src/HOL/Log.thy
author webertj
Fri, 19 Oct 2012 15:12:52 +0200
changeset 49962 a8cc904a6820
parent 47595 836b4c4d7c86
child 50247 491c5c81c2e8
permissions -rw-r--r--
Renamed {left,right}_distrib to distrib_{right,left}.
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
12224
02df7cbe7d25 even more theories from Jacques
paulson
parents:
diff changeset
     1
(*  Title       : Log.thy
02df7cbe7d25 even more theories from Jacques
paulson
parents:
diff changeset
     2
    Author      : Jacques D. Fleuriot
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
     3
                  Additional contributions by Jeremy Avigad
12224
02df7cbe7d25 even more theories from Jacques
paulson
parents:
diff changeset
     4
    Copyright   : 2000,2001 University of Edinburgh
02df7cbe7d25 even more theories from Jacques
paulson
parents:
diff changeset
     5
*)
02df7cbe7d25 even more theories from Jacques
paulson
parents:
diff changeset
     6
14411
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
     7
header{*Logarithms: Standard Version*}
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
     8
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15085
diff changeset
     9
theory Log
15140
322485b816ac import -> imports
nipkow
parents: 15131
diff changeset
    10
imports Transcendental
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15085
diff changeset
    11
begin
12224
02df7cbe7d25 even more theories from Jacques
paulson
parents:
diff changeset
    12
19765
dfe940911617 misc cleanup;
wenzelm
parents: 16819
diff changeset
    13
definition
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 19765
diff changeset
    14
  powr  :: "[real,real] => real"     (infixr "powr" 80) where
14411
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    15
    --{*exponentation with real exponent*}
19765
dfe940911617 misc cleanup;
wenzelm
parents: 16819
diff changeset
    16
  "x powr a = exp(a * ln x)"
12224
02df7cbe7d25 even more theories from Jacques
paulson
parents:
diff changeset
    17
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 19765
diff changeset
    18
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 19765
diff changeset
    19
  log :: "[real,real] => real" where
15053
405be2b48f5b Corrected TeX problems.
nipkow
parents: 14430
diff changeset
    20
    --{*logarithm of @{term x} to base @{term a}*}
19765
dfe940911617 misc cleanup;
wenzelm
parents: 16819
diff changeset
    21
  "log a x = ln x / ln a"
12224
02df7cbe7d25 even more theories from Jacques
paulson
parents:
diff changeset
    22
14411
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    23
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    24
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    25
lemma powr_one_eq_one [simp]: "1 powr a = 1"
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    26
by (simp add: powr_def)
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    27
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    28
lemma powr_zero_eq_one [simp]: "x powr 0 = 1"
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    29
by (simp add: powr_def)
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    30
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    31
lemma powr_one_gt_zero_iff [simp]: "(x powr 1 = x) = (0 < x)"
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    32
by (simp add: powr_def)
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    33
declare powr_one_gt_zero_iff [THEN iffD2, simp]
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    34
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    35
lemma powr_mult: 
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    36
      "[| 0 < x; 0 < y |] ==> (x * y) powr a = (x powr a) * (y powr a)"
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 47595
diff changeset
    37
by (simp add: powr_def exp_add [symmetric] ln_mult distrib_left)
14411
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    38
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    39
lemma powr_gt_zero [simp]: "0 < x powr a"
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    40
by (simp add: powr_def)
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    41
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
    42
lemma powr_ge_pzero [simp]: "0 <= x powr y"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
    43
by (rule order_less_imp_le, rule powr_gt_zero)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
    44
14411
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    45
lemma powr_not_zero [simp]: "x powr a \<noteq> 0"
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    46
by (simp add: powr_def)
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    47
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    48
lemma powr_divide:
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    49
