src/HOL/Ring_and_Field.thy
author obua
Tue, 18 May 2004 10:01:44 +0200
changeset 14754 a080eeeaec14
parent 14738 83f1a514dcb4
child 14770 fe9504ba63d5
permissions -rw-r--r--
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
     1
(*  Title:   HOL/Ring_and_Field.thy
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
     2
    ID:      $Id$
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
     3
    Author:  Gertrud Bauer and Markus Wenzel, TU Muenchen
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
     4
             Lawrence C Paulson, University of Cambridge
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
     5
             Revised and splitted into Ring_and_Field.thy and Group.thy 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
     6
             by Steven Obua, TU Muenchen, in May 2004
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
     7
    License: GPL (GNU GENERAL PUBLIC LICENSE)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
     8
*)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
     9
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    10
header {* (Ordered) Rings and Fields *}
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
    11
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    12
theory Ring_and_Field = OrderedGroup:
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
    13
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    14
text {*
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    15
  The theory of partially ordered rings is taken from the books:
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    16
  \begin{itemize}
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    17
  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    18
  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    19
  \end{itemize}
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    20
  Most of the used notions can also be looked up in 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    21
  \begin{itemize}
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    22
  \item \emph{www.mathworld.com} by Eric Weisstein et. al.
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    23
  \item \emph{Algebra I} by van der Waerden, Springer.
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    24
  \end{itemize}
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    25
*}
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
    26
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    27
axclass semiring \<subseteq> ab_semigroup_add, semigroup_mult
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    28
  left_distrib: "(a + b) * c = a * c + b * c"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    29
  right_distrib: "a * (b + c) = a * b + a * c"
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
    30
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    31
axclass semiring_0 \<subseteq> semiring, comm_monoid_add
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
    32
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    33
axclass comm_semiring \<subseteq> ab_semigroup_add, ab_semigroup_mult  
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    34
  mult_commute: "a * b = b * a"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    35
  distrib: "(a + b) * c = a * c + b * c"
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
    36
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    37
instance comm_semiring \<subseteq> semiring
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    38
proof
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    39
  fix a b c :: 'a
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    40
  show "(a + b) * c = a * c + b * c" by (simp add: distrib)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    41
  have "a * (b + c) = (b + c) * a" by (simp add: mult_ac)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    42
  also have "... = b * a + c * a" by (simp only: distrib)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    43
  also have "... = a * b + a * c" by (simp add: mult_ac)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    44
  finally show "a * (b + c) = a * b + a * c" by blast
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
    45
qed
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
    46
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    47
axclass comm_semiring_0 \<subseteq> comm_semiring, comm_monoid_add
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
    48
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    49
instance comm_semiring_0 \<subseteq> semiring_0 ..
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
    50
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    51
axclass axclass_0_neq_1 \<subseteq> zero, one
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
    52
  zero_neq_one [simp]: "0 \<noteq> 1"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
    53
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    54
axclass semiring_1 \<subseteq> axclass_0_neq_1, semiring_0, monoid_mult
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
    55
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    56
axclass comm_semiring_1 \<subseteq> axclass_0_neq_1, comm_semiring_0, comm_monoid_mult (* previously almost_semiring *)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    57
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    58
instance comm_semiring_1 \<subseteq> semiring_1 ..
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
    59
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    60
axclass axclass_no_zero_divisors \<subseteq> zero, times
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    61
  no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
    62
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    63
axclass comm_semiring_1_cancel \<subseteq> comm_semiring_1, cancel_ab_semigroup_add (* previously semiring *)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    64
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    65
axclass ring \<subseteq> semiring, ab_group_add
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    66
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    67
instance ring \<subseteq> semiring_0 ..
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
    68
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    69
axclass comm_ring \<subseteq> comm_semiring_0, ab_group_add
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    70
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    71
instance comm_ring \<subseteq> ring ..
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
    72
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    73
instance comm_ring \<subseteq> comm_semiring_0 ..
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    74
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    75
axclass ring_1 \<subseteq> ring, semiring_1
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
    76
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    77
axclass comm_ring_1 \<subseteq> comm_ring, comm_semiring_1 (* previously ring *)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    78
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    79
instance comm_ring_1 \<subseteq> ring_1 ..
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
    80
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    81
instance comm_ring_1 \<subseteq> comm_semiring_1_cancel ..
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
    82
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    83
axclass idom \<subseteq> comm_ring_1, axclass_no_zero_divisors
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
    84
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    85
axclass field \<subseteq> comm_ring_1, inverse
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
    86
  left_inverse [simp]: "a \<noteq> 0 ==> inverse a * a = 1"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
    87
  divide_inverse:      "a / b = a * inverse b"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
    88
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    89
lemma mult_zero_left [simp]: "0 * a = (0::'a::{semiring_0, cancel_semigroup_add})"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    90
proof -
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    91
  have "0*a + 0*a = 0*a + 0"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    92
    by (simp add: left_distrib [symmetric])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    93
  thus ?thesis 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    94
    by (simp only: add_left_cancel)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    95
qed
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
    96
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    97
lemma mult_zero_right [simp]: "a * 0 = (0::'a::{semiring_0, cancel_semigroup_add})"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    98
proof -
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
    99
  have "a*0 + a*0 = a*0 + 0"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   100
    by (simp add: right_distrib [symmetric])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   101
  thus ?thesis 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   102
    by (simp only: add_left_cancel)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   103
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   104
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   105
lemma field_mult_eq_0_iff [simp]: "(a*b = (0::'a::field)) = (a = 0 | b = 0)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   106
proof cases
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   107
  assume "a=0" thus ?thesis by simp
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   108
next
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   109
  assume anz [simp]: "a\<noteq>0"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   110
  { assume "a * b = 0"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   111
    hence "inverse a * (a * b) = 0" by simp
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   112
    hence "b = 0"  by (simp (no_asm_use) add: mult_assoc [symmetric])}
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   113
  thus ?thesis by force
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   114
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   115
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   116
instance field \<subseteq> idom
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   117
by (intro_classes, simp)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   118
  
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   119
axclass division_by_zero \<subseteq> zero, inverse
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   120
  inverse_zero [simp]: "inverse 0 = 0"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   121
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   122
subsection {* Distribution rules *}
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   123
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   124
theorems ring_distrib = right_distrib left_distrib
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   125
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   126
text{*For the @{text combine_numerals} simproc*}
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   127
lemma combine_common_factor:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   128
     "a*e + (b*e + c) = (a+b)*e + (c::'a::semiring)"
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   129
by (simp add: left_distrib add_ac)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   130
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   131
lemma minus_mult_left: "- (a * b) = (-a) * (b::'a::ring)"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   132
apply (rule equals_zero_I)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   133
apply (simp add: left_distrib [symmetric]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   134
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   135
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   136
lemma minus_mult_right: "- (a * b) = a * -(b::'a::ring)"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   137
apply (rule equals_zero_I)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   138
apply (simp add: right_distrib [symmetric]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   139
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   140
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   141
lemma minus_mult_minus [simp]: "(- a) * (- b) = a * (b::'a::ring)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   142
  by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   143
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   144
lemma minus_mult_commute: "(- a) * b = a * (- b::'a::ring)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   145
  by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   146
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   147
lemma right_diff_distrib: "a * (b - c) = a * b - a * (c::'a::ring)"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   148
by (simp add: right_distrib diff_minus 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   149
              minus_mult_left [symmetric] minus_mult_right [symmetric]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   150
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   151
lemma left_diff_distrib: "(a - b) * c = a * c - b * (c::'a::ring)"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   152
by (simp add: left_distrib diff_minus 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   153
              minus_mult_left [symmetric] minus_mult_right [symmetric]) 
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   154
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   155
axclass pordered_semiring \<subseteq> semiring_0, pordered_ab_semigroup_add 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   156
  mult_left_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> c * a <= c * b"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   157
  mult_right_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> a * c <= b * c"
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   158
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   159
axclass pordered_cancel_semiring \<subseteq> pordered_semiring, cancel_ab_semigroup_add
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   160
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   161
axclass ordered_semiring_strict \<subseteq> semiring_0, ordered_cancel_ab_semigroup_add
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   162
  mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   163
  mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   164
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   165
instance ordered_semiring_strict \<subseteq> pordered_cancel_semiring
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   166
apply intro_classes
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   167
apply (case_tac "a < b & 0 < c")
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   168
apply (auto simp add: mult_strict_left_mono order_less_le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   169
apply (auto simp add: mult_strict_left_mono order_le_less)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   170
apply (simp add: mult_strict_right_mono)
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   171
done
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   172
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   173
axclass pordered_comm_semiring \<subseteq> comm_semiring_0, pordered_ab_semigroup_add
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   174
  mult_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> c * a <= c * b"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   175
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   176
axclass pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring, cancel_ab_semigroup_add
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   177
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   178
instance pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring ..
