(* Definition 2.1 *)
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
+ 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 (λx,y.hos_excess h y x);
+| 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 ?? z H);
+| intros 4 (x y z H); simplify in H ⊢ %; cases (hos_cotransitive h y x z H);
[right|left] assumption;]
qed.
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.
+
+(* coercion half2full. *)
definition Type_of_ordered_set : ordered_set → Type.
intro o; apply (hos_carr (os_l o)); qed.
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 _)).
+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 _)).
+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:half_ordered_set.λa,b:E. ¬ (a ≰≰ b).
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 "Half ordered set greater or equal than" 'leq_low 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).
+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 hle_reflexive: ∀E.reflexive ? (le E).
-unfold reflexive; intros 3 (E x H); apply (hos_coreflexive ?? H);
+unfold reflexive; intros 3; apply (hos_coreflexive ? x H);
qed.
notation "'le_reflexive'" non associative with precedence 90 for @{'le_reflexive}.
*)
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);
+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.
(* Lemma 2.3 *)
lemma exc_hle_variance:
- ∀O:half_ordered_set.∀a,b,a',b':O.a ≰≰ b → a ≤≤ a' → b' ≤≤ b → a' ≰≰ b'.
+ ∀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 (hos_cotransitive ??? a1 Eab) (H H); [cases (Laa1 H)]
-cases (hos_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.
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_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 (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 (hos_cotransitive ??? z1 H1); [left; left|right;left]assumption;
- |2: cases (hos_cotransitive ??? z2 H1); [left;right|right;right]assumption]]
+[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.