author | haftmann |
Thu, 11 Mar 2010 14:38:20 +0100 | |
changeset 35724 | 178ad68f93ed |
parent 35632 | 61fd75e33137 |
child 36777 | be5461582d0f |
permissions | -rw-r--r-- |
28952
15a4b2cf8c34
made repository layout more coherent with logical distribution structure; stripped some $Id$s
haftmann
parents:
28906
diff
changeset
|
1 |
(* Title : HOL/RealPow.thy |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
2 |
Author : Jacques D. Fleuriot |
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
3 |
Copyright : 1998 University of Cambridge |
20634 | 4 |
*) |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
5 |
|
20634 | 6 |
header {* Natural powers theory *} |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
7 |
|
15131 | 8 |
theory RealPow |
35578
384ad08a1d1b
Added natfloor and floor rules for multiplication and power.
hoelzl
parents:
35442
diff
changeset
|
9 |
imports RealDef RComplete |
15131 | 10 |
begin |
9435
c3a13a7d4424
lemmas [arith_split] = abs_split (*belongs to theory RealAbs*);
wenzelm
parents:
9013
diff
changeset
|
11 |
|
35347 | 12 |
(* FIXME: declare this in Rings.thy or not at all *) |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
13 |
declare abs_mult_self [simp] |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
14 |
|
35347 | 15 |
(* used by Import/HOL/real.imp *) |
16 |
lemma two_realpow_ge_one: "(1::real) \<le> 2 ^ n" |
|
25875 | 17 |
by simp |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
18 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
19 |
lemma two_realpow_gt [simp]: "real (n::nat) < 2 ^ n" |
15251 | 20 |
apply (induct "n") |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
21 |
apply (auto simp add: real_of_nat_Suc) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14352
diff
changeset
|
22 |
apply (subst mult_2) |
35216 | 23 |
apply (erule add_less_le_mono) |
24 |
apply (rule two_realpow_ge_one) |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
25 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
26 |
|
35632
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
27 |
(* TODO: no longer real-specific; rename and move elsewhere *) |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
28 |
lemma realpow_Suc_le_self: |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
29 |
fixes r :: "'a::linordered_semidom" |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
30 |
shows "[| 0 \<le> r; r \<le> 1 |] ==> r ^ Suc n \<le> r" |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
31 |
by (insert power_decreasing [of 1 "Suc n" r], simp) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
32 |
|
35632
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
33 |
(* TODO: no longer real-specific; rename and move elsewhere *) |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
34 |
lemma realpow_minus_mult: |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
35 |
fixes x :: "'a::monoid_mult" |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
36 |
shows "0 < n \<Longrightarrow> x ^ (n - 1) * x = x ^ n" |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
37 |
by (simp add: power_commutes split add: nat_diff_split) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
38 |
|
35632
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
39 |
(* TODO: no longer real-specific; rename and move elsewhere *) |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
40 |
lemma realpow_two_diff: |
35632
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
41 |
fixes x :: "'a::comm_ring_1" |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
42 |
shows "x^Suc (Suc 0) - y^Suc (Suc 0) = (x - y) * (x + y)" |
35347 | 43 |
by (simp add: algebra_simps) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
44 |
|
35632
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
45 |
(* TODO: move elsewhere *) |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
46 |
lemma add_eq_0_iff: |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
47 |
fixes x y :: "'a::group_add" |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
48 |
shows "x + y = 0 \<longleftrightarrow> y = - x" |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
49 |
by (auto dest: minus_unique) |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
50 |
|
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
51 |
(* TODO: no longer real-specific; rename and move elsewhere *) |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
52 |
lemma realpow_two_disj: |
35632
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
53 |
fixes x :: "'a::idom" |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
54 |
shows "(x^Suc (Suc 0) = y^Suc (Suc 0)) = (x = y | x = -y)" |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
55 |
using realpow_two_diff [of x y] |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
56 |
by (auto simp add: add_eq_0_iff) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
57 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
