author | paulson |
Thu, 29 Jul 2004 16:14:42 +0200 | |
changeset 15085 | 5693a977a767 |
parent 15003 | 6145dd7538d7 |
child 15131 | c69542757a4d |
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 |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14352
diff
changeset
|
9 |
theory RealPow = RealDef: |
9435
c3a13a7d4424
lemmas [arith_split] = abs_split (*belongs to theory RealAbs*);
wenzelm
parents:
9013
diff
changeset
|
10 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
11 |
declare abs_mult_self [simp] |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
12 |
|
10309 | 13 |
instance real :: power .. |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
14 |
|
8856 | 15 |
primrec (realpow) |
12018
ec054019c910
Numerals and simprocs for types real and hypreal. The abstract
paulson
parents:
11701
diff
changeset
|
16 |
realpow_0: "r ^ 0 = 1" |
9435
c3a13a7d4424
lemmas [arith_split] = abs_split (*belongs to theory RealAbs*);
wenzelm
parents:
9013
diff
changeset
|
17 |
realpow_Suc: "r ^ (Suc n) = (r::real) * (r ^ n)" |
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
18 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
19 |
|
15003 | 20 |
instance real :: recpower |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
21 |
proof |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
22 |
fix z :: real |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
23 |
fix n :: nat |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
24 |
show "z^0 = 1" by simp |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
25 |
show "z^(Suc n) = z * (z^n)" by simp |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
26 |
qed |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
27 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
28 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
29 |
lemma realpow_not_zero: "r \<noteq> (0::real) ==> r ^ n \<noteq> 0" |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
30 |
by (rule field_power_not_zero) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
31 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
32 |
lemma realpow_zero_zero: "r ^ n = (0::real) ==> r = 0" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
33 |
by simp |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
34 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
35 |
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
|
36 |
by simp |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
37 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
38 |
text{*Legacy: weaker version of the theorem @{text power_strict_mono}, |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
39 |
used 6 times in NthRoot and Transcendental*} |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
40 |
lemma realpow_less: |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
41 |
"[|(0::real) < x; x < y; 0 < n|] ==> x ^ n < y ^ n" |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
42 |
apply (rule power_strict_mono, auto) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
43 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
44 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
45 |
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
|
46 |
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
|
47 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
48 |
lemma abs_realpow_two [simp]: "abs((x::real)^Suc (Suc 0)) = x^Suc (Suc 0)" |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
49 |
by (simp add: abs_mult) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
50 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
51 |
lemma realpow_two_abs [simp]: "abs(x::real)^Suc (Suc 0) = x^Suc (Suc 0)" |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
52 |
by (simp add: power_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
|
53 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
54 |
lemma two_realpow_ge_one [simp]: "(1::real) \<le> 2 ^ n" |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
55 |
by (insert power_increasing [of 0 n "2::real"], simp) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
56 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
57 |
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
|
58 |
apply (induct_tac "n") |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
59 |
apply (auto simp add: real_of_nat_Suc) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14352
diff
changeset
|
60 |
apply (subst mult_2) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
61 |
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
|
62 |
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
|
63 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
64 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
65 |
lemma realpow_Suc_le_self: "[| 0 \<le> r; r \<le> (1::real) |] ==> r ^ Suc n \<le> r" |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
66 |
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
|
67 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
68 |
text{*Used ONCE in Transcendental*} |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
69 |
lemma realpow_Suc_less_one: "[| 0 < r; r < (1::real) |] ==> r ^ Suc n < 1" |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
70 |
by (insert power_strict_decreasing [of 0 "Suc n" r], simp) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
71 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
72 |
text{*Used ONCE in Lim.ML*} |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
73 |
lemma realpow_minus_mult [rule_format]: |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
74 |
"0 < n --> (x::real) ^ (n - 1) * x = x ^ n" |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
75 |
apply (simp 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
|
76 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
77 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
78 |
lemma realpow_two_mult_inverse [simp]: |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
79 |
"r \<noteq> 0 ==> r * inverse r ^Suc (Suc 0) = inverse (r::real)" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
80 |
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
|
81 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
82 |
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
|
83 |
by simp |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
84 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
85 |
lemma realpow_two_diff: |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
86 |
"(x::real)^Suc (Suc 0) - y^Suc (Suc 0) = (x - y) * (x + y)" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
87 |
apply (unfold real_diff_def) |
14334 | 88 |
apply (simp add: right_distrib left_distrib mult_ac) |
14265
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 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
91 |
lemma realpow_two_disj: |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
92 |
"((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
|
93 |
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
|
94 |
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
|
95 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
96 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
97 |
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
|
98 |
apply (induct_tac "n") |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
99 |
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
|
100 |
done |
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_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
