author | bauerg |
Wed, 06 Dec 2000 12:34:40 +0100 | |
changeset 10607 | 352f6f209775 |
parent 10156 | 9d4d5852eb47 |
child 10677 | 36625483213f |
permissions | -rw-r--r-- |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1 |
(* Title : STAR.ML |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
2 |
Author : Jacques D. Fleuriot |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
3 |
Copyright : 1998 University of Cambridge |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
4 |
Description : *-transforms |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
5 |
*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
6 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
7 |
(*-------------------------------------------------------- |
10095 | 8 |
Preamble - Pulling "EX" over "ALL" |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
9 |
---------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
10 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
11 |
(* This proof does not need AC and was suggested by the |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
12 |
referee for the JCM Paper: let f(x) be least y such |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
13 |
that Q(x,y) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
14 |
*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
15 |
Goal "!!Q. ALL x. EX y. Q x y ==> EX (f :: nat => nat). ALL x. Q x (f x)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
16 |
by (res_inst_tac [("x","%x. LEAST y. Q x y")] exI 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
17 |
by (blast_tac (claset() addIs [LeastI]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
18 |
qed "no_choice"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
19 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
20 |
(*------------------------------------------------------------ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
21 |
Properties of the *-transform applied to sets of reals |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
22 |
------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
23 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
24 |
Goalw [starset_def] "*s*(UNIV::real set) = (UNIV::hypreal set)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
25 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
26 |
qed "STAR_real_set"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
27 |
Addsimps [STAR_real_set]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
28 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
29 |
Goalw [starset_def] "*s* {} = {}"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
30 |
by (Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
31 |
by (res_inst_tac [("z","x")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
32 |
by (dres_inst_tac [("x","%n. xa n")] bspec 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
33 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
34 |
qed "STAR_empty_set"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
35 |
Addsimps [STAR_empty_set]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
36 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
37 |
Goalw [starset_def] "*s* (A Un B) = *s* A Un *s* B"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
38 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
39 |
by (REPEAT(blast_tac (claset() addIs [FreeUltrafilterNat_subset]) 2)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
40 |
by (dtac FreeUltrafilterNat_Compl_mem 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
41 |
by (dtac bspec 1 THEN assume_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
42 |
by (res_inst_tac [("z","x")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
43 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
44 |
by (Fuf_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
45 |
qed "STAR_Un"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
46 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
47 |
Goalw [starset_def] "*s* (A Int B) = *s* A Int *s* B"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
48 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
49 |
by (blast_tac (claset() addIs [FreeUltrafilterNat_Int, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
50 |
FreeUltrafilterNat_subset]) 3); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
51 |
by (REPEAT(blast_tac (claset() addIs [FreeUltrafilterNat_subset]) 1)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
52 |
qed "STAR_Int"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
53 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
54 |
Goalw [starset_def] "*s* -A = -(*s* A)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
55 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
56 |
by (res_inst_tac [("z","x")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
57 |
by (res_inst_tac [("z","x")] eq_Abs_hypreal 2); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
58 |
by (REPEAT(Step_tac 1) THEN Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
59 |
by (Fuf_empty_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
60 |
by (dtac FreeUltrafilterNat_Compl_mem 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
61 |
by (Fuf_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
62 |
qed "STAR_Compl"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
63 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
64 |
goal Set.thy "(A - B) = (A Int (- B))"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
65 |
by (Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
66 |
qed "set_diff_iff2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
67 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
68 |
Goal "!!x. x ~: *s* F ==> x : *s* (- F)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
69 |
by (auto_tac (claset(),simpset() addsimps [STAR_Compl])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
70 |
qed "STAR_mem_Compl"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
71 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
72 |
Goal "*s* (A - B) = *s* A - *s* B"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
73 |
by (auto_tac (claset(),simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