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   179
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   180
axclass ordered_comm_semiring_strict \<subseteq> comm_semiring_0, ordered_cancel_ab_semigroup_add
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   181
  mult_strict_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   182
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   183
instance pordered_comm_semiring \<subseteq> pordered_semiring
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   184
by (intro_classes, insert mult_mono, simp_all add: mult_commute, blast+)
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   185
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   186
instance pordered_cancel_comm_semiring \<subseteq> pordered_cancel_semiring ..
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   187
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   188
instance ordered_comm_semiring_strict \<subseteq> ordered_semiring_strict
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   189
by (intro_classes, insert mult_strict_mono, simp_all add: mult_commute, blast+)
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   190
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   191
instance ordered_comm_semiring_strict \<subseteq> pordered_cancel_comm_semiring
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   192
apply (intro_classes)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   193
apply (case_tac "a < b & 0 < c")
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   194
apply (auto simp add: mult_strict_left_mono order_less_le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   195
apply (auto simp add: mult_strict_left_mono order_le_less)
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   196
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   197
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   198
axclass pordered_ring \<subseteq> ring, pordered_semiring 
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   199
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   200
instance pordered_ring \<subseteq> pordered_ab_group_add ..
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   201
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   202
instance pordered_ring \<subseteq> pordered_cancel_semiring ..
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   203
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   204
axclass lordered_ring \<subseteq> pordered_ring, lordered_ab_group_abs
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   205
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   206
axclass axclass_abs_if \<subseteq> minus, ord, zero
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   207
  abs_if: "abs a = (if (a < 0) then (-a) else a)"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   208
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   209
axclass ordered_ring_strict \<subseteq> ring, ordered_semiring_strict, axclass_abs_if
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   210
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   211
instance ordered_ring_strict \<subseteq> lordered_ab_group ..
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   212
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   213
instance ordered_ring_strict \<subseteq> lordered_ring
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   214
by (intro_classes, simp add: abs_if join_eq_if)
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   215
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   216
axclass pordered_comm_ring \<subseteq> comm_ring, pordered_comm_semiring
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   217
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   218
axclass ordered_semidom \<subseteq> comm_semiring_1_cancel, ordered_comm_semiring_strict (* previously ordered_semiring *)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   219
  zero_less_one [simp]: "0 < 1"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   220
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   221
axclass ordered_idom \<subseteq> comm_ring_1, ordered_comm_semiring_strict, axclass_abs_if (* previously ordered_ring *)
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   222
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   223
instance ordered_idom \<subseteq> ordered_ring_strict ..
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   224
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   225
axclass ordered_field \<subseteq> field, ordered_idom
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   226
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   227
lemma eq_add_iff1:
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   228
     "(a*e + c = b*e + d) = ((a-b)*e + c = (d::'a::ring))"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   229
apply (simp add: diff_minus left_distrib)
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   230
apply (simp add: diff_minus left_distrib add_ac)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   231
apply (simp add: compare_rls minus_mult_left [symmetric])
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   232
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   233
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   234
lemma eq_add_iff2:
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   235
     "(a*e + c = b*e + d) = (c = (b-a)*e + (d::'a::ring))"
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   236
apply (simp add: diff_minus left_distrib add_ac)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   237
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   238
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   239
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   240
lemma less_add_iff1:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   241
     "(a*e + c < b*e + d) = ((a-b)*e + c < (d::'a::pordered_ring))"
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   242
apply (simp add: diff_minus left_distrib add_ac)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   243
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   244
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   245
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   246
lemma less_add_iff2:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   247
     "(a*e + c < b*e + d) = (c < (b-a)*e + (d::'a::pordered_ring))"
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   248
apply (simp add: diff_minus left_distrib add_ac)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   249
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   250
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   251
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   252
lemma le_add_iff1:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   253
     "(a*e + c \<le> b*e + d) = ((a-b)*e + c \<le> (d::'a::pordered_ring))"
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   254
apply (simp add: diff_minus left_distrib add_ac)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   255
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   256
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   257
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   258
lemma le_add_iff2:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   259
     "(a*e + c \<le> b*e + d) = (c \<le> (b-a)*e + (d::'a::pordered_ring))"
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   260
apply (simp add: diff_minus left_distrib add_ac)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   261
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   262
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   263
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   264
subsection {* Ordering Rules for Multiplication *}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   265
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   266
lemma mult_left_le_imp_le:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   267
     "[|c*a \<le> c*b; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)"
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   268
  by (force simp add: mult_strict_left_mono linorder_not_less [symmetric])
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   269
 
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   270
lemma mult_right_le_imp_le:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   271
     "[|a*c \<le> b*c; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)"
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   272
  by (force simp add: mult_strict_right_mono linorder_not_less [symmetric])
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   273
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   274
lemma mult_left_less_imp_less:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   275
     "[|c*a < c*b; 0 \<le> c|] ==> a < (b::'a::ordered_semiring_strict)"
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   276
  by (force simp add: mult_left_mono linorder_not_le [symmetric])
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   277
 
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   278
lemma mult_right_less_imp_less:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   279
     "[|a*c < b*c; 0 \<le> c|] ==> a < (b::'a::ordered_semiring_strict)"
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   280
  by (force simp add: mult_right_mono linorder_not_le [symmetric])
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   281
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   282
lemma mult_strict_left_mono_neg:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   283
     "[|b < a; c < 0|] ==> c * a < c * (b::'a::ordered_ring_strict)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   284
apply (drule mult_strict_left_mono [of _ _ "-c"])
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   285
apply (simp_all add: minus_mult_left [symmetric]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   286
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   287
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   288
lemma mult_left_mono_neg:
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   289
     "[|b \<le> a; c \<le> 0|] ==> c * a \<le>  c * (b::'a::pordered_ring)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   290
apply (drule mult_left_mono [of _ _ "-c"])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   291
