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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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17 record ordered_uniform_space_ : Type ≝ {
19 ous_us_: uniform_space;
20 with_ : us_carr ous_us_ = bishop_set_of_ordered_set ous_os
23 lemma ous_unifspace: ordered_uniform_space_ → uniform_space.
24 intro X; apply (mk_uniform_space (bishop_set_of_ordered_set X));
25 unfold bishop_set_OF_ordered_uniform_space_;
26 [1: rewrite < (with_ X); simplify; apply (us_unifbase (ous_us_ X));
27 |2: cases (with_ X); simplify; apply (us_phi1 (ous_us_ X));
28 |3: cases (with_ X); simplify; apply (us_phi2 (ous_us_ X));
29 |4: cases (with_ X); simplify; apply (us_phi3 (ous_us_ X));
30 |5: cases (with_ X); simplify; apply (us_phi4 (ous_us_ X))]
33 coercion ous_unifspace.
35 record ordered_uniform_space : Type ≝ {
36 ous_stuff :> ordered_uniform_space_;
37 ous_convex: ∀U.us_unifbase ous_stuff U → convex ous_stuff U
40 definition half_ordered_set_OF_ordered_uniform_space : ordered_uniform_space → half_ordered_set.
41 intro; compose ordered_set_OF_ordered_uniform_space with os_l. apply (f o);
44 definition invert_os_relation ≝
45 λC:ordered_set.λU:C squareO → Prop.
46 λx:C squareO. U 〈\snd x,\fst x〉.
48 interpretation "relation invertion" 'invert a = (invert_os_relation _ a).
49 interpretation "relation invertion" 'invert_symbol = (invert_os_relation _).
50 interpretation "relation invertion" 'invert_appl a x = (invert_os_relation _ a x).
52 lemma hint_segment: ∀O.
53 segment (Type_of_ordered_set O) →
54 segment (hos_carr (os_l O)).
58 coercion hint_segment nocomposites.
60 lemma segment_square_of_ordered_set_square:
61 ∀O:ordered_set.∀s:‡O.∀x:O squareO.
62 \fst x ∈ s → \snd x ∈ s → {[s]} squareO.
63 intros; split; exists; [1: apply (\fst x) |3: apply (\snd x)] assumption;
66 coercion segment_square_of_ordered_set_square with 0 2 nocomposites.
68 alias symbol "pi1" (instance 4) = "exT \fst".
69 alias symbol "pi1" (instance 2) = "exT \fst".
70 lemma ordered_set_square_of_segment_square :
71 ∀O:ordered_set.∀s:‡O.{[s]} squareO → O squareO ≝
72 λO:ordered_set.λs:‡O.λb:{[s]} squareO.〈\fst(\fst b),\fst(\snd b)〉.
74 coercion ordered_set_square_of_segment_square nocomposites.
76 lemma restriction_agreement :
77 ∀O:ordered_uniform_space.∀s:‡O.∀P:{[s]} squareO → Prop.∀OP:O squareO → Prop.Prop.
78 apply(λO:ordered_uniform_space.λs:‡O.
79 λP:{[s]} squareO → Prop. λOP:O squareO → Prop.
80 ∀b:O squareO.∀H1,H2.(P b → OP b) ∧ (OP b → P b));
81 [5,7: apply H1|6,8:apply H2]skip;
84 lemma unrestrict: ∀O:ordered_uniform_space.∀s:‡O.∀U,u.∀x:{[s]} squareO.
85 restriction_agreement ? s U u → U x → u x.
86 intros 6; cases x (b b1); cases b (w1 H1); cases b1 (w2 H2); clear b b1 x;
87 cases (H 〈w1,w2〉 H1 H2) (L _); intro Uw; apply L; apply Uw;
90 lemma restrict: ∀O:ordered_uniform_space.∀s:‡O.∀U,u.∀x:{[s]} squareO.
91 restriction_agreement ? s U u → u x → U x.
92 intros 5; cases x (b b1); cases b (w1 H1); cases b1 (w2 H2); clear b1 b x;
93 intros (Ra uw); cases (Ra 〈w1,w2〉 H1 H2) (_ R); apply R; apply uw;
96 lemma invert_restriction_agreement:
97 ∀O:ordered_uniform_space.∀s:‡O.
98 ∀U:{[s]} squareO → Prop.∀u:O squareO → Prop.
99 restriction_agreement ? s U u →
100 restriction_agreement ? s (\inv U) (\inv u).
101 intros 8; split; intro;
102 [1: apply (unrestrict ???? (segment_square_of_ordered_set_square ?? 〈\snd b,\fst b〉 H2 H1) H H3);
103 |2: apply (restrict ???? (segment_square_of_ordered_set_square ?? 〈\snd b,\fst b〉 H2 H1) H H3);]
107 ∀O:ordered_set.∀s:‡O.(bishop_set_of_ordered_set {[s]}) squareB → (bishop_set_of_ordered_set O) squareB ≝
108 λO:ordered_set.λs:‡O.λb:{[s]} squareO.〈\fst(\fst b),\fst(\snd b)〉.
110 coercion bs2_of_bss2 nocomposites.
114 ∀O:ordered_uniform_space.∀s:‡(ordered_set_OF_ordered_uniform_space O).
118 (bishop_set_of_ordered_set
120 (ordered_set_OF_ordered_uniform_space O) s))).
121 (\fst x) ≈ (\snd x) →
122 (\fst (bs2_of_bss2 (ordered_set_OF_ordered_uniform_space O) s x))
124 (\snd (bs2_of_bss2 (ordered_set_OF_ordered_uniform_space O) s x)).
