X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Fsupremum.ma;h=a492c8f3299e8255c0b86be2605972da857b3310;hb=8f4162a9db17a597d4fba49eb957009fc0268378;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..a492c8f32 100644
--- a/helm/software/matita/contribs/dama/dama/supremum.ma
+++ b/helm/software/matita/contribs/dama/dama/supremum.ma
@@ -12,8 +12,11 @@
 (*                                                                        *)
 (**************************************************************************)
 
+
+include "datatypes/constructors.ma".
+include "nat/plus.ma".
+include "nat_ordered_set.ma".
 include "sequence.ma".
-include "ordered_set.ma".
 
 (* Definition 2.4 *)
 definition upper_bound ≝ λO:ordered_set.λa:sequence O.λu:O.∀n:nat.a n ≤ u.
@@ -27,46 +30,38 @@ definition infimum ≝
 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.
 
-notation < "x \nbsp 'is_upper_bound' \nbsp s" non associative with precedence 50 
+notation < "x \nbsp 'is_upper_bound' \nbsp s" non associative with precedence 45 
   for @{'upper_bound $s $x}.
-notation < "x \nbsp 'is_lower_bound' \nbsp s" non associative with precedence 50 
+notation < "x \nbsp 'is_lower_bound' \nbsp s" non associative with precedence 45 
   for @{'lower_bound $s $x}.
-notation < "s \nbsp 'is_increasing'" non associative with precedence 50 
+notation < "s \nbsp 'is_increasing'" non associative with precedence 45 
   for @{'increasing $s}.
-notation < "s \nbsp 'is_decreasing'" non associative with precedence 50 
+notation < "s \nbsp 'is_decreasing'" non associative with precedence 45 
   for @{'decreasing $s}.
-notation < "x \nbsp 'is_supremum' \nbsp s" non associative with precedence 50 
+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 50 
+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 50 
+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 50 
+notation > "x 'is_lower_bound' s" non associative with precedence 45 
   for @{'lower_bound $s $x}.
-notation > "s 'is_increasing'"    non associative with precedence 50 
+notation > "s 'is_increasing'"    non associative with precedence 45 
   for @{'increasing $s}.
-notation > "s 'is_decreasing'"    non associative with precedence 50 
+notation > "s 'is_decreasing'"    non associative with precedence 45 
   for @{'decreasing $s}.
-notation > "x 'is_supremum' s"  non associative with precedence 50 
+notation > "x 'is_supremum' s"  non associative with precedence 45 
   for @{'supremum $s $x}.
-notation > "x 'is_infimum' s"  non associative with precedence 50 
+notation > "x 'is_infimum' s"  non associative with precedence 45 
   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).
-  
-include "bishop_set.ma".
+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).
   
 lemma uniq_supremum: 
   ∀O:ordered_set.∀s:sequence O.∀t1,t2:O.
@@ -85,44 +80,75 @@ intros 7 (C s u Hu v Hv H); cases Hu (_ H1); clear Hu;
 cases (H1 ? H) (w Hw); apply Hv; assumption;
 qed.
 
+lemma infimum_is_lower_bound: 
+  ∀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;
+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).
 
- 
 notation < "s \nbsp 'is_strictly_increasing'" non associative with precedence 50 
   for @{'strictly_increasing $s}.
 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).
-  
-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)).
-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)).
+  (strictly_decreasing _ 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.
+     
+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).
+
+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).
 
-include "nat/plus.ma".
-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. m ≰ n → a n ≤ a m.
+  ∀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_decreasing: 
+  ∀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.
+
+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; 
 [1: change in n1 with (os_carr nat_ordered_set); (* canonical structures *) 
@@ -133,6 +159,20 @@ intro; apply H;
     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.
+
+alias symbol "exists" = "CProp exists".
 lemma strictly_increasing_reaches: 
   ∀C:ordered_set.∀m:sequence nat_ordered_set.
    m is_strictly_increasing → ∀w.∃t.m t ≰ w.
@@ -149,133 +189,138 @@ intros; elim w;
         apply (transitive_le ??? H2 Hp);]]
 qed.
      
-lemma selection: 
+lemma selection_uparrow: 
   ∀C:ordered_set.∀m:sequence nat_ordered_set.m is_strictly_increasing →
-    ∀a:sequence C.∀u.a ↑ u → (λx.a (m x)) ↑ u.
+    ∀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_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.     
-    
+
+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.
+
 (* Definition 2.7 *)
-alias symbol "exists" = "CProp exists".
-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 ∧
-    ∀i:nat. (l i) is_infimum (λw.a (w+i)) ∧ (u i) is_supremum (λw.a (w+i)).
+   exT23 (sequence O) (λl.l ↑ x) (λu.u ↓ x)
+     (λl,u:sequence O.∀i:nat. (l i) is_infimum ⌊w,a (w+i)⌋ ∧ 
+                   (u i) is_supremum ⌊w,a (w+i)⌋).
     