apply (simp_all add: minus_mult_left [symmetric]) 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   292
done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   293
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   294
lemma mult_strict_right_mono_neg:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   295
     "[|b < a; c < 0|] ==> a * c < b * (c::'a::ordered_ring_strict)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   296
apply (drule mult_strict_right_mono [of _ _ "-c"])
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   297
apply (simp_all add: minus_mult_right [symmetric]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   298
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   299
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   300
lemma mult_right_mono_neg:
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   301
     "[|b \<le> a; c \<le> 0|] ==> a * c \<le>  (b::'a::pordered_ring) * c"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   302
apply (drule mult_right_mono [of _ _ "-c"])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   303
apply (simp)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   304
apply (simp_all add: minus_mult_right [symmetric]) 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   305
done
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   306
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   307
subsection{* Products of Signs *}
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   308
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   309
lemma mult_pos: "[| (0::'a::ordered_semiring_strict) < a; 0 < b |] ==> 0 < a*b"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   310
by (drule mult_strict_left_mono [of 0 b], auto)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   311
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   312
lemma mult_pos_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; 0 \<le> b |] ==> 0 \<le> a*b"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   313
by (drule mult_left_mono [of 0 b], auto)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   314
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   315
lemma mult_pos_neg: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> a*b < 0"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   316
by (drule mult_strict_left_mono [of b 0], auto)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   317
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   318
lemma mult_pos_neg_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> a*b \<le> 0"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   319
by (drule mult_left_mono [of b 0], auto)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   320
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   321
lemma mult_pos_neg2: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> b*a < 0" 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   322
by (drule mult_strict_right_mono[of b 0], auto)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   323
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   324
lemma mult_pos_neg2_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> b*a \<le> 0" 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   325
by (drule mult_right_mono[of b 0], auto)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   326
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   327
lemma mult_neg: "[| a < (0::'a::ordered_ring_strict); b < 0 |] ==> 0 < a*b"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   328
by (drule mult_strict_right_mono_neg, auto)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   329
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   330
lemma mult_neg_le: "[| a \<le> (0::'a::pordered_ring); b \<le> 0 |] ==> 0 \<le> a*b"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   331
by (drule mult_right_mono_neg[of a 0 b ], auto)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   332
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   333
lemma zero_less_mult_pos:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   334
     "[| 0 < a*b; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   335
apply (case_tac "b\<le>0") 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   336
 apply (auto simp add: order_le_less linorder_not_less)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   337
apply (drule_tac mult_pos_neg [of a b]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   338
 apply (auto dest: order_less_not_sym)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   339
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   340
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   341
lemma zero_less_mult_pos2:
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   342
     "[| 0 < b*a; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   343
apply (case_tac "b\<le>0") 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   344
 apply (auto simp add: order_le_less linorder_not_less)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   345
apply (drule_tac mult_pos_neg2 [of a b]) 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   346
 apply (auto dest: order_less_not_sym)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   347
done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   348
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   349
lemma zero_less_mult_iff:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   350
     "((0::'a::ordered_ring_strict) < a*b) = (0 < a & 0 < b | a < 0 & b < 0)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   351
apply (auto simp add: order_le_less linorder_not_less mult_pos mult_neg)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   352
apply (blast dest: zero_less_mult_pos) 
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   353
apply (blast dest: zero_less_mult_pos2)
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   354
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   355
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   356
text{*A field has no "zero divisors", and this theorem holds without the
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   357
      assumption of an ordering.  See @{text field_mult_eq_0_iff} below.*}
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   358
lemma mult_eq_0_iff [simp]: "(a*b = (0::'a::ordered_ring_strict)) = (a = 0 | b = 0)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   359
apply (case_tac "a < 0")
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   360
apply (auto simp add: linorder_not_less order_le_less linorder_neq_iff)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   361
apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono)+
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   362
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   363
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   364
lemma zero_le_mult_iff:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   365
     "((0::'a::ordered_ring_strict) \<le> a*b) = (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   366
by (auto simp add: eq_commute [of 0] order_le_less linorder_not_less
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   367
                   zero_less_mult_iff)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   368
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   369
lemma mult_less_0_iff:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   370
     "(a*b < (0::'a::ordered_ring_strict)) = (0 < a & b < 0 | a < 0 & 0 < b)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   371
apply (insert zero_less_mult_iff [of "-a" b]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   372
apply (force simp add: minus_mult_left[symmetric]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   373
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   374
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   375
lemma mult_le_0_iff:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   376
     "(a*b \<le> (0::'a::ordered_ring_strict)) = (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   377
apply (insert zero_le_mult_iff [of "-a" b]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   378
apply (force simp add: minus_mult_left[symmetric]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   379
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   380
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   381
lemma split_mult_pos_le: "(0 \<le> a & 0 \<le> b) | (a \<le> 0 & b \<le> 0) \<Longrightarrow> 0 \<le> a * (b::_::pordered_ring)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   382
by (auto simp add: mult_pos_le mult_neg_le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   383
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   384
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> (0::_::pordered_cancel_semiring)" 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   385
by (auto simp add: mult_pos_neg_le mult_pos_neg2_le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   386
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   387
lemma zero_le_square: "(0::'a::ordered_ring_strict) \<le> a*a"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   388
by (simp add: zero_le_mult_iff linorder_linear) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   389
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   390
text{*Proving axiom @{text zero_less_one} makes all @{text ordered_semidom}
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   391
      theorems available to members of @{term ordered_idom} *}
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   392
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   393
instance ordered_idom \<subseteq> ordered_semidom
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   394
proof
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   395
  have "(0::'a) \<le> 1*1" by (rule zero_le_square)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   396
  thus "(0::'a) < 1" by (simp add: order_le_less) 
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   397
qed
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   398
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   399
instance ordered_ring_strict \<subseteq> axclass_no_zero_divisors 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   400
by (intro_classes, simp)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   401
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   402
instance ordered_idom \<subseteq> idom ..
