author | paulson |
Fri, 28 Nov 2003 12:09:37 +0100 | |
changeset 14269 | 502a7c95de73 |
parent 14268 | 5cf13e80be0e |
child 14288 | d149e3cbdb39 |
permissions | -rw-r--r-- |
9435
c3a13a7d4424
lemmas [arith_split] = abs_split (*belongs to theory RealAbs*);
wenzelm
parents:
9013
diff
changeset
|
1 |
(* Title : HOL/Real/RealPow.thy |
7219 | 2 |
ID : $Id$ |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
3 |
Author : Jacques D. Fleuriot |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
4 |
Copyright : 1998 University of Cambridge |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
5 |
Description : Natural powers theory |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
6 |
|
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
7 |
*) |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
8 |
|
14269 | 9 |
theory RealPow = RealArith: |
9435
c3a13a7d4424
lemmas [arith_split] = abs_split (*belongs to theory RealAbs*);
wenzelm
parents:
9013
diff
changeset
|
10 |
|
10309 | 11 |
instance real :: power .. |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
12 |
|
8856 | 13 |
primrec (realpow) |
12018
ec054019c910
Numerals and simprocs for types real and hypreal. The abstract
paulson
parents:
11701
diff
changeset
|
14 |
realpow_0: "r ^ 0 = 1" |
9435
c3a13a7d4424
lemmas [arith_split] = abs_split (*belongs to theory RealAbs*);
wenzelm
parents:
9013
diff
changeset
|
15 |
realpow_Suc: "r ^ (Suc n) = (r::real) * (r ^ n)" |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
16 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
17 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
18 |
lemma realpow_zero [simp]: "(0::real) ^ (Suc n) = 0" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
19 |
by auto |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
20 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
21 |
lemma realpow_not_zero [rule_format]: "r \<noteq> (0::real) --> r ^ n \<noteq> 0" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
22 |
by (induct_tac "n", auto) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
23 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
24 |
lemma realpow_zero_zero: "r ^ n = (0::real) ==> r = 0" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
25 |
apply (rule ccontr) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
26 |
apply (auto dest: realpow_not_zero) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
27 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
28 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
29 |
lemma realpow_inverse: "inverse ((r::real) ^ n) = (inverse r) ^ n" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
30 |
apply (induct_tac "n") |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
31 |
apply (auto simp add: real_inverse_distrib) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
32 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
33 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
34 |
lemma realpow_abs: "abs(r ^ n) = abs(r::real) ^ n" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
35 |
apply (induct_tac "n") |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
36 |
apply (auto simp add: abs_mult) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
37 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
38 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
39 |
lemma realpow_add: "(r::real) ^ (n + m) = (r ^ n) * (r ^ m)" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
40 |
apply (induct_tac "n") |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
41 |
apply (auto simp add: real_mult_ac) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
42 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
43 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
44 |
lemma realpow_one [simp]: "(r::real) ^ 1 = r" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
45 |
by simp |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
46 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
47 |
lemma realpow_two: "(r::real)^ (Suc (Suc 0)) = r * r" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
48 |
by simp |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
49 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
50 |
lemma realpow_gt_zero [rule_format]: "(0::real) < r --> 0 < r ^ n" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
51 |
apply (induct_tac "n") |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
52 |
apply (auto intro: real_mult_order simp add: real_zero_less_one) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
53 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
54 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
55 |
lemma realpow_ge_zero [rule_format]: "(0::real) \<le> r --> 0 \<le> r ^ n" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
56 |
apply (induct_tac "n") |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
57 |
apply (auto simp add: real_0_le_mult_iff) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
58 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
59 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
60 |
lemma realpow_le [rule_format]: "(0::real) \<le> x & x \<le> y --> x ^ n \<le> y ^ n" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
61 |
apply (induct_tac "n") |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
62 |
apply (auto intro!: real_mult_le_mono) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
63 |
apply (simp (no_asm_simp) add: realpow_ge_zero) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
64 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
65 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
66 |
lemma realpow_0_left [rule_format, simp]: |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
67 |
"0 < n --> 0 ^ n = (0::real)" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
68 |
apply (induct_tac "n", auto) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
69 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
70 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
71 |
lemma realpow_less' [rule_format]: |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
72 |
"[|(0::real) \<le> x; x < y |] ==> 0 < n --> x ^ n < y ^ n" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
73 |
apply (induct n) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
74 |
apply (auto simp add: real_mult_less_mono' realpow_ge_zero) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
75 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
76 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
77 |
text{*Legacy: weaker version of the theorem above*} |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
78 |
lemma realpow_less: |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
79 |
"[|(0::real) < x; x < y; 0 < n|] ==> x ^ n < y ^ n" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
80 |
apply (rule realpow_less', auto) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
