src/HOL/Ring_and_Field.thy
 author obua Tue Apr 19 10:59:31 2005 +0200 (2005-04-19) changeset 15769 38c8ea8521e7 parent 15580 900291ee0af8 child 15923 01d5d0c1c078 permissions -rw-r--r--
Removed mult_commute axiom from comm_semiring axclass.
     1 (*  Title:   HOL/Ring_and_Field.thy

     2     ID:      $Id$

     3     Author:  Gertrud Bauer, Steven Obua, Lawrence C Paulson and Markus Wenzel

     4 *)

     5

     6 header {* (Ordered) Rings and Fields *}

     7

     8 theory Ring_and_Field

     9 imports OrderedGroup

    10 begin

    11

    12 text {*

    13   The theory of partially ordered rings is taken from the books:

    14   \begin{itemize}

    15   \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979

    16   \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963

    17   \end{itemize}

    18   Most of the used notions can also be looked up in

    19   \begin{itemize}

    20   \item \url{http://www.mathworld.com} by Eric Weisstein et. al.

    21   \item \emph{Algebra I} by van der Waerden, Springer.

    22   \end{itemize}

    23 *}

    24

    25 axclass semiring \<subseteq> ab_semigroup_add, semigroup_mult

    26   left_distrib: "(a + b) * c = a * c + b * c"

    27   right_distrib: "a * (b + c) = a * b + a * c"

    28

    29 axclass semiring_0 \<subseteq> semiring, comm_monoid_add

    30

    31 axclass semiring_0_cancel \<subseteq> semiring_0, cancel_ab_semigroup_add

    32

    33 axclass comm_semiring \<subseteq> ab_semigroup_add, ab_semigroup_mult

    34   distrib: "(a + b) * c = a * c + b * c"

    35

    36 instance comm_semiring \<subseteq> semiring

    37 proof

    38   fix a b c :: 'a

    39   show "(a + b) * c = a * c + b * c" by (simp add: distrib)

    40   have "a * (b + c) = (b + c) * a" by (simp add: mult_ac)

    41   also have "... = b * a + c * a" by (simp only: distrib)

    42   also have "... = a * b + a * c" by (simp add: mult_ac)

    43   finally show "a * (b + c) = a * b + a * c" by blast

    44 qed

    45

    46 axclass comm_semiring_0 \<subseteq> comm_semiring, comm_monoid_add

    47

    48 instance comm_semiring_0 \<subseteq> semiring_0 ..

    49

    50 axclass comm_semiring_0_cancel \<subseteq> comm_semiring_0, cancel_ab_semigroup_add

    51

    52 instance comm_semiring_0_cancel \<subseteq> semiring_0_cancel ..

    53

    54 axclass axclass_0_neq_1 \<subseteq> zero, one

    55   zero_neq_one [simp]: "0 \<noteq> 1"

    56

    57 axclass semiring_1 \<subseteq> axclass_0_neq_1, semiring_0, monoid_mult

    58

    59 axclass comm_semiring_1 \<subseteq> axclass_0_neq_1, comm_semiring_0, comm_monoid_mult (* previously almost_semiring *)

    60

    61 instance comm_semiring_1 \<subseteq> semiring_1 ..

    62

    63 axclass axclass_no_zero_divisors \<subseteq> zero, times

    64   no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"

    65

    66 axclass semiring_1_cancel \<subseteq> semiring_1, cancel_ab_semigroup_add

    67

    68 instance semiring_1_cancel \<subseteq> semiring_0_cancel ..

    69

    70 axclass comm_semiring_1_cancel \<subseteq> comm_semiring_1, cancel_ab_semigroup_add (* previously semiring *)

    71

    72 instance comm_semiring_1_cancel \<subseteq> semiring_1_cancel ..

    73

    74 instance comm_semiring_1_cancel \<subseteq> comm_semiring_0_cancel ..

    75

    76 axclass ring \<subseteq> semiring, ab_group_add

    77

    78 instance ring \<subseteq> semiring_0_cancel ..

    79

    80 axclass comm_ring \<subseteq> comm_semiring_0, ab_group_add

    81

    82 instance comm_ring \<subseteq> ring ..

    83

    84 instance comm_ring \<subseteq> comm_semiring_0_cancel ..

    85

    86 axclass ring_1 \<subseteq> ring, semiring_1

    87

    88 instance ring_1 \<subseteq> semiring_1_cancel ..

    89

    90 axclass comm_ring_1 \<subseteq> comm_ring, comm_semiring_1 (* previously ring *)

    91

    92 instance comm_ring_1 \<subseteq> ring_1 ..

    93

    94 instance comm_ring_1 \<subseteq> comm_semiring_1_cancel ..

    95

    96 axclass idom \<subseteq> comm_ring_1, axclass_no_zero_divisors

    97

    98 axclass field \<subseteq> comm_ring_1, inverse

    99   left_inverse [simp]: "a \<noteq> 0 ==> inverse a * a = 1"

   100   divide_inverse:      "a / b = a * inverse b"

   101

   102 lemma mult_zero_left [simp]: "0 * a = (0::'a::semiring_0_cancel)"

   103 proof -

   104   have "0*a + 0*a = 0*a + 0"

   105     by (simp add: left_distrib [symmetric])

   106   thus ?thesis

   107     by (simp only: add_left_cancel)

   108 qed

   109

   110 lemma mult_zero_right [simp]: "a * 0 = (0::'a::semiring_0_cancel)"

   111 proof -

   112   have "a*0 + a*0 = a*0 + 0"

   113     by (simp add: right_distrib [symmetric])

   114   thus ?thesis

   115     by (simp only: add_left_cancel)

   116 qed

   117

   118 lemma field_mult_eq_0_iff [simp]: "(a*b = (0::'a::field)) = (a = 0 | b = 0)"

   119 proof cases

   120   assume "a=0" thus ?thesis by simp

   121 next

   122   assume anz [simp]: "a\<noteq>0"

   123   { assume "a * b = 0"

   124     hence "inverse a * (a * b) = 0" by simp

   125     hence "b = 0"  by (simp (no_asm_use) add: mult_assoc [symmetric])}

   126   thus ?thesis by force

   127 qed

   128

   129 instance field \<subseteq> idom

   130 by (intro_classes, simp)

   131

   132 axclass division_by_zero \<subseteq> zero, inverse

   133   inverse_zero [simp]: "inverse 0 = 0"

   134

   135 subsection {* Distribution rules *}

   136

   137 theorems ring_distrib = right_distrib left_distrib

   138

   139 text{*For the @{text combine_numerals} simproc*}

   140 lemma combine_common_factor:

   141      "a*e + (b*e + c) = (a+b)*e + (c::'a::semiring)"

   142 by (simp add: left_distrib add_ac)

   143

   144 lemma minus_mult_left: "- (a * b) = (-a) * (b::'a::ring)"

   145 apply (rule equals_zero_I)

   146 apply (simp add: left_distrib [symmetric])

   147 done

   148

   149 lemma minus_mult_right: "- (a * b) = a * -(b::'a::ring)"

   150 apply (rule equals_zero_I)

   151 apply (simp add: right_distrib [symmetric])

   152 done

   153

   154 lemma minus_mult_minus [simp]: "(- a) * (- b) = a * (b::'a::ring)"

   155   by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])

   156

   157 lemma minus_mult_commute: "(- a) * b = a * (- b::'a::ring)"

   158   by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])

   159

   160 lemma right_diff_distrib: "a * (b - c) = a * b - a * (c::'a::ring)"

   161 by (simp add: right_distrib diff_minus

   162               minus_mult_left [symmetric] minus_mult_right [symmetric])

   163

   164 lemma left_diff_distrib: "(a - b) * c = a * c - b * (c::'a::ring)"

   165 by (simp add: left_distrib diff_minus

   166               minus_mult_left [symmetric] minus_mult_right [symmetric])

   167

   168 axclass pordered_semiring \<subseteq> semiring_0, pordered_ab_semigroup_add

   169   mult_left_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> c * a <= c * b"

   170   mult_right_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> a * c <= b * c"

   171

   172 axclass pordered_cancel_semiring \<subseteq> pordered_semiring, cancel_ab_semigroup_add

   173

   174 instance pordered_cancel_semiring \<subseteq> semiring_0_cancel ..

   175

   176 axclass ordered_semiring_strict \<subseteq> semiring_0, ordered_cancel_ab_semigroup_add

   177   mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"

   178   mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"

   179

   180 instance ordered_semiring_strict \<subseteq> semiring_0_cancel ..

   181

   182 instance ordered_semiring_strict \<subseteq> pordered_cancel_semiring

   183 apply intro_classes

   184 apply (case_tac "a < b & 0 < c")

   185 apply (auto simp add: mult_strict_left_mono order_less_le)

   186 apply (auto simp add: mult_strict_left_mono order_le_less)

   187 apply (simp add: mult_strict_right_mono)

   188 done

   189

   190 axclass pordered_comm_semiring \<subseteq> comm_semiring_0, pordered_ab_semigroup_add

   191   mult_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> c * a <= c * b"

   192

   193 axclass pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring, cancel_ab_semigroup_add

   194

   195 instance pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring ..

   196

   197 axclass ordered_comm_semiring_strict \<subseteq> comm_semiring_0, ordered_cancel_ab_semigroup_add

   198   mult_strict_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"

   199

   200 instance pordered_comm_semiring \<subseteq> pordered_semiring

   201 by (intro_classes, insert mult_mono, simp_all add: mult_commute, blast+)

   202

   203 instance pordered_cancel_comm_semiring \<subseteq> pordered_cancel_semiring ..

   204

   205 instance ordered_comm_semiring_strict \<subseteq> ordered_semiring_strict

   206 by (intro_classes, insert mult_strict_mono, simp_all add: mult_commute, blast+)

   207

   208 instance ordered_comm_semiring_strict \<subseteq> pordered_cancel_comm_semiring

   209 apply (intro_classes)

   210 apply (case_tac "a < b & 0 < c")

   211 apply (auto simp add: mult_strict_left_mono order_less_le)

   212 apply (auto simp add: mult_strict_left_mono order_le_less)

   213 done

   214

   215 axclass pordered_ring \<subseteq> ring, pordered_semiring

   216

   217 instance pordered_ring \<subseteq> pordered_ab_group_add ..

   218

   219 instance pordered_ring \<subseteq> pordered_cancel_semiring ..

   220

   221 axclass lordered_ring \<subseteq> pordered_ring, lordered_ab_group_abs

   222

   223 instance lordered_ring \<subseteq> lordered_ab_group_meet ..

   224

   225 instance lordered_ring \<subseteq> lordered_ab_group_join ..

   226

   227 axclass axclass_abs_if \<subseteq> minus, ord, zero

   228   abs_if: "abs a = (if (a < 0) then (-a) else a)"

   229

   230 axclass ordered_ring_strict \<subseteq> ring, ordered_semiring_strict, axclass_abs_if

   231

   232 instance ordered_ring_strict \<subseteq> lordered_ab_group ..

   233

   234 instance ordered_ring_strict \<subseteq> lordered_ring

   235 by (intro_classes, simp add: abs_if join_eq_if)

   236

   237 axclass pordered_comm_ring \<subseteq> comm_ring, pordered_comm_semiring

   238

   239 axclass ordered_semidom \<subseteq> comm_semiring_1_cancel, ordered_comm_semiring_strict (* previously ordered_semiring *)

   240   zero_less_one [simp]: "0 < 1"

   241

   242 axclass ordered_idom \<subseteq> comm_ring_1, ordered_comm_semiring_strict, axclass_abs_if (* previously ordered_ring *)

   243

   244 instance ordered_idom \<subseteq> ordered_ring_strict ..