     "[| 0 < x; 0 < y |] ==> (x / y) powr a = (x powr a)/(y powr a)"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14411
diff changeset
    50
apply (simp add: divide_inverse positive_imp_inverse_positive powr_mult)
14411
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    51
apply (simp add: powr_def exp_minus [symmetric] exp_add [symmetric] ln_inverse)
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    52
done
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    53
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
    54
lemma powr_divide2: "x powr a / x powr b = x powr (a - b)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
    55
  apply (simp add: powr_def)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
    56
  apply (subst exp_diff [THEN sym])
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
    57
  apply (simp add: left_diff_distrib)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
    58
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
    59
14411
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    60
lemma powr_add: "x powr (a + b) = (x powr a) * (x powr b)"
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 47595
diff changeset
    61
by (simp add: powr_def exp_add [symmetric] distrib_right)
14411
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    62
45930
2a882ef2cd73 add lemmas
noschinl
parents: 45916
diff changeset
    63
lemma powr_mult_base:
2a882ef2cd73 add lemmas
noschinl
parents: 45916
diff changeset
    64
  "0 < x \<Longrightarrow>x * x powr y = x powr (1 + y)"
2a882ef2cd73 add lemmas
noschinl
parents: 45916
diff changeset
    65
using assms by (auto simp: powr_add)
2a882ef2cd73 add lemmas
noschinl
parents: 45916
diff changeset
    66
14411
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    67
lemma powr_powr: "(x powr a) powr b = x powr (a * b)"
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    68
by (simp add: powr_def)
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    69
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    70
lemma powr_powr_swap: "(x powr a) powr b = (x powr b) powr a"
36777
be5461582d0f avoid using real-specific versions of generic lemmas
huffman
parents: 36622
diff changeset
    71
by (simp add: powr_powr mult_commute)
14411
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    72
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    73
lemma powr_minus: "x powr (-a) = inverse (x powr a)"
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    74
by (simp add: powr_def exp_minus [symmetric])
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    75
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    76
lemma powr_minus_divide: "x powr (-a) = 1/(x powr a)"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14411
diff changeset
    77
by (simp add: divide_inverse powr_minus)
14411
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    78
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    79
lemma powr_less_mono: "[| a < b; 1 < x |] ==> x powr a < x powr b"
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    80
by (simp add: powr_def)
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    81
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    82
lemma powr_less_cancel: "[| x powr a < x powr b; 1 < x |] ==> a < b"
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    83
by (simp add: powr_def)
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    84
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    85
lemma powr_less_cancel_iff [simp]: "1 < x ==> (x powr a < x powr b) = (a < b)"
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    86
by (blast intro: powr_less_cancel powr_less_mono)
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    87
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    88
lemma powr_le_cancel_iff [simp]: "1 < x ==> (x powr a \<le> x powr b) = (a \<le> b)"
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    89
by (simp add: linorder_not_less [symmetric])
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    90
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    91
lemma log_ln: "ln x = log (exp(1)) x"
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    92
by (simp add: log_def)
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
    93
45916
758671e966a0 isarfied proof; add log to DERIV_intros
hoelzl
parents: 45915
diff changeset
    94
lemma DERIV_log: assumes "x > 0" shows "DERIV (\<lambda>y. log b y) x :> 1 / (ln b * x)"
758671e966a0 isarfied proof; add log to DERIV_intros
hoelzl
parents: 45915
diff changeset
    95
proof -
758671e966a0 isarfied proof; add log to DERIV_intros
hoelzl
parents: 45915
diff changeset
    96
  def lb \<equiv> "1 / ln b"
758671e966a0 isarfied proof; add log to DERIV_intros
hoelzl
parents: 45915
diff changeset
    97
  moreover have "DERIV (\<lambda>y. lb * ln y) x :> lb / x"
758671e966a0 isarfied proof; add log to DERIV_intros
