1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "datatypes/constructors.ma".
16 include "logic/cprop_connectives.ma".
20 notation "''" non associative with precedence 90 for @{'}.
21 notation "''" non associative with precedence 90 for @{'}.
23 interpretation "" ' = ( (os_l _)).
24 interpretation "" ' = ( (os_r _)).
28 record half_ordered_set: Type ≝ {
30 hos_excess: hos_carr → hos_carr → CProp;
31 hos_coreflexive: coreflexive ? hos_excess;
32 hos_cotransitive: cotransitive ? hos_excess
35 definition dual_hos : half_ordered_set → half_ordered_set.
38 | apply (λx,y.hos_excess h y x);
39 | apply (hos_coreflexive h);
40 | intros 4 (x y z H); simplify in H ⊢ %; cases (hos_cotransitive h ?? z H);
41 [right|left] assumption;]
44 record ordered_set : Type ≝ {
45 os_l_ : half_ordered_set;
46 os_r_ : half_ordered_set;
47 os_with : os_r_ = dual_hos os_l_
50 definition os_l : ordered_set → half_ordered_set.
51 intro h; constructor 1;
52 [ apply (hos_carr (os_l_ h));
53 | apply (λx,y.hos_excess (os_l_ h) x y);
54 | apply (hos_coreflexive (os_l_ h));
55 | apply (hos_cotransitive (os_l_ h));
59 definition os_r : ordered_set → half_ordered_set.
60 intro o; apply (dual_hos (os_l_ o)); qed.
62 lemma half2full : half_ordered_set → ordered_set.
64 constructor 1; [apply hos; | apply (dual_hos hos); | reflexivity]
67 (* coercion half2full. *)
69 definition Type_of_ordered_set : ordered_set → Type.
70 intro o; apply (hos_carr (os_l o)); qed.
72 definition Type_of_ordered_set_dual : ordered_set → Type.
73 intro o; apply (hos_carr (os_r o)); qed.
75 coercion Type_of_ordered_set_dual.
76 coercion Type_of_ordered_set.
78 notation "a ≰≰ b" non associative with precedence 45 for @{'nleq_low $a $b}.
79 interpretation "Ordered half set excess" 'nleq_low a b = (hos_excess _ a b).
81 interpretation "Ordered set excess (dual)" 'ngeq a b = (hos_excess (os_r _) a b).
82 interpretation "Ordered set excess" 'nleq a b = (hos_excess (os_l _) a b).
84 notation "'exc_coreflexive'" non associative with precedence 90 for @{'exc_coreflexive}.
85 notation "'cxe_coreflexive'" non associative with precedence 90 for @{'cxe_coreflexive}.
87 interpretation "exc_coreflexive" 'exc_coreflexive = (hos_coreflexive (os_l _)).
88 interpretation "cxe_coreflexive" 'cxe_coreflexive = (hos_coreflexive (os_r _)).
90 notation "'exc_cotransitive'" non associative with precedence 90 for @{'exc_cotransitive}.
91 notation "'cxe_cotransitive'" non associative with precedence 90 for @{'cxe_cotransitive}.
93 interpretation "exc_cotransitive" 'exc_cotransitive = (hos_cotransitive (os_l _)).
94 interpretation "cxe_cotransitive" 'cxe_cotransitive = (hos_cotransitive (os_r _)).
96 (* Definition 2.2 (3) *)
97 definition le ≝ λE:half_ordered_set.λa,b:E. ¬ (a ≰≰ b).
99 notation "hvbox(a break ≤≤ b)" non associative with precedence 45 for @{ 'leq_low $a $b }.
100 interpretation "Ordered half set less or equal than" 'leq_low a b = (le _ a b).
102 interpretation "Ordered set greater or equal than" 'geq a b = (le (os_r _) a b).
103 interpretation "Ordered set less or equal than" 'leq a b = (le (os_l _) a b).
105 lemma hle_reflexive: ∀E.reflexive ? (le E).
