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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:half_ordered_set.∀s:segment O.∀x:O.
73 seg_l O s (λl.l ≤≤ x) → seg_u O s (λu.x ≤≤ u) → x ∈ s.
74 intros; unfold; cases (wloss_prop O); rewrite < H2;
78 lemma h_uparrow_to_in_segment:
85 intros (C H a H1 x H2); unfold in H2; cases H2; clear H2;unfold in H3 H4; cases H4; clear H4; unfold in H2;
86 cases (wloss_prop C) (W W); apply prove_in_segment; unfold; rewrite <W;simplify;
87 [ apply (hle_transitive ??? x ? (H2 O)); lapply (H1 O) as K; unfold in K; rewrite <W in K;
88 cases K; unfold in H4 H6; rewrite <W in H6 H4; simplify in H4 H6; assumption;
89 | intro; cases (H5 ? H4); clear H5 H4;lapply(H1 w) as K; unfold in K; rewrite<W in K;
90 cases K; unfold in H5 H4; rewrite<W in H4 H5; simplify in H4 H5; apply (H5 H6);
91 | apply (hle_transitive ??? x ? (H2 O)); lapply (H1 0) as K; unfold in K; rewrite <W in K;
92 cases K; unfold in H4 H6; rewrite <W in H4 H6; simplify in H4 H6; assumption;
93 | intro; cases (H5 ? H4); clear H5 H4;lapply(H1 w) as K; unfold in K; rewrite<W in K;
94 cases K; unfold in H5 H4; rewrite<W in H4 H5; simplify in H4 H5; apply (H4 H6);]
97 notation "'uparrow_to_in_segment'" non associative with precedence 90 for @{'uparrow_to_in_segment}.
98 notation "'downarrow_to_in_segment'" non associative with precedence 90 for @{'downarrow_to_in_segment}.
100 interpretation "uparrow_to_in_segment" 'uparrow_to_in_segment = (h_uparrow_to_in_segment (os_l _)).
101 interpretation "downarrow_to_in_segment" 'downarrow_to_in_segment = (h_uparrow_to_in_segment (os_r _)).
103 (* Lemma 3.8 NON DUALIZZATO *)
104 lemma restrict_uniform_convergence_uparrow:
105 ∀C:ordered_uniform_space.property_sigma C →
106 ∀s:segment (os_l C).exhaustive (segment_ordered_uniform_space C s) →
107 ∀a:sequence (segment_ordered_uniform_space C s).
108 ∀x:C. ⌊n,\fst (a n)⌋ ↑ x →
109 in_segment (os_l C) s x ∧ ∀h:x ∈ s.a uniform_converges ≪x,h≫.
111 [1: apply (uparrow_to_in_segment s ⌊n,\fst (a \sub n)⌋ ? x H2);
112 simplify; intros; cases (a i); assumption;
114 lapply (uparrow_upperlocated a ≪x,h≫) as Ha1;
115 [2: apply (segment_preserves_uparrow s); assumption;]
116 lapply (segment_preserves_supremum s a ≪?,h≫ H2) as Ha2;
117 cases Ha2; clear Ha2;
118 cases (H1 a a); lapply (H5 H3 Ha1) as HaC;
119 lapply (segment_cauchy C s ? HaC) as Ha;
120 lapply (sigma_cauchy ? H ? x ? Ha); [left; assumption]
121 apply (restric_uniform_convergence C s ≪x,h≫ a Hletin)]
125 ∀C. Type_OF_ordered_uniform_space1 C → hos_carr (os_r C).
126 intros; assumption; qed.
128 coercion hint_mah1 nocomposites.
131 ∀C. sequence (hos_carr (os_l C)) → sequence (hos_carr (os_r C)).
132 intros; assumption; qed.
134 coercion hint_mah2 nocomposites.
137 ∀C. Type_OF_ordered_uniform_space C → hos_carr (os_r C).
138 intros; assumption; qed.
140 coercion hint_mah3 nocomposites.
143 ∀C. sequence (hos_carr (os_r C)) → sequence (hos_carr (os_l C)).
144 intros; assumption; qed.
146 coercion hint_mah4 nocomposites.
149 ∀C. segment (hos_carr (os_r C)) → segment (hos_carr (os_l C)).
150 intros; assumption; qed.
152 coercion hint_mah5 nocomposites.
155 ∀C. segment (hos_carr (os_l C)) → segment (hos_carr (os_r C)).
156 intros; assumption; qed.
158 coercion hint_mah6 nocomposites.
160 lemma restrict_uniform_convergence_downarrow:
161 ∀C:ordered_uniform_space.property_sigma C →
162 ∀s:segment (os_l C).exhaustive (segment_ordered_uniform_space C s) →
163 ∀a:sequence (segment_ordered_uniform_space C s).
164 ∀x:C. ⌊n,\fst (a n)⌋ ↓ x →
165 in_segment (os_l C) s x ∧ ∀h:x ∈ s.a uniform_converges ≪x,h≫.
167 [1: apply (downarrow_to_in_segment s ⌊n,\fst (a n)⌋ ? x); [2: apply H2];
168 simplify; intros; cases (a i); assumption;
170 lapply (downarrow_lowerlocated a ≪x,h≫) as Ha1;
171 [2: apply (segment_preserves_downarrow s a x h H2);]
172 lapply (segment_preserves_infimum s a ≪?,h≫ H2) as Ha2;
173 cases Ha2; clear Ha2;
174 cases (H1 a a); lapply (H6 H3 Ha1) as HaC;
175 lapply (segment_cauchy C s ? HaC) as Ha;
176 lapply (sigma_cauchy ? H ? x ? Ha); [right; assumption]
177 apply (restric_uniform_convergence C s ≪x,h≫ a Hletin)]