81 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
82 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
83 |
lemma realpow_eq_one [simp]: "1 ^ n = (1::real)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
84 |
by (induct_tac "n", auto) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
85 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
86 |
lemma abs_realpow_minus_one [simp]: "abs((-1) ^ n) = (1::real)" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
87 |
apply (induct_tac "n") |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
88 |
apply (auto simp add: abs_mult) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
89 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
90 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
91 |
lemma realpow_mult: "((r::real) * s) ^ n = (r ^ n) * (s ^ n)" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
92 |
apply (induct_tac "n") |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
93 |
apply (auto simp add: real_mult_ac) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
94 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
95 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
96 |
lemma realpow_two_le [simp]: "(0::real) \<le> r^ Suc (Suc 0)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
97 |
by (simp add: real_le_square) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
98 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
99 |
lemma abs_realpow_two [simp]: "abs((x::real)^Suc (Suc 0)) = x^Suc (Suc 0)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
100 |
by (simp add: abs_eqI1 real_le_square) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
101 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
102 |
lemma realpow_two_abs [simp]: "abs(x::real)^Suc (Suc 0) = x^Suc (Suc 0)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
103 |
by (simp add: realpow_abs [symmetric] abs_eqI1 del: realpow_Suc) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
104 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
105 |
lemma realpow_two_gt_one: "(1::real) < r ==> 1 < r^ (Suc (Suc 0))" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
106 |
apply auto |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
107 |
apply (cut_tac real_zero_less_one) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
108 |
apply (frule_tac x = 0 in order_less_trans, assumption) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
109 |
apply (drule_tac z = r and x = 1 in real_mult_less_mono1) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
110 |
apply (auto intro: order_less_trans) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
111 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
112 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
113 |
lemma realpow_ge_one [rule_format]: "(1::real) < r --> 1 \<le> r ^ n" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
114 |
apply (induct_tac "n", auto) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
115 |
apply (subgoal_tac "1*1 \<le> r * r^n") |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
116 |
apply (rule_tac [2] real_mult_le_mono, auto) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
117 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
118 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
119 |
lemma realpow_ge_one2: "(1::real) \<le> r ==> 1 \<le> r ^ n" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
120 |
apply (drule order_le_imp_less_or_eq) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
121 |
apply (auto dest: realpow_ge_one) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
122 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
123 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
124 |
lemma two_realpow_ge_one [simp]: "(1::real) \<le> 2 ^ n" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
125 |
apply (rule_tac y = "1 ^ n" in order_trans) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
126 |
apply (rule_tac [2] realpow_le) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
127 |
apply (auto intro: order_less_imp_le) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
128 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
129 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
130 |
lemma two_realpow_gt [simp]: "real (n::nat) < 2 ^ n" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
131 |
apply (induct_tac "n") |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
132 |
apply (auto simp add: real_of_nat_Suc) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
133 |
apply (subst real_mult_2) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
134 |
apply (rule real_add_less_le_mono) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
135 |
apply (auto simp add: two_realpow_ge_one) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
136 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
137 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
138 |
lemma realpow_minus_one [simp]: "(-1) ^ (2*n) = (1::real)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
139 |
by (induct_tac "n", auto) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
140 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
141 |
lemma realpow_minus_one_odd [simp]: "(-1) ^ Suc (2*n) = -(1::real)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
142 |
by auto |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
143 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
144 |
lemma realpow_minus_one_even [simp]: "(-1) ^ Suc (Suc (2*n)) = (1::real)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
145 |
by auto |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
146 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
147 |
lemma realpow_Suc_less [rule_format]: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
148 |
"(0::real) < r & r < (1::real) --> r ^ Suc n < r ^ n" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
149 |
by (induct_tac "n", auto) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
150 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
151 |
lemma realpow_Suc_le [rule_format]: "0 \<le> r & r < (1::real) --> r ^ Suc n \<le> r ^ n" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
152 |
apply (induct_tac "n") |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
153 |
apply (auto intro: order_less_imp_le dest!: order_le_imp_less_or_eq) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
154 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
155 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
156 |
lemma realpow_zero_le [simp]: "(0::real) \<le> 0 ^ n" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
157 |
by (case_tac "n", auto) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
158 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
159 |
