a n ≤≤ a m.
 intros 5 (C a Hs n m); elim m; [cases (not_le_Sn_O n H);]
 intro; apply H; 
-[1: change in n1 with (os_carr nat_ordered_set); (* canonical structures *) 
-    change with (n<n1); (* is sort elimination not allowed preserved by delta? *)
+[1: change in n1 with (hos_carr (os_l nat_ordered_set)); 
+    change with (n<n1);
     cases (le_to_or_lt_eq ?? H1); [apply le_S_S_to_le;assumption]
-    cases (Hs n); rewrite < H3 in H2; assumption (* ogni goal di tipo Prop non è anche di tipo CProp *)    
-|2: cases (os_cotransitive ??? (a n1) H2); [assumption]
+    cases (Hs n); rewrite < H3 in H2; assumption;    
+|2: cases (hos_cotransitive ??? (a n1) H2); [assumption]
     cases (Hs n1); assumption;]
 qed.
 
-lemma trans_decreasing_exc: 
-  ∀C:ordered_set.∀a:sequence C.a is_decreasing →
-   ∀n,m:nat_ordered_set. m ≰ n → a m ≤ a n .
-intros 5 (C a Hs n m); elim m; [cases (not_le_Sn_O n H);]
-intro; apply H; 
-[1: change in n1 with (os_carr nat_ordered_set); (* canonical structures *) 
-    change with (n<n1); (* is sort elimination not allowed preserved by delta? *)
-    cases (le_to_or_lt_eq ?? H1); [apply le_S_S_to_le;assumption]
-    cases (Hs n); rewrite < H3 in H2; assumption (* ogni goal di tipo Prop non è anche di tipo CProp *)    
-|2: cases (os_cotransitive ??? (a n1) H2); [2:assumption]
-    cases (Hs n1); assumption;]
-qed.
+notation "'trans_increasing_exc'" non associative with precedence 90 for @{'trans_increasing_exc}.
+notation "'trans_decreasing_exc'" non associative with precedence 90 for @{'trans_decreasing_exc}.
+
+interpretation "trans_increasing_exc" 'trans_increasing_exc = (h_trans_increasing_exc (os_l _)).
+interpretation "trans_decreasing_exc" 'trans_decreasing_exc = (h_trans_increasing_exc (os_r _)).
 
 alias symbol "exists" = "CProp exists".
-lemma strictly_increasing_reaches: 
-  ∀C:ordered_set.∀m:sequence nat_ordered_set.
+lemma nat_strictly_increasing_reaches: 
+  ∀m:sequence nat_ordered_set.
    m is_strictly_increasing → ∀w.∃t.m t ≰ w.
 intros; elim w;
 [1: cases (nat_discriminable O (m O)); [2: cases (not_le_Sn_n O (ltn_to_ltO ?? H1))]
@@ -189,29 +189,24 @@ intros; elim w;
         apply (transitive_le ??? H2 Hp);]]
 qed.
      
-lemma selection_uparrow: 
-  ∀C:ordered_set.∀m:sequence nat_ordered_set.m is_strictly_increasing →
-    ∀a:sequence C.∀u.a ↑ u → ⌊x,a (m x)⌋ ↑ u.
+lemma h_selection_uparrow: 
+  ∀C:half_ordered_set.∀m:sequence nat_ordered_set.
+   m is_strictly_increasing →
+    ∀a:sequence C.∀u.uparrow ? a u → uparrow ? ⌊x,a (m x)⌋ u.
 intros (C m Hm a u Ha); cases Ha (Ia Su); cases Su (Uu Hu); repeat split; 
-[1: intro n; simplify; apply trans_increasing_exc; [assumption] apply (Hm n);
+[1: intro n; simplify; apply (h_trans_increasing_exc ? a Ia); apply (Hm n);
 |2: intro n; simplify; apply Uu;
 |3: intros (y Hy); simplify; cases (Hu ? Hy);
-    cases (strictly_increasing_reaches C ? Hm w); 
-    exists [apply w1]; cases (os_cotransitive ??? (a (m w1)) H); [2:assumption]  
-    cases (trans_increasing_exc C ? Ia ?? H1); assumption;]
+    cases (nat_strictly_increasing_reaches ? Hm w); 
+    exists [apply w1]; cases (hos_cotransitive ??? (a (m w1)) H); [2:assumption]  
+    cases (h_trans_increasing_exc ?? Ia ?? H1); assumption;]
 qed.     
 
