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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 *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "ordered_uniform.ma".
16 include "property_sigma.ma".
18 lemma h_segment_upperbound:
21 ∀a:sequence (half_segment_ordered_set C s).
22 (seg_u C s) (upper_bound ? ⌊n,\fst (a n)⌋).
23 intros; cases (wloss_prop C); unfold; rewrite < H; simplify; intro n;
24 cases (a n); simplify; unfold in H1; rewrite < H in H1; cases H1;
25 simplify in H2 H3; rewrite < H in H2 H3; assumption;
28 notation "'segment_upperbound'" non associative with precedence 90 for @{'segment_upperbound}.
29 notation "'segment_lowerbound'" non associative with precedence 90 for @{'segment_lowerbound}.
31 interpretation "segment_upperbound" 'segment_upperbound = (h_segment_upperbound (os_l _)).
32 interpretation "segment_lowerbound" 'segment_lowerbound = (h_segment_upperbound (os_r _)).
34 lemma h_segment_preserves_uparrow:
35 ∀C:half_ordered_set.∀s:segment C.∀a:sequence (half_segment_ordered_set C s).
36 ∀x,h. uparrow C ⌊n,\fst (a n)⌋ x → uparrow (half_segment_ordered_set C s) a ≪x,h≫.
37 intros; cases H (Ha Hx); split;
38 [ intro n; intro H; apply (Ha n); apply (sx2x ???? H);
40 [ intro n; intro H; apply (H1 n);apply (sx2x ???? H);
41 | intros; cases (H2 (\fst y)); [2: apply (sx2x ???? H3);]
42 exists [apply w] apply (x2sx ?? (a w) y H4);]]
45 notation "'segment_preserves_uparrow'" non associative with precedence 90 for @{'segment_preserves_uparrow}.
46 notation "'segment_preserves_downarrow'" non associative with precedence 90 for @{'segment_preserves_downarrow}.
48 interpretation "segment_preserves_uparrow" 'segment_preserves_uparrow = (h_segment_preserves_uparrow (os_l _)).
49 interpretation "segment_preserves_downarrow" 'segment_preserves_downarrow = (h_segment_preserves_uparrow (os_r _)).
53 ∀C:ordered_uniform_space.∀s:‡C.∀a:sequence {[s]}.
54 a is_cauchy → ⌊n,\fst (a n)⌋ is_cauchy.
56 alias symbol "pi1" (instance 3) = "pair pi1".
57 alias symbol "pi2" = "pair pi2".
58 apply (H (λx:{[s]} squareB.U 〈\fst (\fst x),\fst (\snd x)〉));
59 (unfold segment_ordered_uniform_space; simplify);
60 exists [apply U] split; [assumption;]
61 intro; cases b; intros; simplify; split; intros; assumption;
65 definition exhaustive ≝
66 λC:ordered_uniform_space.
68 (a is_increasing → a is_upper_located → a is_cauchy) ∧
69 (b is_decreasing → b is_lower_located → b is_cauchy).
71 lemma prove_in_segment:
72 ∀O:ordered_set.∀s:segment (os_l O).∀x:O.
73 𝕝_s (λl.l ≤ x) → 𝕦_s (λu.x ≤ u) → x ∈ s.
74 intros; unfold; cases (wloss_prop (os_l O)); rewrite < H2;
78 lemma under_wloss_upperbound:
79 ∀C:half_ordered_set.∀s:segment C.∀a:sequence C.
80 seg_u C s (upper_bound C a) →
81 ∀i.seg_u C s (λu.a i ≤≤ u).
82 intros; unfold in H; unfold;
83 cases (wloss_prop C); rewrite <H1 in H ⊢ %;
88 (* Lemma 3.8 NON DUALIZZATO *)
89 lemma restrict_uniform_convergence_uparrow:
90 ∀C:ordered_uniform_space.property_sigma C →
91 ∀s:segment (os_l C).exhaustive (segment_ordered_uniform_space C s) →
92 ∀a:sequence (segment_ordered_uniform_space C s).
93 ∀x:C. ⌊n,\fst (a n)⌋ ↑ x →
94 in_segment (os_l C) s x ∧ ∀h:x ∈ s.a uniform_converges ≪x,h≫.
