- x ∈ [l,u] ∧
- ∀h:x ∈ [l,u]. (* manca il pullback? *)
- uniform_converge
- (uniform_space_OF_ordered_uniform_space
- (segment_ordered_uniform_space C l u))
- (λn.sig_in C (λx.x∈[l,u]) (a n) (H n))
- (sig_in ?? x h).
-intros; cases H3 (xi H4); cases H4 (yi H5); cases H5; cases H6; cases H8;
-cases H9; cases H10; cases H11; clear H3 H4 H5 H6 H8 H9 H10 H11 H15 H16;
-lapply (uparrow_upperlocated ? xi x)as Ux;[2: split; assumption]
-lapply (downarrow_lowerlocated ? yi x)as Uy;[2: split; assumption]
-cases (restrict_uniform_convergence ? H ?? (H1 l u) (λn:nat.sig_in ?? (a n) (H2 n)) x);
-[ split; assumption]
-split; simplify;
- [1: intro; cases (H7 n); cases H3;
-
-
- lapply (sandwich ? x xi yi a );
- [2: intro; cases (H7 i); cases H3; cases H4; split[apply (H5 0)|apply (H8 0)]
- |3: intros 2;
- cases (restrict_uniform_convergence ? H ?? (H1 l u) ? x);
- [1:
+ x ∈ s ∧
+ ∀h:x ∈ s.
+ uniform_converge {[s]} (⌊n,≪a n,H n≫⌋) ≪x,h≫.
+intros (C S);
+generalize in match (order_converges_bigger_lowsegment ? a s H1 ? H2);
+generalize in match (order_converges_smaller_upsegment ? a s H1 ? H2);
+cases H2 (xi yi Hx Hy Hxy); clear H2; simplify in ⊢ ((?→???%) → (?→???%) → ?); intros;
+cut (∀i.xi i ∈ s) as Hxi; [2:
+ intros; apply (prove_in_segment (os_l C)); [apply (H3 i)] cases (Hxy i) (H5 _); cases H5 (H7 _);
+ lapply (H7 0) as K; cases (cases_in_segment ? s ? (H1 i)) (Pl Pu);
+ simplify in K:(? ? % ?); apply (hle_transitive (os_l C) (xi i) (a i) 𝕦_s K Pu);] clear H3;
+cut (∀i.yi i ∈ s) as Hyi; [2:
+ intros; apply (prove_in_segment (os_l C)); [2:apply (H2 i)] cases (Hxy i) (_ H5); cases H5 (H7 _);
+ lapply (H7 0) as K; cases (cases_in_segment ? s ? (H1 i)) (Pl Pu); simplify in K;
+ apply (le_transitive 𝕝_s ? ? ? K);apply Pl;] clear H2;
+letin Xi ≝ (⌊n,≪xi n, Hxi n≫⌋);
+letin Yi ≝ (⌊n,≪yi n, Hyi n≫⌋);
+cases (restrict_uniform_convergence_uparrow ? S ? (H s) Xi x Hx);
+cases (restrict_uniform_convergence_downarrow ? S ? (H s) Yi x Hy);
+split; [1: assumption]
+intros 3;
+lapply (uparrow_upperlocated xi x Hx)as Ux;
+lapply (downarrow_lowerlocated yi x Hy)as Uy;
+letin Ai ≝ (⌊n,≪a n, H1 n≫⌋);
+apply (sandwich {[s]} ≪x, h≫ Xi Yi Ai); [4: assumption;|2:apply H3;|3:apply H5]
+intro j; cases (Hxy j); cases H7; cases H8; split;
+[apply (l2sl ? s (Xi j) (Ai j) (H9 0));|apply (l2sl ? s (Ai j) (Yi j) (H11 0))]
+qed.
+
+