lemma realpow_Suc_le2 [rule_format]: "0 < r & r < (1::real) --> r ^ Suc n \<le> r ^ n" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
160 |
by (blast intro!: order_less_imp_le realpow_Suc_less) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
161 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
162 |
lemma realpow_Suc_le3: "[| 0 \<le> r; r < (1::real) |] ==> r ^ Suc n \<le> r ^ n" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
163 |
apply (erule order_le_imp_less_or_eq [THEN disjE]) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
164 |
apply (rule realpow_Suc_le2, auto) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
165 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
166 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
167 |
lemma realpow_less_le [rule_format]: "0 \<le> r & r < (1::real) & n < N --> r ^ N \<le> r ^ n" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
168 |
apply (induct_tac "N") |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
169 |
apply (simp_all (no_asm_simp)) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
170 |
apply clarify |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
171 |
apply (subgoal_tac "r * r ^ na \<le> 1 * r ^ n", simp) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
172 |
apply (rule real_mult_le_mono) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
173 |
apply (auto simp add: realpow_ge_zero less_Suc_eq) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
174 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
175 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
176 |
lemma realpow_le_le: "[| 0 \<le> r; r < (1::real); n \<le> N |] ==> r ^ N \<le> r ^ n" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
177 |
apply (drule_tac n = N in le_imp_less_or_eq) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
178 |
apply (auto intro: realpow_less_le) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
179 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
180 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
181 |
lemma realpow_Suc_le_self: "[| 0 < r; r < (1::real) |] ==> r ^ Suc n \<le> r" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
182 |
by (drule_tac n = 1 and N = "Suc n" in order_less_imp_le [THEN realpow_le_le], auto) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
183 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
184 |
lemma realpow_Suc_less_one: "[| 0 < r; r < (1::real) |] ==> r ^ Suc n < 1" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
185 |
by (blast intro: realpow_Suc_le_self order_le_less_trans) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
186 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
187 |
lemma realpow_le_Suc [rule_format]: "(1::real) \<le> r --> r ^ n \<le> r ^ Suc n" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
188 |
by (induct_tac "n", auto) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
189 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
190 |
lemma realpow_less_Suc [rule_format]: "(1::real) < r --> r ^ n < r ^ Suc n" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
191 |
by (induct_tac "n", auto) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
192 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
193 |
lemma realpow_le_Suc2 [rule_format]: "(1::real) < r --> r ^ n \<le> r ^ Suc n" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
194 |
by (blast intro!: order_less_imp_le realpow_less_Suc) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
195 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
196 |
(*One use in RealPow.thy*) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
197 |
lemma real_mult_self_le2: "[| (1::real) \<le> r; (1::real) \<le> x |] ==> x \<le> r * x" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
198 |
apply (subgoal_tac "1 * x \<le> r * x", simp) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
199 |
apply (rule mult_right_mono, auto) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
200 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
201 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
202 |
lemma realpow_gt_ge2 [rule_format]: "(1::real) \<le> r & n < N --> r ^ n \<le> r ^ N" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
203 |
apply (induct_tac "N", auto) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
204 |
apply (frule_tac [!] n = na in realpow_ge_one2) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
205 |
apply (drule_tac [!] real_mult_self_le2, assumption) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
206 |
prefer 2 apply assumption |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
207 |
apply (auto intro: order_trans simp add: less_Suc_eq) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
208 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
209 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
210 |
lemma realpow_ge_ge2: "[| (1::real) \<le> r; n \<le> N |] ==> r ^ n \<le> r ^ N" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
211 |
apply (drule_tac n = N in le_imp_less_or_eq) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
212 |
apply (auto intro: realpow_gt_ge2) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
213 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
214 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
215 |
lemma realpow_Suc_ge_self2: "(1::real) \<le> r ==> r \<le> r ^ Suc n" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
216 |
by (drule_tac n = 1 and N = "Suc n" in realpow_ge_ge2, auto) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
217 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
218 |
(*Used ONCE in Hyperreal/NthRoot.ML*) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
219 |
lemma realpow_ge_self2: "[| (1::real) \<le> r; 0 < n |] ==> r \<le> r ^ n" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
220 |
apply (drule less_not_refl2 [THEN not0_implies_Suc]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
221 |
apply (auto intro!: realpow_Suc_ge_self2) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
222 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
223 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
224 |
lemma realpow_minus_mult [rule_format, simp]: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
225 |
"0 < n --> (x::real) ^ (n - 1) * x = x ^ n" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
226 |
apply (induct_tac "n") |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
227 |
apply (auto simp add: real_mult_commute) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
228 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
229 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
230 |
lemma realpow_two_mult_inverse [simp]: "r \<noteq> 0 ==> r * inverse r ^Suc (Suc 0) = inverse (r::real)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
231 |
by (simp add: realpow_two real_mult_assoc [symmetric]) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