   245

   246 axclass ordered_field \<subseteq> field, ordered_idom

   247

   248 lemma eq_add_iff1:

   249      "(a*e + c = b*e + d) = ((a-b)*e + c = (d::'a::ring))"

   250 apply (simp add: diff_minus left_distrib)

   251 apply (simp add: diff_minus left_distrib add_ac)

   252 apply (simp add: compare_rls minus_mult_left [symmetric])

   253 done

   254

   255 lemma eq_add_iff2:

   256      "(a*e + c = b*e + d) = (c = (b-a)*e + (d::'a::ring))"

   257 apply (simp add: diff_minus left_distrib add_ac)

   258 apply (simp add: compare_rls minus_mult_left [symmetric])

   259 done

   260

   261 lemma less_add_iff1:

   262      "(a*e + c < b*e + d) = ((a-b)*e + c < (d::'a::pordered_ring))"

   263 apply (simp add: diff_minus left_distrib add_ac)

   264 apply (simp add: compare_rls minus_mult_left [symmetric])

   265 done

   266

   267 lemma less_add_iff2:

   268      "(a*e + c < b*e + d) = (c < (b-a)*e + (d::'a::pordered_ring))"

   269 apply (simp add: diff_minus left_distrib add_ac)

   270 apply (simp add: compare_rls minus_mult_left [symmetric])

   271 done

   272

   273 lemma le_add_iff1:

   274      "(a*e + c \<le> b*e + d) = ((a-b)*e + c \<le> (d::'a::pordered_ring))"

   275 apply (simp add: diff_minus left_distrib add_ac)

   276 apply (simp add: compare_rls minus_mult_left [symmetric])

   277 done

   278

   279 lemma le_add_iff2:

   280      "(a*e + c \<le> b*e + d) = (c \<le> (b-a)*e + (d::'a::pordered_ring))"

   281 apply (simp add: diff_minus left_distrib add_ac)

   282 apply (simp add: compare_rls minus_mult_left [symmetric])

   283 done

   284

   285 subsection {* Ordering Rules for Multiplication *}

   286

   287 lemma mult_left_le_imp_le:

   288      "[|c*a \<le> c*b; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)"

   289   by (force simp add: mult_strict_left_mono linorder_not_less [symmetric])

   290

   291 lemma mult_right_le_imp_le:

   292      "[|a*c \<le> b*c; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)"

   293   by (force simp add: mult_strict_right_mono linorder_not_less [symmetric])

   294

   295 lemma mult_left_less_imp_less:

   296      "[|c*a < c*b; 0 \<le> c|] ==> a < (b::'a::ordered_semiring_strict)"

   297   by (force simp add: mult_left_mono linorder_not_le [symmetric])

   298

   299 lemma mult_right_less_imp_less:

   300      "[|a*c < b*c; 0 \<le> c|] ==> a < (b::'a::ordered_semiring_strict)"

   301   by (force simp add: mult_right_mono linorder_not_le [symmetric])

   302

   303 lemma mult_strict_left_mono_neg:

   304      "[|b < a; c < 0|] ==> c * a < c * (b::'a::ordered_ring_strict)"

   305 apply (drule mult_strict_left_mono [of _ _ "-c"])

   306 apply (simp_all add: minus_mult_left [symmetric])

   307 done

   308

   309 lemma mult_left_mono_neg:

   310      "[|b \<le> a; c \<le> 0|] ==> c * a \<le>  c * (b::'a::pordered_ring)"

   311 apply (drule mult_left_mono [of _ _ "-c"])

   312 apply (simp_all add: minus_mult_left [symmetric])

   313 done

   314

   315 lemma mult_strict_right_mono_neg:

   316      "[|b < a; c < 0|] ==> a * c < b * (c::'a::ordered_ring_strict)"

   317 apply (drule mult_strict_right_mono [of _ _ "-c"])

   318 apply (simp_all add: minus_mult_right [symmetric])

   319 done

   320

   321 lemma mult_right_mono_neg:

   322      "[|b \<le> a; c \<le> 0|] ==> a * c \<le>  (b::'a::pordered_ring) * c"

   323 apply (drule mult_right_mono [of _ _ "-c"])

   324 apply (simp)

   325 apply (simp_all add: minus_mult_right [symmetric])

   326 done

   327

   328 subsection{* Products of Signs *}

   329

   330 lemma mult_pos: "[| (0::'a::ordered_semiring_strict) < a; 0 < b |] ==> 0 < a*b"

   331 by (drule mult_strict_left_mono [of 0 b], auto)

   332

   333 lemma mult_pos_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; 0 \<le> b |] ==> 0 \<le> a*b"

   334 by (drule mult_left_mono [of 0 b], auto)

   335

   336 lemma mult_pos_neg: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> a*b < 0"

   337 by (drule mult_strict_left_mono [of b 0], auto)

   338

   339 lemma mult_pos_neg_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> a*b \<le> 0"

   340 by (drule mult_left_mono [of b 0], auto)

   341

   342 lemma mult_pos_neg2: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> b*a < 0"

   343 by (drule mult_strict_right_mono[of b 0], auto)

   344

   345 lemma mult_pos_neg2_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> b*a \<le> 0"

   346 by (drule mult_right_mono[of b 0], auto)

   347

   348 lemma mult_neg: "[| a < (0::'a::ordered_ring_strict); b < 0 |] ==> 0 < a*b"

   349 by (drule mult_strict_right_mono_neg, auto)

   350

   351 lemma mult_neg_le: "[| a \<le> (0::'a::pordered_ring); b \<le> 0 |] ==> 0 \<le> a*b"

   352 by (drule mult_right_mono_neg[of a 0 b ], auto)

   353

   354 lemma zero_less_mult_pos:

   355      "[| 0 < a*b; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"

   356 apply (case_tac "b\<le>0")

   357  apply (auto simp add: order_le_less linorder_not_less)

   358 apply (drule_tac mult_pos_neg [of a b])

   359  apply (auto dest: order_less_not_sym)

   360 done

   361

   362 lemma zero_less_mult_pos2:

   363      "[| 0 < b*a; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"

   364 apply (case_tac "b\<le>0")

   365  apply (auto simp add: order_le_less linorder_not_less)

   366 apply (drule_tac mult_pos_neg2 [of a b])

   367  apply (auto dest: order_less_not_sym)

   368 done

   369

   370 lemma zero_less_mult_iff:

   371      "((0::'a::ordered_ring_strict) < a*b) = (0 < a & 0 < b | a < 0 & b < 0)"

   372 apply (auto simp add: order_le_less linorder_not_less mult_pos mult_neg)

   373 apply (blast dest: zero_less_mult_pos)

   374 apply (blast dest: zero_less_mult_pos2)

   375 done

   376

   377 text{*A field has no "zero divisors", and this theorem holds without the

   378       assumption of an ordering.  See @{text field_mult_eq_0_iff} below.*}

   379 lemma mult_eq_0_iff [simp]: "(a*b = (0::'a::ordered_ring_strict)) = (a = 0 | b = 0)"

   380 apply (case_tac "a < 0")

   381 apply (auto simp add: linorder_not_less order_le_less linorder_neq_iff)

   382 apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono)+

   383 done

   384

   385 lemma zero_le_mult_iff:

   386      "((0::'a::ordered_ring_strict) \<le> a*b) = (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"

   387 by (auto simp add: eq_commute [of 0] order_le_less linorder_not_less

   388                    zero_less_mult_iff)

   389

   390 lemma mult_less_0_iff:

   391      "(a*b < (0::'a::ordered_ring_strict)) = (0 < a & b < 0 | a < 0 & 0 < b)"

   392 apply (insert zero_less_mult_iff [of "-a" b])

   393 apply (force simp add: minus_mult_left[symmetric])

   394 done

   395

   396 lemma mult_le_0_iff:

   397      "(a*b \<le> (0::'a::ordered_ring_strict)) = (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"

   398 apply (insert zero_le_mult_iff [of "-a" b])

   399 apply (force simp add: minus_mult_left[symmetric])

   400 done

   401

   402 lemma split_mult_pos_le: "(0 \<le> a & 0 \<le> b) | (a \<le> 0 & b \<le> 0) \<Longrightarrow> 0 \<le> a * (b::_::pordered_ring)"

   403 by (auto simp add: mult_pos_le mult_neg_le)

   404

   405 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)"

   406 by (auto simp add: mult_pos_neg_le mult_pos_neg2_le)

   407

   408 lemma zero_le_square: "(0::'a::ordered_ring_strict) \<le> a*a"

   409 by (simp add: zero_le_mult_iff linorder_linear)

   410

   411 text{*Proving axiom @{text zero_less_one} makes all @{text ordered_semidom}

   412       theorems available to members of @{term ordered_idom} *}

   413

   414 instance ordered_idom \<subseteq> ordered_semidom

   415 proof

   416   have "(0::'a) \<le> 1*1" by (rule zero_le_square)

   417   thus "(0::'a) < 1" by (simp add: order_le_less)

   418 qed

   419

   420 instance ordered_ring_strict \<subseteq> axclass_no_zero_divisors

   421 by (intro_classes, simp)

   422

   423 instance ordered_idom \<subseteq> idom ..

   424

   425 text{*All three types of comparision involving 0 and 1 are covered.*}

   426

   427 declare zero_neq_one [THEN not_sym, simp]

   428

   429 lemma zero_le_one [simp]: "(0::'a::ordered_semidom) \<le> 1"

   430   by (rule zero_less_one [THEN order_less_imp_le])

   431

   432 lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semidom) \<le> 0"

   433 by (simp add: linorder_not_le)

   434

   435 lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semidom) < 0"

   436 by (simp add: linorder_not_less)

   437

   438 subsection{*More Monotonicity*}

   439

   440 text{*Strict monotonicity in both arguments*}

   441 lemma mult_strict_mono:

   442      "[|a<b; c<d; 0<b; 0\<le>c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"

   443 apply (case_tac "c=0")

   444  apply (simp add: mult_pos)

   445 apply (erule mult_strict_right_mono [THEN order_less_trans])

   446  apply (force simp add: order_le_less)

   447 apply (erule mult_strict_left_mono, assumption)

   448 done

   449

   450 text{*This weaker variant has more natural premises*}

   451 lemma mult_strict_mono':

   452      "[| a<b; c<d; 0 \<le> a; 0 \<le> c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"

   453 apply (rule mult_strict_mono)

   454 apply (blast intro: order_le_less_trans)+

   455 done

   456

   457 lemma mult_mono:

   458      "[|a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c|]

   459       ==> a * c  \<le>  b * (d::'a::pordered_semiring)"

   460 apply (erule mult_right_mono [THEN order_trans], assumption)

   461 apply (erule mult_left_mono, assumption)

   462 done

   463

   464 lemma less_1_mult: "[| 1 < m; 1 < n |] ==> 1 < m*(n::'a::ordered_semidom)"

   465 apply (insert mult_strict_mono [of 1 m 1 n])

   466 apply (simp add:  order_less_trans [OF zero_less_one])

   467 done

   468

   469 subsection{*Cancellation Laws for Relationships With a Common Factor*}

   470

   471 text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},

   472    also with the relations @{text "\<le>"} and equality.*}

   473

   474 text{*These disjunction'' versions produce two cases when the comparison is

   475  an assumption, but effectively four when the comparison is a goal.*}

   476

   477 lemma mult_less_cancel_right_disj:

   478     "(a*c < b*c) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"

   479 apply (case_tac "c = 0")

   480 apply (auto simp add: linorder_neq_iff mult_strict_right_mono

   481                       mult_strict_right_mono_neg)

   482 apply (auto simp add: linorder_not_less

   483                       linorder_not_le [symmetric, of "a*c"]

   484                       linorder_not_le [symmetric, of a])

   485 apply (erule_tac [!] notE)

   486 apply (auto simp add: order_less_imp_le mult_right_mono

   487                       mult_right_mono_neg)

   488 done

   489

   490 lemma mult_less_cancel_left_disj:

   491     "(c*a < c*b) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"

   492 apply (case_tac "c = 0")

   493 apply (auto simp add: linorder_neq_iff mult_strict_left_mono

   494                       mult_strict_left_mono_neg)

   495 apply (auto simp add: linorder_not_less

   496                       linorder_not_le [symmetric, of "c*a"]

   497                       linorder_not_le [symmetric, of a])

   498 apply (erule_tac [!] notE)