hoelzl
parents: 45915
diff changeset
    98
    using `x > 0` by (auto intro!: DERIV_intros)
758671e966a0 isarfied proof; add log to DERIV_intros
hoelzl
parents: 45915
diff changeset
    99
  ultimately show ?thesis
758671e966a0 isarfied proof; add log to DERIV_intros
hoelzl
parents: 45915
diff changeset
   100
    by (simp add: log_def)
758671e966a0 isarfied proof; add log to DERIV_intros
hoelzl
parents: 45915
diff changeset
   101
qed
758671e966a0 isarfied proof; add log to DERIV_intros
hoelzl
parents: 45915
diff changeset
   102
758671e966a0 isarfied proof; add log to DERIV_intros
hoelzl
parents: 45915
diff changeset
   103
lemmas DERIV_log[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
33716
c7b42ad3fadf A few more lemmas from Jeremy
paulson
parents: 31336
diff changeset
   104
14411
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   105
lemma powr_log_cancel [simp]:
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   106
     "[| 0 < a; a \<noteq> 1; 0 < x |] ==> a powr (log a x) = x"
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   107
by (simp add: powr_def log_def)
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   108
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   109
lemma log_powr_cancel [simp]: "[| 0 < a; a \<noteq> 1 |] ==> log a (a powr y) = y"
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   110
by (simp add: log_def powr_def)
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   111
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   112
lemma log_mult: 
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   113
     "[| 0 < a; a \<noteq> 1; 0 < x; 0 < y |]  
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   114
      ==> log a (x * y) = log a x + log a y"
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 47595
diff changeset
   115
by (simp add: log_def ln_mult divide_inverse distrib_right)
14411
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   116
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   117
lemma log_eq_div_ln_mult_log: 
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   118
     "[| 0 < a; a \<noteq> 1; 0 < b; b \<noteq> 1; 0 < x |]  
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   119
      ==> log a x = (ln b/ln a) * log b x"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14411
diff changeset
   120
by (simp add: log_def divide_inverse)
14411
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   121
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   122
text{*Base 10 logarithms*}
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   123
lemma log_base_10_eq1: "0 < x ==> log 10 x = (ln (exp 1) / ln 10) * ln x"
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   124
by (simp add: log_def)
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   125
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   126
lemma log_base_10_eq2: "0 < x ==> log 10 x = (log 10 (exp 1)) * ln x"
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   127
by (simp add: log_def)
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   128
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   129
lemma log_one [simp]: "log a 1 = 0"
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   130
by (simp add: log_def)
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   131
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   132
lemma log_eq_one [simp]: "[| 0 < a; a \<noteq> 1 |] ==> log a a = 1"
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   133
by (simp add: log_def)
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   134
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   135
lemma log_inverse:
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   136
     "[| 0 < a; a \<noteq> 1; 0 < x |] ==> log a (inverse x) = - log a x"
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   137
apply (rule_tac a1 = "log a x" in add_left_cancel [THEN iffD1])
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   138
apply (simp add: log_mult [symmetric])
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   139
done
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   140
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   141
lemma log_divide:
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   142
     "[|0 < a; a \<noteq> 1; 0 < x; 0 < y|] ==> log a (x/y) = log a x - log a y"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14411
diff changeset
   143
by (simp add: log_mult divide_inverse log_inverse)
14411
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   144
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   145
lemma log_less_cancel_iff [simp]:
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   146
     "[| 1 < a; 0 < x; 0 < y |] ==> (log a x < log a y) = (x < y)"
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   147
apply safe
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   148