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   403
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   404
text{*All three types of comparision involving 0 and 1 are covered.*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   405
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   406
declare zero_neq_one [THEN not_sym, simp]
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   407
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   408
lemma zero_le_one [simp]: "(0::'a::ordered_semidom) \<le> 1"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   409
  by (rule zero_less_one [THEN order_less_imp_le]) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   410
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   411
lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semidom) \<le> 0"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   412
by (simp add: linorder_not_le) 
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   413
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   414
lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semidom) < 0"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   415
by (simp add: linorder_not_less) 
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   416
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   417
subsection{*More Monotonicity*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   418
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   419
lemma mult_left_mono_neg:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   420
     "[|b \<le> a; c \<le> 0|] ==> c * a \<le> c * (b::'a::pordered_ring)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   421
apply (drule mult_left_mono [of _ _ "-c"]) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   422
apply (simp_all add: minus_mult_left [symmetric]) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   423
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   424
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   425
lemma mult_right_mono_neg:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   426
     "[|b \<le> a; c \<le> 0|] ==> a * c \<le> b * (c::'a::pordered_ring)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   427
apply (drule mult_right_mono [of _ _ "-c"]) 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   428
apply (simp_all add: minus_mult_right [symmetric]) 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   429
done  
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   430
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   431
text{*Strict monotonicity in both arguments*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   432
lemma mult_strict_mono:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   433
     "[|a<b; c<d; 0<b; 0\<le>c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   434
apply (case_tac "c=0")
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   435
 apply (simp add: mult_pos) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   436
apply (erule mult_strict_right_mono [THEN order_less_trans])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   437
 apply (force simp add: order_le_less) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   438
apply (erule mult_strict_left_mono, assumption)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   439
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   440
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   441
text{*This weaker variant has more natural premises*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   442
lemma mult_strict_mono':
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   443
     "[| a<b; c<d; 0 \<le> a; 0 \<le> c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   444
apply (rule mult_strict_mono)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   445
apply (blast intro: order_le_less_trans)+
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   446
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   447
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   448
lemma mult_mono:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   449
     "[|a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c|] 
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   450
      ==> a * c  \<le>  b * (d::'a::pordered_semiring)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   451
apply (erule mult_right_mono [THEN order_trans], assumption)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   452
apply (erule mult_left_mono, assumption)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   453
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   454
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   455
lemma less_1_mult: "[| 1 < m; 1 < n |] ==> 1 < m*(n::'a::ordered_semidom)"
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   456
apply (insert mult_strict_mono [of 1 m 1 n]) 
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   457
apply (simp add:  order_less_trans [OF zero_less_one]) 
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   458
done
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   459
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   460
subsection{*Cancellation Laws for Relationships With a Common Factor*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   461
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   462
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   463
   also with the relations @{text "\<le>"} and equality.*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   464
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   465
lemma mult_less_cancel_right:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   466
    "(a*c < b*c) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   467
apply (case_tac "c = 0")
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   468
apply (auto simp add: linorder_neq_iff mult_strict_right_mono 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   469
                      mult_strict_right_mono_neg)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   470
apply (auto simp add: linorder_not_less 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   471
                      linorder_not_le [symmetric, of "a*c"]
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   472
                      linorder_not_le [symmetric, of a])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   473
apply (erule_tac [!] notE)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   474
apply (auto simp add: order_less_imp_le mult_right_mono 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   475
                      mult_right_mono_neg)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   476
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   477
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   478
lemma mult_less_cancel_left:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   479
    "(c*a < c*b) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   480
apply (case_tac "c = 0")
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   481
apply (auto simp add: linorder_neq_iff mult_strict_left_mono 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   482
                      mult_strict_left_mono_neg)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   483
apply (auto simp add: linorder_not_less 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   484
                      linorder_not_le [symmetric, of "c*a"]
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   485
                      linorder_not_le [symmetric, of a])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   486
apply (erule_tac [!] notE)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   487
apply (auto simp add: order_less_imp_le mult_left_mono 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   488
                      mult_left_mono_neg)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   489
done
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   490
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   491
lemma mult_le_cancel_right:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   492
     "(a*c \<le> b*c) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   493
by (simp add: linorder_not_less [symmetric] mult_less_cancel_right)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   494
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   495
lemma mult_le_cancel_left:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   496
     "(c*a \<le> c*b) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   497
by (simp add: linorder_not_less [symmetric] mult_less_cancel_left)
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   498
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   499
lemma mult_less_imp_less_left:
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   500
      assumes less: "c*a < c*b" and nonneg: "0 \<le> c"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   501
      shows "a < (b::'a::ordered_semiring_strict)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   502
proof (rule ccontr)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   503
  assume "~ a < b"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   504
  hence "b \<le> a" by (simp add: linorder_not_less)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   505
  hence "c*b \<le> c*a" by (rule mult_left_mono)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   506
  with this and less show False 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   507
    by (simp add: linorder_not_less [symmetric])
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   508
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   509
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   510
lemma mult_less_imp_less_right:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   511
  assumes less: "a*c < b*c" and nonneg: "0 <= c"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   512
  shows "a < (b::'a::ordered_semiring_strict)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   513
proof (rule ccontr)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   514
  assume "~ a < b"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   515
  hence "b \<le> a" by (simp add: linorder_not_less)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   516
  hence "b*c \<le> a*c" by (rule mult_right_mono)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   517
  with this and less show False 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   518
    by (simp add: linorder_not_less [symmetric])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   519
qed  
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   520
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   521
text{*Cancellation of equalities with a common factor*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   522
lemma mult_cancel_right [simp]:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   523
     "(a*c = b*c) = (c = (0::'a::ordered_ring_strict) | a=b)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   524
apply (cut_tac linorder_less_linear [of 0 c])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   525
apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   526
             simp add: linorder_neq_iff)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   527
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   528
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   529
text{*These cancellation theorems require an ordering. Versions are proved
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   530
      below that work for fields without an ordering.*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   531
lemma mult_cancel_left [simp]:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   532
     "(c*a = c*b) = (c = (0::'a::ordered_ring_strict) | a=b)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   533
apply (cut_tac linorder_less_linear [of 0 c])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   534
apply (force dest: mult_strict_left_mono_neg mult_strict_left_mono
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   535
             simp add: linorder_neq_iff)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   536
done
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   537
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   538
text{*This list of rewrites decides ring equalities by ordered rewriting.*}
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   539
lemmas ring_eq_simps =
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   540
  mult_ac
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   541
  left_distrib right_distrib left_diff_distrib right_diff_distrib
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   542
  add_ac
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   543
  add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   544
  diff_eq_eq eq_diff_eq
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   545
    
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   546
thm ring_eq_simps
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   547
subsection {* Fields *}
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   548
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   549
lemma right_inverse [simp]:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   550
      assumes not0: "a \<noteq> 0" shows "a * inverse (a::'a::field) = 1"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   551
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   552
  have "a * inverse a = inverse a * a" by (simp add: mult_ac)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   553
  also have "... = 1" using not0 by simp
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   554
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   555
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   556
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   557
lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   558
proof
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   559
  assume neq: "b \<noteq> 0"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   560
  {
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   561
    hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   562
    also assume "a / b = 1"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   563
    finally show "a = b" by simp
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   564
  next
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   565
    assume "a = b"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   566
    with neq show "a / b = 1" by (simp add: divide_inverse)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   567
  }