232 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
233 |
(* 05/00 *) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
234 |
lemma realpow_two_minus [simp]: "(-x)^Suc (Suc 0) = (x::real)^Suc (Suc 0)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
235 |
by simp |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
236 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
237 |
lemma realpow_two_diff: "(x::real)^Suc (Suc 0) - y^Suc (Suc 0) = (x - y) * (x + y)" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
238 |
apply (unfold real_diff_def) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
239 |
apply (simp add: real_add_mult_distrib2 real_add_mult_distrib real_mult_ac) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
240 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
241 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
242 |
lemma realpow_two_disj: "((x::real)^Suc (Suc 0) = y^Suc (Suc 0)) = (x = y | x = -y)" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
243 |
apply (cut_tac x = x and y = y in realpow_two_diff) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
244 |
apply (auto simp del: realpow_Suc) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
245 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
246 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
247 |
(* used in Transc *) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
248 |
lemma realpow_diff: "[|(x::real) \<noteq> 0; m \<le> n |] ==> x ^ (n - m) = x ^ n * inverse (x ^ m)" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
249 |
by (auto simp add: le_eq_less_or_eq less_iff_Suc_add realpow_add realpow_not_zero real_mult_ac) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
250 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
251 |
lemma realpow_real_of_nat: "real (m::nat) ^ n = real (m ^ n)" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
252 |
apply (induct_tac "n") |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
253 |
apply (auto simp add: real_of_nat_one real_of_nat_mult) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
254 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
255 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
256 |
lemma realpow_real_of_nat_two_pos [simp] : "0 < real (Suc (Suc 0) ^ n)" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
257 |
apply (induct_tac "n") |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
258 |
apply (auto simp add: real_of_nat_mult real_0_less_mult_iff) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
259 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
260 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
261 |
lemma realpow_increasing: |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
262 |
assumes xnonneg: "(0::real) \<le> x" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
263 |
and ynonneg: "0 \<le> y" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
264 |
and le: "x ^ Suc n \<le> y ^ Suc n" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
265 |
shows "x \<le> y" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
266 |
proof (rule ccontr) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
267 |
assume "~ x \<le> y" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
268 |
then have "y<x" by simp |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
269 |
then have "y ^ Suc n < x ^ Suc n" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
270 |
by (simp only: prems realpow_less') |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
271 |
from le and this show "False" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
272 |
by simp |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
273 |
qed |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
274 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
275 |
lemma realpow_Suc_cancel_eq: "[| (0::real) \<le> x; 0 \<le> y; x ^ Suc n = y ^ Suc n |] ==> x = y" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
276 |
by (blast intro: realpow_increasing order_antisym order_eq_refl sym) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
277 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
278 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
279 |
(*** Logical equivalences for inequalities ***) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
280 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
281 |
lemma realpow_eq_0_iff [simp]: "(x^n = 0) = (x = (0::real) & 0<n)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
282 |
by (induct_tac "n", auto) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
283 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
284 |
lemma zero_less_realpow_abs_iff [simp]: "(0 < (abs x)^n) = (x \<noteq> (0::real) | n=0)" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
285 |
apply (induct_tac "n") |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
286 |
apply (auto simp add: real_0_less_mult_iff) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
287 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
288 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
289 |
lemma zero_le_realpow_abs [simp]: "(0::real) \<le> (abs x)^n" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
290 |
apply (induct_tac "n") |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
291 |
apply (auto simp add: real_0_le_mult_iff) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
292 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
293 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
294 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
295 |
(*** Literal arithmetic involving powers, type real ***) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
296 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
297 |
lemma real_of_int_power: "real (x::int) ^ n = real (x ^ n)" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
298 |
apply (induct_tac "n") |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
299 |
apply (simp_all (no_asm_simp) add: nat_mult_distrib) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
300 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
301 |
declare real_of_int_power [symmetric, simp] |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
302 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
303 |
lemma power_real_number_of: "(number_of v :: real) ^ n = real ((number_of v :: int) ^ n)" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
304 |
by (simp only: real_number_of_def real_of_int_power) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
305 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
306 |
declare power_real_number_of [of _ "number_of w", standard, simp] |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
307 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
308 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
309 |
