X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Fsupremum.ma;h=50b1ebedae5d64b9f1a8842fea85e20c440410af;hb=9eabe046c1182960de8cfdba96c5414224e3a61e;hp=0c6d545b8b9f5d762c6a17de39c254f8529f2747;hpb=695cc9c1ead293e380709ebcd98818e019b8e49e;p=helm.git

diff --git a/helm/software/matita/contribs/dama/dama/supremum.ma b/helm/software/matita/contribs/dama/dama/supremum.ma
index 0c6d545b8..50b1ebeda 100644
--- a/helm/software/matita/contribs/dama/dama/supremum.ma
+++ b/helm/software/matita/contribs/dama/dama/supremum.ma
@@ -53,18 +53,12 @@ notation > "x 'is_supremum' s"  non associative with precedence 50
 notation > "x 'is_infimum' s"  non associative with precedence 50 
   for @{'infimum $s $x}.
 
-interpretation "Ordered set upper bound" 'upper_bound s x = 
-  (cic:/matita/dama/supremum/upper_bound.con _ s x).
-interpretation "Ordered set lower bound" 'lower_bound s x = 
-  (cic:/matita/dama/supremum/lower_bound.con _ s x).
-interpretation "Ordered set increasing"  'increasing s    = 
-  (cic:/matita/dama/supremum/increasing.con _ s).
-interpretation "Ordered set decreasing"  'decreasing s    = 
-  (cic:/matita/dama/supremum/decreasing.con _ s).
-interpretation "Ordered set strong sup"  'supremum s x  = 
-  (cic:/matita/dama/supremum/supremum.con _ s x).
-interpretation "Ordered set strong inf"  'infimum s x  = 
-  (cic:/matita/dama/supremum/infimum.con _ s x).
+interpretation "Ordered set upper bound" 'upper_bound s x = (upper_bound _ s x).
+interpretation "Ordered set lower bound" 'lower_bound s x = (lower_bound _ s x).
+interpretation "Ordered set increasing"  'increasing s    = (increasing _ s).
+interpretation "Ordered set decreasing"  'decreasing s    = (decreasing _ s).
+interpretation "Ordered set strong sup"  'supremum s x  = (supremum _ s x).
+interpretation "Ordered set strong inf"  'infimum s x  = (infimum _ s x).
   
 include "bishop_set.ma".
   
@@ -97,24 +91,24 @@ notation < "s \nbsp 'is_strictly_increasing'" non associative with precedence 50
 notation > "s 'is_strictly_increasing'" non associative with precedence 50 
   for @{'strictly_increasing $s}.
 interpretation "Ordered set strict increasing"  'strictly_increasing s    = 
-  (cic:/matita/dama/supremum/strictly_increasing.con _ s).
+  (strictly_increasing _ s).
 notation < "s \nbsp 'is_strictly_decreasing'" non associative with precedence 50 
   for @{'strictly_decreasing $s}.
 notation > "s 'is_strictly_decreasing'" non associative with precedence 50 
   for @{'strictly_decreasing $s}.
 interpretation "Ordered set strict decreasing"  'strictly_decreasing s    = 
-  (cic:/matita/dama/supremum/strictly_decreasing.con _ s).
+  (strictly_decreasing _ s).
   
 notation "a \uparrow u" non associative with precedence 50 for @{'sup_inc $a $u}.
 interpretation "Ordered set supremum of increasing" 'sup_inc s u =
  (cic:/matita/dama/cprop_connectives/And.ind#xpointer(1/1) 
-  (cic:/matita/dama/supremum/increasing.con _ s)
-  (cic:/matita/dama/supremum/supremum.con _ s u)).
+  (increasing _ s)
+  (supremum _ s u)).
 notation "a \downarrow u" non associative with precedence 50 for @{'inf_dec $a $u}.
 interpretation "Ordered set supremum of increasing" 'inf_dec s u =
  (cic:/matita/dama/cprop_connectives/And.ind#xpointer(1/1) 
-  (cic:/matita/dama/supremum/decreasing.con _ s)
-  (cic:/matita/dama/supremum/infimum.con _ s u)).
+  (decreasing _ s)
+  (infimum _ s u)).
 