   499 apply (auto simp add: order_less_imp_le mult_left_mono

   500                       mult_left_mono_neg)

   501 done

   502

   503

   504 text{*The conjunction of implication'' lemmas produce two cases when the

   505 comparison is a goal, but give four when the comparison is an assumption.*}

   506

   507 lemma mult_less_cancel_right:

   508   fixes c :: "'a :: ordered_ring_strict"

   509   shows      "(a*c < b*c) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))"

   510 by (insert mult_less_cancel_right_disj [of a c b], auto)

   511

   512 lemma mult_less_cancel_left:

   513   fixes c :: "'a :: ordered_ring_strict"

   514   shows      "(c*a < c*b) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))"

   515 by (insert mult_less_cancel_left_disj [of c a b], auto)

   516

   517 lemma mult_le_cancel_right:

   518      "(a*c \<le> b*c) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"

   519 by (simp add: linorder_not_less [symmetric] mult_less_cancel_right_disj)

   520

   521 lemma mult_le_cancel_left:

   522      "(c*a \<le> c*b) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"

   523 by (simp add: linorder_not_less [symmetric] mult_less_cancel_left_disj)

   524

   525 lemma mult_less_imp_less_left:

   526       assumes less: "c*a < c*b" and nonneg: "0 \<le> c"

   527       shows "a < (b::'a::ordered_semiring_strict)"

   528 proof (rule ccontr)

   529   assume "~ a < b"

   530   hence "b \<le> a" by (simp add: linorder_not_less)

   531   hence "c*b \<le> c*a" by (rule mult_left_mono)

   532   with this and less show False

   533     by (simp add: linorder_not_less [symmetric])

   534 qed

   535

   536 lemma mult_less_imp_less_right:

   537   assumes less: "a*c < b*c" and nonneg: "0 <= c"

   538   shows "a < (b::'a::ordered_semiring_strict)"

   539 proof (rule ccontr)

   540   assume "~ a < b"

   541   hence "b \<le> a" by (simp add: linorder_not_less)

   542   hence "b*c \<le> a*c" by (rule mult_right_mono)

   543   with this and less show False

   544     by (simp add: linorder_not_less [symmetric])

   545 qed

   546

   547 text{*Cancellation of equalities with a common factor*}

   548 lemma mult_cancel_right [simp]:

   549      "(a*c = b*c) = (c = (0::'a::ordered_ring_strict) | a=b)"

   550 apply (cut_tac linorder_less_linear [of 0 c])

   551 apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono

   552              simp add: linorder_neq_iff)

   553 done

   554

   555 text{*These cancellation theorems require an ordering. Versions are proved

   556       below that work for fields without an ordering.*}

   557 lemma mult_cancel_left [simp]:

   558      "(c*a = c*b) = (c = (0::'a::ordered_ring_strict) | a=b)"

   559 apply (cut_tac linorder_less_linear [of 0 c])

   560 apply (force dest: mult_strict_left_mono_neg mult_strict_left_mono

   561              simp add: linorder_neq_iff)

   562 done

   563

   564

   565 subsubsection{*Special Cancellation Simprules for Multiplication*}

   566

   567 text{*These also produce two cases when the comparison is a goal.*}

   568

   569 lemma mult_le_cancel_right1:

   570   fixes c :: "'a :: ordered_idom"

   571   shows "(c \<le> b*c) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))"

   572 by (insert mult_le_cancel_right [of 1 c b], simp)

   573

   574 lemma mult_le_cancel_right2:

   575   fixes c :: "'a :: ordered_idom"

   576   shows "(a*c \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))"

   577 by (insert mult_le_cancel_right [of a c 1], simp)

   578

   579 lemma mult_le_cancel_left1:

   580   fixes c :: "'a :: ordered_idom"

   581   shows "(c \<le> c*b) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))"

   582 by (insert mult_le_cancel_left [of c 1 b], simp)

   583

   584 lemma mult_le_cancel_left2:

   585   fixes c :: "'a :: ordered_idom"

   586   shows "(c*a \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))"

   587 by (insert mult_le_cancel_left [of c a 1], simp)

   588

   589 lemma mult_less_cancel_right1:

   590   fixes c :: "'a :: ordered_idom"

   591   shows "(c < b*c) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))"

   592 by (insert mult_less_cancel_right [of 1 c b], simp)

   593

   594 lemma mult_less_cancel_right2:

   595   fixes c :: "'a :: ordered_idom"

   596   shows "(a*c < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))"

   597 by (insert mult_less_cancel_right [of a c 1], simp)

   598

   599 lemma mult_less_cancel_left1:

   600   fixes c :: "'a :: ordered_idom"

   601   shows "(c < c*b) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))"

   602 by (insert mult_less_cancel_left [of c 1 b], simp)

   603

   604 lemma mult_less_cancel_left2:

   605   fixes c :: "'a :: ordered_idom"

   606   shows "(c*a < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))"

   607 by (insert mult_less_cancel_left [of c a 1], simp)

   608

   609 lemma mult_cancel_right1 [simp]:

   610 fixes c :: "'a :: ordered_idom"

   611   shows "(c = b*c) = (c = 0 | b=1)"

   612 by (insert mult_cancel_right [of 1 c b], force)

   613

   614 lemma mult_cancel_right2 [simp]:

   615 fixes c :: "'a :: ordered_idom"

   616   shows "(a*c = c) = (c = 0 | a=1)"

   617 by (insert mult_cancel_right [of a c 1], simp)

   618

   619 lemma mult_cancel_left1 [simp]:

   620 fixes c :: "'a :: ordered_idom"

   621   shows "(c = c*b) = (c = 0 | b=1)"

   622 by (insert mult_cancel_left [of c 1 b], force)

   623

   624 lemma mult_cancel_left2 [simp]:

   625 fixes c :: "'a :: ordered_idom"

   626   shows "(c*a = c) = (c = 0 | a=1)"

   627 by (insert mult_cancel_left [of c a 1], simp)

   628

   629

   630 text{*Simprules for comparisons where common factors can be cancelled.*}

   631 lemmas mult_compare_simps =

   632     mult_le_cancel_right mult_le_cancel_left

   633     mult_le_cancel_right1 mult_le_cancel_right2

   634     mult_le_cancel_left1 mult_le_cancel_left2

   635     mult_less_cancel_right mult_less_cancel_left

   636     mult_less_cancel_right1 mult_less_cancel_right2

   637     mult_less_cancel_left1 mult_less_cancel_left2

   638     mult_cancel_right mult_cancel_left

   639     mult_cancel_right1 mult_cancel_right2

   640     mult_cancel_left1 mult_cancel_left2

   641

   642

   643 text{*This list of rewrites decides ring equalities by ordered rewriting.*}

   644 lemmas ring_eq_simps =

   645 (*  mult_ac*)

   646   left_distrib right_distrib left_diff_distrib right_diff_distrib

   647   group_eq_simps

   648 (*  add_ac

   649   add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2

   650   diff_eq_eq eq_diff_eq *)

   651

   652 subsection {* Fields *}

   653

   654 lemma right_inverse [simp]:

   655       assumes not0: "a \<noteq> 0" shows "a * inverse (a::'a::field) = 1"

   656 proof -

   657   have "a * inverse a = inverse a * a" by (simp add: mult_ac)

   658   also have "... = 1" using not0 by simp

   659   finally show ?thesis .

   660 qed

   661

   662 lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))"

   663 proof

   664   assume neq: "b \<noteq> 0"

   665   {

   666     hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac)

   667     also assume "a / b = 1"

   668     finally show "a = b" by simp

   669   next

   670     assume "a = b"

   671     with neq show "a / b = 1" by (simp add: divide_inverse)

   672   }

   673 qed

   674

   675 lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a"

   676 by (simp add: divide_inverse)

   677

   678 lemma divide_self: "a \<noteq> 0 ==> a / (a::'a::field) = 1"

   679   by (simp add: divide_inverse)

   680

   681 lemma divide_zero [simp]: "a / 0 = (0::'a::{field,division_by_zero})"

   682 by (simp add: divide_inverse)

   683

   684 lemma divide_self_if [simp]:

   685      "a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)"

   686   by (simp add: divide_self)

   687

   688 lemma divide_zero_left [simp]: "0/a = (0::'a::field)"

   689 by (simp add: divide_inverse)

   690

   691 lemma inverse_eq_divide: "inverse (a::'a::field) = 1/a"

   692 by (simp add: divide_inverse)

   693

   694 lemma add_divide_distrib: "(a+b)/(c::'a::field) = a/c + b/c"

   695 by (simp add: divide_inverse left_distrib)

   696

   697

   698 text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement

   699       of an ordering.*}

   700 lemma field_mult_eq_0_iff [simp]: "(a*b = (0::'a::field)) = (a = 0 | b = 0)"

   701 proof cases

   702   assume "a=0" thus ?thesis by simp

   703 next

   704   assume anz [simp]: "a\<noteq>0"

   705   { assume "a * b = 0"

   706     hence "inverse a * (a * b) = 0" by simp

   707     hence "b = 0"  by (simp (no_asm_use) add: mult_assoc [symmetric])}

   708   thus ?thesis by force

   709 qed

   710

   711 text{*Cancellation of equalities with a common factor*}

   712 lemma field_mult_cancel_right_lemma:

   713       assumes cnz: "c \<noteq> (0::'a::field)"

   714 	  and eq:  "a*c = b*c"

   715 	 shows "a=b"

   716 proof -

   717   have "(a * c) * inverse c = (b * c) * inverse c"

   718     by (simp add: eq)

   719   thus "a=b"

   720     by (simp add: mult_assoc cnz)

   721 qed

   722

   723 lemma field_mult_cancel_right [simp]:

   724      "(a*c = b*c) = (c = (0::'a::field) | a=b)"

   725 proof cases

   726   assume "c=0" thus ?thesis by simp

   727 next

   728   assume "c\<noteq>0"

   729   thus ?thesis by (force dest: field_mult_cancel_right_lemma)

   730 qed

   731

   732 lemma field_mult_cancel_left [simp]:

   733      "(c*a = c*b) = (c = (0::'a::field) | a=b)"

   734   by (simp add: mult_commute [of c] field_mult_cancel_right)

   735

   736 lemma nonzero_imp_inverse_nonzero: "a \<noteq> 0 ==> inverse a \<noteq> (0::'a::field)"

   737 proof

   738   assume ianz: "inverse a = 0"

   739   assume "a \<noteq> 0"

   740   hence "1 = a * inverse a" by simp

   741   also have "... = 0" by (simp add: ianz)

   742   finally have "1 = (0::'a::field)" .