58 |
|
22970 | 59 |
subsection{* Squares of Reals *} |
60 |
||
35632
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
61 |
(* FIXME: declare this [simp] for all types, or not at all *) |
22970 | 62 |
lemma real_two_squares_add_zero_iff [simp]: |
63 |
"(x * x + y * y = 0) = ((x::real) = 0 \<and> y = 0)" |
|
64 |
by (rule sum_squares_eq_zero_iff) |
|
65 |
||
35632
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
66 |
(* TODO: no longer real-specific; rename and move elsewhere *) |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
67 |
lemma real_squared_diff_one_factored: |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
68 |
fixes x :: "'a::ring_1" |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
69 |
shows "x * x - 1 = (x + 1) * (x - 1)" |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
70 |
by (simp add: algebra_simps) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
71 |
|
35632
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
72 |
(* TODO: no longer real-specific; rename and move elsewhere *) |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
73 |
lemma real_mult_is_one [simp]: |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
74 |
fixes x :: "'a::ring_1_no_zero_divisors" |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
75 |
shows "x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1" |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
76 |
proof - |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
77 |
have "x * x = 1 \<longleftrightarrow> (x + 1) * (x - 1) = 0" |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
78 |
by (simp add: algebra_simps) |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
79 |
also have "\<dots> \<longleftrightarrow> x = 1 \<or> x = - 1" |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
80 |
by (auto simp add: add_eq_0_iff minus_equation_iff [of _ 1]) |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
81 |
finally show ?thesis . |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
82 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
83 |
|
35632
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
84 |
(* FIXME: declare this [simp] for all types, or not at all *) |
22970 | 85 |
lemma realpow_two_sum_zero_iff [simp]: |
86 |
"(x ^ 2 + y ^ 2 = (0::real)) = (x = 0 & y = 0)" |
|
87 |
by (rule sum_power2_eq_zero_iff) |
|
88 |
||
35632
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
89 |
(* FIXME: declare this [simp] for all types, or not at all *) |
22970 | 90 |
lemma realpow_two_le_add_order [simp]: "(0::real) \<le> u ^ 2 + v ^ 2" |
91 |
by (rule sum_power2_ge_zero) |
|
92 |
||
35632
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
93 |
(* FIXME: declare this [simp] for all types, or not at all *) |
22970 | 94 |
lemma realpow_two_le_add_order2 [simp]: "(0::real) \<le> u ^ 2 + v ^ 2 + w ^ 2" |
95 |
by (intro add_nonneg_nonneg zero_le_power2) |
|
96 |
||
97 |
lemma real_minus_mult_self_le [simp]: "-(u * u) \<le> (x * (x::real))" |
|
98 |
by (rule_tac j = 0 in real_le_trans, auto) |
|
99 |
||
100 |
lemma realpow_square_minus_le [simp]: "-(u ^ 2) \<le> (x::real) ^ 2" |
|
101 |
by (auto simp add: power2_eq_square) |
|
102 |
||
103 |
(* The following theorem is by Benjamin Porter *) |
|
35632
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
104 |
(* TODO: no longer real-specific; rename and move elsewhere *) |
22970 | 105 |
lemma real_sq_order: |
35632
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
106 |
fixes x :: "'a::linordered_semidom" |
22970 | 107 |
assumes xgt0: "0 \<le> x" and ygt0: "0 \<le> y" and sq: "x^2 \<le> y^2" |
108 |
shows "x \<le> y" |
|
109 |
proof - |
|
110 |
from sq have "x ^ Suc (Suc 0) \<le> y ^ Suc (Suc 0)" |
|
111 |
by (simp only: numeral_2_eq_2) |
|
112 |
thus "x \<le> y" using ygt0 |
|
113 |
by (rule power_le_imp_le_base) |
|
114 |
qed |
|
115 |
||
35578
384ad08a1d1b
Added natfloor and floor rules for multiplication and power.
hoelzl
parents:
35442
diff
changeset
|
116 |
subsection {*Floor*} |
384ad08a1d1b
Added natfloor and floor rules for multiplication and power.
hoelzl
parents:
35442
diff
changeset
|
117 |
|
384ad08a1d1b
Added natfloor and floor rules for multiplication and power.
hoelzl
parents:
35442
diff
changeset
|
118 |
lemma floor_power: |
384ad08a1d1b
Added natfloor and floor rules for multiplication and power.
hoelzl
parents:
35442
diff
changeset
|
119 |
assumes "x = real (floor x)" |
384ad08a1d1b
Added natfloor and floor rules for multiplication and power.
hoelzl
parents:
35442
diff
changeset
|
120 |
shows "floor (x ^ n) = floor x ^ n" |
384ad08a1d1b
Added natfloor and floor rules for multiplication and power.
hoelzl
parents:
35442
diff
changeset
|
121 |
proof - |
384ad08a1d1b
Added natfloor and floor rules for multiplication and power.
hoelzl
parents:
35442
diff
changeset
|
122 |
have *: "x ^ n = real (floor x ^ n)" |
384ad08a1d1b
Added natfloor and floor rules for multiplication and power.