lemma real_power_two: "(r::real)\<twosuperior> = r * r" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
310 |
by (simp add: numeral_2_eq_2) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
311 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
312 |
lemma real_sqr_ge_zero [iff]: "0 \<le> (r::real)\<twosuperior>" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
313 |
by (simp add: real_power_two) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
314 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
315 |
lemma real_sqr_gt_zero: "(r::real) \<noteq> 0 ==> 0 < r\<twosuperior>" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
316 |
proof - |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
317 |
assume "r \<noteq> 0" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
318 |
hence "0 \<noteq> r\<twosuperior>" by simp |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
319 |
also have "0 \<le> r\<twosuperior>" by (simp add: real_sqr_ge_zero) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
320 |
finally show ?thesis . |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
321 |
qed |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
322 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
323 |
lemma real_sqr_not_zero: "r \<noteq> 0 ==> (r::real)\<twosuperior> \<noteq> 0" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
324 |
by simp |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
325 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
326 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
327 |
subsection{*Various Other Theorems*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
328 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
329 |
text{*Used several times in Hyperreal/Transcendental.ML*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
330 |
lemma real_sum_squares_cancel_a: "x * x = -(y * y) ==> x = (0::real) & y=0" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
331 |
by (auto intro: real_sum_squares_cancel) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
332 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
333 |
lemma real_squared_diff_one_factored: "x*x - (1::real) = (x + 1)*(x - 1)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
334 |
apply (auto simp add: real_add_mult_distrib real_add_mult_distrib2 real_diff_def) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
335 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
336 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
337 |
lemma real_mult_is_one: "(x*x = (1::real)) = (x = 1 | x = - 1)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
338 |
apply auto |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
339 |
apply (drule right_minus_eq [THEN iffD2]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
340 |
apply (auto simp add: real_squared_diff_one_factored) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
341 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
342 |
declare real_mult_is_one [iff] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
343 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
344 |
lemma real_le_add_half_cancel: "(x + y/2 <= (y::real)) = (x <= y /2)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
345 |
apply auto |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
346 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
347 |
declare real_le_add_half_cancel [simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
348 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
349 |
lemma real_minus_half_eq: "(x::real) - x/2 = x/2" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
350 |
apply auto |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
351 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
352 |
declare real_minus_half_eq [simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
353 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
354 |
lemma real_mult_inverse_cancel: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
355 |
"[|(0::real) < x; 0 < x1; x1 * y < x * u |] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
356 |
==> inverse x * y < inverse x1 * u" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
357 |
apply (rule_tac c=x in mult_less_imp_less_left) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
358 |
apply (auto simp add: real_mult_assoc [symmetric]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
359 |
apply (simp (no_asm) add: real_mult_ac) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
360 |
apply (rule_tac c=x1 in mult_less_imp_less_right) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
361 |
apply (auto simp add: real_mult_ac) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
362 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
363 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
364 |
text{*Used once: in Hyperreal/Transcendental.ML*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
365 |
lemma real_mult_inverse_cancel2: "[|(0::real) < x;0 < x1; x1 * y < x * u |] ==> y * inverse x < u * inverse x1" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
366 |
apply (auto dest: real_mult_inverse_cancel simp add: real_mult_ac) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
367 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
368 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
369 |
lemma inverse_real_of_nat_gt_zero: "0 < inverse (real (Suc n))" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
370 |
apply auto |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
371 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
372 |
declare inverse_real_of_nat_gt_zero [simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
373 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
374 |
lemma inverse_real_of_nat_ge_zero: "0 <= inverse (real (Suc n))" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
375 |
apply auto |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
376 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
377 |
declare inverse_real_of_nat_ge_zero [simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
378 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
379 |
lemma real_sum_squares_not_zero: "x ~= 0 ==> x * x + y * y ~= (0::real)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
380 |
apply (blast dest!: real_sum_squares_cancel) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
381 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
382 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
383 |
lemma real_sum_squares_not_zero2: "y ~= 0 ==> x * x + y * y ~= (0::real)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
384 |
apply (blast dest!: real_sum_squares_cancel2) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
385 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
386 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
387 |
(* nice theorem *) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