apply (rule_tac [2] powr_less_cancel)
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   149
apply (drule_tac a = "log a x" in powr_less_mono, auto)
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   150
done
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   151
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 33716
diff changeset
   152
lemma log_inj: assumes "1 < b" shows "inj_on (log b) {0 <..}"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 33716
diff changeset
   153
proof (rule inj_onI, simp)
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 33716
diff changeset
   154
  fix x y assume pos: "0 < x" "0 < y" and *: "log b x = log b y"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 33716
diff changeset
   155
  show "x = y"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 33716
diff changeset
   156
  proof (cases rule: linorder_cases)
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 33716
diff changeset
   157
    assume "x < y" hence "log b x < log b y"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 33716
diff changeset
   158
      using log_less_cancel_iff[OF `1 < b`] pos by simp
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 33716
diff changeset
   159
    thus ?thesis using * by simp
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 33716
diff changeset
   160
  next
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 33716
diff changeset
   161
    assume "y < x" hence "log b y < log b x"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 33716
diff changeset
   162
      using log_less_cancel_iff[OF `1 < b`] pos by simp
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 33716
diff changeset
   163
    thus ?thesis using * by simp
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 33716
diff changeset
   164
  qed simp
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 33716
diff changeset
   165
qed
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 33716
diff changeset
   166
14411
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   167
lemma log_le_cancel_iff [simp]:
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   168
     "[| 1 < a; 0 < x; 0 < y |] ==> (log a x \<le> log a y) = (x \<le> y)"
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   169
by (simp add: linorder_not_less [symmetric])
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   170
47593
69f0af2b7d54 add lemmas to compare log with 0 and 1
hoelzl
parents: 45930
diff changeset
   171
lemma zero_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < log a x \<longleftrightarrow> 1 < x"
69f0af2b7d54 add lemmas to compare log with 0 and 1
hoelzl
parents: 45930
diff changeset
   172
  using log_less_cancel_iff[of a 1 x] by simp
69f0af2b7d54 add lemmas to compare log with 0 and 1
hoelzl
parents: 45930
diff changeset
   173
69f0af2b7d54 add lemmas to compare log with 0 and 1
hoelzl
parents: 45930
diff changeset
   174
lemma zero_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 \<le> log a x \<longleftrightarrow> 1 \<le> x"
69f0af2b7d54 add lemmas to compare log with 0 and 1
hoelzl
parents: 45930
diff changeset
   175
  using log_le_cancel_iff[of a 1 x] by simp
69f0af2b7d54 add lemmas to compare log with 0 and 1
hoelzl
parents: 45930
diff changeset
   176
69f0af2b7d54 add lemmas to compare log with 0 and 1
hoelzl
parents: 45930
diff changeset
   177
lemma log_less_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 0 \<longleftrightarrow> x < 1"
69f0af2b7d54 add lemmas to compare log with 0 and 1
hoelzl
parents: 45930
diff changeset
   178
  using log_less_cancel_iff[of a x 1] by simp
69f0af2b7d54 add lemmas to compare log with 0 and 1
hoelzl
parents: 45930
diff changeset
   179
69f0af2b7d54 add lemmas to compare log with 0 and 1
hoelzl
parents: 45930
diff changeset
   180
lemma log_le_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 0 \<longleftrightarrow> x \<le> 1"
69f0af2b7d54 add lemmas to compare log with 0 and 1
hoelzl
parents: 45930
diff changeset
   181
  using log_le_cancel_iff[of a x 1] by simp
69f0af2b7d54 add lemmas to compare log with 0 and 1
hoelzl
parents: 45930
diff changeset
   182
69f0af2b7d54 add lemmas to compare log with 0 and 1
hoelzl
parents: 45930
diff changeset
   183
lemma one_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 < log a x \<longleftrightarrow> a < x"
69f0af2b7d54 add lemmas to compare log with 0 and 1
hoelzl
parents: 45930
diff changeset
   184
  using log_less_cancel_iff[of a a x] by simp
69f0af2b7d54 add lemmas to compare log with 0 and 1
hoelzl
parents: 45930
diff changeset
   185
69f0af2b7d54 add lemmas to compare log with 0 and 1
hoelzl
parents: 45930
diff changeset
   186
lemma one_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> log a x \<longleftrightarrow> a \<le> x"
69f0af2b7d54 add lemmas to compare log with 0 and 1
hoelzl
parents: 45930
diff changeset
   187
