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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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11 (* v GNU General Public License Version 2 *)
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15 (* manca un pezzo del pullback, se inverto poi non tipa *)
16 include "sandwich.ma".
17 include "property_exhaustivity.ma".
20 alias symbol "low" = "lower".
21 alias id "le" = "cic:/matita/dama/ordered_set/le.con".
22 lemma order_converges_bigger_lowsegment:
24 ∀a:sequence (os_l C).∀s:segment C.∀H:∀i:nat.a i ∈ s.
25 ∀x:C.∀p:order_converge C a x.
26 ∀j. seg_l (os_l C) s (λl.le (os_l C) l (pi1exT23 ???? p j)).
27 intros; cases p (xi yi Ux Dy Hxy); clear p; simplify;
28 cases Ux (Ixi Sxi); clear Ux; cases Dy (Dyi Iyi); clear Dy;
29 cases (Hxy j) (Ia Sa); clear Hxy; cases Ia (Da SSa); cases Sa (Inca SIa); clear Ia Sa;
30 cases (wloss_prop (os_l C))(W W);rewrite <W;
31 [ intro H2; cases (SSa (seg_l_ C s) H2) (w Hw); simplify in Hw;
32 lapply (H (w+j)) as K; unfold in K;
33 whd in K:(? % ? ? ? ?); simplify in K:(%); rewrite <W in K; cases K;
34 whd in H1:(? % ? ? ? ?); simplify in H1:(%); rewrite <W in H1;
35 simplify in H1; apply (H1 Hw);
36 | intro H2; cases (SSa (seg_u_ C s) H2) (w Hw); simplify in Hw;
37 lapply (H (w+j)) as K; unfold in K;
38 whd in K:(? % ? ? ? ?);simplify in K:(%); rewrite <W in K; cases K;
39 whd in H3:(? % ? ? ? ?);simplify in H3:(%); rewrite <W in H3;
40 simplify in H3; apply (H3 Hw);]
43 lemma order_converges_smaller_upsegment:
45 ∀a:sequence (os_l C).∀s:segment C.∀H:∀i:nat.a i ∈ s.
46 ∀x:C.∀p:order_converge C a x.
47 ∀j. seg_u (os_l C) s (λu.le (os_l C) (pi2exT23 ???? p j) u).
48 intros; cases p (xi yi Ux Dy Hxy); clear p; simplify;
49 cases Ux (Ixi Sxi); clear Ux; cases Dy (Dyi Iyi); clear Dy;
50 cases (Hxy j) (Ia Sa); clear Hxy; cases Ia (Da SSa); cases Sa (Inca SIa); clear Ia Sa;
51 cases (wloss_prop (os_l C))(W W); unfold os_r; unfold dual_hos; simplify;rewrite <W;
52 [ intro H2; cases (SIa (seg_u_ (os_l C) s) H2) (w Hw); simplify in Hw;
53 lapply (H (w+j)) as K; unfold in K; whd in K:(? % ? ? ? ?); simplify in K:(%);
54 rewrite <W in K; cases K; whd in H3:(? % ? ? ? ?); simplify in H3:(%); rewrite <W in H3;
55 simplify in H3; apply (H3 Hw);
56 | intro H2; cases (SIa (seg_l_ C s) H2) (w Hw); simplify in Hw;
57 lapply (H (w+j)) as K; unfold in K; whd in K:(? % ? ? ? ?); simplify in K:(%);
58 rewrite <W in K; cases K; whd in H1:(? % ? ? ? ?); simplify in H1:(%);
59 rewrite <W in H1; simplify in H1; apply (H1 Hw);]
62 lemma trans_under_upp:
63 ∀O:ordered_set.∀s:‡O.∀x,y:O.
64 x ≤ y → 𝕦_s (λu.y ≤ u) → 𝕦_s (λu.x ≤ u).
65 intros; cases (wloss_prop (os_l O)) (W W); unfold; unfold in H1; rewrite<W in H1 ⊢ %;
66 apply (le_transitive ??? H H1);
69 lemma trans_under_low:
70 ∀O:ordered_set.∀s:‡O.∀x,y:O.
71 y ≤ x → 𝕝_s (λl.l ≤ y) → 𝕝_s (λl.l ≤ x).
72 intros; cases (wloss_prop (os_l O)) (W W); unfold; unfold in H1; rewrite<W in H1 ⊢ %;
73 apply (le_transitive ??? H1 H);
77 ∀C,s.∀x,y:half_segment_ordered_set C s. \fst x ≤≤ \fst y → x ≤≤ y.
78 intros; unfold in H ⊢ %; intro; apply H; clear H; unfold in H1 ⊢ %;
79 cases (wloss_prop C) (W W); whd in H1:(? (% ? ?) ? ? ? ?); simplify in H1:(%);
80 rewrite < W in H1 ⊢ %; apply H1;
85 ∀C:ordered_uniform_space.
86 (∀s:‡C.order_continuity {[s]}) →
87 ∀a:sequence C.∀s:‡C.∀H:∀i:nat.a i ∈ s.
88 ∀x:C.a order_converges x →
91 uniform_converge {[s]} (⌊n,≪a n,H n≫⌋) ≪x,h≫.