 include "nat/plus.ma".
 include "nat_ordered_set.ma".
@@ -122,6 +116,17 @@ include "nat_ordered_set.ma".
 alias symbol "nleq" = "Ordered set excess".
 alias symbol "leq" = "Ordered set less or equal than".
 lemma trans_increasing: 
+  ∀C:ordered_set.∀a:sequence C.a is_increasing → ∀n,m:nat_ordered_set. n ≤ m → a n ≤ a m.
+intros 5 (C a Hs n m); elim m; [
+  rewrite > (le_n_O_to_eq n (not_lt_to_le O n H));
+  intro X; cases (os_coreflexive ?? X);]
+cases (le_to_or_lt_eq ?? (not_lt_to_le (S n1) n H1)); clear H1;
+[2: rewrite > H2; intro; cases (os_coreflexive ?? H1);
+|1: apply (le_transitive ???? (H ?) (Hs ?));
+    intro; whd in H1; apply (not_le_Sn_n n); apply (transitive_le ??? H2 H1);]
+qed.
+
+lemma trans_increasing_exc: 
   ∀C:ordered_set.∀a:sequence C.a is_increasing → ∀n,m:nat_ordered_set. m ≰ n → a n ≤ a m.
 intros 5 (C a Hs n m); elim m; [cases (not_le_Sn_O n H);]
 intro; apply H; 
@@ -153,12 +158,12 @@ lemma selection:
   ∀C:ordered_set.∀m:sequence nat_ordered_set.m is_strictly_increasing →
     ∀a:sequence C.∀u.a ↑ u → (λx.a (m x)) ↑ u.
 intros (C m Hm a u Ha); cases Ha (Ia Su); cases Su (Uu Hu); repeat split; 
-[1: intro n; simplify; apply trans_increasing; [assumption] apply (Hm n);
+[1: intro n; simplify; apply trans_increasing_exc; [assumption] apply (Hm n);
 |2: intro n; simplify; apply Uu;
 |3: intros (y Hy); simplify; cases (Hu ? Hy);
     cases (strictly_increasing_reaches C ? Hm w); 
     exists [apply w1]; cases (os_cotransitive ??? (a (m w1)) H); [2:assumption]  
-    cases (trans_increasing C ? Ia ?? H1); assumption;]
+    cases (trans_increasing_exc C ? Ia ?? H1); assumption;]
 qed.     
     
 (* Definition 2.7 *)
@@ -167,15 +172,14 @@ alias symbol "and" = "constructive and".
 definition order_converge ≝
   λO:ordered_set.λa:sequence O.λx:O.
    ∃l:sequence O.∃u:sequence O.
-    l is_increasing ∧ u is_decreasing ∧ l ↑ x ∧ u ↓ x ∧
+    (*l is_increasing ∧ u is_decreasing ∧*) l ↑ x ∧ u ↓ x ∧
     ∀i:nat. (l i) is_infimum (λw.a (w+i)) ∧ (u i) is_supremum (λw.a (w+i)).
     
 notation < "a \nbsp (\circ \atop (\horbar\triangleright)) \nbsp x" non associative with precedence 50 
   for @{'order_converge $a $x}.
 notation > "a 'order_converges' x" non associative with precedence 50 
   for @{'order_converge $a $x}.
-interpretation "Order convergence" 'order_converge s u =
- (cic:/matita/dama/supremum/order_converge.con _ s u).   
+interpretation "Order convergence" 'order_converge s u = (order_converge _ s u).   
     
 (* Definition 2.8 *)
 
@@ -184,25 +188,22 @@ definition segment ≝ λO:ordered_set.λa,b:O.λx:O.
 
 notation "[a,b]" non associative with precedence 50 
   for @{'segment $a $b}.
-interpretation "Ordered set sergment" 'segment a b =
-  (cic:/matita/dama/supremum/segment.con _ a b).
+interpretation "Ordered set sergment" 'segment a b = (segment _ a b).
 