74 |
[set_diff_iff2,STAR_Int,STAR_Compl])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
75 |
qed "STAR_diff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
76 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
77 |
Goalw [starset_def] "!!A. A <= B ==> *s* A <= *s* B"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
78 |
by (REPEAT(blast_tac (claset() addIs [FreeUltrafilterNat_subset]) 1)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
79 |
qed "STAR_subset"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
80 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
81 |
Goalw [starset_def,hypreal_of_real_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
82 |
"!!A. a : A ==> hypreal_of_real a : *s* A"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
83 |
by (auto_tac (claset() addIs [FreeUltrafilterNat_subset],simpset())); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
84 |
qed "STAR_mem"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
85 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
86 |
Goalw [starset_def] "hypreal_of_real `` A <= *s* A"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
87 |
by (auto_tac (claset(),simpset() addsimps [hypreal_of_real_def])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
88 |
by (blast_tac (claset() addIs [FreeUltrafilterNat_subset]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
89 |
qed "STAR_hypreal_of_real_image_subset"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
90 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
91 |
Goalw [starset_def] "*s* X Int SReal = hypreal_of_real `` X"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
92 |
by (auto_tac (claset(),simpset() addsimps [hypreal_of_real_def,SReal_def])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
93 |
by (fold_tac [hypreal_of_real_def]); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
94 |
by (rtac imageI 1 THEN rtac ccontr 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
95 |
by (dtac bspec 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
96 |
by (rtac lemma_hyprel_refl 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
97 |
by (blast_tac (claset() addIs [FreeUltrafilterNat_subset]) 2); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
98 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
99 |
qed "STAR_hypreal_of_real_Int"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
100 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
101 |
Goal "!!x. x ~: hypreal_of_real `` A ==> ALL y: A. x ~= hypreal_of_real y"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
102 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
103 |
qed "lemma_not_hyprealA"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
104 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
105 |
Goal "- {n. X n = xa} = {n. X n ~= xa}"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
106 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
107 |
qed "lemma_Compl_eq"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
108 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
109 |
Goalw [starset_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
110 |
"!!M. ALL n. (X n) ~: M \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
111 |
\ ==> Abs_hypreal(hyprel^^{X}) ~: *s* M"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
112 |
by (Auto_tac THEN rtac bexI 1 THEN rtac lemma_hyprel_refl 2); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
113 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
114 |
qed "STAR_real_seq_to_hypreal"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
115 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
116 |
Goalw [starset_def] "*s* {x} = {hypreal_of_real x}"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
117 |
by (auto_tac (claset(),simpset() addsimps [hypreal_of_real_def])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
118 |
by (res_inst_tac [("z","xa")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
119 |
by (auto_tac (claset() addIs [FreeUltrafilterNat_subset],simpset())); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
120 |
qed "STAR_singleton"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
121 |
Addsimps [STAR_singleton]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
122 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
123 |
Goal "!!x. x ~: F ==> hypreal_of_real x ~: *s* F"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
124 |
by (auto_tac (claset(),simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
125 |
[starset_def,hypreal_of_real_def])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
126 |
by (rtac bexI 1 THEN rtac lemma_hyprel_refl 2); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
127 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
128 |
qed "STAR_not_mem"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
129 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
130 |
Goal "!!x. [| x : *s* A; A <= B |] ==> x : *s* B"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
131 |
by (blast_tac (claset() addDs [STAR_subset]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
132 |
qed "STAR_subset_closed"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
133 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
134 |
(*------------------------------------------------------------------ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
135 |
Nonstandard extension of a set (defined using a constant |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
136 |
sequence) as a special case of an internal set |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
137 |
-----------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
138 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
139 |
Goalw [starset_n_def,starset_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
140 |
"!!A. ALL n. (As n = A) ==> *sn* As = *s* A"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
141 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
142 |
qed "starset_n_starset"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
143 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
