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1 (**************************************************************************)
2 (*       ___                                                              *)
3 (*      ||M||                                                             *)
4 (*      ||A||       A project by Andrea Asperti                           *)
5 (*      ||T||                                                             *)
6 (*      ||I||       Developers:                                           *)
7 (*      ||T||         The HELM team.                                      *)
8 (*      ||A||         http://helm.cs.unibo.it                             *)
9 (*      \   /                                                             *)
10 (*       \ /        This file is distributed under the terms of the       *)
11 (*        v         GNU General Public License Version 2                  *)
12 (*                                                                        *)
13 (**************************************************************************)
14
15 (* This file was automatically generated: do not edit *********************)
16
17 include "CoRN.ma".
18
19 (* $Id: StrongIVT.v,v 1.5 2004/04/23 10:01:01 lcf Exp $ *)
20
21 include "ftc/WeakIVT.ma".
22
23 include "ftc/CalculusTheorems.ma".
24
25 (* UNEXPORTED
26 Section IVT'
27 *)
28
29 (*#* ** Strong IVT for partial functions
30
31 The IVT can be generalized to arbitrary partial functions; in the first
32 part, we will simply do that, repeating the previous construction.
33
34 The same notations and conventions apply as before.
35 *)
36
37 (* UNEXPORTED
38 cic:/CoRN/ftc/StrongIVT/IVT'/a.var
39 *)
40
41 (* UNEXPORTED
42 cic:/CoRN/ftc/StrongIVT/IVT'/b.var
43 *)
44
45 (* UNEXPORTED
46 cic:/CoRN/ftc/StrongIVT/IVT'/Hab'.var
47 *)
48
49 (* UNEXPORTED
50 cic:/CoRN/ftc/StrongIVT/IVT'/Hab.var
51 *)
52
53 (* begin hide *)
54
55 inline procedural "cic:/CoRN/ftc/StrongIVT/IVT'/I.con" "IVT'__" as definition.
56
57 inline procedural "cic:/CoRN/ftc/StrongIVT/IVT'/I'.con" "IVT'__" as definition.
58
59 (* end hide *)
60
61 (* UNEXPORTED
62 cic:/CoRN/ftc/StrongIVT/IVT'/F.var
63 *)
64
65 (* UNEXPORTED
66 cic:/CoRN/ftc/StrongIVT/IVT'/contF.var
67 *)
68
69 (* begin hide *)
70
71 inline procedural "cic:/CoRN/ftc/StrongIVT/IVT'/incF.con" "IVT'__" as definition.
72
73 (* end hide *)
74
75 (* begin show *)
76
77 (* UNEXPORTED
78 cic:/CoRN/ftc/StrongIVT/IVT'/incrF.var
79 *)
80
81 (* end show *)
82
83 (* begin hide *)
84
85 inline procedural "cic:/CoRN/ftc/StrongIVT/IVT'/Ha.con" "IVT'__" as definition.
86
87 inline procedural "cic:/CoRN/ftc/StrongIVT/IVT'/Hb.con" "IVT'__" as definition.
88
89 inline procedural "cic:/CoRN/ftc/StrongIVT/IVT'/HFab'.con" "IVT'__" as definition.
90
91 (* end hide *)
92
93 (* begin show *)
94
95 (* UNEXPORTED
96 cic:/CoRN/ftc/StrongIVT/IVT'/z.var
97 *)
98
99 (* UNEXPORTED
100 cic:/CoRN/ftc/StrongIVT/IVT'/Haz.var
101 *)
102
103 (* UNEXPORTED
104 cic:/CoRN/ftc/StrongIVT/IVT'/Hzb.var
105 *)
106
107 (* end show *)
108
109 inline procedural "cic:/CoRN/ftc/StrongIVT/IVT'_seq_lemma.con" as lemma.
110
111 (* end hide *)
112
113 inline procedural "cic:/CoRN/ftc/StrongIVT/IVT'_aux_seq_type.ind".
114
115 inline procedural "cic:/CoRN/ftc/StrongIVT/IVT'_iter.con" as definition.
116
117 inline procedural "cic:/CoRN/ftc/StrongIVT/IVT'_seq.con" as definition.
118
119 inline procedural "cic:/CoRN/ftc/StrongIVT/a'_seq.con" as definition.
120
121 inline procedural "cic:/CoRN/ftc/StrongIVT/b'_seq.con" as definition.
122
123 inline procedural "cic:/CoRN/ftc/StrongIVT/a'_seq_I.con" as definition.
124
125 inline procedural "cic:/CoRN/ftc/StrongIVT/b'_seq_I.con" as definition.
126
127 inline procedural "cic:/CoRN/ftc/StrongIVT/a'_seq_less_b'_seq.con" as lemma.
128
129 inline procedural "cic:/CoRN/ftc/StrongIVT/a'_seq_less_z.con" as lemma.
130
131 inline procedural "cic:/CoRN/ftc/StrongIVT/z_less_b'_seq.con" as lemma.
132
133 inline procedural "cic:/CoRN/ftc/StrongIVT/a'_seq_mon.con" as lemma.
134
135 inline procedural "cic:/CoRN/ftc/StrongIVT/b'_seq_mon.con" as lemma.
136
137 inline procedural "cic:/CoRN/ftc/StrongIVT/a'_seq_b'_seq_dist_n.con" as lemma.
138
139 inline procedural "cic:/CoRN/ftc/StrongIVT/a'_seq_b'_seq_dist.con" as lemma.
140
141 inline procedural "cic:/CoRN/ftc/StrongIVT/a'_seq_Cauchy.con" as lemma.
142
143 inline procedural "cic:/CoRN/ftc/StrongIVT/b'_seq_Cauchy.con" as lemma.
144
145 inline procedural "cic:/CoRN/ftc/StrongIVT/IVT'/xa.con" "IVT'__" as definition.
146
147 inline procedural "cic:/CoRN/ftc/StrongIVT/IVT'/xb.con" "IVT'__" as definition.
148
149 inline procedural "cic:/CoRN/ftc/StrongIVT/a'_seq_b'_seq_lim.con" as lemma.
150
151 inline procedural "cic:/CoRN/ftc/StrongIVT/xa'_in_interval.con" as lemma.
152
153 inline procedural "cic:/CoRN/ftc/StrongIVT/IVT'_I.con" as lemma.
154
155 (* UNEXPORTED
156 End IVT'
157 *)
158
159 (*#* **Other formulations
160
161 We now generalize the various statements of the intermediate value
162 theorem to more widely applicable forms.
163 *)
164
165 inline procedural "cic:/CoRN/ftc/StrongIVT/Weak_IVT.con" as lemma.
166
167 inline procedural "cic:/CoRN/ftc/StrongIVT/IVT_inc.con" as lemma.
168
169 (* UNEXPORTED
170 Transparent Min.
171 *)
172
173 inline procedural "cic:/CoRN/ftc/StrongIVT/IVT_dec.con" as lemma.
174
175 inline procedural "cic:/CoRN/ftc/StrongIVT/IVT'_inc.con" as lemma.
176
177 (* UNEXPORTED
178 Transparent Min.
179 *)
180
181 inline procedural "cic:/CoRN/ftc/StrongIVT/IVT'_dec.con" as lemma.
182