106 unfold reflexive; intros 3 (E x H); apply (hos_coreflexive ?? H);
109 notation "'le_reflexive'" non associative with precedence 90 for @{'le_reflexive}.
110 notation "'ge_reflexive'" non associative with precedence 90 for @{'ge_reflexive}.
112 interpretation "le reflexive" 'le_reflexive = (hle_reflexive (os_l _)).
113 interpretation "ge reflexive" 'ge_reflexive = (hle_reflexive (os_r _)).
116 lemma test_le_ge_convertible :∀o:ordered_set.∀x,y:o. x ≤ y → y ≥ x.
117 intros; assumption; qed.
119 lemma test_ge_reflexive :∀o:ordered_set.∀x:o. x ≥ x.
120 intros; apply ge_reflexive. qed.
122 lemma test_le_reflexive :∀o:ordered_set.∀x:o. x ≤ x.
123 intros; apply le_reflexive. qed.
126 lemma hle_transitive: ∀E.transitive ? (le E).
127 unfold transitive; intros 7 (E x y z H1 H2 H3); cases (hos_cotransitive ??? y H3) (H4 H4);
128 [cases (H1 H4)|cases (H2 H4)]
131 notation "'le_transitive'" non associative with precedence 90 for @{'le_transitive}.
132 notation "'ge_transitive'" non associative with precedence 90 for @{'ge_transitive}.
134 interpretation "le transitive" 'le_transitive = (hle_transitive (os_l _)).
135 interpretation "ge transitive" 'ge_transitive = (hle_transitive (os_r _)).
138 lemma exc_hle_variance:
139 ∀O:half_ordered_set.∀a,b,a',b':O.a ≰≰ b → a ≤≤ a' → b' ≤≤ b → a' ≰≰ b'.
140 intros (O a b a1 b1 Eab Laa1 Lb1b);
141 cases (hos_cotransitive ??? a1 Eab) (H H); [cases (Laa1 H)]
142 cases (hos_cotransitive ??? b1 H) (H1 H1); [assumption]
146 notation "'exc_le_variance'" non associative with precedence 90 for @{'exc_le_variance}.
147 notation "'exc_ge_variance'" non associative with precedence 90 for @{'exc_ge_variance}.
149 interpretation "exc_le_variance" 'exc_le_variance = (exc_hle_variance (os_l _)).
150 interpretation "exc_ge_variance" 'exc_ge_variance = (exc_hle_variance (os_r _)).
152 lemma square_half_ordered_set: half_ordered_set → half_ordered_set.
154 apply (mk_half_ordered_set (O × O));
155 [1: intros (x y); apply (\fst x ≰≰ \fst y ∨ \snd x ≰≰ \snd y);
156 |2: intro x0; cases x0 (x y); clear x0; simplify; intro H;
157 cases H (X X); apply (hos_coreflexive ?? X);
158 |3: intros 3 (x0 y0 z0); cases x0 (x1 x2); cases y0 (y1 y2) ; cases z0 (z1 z2);
159 clear x0 y0 z0; simplify; intro H; cases H (H1 H1); clear H;
160 [1: cases (hos_cotransitive ??? z1 H1); [left; left|right;left]assumption;
161 |2: cases (hos_cotransitive ??? z2 H1); [left;right|right;right]assumption]]
164 lemma square_ordered_set: ordered_set → ordered_set.
165 intro O; constructor 1;
166 [ apply (square_half_ordered_set (os_l O));
167 | apply (dual_hos (square_half_ordered_set (os_l O)));
171 notation "s 2 \atop \nleq" non associative with precedence 90
172 for @{ 'square_os $s }.
173 notation > "s 'squareO'" non associative with precedence 90
174 for @{ 'squareO $s }.
175 interpretation "ordered set square" 'squareO s = (square_ordered_set s).
176 interpretation "ordered set square" 'square_os s = (square_ordered_set s).
178 definition os_subset ≝ λO:ordered_set.λP,Q:O→Prop.∀x:O.P x → Q x.
180 interpretation "ordered set subset" 'subseteq a b = (os_subset _ a b).