  using log_le_cancel_iff[of a a x] by simp
69f0af2b7d54 add lemmas to compare log with 0 and 1
hoelzl
parents: 45930
diff changeset
   188
69f0af2b7d54 add lemmas to compare log with 0 and 1
hoelzl
parents: 45930
diff changeset
   189
lemma log_less_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 1 \<longleftrightarrow> x < a"
69f0af2b7d54 add lemmas to compare log with 0 and 1
hoelzl
parents: 45930
diff changeset
   190
  using log_less_cancel_iff[of a x a] by simp
69f0af2b7d54 add lemmas to compare log with 0 and 1
hoelzl
parents: 45930
diff changeset
   191
69f0af2b7d54 add lemmas to compare log with 0 and 1
hoelzl
parents: 45930
diff changeset
   192
lemma log_le_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 1 \<longleftrightarrow> x \<le> a"
69f0af2b7d54 add lemmas to compare log with 0 and 1
hoelzl
parents: 45930
diff changeset
   193
  using log_le_cancel_iff[of a x a] by simp
14411
7851e526b8b7 converted Hyperreal/Log and Hyperreal/HLog to Isar scripts
paulson
parents: 12224
diff changeset
   194
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   195
lemma powr_realpow: "0 < x ==> x powr (real n) = x^n"
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   196
  apply (induct n, simp)
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   197
  apply (subgoal_tac "real(Suc n) = real n + 1")
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   198
  apply (erule ssubst)
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   199
  apply (subst powr_add, simp, simp)
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   200
done
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   201
47594
be2ac449488c add lemmas to rewrite powr to power
hoelzl
parents: 47593
diff changeset
   202
lemma powr_realpow2: "0 <= x ==> 0 < n ==> x^n = (if (x = 0) then 0 else x powr (real n))"
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   203
  apply (case_tac "x = 0", simp, simp)
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   204
  apply (rule powr_realpow [THEN sym], simp)
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   205
done
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   206
47594
be2ac449488c add lemmas to rewrite powr to power
hoelzl
parents: 47593
diff changeset
   207
lemma powr_int:
be2ac449488c add lemmas to rewrite powr to power
hoelzl
parents: 47593
diff changeset
   208
  assumes "x > 0"
be2ac449488c add lemmas to rewrite powr to power
hoelzl
parents: 47593
diff changeset
   209
  shows "x powr i = (if i \<ge> 0 then x ^ nat i else 1 / x ^ nat (-i))"
be2ac449488c add lemmas to rewrite powr to power
hoelzl
parents: 47593
diff changeset
   210
proof cases
be2ac449488c add lemmas to rewrite powr to power
hoelzl
parents: 47593
diff changeset
   211
  assume "i < 0"
be2ac449488c add lemmas to rewrite powr to power
hoelzl
parents: 47593
diff changeset
   212
  have r: "x powr i = 1 / x powr (-i)" by (simp add: powr_minus field_simps)
be2ac449488c add lemmas to rewrite powr to power
hoelzl
parents: 47593
diff changeset
   213
  show ?thesis using `i < 0` `x > 0` by (simp add: r field_simps powr_realpow[symmetric])
be2ac449488c add lemmas to rewrite powr to power
hoelzl
parents: 47593
diff changeset
   214
qed (simp add: assms powr_realpow[symmetric])
be2ac449488c add lemmas to rewrite powr to power
hoelzl
parents: 47593
diff changeset
   215
be2ac449488c add lemmas to rewrite powr to power
hoelzl
parents: 47593
diff changeset
   216
lemma powr_numeral: "0 < x \<Longrightarrow> x powr numeral n = x^numeral n"
be2ac449488c add lemmas to rewrite powr to power
hoelzl
parents: 47593
diff changeset
   217
  using powr_realpow[of x "numeral n"] by simp
be2ac449488c add lemmas to rewrite powr to power
hoelzl
parents: 47593
diff changeset
   218
be2ac449488c add lemmas to rewrite powr to power
hoelzl
parents: 47593
diff changeset
   219
lemma powr_neg_numeral: "0 < x \<Longrightarrow> x powr neg_numeral n = 1 / x^numeral n"
be2ac449488c add lemmas to rewrite powr to power
hoelzl
parents: 47593
diff changeset
   220
  using powr_int[of x "neg_numeral n"] by simp
be2ac449488c add lemmas to rewrite powr to power
hoelzl
parents: 47593
diff changeset
   221
45930
2a882ef2cd73 add lemmas
noschinl
parents: 45916
diff changeset
   222
lemma root_powr_inverse:
2a882ef2cd73 add lemmas
noschinl
parents: 45916
diff changeset
   223
  "0 < n \<Longrightarrow> 0 < x \<Longrightarrow> root n x = x powr (1/n)"
2a882ef2cd73 add lemmas
noschinl
parents: 45916
diff changeset
   224
by (auto simp: root_def powr_realpow[symmetric] powr_powr)
2a882ef2cd73 add lemmas
noschinl
parents: 45916
diff changeset
   225
33716
c7b42ad3fadf A few more lemmas from Jeremy
paulson
parents: 31336
diff changeset
   226
lemma ln_powr: "0 < x ==> 0 < y ==> ln(x powr y) = y * ln x"
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   227
by (unfold powr_def, simp)
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   228
33716
