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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 "ordered_set.ma".
17 (* Definition 2.2, setoid *)
18 record bishop_set: Type ≝ {
20 bs_apart: bs_carr → bs_carr → CProp;
21 bs_coreflexive: coreflexive ? bs_apart;
22 bs_symmetric: symmetric ? bs_apart;
23 bs_cotransitive: cotransitive ? bs_apart
26 notation "hvbox(a break # b)" non associative with precedence 50
29 interpretation "bishop_setapartness" 'apart x y = (bs_apart _ x y).
31 definition bishop_set_of_ordered_set: ordered_set → bishop_set.
32 intros (E); apply (mk_bishop_set E (λa,b:E. a ≰ b ∨ b ≰ a));
33 [1: unfold; cases E; simplify; clear E; intros (x); unfold;
34 intros (H1); apply (H x); cases H1; assumption;
35 |2: unfold; intros(x y H); cases H; clear H; [right|left] assumption;
36 |3: intros (E); unfold; cases E (T f _ cTf); simplify; intros (x y z Axy);
37 cases Axy (H H); lapply (cTf ? ? z H) as H1; cases H1; clear Axy H1;
38 [left; left|right; left|right; right|left; right] assumption]
41 (* Definition 2.2 (2) *)
42 definition eq ≝ λA:bishop_set.λa,b:A. ¬ (a # b).
44 notation "hvbox(a break ≈ b)" non associative with precedence 50
45 for @{ 'napart $a $b}.
47 interpretation "Bishop set alikeness" 'napart a b = (eq _ a b).
49 lemma eq_reflexive:∀E:bishop_set. reflexive ? (eq E).
50 intros (E); unfold; intros (x); apply bs_coreflexive;
53 lemma eq_sym_:∀E:bishop_set.symmetric ? (eq E).
54 intros 4 (E x y H); intro T; cases (H (bs_symmetric ??? T));
57 lemma eq_sym:∀E:bishop_set.∀x,y:E.x ≈ y → y ≈ x ≝ eq_sym_.
59 lemma eq_trans_: ∀E:bishop_set.transitive ? (eq E).
60 (* bug. intros k deve fare whd quanto basta *)
61 intros 6 (E x y z Exy Eyz); intro Axy; cases (bs_cotransitive ???y Axy);
62 [apply Exy|apply Eyz] assumption.
65 coercion cic:/matita/dama/bishop_set/bishop_set_of_ordered_set.con.
67 lemma le_antisymmetric:
68 ∀E:ordered_set.antisymmetric E (le E) (eq E).
69 intros 5 (E x y Lxy Lyx); intro H;
70 cases H; [apply Lxy;|apply Lyx] assumption;
73 lemma le_le_eq: ∀E:ordered_set.∀a,b:E. a ≤ b → b ≤ a → a ≈ b.
74 intros (E x y L1 L2); intro H; cases H; [apply L1|apply L2] assumption;
77 definition lt ≝ λE:ordered_set.λa,b:E. a ≤ b ∧ a # b.
79 interpretation "ordered sets less than" 'lt a b = (lt _ a b).
81 lemma lt_coreflexive: ∀E.coreflexive ? (lt E).
82 intros 2 (E x); intro H; cases H (_ ABS);
83 apply (bs_coreflexive ? x ABS);
86 lemma lt_transitive: ∀E.transitive ? (lt E).
87 intros (E); unfold; intros (x y z H1 H2); cases H1 (Lxy Axy); cases H2 (Lyz Ayz);
88 split; [apply (le_transitive ???? Lxy Lyz)] clear H1 H2;
89 cases Axy (H1 H1); cases Ayz (H2 H2); [1:cases (Lxy H1)|3:cases (Lyz H2)]clear Axy Ayz;
90 [1: cases (os_cotransitive ??? y H1) (X X); [cases (Lxy X)|cases (os_coreflexive ?? X)]
91 |2: cases (os_cotransitive ??? x H2) (X X); [right;assumption|cases (Lxy X)]]
94 theorem lt_to_excess: ∀E:ordered_set.∀a,b:E. (a < b) → (b ≰ a).
95 intros (E a b Lab); cases Lab (LEab Aab); cases Aab (H H);[cases (LEab H)]