X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Fsupremum.ma;h=4ba3ca3afa678e4e9878ebd20e166e582606b068;hb=82d281529c1a9450ac213a058e7f8c0e228026fa;hp=b4430d6edff7d4daaa16eaa9a899786a4dea2498;hpb=1c2cadee5d666f0e31085a4ff358d667379c4f25;p=helm.git

diff --git a/helm/software/matita/contribs/dama/dama/supremum.ma b/helm/software/matita/contribs/dama/dama/supremum.ma
index b4430d6ed..4ba3ca3af 100644
--- a/helm/software/matita/contribs/dama/dama/supremum.ma
+++ b/helm/software/matita/contribs/dama/dama/supremum.ma
@@ -19,16 +19,15 @@ include "nat_ordered_set.ma".
 include "sequence.ma".
 
 (* Definition 2.4 *)
-definition upper_bound ≝ λO:ordered_set.λa:sequence O.λu:O.∀n:nat.a n ≤ u.
-definition lower_bound ≝ λO:ordered_set.λa:sequence O.λu:O.∀n:nat.u ≤ a n.
+definition upper_bound ≝ 
+  λO:half_ordered_set.λa:sequence O.λu:O.∀n:nat.a n ≤≤ u.
 
 definition supremum ≝
-  λO:ordered_set.λs:sequence O.λx.upper_bound ? s x ∧ (∀y:O.x ≰ y → ∃n.s n ≰ y).
-definition infimum ≝
-  λO:ordered_set.λs:sequence O.λx.lower_bound ? s x ∧ (∀y:O.y ≰ x → ∃n.y ≰ s n).
+  λO:half_ordered_set.λs:sequence O.λx.
+    upper_bound ? s x ∧ (∀y:O.x ≰≰ y → ∃n.s n ≰≰ y).
 
-definition increasing ≝ λO:ordered_set.λa:sequence O.∀n:nat.a n ≤ a (S n).
-definition decreasing ≝ λO:ordered_set.λa:sequence O.∀n:nat.a (S n) ≤ a n.
+definition increasing ≝ 
+  λO:half_ordered_set.λa:sequence O.∀n:nat.a n ≤≤ a (S n).
 
 notation < "x \nbsp 'is_upper_bound' \nbsp s" non associative with precedence 45 
   for @{'upper_bound $s $x}.
@@ -42,7 +41,6 @@ notation < "x \nbsp 'is_supremum' \nbsp s" non associative with precedence 45
   for @{'supremum $s $x}.
 notation < "x \nbsp 'is_infimum' \nbsp s" non associative with precedence 45 
   for @{'infimum $s $x}.
-
 notation > "x 'is_upper_bound' s" non associative with precedence 45 
   for @{'upper_bound $s $x}.
 notation > "x 'is_lower_bound' s" non associative with precedence 45 
@@ -56,13 +54,16 @@ notation > "x 'is_supremum' s"  non associative with precedence 45
 notation > "x 'is_infimum' s"  non associative with precedence 45 
   for @{'infimum $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).
+interpretation "Ordered set upper bound" 'upper_bound s x = (upper_bound (os_l _) s x).
+interpretation "Ordered set lower bound" 'lower_bound s x = (upper_bound (os_r _) s x).
+
+interpretation "Ordered set increasing"  'increasing s = (increasing (os_l _) s).
+interpretation "Ordered set decreasing"  'decreasing s = (increasing (os_r _) s).
+
+interpretation "Ordered set strong sup"  'supremum s x = (supremum (os_l _) s x).
+interpretation "Ordered set strong inf"  'infimum s x = (supremum (os_r _) s x).
   
+(* se non faccio il bs_of_hos perdo dualità qui *)  
 lemma uniq_supremum: 
   ∀O:ordered_set.∀s:sequence O.∀t1,t2:O.
     t1 is_supremum s → t2 is_supremum s → t1 ≈ t2.
@@ -73,108 +74,107 @@ apply le_le_eq; intro X;
 qed.
 
 (* Fact 2.5 *)
-lemma supremum_is_upper_bound: 
-  ∀C:ordered_set.∀a:sequence C.∀u:C.
-   u is_supremum a → ∀v.v is_upper_bound a → u ≤ v.
+lemma h_supremum_is_upper_bound: 
+  ∀C:half_ordered_set.∀a:sequence C.∀u:C.
+   supremum ? a u → ∀v.upper_bound ? a v → u ≤≤ v.
 intros 7 (C s u Hu v Hv H); cases Hu (_ H1); clear Hu;
-cases (H1 ? H) (w Hw); apply Hv; assumption;
+cases (H1 ? H) (w Hw); apply Hv; [apply w] assumption;
 qed.
 