388 |
lemma abs_mult_abs: "abs x * abs x = x * (x::real)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
389 |
apply (insert linorder_less_linear [of x 0]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
390 |
apply (auto simp add: abs_eqI2 abs_minus_eqI2) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
391 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
392 |
declare abs_mult_abs [simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
393 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
394 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
395 |
subsection {*Various Other Theorems*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
396 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
397 |
lemma realpow_divide: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
398 |
"(x/y) ^ n = ((x::real) ^ n/ y ^ n)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
399 |
apply (unfold real_divide_def) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
400 |
apply (auto simp add: realpow_mult realpow_inverse) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
401 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
402 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
403 |
lemma realpow_ge_zero2 [rule_format (no_asm)]: "(0::real) <= r --> 0 <= r ^ n" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
404 |
apply (induct_tac "n") |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
405 |
apply (auto simp add: real_0_le_mult_iff) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
406 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
407 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
408 |
lemma realpow_le2 [rule_format (no_asm)]: "(0::real) <= x & x <= y --> x ^ n <= y ^ n" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
409 |
apply (induct_tac "n") |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
410 |
apply (auto intro!: real_mult_le_mono simp add: realpow_ge_zero2) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
411 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
412 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
413 |
lemma realpow_Suc_gt_one: "(1::real) < r ==> 1 < r ^ (Suc n)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
414 |
apply (frule_tac n = "n" in realpow_ge_one) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
415 |
apply (drule real_le_imp_less_or_eq, safe) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
416 |
apply (frule real_zero_less_one [THEN real_less_trans]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
417 |
apply (drule_tac y = "r ^ n" in real_mult_less_mono2) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
418 |
apply assumption |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
419 |
apply (auto dest: real_less_trans) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
420 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
421 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
422 |
lemma realpow_two_sum_zero_iff: "(x ^ 2 + y ^ 2 = (0::real)) = (x = 0 & y = 0)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
423 |
apply (auto intro: real_sum_squares_cancel real_sum_squares_cancel2 simp add: numeral_2_eq_2) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
424 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
425 |
declare realpow_two_sum_zero_iff [simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
426 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
427 |
lemma realpow_two_le_add_order: "(0::real) <= u ^ 2 + v ^ 2" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
428 |
apply (rule real_le_add_order) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
429 |
apply (auto simp add: numeral_2_eq_2) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
430 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
431 |
declare realpow_two_le_add_order [simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
432 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
433 |
lemma realpow_two_le_add_order2: "(0::real) <= u ^ 2 + v ^ 2 + w ^ 2" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
434 |
apply (rule real_le_add_order)+ |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
435 |
apply (auto simp add: numeral_2_eq_2) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
436 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
437 |
declare realpow_two_le_add_order2 [simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
438 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
439 |
lemma real_mult_self_sum_ge_zero: "(0::real) <= x*x + y*y" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
440 |
apply (cut_tac u = "x" and v = "y" in realpow_two_le_add_order) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
441 |
apply (auto simp add: numeral_2_eq_2) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
442 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
443 |
declare real_mult_self_sum_ge_zero [simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
444 |
declare real_mult_self_sum_ge_zero [THEN abs_eqI1, simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
445 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
446 |
lemma real_sum_square_gt_zero: "x ~= 0 ==> (0::real) < x * x + y * y" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
447 |
apply (cut_tac x = "x" and y = "y" in real_mult_self_sum_ge_zero) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
448 |
apply (drule real_le_imp_less_or_eq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
449 |
apply (drule_tac y = "y" in real_sum_squares_not_zero) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
450 |
apply auto |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
451 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
452 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
453 |
lemma real_sum_square_gt_zero2: "y ~= 0 ==> (0::real) < x * x + y * y" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
454 |
apply (rule real_add_commute [THEN subst]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
455 |
apply (erule real_sum_square_gt_zero) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
456 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
457 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
458 |
lemma real_minus_mult_self_le: "-(u * u) <= (x * (x::real))" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
459 |
apply (rule_tac j = "0" in real_le_trans) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
460 |
apply auto |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
461 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
462 |
declare real_minus_mult_self_le [simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
463 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
464 |
lemma realpow_square_minus_le: "-(u ^ 2) <= (x::real) ^ 2" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