-lemma selection_downarrow: 
-  ∀C:ordered_set.∀m:sequence nat_ordered_set.m is_strictly_increasing →
-    ∀a:sequence C.∀u.a ↓ u → ⌊x,a (m x)⌋ ↓ u.
-intros (C m Hm a u Ha); cases Ha (Ia Su); cases Su (Uu Hu); repeat split; 
-[1: intro n; simplify; apply trans_decreasing_exc; [assumption] apply (Hm n);
-|2: intro n; simplify; apply Uu;
-|3: intros (y Hy); simplify; cases (Hu ? Hy);
-    cases (strictly_increasing_reaches C ? Hm w); 
-    exists [apply w1]; cases (os_cotransitive ??? (a (m w1)) H); [assumption]  
-    cases (trans_decreasing_exc C ? Ia ?? H1); assumption;]
-qed.
+notation "'selection_uparrow'" non associative with precedence 90 for @{'selection_uparrow}.
+notation "'selection_downarrow'" non associative with precedence 90 for @{'selection_downarrow}.
+
+interpretation "selection_uparrow" 'selection_uparrow = (h_selection_uparrow (os_l _)).
+interpretation "selection_downarrow" 'selection_downarrow = (h_selection_uparrow (os_r _)).
 
 (* Definition 2.7 *)
 definition order_converge ≝
@@ -227,33 +222,52 @@ notation > "a 'order_converges' x" non associative with precedence 45
 interpretation "Order convergence" 'order_converge s u = (order_converge _ s u).   
     
 (* Definition 2.8 *)
-definition segment ≝ λO:ordered_set.λa,b:O.λx:O.(x ≤ b) ∧ (a ≤ x).
+definition segment ≝ λO:half_ordered_set.λa,b:O.λx:O.(x ≤≤ b) ∧ (a ≤≤ x).
 
 notation "[term 19 a,term 19 b]" non associative with precedence 90 for @{'segment $a $b}.
-interpretation "Ordered set sergment" 'segment a b = (segment _ a b).
+interpretation "Ordered set sergment" 'segment a b = (segment (os_l _) a b).
 
 notation "hvbox(x \in break [term 19 a, term 19 b])" non associative with precedence 45 
   for @{'segment_in $a $b $x}.
-interpretation "Ordered set sergment in" 'segment_in a b x= (segment _ a b x).
+interpretation "Ordered set sergment in" 'segment_in a b x= (segment (os_l _) a b x).
+
+definition segment_ordered_set_carr ≝
+  λO:half_ordered_set.λu,v:O.∃x.segment ? u v x.
+definition segment_ordered_set_exc ≝ 
+  λO:half_ordered_set.λu,v:O.
+   λx,y:segment_ordered_set_carr ? u v.\fst x ≰≰ \fst y.
+lemma segment_ordered_set_corefl:
+ ∀O,u,v. coreflexive ? (segment_ordered_set_exc O u v).
+intros 4; cases x; simplify; apply hos_coreflexive; qed.
+lemma segment_ordered_set_cotrans : 
+  ∀O,u,v. cotransitive ? (segment_ordered_set_exc O u v).
+intros 6 (O u v x y z); cases x; cases y ; cases z; simplify; apply hos_cotransitive;
+qed.  
+  
+lemma half_segment_ordered_set: 
+  ∀O:half_ordered_set.∀u,v:O.half_ordered_set.
+intros (O u v); apply (mk_half_ordered_set ?? (segment_ordered_set_corefl O u v) (segment_ordered_set_cotrans ???));
+qed.
 
 lemma segment_ordered_set: 
   ∀O:ordered_set.∀u,v:O.ordered_set.
-intros (O u v); apply (mk_ordered_set (∃x.x ∈ [u,v]));
-[1: intros (x y); apply (\fst x ≰ \fst y);
-|2: intro x; cases x; simplify; apply os_coreflexive;
-|3: intros 3 (x y z); cases x; cases y ; cases z; simplify; apply os_cotransitive]
+intros (O u v); letin hos ≝ (half_segment_ordered_set (os_l O) u v);
+constructor 1; [apply hos; | apply (dual_hos hos); | reflexivity] 
 qed.
 
 notation "hvbox({[a, break b]})" non associative with precedence 90 
   for @{'segment_set $a $b}.
+interpretation "Ordered set segment" 'segment_set a b = 
+  (half_segment_ordered_set _ a b).
 interpretation "Ordered set segment" 'segment_set a b = 
   (segment_ordered_set _ a b).
 
+
 (* Lemma 2.9 *)
-lemma segment_preserves_supremum:
-  ∀O:ordered_set.∀l,u:O.∀a:sequence {[l,u]}.∀x:{[l,u]}. 
-    ⌊n,\fst (a n)⌋ is_increasing ∧ 
-    (\fst x) is_supremum ⌊n,\fst (a n)⌋ → a ↑ x.
+lemma h_segment_preserves_supremum:
+  ∀O:half_ordered_set.∀l,u:O.∀a:sequence {[l,u]}.∀x:{[l,u]}. 
+    increasing ? ⌊n,\fst (a n)⌋ ∧ 
+    supremum ? ⌊n,\fst (a n)⌋ (\fst x) → uparrow ? a x.
 intros; split; cases H; clear H; 
 [1: apply H1;
 |2: cases H2; split; clear H2;
@@ -261,23 +275,18 @@ intros; split; cases H; clear H;
     |2: clear H; intro y0; apply (H3 (\fst y0));]]
 qed.
 