-lemma infimum_is_lower_bound: 
+notation "'supremum_is_upper_bound'" non associative with precedence 90 for @{'supremum_is_upper_bound}.
+notation "'infimum_is_lower_bound'" non associative with precedence 90 for @{'infimum_is_lower_bound}.
+
+interpretation "supremum_is_upper_bound" 'supremum_is_upper_bound = (h_supremum_is_upper_bound (os_l _)).
+interpretation "infimum_is_lower_bound" 'infimum_is_lower_bound = (h_supremum_is_upper_bound (os_r _)).
+
+(* TEST DUALITY
+lemma test_infimum_is_lower_bound_duality: 
   ∀C:ordered_set.∀a:sequence C.∀u:C.
-   u is_infimum a → ∀v.v is_lower_bound a → v ≤ u.
-intros 7 (C s u Hu v Hv H); cases Hu (_ H1); clear Hu;
-cases (H1 ? H) (w Hw); apply Hv; assumption;
+   u is_infimum a → ∀v.v is_lower_bound a → u ≥ v. 
+intros; lapply (infimum_is_lower_bound a u H v H1); assumption;
 qed.
+*)
 
 (* Lemma 2.6 *)
 definition strictly_increasing ≝ 
-  λC:ordered_set.λa:sequence C.∀n:nat.a (S n) ≰ a n.
-definition strictly_decreasing ≝ 
-  λC:ordered_set.λa:sequence C.∀n:nat.a n ≰ a (S n).
+  λC:half_ordered_set.λa:sequence C.∀n:nat.a (S n) ≰≰ a n.
 
-notation < "s \nbsp 'is_strictly_increasing'" non associative with precedence 50 
+notation < "s \nbsp 'is_strictly_increasing'" non associative with precedence 45 
   for @{'strictly_increasing $s}.
-notation > "s 'is_strictly_increasing'" non associative with precedence 50 
+notation > "s 'is_strictly_increasing'" non associative with precedence 45 
   for @{'strictly_increasing $s}.
 interpretation "Ordered set strict increasing"  'strictly_increasing s    = 
-  (strictly_increasing _ s).
-notation < "s \nbsp 'is_strictly_decreasing'" non associative with precedence 50 
+  (strictly_increasing (os_l _) s).
+  
+notation < "s \nbsp 'is_strictly_decreasing'" non associative with precedence 45 
   for @{'strictly_decreasing $s}.
-notation > "s 'is_strictly_decreasing'" non associative with precedence 50 
+notation > "s 'is_strictly_decreasing'" non associative with precedence 45 
   for @{'strictly_decreasing $s}.
 interpretation "Ordered set strict decreasing"  'strictly_decreasing s    = 
-  (strictly_decreasing _ s).
+  (strictly_increasing (os_r _) s).
 
 definition uparrow ≝
-  λC:ordered_set.λs:sequence C.λu:C.
-   s is_increasing ∧ u is_supremum s.
-   
-definition downarrow ≝
-  λC:ordered_set.λs:sequence C.λu:C.
-   s is_decreasing ∧ u is_infimum s.
-     
+  λC:half_ordered_set.λs:sequence C.λu:C.
+   increasing ? s ∧ supremum ? s u.
+(*   
 notation < "a \uparrow \nbsp u" non associative with precedence 45 for @{'sup_inc $a $u}.
 notation > "a \uparrow u" non associative with precedence 45 for @{'sup_inc $a $u}.
-interpretation "Ordered set uparrow" 'sup_inc s u = (uparrow _ s u).
+*)
+interpretation "Ordered set uparrow" 'funion s u = (uparrow (os_l _) s u).
 
+(*
 notation < "a \downarrow \nbsp u" non associative with precedence 45 for @{'inf_dec $a $u}.
 notation > "a \downarrow u" non associative with precedence 45 for @{'inf_dec $a $u}.
-interpretation "Ordered set downarrow" 'inf_dec s u = (downarrow _ s u).
+*)
+interpretation "Ordered set downarrow" 'fintersects s u = (uparrow (os_r _) s u).
 
-lemma trans_increasing: 
-  ∀C:ordered_set.∀a:sequence C.a is_increasing → 
-   ∀n,m:nat_ordered_set. n ≤ m → a n ≤ a m.
+lemma h_trans_increasing: 
+  ∀C:half_ordered_set.∀a:sequence C.increasing ? a → 
+   ∀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);]
+  intro X; cases (hos_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 ?));
+[2: rewrite > H2; intro; cases (hos_coreflexive ?? H1);
+|1: apply (hle_transitive ???? (H ?) (Hs ?));
     intro; whd in H1; apply (not_le_Sn_n n); apply (transitive_le ??? H2 H1);]
 qed.
 