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   568
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   569
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   570
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   571
by (simp add: divide_inverse)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   572
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   573
lemma divide_self [simp]: "a \<noteq> 0 ==> a / (a::'a::field) = 1"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   574
  by (simp add: divide_inverse)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   575
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   576
lemma divide_zero [simp]: "a / 0 = (0::'a::{field,division_by_zero})"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   577
by (simp add: divide_inverse)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   578
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   579
lemma divide_zero_left [simp]: "0/a = (0::'a::field)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   580
by (simp add: divide_inverse)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   581
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   582
lemma inverse_eq_divide: "inverse (a::'a::field) = 1/a"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   583
by (simp add: divide_inverse)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   584
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   585
lemma add_divide_distrib: "(a+b)/(c::'a::field) = a/c + b/c"
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   586
by (simp add: divide_inverse left_distrib) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   587
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   588
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   589
text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   590
      of an ordering.*}
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   591
lemma field_mult_eq_0_iff [simp]: "(a*b = (0::'a::field)) = (a = 0 | b = 0)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   592
proof cases
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   593
  assume "a=0" thus ?thesis by simp
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   594
next
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   595
  assume anz [simp]: "a\<noteq>0"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   596
  { assume "a * b = 0"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   597
    hence "inverse a * (a * b) = 0" by simp
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   598
    hence "b = 0"  by (simp (no_asm_use) add: mult_assoc [symmetric])}
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   599
  thus ?thesis by force
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   600
qed
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   601
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   602
text{*Cancellation of equalities with a common factor*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   603
lemma field_mult_cancel_right_lemma:
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   604
      assumes cnz: "c \<noteq> (0::'a::field)"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   605
	  and eq:  "a*c = b*c"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   606
	 shows "a=b"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   607
proof -
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   608
  have "(a * c) * inverse c = (b * c) * inverse c"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   609
    by (simp add: eq)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   610
  thus "a=b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   611
    by (simp add: mult_assoc cnz)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   612
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   613
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   614
lemma field_mult_cancel_right [simp]:
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   615
     "(a*c = b*c) = (c = (0::'a::field) | a=b)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   616
proof cases
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   617
  assume "c=0" thus ?thesis by simp
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   618
next
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   619
  assume "c\<noteq>0" 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   620
  thus ?thesis by (force dest: field_mult_cancel_right_lemma)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   621
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   622
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   623
lemma field_mult_cancel_left [simp]:
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   624
     "(c*a = c*b) = (c = (0::'a::field) | a=b)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   625
  by (simp add: mult_commute [of c] field_mult_cancel_right) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   626
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   627
lemma nonzero_imp_inverse_nonzero: "a \<noteq> 0 ==> inverse a \<noteq> (0::'a::field)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   628
proof
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   629
  assume ianz: "inverse a = 0"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   630
  assume "a \<noteq> 0"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   631
  hence "1 = a * inverse a" by simp
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   632
  also have "... = 0" by (simp add: ianz)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   633
  finally have "1 = (0::'a::field)" .
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   634
  thus False by (simp add: eq_commute)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   635
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   636
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   637
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   638
subsection{*Basic Properties of @{term inverse}*}
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   639
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   640
lemma inverse_zero_imp_zero: "inverse a = 0 ==> a = (0::'a::field)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   641
apply (rule ccontr) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   642
apply (blast dest: nonzero_imp_inverse_nonzero) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   643
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   644
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   645
lemma inverse_nonzero_imp_nonzero:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   646
   "inverse a = 0 ==> a = (0::'a::field)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   647
apply (rule ccontr) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   648
apply (blast dest: nonzero_imp_inverse_nonzero) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   649
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   650
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   651
lemma inverse_nonzero_iff_nonzero [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   652
   "(inverse a = 0) = (a = (0::'a::{field,division_by_zero}))"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   653
by (force dest: inverse_nonzero_imp_nonzero) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   654
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   655
lemma nonzero_inverse_minus_eq:
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   656
      assumes [simp]: "a\<noteq>0"  shows "inverse(-a) = -inverse(a::'a::field)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   657
proof -
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   658
  have "-a * inverse (- a) = -a * - inverse a"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   659
    by simp
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   660
  thus ?thesis 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   661
    by (simp only: field_mult_cancel_left, simp)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   662
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   663
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   664
lemma inverse_minus_eq [simp]:
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   665
   "inverse(-a) = -inverse(a::'a::{field,division_by_zero})";
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   666
proof cases
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   667
  assume "a=0" thus ?thesis by (simp add: inverse_zero)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   668
next
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   669
  assume "a\<noteq>0" 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   670
  thus ?thesis by (simp add: nonzero_inverse_minus_eq)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   671
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   672
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   673
lemma nonzero_inverse_eq_imp_eq:
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   674
      assumes inveq: "inverse a = inverse b"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   675
	  and anz:  "a \<noteq> 0"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   676
	  and bnz:  "b \<noteq> 0"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   677
	 shows "a = (b::'a::field)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   678
proof -
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   679
  have "a * inverse b = a * inverse a"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   680
    by (simp add: inveq)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   681
  hence "(a * inverse b) * b = (a * inverse a) * b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   682
    by simp
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   683
  thus "a = b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   684
    by (simp add: mult_assoc anz bnz)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   685
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   686
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   687
lemma inverse_eq_imp_eq:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   688
     "inverse a = inverse b ==> a = (b::'a::{field,division_by_zero})"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   689
apply (case_tac "a=0 | b=0") 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   690
 apply (force dest!: inverse_zero_imp_zero
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   691
              simp add: eq_commute [of "0::'a"])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   692
apply (force dest!: nonzero_inverse_eq_imp_eq) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   693
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   694
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   695
lemma inverse_eq_iff_eq [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   696
     "(inverse a = inverse b) = (a = (b::'a::{field,division_by_zero}))"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   697
by (force dest!: inverse_eq_imp_eq) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   698
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   699
lemma nonzero_inverse_inverse_eq:
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   700
      assumes [simp]: "a \<noteq> 0"  shows "inverse(inverse (a::'a::field)) = a"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   701
  proof -
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   702
  have "(inverse (inverse a) * inverse a) * a = a" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   703
    by (simp add: nonzero_imp_inverse_nonzero)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   704
  thus ?thesis
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   705
    by (simp add: mult_assoc)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   706
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   707
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   708
lemma inverse_inverse_eq [simp]:
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   709
     "inverse(inverse (a::'a::{field,division_by_zero})) = a"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   710
  proof cases
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   711
    assume "a=0" thus ?thesis by simp
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   712
  next
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   713
    assume "a\<noteq>0" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   714
    thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   715
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   716
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   717
lemma inverse_1 [simp]: "inverse 1 = (1::'a::field)"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   718
  proof -
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   719
  have "inverse 1 * 1 = (1::'a::field)" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   720
    by (rule left_inverse [OF zero_neq_one [symmetric]])
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   721
  thus ?thesis  by simp
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   722
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   723
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   724
lemma nonzero_inverse_mult_distrib: 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   725
      assumes anz: "a \<noteq> 0"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   726
          and bnz: "b \<noteq> 0"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   727
      shows "inverse(a*b) = inverse(b) * inverse(a::'a::field)"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   728
  proof -
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   729
  have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   730
    by (simp add: field_mult_eq_0_iff anz bnz)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   731
  hence "inverse(a*b) * a = inverse(b)" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   732
    by (simp add: mult_assoc bnz)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   733
  hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   734
    by simp
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   735
  thus ?thesis
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   736
    by (simp add: mult_assoc anz)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   737
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   738
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   739
text{*This version builds in division by zero while also re-orienting