-lemma segment_preserves_infimum:
-  ∀O:ordered_set.∀l,u:O.∀a:sequence {[l,u]}.∀x:{[l,u]}. 
-    ⌊n,\fst (a n)⌋ is_decreasing ∧ 
-    (\fst x) is_infimum ⌊n,\fst (a n)⌋ → a ↓ x.
-intros; split; cases H; clear H; 
-[1: apply H1;
-|2: cases H2; split; clear H2;
-    [1: apply H;
-    |2: clear H; intro y0; apply (H3 (\fst y0));]]
-qed.
+notation "'segment_preserves_supremum'" non associative with precedence 90 for @{'segment_preserves_supremum}.
+notation "'segment_preserves_infimum'" non associative with precedence 90 for @{'segment_preserves_infimum}.
+
+interpretation "segment_preserves_supremum" 'segment_preserves_supremum = (h_segment_preserves_supremum (os_l _)).
+interpretation "segment_preserves_infimum" 'segment_preserves_infimum = (h_segment_preserves_supremum (os_r _)).
 
 (* Definition 2.10 *)
 alias symbol "square" = "ordered set square".
 alias symbol "pi2" = "pair pi2".
 alias symbol "pi1" = "pair pi1".
 definition square_segment ≝ 
-  λO:ordered_set.λa,b:O.λx:O square.
+  λO:ordered_set.λa,b:O.λx: O square.
     And4 (\fst x ≤ b) (a ≤ \fst x) (\snd x ≤ b) (a ≤ \snd x).
  
 definition convex ≝
@@ -286,40 +295,34 @@ definition convex ≝
   
 (* Definition 2.11 *)  
 definition upper_located ≝
-  λO:ordered_set.λa:sequence O.∀x,y:O. y ≰ x → 
-    (∃i:nat.a i ≰ x) ∨ (∃b:O.y≰b ∧ ∀i:nat.a i ≤ b).
-
-definition lower_located ≝
-  λO:ordered_set.λa:sequence O.∀x,y:O. x ≰ y → 
-    (∃i:nat.x ≰ a i) ∨ (∃b:O.b≰y ∧ ∀i:nat.b ≤ a i).
+  λO:half_ordered_set.λa:sequence O.∀x,y:O. y ≰≰ x → 
+    (∃i:nat.a i ≰≰ x) ∨ (∃b:O.y ≰≰ b ∧ ∀i:nat.a i ≤≤ b).
 
 notation < "s \nbsp 'is_upper_located'" non associative with precedence 45 
   for @{'upper_located $s}.
 notation > "s 'is_upper_located'" non associative with precedence 45 
   for @{'upper_located $s}.
 interpretation "Ordered set upper locatedness" 'upper_located s = 
-  (upper_located _ s).
+  (upper_located (os_l _) s).
 
 notation < "s \nbsp 'is_lower_located'" non associative with precedence 45 
   for @{'lower_located $s}.
 notation > "s 'is_lower_located'" non associative with precedence 45
   for @{'lower_located $s}.
 interpretation "Ordered set lower locatedness" 'lower_located s = 
-  (lower_located _ s).
-    
+  (upper_located (os_r _) s).
+      
 (* Lemma 2.12 *)    
-lemma uparrow_upperlocated:
-  ∀C:ordered_set.∀a:sequence C.∀u:C.a ↑ u → a is_upper_located.
+lemma h_uparrow_upperlocated:
+  ∀C:half_ordered_set.∀a:sequence C.∀u:C.uparrow ? a u → upper_located ? a.
 intros (C a u H); cases H (H2 H3); clear H; intros 3 (x y Hxy);
-cases H3 (H4 H5); clear H3; cases (os_cotransitive ??? u Hxy) (W W);
+cases H3 (H4 H5); clear H3; cases (hos_cotransitive ??? u Hxy) (W W);
 [2: cases (H5 ? W) (w Hw); left; exists [apply w] assumption;
 |1: right; exists [apply u]; split; [apply W|apply H4]]
-qed. 
+qed.
 
-lemma downarrow_lowerlocated:
-  ∀C:ordered_set.∀a:sequence C.∀u:C.a ↓ u → a is_lower_located.
-intros (C a u H); cases H (H2 H3); clear H; intros 3 (x y Hxy);
-cases H3 (H4 H5); clear H3; cases (os_cotransitive ??? u Hxy) (W W);
-[1: cases (H5 ? W) (w Hw); left; exists [apply w] assumption;
-|2: right; exists [apply u]; split; [apply W|apply H4]]
-qed. 
+notation "'uparrow_upperlocated'" non associative with precedence 90 for @{'uparrow_upperlocated}.
+notation "'downarrow_lowerlocated'" non associative with precedence 90 for @{'downarrow_lowerlocated}.
+
+interpretation "uparrow_upperlocated" 'uparrow_upperlocated = (h_uparrow_upperlocated (os_l _)).
+interpretation "downarrow_lowerlocated" 'downarrow_lowerlocated = (h_uparrow_upperlocated (os_r _)).