|
103 |
apply (induct_tac "n") |
14334 | 104 |
apply (auto simp add: real_of_nat_mult zero_less_mult_iff) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
105 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
106 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
107 |
lemma realpow_increasing: |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
108 |
"[|(0::real) \<le> x; 0 \<le> y; x ^ Suc n \<le> y ^ Suc n|] ==> x \<le> y" |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
109 |
by (rule power_le_imp_le_base) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
110 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
111 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
112 |
lemma zero_less_realpow_abs_iff [simp]: |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
113 |
"(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
|
114 |
apply (induct_tac "n") |
14334 | 115 |
apply (auto simp add: zero_less_mult_iff) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
116 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
117 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
118 |
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
|
119 |
apply (induct_tac "n") |
14334 | 120 |
apply (auto simp add: zero_le_mult_iff) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
121 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
122 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
123 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
124 |
subsection{*Literal Arithmetic Involving Powers, Type @{typ real}*} |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
125 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
126 |
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
|
127 |
apply (induct_tac "n") |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14352
diff
changeset
|
128 |
apply (simp_all add: nat_mult_distrib) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
129 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
130 |
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
|
131 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
132 |
lemma power_real_number_of: |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
133 |
"(number_of v :: real) ^ n = real ((number_of v :: int) ^ n)" |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14352
diff
changeset
|
134 |
by (simp only: real_number_of [symmetric] real_of_int_power) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
135 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
136 |
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
|
137 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
138 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
139 |
subsection{*Various Other Theorems*} |
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 |
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
|
142 |
lemma real_sum_squares_cancel_a: "x * x = -(y * y) ==> x = (0::real) & y=0" |
15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15003
diff
changeset
|
143 |
apply (auto dest: real_sum_squares_cancel simp add: real_add_eq_0_iff [symmetric]) |
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15003
diff
changeset
|
144 |
apply (auto dest: real_sum_squares_cancel simp add: add_commute) |
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15003
diff
changeset
|
145 |
done |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
146 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
147 |
lemma real_squared_diff_one_factored: "x*x - (1::real) = (x + 1)*(x - 1)" |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
148 |
by (auto simp add: left_distrib right_distrib real_diff_def) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
149 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
150 |
lemma real_mult_is_one [simp]: "(x*x = (1::real)) = (x = 1 | x = - 1)" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
151 |
apply auto |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
152 |
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
|
153 |
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
|
154 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
155 |
|
14304 | 156 |
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
|
157 |
by auto |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
158 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
159 |
lemma real_minus_half_eq [simp]: "(x::real) - x/2 = x/2" |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
160 |
by auto |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
161 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
162 |
lemma real_mult_inverse_cancel: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
163 |
"[|(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
|
164 |
==> inverse x * y < inverse x1 * u" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
165 |
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
|
166 |
apply (auto simp add: real_mult_assoc [symmetric]) |
14334 | 167 |
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
|
168 |
apply (rule_tac c=x1 in mult_less_imp_less_right) |
14334 | 169 |
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
|
170 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
171 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
172 |
text{*Used once: in Hyperreal/Transcendental.ML*} |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
173 |
lemma real_mult_inverse_cancel2: |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
174 |
"[|(0::real) < x;0 < x1; x1 * y < x * u |] ==> y * inverse x < u * inverse x1" |
14334 | 175 |
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
|
176 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
177 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
178 |
lemma inverse_real_of_nat_gt_zero [simp]: "0 < inverse (real (Suc n))" |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
179 |
by auto |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
180 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
181 |
lemma inverse_real_of_nat_ge_zero [simp]: "0 \<le> inverse (real (Suc n))" |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
182 |
by auto |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
183 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
184 |
lemma real_sum_squares_not_zero: "x ~= 0 ==> x * x + y * y ~= (0::real)" |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
185 |
by (blast dest!: real_sum_squares_cancel) |
14268
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 real_sum_squares_not_zero2: "y ~= 0 ==> x * x + y * y ~= (0::real)" |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
188 |
by (blast dest!: real_sum_squares_cancel2) |
14268
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 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
191 |
subsection {*Various Other Theorems*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
192 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
193 |
lemma realpow_divide: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
194 |
"(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
|
195 |
apply (unfold real_divide_def) |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
196 |
apply (auto simp add: power_mult_distrib power_inverse) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
197 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
198 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
199 |