c7b42ad3fadf A few more lemmas from Jeremy
paulson
parents: 31336
diff changeset
   229
lemma log_powr: "0 < x ==> 0 \<le> y ==> log b (x powr y) = y * log b x"
c7b42ad3fadf A few more lemmas from Jeremy
paulson
parents: 31336
diff changeset
   230
  apply (case_tac "y = 0")
c7b42ad3fadf A few more lemmas from Jeremy
paulson
parents: 31336
diff changeset
   231
  apply force
c7b42ad3fadf A few more lemmas from Jeremy
paulson
parents: 31336
diff changeset
   232
  apply (auto simp add: log_def ln_powr field_simps)
c7b42ad3fadf A few more lemmas from Jeremy
paulson
parents: 31336
diff changeset
   233
done
c7b42ad3fadf A few more lemmas from Jeremy
paulson
parents: 31336
diff changeset
   234
c7b42ad3fadf A few more lemmas from Jeremy
paulson
parents: 31336
diff changeset
   235
lemma log_nat_power: "0 < x ==> log b (x^n) = real n * log b x"
c7b42ad3fadf A few more lemmas from Jeremy
paulson
parents: 31336
diff changeset
   236
  apply (subst powr_realpow [symmetric])
c7b42ad3fadf A few more lemmas from Jeremy
paulson
parents: 31336
diff changeset
   237
  apply (auto simp add: log_powr)
c7b42ad3fadf A few more lemmas from Jeremy
paulson
parents: 31336
diff changeset
   238
done
c7b42ad3fadf A few more lemmas from Jeremy
paulson
parents: 31336
diff changeset
   239
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   240
lemma ln_bound: "1 <= x ==> ln x <= x"
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   241
  apply (subgoal_tac "ln(1 + (x - 1)) <= x - 1")
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   242
  apply simp
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   243
  apply (rule ln_add_one_self_le_self, simp)
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   244
done
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   245
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   246
lemma powr_mono: "a <= b ==> 1 <= x ==> x powr a <= x powr b"
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   247
  apply (case_tac "x = 1", simp)
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   248
  apply (case_tac "a = b", simp)
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   249
  apply (rule order_less_imp_le)
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   250
  apply (rule powr_less_mono, auto)
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   251
done
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   252
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   253
lemma ge_one_powr_ge_zero: "1 <= x ==> 0 <= a ==> 1 <= x powr a"
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   254
  apply (subst powr_zero_eq_one [THEN sym])
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   255
  apply (rule powr_mono, assumption+)
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   256
done
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   257
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   258
lemma powr_less_mono2: "0 < a ==> 0 < x ==> x < y ==> x powr a <
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   259
    y powr a"
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   260
  apply (unfold powr_def)
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   261
  apply (rule exp_less_mono)
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   262
  apply (rule mult_strict_left_mono)
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   263
  apply (subst ln_less_cancel_iff, assumption)
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   264
  apply (rule order_less_trans)
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   265
  prefer 2
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   266
  apply assumption+
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   267
done
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   268
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   269
lemma powr_less_mono2_neg: "a < 0 ==> 0 < x ==> x < y ==> y powr a <
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   270
    x powr a"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   271
  apply (unfold powr_def)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   272
  apply (rule exp_less_mono)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   273
  apply (rule mult_strict_left_mono_neg)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   274
  apply (subst ln_less_cancel_iff)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   275
  apply assumption
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   276
  apply (rule order_less_trans)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   277
  prefer 2
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   278
  apply assumption+
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   279
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   280
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   281
lemma powr_mono2: "0 <= a ==> 0 < x ==> x <= y ==> x powr a <= y powr a"