93 generalize in match (order_converges_bigger_lowsegment ? a s H1 ? H2);
94 generalize in match (order_converges_smaller_upsegment ? a s H1 ? H2);
95 cases H2 (xi yi Hx Hy Hxy); clear H2; simplify in ⊢ ((?→???%) → (?→???%) → ?); intros;
96 cut (∀i.xi i ∈ s) as Hxi; [2:
97 intros; apply (prove_in_segment (os_l C)); [apply (H3 i)] cases (Hxy i) (H5 _); cases H5 (H7 _);
98 lapply (H7 0) as K; cases (cases_in_segment ? s ? (H1 i)) (Pl Pu); simplify in K;
99 apply (trans_under_upp ?? (xi i) (a i) K Pu);] clear H3;
100 cut (∀i.yi i ∈ s) as Hyi; [2:
101 intros; apply (prove_in_segment (os_l C)); [2:apply (H2 i)] cases (Hxy i) (_ H5); cases H5 (H7 _);
102 lapply (H7 0) as K; cases (cases_in_segment ? s ? (H1 i)) (Pl Pu); simplify in K;
103 apply (trans_under_low ?? (yi i) (a i) K Pl);] clear H2;
105 [1: apply (uparrow_to_in_segment s ? Hxi ? Hx);
107 letin Xi ≝ (⌊n,≪xi n, Hxi n≫⌋);
108 letin Yi ≝ (⌊n,≪yi n, Hyi n≫⌋);
109 letin Ai ≝ (⌊n,≪a n, H1 n≫⌋);
110 apply (sandwich {[s]} ≪x, h≫ Xi Yi Ai); [4: assumption;]
111 [1: intro j; cases (Hxy j); cases H3; cases H4; split; clear H3 H4; simplify in H5 H7;
112 [apply (l2sl ? s (Xi j) (Ai j) (H5 0));|apply (l2sl ? s (Ai j) (Yi j) (H7 0))]
113 |2: cases (H s Xi ≪?,h≫) (Ux Uy); apply Ux; cases Hx; split; [intro i; apply (l2sl ? s (Xi i) (Xi (S i)) (H3 i));]
114 cases H4; split; [intro i; apply (l2sl ? s (Xi i) ≪x,h≫ (H5 i))]
115 intros (y Hy);cases (H6 (\fst y));[2:apply (sx2x ? s ? y Hy)]
116 exists [apply w] apply (x2sx ? s (Xi w) y H7);
117 |3: cases (H s Yi ≪?,h≫) (Ux Uy); apply Uy; cases Hy; split; [intro i; apply (l2sl ? s (Yi (S i)) (Yi i) (H3 i));]
118 cases H4; split; [intro i; apply (l2sl ? s ≪x,h≫ (Yi i) (H5 i))]
119 intros (y Hy);cases (H6 (\fst y));[2:apply (sx2x ? s y ≪x,h≫ Hy)]
120 exists [apply w] apply (x2sx ? s y (Yi w) H7);]]
126 ∀C:ordered_uniform_space.property_sigma C →
127 (∀s:‡C.exhaustive {[s]}) →
128 ∀a:sequence C.∀s:‡C.∀H:∀i:nat.a i ∈ s.
129 ∀x:C.a order_converges x →
132 uniform_converge {[s]} (⌊n,≪a n,H n≫⌋) ≪x,h≫.
134 generalize in match (order_converges_bigger_lowsegment ? a s H1 ? H2);
135 generalize in match (order_converges_smaller_upsegment ? a s H1 ? H2);
136 cases H2 (xi yi Hx Hy Hxy); clear H2; simplify in ⊢ ((?→???%) → (?→???%) → ?); intros;
137 cut (∀i.xi i ∈ s) as Hxi; [2:
138 intros; apply (prove_in_segment (os_l C)); [apply (H3 i)] cases (Hxy i) (H5 _); cases H5 (H7 _);
139 lapply (H7 0) as K; cases (cases_in_segment ? s ? (H1 i)) (Pl Pu); simplify in K;
140 apply (trans_under_upp ?? (xi i) (a i) K Pu);] clear H3;
141 cut (∀i.yi i ∈ s) as Hyi; [2:
142 intros; apply (prove_in_segment (os_l C)); [2:apply (H2 i)] cases (Hxy i) (_ H5); cases H5 (H7 _);
143 lapply (H7 0) as K; cases (cases_in_segment ? s ? (H1 i)) (Pl Pu); simplify in K;
144 apply (trans_under_low ?? (yi i) (a i) K Pl);] clear H2;
145 letin Xi ≝ (⌊n,≪xi n, Hxi n≫⌋);
146 letin Yi ≝ (⌊n,≪yi n, Hyi n≫⌋);
147 cases (restrict_uniform_convergence_uparrow ? S ? (H s) Xi x Hx);
148 cases (restrict_uniform_convergence_downarrow ? S ? (H s) Yi x Hy);
149 split; [1: assumption]
151 lapply (uparrow_upperlocated xi x Hx)as Ux;
152 lapply (downarrow_lowerlocated yi x Hy)as Uy;
153 letin Ai ≝ (⌊n,≪a n, H1 n≫⌋);
154 apply (sandwich {[s]} ≪x, h≫ Xi Yi Ai); [4: assumption;|2:apply H3;|3:apply H5]
155 intro j; cases (Hxy j); cases H7; cases H8; split;
156 [apply (l2sl ? s (Xi j) (Ai j) (H9 0));|apply (l2sl ? s (Ai j) (Yi j) (H11 0))]