-lemma trans_decreasing: 
+notation "'trans_increasing'" non associative with precedence 90 for @{'trans_increasing}.
+notation "'trans_decreasing'" non associative with precedence 90 for @{'trans_decreasing}.
+
+interpretation "trans_increasing" 'trans_increasing = (h_trans_increasing (os_l _)).
+interpretation "trans_decreasing" 'trans_decreasing = (h_trans_increasing (os_r _)).
+
+(* TEST DUALITY
+lemma test_trans_decreasing_duality: 
   ∀C:ordered_set.∀a:sequence C.a is_decreasing → 
    ∀n,m:nat_ordered_set. n ≤ m → a m ≤ a n.
-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 ???? (Hs ?) (H ?));
-    intro; whd in H1; apply (not_le_Sn_n n); apply (transitive_le ??? H2 H1);]
-qed.
+intros; apply (trans_decreasing ? H ?? 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.
+lemma h_trans_increasing_exc: 
+  ∀C:half_ordered_set.∀a:sequence C.increasing ? a → 
+   ∀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; 
-[1: change in n1 with (os_carr nat_ordered_set); (* canonical structures *) 
-    change with (n<n1); (* is sort elimination not allowed preserved by delta? *)
+[1: change in n1 with (hos_carr (os_l nat_ordered_set)); 
+    change with (n<n1);
     cases (le_to_or_lt_eq ?? H1); [apply le_S_S_to_le;assumption]
-    cases (Hs n); rewrite < H3 in H2; assumption (* ogni goal di tipo Prop non è anche di tipo CProp *)    
-|2: cases (os_cotransitive ??? (a n1) H2); [assumption]
+    cases (Hs n); rewrite < H3 in H2; assumption;    
+|2: cases (hos_cotransitive ??? (a n1) H2); [assumption]
     cases (Hs n1); assumption;]
 qed.
 
-lemma trans_decreasing_exc: 
-  ∀C:ordered_set.∀a:sequence C.a is_decreasing →
-   ∀n,m:nat_ordered_set. m ≰ n → a m ≤ a n .
-intros 5 (C a Hs n m); elim m; [cases (not_le_Sn_O n H);]
-intro; apply H; 
-[1: change in n1 with (os_carr nat_ordered_set); (* canonical structures *) 
-    change with (n<n1); (* is sort elimination not allowed preserved by delta? *)
-    cases (le_to_or_lt_eq ?? H1); [apply le_S_S_to_le;assumption]
-    cases (Hs n); rewrite < H3 in H2; assumption (* ogni goal di tipo Prop non è anche di tipo CProp *)    
-|2: cases (os_cotransitive ??? (a n1) H2); [2:assumption]
-    cases (Hs n1); assumption;]
-qed.
+notation "'trans_increasing_exc'" non associative with precedence 90 for @{'trans_increasing_exc}.
+notation "'trans_decreasing_exc'" non associative with precedence 90 for @{'trans_decreasing_exc}.
+
+interpretation "trans_increasing_exc" 'trans_increasing_exc = (h_trans_increasing_exc (os_l _)).
+interpretation "trans_decreasing_exc" 'trans_decreasing_exc = (h_trans_increasing_exc (os_r _)).
 
 alias symbol "exists" = "CProp exists".
-lemma strictly_increasing_reaches: 
-  ∀C:ordered_set.∀m:sequence nat_ordered_set.
+lemma nat_strictly_increasing_reaches: 
+  ∀m:sequence nat_ordered_set.
    m is_strictly_increasing → ∀w.∃t.m t ≰ w.
 intros; elim w;
 [1: cases (nat_discriminable O (m O)); [2: cases (not_le_Sn_n O (ltn_to_ltO ?? H1))]
@@ -189,29 +189,24 @@ intros; elim w;
         apply (transitive_le ??? H2 Hp);]]
 qed.
      
-lemma selection_uparrow: 
-  ∀C:ordered_set.∀m:sequence nat_ordered_set.m is_strictly_increasing →
-    ∀a:sequence C.∀u.a ↑ u → ⌊x,a (m x)⌋ ↑ u.
+lemma h_selection_uparrow: 
+  ∀C:half_ordered_set.∀m:sequence nat_ordered_set.
+   m is_strictly_increasing →
+    ∀a:sequence C.∀u.uparrow ? a u → uparrow ? ⌊x,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_exc; [assumption] apply (Hm n);
+[1: intro n; simplify; apply (h_trans_increasing_exc ? a Ia); 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_exc C ? Ia ?? H1); assumption;]
+    cases (nat_strictly_increasing_reaches ? Hm w); 
+    exists [apply w1]; cases (hos_cotransitive ??? (a (m w1)) H); [2:assumption]  
+    cases (h_trans_increasing_exc ?? Ia ?? H1); assumption;]
 qed.     
 