   743   thus False by (simp add: eq_commute)

   744 qed

   745

   746

   747 subsection{*Basic Properties of @{term inverse}*}

   748

   749 lemma inverse_zero_imp_zero: "inverse a = 0 ==> a = (0::'a::field)"

   750 apply (rule ccontr)

   751 apply (blast dest: nonzero_imp_inverse_nonzero)

   752 done

   753

   754 lemma inverse_nonzero_imp_nonzero:

   755    "inverse a = 0 ==> a = (0::'a::field)"

   756 apply (rule ccontr)

   757 apply (blast dest: nonzero_imp_inverse_nonzero)

   758 done

   759

   760 lemma inverse_nonzero_iff_nonzero [simp]:

   761    "(inverse a = 0) = (a = (0::'a::{field,division_by_zero}))"

   762 by (force dest: inverse_nonzero_imp_nonzero)

   763

   764 lemma nonzero_inverse_minus_eq:

   765       assumes [simp]: "a\<noteq>0"  shows "inverse(-a) = -inverse(a::'a::field)"

   766 proof -

   767   have "-a * inverse (- a) = -a * - inverse a"

   768     by simp

   769   thus ?thesis

   770     by (simp only: field_mult_cancel_left, simp)

   771 qed

   772

   773 lemma inverse_minus_eq [simp]:

   774    "inverse(-a) = -inverse(a::'a::{field,division_by_zero})";

   775 proof cases

   776   assume "a=0" thus ?thesis by (simp add: inverse_zero)

   777 next

   778   assume "a\<noteq>0"

   779   thus ?thesis by (simp add: nonzero_inverse_minus_eq)

   780 qed

   781

   782 lemma nonzero_inverse_eq_imp_eq:

   783       assumes inveq: "inverse a = inverse b"

   784 	  and anz:  "a \<noteq> 0"

   785 	  and bnz:  "b \<noteq> 0"

   786 	 shows "a = (b::'a::field)"

   787 proof -

   788   have "a * inverse b = a * inverse a"

   789     by (simp add: inveq)

   790   hence "(a * inverse b) * b = (a * inverse a) * b"

   791     by simp

   792   thus "a = b"

   793     by (simp add: mult_assoc anz bnz)

   794 qed

   795

   796 lemma inverse_eq_imp_eq:

   797      "inverse a = inverse b ==> a = (b::'a::{field,division_by_zero})"

   798 apply (case_tac "a=0 | b=0")

   799  apply (force dest!: inverse_zero_imp_zero

   800               simp add: eq_commute [of "0::'a"])

   801 apply (force dest!: nonzero_inverse_eq_imp_eq)

   802 done

   803

   804 lemma inverse_eq_iff_eq [simp]:

   805      "(inverse a = inverse b) = (a = (b::'a::{field,division_by_zero}))"

   806 by (force dest!: inverse_eq_imp_eq)

   807

   808 lemma nonzero_inverse_inverse_eq:

   809       assumes [simp]: "a \<noteq> 0"  shows "inverse(inverse (a::'a::field)) = a"

   810   proof -

   811   have "(inverse (inverse a) * inverse a) * a = a"

   812     by (simp add: nonzero_imp_inverse_nonzero)

   813   thus ?thesis

   814     by (simp add: mult_assoc)

   815   qed

   816

   817 lemma inverse_inverse_eq [simp]:

   818      "inverse(inverse (a::'a::{field,division_by_zero})) = a"

   819   proof cases

   820     assume "a=0" thus ?thesis by simp

   821   next

   822     assume "a\<noteq>0"

   823     thus ?thesis by (simp add: nonzero_inverse_inverse_eq)

   824   qed

   825

   826 lemma inverse_1 [simp]: "inverse 1 = (1::'a::field)"

   827   proof -

   828   have "inverse 1 * 1 = (1::'a::field)"

   829     by (rule left_inverse [OF zero_neq_one [symmetric]])

   830   thus ?thesis  by simp

   831   qed

   832

   833 lemma inverse_unique:

   834   assumes ab: "a*b = 1"

   835   shows "inverse a = (b::'a::field)"

   836 proof -

   837   have "a \<noteq> 0" using ab by auto

   838   moreover have "inverse a * (a * b) = inverse a" by (simp add: ab)

   839   ultimately show ?thesis by (simp add: mult_assoc [symmetric])

   840 qed

   841

   842 lemma nonzero_inverse_mult_distrib:

   843       assumes anz: "a \<noteq> 0"

   844           and bnz: "b \<noteq> 0"

   845       shows "inverse(a*b) = inverse(b) * inverse(a::'a::field)"

   846   proof -

   847   have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)"

   848     by (simp add: field_mult_eq_0_iff anz bnz)

   849   hence "inverse(a*b) * a = inverse(b)"

   850     by (simp add: mult_assoc bnz)

   851   hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)"

   852     by simp

   853   thus ?thesis

   854     by (simp add: mult_assoc anz)

   855   qed

   856

   857 text{*This version builds in division by zero while also re-orienting

   858       the right-hand side.*}

   859 lemma inverse_mult_distrib [simp]:

   860      "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"

   861   proof cases

   862     assume "a \<noteq> 0 & b \<noteq> 0"

   863     thus ?thesis  by (simp add: nonzero_inverse_mult_distrib mult_commute)

   864   next

   865     assume "~ (a \<noteq> 0 & b \<noteq> 0)"

   866     thus ?thesis  by force

   867   qed

   868

   869 text{*There is no slick version using division by zero.*}

   870 lemma inverse_add:

   871      "[|a \<noteq> 0;  b \<noteq> 0|]

   872       ==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)"

   873 apply (simp add: left_distrib mult_assoc)

   874 apply (simp add: mult_commute [of "inverse a"])

   875 apply (simp add: mult_assoc [symmetric] add_commute)

   876 done

   877

   878 lemma inverse_divide [simp]:

   879       "inverse (a/b) = b / (a::'a::{field,division_by_zero})"

   880   by (simp add: divide_inverse mult_commute)

   881

   882 lemma nonzero_mult_divide_cancel_left:

   883   assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0"

   884     shows "(c*a)/(c*b) = a/(b::'a::field)"

   885 proof -

   886   have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"

   887     by (simp add: field_mult_eq_0_iff divide_inverse

   888                   nonzero_inverse_mult_distrib)

   889   also have "... =  a * inverse b * (inverse c * c)"

   890     by (simp only: mult_ac)

   891   also have "... =  a * inverse b"

   892     by simp

   893     finally show ?thesis

   894     by (simp add: divide_inverse)

   895 qed

   896

   897 lemma mult_divide_cancel_left:

   898      "c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"

   899 apply (case_tac "b = 0")

   900 apply (simp_all add: nonzero_mult_divide_cancel_left)

   901 done

   902

   903 lemma nonzero_mult_divide_cancel_right:

   904      "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)"

   905 by (simp add: mult_commute [of _ c] nonzero_mult_divide_cancel_left)

   906

   907 lemma mult_divide_cancel_right:

   908      "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"

   909 apply (case_tac "b = 0")

   910 apply (simp_all add: nonzero_mult_divide_cancel_right)

   911 done

   912

   913 (*For ExtractCommonTerm*)

   914 lemma mult_divide_cancel_eq_if:

   915      "(c*a) / (c*b) =

   916       (if c=0 then 0 else a / (b::'a::{field,division_by_zero}))"

   917   by (simp add: mult_divide_cancel_left)

   918

   919 lemma divide_1 [simp]: "a/1 = (a::'a::field)"

   920   by (simp add: divide_inverse)

   921

   922 lemma times_divide_eq_right: "a * (b/c) = (a*b) / (c::'a::field)"

   923 by (simp add: divide_inverse mult_assoc)

   924

   925 lemma times_divide_eq_left: "(b/c) * a = (b*a) / (c::'a::field)"

   926 by (simp add: divide_inverse mult_ac)

   927

   928 lemma divide_divide_eq_right [simp]:

   929      "a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"

   930 by (simp add: divide_inverse mult_ac)

   931

   932 lemma divide_divide_eq_left [simp]:

   933      "(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"

   934 by (simp add: divide_inverse mult_assoc)

   935

   936

   937 subsubsection{*Special Cancellation Simprules for Division*}

   938

   939 lemma mult_divide_cancel_left_if [simp]:

   940   fixes c :: "'a :: {field,division_by_zero}"

   941   shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)"

   942 by (simp add: mult_divide_cancel_left)

   943

   944 lemma mult_divide_cancel_right_if [simp]:

   945   fixes c :: "'a :: {field,division_by_zero}"

   946   shows "(a*c) / (b*c) = (if c=0 then 0 else a/b)"

   947 by (simp add: mult_divide_cancel_right)

   948

   949 lemma mult_divide_cancel_left_if1 [simp]:

   950   fixes c :: "'a :: {field,division_by_zero}"

   951   shows "c / (c*b) = (if c=0 then 0 else 1/b)"

   952 apply (insert mult_divide_cancel_left_if [of c 1 b])

   953 apply (simp del: mult_divide_cancel_left_if)

   954 done

   955

   956 lemma mult_divide_cancel_left_if2 [simp]:

   957   fixes c :: "'a :: {field,division_by_zero}"

   958   shows "(c*a) / c = (if c=0 then 0 else a)"

   959 apply (insert mult_divide_cancel_left_if [of c a 1])

   960 apply (simp del: mult_divide_cancel_left_if)

   961 done

   962

   963 lemma mult_divide_cancel_right_if1 [simp]:

   964   fixes c :: "'a :: {field,division_by_zero}"

   965   shows "c / (b*c) = (if c=0 then 0 else 1/b)"

   966 apply (insert mult_divide_cancel_right_if [of 1 c b])

   967 apply (simp del: mult_divide_cancel_right_if)

   968 done

   969

   970 lemma mult_divide_cancel_right_if2 [simp]:

   971   fixes c :: "'a :: {field,division_by_zero}"

   972   shows "(a*c) / c = (if c=0 then 0 else a)"

   973 apply (insert mult_divide_cancel_right_if [of a c 1])

   974 apply (simp del: mult_divide_cancel_right_if)

   975 done

   976

   977 text{*Two lemmas for cancelling the denominator*}

   978

   979 lemma times_divide_self_right [simp]:

   980   fixes a :: "'a :: {field,division_by_zero}"

   981   shows "a * (b/a) = (if a=0 then 0 else b)"

   982 by (simp add: times_divide_eq_right)

   983

   984 lemma times_divide_self_left [simp]:

   985   fixes a :: "'a :: {field,division_by_zero}"

   986   shows "(b/a) * a = (if a=0 then 0 else b)"

   987 by (simp add: times_divide_eq_left)

   988

   989

   990 subsection {* Division and Unary Minus *}

   991

   992 lemma nonzero_minus_divide_left: "b \<noteq> 0 ==> - (a/b) = (-a) / (b::'a::field)"

   993 by (simp add: divide_inverse minus_mult_left)

   994

   995 lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a/b) = a / -(b::'a::field)"

   996 by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right)

   997

   998 lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a)/(-b) = a / (b::'a::field)"

   999 by (simp add: divide_inverse nonzero_inverse_minus_eq)

  1000

  1001 lemma minus_divide_left: "- (a/b) = (-a) / (b::'a::field)"

  1002 by (simp add: divide_inverse minus_mult_left [symmetric])

  1003

  1004 lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"

  1005 by (simp add: divide_inverse minus_mult_right [symmetric])

  1006

  1007

  1008 text{*The effect is to extract signs from divisions*}

  1009 declare minus_divide_left  [symmetric, simp]

  1010 declare minus_divide_right [symmetric, simp]

  1011

  1012 text{*Also, extract signs from products*}

  1013 declare minus_mult_left [symmetric, simp]

  1014 declare minus_mult_right [symmetric, simp]

  1015

  1016 lemma minus_divide_divide [simp]:

  1017      "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"

  1018 apply (case_tac "b=0", simp)

  1019 apply (simp add: nonzero_minus_divide_divide)

  1020 done

  1021

  1022 lemma diff_divide_distrib: "(a-b)/(c::'a::field) = a/c - b/c"

  1023 by (simp add: diff_minus add_divide_distrib)

  1024

  1025

  1026 subsection {* Ordered Fields *}

  1027

  1028 lemma positive_imp_inverse_positive:

  1029       assumes a_gt_0: "0 < a"  shows "0 < inverse (a::'a::ordered_field)"

  1030   proof -

  1031   have "0 < a * inverse a"

  1032     by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one)

  1033   thus "0 < inverse a"

  1034     by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff)

  1035   qed

  1036

  1037 lemma negative_imp_inverse_negative:

  1038      "a < 0 ==> inverse a < (0::'a::ordered_field)"

  1039   by (insert positive_imp_inverse_positive [of "-a"],

  1040       simp add: nonzero_inverse_minus_eq order_less_imp_not_eq)

  1041

  1042 lemma inverse_le_imp_le:

  1043       assumes invle: "inverse a \<le> inverse b"

  1044 	  and apos:  "0 < a"

  1045 	 shows "b \<le> (a::'a::ordered_field)"

  1046   proof (rule classical)

  1047   assume "~ b \<le> a"

  1048   hence "a < b"

  1049     by (simp add: linorder_not_le)

  1050   hence bpos: "0 < b"

  1051     by (blast intro: apos order_less_trans)

  1052   hence "a * inverse a \<le> a * inverse b"

  1053     by (simp add: apos invle order_less_imp_le mult_left_mono)

  1054   hence "(a * inverse a) * b \<le> (a * inverse b) * b"