lemma realpow_two_sum_zero_iff [simp]: |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
200 |
"(x ^ 2 + y ^ 2 = (0::real)) = (x = 0 & y = 0)" |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
201 |
apply (auto intro: real_sum_squares_cancel real_sum_squares_cancel2 |
14352 | 202 |
simp add: power2_eq_square) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
203 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
204 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
205 |
lemma realpow_two_le_add_order [simp]: "(0::real) \<le> u ^ 2 + v ^ 2" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
206 |
apply (rule real_le_add_order) |
14352 | 207 |
apply (auto simp add: power2_eq_square) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
208 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
209 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
210 |
lemma realpow_two_le_add_order2 [simp]: "(0::real) \<le> u ^ 2 + v ^ 2 + w ^ 2" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
211 |
apply (rule real_le_add_order)+ |
14352 | 212 |
apply (auto simp add: power2_eq_square) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
213 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
214 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
215 |
lemma real_sum_square_gt_zero: "x ~= 0 ==> (0::real) < x * x + y * y" |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
216 |
apply (cut_tac x = x and y = y in real_mult_self_sum_ge_zero) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
217 |
apply (drule real_le_imp_less_or_eq) |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
218 |
apply (drule_tac y = y in real_sum_squares_not_zero, auto) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
219 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
220 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
221 |
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
|
222 |
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
|
223 |
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
|
224 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
225 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
226 |
lemma real_minus_mult_self_le [simp]: "-(u * u) \<le> (x * (x::real))" |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
227 |
by (rule_tac j = 0 in real_le_trans, auto) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
228 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
229 |
lemma realpow_square_minus_le [simp]: "-(u ^ 2) \<le> (x::real) ^ 2" |
14352 | 230 |
by (auto simp add: power2_eq_square) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
231 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
232 |
lemma realpow_num_eq_if: "(m::real) ^ n = (if n=0 then 1 else m * m ^ (n - 1))" |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
233 |
by (case_tac "n", auto) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
234 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
235 |
lemma real_num_zero_less_two_pow [simp]: "0 < (2::real) ^ (4*d)" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
236 |
apply (induct_tac "d") |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
237 |
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
|
238 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
239 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
240 |
lemma lemma_realpow_num_two_mono: |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14334
diff
changeset
|
241 |
"x * (4::real) < y ==> x * (2 ^ 8) < y * (2 ^ 6)" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
242 |
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
|
243 |
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
|
244 |
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
|
245 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
246 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
247 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
248 |
ML |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
249 |
{* |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
250 |
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
|
251 |
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
|
252 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
253 |
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
|
254 |
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
|
255 |
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
|
256 |
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
|
257 |
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
|
258 |
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
|
259 |
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
|
260 |
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
|
261 |
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
|
262 |
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
|
263 |
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
|
264 |
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
|
265 |
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
|
266 |
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
|
267 |
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
|
268 |
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
|
269 |
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
|
270 |
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
|
271 |
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
|
272 |
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
|
273 |
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
|
274 |
val power_real_number_of = thm "power_real_number_of"; |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
275 |
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
|
276 |
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
|
277 |
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
|
278 |
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
|
279 |
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
|
280 |
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
|
281 |
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
|
282 |
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
|
283 |
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
|
284 |
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
|
285 |
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
|
286 |
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
|
287 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
288 |
val realpow_divide = thm "realpow_divide"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14265
diff
changeset
|
289 |
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
|
290 |
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
|
291 |
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
|
292 |
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
|
293 |
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
|
294 |
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
|
295 |
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
|
296 |
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
|
297 |
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
|
298 |
val lemma_realpow_num_two_mono = thm "lemma_realpow_num_two_mono"; |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
299 |
*} |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
300 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
12018
diff
changeset
|
301 |
|
7077
60b098bb8b8a
heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff
changeset
|
302 |
end |