hoelzl
parents:
35442
diff
changeset
|
123 |
using assms by (induct n arbitrary: x) simp_all |
384ad08a1d1b
Added natfloor and floor rules for multiplication and power.
hoelzl
parents:
35442
diff
changeset
|
124 |
show ?thesis unfolding real_of_int_inject[symmetric] |
384ad08a1d1b
Added natfloor and floor rules for multiplication and power.
hoelzl
parents:
35442
diff
changeset
|
125 |
unfolding * floor_real_of_int .. |
384ad08a1d1b
Added natfloor and floor rules for multiplication and power.
hoelzl
parents:
35442
diff
changeset
|
126 |
qed |
384ad08a1d1b
Added natfloor and floor rules for multiplication and power.
hoelzl
parents:
35442
diff
changeset
|
127 |
|
384ad08a1d1b
Added natfloor and floor rules for multiplication and power.
hoelzl
parents:
35442
diff
changeset
|
128 |
lemma natfloor_power: |
384ad08a1d1b
Added natfloor and floor rules for multiplication and power.
hoelzl
parents:
35442
diff
changeset
|
129 |
assumes "x = real (natfloor x)" |
384ad08a1d1b
Added natfloor and floor rules for multiplication and power.
hoelzl
parents:
35442
diff
changeset
|
130 |
shows "natfloor (x ^ n) = natfloor x ^ n" |
384ad08a1d1b
Added natfloor and floor rules for multiplication and power.
hoelzl
parents:
35442
diff
changeset
|
131 |
proof - |
384ad08a1d1b
Added natfloor and floor rules for multiplication and power.
hoelzl
parents:
35442
diff
changeset
|
132 |
from assms have "0 \<le> floor x" by auto |
384ad08a1d1b
Added natfloor and floor rules for multiplication and power.
hoelzl
parents:
35442
diff
changeset
|
133 |
note assms[unfolded natfloor_def real_nat_eq_real[OF `0 \<le> floor x`]] |
384ad08a1d1b
Added natfloor and floor rules for multiplication and power.
hoelzl
parents:
35442
diff
changeset
|
134 |
from floor_power[OF this] |
384ad08a1d1b
Added natfloor and floor rules for multiplication and power.
hoelzl
parents:
35442
diff
changeset
|
135 |
show ?thesis unfolding natfloor_def nat_power_eq[OF `0 \<le> floor x`, symmetric] |
384ad08a1d1b
Added natfloor and floor rules for multiplication and power.
hoelzl
parents:
35442
diff
changeset
|
136 |
by simp |
384ad08a1d1b
Added natfloor and floor rules for multiplication and power.
hoelzl
parents:
35442
diff
changeset
|
137 |
qed |
22970 | 138 |
|
139 |
subsection {*Various Other Theorems*} |
|
140 |
||
14304 | 141 |
lemma real_le_add_half_cancel: "(x + y/2 \<le> (y::real)) = (x \<le> y /2)" |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
142 |
by auto |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
143 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
144 |
lemma real_mult_inverse_cancel: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
145 |
"[|(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
|
146 |
==> inverse x * y < inverse x1 * u" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
147 |
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
|
148 |
apply (auto simp add: real_mult_assoc [symmetric]) |
14334 | 149 |
apply (simp (no_asm) add: mult_ac) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
150 |
apply (rule_tac c=x1 in mult_less_imp_less_right) |
14334 | 151 |
apply (auto simp add: mult_ac) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
152 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
153 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
154 |
lemma real_mult_inverse_cancel2: |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
155 |
"[|(0::real) < x;0 < x1; x1 * y < x * u |] ==> y * inverse x < u * inverse x1" |
14334 | 156 |
apply (auto dest: real_mult_inverse_cancel simp add: mult_ac) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
157 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
158 |
|
35632
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
159 |
(* TODO: no longer real-specific; rename and move elsewhere *) |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
160 |
lemma realpow_num_eq_if: |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
161 |
fixes m :: "'a::power" |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
162 |
shows "m ^ n = (if n=0 then 1 else m * m ^ (n - 1))" |
61fd75e33137
generalize some lemmas, and remove a few unnecessary ones
huffman
parents:
35578
diff
changeset
|
163 |
by (cases n, auto) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
164 |
|
35578
384ad08a1d1b
Added natfloor and floor rules for multiplication and power.
hoelzl
parents:
35442
diff
changeset
|
165 |
|
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
166 |
end |