-notation "x \in [a,b]" non associative with precedence 50 
+notation "hvbox(x \in break [a,b])" non associative with precedence 50 
   for @{'segment2 $a $b $x}.
-interpretation "Ordered set sergment in" 'segment2 a b x=
-  (cic:/matita/dama/supremum/segment.con _ a b x).
+interpretation "Ordered set sergment in" 'segment2 a b x= (segment _ a b x).
 
 coinductive sigma (A:Type) (P:A→Prop) : Type ≝ sig_in : ∀x.P x → sigma A P.
 
 definition pi1 : ∀A.∀P.sigma A P → A ≝ λA,P,s.match s with [sig_in x _ ⇒ x].  
 
-notation < "\pi \sub 1 x" non associative with precedence 50 for @{'pi1 $x}.
-notation < "\pi \sub 2 x" non associative with precedence 50 for @{'pi2 $x}.
+notation < "'fst' \nbsp x" non associative with precedence 50 for @{'pi1 $x}.
+notation < "'snd' \nbsp x" non associative with precedence 50 for @{'pi2 $x}.
 notation > "'fst' x" non associative with precedence 50 for @{'pi1 $x}.
 notation > "'snd' x" non associative with precedence 50 for @{'pi2 $x}.
-interpretation "sigma pi1" 'pi1 x = 
- (cic:/matita/dama/supremum/pi1.con _ _ x).
-
+interpretation "sigma pi1" 'pi1 x = (pi1 _ _ x).
+ 
 interpretation "Type exists" 'exists \eta.x =
   (cic:/matita/dama/supremum/sigma.ind#xpointer(1/1) _ x).
 
@@ -214,15 +215,14 @@ intros (O u v); apply (mk_ordered_set (∃x.x ∈ [u,v]));
 |3: intros 3 (x y z); cases x; cases y ; cases z; simplify; apply os_cotransitive]
 qed.
 
-notation < "{\x|\x \in [a,b]}" non associative with precedence 90 
+notation "hvbox({[a, break b]})" non associative with precedence 90 
   for @{'segment_set $a $b}.
 interpretation "Ordered set segment" 'segment_set a b = 
- (cic:/matita/dama/supremum/segment_ordered_set.con _ a b).
+ (segment_ordered_set _ a b).
 
 (* Lemma 2.9 *)
 lemma segment_preserves_supremum:
-  ∀O:ordered_set.∀l,u:O.∀a:sequence (segment_ordered_set ? l u).
-    ∀x:(segment_ordered_set ? l u). 
+  ∀O:ordered_set.∀l,u:O.∀a:sequence {[l,u]}.∀x:{[l,u]}. 
     (λn.fst (a n)) is_increasing ∧ 
     (fst x) is_supremum (λn.fst (a n)) → a ↑ x.
 intros; split; cases H; clear H; 
@@ -237,13 +237,17 @@ coinductive pair (A,B:Type) : Type ≝ prod : ∀a:A.∀b:B.pair A B.
 definition first : ∀A.∀P.pair A P → A ≝ λA,P,s.match s with [prod x _ ⇒ x].
 definition second : ∀A.∀P.pair A P → P ≝ λA,P,s.match s with [prod _ y ⇒ y].
   
-interpretation "pair pi1" 'pi1 x = 
- (cic:/matita/dama/supremum/first.con _ _ x).
-interpretation "pair pi2" 'pi2 x = 
- (cic:/matita/dama/supremum/second.con _ _ x).
+interpretation "pair pi1" 'pi1 x = (first _ _ x).
+interpretation "pair pi2" 'pi2 x = (second _ _ x).
+
+notation "hvbox(\langle a, break b\rangle)" non associative with precedence 91 for @{ 'pair $a $b}.
+interpretation "pair" 'pair a b = (prod _ _ a b).
+ 
+notation "a \times b" left associative with precedence 60 for @{'prod $a $b}.
+interpretation "prod" 'prod a b = (pair a b).
  