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   740
      the right-hand side.*}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   741
lemma inverse_mult_distrib [simp]:
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   742
     "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   743
  proof cases
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   744
    assume "a \<noteq> 0 & b \<noteq> 0" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   745
    thus ?thesis  by (simp add: nonzero_inverse_mult_distrib mult_commute)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   746
  next
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   747
    assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   748
    thus ?thesis  by force
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   749
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   750
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   751
text{*There is no slick version using division by zero.*}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   752
lemma inverse_add:
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   753
     "[|a \<noteq> 0;  b \<noteq> 0|]
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   754
      ==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   755
apply (simp add: left_distrib mult_assoc)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   756
apply (simp add: mult_commute [of "inverse a"]) 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   757
apply (simp add: mult_assoc [symmetric] add_commute)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   758
done
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   759
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   760
lemma inverse_divide [simp]:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   761
      "inverse (a/b) = b / (a::'a::{field,division_by_zero})"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   762
  by (simp add: divide_inverse mult_commute)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   763
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   764
lemma nonzero_mult_divide_cancel_left:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   765
  assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" 
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   766
    shows "(c*a)/(c*b) = a/(b::'a::field)"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   767
proof -
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   768
  have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   769
    by (simp add: field_mult_eq_0_iff divide_inverse 
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   770
                  nonzero_inverse_mult_distrib)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   771
  also have "... =  a * inverse b * (inverse c * c)"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   772
    by (simp only: mult_ac)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   773
  also have "... =  a * inverse b"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   774
    by simp
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   775
    finally show ?thesis 
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   776
    by (simp add: divide_inverse)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   777
qed
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   778
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   779
lemma mult_divide_cancel_left:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   780
     "c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   781
apply (case_tac "b = 0")
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   782
apply (simp_all add: nonzero_mult_divide_cancel_left)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   783
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   784
14321
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   785
lemma nonzero_mult_divide_cancel_right:
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   786
     "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)"
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   787
by (simp add: mult_commute [of _ c] nonzero_mult_divide_cancel_left) 
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   788
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   789
lemma mult_divide_cancel_right:
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   790
     "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   791
apply (case_tac "b = 0")
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   792
apply (simp_all add: nonzero_mult_divide_cancel_right)
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   793
done
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   794
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   795
(*For ExtractCommonTerm*)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   796
lemma mult_divide_cancel_eq_if:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   797
     "(c*a) / (c*b) = 
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   798
      (if c=0 then 0 else a / (b::'a::{field,division_by_zero}))"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   799
  by (simp add: mult_divide_cancel_left)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   800
14284
f1abe67c448a re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents: 14277
diff changeset
   801
lemma divide_1 [simp]: "a/1 = (a::'a::field)"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   802
  by (simp add: divide_inverse)
14284
f1abe67c448a re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents: 14277
diff changeset
   803
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   804
lemma times_divide_eq_right [simp]: "a * (b/c) = (a*b) / (c::'a::field)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   805
by (simp add: divide_inverse mult_assoc)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   806
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   807
lemma times_divide_eq_left: "(b/c) * a = (b*a) / (c::'a::field)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   808
by (simp add: divide_inverse mult_ac)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   809
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   810
lemma divide_divide_eq_right [simp]:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   811
     "a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   812
by (simp add: divide_inverse mult_ac)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   813
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   814
lemma divide_divide_eq_left [simp]:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   815
     "(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   816
by (simp add: divide_inverse mult_assoc)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   817
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   818
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   819
subsection {* Division and Unary Minus *}
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   820
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   821
lemma nonzero_minus_divide_left: "b \<noteq> 0 ==> - (a/b) = (-a) / (b::'a::field)"
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   822
by (simp add: divide_inverse minus_mult_left)
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   823
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   824
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a/b) = a / -(b::'a::field)"
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   825
by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right)
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   826
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   827
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a)/(-b) = a / (b::'a::field)"
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   828
by (simp add: divide_inverse nonzero_inverse_minus_eq)
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   829
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   830
lemma minus_divide_left: "- (a/b) = (-a) / (b::'a::field)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   831
by (simp add: divide_inverse minus_mult_left [symmetric])
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   832
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   833
lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   834
by (simp add: divide_inverse minus_mult_right [symmetric])
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   835
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   836
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   837
text{*The effect is to extract signs from divisions*}
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   838
declare minus_divide_left  [symmetric, simp]
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   839
declare minus_divide_right [symmetric, simp]
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   840
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   841
text{*Also, extract signs from products*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   842
declare minus_mult_left [symmetric, simp]
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   843
declare minus_mult_right [symmetric, simp]
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   844
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   845
lemma minus_divide_divide [simp]:
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   846
     "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   847
apply (case_tac "b=0", simp) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   848
apply (simp add: nonzero_minus_divide_divide) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   849
done
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   850
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   851
lemma diff_divide_distrib: "(a-b)/(c::'a::field) = a/c - b/c"
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   852
by (simp add: diff_minus add_divide_distrib) 
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   853
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   854
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   855
subsection {* Ordered Fields *}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   856
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   857
lemma positive_imp_inverse_positive: 
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   858
      assumes a_gt_0: "0 < a"  shows "0 < inverse (a::'a::ordered_field)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   859
  proof -
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   860
  have "0 < a * inverse a" 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   861
    by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   862
  thus "0 < inverse a" 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   863
    by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   864
  qed
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   865
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   866
lemma negative_imp_inverse_negative:
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   867
     "a < 0 ==> inverse a < (0::'a::ordered_field)"
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   868
  by (insert positive_imp_inverse_positive [of "-a"], 
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   869
      simp add: nonzero_inverse_minus_eq order_less_imp_not_eq) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   870
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   871
lemma inverse_le_imp_le:
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   872
      assumes invle: "inverse a \<le> inverse b"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   873
	  and apos:  "0 < a"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   874
	 shows "b \<le> (a::'a::ordered_field)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   875
  proof (rule classical)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   876
  assume "~ b \<le> a"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   877
  hence "a < b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   878
    by (simp add: linorder_not_le)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   879
  hence bpos: "0 < b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   880
    by (blast intro: apos order_less_trans)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   881
  hence "a * inverse a \<le> a * inverse b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   882
    by (simp add: apos invle order_less_imp_le mult_left_mono)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   883
  hence "(a * inverse a) * b \<le> (a * inverse b) * b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   884
    by (simp add: bpos order_less_imp_le mult_right_mono)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   885
  thus "b \<le> a"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   886
    by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   887
  qed
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   888
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   889
lemma inverse_positive_imp_positive:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   890
      assumes inv_gt_0: "0 < inverse a"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   891
          and [simp]:   "a \<noteq> 0"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   892
        shows "0 < (a::'a::ordered_field)"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   893
  proof -
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   894
  have "0 < inverse (inverse a)"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   895
    by (rule positive_imp_inverse_positive)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   896
  thus "0 < a"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   897
    by (simp add: nonzero_inverse_inverse_eq)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   898
  qed
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   899
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   900
lemma inverse_positive_iff_positive [simp]:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   901
      "(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   902
apply (case_tac "a = 0", simp)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   903
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   904
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   905
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   906
lemma inverse_negative_imp_negative:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   907
      assumes inv_less_0: "inverse a < 0"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   908
