(* *)
(**************************************************************************)
+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;
+ wloss: ∀A:Type. (A → A → CProp) → A → A → CProp;
+ wloss_prop: (∀T,P,x,y.P x y = wloss T P x y) ∨ (∀T,P,x,y.P y x = wloss T P x y);
+ hos_excess_: hos_carr → hos_carr → CProp;
+ hos_coreflexive: coreflexive ? (wloss ? hos_excess_);
+ hos_cotransitive: cotransitive ? (wloss ? hos_excess_)
+}.
+
+definition hos_excess ≝ λO:half_ordered_set.wloss O ? (hos_excess_ O).
+
+(*
+lemma find_leq : half_ordered_set → half_ordered_set.
+intro O; constructor 1;
+[1: apply (hos_carr O);
+|2: apply (λT:Type.λf:T→T→CProp.f);
+|3: intros; left; intros; reflexivity;
+|4: apply (hos_excess_ O);
+|5: intro x; lapply (hos_coreflexive O x) as H; cases (wloss_prop O);
+ rewrite < H1 in H; apply H;
+|6: intros 4 (x y z H); cases (wloss_prop O);
+ rewrite > (H1 ? (hos_excess_ O)) in H ⊢ %;
+ rewrite > (H1 ? (hos_excess_ O)); lapply (hos_cotransitive O ?? z H);
+ [assumption] cases Hletin;[right|left]assumption;]
+qed.
+*)
+
+definition dual_hos : half_ordered_set → half_ordered_set.
+intro; constructor 1;
+[ apply (hos_carr h);
+| apply (λT,f,x,y.wloss h T f y x);
+| intros; cases (wloss_prop h);[right|left]intros;apply H;
+| apply (hos_excess_ h);
+| apply (hos_coreflexive h);
+| intros 4 (x y z H); simplify in H ⊢ %; cases (hos_cotransitive h y x 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.
+
+lemma half2full : half_ordered_set → ordered_set.
+intro hos;
+constructor 1; [apply hos; | apply (dual_hos hos); | reflexivity]
+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 "Half ordered set greater 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; apply (hos_coreflexive ? x 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 E x z 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 ? a b 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 _)).
+
+definition square_exc ≝
+ λO:half_ordered_set.λx,y:O×O.\fst x ≰≰ \fst y ∨ \snd x ≰≰ \snd y.
+
+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);
-|2: intro x0; cases x0 (x y); clear x0; simplify; intro H;
- cases H (X X); apply (os_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]]
+apply (mk_half_ordered_set (O × O));
+[1: apply (wloss O);
+|2: intros; cases (wloss_prop O); [left|right] intros; apply H;
+|3: apply (square_exc O);
+|4: intro x; cases (wloss_prop O); rewrite < (H ? (square_exc O) x x); clear H;
+ cases x; clear x; unfold square_exc; intro H; cases H; clear H; simplify in H1;
+ [1,3: apply (hos_coreflexive O h H1);
+ |*: apply (hos_coreflexive O h1 H1);]
+|5: intros 3 (x0 y0 z0); cases (wloss_prop O);
+ do 3 rewrite < (H ? (square_exc O)); clear H; cases x0; cases y0; cases z0; clear x0 y0 z0;
+ simplify; intro H; cases H; clear H;
+ [1: cases (hos_cotransitive ? h h2 h4 H1); [left;left|right;left] assumption;
+ |2: cases (hos_cotransitive ? h1 h3 h5 H1); [left;right|right;right] assumption;
+ |3: cases (hos_cotransitive ? h2 h h4 H1); [right;left|left;left] assumption;
+ |4: cases (hos_cotransitive ? h3 h1 h5 H1); [right;right|left;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
for @{ 'square_os $s }.
-notation > "s 'square'" non associative with precedence 90
- for @{ 'square $s }.
-interpretation "ordered set square" 'square s = (square_ordered_set s).
+notation > "s 'squareO'" non associative with precedence 90
+ for @{ 'squareO $s }.
+interpretation "ordered set square" 'squareO s = (square_ordered_set s).
interpretation "ordered set square" 'square_os s = (square_ordered_set s).
definition os_subset ≝ λO:ordered_set.λP,Q:O→Prop.∀x:O.P x → Q x.
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