 lemma square_ordered_set: ordered_set → ordered_set.
-intro O; apply (mk_ordered_set (pair O O));
+intro O; apply (mk_ordered_set (O × O));
 [1: intros (x y); apply (fst x ≰ fst y ∨ snd x ≰ snd y);
 |2: intro x0; cases x0 (x y); clear x0; simplify; intro H;
     cases H (X X); apply (os_coreflexive ?? X);
@@ -252,6 +256,12 @@ intro O; apply (mk_ordered_set (pair O O));
     [1: cases (os_cotransitive ??? z1 H1); [left; left|right;left]assumption;
     |2: cases (os_cotransitive ??? z2 H1); [left;right|right;right]assumption]]
 qed.
+
+notation < "s  2 \atop \nleq" non associative with precedence 90
+  for @{ 'square $s }.
+notation > "s 'square'" non associative with precedence 90
+  for @{ 'square $s }.
+interpretation "ordered set square" 'square s = (square_ordered_set s).
  
 definition square_segment ≝ 
   λO:ordered_set.λa,b:O.λx:square_ordered_set O.
@@ -260,22 +270,47 @@ definition square_segment ≝
    (cic:/matita/logic/connectives/And.ind#xpointer(1/1) (snd x ≤ b) (a ≤ snd x))).
  
 definition convex ≝
-  λO:ordered_set.λU:square_ordered_set O → Prop.
+  λO:ordered_set.λU:O square → Prop.
   ∀p.U p → fst p ≤ snd p → ∀y. square_segment ? (fst p) (snd p) y → U y.
   
 (* Definition 2.11 *)  
 definition upper_located ≝
   λO:ordered_set.λa:sequence O.∀x,y:O. y ≰ x → 
     (∃i:nat.a i ≰ x) ∨ (∃b:O.y≰b ∧ ∀i:nat.a i ≤ b).
+
+definition lower_located ≝
+  λO:ordered_set.λa:sequence O.∀x,y:O. x ≰ y → 
+    (∃i:nat.x ≰ a i) ∨ (∃b:O.b≰y ∧ ∀i:nat.b ≤ a i).
+
+notation < "s \nbsp 'is_upper_located'" non associative with precedence 50 
+  for @{'upper_located $s}.
+notation > "s 'is_upper_located'" non associative with precedence 50 
+  for @{'upper_located $s}.
+interpretation "Ordered set upper locatedness" 'upper_located s    = 
+  (upper_located _ s).
+
+notation < "s \nbsp 'is_lower_located'" non associative with precedence 50 
+  for @{'lower_located $s}.
+notation > "s 'is_lower_located'" non associative with precedence 50 
+  for @{'lower_located $s}.
+interpretation "Ordered set lower locatedness" 'lower_located s    = 
+  (lower_located _ s).
     
 (* Lemma 2.12 *)    
-lemma uparrow_located:
-  ∀C:ordered_set.∀a:sequence C.∀u:C.a ↑ u → upper_located ? a.
+lemma uparrow_upperlocated:
+  ∀C:ordered_set.∀a:sequence C.∀u:C.a ↑ u → a is_upper_located.
 intros (C a u H); cases H (H2 H3); clear H; intros 3 (x y Hxy);
 cases H3 (H4 H5); clear H3; cases (os_cotransitive ??? u Hxy) (W W);
 [2: cases (H5 ? W) (w Hw); left; exists [apply w] assumption;
 |1: right; exists [apply u]; split; [apply W|apply H4]]
 qed. 
 
+lemma downarrow_lowerlocated:
+  ∀C:ordered_set.∀a:sequence C.∀u:C.a ↓ u → a is_lower_located.
+intros (C a u H); cases H (H2 H3); clear H; intros 3 (x y Hxy);
+cases H3 (H4 H5); clear H3; cases (os_cotransitive ??? u Hxy) (W W);
+[1: cases (H5 ? W) (w Hw); left; exists [apply w] assumption;
+|2: right; exists [apply u]; split; [apply W|apply H4]]
+qed. 
      
-    
\ No newline at end of file
+