bs_cotransitive: cotransitive ? bs_apart
}.
-notation "hvbox(a break # b)" non associative with precedence 50
- for @{ 'apart $a $b}.
-
-interpretation "bishop_setapartness" 'apart x y =
- (cic:/matita/dama/bishop_set/bs_apart.con _ x y).
+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) *)
definition eq ≝ λA:bishop_set.λa,b:A. ¬ (a # b).
-notation "hvbox(a break ≈ b)" non associative with precedence 50
- for @{ 'napart $a $b}.
-
-interpretation "Bishop set alikeness" 'napart a b =
- (cic:/matita/dama/bishop_set/eq.con _ a b).
+interpretation "Bishop set alikeness" 'napart a b = (eq _ a b).
lemma eq_reflexive:∀E:bishop_set. reflexive ? (eq E).
intros (E); unfold; intros (x); apply bs_coreflexive;
lemma eq_sym:∀E:bishop_set.∀x,y:E.x ≈ y → y ≈ x ≝ eq_sym_.
lemma eq_trans_: ∀E:bishop_set.transitive ? (eq E).
-(* bug. intros k deve fare whd quanto basta *)
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.
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 bs_subset ≝ λO:bishop_set.λP,Q:O→Prop.∀x:O.P x → Q x.
+
+interpretation "bishop set subset" 'subseteq a b = (bs_subset _ a b).
+
+definition square_bishop_set : bishop_set → bishop_set.
+intro S; apply (mk_bishop_set (S × S));
+[1: intros (x y); apply ((\fst x # \fst y) ∨ (\snd x # \snd y));
+|2: intro x; simplify; intro; cases H (X X); clear H; apply (bs_coreflexive ?? X);
+|3: intros 2 (x y); simplify; intro H; cases H (X X); clear H; [left|right] apply (bs_symmetric ??? X);
+|4: intros 3 (x y z); simplify; intro H; cases H (X X); clear H;
+ [1: cases (bs_cotransitive ??? (\fst z) X); [left;left|right;left]assumption;
+ |2: cases (bs_cotransitive ??? (\snd z) X); [left;right|right;right]assumption;]]
+qed.
+
+notation "s 2 \atop \neq" non associative with precedence 90
+ for @{ 'square_bs $s }.
+notation > "s 'squareB'" non associative with precedence 90
+ for @{ 'squareB $s }.
+interpretation "bishop set square" 'squareB x = (square_bishop_set x).
+interpretation "bishop set square" 'square_bs x = (square_bishop_set x).
\ No newline at end of file