more notation

[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 1c68695d3038dbc8806519257c1ae3d867330f02..78176d776816c85c513348d356568924e90c77f0 100644 (file)
--- a/helm/software/matita/contribs/dama/dama/bishop_set.ma
+++ b/helm/software/matita/contribs/dama/dama/bishop_set.ma
@@ -23,11 +23,10 @@ record bishop_set: Type ≝ {
   bs_cotransitive: cotransitive ? bs_apart
 }.
 
-notation "hvbox(a break # b)" non associative with precedence 50 
+notation "hvbox(a break # b)" non associative with precedence 45 
   for @{ 'apart $a $b}.
   
-interpretation "bishop_setapartness" 'apart x y = 
-  (cic:/matita/dama/bishop_set/bs_apart.con _ x y).
+interpretation "bishop_setapartness" '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));  
@@ -42,11 +41,10 @@ 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 
+notation "hvbox(a break \approx b)" non associative with precedence 45 
   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; 
@@ -59,7 +57,6 @@ 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.
@@ -75,3 +72,25 @@ 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 lt ≝ λE:ordered_set.λa,b:E. a ≤ b ∧ a # b.
+
+interpretation "ordered sets less than" 'lt a b = (lt _ a b).
+
+lemma lt_coreflexive: ∀E.coreflexive ? (lt E).
+intros 2 (E x); intro H; cases H (_ ABS);
+apply (bs_coreflexive ? x ABS);
+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.