  1055     by (simp add: bpos order_less_imp_le mult_right_mono)

  1056   thus "b \<le> a"

  1057     by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)

  1058   qed

  1059

  1060 lemma inverse_positive_imp_positive:

  1061       assumes inv_gt_0: "0 < inverse a"

  1062           and [simp]:   "a \<noteq> 0"

  1063         shows "0 < (a::'a::ordered_field)"

  1064   proof -

  1065   have "0 < inverse (inverse a)"

  1066     by (rule positive_imp_inverse_positive)

  1067   thus "0 < a"

  1068     by (simp add: nonzero_inverse_inverse_eq)

  1069   qed

  1070

  1071 lemma inverse_positive_iff_positive [simp]:

  1072       "(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))"

  1073 apply (case_tac "a = 0", simp)

  1074 apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)

  1075 done

  1076

  1077 lemma inverse_negative_imp_negative:

  1078       assumes inv_less_0: "inverse a < 0"

  1079           and [simp]:   "a \<noteq> 0"

  1080         shows "a < (0::'a::ordered_field)"

  1081   proof -

  1082   have "inverse (inverse a) < 0"

  1083     by (rule negative_imp_inverse_negative)

  1084   thus "a < 0"

  1085     by (simp add: nonzero_inverse_inverse_eq)

  1086   qed

  1087

  1088 lemma inverse_negative_iff_negative [simp]:

  1089       "(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))"

  1090 apply (case_tac "a = 0", simp)

  1091 apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)

  1092 done

  1093

  1094 lemma inverse_nonnegative_iff_nonnegative [simp]:

  1095       "(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))"

  1096 by (simp add: linorder_not_less [symmetric])

  1097

  1098 lemma inverse_nonpositive_iff_nonpositive [simp]:

  1099       "(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))"

  1100 by (simp add: linorder_not_less [symmetric])

  1101

  1102

  1103 subsection{*Anti-Monotonicity of @{term inverse}*}

  1104

  1105 lemma less_imp_inverse_less:

  1106       assumes less: "a < b"

  1107 	  and apos:  "0 < a"

  1108 	shows "inverse b < inverse (a::'a::ordered_field)"

  1109   proof (rule ccontr)

  1110   assume "~ inverse b < inverse a"

  1111   hence "inverse a \<le> inverse b"

  1112     by (simp add: linorder_not_less)

  1113   hence "~ (a < b)"

  1114     by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos])

  1115   thus False

  1116     by (rule notE [OF _ less])

  1117   qed

  1118

  1119 lemma inverse_less_imp_less:

  1120    "[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::ordered_field)"

  1121 apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"])

  1122 apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq)

  1123 done

  1124

  1125 text{*Both premises are essential. Consider -1 and 1.*}

  1126 lemma inverse_less_iff_less [simp]:

  1127      "[|0 < a; 0 < b|]

  1128       ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"

  1129 by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less)

  1130

  1131 lemma le_imp_inverse_le:

  1132    "[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::ordered_field)"

  1133   by (force simp add: order_le_less less_imp_inverse_less)

  1134

  1135 lemma inverse_le_iff_le [simp]:

  1136      "[|0 < a; 0 < b|]

  1137       ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"

  1138 by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le)

  1139

  1140

  1141 text{*These results refer to both operands being negative.  The opposite-sign

  1142 case is trivial, since inverse preserves signs.*}

  1143 lemma inverse_le_imp_le_neg:

  1144    "[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::ordered_field)"

  1145   apply (rule classical)

  1146   apply (subgoal_tac "a < 0")

  1147    prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans)

  1148   apply (insert inverse_le_imp_le [of "-b" "-a"])

  1149   apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq)

  1150   done

  1151

  1152 lemma less_imp_inverse_less_neg:

  1153    "[|a < b; b < 0|] ==> inverse b < inverse (a::'a::ordered_field)"

  1154   apply (subgoal_tac "a < 0")

  1155    prefer 2 apply (blast intro: order_less_trans)

  1156   apply (insert less_imp_inverse_less [of "-b" "-a"])

  1157   apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq)

  1158   done

  1159

  1160 lemma inverse_less_imp_less_neg:

  1161    "[|inverse a < inverse b; b < 0|] ==> b < (a::'a::ordered_field)"

  1162   apply (rule classical)

  1163   apply (subgoal_tac "a < 0")

  1164    prefer 2

  1165    apply (force simp add: linorder_not_less intro: order_le_less_trans)

  1166   apply (insert inverse_less_imp_less [of "-b" "-a"])

  1167   apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq)

  1168   done

  1169

  1170 lemma inverse_less_iff_less_neg [simp]:

  1171      "[|a < 0; b < 0|]

  1172       ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"

  1173   apply (insert inverse_less_iff_less [of "-b" "-a"])

  1174   apply (simp del: inverse_less_iff_less

  1175 	      add: order_less_imp_not_eq nonzero_inverse_minus_eq)

  1176   done

  1177

  1178 lemma le_imp_inverse_le_neg:

  1179    "[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)"

  1180   by (force simp add: order_le_less less_imp_inverse_less_neg)

  1181

  1182 lemma inverse_le_iff_le_neg [simp]:

  1183      "[|a < 0; b < 0|]

  1184       ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"

  1185 by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg)

  1186

  1187

  1188 subsection{*Inverses and the Number One*}

  1189

  1190 lemma one_less_inverse_iff:

  1191     "(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"proof cases

  1192   assume "0 < x"

  1193     with inverse_less_iff_less [OF zero_less_one, of x]

  1194     show ?thesis by simp

  1195 next

  1196   assume notless: "~ (0 < x)"

  1197   have "~ (1 < inverse x)"

  1198   proof

  1199     assume "1 < inverse x"

  1200     also with notless have "... \<le> 0" by (simp add: linorder_not_less)

  1201     also have "... < 1" by (rule zero_less_one)

  1202     finally show False by auto

  1203   qed

  1204   with notless show ?thesis by simp

  1205 qed

  1206

  1207 lemma inverse_eq_1_iff [simp]:

  1208     "(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))"

  1209 by (insert inverse_eq_iff_eq [of x 1], simp)

  1210

  1211 lemma one_le_inverse_iff:

  1212    "(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))"

  1213 by (force simp add: order_le_less one_less_inverse_iff zero_less_one

  1214                     eq_commute [of 1])

  1215

  1216 lemma inverse_less_1_iff:

  1217    "(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{ordered_field,division_by_zero}))"

  1218 by (simp add: linorder_not_le [symmetric] one_le_inverse_iff)

  1219

  1220 lemma inverse_le_1_iff:

  1221    "(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{ordered_field,division_by_zero}))"

  1222 by (simp add: linorder_not_less [symmetric] one_less_inverse_iff)

  1223

  1224

  1225 subsection{*Division and Signs*}

  1226

  1227 lemma zero_less_divide_iff:

  1228      "((0::'a::{ordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"

  1229 by (simp add: divide_inverse zero_less_mult_iff)

  1230

  1231 lemma divide_less_0_iff:

  1232      "(a/b < (0::'a::{ordered_field,division_by_zero})) =

  1233       (0 < a & b < 0 | a < 0 & 0 < b)"

  1234 by (simp add: divide_inverse mult_less_0_iff)

  1235

  1236 lemma zero_le_divide_iff:

  1237      "((0::'a::{ordered_field,division_by_zero}) \<le> a/b) =

  1238       (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"

  1239 by (simp add: divide_inverse zero_le_mult_iff)

  1240

  1241 lemma divide_le_0_iff:

  1242      "(a/b \<le> (0::'a::{ordered_field,division_by_zero})) =

  1243       (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"

  1244 by (simp add: divide_inverse mult_le_0_iff)

  1245

  1246 lemma divide_eq_0_iff [simp]:

  1247      "(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))"

  1248 by (simp add: divide_inverse field_mult_eq_0_iff)

  1249

  1250

  1251 subsection{*Simplification of Inequalities Involving Literal Divisors*}

  1252

  1253 lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"

  1254 proof -

  1255   assume less: "0<c"

  1256   hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"

  1257     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])

  1258   also have "... = (a*c \<le> b)"

  1259     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc)

  1260   finally show ?thesis .

  1261 qed

  1262

  1263

  1264 lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"

  1265 proof -

  1266   assume less: "c<0"

  1267   hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"

  1268     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])

  1269   also have "... = (b \<le> a*c)"

  1270     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc)

  1271   finally show ?thesis .

  1272 qed

  1273

  1274 lemma le_divide_eq:

  1275   "(a \<le> b/c) =

  1276    (if 0 < c then a*c \<le> b

  1277              else if c < 0 then b \<le> a*c

  1278              else  a \<le> (0::'a::{ordered_field,division_by_zero}))"

  1279 apply (case_tac "c=0", simp)

  1280 apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff)

  1281 done

  1282

  1283 lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)"

  1284 proof -

  1285   assume less: "0<c"

  1286   hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"

  1287     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])

  1288   also have "... = (b \<le> a*c)"

  1289     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc)

  1290   finally show ?thesis .

  1291 qed

  1292

  1293 lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)"

  1294 proof -

  1295   assume less: "c<0"

  1296   hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"

  1297     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])

  1298   also have "... = (a*c \<le> b)"

  1299     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc)

  1300   finally show ?thesis .

  1301 qed

  1302

  1303 lemma divide_le_eq:

  1304   "(b/c \<le> a) =

  1305    (if 0 < c then b \<le> a*c

  1306              else if c < 0 then a*c \<le> b

  1307              else 0 \<le> (a::'a::{ordered_field,division_by_zero}))"

  1308 apply (case_tac "c=0", simp)

  1309 apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff)

  1310 done

  1311

  1312

  1313 lemma pos_less_divide_eq:

  1314      "0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)"

  1315 proof -

  1316   assume less: "0<c"

  1317   hence "(a < b/c) = (a*c < (b/c)*c)"

  1318     by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])

  1319   also have "... = (a*c < b)"

  1320     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc)

  1321   finally show ?thesis .

  1322 qed

  1323

  1324 lemma neg_less_divide_eq:

  1325  "c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)"

  1326 proof -

  1327   assume less: "c<0"

  1328   hence "(a < b/c) = ((b/c)*c < a*c)"

  1329     by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])

  1330   also have "... = (b < a*c)"

  1331     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc)

  1332   finally show ?thesis .

  1333 qed

  1334

  1335 lemma less_divide_eq:

  1336   "(a < b/c) =

  1337    (if 0 < c then a*c < b

  1338              else if c < 0 then b < a*c

  1339              else  a < (0::'a::{ordered_field,division_by_zero}))"

  1340 apply (case_tac "c=0", simp)

  1341 apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff)

  1342 done

  1343

  1344 lemma pos_divide_less_eq:

  1345      "0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)"

  1346 proof -

  1347   assume less: "0<c"

  1348   hence "(b/c < a) = ((b/c)*c < a*c)"

  1349     by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])

  1350   also have "... = (b < a*c)"

  1351     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc)

  1352   finally show ?thesis .

  1353 qed

  1354

  1355 lemma neg_divide_less_eq:

  1356  "c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)"

  1357 proof -

  1358   assume less: "c<0"

  1359   hence "(b/c < a) = (a*c < (b/c)*c)"

  1360     by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])

  1361   also have "... = (a*c < b)"

  1362     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc)

  1363   finally show ?thesis .

  1364 qed

  1365

  1366 lemma divide_less_eq:

  1367   "(b/c < a) =

  1368    (if 0 < c then b < a*c

  1369              else if c < 0 then a*c < b

  1370              else 0 < (a::'a::{ordered_field,division_by_zero}))"

  1371 apply (case_tac "c=0", simp)

  1372 apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff)

  1373 done

  1374

  1375 lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)"

  1376 proof -

  1377   assume [simp]: "c\<noteq>0"

  1378   have "(a = b/c) = (a*c = (b/c)*c)"

  1379     by (simp add: field_mult_cancel_right)

  1380   also have "... = (a*c = b)"

  1381     by (simp add: divide_inverse mult_assoc)

  1382   finally show ?thesis .