-lemma selection_downarrow: 
-  ∀C:ordered_set.∀m:sequence nat_ordered_set.m is_strictly_increasing →
-    ∀a:sequence C.∀u.a ↓ u → ⌊x,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_decreasing_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); [assumption]  
-    cases (trans_decreasing_exc C ? Ia ?? H1); assumption;]
-qed.
+notation "'selection_uparrow'" non associative with precedence 90 for @{'selection_uparrow}.
+notation "'selection_downarrow'" non associative with precedence 90 for @{'selection_downarrow}.
+
+interpretation "selection_uparrow" 'selection_uparrow = (h_selection_uparrow (os_l _)).
+interpretation "selection_downarrow" 'selection_downarrow = (h_selection_uparrow (os_r _)).
 
 (* Definition 2.7 *)
 definition order_converge ≝
@@ -227,29 +222,85 @@ notation > "a 'order_converges' x" non associative with precedence 45
 interpretation "Order convergence" 'order_converge s u = (order_converge _ s u).   
     
 (* Definition 2.8 *)
-definition segment ≝ λO:ordered_set.λa,b:O.λx:O.(x ≤ b) ∧ (a ≤ x).
+definition segment ≝ λO:half_ordered_set.λa,b:O.λx:O.(x ≤≤ b) ∧ (a ≤≤ x).
 
-notation "[a,b]" left associative with precedence 70 for @{'segment $a $b}.
-interpretation "Ordered set sergment" 'segment a b = (segment _ a b).
+notation "[term 19 a,term 19 b]" non associative with precedence 90 for @{'segment $a $b}.
+interpretation "Ordered set sergment" 'segment a b = (segment (os_l _) a b).
 
-notation "hvbox(x \in break [a,b])" non associative with precedence 45 
+notation "hvbox(x \in break [term 19 a, term 19 b])" non associative with precedence 45 
   for @{'segment_in $a $b $x}.
-interpretation "Ordered set sergment in" 'segment_in a b x= (segment _ a b x).
+interpretation "Ordered set sergment in" 'segment_in a b x= (segment (os_l _) a b x).
+
+definition segment_ordered_set_carr ≝
+  λO:half_ordered_set.λu,v:O.∃x.segment ? u v x.
+definition segment_ordered_set_exc ≝ 
+  λO:half_ordered_set.λu,v:O.
+   λx,y:segment_ordered_set_carr ? u v.\fst x ≰≰ \fst y.
+lemma segment_ordered_set_corefl:
+ ∀O,u,v. coreflexive ? (segment_ordered_set_exc O u v).
+intros 4; cases x; simplify; apply hos_coreflexive; qed.
+lemma segment_ordered_set_cotrans : 
+  ∀O,u,v. cotransitive ? (segment_ordered_set_exc O u v).
+intros 6 (O u v x y z); cases x; cases y ; cases z; simplify; apply hos_cotransitive;
+qed.  
+  
+lemma half_segment_ordered_set: 
+  ∀O:half_ordered_set.∀u,v:O.half_ordered_set.
+intros (O u v); apply (mk_half_ordered_set ?? (segment_ordered_set_corefl O u v) (segment_ordered_set_cotrans ???));
+qed.
 
 lemma segment_ordered_set: 
   ∀O:ordered_set.∀u,v:O.ordered_set.
-intros (O u v); apply (mk_ordered_set (sigT ? (λx.x ∈ [u,v])));
-[1: intros (x y); apply (\fst x ≰ \fst y);
-|2: intro x; cases x; simplify; apply os_coreflexive;
-|3: intros 3 (x y z); cases x; cases y ; cases z; simplify; apply os_cotransitive]
+intros (O u v); 
+apply half2full; apply (half_segment_ordered_set (os_l O) u v); 
 qed.
 