465 |
apply (auto simp add: numeral_2_eq_2) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
466 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
467 |
declare realpow_square_minus_le [simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
468 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
469 |
lemma realpow_num_eq_if: "(m::real) ^ n = (if n=0 then 1 else m * m ^ (n - 1))" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
470 |
apply (case_tac "n") |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
471 |
apply auto |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
472 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
473 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
474 |
lemma real_num_zero_less_two_pow: "0 < (2::real) ^ (4*d)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
475 |
apply (induct_tac "d") |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
476 |
apply (auto simp add: realpow_num_eq_if) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
477 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
478 |
declare real_num_zero_less_two_pow [simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
479 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
480 |
lemma lemma_realpow_num_two_mono: "x * (4::real) < y ==> x * (2 ^ 8) < y * (2 ^ 6)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
481 |
apply (subgoal_tac " (2::real) ^ 8 = 4 * (2 ^ 6) ") |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
482 |
apply (simp (no_asm_simp) add: real_mult_assoc [symmetric]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
483 |
apply (auto simp add: realpow_num_eq_if) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
484 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
485 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
486 |
lemma lemma_realpow_4: "2 ^ 2 = (4::real)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
487 |
apply (simp (no_asm) add: realpow_num_eq_if) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
488 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
489 |
declare lemma_realpow_4 [simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
490 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
491 |
lemma lemma_realpow_16: "2 ^ 4 = (16::real)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
492 |
apply (simp (no_asm) add: realpow_num_eq_if) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
493 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
494 |
declare lemma_realpow_16 [simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
495 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
496 |
lemma zero_le_x_squared: "(0::real) <= x^2" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
497 |
apply (simp add: numeral_2_eq_2) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
498 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
499 |
declare zero_le_x_squared [simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
500 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
501 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
502 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
503 |
ML |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
504 |
{* |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
505 |
val realpow_0 = thm "realpow_0"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
506 |
val realpow_Suc = thm "realpow_Suc"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
507 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
508 |
val realpow_zero = thm "realpow_zero"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
509 |
val realpow_not_zero = thm "realpow_not_zero"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
510 |
val realpow_zero_zero = thm "realpow_zero_zero"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
511 |
val realpow_inverse = thm "realpow_inverse"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
512 |
val realpow_abs = thm "realpow_abs"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
513 |
val realpow_add = thm "realpow_add"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
514 |
val realpow_one = thm "realpow_one"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
515 |
val realpow_two = thm "realpow_two"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
516 |
val realpow_gt_zero = thm "realpow_gt_zero"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
517 |
val realpow_ge_zero = thm "realpow_ge_zero"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
518 |
val realpow_le = thm "realpow_le"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
519 |
val realpow_0_left = thm "realpow_0_left"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
520 |
val realpow_less = thm "realpow_less"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
521 |
val realpow_eq_one = thm "realpow_eq_one"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
522 |
val abs_realpow_minus_one = thm "abs_realpow_minus_one"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
523 |
val realpow_mult = thm "realpow_mult"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
524 |
val realpow_two_le = thm "realpow_two_le"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
525 |
val abs_realpow_two = thm "abs_realpow_two"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
526 |
val realpow_two_abs = thm "realpow_two_abs"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
527 |
val realpow_two_gt_one = thm "realpow_two_gt_one"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
528 |
val realpow_ge_one = thm "realpow_ge_one"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
529 |
val realpow_ge_one2 = thm "realpow_ge_one2"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
530 |
val two_realpow_ge_one = thm "two_realpow_ge_one"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
531 |
val two_realpow_gt = thm "two_realpow_gt"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
532 |
val realpow_minus_one = thm "realpow_minus_one"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
533 |
val realpow_minus_one_odd = thm "realpow_minus_one_odd"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
534 |
val realpow_minus_one_even = thm "realpow_minus_one_even"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
535 |
val realpow_Suc_less = thm "realpow_Suc_less"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
536 |
val realpow_Suc_le = thm "realpow_Suc_le"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
537 |
val realpow_zero_le = thm "realpow_zero_le"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
538 |
val realpow_Suc_le2 = thm "realpow_Suc_le2"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
539 |
val realpow_Suc_le3 = thm "realpow_Suc_le3"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
540 |
val realpow_less_le = thm "realpow_less_le"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