125 intros 3; cases x (a b); clear x; cases a (x Hx); cases b (y Hy); clear a b;
126 simplify; intros 2 (K H); apply K; clear K; whd; whd in H; cases H;
127 [left|right] apply x2sx; assumption;
131 lemma segment_ordered_uniform_space:
132 ∀O:ordered_uniform_space.∀s:‡O.ordered_uniform_space.
133 intros (O s); apply mk_ordered_uniform_space;
134 [1: apply (mk_ordered_uniform_space_ {[s]});
135 [1: alias symbol "and" = "constructive and".
136 letin f ≝ (λP:{[s]} squareO → Prop. ∃OP:O squareO → Prop.
137 (us_unifbase O OP) ∧ restriction_agreement ?? P OP);
138 apply (mk_uniform_space (bishop_set_of_ordered_set {[s]}) f);
139 [1: intros (U H); intro x; simplify;
140 cases H (w Hw); cases Hw (Gw Hwp); clear H Hw; intro Hm;
141 lapply (us_phi1 O w Gw x (a2sa ??? Hm)) as IH;
142 apply (restrict ? s ??? Hwp IH);
143 |2: intros (U V HU HV); cases HU (u Hu); cases HV (v Hv); clear HU HV;
144 cases Hu (Gu HuU); cases Hv (Gv HvV); clear Hu Hv;
145 cases (us_phi2 O u v Gu Gv) (w HW); cases HW (Gw Hw); clear HW;
146 exists; [apply (λb:{[s]} squareB.w b)] split;
147 [1: unfold f; simplify; clearbody f;
148 exists; [apply w]; split; [assumption] intro b; simplify;
149 unfold segment_square_of_ordered_set_square;
150 cases b; intros; split; intros; assumption;
151 |2: intros 2 (x Hx); cases (Hw ? Hx); split;
152 [apply (restrict O s ??? HuU H)|apply (restrict O s ??? HvV H1);]]
153 |3: intros (U Hu); cases Hu (u HU); cases HU (Gu HuU); clear Hu HU;
154 cases (us_phi3 O u Gu) (w HW); cases HW (Gw Hwu); clear HW;
155 exists; [apply (λx:{[s]} squareB.w x)] split;
156 [1: exists;[apply w];split;[assumption] intros; simplify; intro;
157 unfold segment_square_of_ordered_set_square;
158 cases b; intros; split; intro; assumption;
159 |2: intros 2 (x Hx); apply (restrict O s ??? HuU); apply Hwu;
160 cases Hx (m Hm); exists[apply (\fst m)] apply Hm;]
161 |4: intros (U HU x); cases HU (u Hu); cases Hu (Gu HuU); clear HU Hu;
162 cases (us_phi4 O u Gu x) (Hul Hur);
164 [1: lapply (invert_restriction_agreement O s ?? HuU) as Ra;
165 apply (restrict O s ?? x Ra);
166 apply Hul; apply (unrestrict O s ??? HuU H);
167 |2: apply (restrict O s ??? HuU); apply Hur;
168 apply (unrestrict O s ??? (invert_restriction_agreement O s ?? HuU) H);]]
169 |2: simplify; reflexivity;]
170 |2: simplify; unfold convex; intros;
171 cases H (u HU); cases HU (Gu HuU); clear HU H;
174 ∀O:ordered_set.∀s:‡O.∀x,y:os_l (square_ordered_set (segment_ordered_set O s)).
175 le (os_l (square_ordered_set (segment_ordered_set O s))) x y →
177 intros 4; cases x (a1 a2); cases y (b1 b2); clear x y;
178 intros 2 (H K); apply H; clear H;
179 simplify in K:(? ? ? %);
180 simplify in K:(? ? % ?);
181 whd in ⊢ (? (? (% ?)) ? ?);
182 whd in ⊢ (? (? ((λ_:?.? ? ? (? ? (? (% ?)))) ?)) ? ?);
186 whd in ⊢ (? (% ?) ? ? ? ?);
187 simplify in ⊢ (? (% ?) ? ? ? ?);
188 simplify in ⊢ (? % ? ? ? ?);
190 cases (wloss_prop (os_l (segment_ordered_set O s)));
194 unfold hos_excess; do 2 rewrite <H;
196 SQUARE SEEMS NOT DUAL!
198 cases a1 (a _); cases a2 (b _);
199 cases b1 (c _); cases b2 (d _); clear a1 a2 b1 b2; simplify;
200 intros 2 (H K); apply H; clear H; apply K;
202 lapply (ous_convex ?? Gu p ? H2 y H3) as Cu;
203 [1: apply (unrestrict O s ??? HuU); apply H1;
204 |2: apply (restrict O s ??? HuU Cu);]]
207 interpretation "Ordered uniform space segment" 'segment_set a b =
208 (segment_ordered_uniform_space _ a b).
211 alias symbol "pi1" = "exT \fst".
212 lemma restric_uniform_convergence:
213 ∀O:ordered_uniform_space.∀l,u:O.
216 (⌊n, \fst (a n)⌋ : sequence O) uniform_converges (\fst x) →
217 a uniform_converges x.
218 intros 8; cases H1; cases H2; clear H2 H1;
219 cases (H ? H3) (m Hm); exists [apply m]; intros;
220 apply (restrict ? l u ??? H4); apply (Hm ? H1);
223 definition order_continuity ≝
224 λC:ordered_uniform_space.∀a:sequence C.∀x:C.
225 (a ↑ x → a uniform_converges x) ∧ (a ↓ x → a uniform_converges x).
227 lemma hint_boh1: ∀O. Type_OF_ordered_uniform_space O → hos_carr (os_l O).
231 coercion hint_boh1 nocomposites.
233 lemma hint_boh2: ∀O:ordered_uniform_space. hos_carr (os_l O) → Type_OF_ordered_uniform_space O.
237 coercion hint_boh2 nocomposites.