96 [1: unfold in H2; cases H2; clear H2;unfold in H3 H4; cases H4; clear H4; unfold in H2;
97 cases (wloss_prop (os_l C)) (W W); apply prove_in_segment; unfold; rewrite <W;
99 [ apply (le_transitive ?? x ? (H2 O));
100 lapply (under_wloss_upperbound (os_r C) ?? (h_segment_upperbound (os_r C) s a) O) as K;
101 unfold in K; whd in K:(?%????); simplify in K; rewrite <W in K; apply K;
102 | intro; cases (H5 ? H4); clear H5 H4;
103 lapply (under_wloss_upperbound (os_l C) ?? (segment_upperbound s a) w) as K;
104 unfold in K; whd in K:(?%????); simplify in K; rewrite <W in K;
106 | intro; unfold in H4; rewrite <W in H4;
107 lapply depth=0 (H5 (seg_u_ (os_l C) s)) as k; unfold in k:(%???→?);
108 simplify in k; rewrite <W in k; lapply (k
109 simplify;intro; cases (H5 ? H4); clear H5 H4;
110 lapply (under_wloss_upperbound (os_l C) ?? (segment_upperbound s a) w) as K;
111 unfold in K; whd in K:(?%????); simplify in K; rewrite <W in K;
116 cases H2 (Ha Hx); clear H2; cases Hx; split;
117 lapply depth=0 (under_wloss_upperbound (os_l C) ?? (segment_upperbound s a) O) as W1;
118 lapply depth=0 (under_wloss_upperbound (os_r C) ?? (h_segment_upperbound (os_r C) s a) O) as W2;
119 lapply (H2 O); simplify in Hletin; simplify in W2 W1;
120 cases a in Hletin W2 W1; simplify; cases (f O); simplify; intros;
121 whd in H6:(? % ? ? ? ?);
123 cases (wloss_prop (os_l C)); rewrite <H8 in H5 H6 ⊢ %;
124 [ change in H6 with (le (os_l C) (seg_l_ (os_l C) s) w);
125 apply (le_transitive ??? H6 H7);
126 | apply (le_transitive (seg_u_ (os_l C) s) w x H6 H7);
128 lapply depth=0 (supremum_is_upper_bound ? x Hx (seg_u_ (os_l C) s)) as K;
129 lapply depth=0 (under_wloss_upperbound (os_l C) ?? (segment_upperbound s a));
130 apply K; intro; lapply (Hletin n); unfold; unfold in Hletin1;
131 rewrite < H8 in Hletin1; intro; apply Hletin1; clear Hletin1; apply H9;
132 | 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;
133 lapply depth=0 (under_wloss_upperbound (os_r C) ?? (h_segment_upperbound (os_r C) s a));
134 apply K; intro; lapply (Hletin n); unfold; unfold in Hletin1;
135 whd in Hletin1:(? % ? ? ? ?);
136 simplify in Hletin1:(%);
137 rewrite < H8 in Hletin1; intro; apply Hletin1; clear Hletin1; apply H9;
140 apply (segment_upperbound ? l);
141 generalize in match (H2 O); generalize in match Hx; unfold supremum;
142 unfold upper_bound; whd in ⊢ (?→%→?); rewrite < H4;
143 split; unfold; rewrite < H4; simplify;
144 [1: lapply (infimum_is_lower_bound ? ? Hx u);
149 [1: apply (supremum_is_upper_bound ? x Hx u);
150 apply (segment_upperbound ? l);
151 |2: apply (le_transitive l ? x ? (H2 O));
152 apply (segment_lowerbound ? l u a 0);]
154 lapply (uparrow_upperlocated a ≪x,h≫) as Ha1;
155 [2: apply (segment_preserves_uparrow C l u);split; assumption;]
156 lapply (segment_preserves_supremum C l u a ≪?,h≫) as Ha2;
157 [2:split; assumption]; cases Ha2; clear Ha2;
158 cases (H1 a a); lapply (H6 H4 Ha1) as HaC;
159 lapply (segment_cauchy ? l u ? HaC) as Ha;
160 lapply (sigma_cauchy ? H ? x ? Ha); [left; split; assumption]
161 apply restric_uniform_convergence; assumption;]
165 ∀C. Type_OF_ordered_uniform_space1 C → hos_carr (os_r C).
166 intros; assumption; qed.
168 coercion hint_mah1 nocomposites.
171 ∀C. sequence (hos_carr (os_l C)) → sequence (hos_carr (os_r C)).
172 intros; assumption; qed.
174 coercion hint_mah2 nocomposites.
177 ∀C. Type_OF_ordered_uniform_space C → hos_carr (os_r C).
178 intros; assumption; qed.
180 coercion hint_mah3 nocomposites.
183 ∀C. sequence (hos_carr (os_r C)) → sequence (hos_carr (os_l C)).
184 intros; assumption; qed.
186 coercion hint_mah4 nocomposites.
188 lemma restrict_uniform_convergence_downarrow:
189 ∀C:ordered_uniform_space.property_sigma C →
190 ∀l,u:C.exhaustive {[l,u]} →
191 ∀a:sequence {[l,u]}.∀x: C. ⌊n,\fst (a n)⌋ ↓ x →
192 x∈[l,u] ∧ ∀h:x ∈ [l,u].a uniform_converges ≪x,h≫.
193 intros; cases H2 (Ha Hx); clear H2; cases Hx; split;
195 [2: apply (infimum_is_lower_bound ? x Hx l);
196 apply (segment_lowerbound ? l u);
197 |1: lapply (ge_transitive ? ? x ? (H2 O)); [apply u||assumption]
198 apply (segment_upperbound ? l u a 0);]
200 lapply (downarrow_lowerlocated a ≪x,h≫) as Ha1;
201 [2: apply (segment_preserves_downarrow ? l u);split; assumption;]
202 lapply (segment_preserves_infimum C l u a ≪x,h≫) as Ha2;
203 [2:split; assumption]; cases Ha2; clear Ha2;
204 cases (H1 a a); lapply (H7 H4 Ha1) as HaC;
205 lapply (segment_cauchy ? l u ? HaC) as Ha;
206 lapply (sigma_cauchy ? H ? x ? Ha); [right; split; assumption]
207 apply restric_uniform_convergence; assumption;]