144 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
145 |
(*----------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
146 |
(* Theorems about nonstandard extensions of functions *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
147 |
(*----------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
148 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
149 |
(*----------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
150 |
(* Nonstandard extension of a function (defined using a *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
151 |
(* constant sequence) as a special case of an internal function *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
152 |
(*----------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
153 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
154 |
Goalw [starfun_n_def,starfun_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
155 |
"!!A. ALL n. (F n = f) ==> *fn* F = *f* f"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
156 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
157 |
qed "starfun_n_starfun"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
158 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
159 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
160 |
(* |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
161 |
Prove that hrabs is a nonstandard extension of rabs without |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
162 |
use of congruence property (proved after this for general |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
163 |
nonstandard extensions of real valued functions). This makes |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
164 |
proof much longer- see comments at end of HREALABS.thy where |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
165 |
we proved a congruence theorem for hrabs. |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
166 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
167 |
NEW!!! No need to prove all the lemmas anymore. Use the ultrafilter |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
168 |
tactic! |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
169 |
*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
170 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
171 |
Goalw [is_starext_def] "is_starext abs abs"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
172 |
by (Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
173 |
by (res_inst_tac [("z","x")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
174 |
by (res_inst_tac [("z","y")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
175 |
by Auto_tac; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
176 |
by (rtac bexI 1 THEN rtac lemma_hyprel_refl 2); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
177 |
by (rtac bexI 1 THEN rtac lemma_hyprel_refl 2); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
178 |
by (auto_tac (claset() addSDs [spec],simpset() addsimps [hypreal_minus,hrabs_def, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
179 |
hypreal_zero_def,hypreal_le_def,hypreal_less_def])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
180 |
by (TRYALL(Ultra_tac)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
181 |
by (TRYALL(arith_tac)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
182 |
qed "hrabs_is_starext_rabs"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
183 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
184 |
Goal "!!z. [| X: Rep_hypreal z; Y: Rep_hypreal z |] \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
185 |
\ ==> {n. X n = Y n} : FreeUltrafilterNat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
186 |
by (res_inst_tac [("z","z")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
187 |
by (Auto_tac THEN Fuf_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
188 |
qed "Rep_hypreal_FreeUltrafilterNat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
189 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
190 |
(*----------------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
191 |
Nonstandard extension of functions- congruence |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
192 |
-----------------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
193 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
194 |
Goalw [congruent_def] "congruent hyprel (%X. hyprel^^{%n. f (X n)})"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
195 |
by (safe_tac (claset())); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
196 |
by (ALLGOALS(Fuf_tac)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
197 |
qed "starfun_congruent"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
198 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
199 |
Goalw [starfun_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
200 |
"(*f* f) (Abs_hypreal(hyprel^^{%n. X n})) = \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
201 |
\ Abs_hypreal(hyprel ^^ {%n. f (X n)})"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
202 |
by (res_inst_tac [("f","Abs_hypreal")] arg_cong 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
203 |
by (simp_tac (simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
204 |
[hyprel_in_hypreal RS Abs_hypreal_inverse,[equiv_hyprel, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
205 |
starfun_congruent] MRS UN_equiv_class]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
206 |
qed "starfun"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
207 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
208 |
(*------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
209 |
multiplication: ( *f ) x ( *g ) = *(f x g) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
210 |
------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
211 |
Goal "(*f* f) xa * (*f* g) xa = (*f* (%x. f x * g x)) xa"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
212 |
by (res_inst_tac [("z","xa")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
213 |
by (auto_tac (claset(),simpset() addsimps [starfun,hypreal_mult])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
214 |
qed "starfun_mult"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