-notation < "a \nbsp (\circ \atop (\horbar\triangleright)) \nbsp x" non associative with precedence 50 
+notation < "a \nbsp (\cir \atop (\horbar\triangleright)) \nbsp x" non associative with precedence 45 
   for @{'order_converge $a $x}.
-notation > "a 'order_converges' x" non associative with precedence 50 
+notation > "a 'order_converges' x" non associative with precedence 45 
   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 *)
+alias symbol "and" = "constructive and".
+definition segment ≝ λO:ordered_set.λa,b:O.λx:O.(x ≤ b) ∧ (a ≤ x).
 
-definition segment ≝ λO:ordered_set.λa,b:O.λx:O.
-  (cic:/matita/logic/connectives/And.ind#xpointer(1/1) (x ≤ b) (a ≤ x)).
-
-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).
-
-notation "x \in [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).
-
-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' 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).
+notation "[a,b]" left associative with precedence 70 for @{'segment $a $b}.
+interpretation "Ordered set sergment" 'segment a b = (segment _ a b).
 
-interpretation "Type exists" 'exists \eta.x =
-  (cic:/matita/dama/supremum/sigma.ind#xpointer(1/1) _ x).
+notation "hvbox(x \in break [a,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).
 
 lemma segment_ordered_set: 
   ∀O:ordered_set.∀u,v:O.ordered_set.
 intros (O u v); apply (mk_ordered_set (∃x.x ∈ [u,v]));
-[1: intros (x y); apply (fst x ≰ fst y);
+[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]
 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). 
-    (λn.fst (a n)) is_increasing ∧ 
-    (fst x) is_supremum (λn.fst (a n)) → a ↑ x.
+  ∀O:ordered_set.∀l,u:O.∀a:sequence {[l,u]}.∀x:{[l,u]}. 
+    ⌊n,\fst (a n)⌋ is_increasing ∧ 
+    (\fst x) is_supremum ⌊n,\fst (a n)⌋ → a ↑ x.
 intros; split; cases H; clear H; 
 [1: apply H1;
 |2: cases H2; split; clear H2;
     [1: apply H;
-    |2: clear H; intro y0; apply (H3 (fst y0));]]
+    |2: clear H; intro y0; apply (H3 (\fst y0));]]
 qed.
 
-(* Definition 2.10 *)
-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).
- 
-lemma square_ordered_set: ordered_set → ordered_set.
-intro O; apply (mk_ordered_set (pair 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);
-|3: intros 3 (x0 y0 z0); cases x0 (x1 x2); cases y0 (y1 y2) ; cases z0 (z1 z2); 
-    clear x0 y0 z0; simplify; intro H; cases H (H1 H1); clear H;
-    [1: cases (os_cotransitive ??? z1 H1); [left; left|right;left]assumption;
-    |2: cases (os_cotransitive ??? z2 H1); [left;right|right;right]assumption]]
+lemma segment_preserves_infimum:
+  ∀O:ordered_set.∀l,u:O.∀a:sequence {[l,u]}.∀x:{[l,u]}. 
+    ⌊n,\fst (a n)⌋ is_decreasing ∧ 
+    (\fst x) is_infimum ⌊n,\fst (a n)⌋ → a ↓ x.
+intros; split; cases H; clear H; 
+[1: apply H1;
+|2: cases H2; split; clear H2;
+    [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:square_ordered_set O.
-  (cic:/matita/logic/connectives/And.ind#xpointer(1/1)
-   (cic:/matita/logic/connectives/And.ind#xpointer(1/1) (fst x ≤ b) (a ≤ fst x))
-   (cic:/matita/logic/connectives/And.ind#xpointer(1/1) (snd x ≤ b) (a ≤ snd x))).
+  λO:ordered_set.λa,b:O.λx:O square.
+    And4 (\fst x ≤ b) (a ≤ \fst x) (\snd x ≤ b) (a ≤ \snd x).
  
 definition convex ≝
-  λO:ordered_set.λU:square_ordered_set O → Prop.
-  ∀p.U p → fst p ≤ snd p → ∀y. square_segment ? (fst p) (snd p) y → U y.
+  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 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).
+
+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).
     
 (* 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. 
 
-     
-    
\ No newline at end of file
+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.