  1383 qed

  1384

  1385 lemma eq_divide_eq:

  1386   "((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)"

  1387 by (simp add: nonzero_eq_divide_eq)

  1388

  1389 lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)"

  1390 proof -

  1391   assume [simp]: "c\<noteq>0"

  1392   have "(b/c = a) = ((b/c)*c = a*c)"

  1393     by (simp add: field_mult_cancel_right)

  1394   also have "... = (b = a*c)"

  1395     by (simp add: divide_inverse mult_assoc)

  1396   finally show ?thesis .

  1397 qed

  1398

  1399 lemma divide_eq_eq:

  1400   "(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)"

  1401 by (force simp add: nonzero_divide_eq_eq)

  1402

  1403

  1404 subsection{*Cancellation Laws for Division*}

  1405

  1406 lemma divide_cancel_right [simp]:

  1407      "(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))"

  1408 apply (case_tac "c=0", simp)

  1409 apply (simp add: divide_inverse field_mult_cancel_right)

  1410 done

  1411

  1412 lemma divide_cancel_left [simp]:

  1413      "(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))"

  1414 apply (case_tac "c=0", simp)

  1415 apply (simp add: divide_inverse field_mult_cancel_left)

  1416 done

  1417

  1418 subsection {* Division and the Number One *}

  1419

  1420 text{*Simplify expressions equated with 1*}

  1421 lemma divide_eq_1_iff [simp]:

  1422      "(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"

  1423 apply (case_tac "b=0", simp)

  1424 apply (simp add: right_inverse_eq)

  1425 done

  1426

  1427

  1428 lemma one_eq_divide_iff [simp]:

  1429      "(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"

  1430 by (simp add: eq_commute [of 1])

  1431

  1432 lemma zero_eq_1_divide_iff [simp]:

  1433      "((0::'a::{ordered_field,division_by_zero}) = 1/a) = (a = 0)"

  1434 apply (case_tac "a=0", simp)

  1435 apply (auto simp add: nonzero_eq_divide_eq)

  1436 done

  1437

  1438 lemma one_divide_eq_0_iff [simp]:

  1439      "(1/a = (0::'a::{ordered_field,division_by_zero})) = (a = 0)"

  1440 apply (case_tac "a=0", simp)

  1441 apply (insert zero_neq_one [THEN not_sym])

  1442 apply (auto simp add: nonzero_divide_eq_eq)

  1443 done

  1444

  1445 text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}

  1446 declare zero_less_divide_iff [of "1", simp]

  1447 declare divide_less_0_iff [of "1", simp]

  1448 declare zero_le_divide_iff [of "1", simp]

  1449 declare divide_le_0_iff [of "1", simp]

  1450

  1451

  1452 subsection {* Ordering Rules for Division *}

  1453

  1454 lemma divide_strict_right_mono:

  1455      "[|a < b; 0 < c|] ==> a / c < b / (c::'a::ordered_field)"

  1456 by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono

  1457               positive_imp_inverse_positive)

  1458

  1459 lemma divide_right_mono:

  1460      "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{ordered_field,division_by_zero})"

  1461   by (force simp add: divide_strict_right_mono order_le_less)

  1462

  1463

  1464 text{*The last premise ensures that @{term a} and @{term b}

  1465       have the same sign*}

  1466 lemma divide_strict_left_mono:

  1467        "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"

  1468 by (force simp add: zero_less_mult_iff divide_inverse mult_strict_left_mono

  1469       order_less_imp_not_eq order_less_imp_not_eq2

  1470       less_imp_inverse_less less_imp_inverse_less_neg)

  1471

  1472 lemma divide_left_mono:

  1473      "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::ordered_field)"

  1474   apply (subgoal_tac "a \<noteq> 0 & b \<noteq> 0")

  1475    prefer 2

  1476    apply (force simp add: zero_less_mult_iff order_less_imp_not_eq)

  1477   apply (case_tac "c=0", simp add: divide_inverse)

  1478   apply (force simp add: divide_strict_left_mono order_le_less)

  1479   done

  1480

  1481 lemma divide_strict_left_mono_neg:

  1482      "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"

  1483   apply (subgoal_tac "a \<noteq> 0 & b \<noteq> 0")

  1484    prefer 2

  1485    apply (force simp add: zero_less_mult_iff order_less_imp_not_eq)

  1486   apply (drule divide_strict_left_mono [of _ _ "-c"])

  1487    apply (simp_all add: mult_commute nonzero_minus_divide_left [symmetric])

  1488   done

  1489

  1490 lemma divide_strict_right_mono_neg:

  1491      "[|b < a; c < 0|] ==> a / c < b / (c::'a::ordered_field)"

  1492 apply (drule divide_strict_right_mono [of _ _ "-c"], simp)

  1493 apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric])

  1494 done

  1495

  1496

  1497 subsection {* Ordered Fields are Dense *}

  1498

  1499 lemma less_add_one: "a < (a+1::'a::ordered_semidom)"

  1500 proof -

  1501   have "a+0 < (a+1::'a::ordered_semidom)"

  1502     by (blast intro: zero_less_one add_strict_left_mono)

  1503   thus ?thesis by simp

  1504 qed

  1505

  1506 lemma zero_less_two: "0 < (1+1::'a::ordered_semidom)"

  1507   by (blast intro: order_less_trans zero_less_one less_add_one)

  1508

  1509 lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::ordered_field)"

  1510 by (simp add: zero_less_two pos_less_divide_eq right_distrib)

  1511

  1512 lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::ordered_field) < b"

  1513 by (simp add: zero_less_two pos_divide_less_eq right_distrib)

  1514

  1515 lemma dense: "a < b ==> \<exists>r::'a::ordered_field. a < r & r < b"

  1516 by (blast intro!: less_half_sum gt_half_sum)

  1517

  1518

  1519 lemmas times_divide_eq = times_divide_eq_right times_divide_eq_left

  1520

  1521 text{*It's not obvious whether these should be simprules or not.

  1522   Their effect is to gather terms into one big fraction, like

  1523   a*b*c / x*y*z. The rationale for that is unclear, but many proofs

  1524   seem to need them.*}

  1525

  1526 declare times_divide_eq [simp]

  1527

  1528

  1529 subsection {* Absolute Value *}

  1530

  1531 lemma abs_one [simp]: "abs 1 = (1::'a::ordered_idom)"

  1532   by (simp add: abs_if zero_less_one [THEN order_less_not_sym])

  1533

  1534 lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lordered_ring))"

  1535 proof -

  1536   let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"

  1537   let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"

  1538   have a: "(abs a) * (abs b) = ?x"

  1539     by (simp only: abs_prts[of a] abs_prts[of b] ring_eq_simps)

  1540   {

  1541     fix u v :: 'a

  1542     have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow>

  1543               u * v = pprt a * pprt b + pprt a * nprt b +

  1544                       nprt a * pprt b + nprt a * nprt b"

  1545       apply (subst prts[of u], subst prts[of v])

  1546       apply (simp add: left_distrib right_distrib add_ac)

  1547       done

  1548   }

  1549   note b = this[OF refl[of a] refl[of b]]

  1550   note addm = add_mono[of "0::'a" _ "0::'a", simplified]

  1551   note addm2 = add_mono[of _ "0::'a" _ "0::'a", simplified]

  1552   have xy: "- ?x <= ?y"

  1553     apply (simp)

  1554     apply (rule_tac y="0::'a" in order_trans)

  1555     apply (rule addm2)+

  1556     apply (simp_all add: mult_pos_le mult_neg_le)

  1557     apply (rule addm)+

  1558     apply (simp_all add: mult_pos_le mult_neg_le)

  1559     done

  1560   have yx: "?y <= ?x"

  1561     apply (simp add: add_ac)

  1562     apply (rule_tac y=0 in order_trans)

  1563     apply (rule addm2, (simp add: mult_pos_neg_le mult_pos_neg2_le)+)

  1564     apply (rule addm, (simp add: mult_pos_neg_le mult_pos_neg2_le)+)

  1565     done

  1566   have i1: "a*b <= abs a * abs b" by (simp only: a b yx)

  1567   have i2: "- (abs a * abs b) <= a*b" by (simp only: a b xy)

  1568   show ?thesis

  1569     apply (rule abs_leI)

  1570     apply (simp add: i1)

  1571     apply (simp add: i2[simplified minus_le_iff])

  1572     done

  1573 qed

  1574

  1575 lemma abs_eq_mult:

  1576   assumes "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)"

  1577   shows "abs (a*b) = abs a * abs (b::'a::lordered_ring)"

  1578 proof -

  1579   have s: "(0 <= a*b) | (a*b <= 0)"

  1580     apply (auto)

  1581     apply (rule_tac split_mult_pos_le)

  1582     apply (rule_tac contrapos_np[of "a*b <= 0"])

  1583     apply (simp)

  1584     apply (rule_tac split_mult_neg_le)

  1585     apply (insert prems)

  1586     apply (blast)

  1587     done

  1588   have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)"

  1589     by (simp add: prts[symmetric])

  1590   show ?thesis

  1591   proof cases

  1592     assume "0 <= a * b"

  1593     then show ?thesis

  1594       apply (simp_all add: mulprts abs_prts)

  1595       apply (insert prems)

  1596       apply (auto simp add:

  1597 	ring_eq_simps

  1598 	iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_zero_pprt]

  1599 	iff2imp[OF le_zero_iff_pprt_id] iff2imp[OF zero_le_iff_nprt_id])

  1600 	apply(drule (1) mult_pos_neg_le[of a b], simp)

  1601 	apply(drule (1) mult_pos_neg2_le[of b a], simp)

  1602       done

  1603   next

  1604     assume "~(0 <= a*b)"

  1605     with s have "a*b <= 0" by simp

  1606     then show ?thesis

  1607       apply (simp_all add: mulprts abs_prts)

  1608       apply (insert prems)

  1609       apply (auto simp add: ring_eq_simps)

  1610       apply(drule (1) mult_pos_le[of a b],simp)

  1611       apply(drule (1) mult_neg_le[of a b],simp)

  1612       done

  1613   qed

  1614 qed

  1615

  1616 lemma abs_mult: "abs (a * b) = abs a * abs (b::'a::ordered_idom)"

  1617 by (simp add: abs_eq_mult linorder_linear)

  1618

  1619 lemma abs_mult_self: "abs a * abs a = a * (a::'a::ordered_idom)"

  1620 by (simp add: abs_if)

  1621

  1622 lemma nonzero_abs_inverse:

  1623      "a \<noteq> 0 ==> abs (inverse (a::'a::ordered_field)) = inverse (abs a)"

  1624 apply (auto simp add: linorder_neq_iff abs_if nonzero_inverse_minus_eq

  1625                       negative_imp_inverse_negative)

  1626 apply (blast intro: positive_imp_inverse_positive elim: order_less_asym)

  1627 done

  1628

  1629 lemma abs_inverse [simp]:

  1630      "abs (inverse (a::'a::{ordered_field,division_by_zero})) =

  1631       inverse (abs a)"

  1632 apply (case_tac "a=0", simp)

  1633 apply (simp add: nonzero_abs_inverse)

  1634 done

  1635

  1636 lemma nonzero_abs_divide:

  1637      "b \<noteq> 0 ==> abs (a / (b::'a::ordered_field)) = abs a / abs b"

  1638 by (simp add: divide_inverse abs_mult nonzero_abs_inverse)

  1639

  1640 lemma abs_divide [simp]:

  1641      "abs (a / (b::'a::{ordered_field,division_by_zero})) = abs a / abs b"

  1642 apply (case_tac "b=0", simp)

  1643 apply (simp add: nonzero_abs_divide)

  1644 done

  1645

  1646 lemma abs_mult_less:

  1647      "[| abs a < c; abs b < d |] ==> abs a * abs b < c*(d::'a::ordered_idom)"

  1648 proof -

  1649   assume ac: "abs a < c"

  1650   hence cpos: "0<c" by (blast intro: order_le_less_trans abs_ge_zero)

