src/HOL/Real/RealPow.thy
author paulson
Wed, 18 Feb 2004 10:40:29 +0100
changeset 14392 386760e88462
parent 14387 e96d5c42c4b0
child 14443 75910c7557c5
permissions -rw-r--r--
removed obsolete theorem
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
9435
c3a13a7d4424 lemmas [arith_split] = abs_split (*belongs to theory RealAbs*);
wenzelm
parents: 9013
diff changeset
     1
(*  Title       : HOL/Real/RealPow.thy
7219
4e3f386c2e37 inserted Id: lines
paulson
parents: 7077
diff changeset
     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744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for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744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for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
a7f961fb62c6 intro_classes by default;
wenzelm
parents: 9435
diff changeset
    13
instance real :: power ..
7077
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff changeset
    14
8856
435187ffc64e fixed theory deps;
wenzelm
parents: 7219
diff changeset
    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
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
    20
instance real :: ringpower
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 abs_realpow_minus_one [simp]: "abs((-1) ^ n) = (1::real)"
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
    46
by (simp add: power_abs)
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 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
    49
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
    50
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
    51
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
    52
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
    53
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
    54
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
    55
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
    56
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
    57
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
    58
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
    59
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
    60
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
    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 simp add: real_of_nat_Suc)
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14352
diff changeset
    63
apply (subst mult_2)
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents: 12018
diff changeset
    64
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
    65
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
    66
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents: 12018
diff changeset
    67
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
    68
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
    69
by (induct_tac "n", auto)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
    70
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
    71
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
    72
by auto
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents: 12018
diff changeset
    73
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
    74
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
    75
by auto
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents: 12018
diff changeset
    76
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
    77
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
    78
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
    79
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
    80
text{*Used ONCE in Transcendental*}
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
    81
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
    82
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
    83
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
    84
text{*Used ONCE in Lim.ML*}
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
    85
lemma realpow_minus_mult [rule_format]:
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
    86
     "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
    87
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
    88
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents: 12018
diff changeset
    89
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
    90
lemma realpow_two_mult_inverse [simp]:
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
    91
     "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
    92
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
    93
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
    94
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
    95
by simp
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents: 12018
diff changeset
    96
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
    97
lemma realpow_two_diff:
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
    98
     "(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
    99
apply (unfold real_diff_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14304
diff changeset
   100
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
   101
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents: 12018
diff changeset
   102
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
   103
lemma realpow_two_disj:
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
   104
     "((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
   105
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
   106
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
   107
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents: 12018
diff changeset
   108
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents: 12018
diff changeset
   109
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
   110
apply (induct_tac "n")
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents: 12018
diff changeset
   111
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
   112
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents: 12018
diff changeset
   113
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   114
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
   115
apply (induct_tac "n")
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14304
diff changeset
   116
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
   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_increasing:
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
   120
     "[|(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
   121
  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
   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
lemma zero_less_realpow_abs_iff [simp]:
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
   125
     "(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
   126
apply (induct_tac "n")
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14304
diff changeset
   127
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
   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 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
   131
apply (induct_tac "n")
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14304
diff changeset
   132
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
   133
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents: 12018
diff changeset
   134
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents: 12018
diff changeset
   135
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
   136
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
   137
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents: 12018
diff changeset
   138
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
   139
apply (induct_tac "n")
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14352
diff changeset
   140
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
   141
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents: 12018
diff changeset
   142
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
   143
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
   144
lemma power_real_number_of:
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
   145
     "(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
   146
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
   147
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents: 12018
diff changeset
   148
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
   149
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
subsection{*Various Other Theorems*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   152
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   153
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
   154
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
   155
  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
   156
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   157
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
   158
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
   159
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
   160
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
   161
apply auto
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   162
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
   163
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
   164
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   165
14304
cc0b4bbfbc43 minor tweaks
paulson
parents: 14288
diff changeset
   166
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
   167
by auto
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   168
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
   169
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
   170
by auto
14268
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
lemma real_mult_inverse_cancel:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   173
     "[|(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
   174
      ==> inverse x * y < inverse x1 * u"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   175
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
   176
apply (auto simp add: real_mult_assoc [symmetric])
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14304
diff changeset
   177
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
   178
apply (rule_tac c=x1 in mult_less_imp_less_right) 
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14304
diff changeset
   179
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
   180
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   181
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   182
text{*Used once: in Hyperreal/Transcendental.ML*}
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
   183
lemma real_mult_inverse_cancel2:
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
   184
     "[|(0::real) < x;0 < x1; x1 * y < x * u |] ==> y * inverse x < u * inverse x1"
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14304
diff changeset
   185
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
   186
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   187
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
   188
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
   189
by auto
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   190
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
   191
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
   192
by auto
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   193
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   194
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
   195
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
   196
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   197
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
   198
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
   199
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   200
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   201
subsection {*Various Other Theorems*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   202
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   203
lemma realpow_divide: 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   204
    "(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
   205
apply (unfold real_divide_def)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
   206
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
   207
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   208
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
   209
lemma realpow_two_sum_zero_iff [simp]:
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
   210
     "(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
   211
apply (auto intro: real_sum_squares_cancel real_sum_squares_cancel2 
14352
a8b1a44d8264 Modified real arithmetic simplification
paulson
parents: 14348
diff changeset
   212
                   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
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
   215
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
   216
apply (rule real_le_add_order)
14352
a8b1a44d8264 Modified real arithmetic simplification
paulson
parents: 14348
diff changeset
   217
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
   218
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   219
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
   220
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
   221
apply (rule real_le_add_order)+
14352
a8b1a44d8264 Modified real arithmetic simplification
paulson
parents: 14348
diff changeset
   222
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
   223
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   224
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   225
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
   226
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
   227
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
   228
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
   229
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   230
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   231
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
   232
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
   233
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
   234
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   235
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
   236
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
   237
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
   238
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
   239
lemma realpow_square_minus_le [simp]: "-(u ^ 2) \<le> (x::real) ^ 2"
14352
a8b1a44d8264 Modified real arithmetic simplification
paulson
parents: 14348
diff changeset
   240
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
   241
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   242
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
   243
by (case_tac "n", auto)
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   244
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
   245
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
   246
apply (induct_tac "d")
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   247
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
   248
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   249
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
   250
lemma lemma_realpow_num_two_mono:
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14334
diff changeset
   251
     "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
   252
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
   253
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
   254
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
   255
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   256
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   257
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents: 12018
diff changeset
   258
ML
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents: 12018
diff changeset
   259
{*
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents: 12018
diff changeset
   260
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
   261
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
   262
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents: 12018
diff changeset
   263
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
   264
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
   265
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
   266
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
   267
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
   268
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
   269
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
   270
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
   271
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
   272
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
   273
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
   274
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
   275
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
   276
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
   277
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
   278
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
   279
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
   280
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
   281
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
   282
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
   283
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
   284
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
   285
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
   286
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
   287
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
   288
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
   289
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
   290
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
   291
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
   292
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
   293
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
   294
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
   295
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
   296
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
   297
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
   298
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
   299
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
   300
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
   301
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   302
val realpow_divide = thm "realpow_divide";
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14265
diff changeset
   303
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
   304
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
   305
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
   306
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
   307
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
   308
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
   309
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
   310
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
   311
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
   312
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
   313
*}
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
7077
60b098bb8b8a heavily revised by Jacques: coercions have alphabetic names;
paulson
parents:
diff changeset
   316
end