-[1: unfold in H2; cases H2; clear H2;unfold in H3 H4; cases H4; clear H4; unfold in H2;
- cases (wloss_prop (os_l C)) (W W); apply prove_in_segment; unfold; rewrite <W;
- simplify;
- [ apply (le_transitive ?? x ? (H2 O));
- lapply (under_wloss_upperbound (os_r C) ?? (h_segment_upperbound (os_r C) s a) O) as K;
- unfold in K; whd in K:(?%????); simplify in K; rewrite <W in K; apply K;
- | intro; cases (H5 ? H4); clear H5 H4;
- lapply (under_wloss_upperbound (os_l C) ?? (segment_upperbound s a) w) as K;
- unfold in K; whd in K:(?%????); simplify in K; rewrite <W in K;
- apply K; apply H6;
- | intro; unfold in H4; rewrite <W in H4;
- lapply depth=0 (H5 (seg_u_ (os_l C) s)) as k; unfold in k:(%???→?);
- simplify in k; rewrite <W in k; lapply (k
- simplify;intro; cases (H5 ? H4); clear H5 H4;
- lapply (under_wloss_upperbound (os_l C) ?? (segment_upperbound s a) w) as K;
- unfold in K; whd in K:(?%????); simplify in K; rewrite <W in K;
- apply K; apply H6;
-
-
-
- cases H2 (Ha Hx); clear H2; cases Hx; split;
- lapply depth=0 (under_wloss_upperbound (os_l C) ?? (segment_upperbound s a) O) as W1;
- lapply depth=0 (under_wloss_upperbound (os_r C) ?? (h_segment_upperbound (os_r C) s a) O) as W2;
- lapply (H2 O); simplify in Hletin; simplify in W2 W1;
- cases a in Hletin W2 W1; simplify; cases (f O); simplify; intros;
- whd in H6:(? % ? ? ? ?);
- simplify in H6:(%);
- cases (wloss_prop (os_l C)); rewrite <H8 in H5 H6 ⊢ %;
- [ change in H6 with (le (os_l C) (seg_l_ (os_l C) s) w);
- apply (le_transitive ??? H6 H7);
- | apply (le_transitive (seg_u_ (os_l C) s) w x H6 H7);
- |
- lapply depth=0 (supremum_is_upper_bound ? x Hx (seg_u_ (os_l C) s)) as K;
- lapply depth=0 (under_wloss_upperbound (os_l C) ?? (segment_upperbound s a));
- apply K; intro; lapply (Hletin n); unfold; unfold in Hletin1;
- rewrite < H8 in Hletin1; intro; apply Hletin1; clear Hletin1; apply H9;
- | lapply depth=0 (h_supremum_is_upper_bound (os_r C) ⌊n,\fst (a n)⌋ x Hx (seg_l_ (os_r C) s)) as K;
- lapply depth=0 (under_wloss_upperbound (os_r C) ?? (h_segment_upperbound (os_r C) s a));
- apply K; intro; lapply (Hletin n); unfold; unfold in Hletin1;
-whd in Hletin1:(? % ? ? ? ?);
-simplify in Hletin1:(%);
- rewrite < H8 in Hletin1; intro; apply Hletin1; clear Hletin1; apply H9;
-
-
- apply (segment_upperbound ? l);
- generalize in match (H2 O); generalize in match Hx; unfold supremum;
- unfold upper_bound; whd in ⊢ (?→%→?); rewrite < H4;
- split; unfold; rewrite < H4; simplify;
- [1: lapply (infimum_is_lower_bound ? ? Hx u);
-
-
-
-split;
- [1: apply (supremum_is_upper_bound ? x Hx u);
- apply (segment_upperbound ? l);
- |2: apply (le_transitive l ? x ? (H2 O));
- apply (segment_lowerbound ? l u a 0);]