215 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
216 |
(*--------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
217 |
addition: ( *f ) + ( *g ) = *(f + g) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
218 |
---------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
219 |
Goal "(*f* f) xa + (*f* g) xa = (*f* (%x. f x + g x)) xa"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
220 |
by (res_inst_tac [("z","xa")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
221 |
by (auto_tac (claset(),simpset() addsimps [starfun,hypreal_add])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
222 |
qed "starfun_add"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
223 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
224 |
(*-------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
225 |
subtraction: ( *f ) + -( *g ) = *(f + -g) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
226 |
-------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
227 |
Goal "(*f* f) xa + -(*f* g) xa = (*f* (%x. f x + -g x)) xa"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
228 |
by (res_inst_tac [("z","xa")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
229 |
by (auto_tac (claset(),simpset() addsimps [starfun, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
230 |
hypreal_minus,hypreal_add])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
231 |
qed "starfun_add_minus"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
232 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
233 |
Goalw [hypreal_diff_def,real_diff_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
234 |
"(*f* f) xa - (*f* g) xa = (*f* (%x. f x - g x)) xa"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
235 |
by (rtac starfun_add_minus 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
236 |
qed "starfun_diff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
237 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
238 |
(*-------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
239 |
composition: ( *f ) o ( *g ) = *(f o g) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
240 |
---------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
241 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
242 |
Goal "(%x. (*f* f) ((*f* g) x)) = *f* (%x. f (g x))"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
243 |
by (rtac ext 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
244 |
by (res_inst_tac [("z","x")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
245 |
by (auto_tac (claset(),simpset() addsimps [starfun])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
246 |
qed "starfun_o2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
247 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
248 |
Goalw [o_def] "(*f* f) o (*f* g) = (*f* (f o g))"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
249 |
by (simp_tac (simpset() addsimps [starfun_o2]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
250 |
qed "starfun_o"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
251 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
252 |
(*-------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
253 |
NS extension of constant function |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
254 |
--------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
255 |
Goal "(*f* (%x. k)) xa = hypreal_of_real k"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
256 |
by (res_inst_tac [("z","xa")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
257 |
by (auto_tac (claset(),simpset() addsimps [starfun, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
258 |
hypreal_of_real_def])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
259 |
qed "starfun_const_fun"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
260 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
261 |
Addsimps [starfun_const_fun]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
262 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
263 |
Goal "- (*f* f) x = (*f* (%x. - f x)) x"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
264 |
by (res_inst_tac [("z","x")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
265 |
by (auto_tac (claset(),simpset() addsimps [starfun, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
266 |
hypreal_minus])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
267 |
qed "starfun_minus"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
268 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
269 |
(*---------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
270 |
the NS extension of the identity function |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
271 |
----------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
272 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
273 |
Goal "!!x. x @= hypreal_of_real a ==> (*f* (%x. x)) x @= hypreal_of_real a"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
274 |
by (res_inst_tac [("z","x")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
275 |
by (auto_tac (claset(),simpset() addsimps [starfun])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
276 |
qed "starfun_Idfun_inf_close"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
277 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
278 |
Goal "(*f* (%x. x)) x = x"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
279 |
by (res_inst_tac [("z","x")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
280 |
by (auto_tac (claset(),simpset() addsimps [starfun])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
281 |
qed "starfun_Id"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
282 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
283 |
(*---------------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
284 |
the *-function is a (nonstandard) extension of the function |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
285 |
----------------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
286 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
287 |
Goalw [is_starext_def] "is_starext (*f* f) f"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
288 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