  1651   assume "abs b < d"

  1652   thus ?thesis by (simp add: ac cpos mult_strict_mono)

  1653 qed

  1654

  1655 lemma eq_minus_self_iff: "(a = -a) = (a = (0::'a::ordered_idom))"

  1656 by (force simp add: order_eq_iff le_minus_self_iff minus_le_self_iff)

  1657

  1658 lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::ordered_idom))"

  1659 by (simp add: order_less_le le_minus_self_iff eq_minus_self_iff)

  1660

  1661 lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::ordered_idom))"

  1662 apply (simp add: order_less_le abs_le_iff)

  1663 apply (auto simp add: abs_if minus_le_self_iff eq_minus_self_iff)

  1664 apply (simp add: le_minus_self_iff linorder_neq_iff)

  1665 done

  1666

  1667 lemma linprog_dual_estimate:

  1668   assumes

  1669   "A * x \<le> (b::'a::lordered_ring)"

  1670   "0 \<le> y"

  1671   "abs (A - A') \<le> \<delta>A"

  1672   "b \<le> b'"

  1673   "abs (c - c') \<le> \<delta>c"

  1674   "abs x \<le> r"

  1675   shows

  1676   "c * x \<le> y * b' + (y * \<delta>A + abs (y * A' - c') + \<delta>c) * r"

  1677 proof -

  1678   from prems have 1: "y * b <= y * b'" by (simp add: mult_left_mono)

  1679   from prems have 2: "y * (A * x) <= y * b" by (simp add: mult_left_mono)

  1680   have 3: "y * (A * x) = c * x + (y * (A - A') + (y * A' - c') + (c'-c)) * x" by (simp add: ring_eq_simps)

  1681   from 1 2 3 have 4: "c * x + (y * (A - A') + (y * A' - c') + (c'-c)) * x <= y * b'" by simp

  1682   have 5: "c * x <= y * b' + abs((y * (A - A') + (y * A' - c') + (c'-c)) * x)"

  1683     by (simp only: 4 estimate_by_abs)

  1684   have 6: "abs((y * (A - A') + (y * A' - c') + (c'-c)) * x) <= abs (y * (A - A') + (y * A' - c') + (c'-c)) * abs x"

  1685     by (simp add: abs_le_mult)

  1686   have 7: "(abs (y * (A - A') + (y * A' - c') + (c'-c))) * abs x <= (abs (y * (A-A') + (y*A'-c')) + abs(c'-c)) * abs x"

  1687     by (simp add: abs_triangle_ineq mult_right_mono)

  1688   have 8: " (abs (y * (A-A') + (y*A'-c')) + abs(c'-c)) * abs x <=  (abs (y * (A-A')) + abs (y*A'-c') + abs(c'-c)) * abs x"

  1689     by (simp add: abs_triangle_ineq mult_right_mono)

  1690   have 9: "(abs (y * (A-A')) + abs (y*A'-c') + abs(c'-c)) * abs x <= (abs y * abs (A-A') + abs (y*A'-c') + abs (c'-c)) * abs x"

  1691     by (simp add: abs_le_mult mult_right_mono)

  1692   have 10: "c'-c = -(c-c')" by (simp add: ring_eq_simps)

  1693   have 11: "abs (c'-c) = abs (c-c')"

  1694     by (subst 10, subst abs_minus_cancel, simp)

  1695   have 12: "(abs y * abs (A-A') + abs (y*A'-c') + abs (c'-c)) * abs x <= (abs y * abs (A-A') + abs (y*A'-c') + \<delta>c) * abs x"

  1696     by (simp add: 11 prems mult_right_mono)

  1697   have 13: "(abs y * abs (A-A') + abs (y*A'-c') + \<delta>c) * abs x <= (abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * abs x"

  1698     by (simp add: prems mult_right_mono mult_left_mono)

  1699   have r: "(abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * abs x <=  (abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * r"

  1700     apply (rule mult_left_mono)

  1701     apply (simp add: prems)

  1702     apply (rule_tac add_mono[of "0::'a" _ "0", simplified])+

  1703     apply (rule mult_left_mono[of "0" "\<delta>A", simplified])

  1704     apply (simp_all)

  1705     apply (rule order_trans[where y="abs (A-A')"], simp_all add: prems)

  1706     apply (rule order_trans[where y="abs (c-c')"], simp_all add: prems)

  1707     done

  1708   from 6 7 8 9 12 13 r have 14:" abs((y * (A - A') + (y * A' - c') + (c'-c)) * x) <=(abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * r"

  1709     by (simp)

  1710   show ?thesis

  1711     apply (rule_tac le_add_right_mono[of _ _ "abs((y * (A - A') + (y * A' - c') + (c'-c)) * x)"])

  1712     apply (simp_all add: 5 14[simplified abs_of_ge_0[of y, simplified prems]])

  1713     done

  1714 qed

  1715

  1716 lemma le_ge_imp_abs_diff_1:

  1717   assumes

  1718   "A1 <= (A::'a::lordered_ring)"

  1719   "A <= A2"

  1720   shows "abs (A-A1) <= A2-A1"

  1721 proof -

  1722   have "0 <= A - A1"

  1723   proof -

  1724     have 1: "A - A1 = A + (- A1)" by simp

  1725     show ?thesis by (simp only: 1 add_right_mono[of A1 A "-A1", simplified, simplified prems])

  1726   qed

  1727   then have "abs (A-A1) = A-A1" by (rule abs_of_ge_0)

  1728   with prems show "abs (A-A1) <= (A2-A1)" by simp

  1729 qed

  1730

  1731 lemma mult_le_prts:

  1732   assumes

  1733   "a1 <= (a::'a::lordered_ring)"

  1734   "a <= a2"

  1735   "b1 <= b"

  1736   "b <= b2"

  1737   shows

  1738   "a * b <= pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"

  1739 proof -

  1740   have "a * b = (pprt a + nprt a) * (pprt b + nprt b)"

  1741     apply (subst prts[symmetric])+

  1742     apply simp

  1743     done

  1744   then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"

  1745     by (simp add: ring_eq_simps)

  1746   moreover have "pprt a * pprt b <= pprt a2 * pprt b2"

  1747     by (simp_all add: prems mult_mono)

  1748   moreover have "pprt a * nprt b <= pprt a1 * nprt b2"

  1749   proof -

  1750     have "pprt a * nprt b <= pprt a * nprt b2"

  1751       by (simp add: mult_left_mono prems)

  1752     moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2"

  1753       by (simp add: mult_right_mono_neg prems)

  1754     ultimately show ?thesis

  1755       by simp

  1756   qed

  1757   moreover have "nprt a * pprt b <= nprt a2 * pprt b1"

  1758   proof -

  1759     have "nprt a * pprt b <= nprt a2 * pprt b"

  1760       by (simp add: mult_right_mono prems)

  1761     moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"

  1762       by (simp add: mult_left_mono_neg prems)

  1763     ultimately show ?thesis

  1764       by simp

  1765   qed

  1766   moreover have "nprt a * nprt b <= nprt a1 * nprt b1"

  1767   proof -

  1768     have "nprt a * nprt b <= nprt a * nprt b1"

  1769       by (simp add: mult_left_mono_neg prems)

  1770     moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1"

  1771       by (simp add: mult_right_mono_neg prems)

  1772     ultimately show ?thesis

  1773       by simp

  1774   qed

  1775   ultimately show ?thesis

  1776     by - (rule add_mono | simp)+

  1777 qed

  1778

  1779 lemma mult_le_dual_prts:

  1780   assumes

  1781   "A * x \<le> (b::'a::lordered_ring)"

  1782   "0 \<le> y"

  1783   "A1 \<le> A"

  1784   "A \<le> A2"

  1785   "c1 \<le> c"

  1786   "c \<le> c2"

  1787   "r1 \<le> x"

  1788   "x \<le> r2"

  1789   shows

  1790   "c * x \<le> y * b + (let s1 = c1 - y * A2; s2 = c2 - y * A1 in pprt s2 * pprt r2 + pprt s1 * nprt r2 + nprt s2 * pprt r1 + nprt s1 * nprt r1)"

  1791   (is "_ <= _ + ?C")

  1792 proof -

  1793   from prems have "y * (A * x) <= y * b" by (simp add: mult_left_mono)

  1794   moreover have "y * (A * x) = c * x + (y * A - c) * x" by (simp add: ring_eq_simps)

  1795   ultimately have "c * x + (y * A - c) * x <= y * b" by simp

  1796   then have "c * x <= y * b - (y * A - c) * x" by (simp add: le_diff_eq)

  1797   then have cx: "c * x <= y * b + (c - y * A) * x" by (simp add: ring_eq_simps)

  1798   have s2: "c - y * A <= c2 - y * A1"

  1799     by (simp add: diff_def prems add_mono mult_left_mono)

  1800   have s1: "c1 - y * A2 <= c - y * A"

  1801     by (simp add: diff_def prems add_mono mult_left_mono)

  1802   have prts: "(c - y * A) * x <= ?C"

  1803     apply (simp add: Let_def)

  1804     apply (rule mult_le_prts)

  1805     apply (simp_all add: prems s1 s2)

  1806     done

  1807   then have "y * b + (c - y * A) * x <= y * b + ?C"

  1808     by simp

  1809   with cx show ?thesis

  1810     by(simp only:)

  1811 qed

  1812

  1813 ML {*

  1814 val left_distrib = thm "left_distrib";

  1815 val right_distrib = thm "right_distrib";

  1816 val mult_commute = thm "mult_commute";

  1817 val distrib = thm "distrib";

  1818 val zero_neq_one = thm "zero_neq_one";

  1819 val no_zero_divisors = thm "no_zero_divisors";

  1820 val left_inverse = thm "left_inverse";

  1821 val divide_inverse = thm "divide_inverse";

  1822 val mult_zero_left = thm "mult_zero_left";

  1823 val mult_zero_right = thm "mult_zero_right";

  1824 val field_mult_eq_0_iff = thm "field_mult_eq_0_iff";

  1825 val inverse_zero = thm "inverse_zero";

  1826 val ring_distrib = thms "ring_distrib";

  1827 val combine_common_factor = thm "combine_common_factor";

  1828 val minus_mult_left = thm "minus_mult_left";

  1829 val minus_mult_right = thm "minus_mult_right";

  1830 val minus_mult_minus = thm "minus_mult_minus";

  1831 val minus_mult_commute = thm "minus_mult_commute";

  1832 val right_diff_distrib = thm "right_diff_distrib";

  1833 val left_diff_distrib = thm "left_diff_distrib";

  1834 val mult_left_mono = thm "mult_left_mono";

  1835 val mult_right_mono = thm "mult_right_mono";

  1836 val mult_strict_left_mono = thm "mult_strict_left_mono";

  1837 val mult_strict_right_mono = thm "mult_strict_right_mono";

  1838 val mult_mono = thm "mult_mono";

  1839 val mult_strict_mono = thm "mult_strict_mono";

  1840 val abs_if = thm "abs_if";

  1841 val zero_less_one = thm "zero_less_one";

  1842 val eq_add_iff1 = thm "eq_add_iff1";

  1843 val eq_add_iff2 = thm "eq_add_iff2";

  1844 val less_add_iff1 = thm "less_add_iff1";

  1845 val less_add_iff2 = thm "less_add_iff2";

  1846 val le_add_iff1 = thm "le_add_iff1";

  1847 val le_add_iff2 = thm "le_add_iff2";

  1848 val mult_left_le_imp_le = thm "mult_left_le_imp_le";

  1849 val mult_right_le_imp_le = thm "mult_right_le_imp_le";

  1850 val mult_left_less_imp_less = thm "mult_left_less_imp_less";

  1851 val mult_right_less_imp_less = thm "mult_right_less_imp_less";

  1852 val mult_strict_left_mono_neg = thm "mult_strict_left_mono_neg";

  1853 val mult_left_mono_neg = thm "mult_left_mono_neg";

  1854 val mult_strict_right_mono_neg = thm "mult_strict_right_mono_neg";