-notation "hvbox({[a, break b]})" non associative with precedence 90 
+(*
+notation < "hvbox({[a, break b]/})" non associative with precedence 90 
+  for @{'h_segment_set $a $b}.
+notation > "hvbox({[a, break b]/})" non associative with precedence 90 
+  for @{'h_segment_set $a $b}.
+interpretation "Half ordered set segment" 'h_segment_set a b = 
+  (half_segment_ordered_set _ a b).
+*)
+
+notation < "hvbox({[a, break b]})" non associative with precedence 90 
+  for @{'segment_set $a $b}.
+notation > "hvbox({[a, break b]})" non associative with precedence 90 
   for @{'segment_set $a $b}.
 interpretation "Ordered set segment" 'segment_set a b = 
   (segment_ordered_set _ a b).
+  
+definition hint_sequence: 
+  ∀C:ordered_set.
+    sequence (hos_carr (os_l C)) → sequence (Type_of_ordered_set C).
+intros;assumption;
+qed.
+
+definition hint_sequence1: 
+  ∀C:ordered_set.
+    sequence (hos_carr (os_r C)) → sequence (Type_of_ordered_set_dual C).
+intros;assumption;
+qed.
 
-(* Lemma 2.9 *)
+definition hint_sequence2: 
+  ∀C:ordered_set.
+    sequence (Type_of_ordered_set C) → sequence (hos_carr (os_l C)).
+intros;assumption;
+qed.
+
+definition hint_sequence3: 
+  ∀C:ordered_set.
+    sequence (Type_of_ordered_set_dual C) → sequence (hos_carr (os_r C)).
+intros;assumption;
+qed.
+
+coercion hint_sequence nocomposites.
+coercion hint_sequence1 nocomposites.
+coercion hint_sequence2 nocomposites.
+coercion hint_sequence3 nocomposites.
+
+(* Lemma 2.9 - non easily dualizable *)
 lemma segment_preserves_supremum:
   ∀O:ordered_set.∀l,u:O.∀a:sequence {[l,u]}.∀x:{[l,u]}. 
     ⌊n,\fst (a n)⌋ is_increasing ∧ 
@@ -271,55 +322,49 @@ intros; split; cases H; clear H;
     [1: apply H;
     |2: clear H; intro y0; apply (H3 (\fst y0));]]
 qed.
-
+           
 (* Definition 2.10 *)
-alias symbol "square" = "ordered set square".
 alias symbol "pi2" = "pair pi2".
 alias symbol "pi1" = "pair pi1".
 definition square_segment ≝ 
-  λO:ordered_set.λa,b:O.λx:O square.
+  λO:ordered_set.λa,b:O.λx: O squareO.
     And4 (\fst x ≤ b) (a ≤ \fst x) (\snd x ≤ b) (a ≤ \snd x).
  
 definition convex ≝
-  λO:ordered_set.λU:O square → Prop.
-    ∀p.U p → \fst p ≤ \snd p → ∀y. square_segment ? (\fst p) (\snd p) y → U y.
+  λO:ordered_set.λU:O squareO → Prop.
+    ∀p.U p → \fst p ≤ \snd p → ∀y. 
+      square_segment O (\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).
+  λO:half_ordered_set.λa:sequence O.∀x,y:O. y ≰≰ x → 
+    (∃i:nat.a i ≰≰ x) ∨ (∃b:O.y ≰≰ b ∧ ∀i:nat.a i ≤≤ b).
 
 notation < "s \nbsp 'is_upper_located'" non associative with precedence 45 
   for @{'upper_located $s}.
 notation > "s 'is_upper_located'" non associative with precedence 45 
   for @{'upper_located $s}.
 interpretation "Ordered set upper locatedness" 'upper_located s = 
-  (upper_located _ s).
+  (upper_located (os_l _) s).
 
 notation < "s \nbsp 'is_lower_located'" non associative with precedence 45 
   for @{'lower_located $s}.
 notation > "s 'is_lower_located'" non associative with precedence 45
   for @{'lower_located $s}.
 interpretation "Ordered set lower locatedness" 'lower_located s = 
-  (lower_located _ s).
-    
+  (upper_located (os_r _) s).
+      
 (* Lemma 2.12 *)    
-lemma uparrow_upperlocated:
-  ∀C:ordered_set.∀a:sequence C.∀u:C.a ↑ u → a is_upper_located.
+lemma h_uparrow_upperlocated:
+  ∀C:half_ordered_set.∀a:sequence C.∀u:C.uparrow ? a u → upper_located ? a.
 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);
+cases H3 (H4 H5); clear H3; cases (hos_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. 
+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. 
+notation "'uparrow_upperlocated'" non associative with precedence 90 for @{'uparrow_upperlocated}.
+notation "'downarrow_lowerlocated'" non associative with precedence 90 for @{'downarrow_lowerlocated}.
+
+interpretation "uparrow_upperlocated" 'uparrow_upperlocated = (h_uparrow_upperlocated (os_l _)).
+interpretation "downarrow_lowerlocated" 'downarrow_lowerlocated = (h_uparrow_upperlocated (os_r _)).