289 |
by (res_inst_tac [("z","x")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
290 |
by (res_inst_tac [("z","y")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
291 |
by (auto_tac (claset() addSIs [bexI] ,simpset() addsimps [starfun])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
292 |
qed "is_starext_starfun"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
293 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
294 |
(*---------------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
295 |
Any nonstandard extension is in fact the *-function |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
296 |
----------------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
297 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
298 |
Goalw [is_starext_def] "!!f. is_starext F f ==> F = *f* f"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
299 |
by (rtac ext 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
300 |
by (res_inst_tac [("z","x")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
301 |
by (dres_inst_tac [("x","x")] spec 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
302 |
by (dres_inst_tac [("x","(*f* f) x")] spec 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
303 |
by (auto_tac (claset() addSDs [FreeUltrafilterNat_Compl_mem], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
304 |
simpset() addsimps [starfun])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
305 |
by (Fuf_empty_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
306 |
qed "is_starfun_starext"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
307 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
308 |
Goal "(is_starext F f) = (F = *f* f)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
309 |
by (blast_tac (claset() addIs [is_starfun_starext,is_starext_starfun]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
310 |
qed "is_starext_starfun_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
311 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
312 |
(*-------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
313 |
extented function has same solution as its standard |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
314 |
version for real arguments. i.e they are the same |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
315 |
for all real arguments |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
316 |
-------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
317 |
Goal "(*f* f) (hypreal_of_real a) = hypreal_of_real (f a)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
318 |
by (auto_tac (claset(),simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
319 |
[starfun,hypreal_of_real_def])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
320 |
qed "starfun_eq"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
321 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
322 |
Addsimps [starfun_eq]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
323 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
324 |
Goal "(*f* f) (hypreal_of_real a) @= hypreal_of_real (f a)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
325 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
326 |
qed "starfun_inf_close"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
327 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
328 |
(* useful for NS definition of derivatives *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
329 |
Goal "(*f* (%h. f (x + h))) xa = (*f* f) (hypreal_of_real x + xa)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
330 |
by (res_inst_tac [("z","xa")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
331 |
by (auto_tac (claset(),simpset() addsimps [starfun, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
332 |
hypreal_of_real_def,hypreal_add])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
333 |
qed "starfun_lambda_cancel"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
334 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
335 |
Goal "(*f* (%h. f(g(x + h)))) xa = (*f* (f o g)) (hypreal_of_real x + xa)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
336 |
by (res_inst_tac [("z","xa")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
337 |
by (auto_tac (claset(),simpset() addsimps [starfun, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
338 |
hypreal_of_real_def,hypreal_add])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
339 |
qed "starfun_lambda_cancel2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
340 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
341 |
Goal "!!f. [| (*f* f) xa @= l; (*f* g) xa @= m; \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
342 |
\ l: HFinite; m: HFinite \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
343 |
\ |] ==> (*f* (%x. f x * g x)) xa @= l * m"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
344 |
by (dtac inf_close_mult_HFinite 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
345 |
by (REPEAT(assume_tac 1)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
346 |
by (auto_tac (claset() addIs [inf_close_sym RSN (2,inf_close_HFinite)], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
347 |
simpset() addsimps [starfun_mult])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
348 |
qed "starfun_mult_HFinite_inf_close"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
349 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
350 |
Goal "!!f. [| (*f* f) xa @= l; (*f* g) xa @= m \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
351 |
\ |] ==> (*f* (%x. f x + g x)) xa @= l + m"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
352 |
by (auto_tac (claset() addIs [inf_close_add], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
353 |
simpset() addsimps [starfun_add RS sym])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
354 |
qed "starfun_add_inf_close"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
355 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
356 |
(*---------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
357 |
Examples: hrabs is nonstandard extension of rabs |
10607 | 358 |
