bs_cotransitive: cotransitive ? bs_apart
}.
-notation "hvbox(a break # b)" non associative with precedence 45
- for @{ 'apart $a $b}.
-
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 \approx b)" non associative with precedence 45
- for @{ 'napart $a $b}.
-
interpretation "Bishop set alikeness" 'napart a b = (eq _ a b).
lemma eq_reflexive:∀E:bishop_set. reflexive ? (eq E).
[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.
intros (E x y L1 L2); intro H; cases H; [apply L1|apply L2] assumption;
qed.
-definition lt ≝ λE:ordered_set.λa,b:E. a ≤ b ∧ a # b.
+(*
+definition lt ≝ λE:half_ordered_set.λa,b:E. a ≤ b ∧ a # b.
interpretation "ordered sets less than" 'lt a b = (lt _ a b).
lemma lt_transitive: ∀E.transitive ? (lt E).
intros (E); unfold; intros (x y z H1 H2); cases H1 (Lxy Axy); cases H2 (Lyz Ayz);
-split; [apply (le_transitive ???? Lxy Lyz)] clear H1 H2;
+split; [apply (le_transitive E ??? Lxy Lyz)] clear H1 H2;
cases Axy (H1 H1); cases Ayz (H2 H2); [1:cases (Lxy H1)|3:cases (Lyz H2)]clear Axy Ayz;
-[1: cases (os_cotransitive ??? y H1) (X X); [cases (Lxy X)|cases (os_coreflexive ?? X)]
-|2: cases (os_cotransitive ??? x H2) (X X); [right;assumption|cases (Lxy X)]]
+[1: cases (hos_cotransitive E ?? y H1) (X X); [cases (Lxy X)|cases (hos_coreflexive E ? X)]
+|2: cases (hos_cotransitive E ?? x H2) (X X); [right;assumption|cases (Lxy X)]]
qed.
theorem lt_to_excess: ∀E:ordered_set.∀a,b:E. (a < b) → (b ≰ a).
intros (E a b Lab); cases Lab (LEab Aab); cases Aab (H H);[cases (LEab H)]
assumption;
qed.
+*)
definition bs_subset ≝ λO:bishop_set.λP,Q:O→Prop.∀x:O.P x → Q x.
-interpretation "bishop set subset" 'subset a b = (bs_subset _ a b).
+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));
notation "s 2 \atop \neq" non associative with precedence 90
for @{ 'square_bs $s }.
-interpretation "bishop set square" 'square x = (square_bishop_set x).
+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