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   282
  apply (case_tac "a = 0", simp)
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   283
  apply (case_tac "x = y", simp)
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   284
  apply (rule order_less_imp_le)
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   285
  apply (rule powr_less_mono2, auto)
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   286
done
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   287
47595
836b4c4d7c86 add powr_inj
hoelzl
parents: 47594
diff changeset
   288
lemma powr_inj:
836b4c4d7c86 add powr_inj
hoelzl
parents: 47594
diff changeset
   289
  "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> a powr x = a powr y \<longleftrightarrow> x = y"
836b4c4d7c86 add powr_inj
hoelzl
parents: 47594
diff changeset
   290
  unfolding powr_def exp_inj_iff by simp
836b4c4d7c86 add powr_inj
hoelzl
parents: 47594
diff changeset
   291
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   292
lemma ln_powr_bound: "1 <= x ==> 0 < a ==> ln x <= (x powr a) / a"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   293
  apply (rule mult_imp_le_div_pos)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   294
  apply (assumption)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   295
  apply (subst mult_commute)
33716
c7b42ad3fadf A few more lemmas from Jeremy
paulson
parents: 31336
diff changeset
   296
  apply (subst ln_powr [THEN sym])
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   297
  apply auto
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   298
  apply (rule ln_bound)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   299
  apply (erule ge_one_powr_ge_zero)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   300
  apply (erule order_less_imp_le)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   301
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   302
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 36777
diff changeset
   303
lemma ln_powr_bound2:
efa734d9b221 eliminated global prems;
wenzelm
parents: 36777
diff changeset
   304
  assumes "1 < x" and "0 < a"
efa734d9b221 eliminated global prems;
wenzelm
parents: 36777
diff changeset
   305
  shows "(ln x) powr a <= (a powr a) * x"
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   306
proof -
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 36777
diff changeset
   307
  from assms have "ln x <= (x powr (1 / a)) / (1 / a)"
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   308
    apply (intro ln_powr_bound)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   309
    apply (erule order_less_imp_le)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   310
    apply (rule divide_pos_pos)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   311
    apply simp_all
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   312
    done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   313
  also have "... = a * (x powr (1 / a))"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   314
    by simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   315
  finally have "(ln x) powr a <= (a * (x powr (1 / a))) powr a"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   316
    apply (intro powr_mono2)
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 36777
diff changeset
   317
    apply (rule order_less_imp_le, rule assms)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   318
    apply (rule ln_gt_zero)
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 36777
diff changeset
   319
    apply (rule assms)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   320
    apply assumption
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   321
    done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   322
  also have "... = (a powr a) * ((x powr (1 / a)) powr a)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   323
    apply (rule powr_mult)
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 36777
diff changeset
   324
    apply (rule assms)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   325
    apply (rule powr_gt_zero)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   326
    done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   327
  also have "(x powr (1 / a)) powr a = x powr ((1 / a) * a)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   328
    by (rule powr_powr)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   329
  also have "... = x"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   330
    apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   331
    apply (subgoal_tac "a ~= 0")
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 36777
diff changeset
   332
    using assms apply auto
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   333
    done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   334
  finally show ?thesis .
