duality is a joke

[helm.git] / helm / software / matita / contribs / dama / dama / bishop_set.ma

diff --git a/helm/software/matita/contribs/dama/dama/bishop_set.ma b/helm/software/matita/contribs/dama/dama/bishop_set.ma
index 1e7436af9a3e97d347a47dc259c726537e2a06be..d69bb2732b3f1a4a023aa21bdd54fc4635c3b7b7 100644 (file)
--- a/helm/software/matita/contribs/dama/dama/bishop_set.ma
+++ b/helm/software/matita/contribs/dama/dama/bishop_set.ma
@@ -23,30 +23,22 @@ record bishop_set: Type ≝ {
   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; 
@@ -59,15 +51,14 @@ qed.
 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.
@@ -76,25 +67,23 @@ 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 lt ≝ λE:ordered_set.λa,b:E. a ≤ b ∧ a # b.
+definition bs_subset ≝ λO:bishop_set.λP,Q:O→Prop.∀x:O.P x → Q x.
 
-interpretation "ordered sets less than" 'lt a b = 
-  (cic:/matita/dama/bishop_set/lt.con _ a b).
+interpretation "bishop set subset" 'subseteq a b = (bs_subset _ a b). 
 
-lemma lt_coreflexive: ∀E.coreflexive ? (lt E).
-intros 2 (E x); intro H; cases H (_ ABS);
-apply (bs_coreflexive ? x ABS);
+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.
 
-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;
-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)]]
-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.
+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