          and [simp]:   "a \<noteq> 0"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   909
        shows "a < (0::'a::ordered_field)"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   910
  proof -
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   911
  have "inverse (inverse a) < 0"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   912
    by (rule negative_imp_inverse_negative)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   913
  thus "a < 0"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   914
    by (simp add: nonzero_inverse_inverse_eq)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   915
  qed
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   916
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   917
lemma inverse_negative_iff_negative [simp]:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   918
      "(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   919
apply (case_tac "a = 0", simp)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   920
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   921
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   922
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   923
lemma inverse_nonnegative_iff_nonnegative [simp]:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   924
      "(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   925
by (simp add: linorder_not_less [symmetric])
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   926
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   927
lemma inverse_nonpositive_iff_nonpositive [simp]:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   928
      "(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   929
by (simp add: linorder_not_less [symmetric])
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   930
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   931
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   932
subsection{*Anti-Monotonicity of @{term inverse}*}
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   933
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   934
lemma less_imp_inverse_less:
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   935
      assumes less: "a < b"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   936
	  and apos:  "0 < a"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   937
	shows "inverse b < inverse (a::'a::ordered_field)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   938
  proof (rule ccontr)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   939
  assume "~ inverse b < inverse a"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   940
  hence "inverse a \<le> inverse b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   941
    by (simp add: linorder_not_less)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   942
  hence "~ (a < b)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   943
    by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   944
  thus False
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   945
    by (rule notE [OF _ less])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   946
  qed
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   947
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   948
lemma inverse_less_imp_less:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   949
   "[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::ordered_field)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   950
apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   951
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   952
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   953
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   954
text{*Both premises are essential. Consider -1 and 1.*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   955
lemma inverse_less_iff_less [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   956
     "[|0 < a; 0 < b|] 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   957
      ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   958
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   959
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   960
lemma le_imp_inverse_le:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   961
   "[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   962
  by (force simp add: order_le_less less_imp_inverse_less)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   963
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   964
lemma inverse_le_iff_le [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   965
     "[|0 < a; 0 < b|] 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   966
      ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   967
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   968
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   969
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   970
text{*These results refer to both operands being negative.  The opposite-sign
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   971
case is trivial, since inverse preserves signs.*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   972
lemma inverse_le_imp_le_neg:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   973
   "[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::ordered_field)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   974
  apply (rule classical) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   975
  apply (subgoal_tac "a < 0") 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   976
   prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   977
  apply (insert inverse_le_imp_le [of "-b" "-a"])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   978
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   979
  done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   980
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   981
lemma less_imp_inverse_less_neg:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   982
   "[|a < b; b < 0|] ==> inverse b < inverse (a::'a::ordered_field)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   983
  apply (subgoal_tac "a < 0") 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   984
   prefer 2 apply (blast intro: order_less_trans) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   985
  apply (insert less_imp_inverse_less [of "-b" "-a"])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   986
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   987
  done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   988
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   989
lemma inverse_less_imp_less_neg:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   990
   "[|inverse a < inverse b; b < 0|] ==> b < (a::'a::ordered_field)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   991
  apply (rule classical) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   992
  apply (subgoal_tac "a < 0") 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   993
   prefer 2
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   994
   apply (force simp add: linorder_not_less intro: order_le_less_trans) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   995
  apply (insert inverse_less_imp_less [of "-b" "-a"])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   996
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   997
  done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   998
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   999
lemma inverse_less_iff_less_neg [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1000
     "[|a < 0; b < 0|] 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1001
      ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1002
  apply (insert inverse_less_iff_less [of "-b" "-a"])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1003
  apply (simp del: inverse_less_iff_less 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1004
	      add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1005
  done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1006
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1007
lemma le_imp_inverse_le_neg:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1008
   "[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1009
  by (force simp add: order_le_less less_imp_inverse_less_neg)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1010
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1011
lemma inverse_le_iff_le_neg [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1012
     "[|a < 0; b < 0|] 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1013
      ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1014
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
  1015
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1016
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1017
subsection{*Inverses and the Number One*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1018
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1019
lemma one_less_inverse_iff:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1020
    "(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"proof cases
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1021
  assume "0 < x"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1022
    with inverse_less_iff_less [OF zero_less_one, of x]
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1023
    show ?thesis by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1024
next
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1025
  assume notless: "~ (0 < x)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1026
  have "~ (1 < inverse x)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1027
  proof
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1028
    assume "1 < inverse x"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1029
    also with notless have "... \<le> 0" by (simp add: linorder_not_less)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1030
    also have "... < 1" by (rule zero_less_one) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1031
    finally show False by auto
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1032
  qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1033
  with notless show ?thesis by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1034
qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1035
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1036
lemma inverse_eq_1_iff [simp]:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1037
    "(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1038
by (insert inverse_eq_iff_eq [of x 1], simp) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1039
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1040
lemma one_le_inverse_iff:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1041
   "(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1042
by (force simp add: order_le_less one_less_inverse_iff zero_less_one 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1043
                    eq_commute [of 1]) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1044
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1045
lemma inverse_less_1_iff:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1046
   "(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{ordered_field,division_by_zero}))"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1047
by (simp add: linorder_not_le [symmetric] one_le_inverse_iff) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1048
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1049
lemma inverse_le_1_iff:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1050
   "(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{ordered_field,division_by_zero}))"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1051
by (simp add: linorder_not_less [symmetric] one_less_inverse_iff) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1052
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1053
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1054
subsection{*Division and Signs*}
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1055
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1056
lemma zero_less_divide_iff:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1057
     "((0::'a::{ordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1058
by (simp add: divide_inverse zero_less_mult_iff)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1059
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1060
lemma divide_less_0_iff:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1061
     "(a/b < (0::'a::{ordered_field,division_by_zero})) = 
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1062
      (0 < a & b < 0 | a < 0 & 0 < b)"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1063
by (simp add: divide_inverse mult_less_0_iff)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1064
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1065
lemma zero_le_divide_iff:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1066
     "((0::'a::{ordered_field,division_by_zero}) \<le> a/b) =
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1067
      (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1068
by (simp add: divide_inverse zero_le_mult_iff)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1069
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1070
lemma divide_le_0_iff:
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1071
     "(a/b \<le> (0::'a::{ordered_field,division_by_zero})) =
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1072
      (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1073
by (simp add: divide_inverse mult_le_0_iff)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1074
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1075
lemma divide_eq_0_iff [simp]:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1076
     "(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1077
by (simp add: divide_inverse field_mult_eq_0_iff)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1078
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1079
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1080
subsection{*Simplification of Inequalities Involving Literal Divisors*}
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1081
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1082
lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1083
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1084
  assume less: "0<c"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1085
  hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1086
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1087