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   335
qed
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   336
45915
0e5a87b772f9 tendsto lemmas for ln and powr
huffman
parents: 45892
diff changeset
   337
lemma tendsto_powr [tendsto_intros]:
0e5a87b772f9 tendsto lemmas for ln and powr
huffman
parents: 45892
diff changeset
   338
  "\<lbrakk>(f ---> a) F; (g ---> b) F; 0 < a\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x powr g x) ---> a powr b) F"
0e5a87b772f9 tendsto lemmas for ln and powr
huffman
parents: 45892
diff changeset
   339
  unfolding powr_def by (intro tendsto_intros)
0e5a87b772f9 tendsto lemmas for ln and powr
huffman
parents: 45892
diff changeset
   340
45892
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   341
(* FIXME: generalize by replacing d by with g x and g ---> d? *)
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   342
lemma tendsto_zero_powrI:
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   343
  assumes "eventually (\<lambda>x. 0 < f x ) F" and "(f ---> 0) F"
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   344
  assumes "0 < d"
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   345
  shows "((\<lambda>x. f x powr d) ---> 0) F"
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   346
proof (rule tendstoI)
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   347
  fix e :: real assume "0 < e"
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   348
  def Z \<equiv> "e powr (1 / d)"
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   349
  with `0 < e` have "0 < Z" by simp
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   350
  with assms have "eventually (\<lambda>x. 0 < f x \<and> dist (f x) 0 < Z) F"
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   351
    by (intro eventually_conj tendstoD)
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   352
  moreover
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   353
  from assms have "\<And>x. 0 < x \<and> dist x 0 < Z \<Longrightarrow> x powr d < Z powr d"
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   354
    by (intro powr_less_mono2) (auto simp: dist_real_def)
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   355
  with assms `0 < e` have "\<And>x. 0 < x \<and> dist x 0 < Z \<Longrightarrow> dist (x powr d) 0 < e"
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   356
    unfolding dist_real_def Z_def by (auto simp: powr_powr)
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   357
  ultimately
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   358
  show "eventually (\<lambda>x. dist (f x powr d) 0 < e) F" by (rule eventually_elim1)
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   359
qed
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   360
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   361
lemma tendsto_neg_powr:
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   362
  assumes "s < 0" and "real_tendsto_inf f F"
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   363
  shows "((\<lambda>x. f x powr s) ---> 0) F"
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   364
proof (rule tendstoI)
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   365
  fix e :: real assume "0 < e"
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   366
  def Z \<equiv> "e powr (1 / s)"
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   367
  from assms have "eventually (\<lambda>x. Z < f x) F" by (simp add: real_tendsto_inf_def)
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   368
  moreover
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   369
  from assms have "\<And>x. Z < x \<Longrightarrow> x powr s < Z powr s"
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   370
    by (auto simp: Z_def intro!: powr_less_mono2_neg)
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   371
  with assms `0 < e` have "\<And>x. Z < x \<Longrightarrow> dist (x powr s) 0 < e"
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   372
    by (simp add: powr_powr Z_def dist_real_def)
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   373
  ultimately
8dcf6692433f add lemmas about limits
noschinl
parents: 41550
diff changeset
   374
  show "eventually (\<lambda>x. dist (f x powr s) 0 < e) F" by (rule eventually_elim1)
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 36777
diff changeset
   375
qed
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 15140
diff changeset
   376
12224
02df7cbe7d25 even more theories from Jacques
paulson
parents:
diff changeset
   377
end