interpretation "segment_preserves_supremum" 'segment_preserves_supremum = (h_segment_preserves_supremum (os_l _)).
interpretation "segment_preserves_infimum" 'segment_preserves_infimum = (h_segment_preserves_supremum (os_r _)).
-(* TEST, ma quanto godo! *)
-lemma segment_preserves_infimum2:
+(*
+test segment_preserves_infimum2:
∀O:ordered_set.∀s:‡O.∀a:sequence {[s]}.∀x:{[s]}.
⌊n,\fst (a n)⌋ is_decreasing ∧
(\fst x) is_infimum ⌊n,\fst (a n)⌋ → a ↓ x.
intros; apply (segment_preserves_infimum s a x H);
qed.
*)
-
+
(* Definition 2.10 *)
alias symbol "pi2" = "pair pi2".
alias symbol "pi1" = "pair pi1".
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 (hos_cotransitive ??? u Hxy) (W W);
-[2: cases (H5 ? W) (w Hw); left; exists [apply w] assumption;
+cases H3 (H4 H5); clear H3; cases (hos_cotransitive C y x u Hxy) (W W);
+[2: cases (H5 x W) (w Hw); left; exists [apply w] assumption;
|1: right; exists [apply u]; split; [apply W|apply H4]]
qed.