  1855 val mult_right_mono_neg = thm "mult_right_mono_neg";

  1856 val mult_pos = thm "mult_pos";

  1857 val mult_pos_le = thm "mult_pos_le";

  1858 val mult_pos_neg = thm "mult_pos_neg";

  1859 val mult_pos_neg_le = thm "mult_pos_neg_le";

  1860 val mult_pos_neg2 = thm "mult_pos_neg2";

  1861 val mult_pos_neg2_le = thm "mult_pos_neg2_le";

  1862 val mult_neg = thm "mult_neg";

  1863 val mult_neg_le = thm "mult_neg_le";

  1864 val zero_less_mult_pos = thm "zero_less_mult_pos";

  1865 val zero_less_mult_pos2 = thm "zero_less_mult_pos2";

  1866 val zero_less_mult_iff = thm "zero_less_mult_iff";

  1867 val mult_eq_0_iff = thm "mult_eq_0_iff";

  1868 val zero_le_mult_iff = thm "zero_le_mult_iff";

  1869 val mult_less_0_iff = thm "mult_less_0_iff";

  1870 val mult_le_0_iff = thm "mult_le_0_iff";

  1871 val split_mult_pos_le = thm "split_mult_pos_le";

  1872 val split_mult_neg_le = thm "split_mult_neg_le";

  1873 val zero_le_square = thm "zero_le_square";

  1874 val zero_le_one = thm "zero_le_one";

  1875 val not_one_le_zero = thm "not_one_le_zero";

  1876 val not_one_less_zero = thm "not_one_less_zero";

  1877 val mult_left_mono_neg = thm "mult_left_mono_neg";

  1878 val mult_right_mono_neg = thm "mult_right_mono_neg";

  1879 val mult_strict_mono = thm "mult_strict_mono";

  1880 val mult_strict_mono' = thm "mult_strict_mono'";

  1881 val mult_mono = thm "mult_mono";

  1882 val less_1_mult = thm "less_1_mult";

  1883 val mult_less_cancel_right_disj = thm "mult_less_cancel_right_disj";

  1884 val mult_less_cancel_left_disj = thm "mult_less_cancel_left_disj";

  1885 val mult_less_cancel_right = thm "mult_less_cancel_right";

  1886 val mult_less_cancel_left = thm "mult_less_cancel_left";

  1887 val mult_le_cancel_right = thm "mult_le_cancel_right";

  1888 val mult_le_cancel_left = thm "mult_le_cancel_left";

  1889 val mult_less_imp_less_left = thm "mult_less_imp_less_left";

  1890 val mult_less_imp_less_right = thm "mult_less_imp_less_right";

  1891 val mult_cancel_right = thm "mult_cancel_right";

  1892 val mult_cancel_left = thm "mult_cancel_left";

  1893 val ring_eq_simps = thms "ring_eq_simps";

  1894 val right_inverse = thm "right_inverse";

  1895 val right_inverse_eq = thm "right_inverse_eq";

  1896 val nonzero_inverse_eq_divide = thm "nonzero_inverse_eq_divide";

  1897 val divide_self = thm "divide_self";

  1898 val divide_zero = thm "divide_zero";

  1899 val divide_zero_left = thm "divide_zero_left";

  1900 val inverse_eq_divide = thm "inverse_eq_divide";

  1901 val add_divide_distrib = thm "add_divide_distrib";

  1902 val field_mult_eq_0_iff = thm "field_mult_eq_0_iff";

  1903 val field_mult_cancel_right_lemma = thm "field_mult_cancel_right_lemma";

  1904 val field_mult_cancel_right = thm "field_mult_cancel_right";

  1905 val field_mult_cancel_left = thm "field_mult_cancel_left";

  1906 val nonzero_imp_inverse_nonzero = thm "nonzero_imp_inverse_nonzero";

  1907 val inverse_zero_imp_zero = thm "inverse_zero_imp_zero";

  1908 val inverse_nonzero_imp_nonzero = thm "inverse_nonzero_imp_nonzero";

  1909 val inverse_nonzero_iff_nonzero = thm "inverse_nonzero_iff_nonzero";

  1910 val nonzero_inverse_minus_eq = thm "nonzero_inverse_minus_eq";

  1911 val inverse_minus_eq = thm "inverse_minus_eq";

  1912 val nonzero_inverse_eq_imp_eq = thm "nonzero_inverse_eq_imp_eq";

  1913 val inverse_eq_imp_eq = thm "inverse_eq_imp_eq";

  1914 val inverse_eq_iff_eq = thm "inverse_eq_iff_eq";

  1915 val nonzero_inverse_inverse_eq = thm "nonzero_inverse_inverse_eq";

  1916 val inverse_inverse_eq = thm "inverse_inverse_eq";

  1917 val inverse_1 = thm "inverse_1";

  1918 val nonzero_inverse_mult_distrib = thm "nonzero_inverse_mult_distrib";

  1919 val inverse_mult_distrib = thm "inverse_mult_distrib";

  1920 val inverse_add = thm "inverse_add";

  1921 val inverse_divide = thm "inverse_divide";

  1922 val nonzero_mult_divide_cancel_left = thm "nonzero_mult_divide_cancel_left";

  1923 val mult_divide_cancel_left = thm "mult_divide_cancel_left";

  1924 val nonzero_mult_divide_cancel_right = thm "nonzero_mult_divide_cancel_right";

  1925 val mult_divide_cancel_right = thm "mult_divide_cancel_right";

  1926 val mult_divide_cancel_eq_if = thm "mult_divide_cancel_eq_if";

  1927 val divide_1 = thm "divide_1";

  1928 val times_divide_eq_right = thm "times_divide_eq_right";

  1929 val times_divide_eq_left = thm "times_divide_eq_left";

  1930 val divide_divide_eq_right = thm "divide_divide_eq_right";

  1931 val divide_divide_eq_left = thm "divide_divide_eq_left";

  1932 val nonzero_minus_divide_left = thm "nonzero_minus_divide_left";

  1933 val nonzero_minus_divide_right = thm "nonzero_minus_divide_right";

  1934 val nonzero_minus_divide_divide = thm "nonzero_minus_divide_divide";

  1935 val minus_divide_left = thm "minus_divide_left";

  1936 val minus_divide_right = thm "minus_divide_right";

  1937 val minus_divide_divide = thm "minus_divide_divide";

  1938 val diff_divide_distrib = thm "diff_divide_distrib";

  1939 val positive_imp_inverse_positive = thm "positive_imp_inverse_positive";

  1940 val negative_imp_inverse_negative = thm "negative_imp_inverse_negative";

  1941 val inverse_le_imp_le = thm "inverse_le_imp_le";

  1942 val inverse_positive_imp_positive = thm "inverse_positive_imp_positive";

  1943 val inverse_positive_iff_positive = thm "inverse_positive_iff_positive";

  1944 val inverse_negative_imp_negative = thm "inverse_negative_imp_negative";

  1945 val inverse_negative_iff_negative = thm "inverse_negative_iff_negative";

  1946 val inverse_nonnegative_iff_nonnegative = thm "inverse_nonnegative_iff_nonnegative";

  1947 val inverse_nonpositive_iff_nonpositive = thm "inverse_nonpositive_iff_nonpositive";

  1948 val less_imp_inverse_less = thm "less_imp_inverse_less";

  1949 val inverse_less_imp_less = thm "inverse_less_imp_less";

  1950 val inverse_less_iff_less = thm "inverse_less_iff_less";

  1951 val le_imp_inverse_le = thm "le_imp_inverse_le";

  1952 val inverse_le_iff_le = thm "inverse_le_iff_le";

  1953 val inverse_le_imp_le_neg = thm "inverse_le_imp_le_neg";

  1954 val less_imp_inverse_less_neg = thm "less_imp_inverse_less_neg";

  1955 val inverse_less_imp_less_neg = thm "inverse_less_imp_less_neg";

  1956 val inverse_less_iff_less_neg = thm "inverse_less_iff_less_neg";

  1957 val le_imp_inverse_le_neg = thm "le_imp_inverse_le_neg";

  1958 val inverse_le_iff_le_neg = thm "inverse_le_iff_le_neg";

  1959 val one_less_inverse_iff = thm "one_less_inverse_iff";

  1960 val inverse_eq_1_iff = thm "inverse_eq_1_iff";

  1961 val one_le_inverse_iff = thm "one_le_inverse_iff";

  1962 val inverse_less_1_iff = thm "inverse_less_1_iff";

  1963 val inverse_le_1_iff = thm "inverse_le_1_iff";

  1964 val zero_less_divide_iff = thm "zero_less_divide_iff";

  1965 val divide_less_0_iff = thm "divide_less_0_iff";

  1966 val zero_le_divide_iff = thm "zero_le_divide_iff";

  1967 val divide_le_0_iff = thm "divide_le_0_iff";

  1968 val divide_eq_0_iff = thm "divide_eq_0_iff";

  1969 val pos_le_divide_eq = thm "pos_le_divide_eq";

  1970 val neg_le_divide_eq = thm "neg_le_divide_eq";

  1971 val le_divide_eq = thm "le_divide_eq";

  1972 val pos_divide_le_eq = thm "pos_divide_le_eq";

  1973 val neg_divide_le_eq = thm "neg_divide_le_eq";

  1974 val divide_le_eq = thm "divide_le_eq";

  1975 val pos_less_divide_eq = thm "pos_less_divide_eq";

  1976 val neg_less_divide_eq = thm "neg_less_divide_eq";

  1977 val less_divide_eq = thm "less_divide_eq";

  1978 val pos_divide_less_eq = thm "pos_divide_less_eq";

  1979 val neg_divide_less_eq = thm "neg_divide_less_eq";

  1980 val divide_less_eq = thm "divide_less_eq";

  1981 val nonzero_eq_divide_eq = thm "nonzero_eq_divide_eq";

  1982 val eq_divide_eq = thm "eq_divide_eq";

  1983 val nonzero_divide_eq_eq = thm "nonzero_divide_eq_eq";

  1984 val divide_eq_eq = thm "divide_eq_eq";

  1985 val divide_cancel_right = thm "divide_cancel_right";

  1986 val divide_cancel_left = thm "divide_cancel_left";

  1987 val divide_eq_1_iff = thm "divide_eq_1_iff";

  1988 val one_eq_divide_iff = thm "one_eq_divide_iff";

  1989 val zero_eq_1_divide_iff = thm "zero_eq_1_divide_iff";

  1990 val one_divide_eq_0_iff = thm "one_divide_eq_0_iff";

  1991 val divide_strict_right_mono = thm "divide_strict_right_mono";

  1992 val divide_right_mono = thm "divide_right_mono";

  1993 val divide_strict_left_mono = thm "divide_strict_left_mono";

  1994 val divide_left_mono = thm "divide_left_mono";

  1995 val divide_strict_left_mono_neg = thm "divide_strict_left_mono_neg";

  1996 val divide_strict_right_mono_neg = thm "divide_strict_right_mono_neg";

  1997 val less_add_one = thm "less_add_one";

  1998 val zero_less_two = thm "zero_less_two";

  1999 val less_half_sum = thm "less_half_sum";

  2000 val gt_half_sum = thm "gt_half_sum";

  2001 val dense = thm "dense";

  2002 val abs_one = thm "abs_one";

  2003 val abs_le_mult = thm "abs_le_mult";

  2004 val abs_eq_mult = thm "abs_eq_mult";

  2005 val abs_mult = thm "abs_mult";

  2006 val abs_mult_self = thm "abs_mult_self";

  2007 val nonzero_abs_inverse = thm "nonzero_abs_inverse";

  2008 val abs_inverse = thm "abs_inverse";

  2009 val nonzero_abs_divide = thm "nonzero_abs_divide";

  2010 val abs_divide = thm "abs_divide";

  2011 val abs_mult_less = thm "abs_mult_less";

  2012 val eq_minus_self_iff = thm "eq_minus_self_iff";

  2013 val less_minus_self_iff = thm "less_minus_self_iff";

  2014 val abs_less_iff = thm "abs_less_iff";

  2015 *}

  2016

  2017 end