inverse is nonstandard extension of inverse |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
359 |
---------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
360 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
361 |
(* can be proved easily using theorem "starfun" and *) |
10607 | 362 |
(* properties of ultrafilter as for inverse below we *) |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
363 |
(* use the theorem we proved above instead *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
364 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
365 |
Goal "*f* abs = abs"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
366 |
by (rtac (hrabs_is_starext_rabs RS |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
367 |
(is_starext_starfun_iff RS iffD1) RS sym) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
368 |
qed "starfun_rabs_hrabs"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
369 |
|
10607 | 370 |
Goal "!!x. x ~= 0 ==> (*f* inverse) x = inverse(x)"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
371 |
by (res_inst_tac [("z","x")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
372 |
by (auto_tac (claset(),simpset() addsimps [starfun, |
10607 | 373 |
hypreal_inverse,hypreal_zero_def])); |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
374 |
by (dtac FreeUltrafilterNat_Compl_mem 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
375 |
by (auto_tac (claset() addEs [FreeUltrafilterNat_subset],simpset())); |
10607 | 376 |
qed "starfun_inverse_inverse"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
377 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
378 |
(* more specifically *) |
10607 | 379 |
Goal "(*f* inverse) ehr = inverse (ehr)"; |
380 |
by (rtac (hypreal_epsilon_not_zero RS starfun_inverse_inverse) 1); |
|
381 |
qed "starfun_inverse_epsilon"; |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
382 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
383 |
Goal "ALL x. f x ~= 0 ==> \ |
10607 | 384 |
\ inverse ((*f* f) x) = (*f* (%x. inverse (f x))) x"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
385 |
by (res_inst_tac [("z","x")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
386 |
by (auto_tac (claset(),simpset() addsimps [starfun, |
10607 | 387 |
hypreal_inverse])); |
388 |
qed "starfun_inverse"; |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
389 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
390 |
Goal "(*f* f) x ~= 0 ==> \ |
10607 | 391 |
\ inverse ((*f* f) x) = (*f* (%x. inverse (f x))) x"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
392 |
by (res_inst_tac [("z","x")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
393 |
by (auto_tac (claset() addIs [FreeUltrafilterNat_subset] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
394 |
addSDs [FreeUltrafilterNat_Compl_mem], |
10607 | 395 |
simpset() addsimps [starfun,hypreal_inverse, |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
396 |
hypreal_zero_def])); |
10607 | 397 |
qed "starfun_inverse2"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
398 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
399 |
Goal "a ~= hypreal_of_real b ==> \ |
10607 | 400 |
\ (*f* (%z. inverse (z + -b))) a = inverse(a + -hypreal_of_real b)"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
401 |
by (res_inst_tac [("z","a")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
402 |
by (auto_tac (claset() addIs [FreeUltrafilterNat_subset] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
403 |
addSDs [FreeUltrafilterNat_Compl_mem], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
404 |
simpset() addsimps [starfun,hypreal_of_real_def,hypreal_add, |
10607 | 405 |
hypreal_minus,hypreal_inverse,rename_numerals |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
406 |
(real_eq_minus_iff2 RS sym)])); |
10607 | 407 |
qed "starfun_inverse3"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
408 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
409 |
Goal |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
410 |
"!!f. a + hypreal_of_real b ~= 0 ==> \ |
10607 | 411 |
\ (*f* (%z. inverse (z + b))) a = inverse(a + hypreal_of_real b)"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
412 |
by (res_inst_tac [("z","a")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
413 |
by (auto_tac (claset() addIs [FreeUltrafilterNat_subset] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
414 |
addSDs [FreeUltrafilterNat_Compl_mem], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
415 |
simpset() addsimps [starfun,hypreal_of_real_def,hypreal_add, |
10607 | 416 |
hypreal_inverse,hypreal_zero_def])); |
417 |
qed "starfun_inverse4"; |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
418 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
419 |
(*------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
420 |
General lemma/theorem needed for proofs in elementary |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
421 |
topology of the reals |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
422 |
------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
423 |
Goalw [starset_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
424 |
"!!A. (*f* f) x : *s* A ==> x : *s* {x. f x : A}"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
425 |
by (res_inst_tac [("z","x")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
426 |
by (auto_tac (claset(),simpset() addsimps [starfun])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
427 |
by (dres_inst_tac [("x","%n. f (Xa n)")] bspec 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
428 |
by (Auto_tac THEN Fuf_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
429 |
qed "starfun_mem_starset"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
430 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
431 |
(*------------------------------------------------------------ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
432 |
Alternative definition for hrabs with rabs function |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
433 |
applied entrywise to equivalence class representative. |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
434 |
This is easily proved using starfun and ns extension thm |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