541 |
val realpow_le_le = thm "realpow_le_le"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
542 |
val realpow_Suc_le_self = thm "realpow_Suc_le_self"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
543 |
val realpow_Suc_less_one = thm "realpow_Suc_less_one"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
544 |
val realpow_le_Suc = thm "realpow_le_Suc"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
545 |
val realpow_less_Suc = thm "realpow_less_Suc"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
546 |
val realpow_le_Suc2 = thm "realpow_le_Suc2"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
547 |
val realpow_gt_ge2 = thm "realpow_gt_ge2"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
548 |
val realpow_ge_ge2 = thm "realpow_ge_ge2"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
549 |
val realpow_Suc_ge_self2 = thm "realpow_Suc_ge_self2"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
550 |
val realpow_ge_self2 = thm "realpow_ge_self2"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
551 |
val realpow_minus_mult = thm "realpow_minus_mult"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
552 |
val realpow_two_mult_inverse = thm "realpow_two_mult_inverse"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
553 |
val realpow_two_minus = thm "realpow_two_minus"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
554 |
val realpow_two_disj = thm "realpow_two_disj"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
555 |
val realpow_diff = thm "realpow_diff"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
556 |
val realpow_real_of_nat = thm "realpow_real_of_nat"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
557 |
val realpow_real_of_nat_two_pos = thm "realpow_real_of_nat_two_pos"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
558 |
val realpow_increasing = thm "realpow_increasing"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
559 |
val realpow_Suc_cancel_eq = thm "realpow_Suc_cancel_eq"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
560 |
val realpow_eq_0_iff = thm "realpow_eq_0_iff"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
561 |
val zero_less_realpow_abs_iff = thm "zero_less_realpow_abs_iff"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
562 |
val zero_le_realpow_abs = thm "zero_le_realpow_abs"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
563 |
val real_of_int_power = thm "real_of_int_power"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
564 |
val power_real_number_of = thm "power_real_number_of"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
565 |
val real_power_two = thm "real_power_two"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
566 |
val real_sqr_ge_zero = thm "real_sqr_ge_zero"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
567 |
val real_sqr_gt_zero = thm "real_sqr_gt_zero"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
568 |
val real_sqr_not_zero = thm "real_sqr_not_zero"; |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
569 |
val real_sum_squares_cancel_a = thm "real_sum_squares_cancel_a"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
570 |
val real_mult_inverse_cancel2 = thm "real_mult_inverse_cancel2"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
571 |
val real_squared_diff_one_factored = thm "real_squared_diff_one_factored"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
572 |
val real_mult_is_one = thm "real_mult_is_one"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
573 |
val real_le_add_half_cancel = thm "real_le_add_half_cancel"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
574 |
val real_minus_half_eq = thm "real_minus_half_eq"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
575 |
val real_mult_inverse_cancel = thm "real_mult_inverse_cancel"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
576 |
val real_mult_inverse_cancel2 = thm "real_mult_inverse_cancel2"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
577 |
val inverse_real_of_nat_gt_zero = thm "inverse_real_of_nat_gt_zero"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
578 |
val inverse_real_of_nat_ge_zero = thm "inverse_real_of_nat_ge_zero"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
579 |
val real_sum_squares_not_zero = thm "real_sum_squares_not_zero"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
580 |
val real_sum_squares_not_zero2 = thm "real_sum_squares_not_zero2"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
581 |
val abs_mult_abs = thm "abs_mult_abs"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
582 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
583 |
val realpow_divide = thm "realpow_divide"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
584 |
val realpow_ge_zero2 = thm "realpow_ge_zero2"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
585 |
val realpow_le2 = thm "realpow_le2"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
586 |
val realpow_Suc_gt_one = thm "realpow_Suc_gt_one"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
587 |
val realpow_two_sum_zero_iff = thm "realpow_two_sum_zero_iff"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
588 |
val realpow_two_le_add_order = thm "realpow_two_le_add_order"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
589 |
val realpow_two_le_add_order2 = thm "realpow_two_le_add_order2"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
590 |
val real_mult_self_sum_ge_zero = thm "real_mult_self_sum_ge_zero"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
591 |
val real_sum_square_gt_zero = thm "real_sum_square_gt_zero"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
592 |
val real_sum_square_gt_zero2 = thm "real_sum_square_gt_zero2"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
593 |
val real_minus_mult_self_le = thm "real_minus_mult_self_le"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
594 |
val realpow_square_minus_le = thm "realpow_square_minus_le"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
595 |
val realpow_num_eq_if = thm "realpow_num_eq_if"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
596 |
val real_num_zero_less_two_pow = thm "real_num_zero_less_two_pow"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
597 |
val lemma_realpow_num_two_mono = thm "lemma_realpow_num_two_mono"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
598 |
val lemma_realpow_4 = thm "lemma_realpow_4"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
599 |
val lemma_realpow_16 = thm "lemma_realpow_16"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
600 |
val zero_le_x_squared = thm "zero_le_x_squared"; |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
601 |
*} |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
602 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
603 |
|
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
604 |
end |