  also have "... = (a*c \<le> b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1088
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1089
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1090
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1091
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1092
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1093
lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1094
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1095
  assume less: "c<0"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1096
  hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1097
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1098
  also have "... = (b \<le> a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1099
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1100
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1101
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1102
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1103
lemma le_divide_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1104
  "(a \<le> b/c) = 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1105
   (if 0 < c then a*c \<le> b
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1106
             else if c < 0 then b \<le> a*c
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1107
             else  a \<le> (0::'a::{ordered_field,division_by_zero}))"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1108
apply (case_tac "c=0", simp) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1109
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1110
done
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1111
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1112
lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1113
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1114
  assume less: "0<c"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1115
  hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1116
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1117
  also have "... = (b \<le> a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1118
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1119
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1120
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1121
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1122
lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1123
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1124
  assume less: "c<0"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1125
  hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1126
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1127
  also have "... = (a*c \<le> b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1128
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1129
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1130
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1131
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1132
lemma divide_le_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1133
  "(b/c \<le> a) = 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1134
   (if 0 < c then b \<le> a*c
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1135
             else if c < 0 then a*c \<le> b
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1136
             else 0 \<le> (a::'a::{ordered_field,division_by_zero}))"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1137
apply (case_tac "c=0", simp) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1138
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1139
done
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1140
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1141
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1142
lemma pos_less_divide_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1143
     "0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1144
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1145
  assume less: "0<c"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1146
  hence "(a < b/c) = (a*c < (b/c)*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1147
    by (simp add: mult_less_cancel_right order_less_not_sym [OF less])
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1148
  also have "... = (a*c < b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1149
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1150
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1151
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1152
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1153
lemma neg_less_divide_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1154
 "c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1155
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1156
  assume less: "c<0"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1157
  hence "(a < b/c) = ((b/c)*c < a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1158
    by (simp add: mult_less_cancel_right order_less_not_sym [OF less])
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1159
  also have "... = (b < a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1160
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1161
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1162
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1163
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1164
lemma less_divide_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1165
  "(a < b/c) = 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1166
   (if 0 < c then a*c < b
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1167
             else if c < 0 then b < a*c
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1168
             else  a < (0::'a::{ordered_field,division_by_zero}))"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1169
apply (case_tac "c=0", simp) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1170
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1171
done
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1172
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1173
lemma pos_divide_less_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1174
     "0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1175
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1176
  assume less: "0<c"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1177
  hence "(b/c < a) = ((b/c)*c < a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1178
    by (simp add: mult_less_cancel_right order_less_not_sym [OF less])
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1179
  also have "... = (b < a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1180
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1181
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1182
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1183
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1184
lemma neg_divide_less_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1185
 "c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1186
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1187
  assume less: "c<0"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1188
  hence "(b/c < a) = (a*c < (b/c)*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1189
    by (simp add: mult_less_cancel_right order_less_not_sym [OF less])
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1190
  also have "... = (a*c < b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1191
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1192
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1193
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1194
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1195
lemma divide_less_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1196
  "(b/c < a) = 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1197
   (if 0 < c then b < a*c
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1198
             else if c < 0 then a*c < b
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1199
             else 0 < (a::'a::{ordered_field,division_by_zero}))"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1200
apply (case_tac "c=0", simp) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1201
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1202
done
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1203
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1204
lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1205
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1206
  assume [simp]: "c\<noteq>0"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1207
  have "(a = b/c) = (a*c = (b/c)*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1208
    by (simp add: field_mult_cancel_right)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1209
  also have "... = (a*c = b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1210
    by (simp add: divide_inverse mult_assoc) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1211
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1212
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1213
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1214
lemma eq_divide_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1215
  "((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1216
by (simp add: nonzero_eq_divide_eq) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1217
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1218
lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1219
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1220
  assume [simp]: "c\<noteq>0"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1221
  have "(b/c = a) = ((b/c)*c = a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1222
    by (simp add: field_mult_cancel_right)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1223
  also have "... = (b = a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1224
    by (simp add: divide_inverse mult_assoc) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1225
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1226
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1227
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1228
lemma divide_eq_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1229
  "(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1230
by (force simp add: nonzero_divide_eq_eq) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1231
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1232
subsection{*Cancellation Laws for Division*}
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1233
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1234
lemma divide_cancel_right [simp]:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1235
     "(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1236
apply (case_tac "c=0", simp) 
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1237
apply (simp add: divide_inverse field_mult_cancel_right) 
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1238
done
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1239
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1240
lemma divide_cancel_left [simp]:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1241
     "(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))" 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1242
apply (case_tac "c=0", simp) 
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1243
apply (simp add: divide_inverse field_mult_cancel_left) 
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1244
done
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1245
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1246
subsection {* Division and the Number One *}
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1247
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1248
text{*Simplify expressions equated with 1*}
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1249
lemma divide_eq_1_iff [simp]:
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1250
     "(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1251
apply (case_tac "b=0", simp) 
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1252
apply (simp add: right_inverse_eq) 
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1253
done
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1254
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1255
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1256
lemma one_eq_divide_iff [simp]:
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1257
     "(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1258
by (simp add: eq_commute [of 1])  
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1259
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1260
lemma zero_eq_1_divide_iff [simp]:
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1261
     "((0::'a::{ordered_field,division_by_zero}) = 1/a) = (a = 0)"
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1262
apply (case_tac "a=0", simp) 
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1263
apply (auto simp add: nonzero_eq_divide_eq) 
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1264
done
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1265
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1266
lemma one_divide_eq_0_iff [simp]:
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1267
     "(1/a = (0::'a::{ordered_field,division_by_zero})) = (a = 0)"
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1268
apply (case_tac "a=0", simp) 
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1269
apply (insert zero_neq_one [THEN not_sym]) 
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1270
apply (auto simp add: nonzero_divide_eq_eq) 
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1271
done
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1272
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1273
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1274
declare zero_less_divide_iff [of "1", simp]
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1275
declare divide_less_0_iff [of "1", simp]
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1276
declare zero_le_divide_iff [of "1", simp]
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1277
declare divide_le_0_iff [of "1", simp]
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1278
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1279
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1280
subsection {* Ordering Rules for Division *}
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1281
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1282
lemma divide_strict_right_mono:
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1283
     "[|a < b; 0 < c|] ==> a / c < b / (c::'a::ordered_field)"