435 |
------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
436 |
Goal "abs (Abs_hypreal (hyprel ^^ {X})) = \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
437 |
\ Abs_hypreal(hyprel ^^ {%n. abs (X n)})"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
438 |
by (simp_tac (simpset() addsimps [starfun_rabs_hrabs RS sym,starfun]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
439 |
qed "hypreal_hrabs"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
440 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
441 |
(*---------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
442 |
nonstandard extension of set through nonstandard extension |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
443 |
of rabs function i.e hrabs. A more general result should be |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
444 |
where we replace rabs by some arbitrary function f and hrabs |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
445 |
by its NS extenson ( *f* f). See second NS set extension below. |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
446 |
----------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
447 |
Goalw [starset_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
448 |
"*s* {x. abs (x + - y) < r} = {x. abs(x + -hypreal_of_real y) < hypreal_of_real r}"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
449 |
by (Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
450 |
by (ALLGOALS(res_inst_tac [("z","x")] eq_Abs_hypreal)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
451 |
by (auto_tac (claset() addSIs [exI] addSDs [bspec], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
452 |
simpset() addsimps [hypreal_minus, hypreal_of_real_def,hypreal_add, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
453 |
hypreal_hrabs,hypreal_less_def])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
454 |
by (Fuf_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
455 |
qed "STAR_rabs_add_minus"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
456 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
457 |
Goalw [starset_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
458 |
"*s* {x. abs (f x + - y) < r} = \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
459 |
\ {x. abs((*f* f) x + -hypreal_of_real y) < hypreal_of_real r}"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
460 |
by (Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
461 |
by (ALLGOALS(res_inst_tac [("z","x")] eq_Abs_hypreal)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
462 |
by (auto_tac (claset() addSIs [exI] addSDs [bspec], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
463 |
simpset() addsimps [hypreal_minus, hypreal_of_real_def,hypreal_add, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
464 |
hypreal_hrabs,hypreal_less_def,starfun])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
465 |
by (Fuf_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
466 |
qed "STAR_starfun_rabs_add_minus"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
467 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
468 |
(*------------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
469 |
Another charaterization of Infinitesimal and one of @= relation. |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
470 |
In this theory since hypreal_hrabs proved here. (To Check:) Maybe |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
471 |
move both if possible? |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
472 |
-------------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
473 |
Goal "(x:Infinitesimal) = (EX X:Rep_hypreal(x). \ |
10607 | 474 |
\ ALL m. {n. abs(X n) < inverse(real_of_posnat m)}:FreeUltrafilterNat)"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
475 |
by (res_inst_tac [("z","x")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
476 |
by (auto_tac (claset() addSIs [bexI,lemma_hyprel_refl], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
477 |
simpset() addsimps [Infinitesimal_hypreal_of_posnat_iff, |
10607 | 478 |
hypreal_of_real_of_posnat,hypreal_of_real_def,hypreal_inverse, |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
479 |
hypreal_hrabs,hypreal_less])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
480 |
by (dres_inst_tac [("x","n")] spec 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
481 |
by (Fuf_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
482 |
qed "Infinitesimal_FreeUltrafilterNat_iff2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
483 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
484 |
Goal "(Abs_hypreal(hyprel^^{X}) @= Abs_hypreal(hyprel^^{Y})) = \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
485 |
\ (ALL m. {n. abs (X n + - Y n) < \ |
10607 | 486 |
\ inverse(real_of_posnat m)} : FreeUltrafilterNat)"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
487 |
by (rtac (inf_close_minus_iff RS ssubst) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
488 |
by (rtac (mem_infmal_iff RS subst) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
489 |
by (auto_tac (claset(), simpset() addsimps [hypreal_minus, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
490 |
hypreal_add,Infinitesimal_FreeUltrafilterNat_iff2])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
491 |
by (dres_inst_tac [("x","m")] spec 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
492 |
by (Fuf_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
493 |
qed "inf_close_FreeUltrafilterNat_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
494 |
|
10156 | 495 |
Goal "inj starfun"; |
496 |
by (rtac injI 1); |
|
497 |
by (rtac ext 1 THEN rtac ccontr 1); |
|
498 |
by (dres_inst_tac [("x","Abs_hypreal(hyprel ^^{%n. xa})")] fun_cong 1); |
|
499 |
by (auto_tac (claset(),simpset() addsimps [starfun])); |
|
500 |
qed "inj_starfun"; |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
501 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
502 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
503 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
504 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
505 |