- ∀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:
-
-lapply (restrict_uniform_convergence ? H ?? (H1 l u)
- (λn:nat.sig_in ?? (a n) (H2 n)) x);
- [2:split; assumption]
\ No newline at end of file
+ ∀h:x ∈ [l,u].
+ uniform_converge {[l,u]} (⌊n,≪a n,H n≫⌋) ≪x,h≫.
+intros (C S);
+generalize in match (order_converges_bigger_lowsegment ???? H1 ? H2);
+generalize in match (order_converges_smaller_upsegment ???? H1 ? H2);
+cases H2 (xi yi Hx Hy Hxy); clear H2; simplify in ⊢ (% → % → ?); intros;
+cut (∀i.xi i ∈ [l,u]) as Hxi; [2:
+ intros; split; [2:apply H3] cases (Hxy i) (H5 _); cases H5 (H7 _);
+ apply (le_transitive ???? (H7 0)); simplify;
+ cases (H1 i); assumption;] clear H3;
+cut (∀i.yi i ∈ [l,u]) as Hyi; [2:
+ intros; split; [apply H2] cases (Hxy i) (_ H5); cases H5 (H7 _);
+ apply (le_transitive ????? (H7 0)); simplify;
+ cases (H1 i); assumption;] clear H2;
+split;
+[1: cases Hx; cases H3; cases Hy; cases H7; split;
+ [1: apply (le_transitive ???? (H8 0)); cases (Hyi 0); assumption
+ |2: apply (le_transitive ????? (H4 0)); cases (Hxi 0); assumption]
+|2: intros 3;
+ lapply (uparrow_upperlocated ? xi x Hx)as Ux;
+ lapply (downarrow_lowerlocated ? yi x Hy)as Uy;
+ letin Xi ≝ (⌊n,≪xi n,Hxi n≫⌋);
+ letin Yi ≝ (⌊n,≪yi n,Hyi n≫⌋);
+ letin Ai ≝ (⌊n,≪a n,H1 n≫⌋);
+ apply (sandwich {[l,u]} ≪x,h≫ Xi Yi Ai); try assumption;
+ [1: intro j; cases (Hxy j); cases H3; cases H4; split;
+ [apply (H5 0);|apply (H7 0)]
+ |2: cases (restrict_uniform_convergence_uparrow ? S ?? (H l u) Xi x Hx);
+ apply (H4 h);
+ |3: cases (restrict_uniform_convergence_downarrow ? S ?? (H l u) Yi x Hy);
+ apply (H4 h);]]
+qed.