src/HOL/Ring_and_Field.thy
 author nipkow Sat Jun 25 16:06:17 2005 +0200 (2005-06-25) changeset 16568 e02fe7ae212b parent 15923 01d5d0c1c078 child 16775 c1b87ef4a1c3 permissions -rw-r--r--
Changes due to new abel_cancel.ML
     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 lemmas linorder_neqE_ordered_idom =

   249  linorder_neqE[where 'a = "?'b::ordered_idom"]

   250

   251 lemma eq_add_iff1:

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

   253 apply (simp add: diff_minus left_distrib)

   254 apply (simp add: diff_minus left_distrib add_ac)

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

   256 done

   257

   258 lemma eq_add_iff2:

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

   260 apply (simp add: diff_minus left_distrib add_ac)

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

   262 done

   263

   264 lemma less_add_iff1:

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

   266 apply (simp add: diff_minus left_distrib add_ac)

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

   268 done

   269

   270 lemma less_add_iff2:

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

   272 apply (simp add: diff_minus left_distrib add_ac)

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

   274 done

   275

   276 lemma le_add_iff1:

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

   278 apply (simp add: diff_minus left_distrib add_ac)

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

   280 done

   281

   282 lemma le_add_iff2:

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

   284 apply (simp add: diff_minus left_distrib add_ac)

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

   286 done

   287

   288 subsection {* Ordering Rules for Multiplication *}

   289

   290 lemma mult_left_le_imp_le:

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

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

   293

   294 lemma mult_right_le_imp_le:

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

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

   297

   298 lemma mult_left_less_imp_less:

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

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

   301

   302 lemma mult_right_less_imp_less:

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

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

   305

   306 lemma mult_strict_left_mono_neg:

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

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

   309 apply (simp_all add: minus_mult_left [symmetric])

   310 done

   311

   312 lemma mult_left_mono_neg:

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

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

   315 apply (simp_all add: minus_mult_left [symmetric])

   316 done

   317

   318 lemma mult_strict_right_mono_neg:

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

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

   321 apply (simp_all add: minus_mult_right [symmetric])

   322 done

   323

   324 lemma mult_right_mono_neg:

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

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

   327 apply (simp)

   328 apply (simp_all add: minus_mult_right [symmetric])

   329 done

   330

   331 subsection{* Products of Signs *}

   332

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

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

   335

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

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

   338

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

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

   341

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

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

   344

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

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

   347

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

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

   350

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

   352 by (drule mult_strict_right_mono_neg, auto)

   353

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

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

   356

   357 lemma zero_less_mult_pos:

   358      "[| 0 < a*b; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"

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

   360  apply (auto simp add: order_le_less linorder_not_less)

   361 apply (drule_tac mult_pos_neg [of a b])

   362  apply (auto dest: order_less_not_sym)

   363 done

   364

   365 lemma zero_less_mult_pos2:

   366      "[| 0 < b*a; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"

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

   368  apply (auto simp add: order_le_less linorder_not_less)

   369 apply (drule_tac mult_pos_neg2 [of a b])

   370  apply (auto dest: order_less_not_sym)

   371 done

   372

   373 lemma zero_less_mult_iff:

   374      "((0::'a::ordered_ring_strict) < a*b) = (0 < a & 0 < b | a < 0 & b < 0)"

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

   376 apply (blast dest: zero_less_mult_pos)

   377 apply (blast dest: zero_less_mult_pos2)

   378 done

   379

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

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

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

   383 apply (case_tac "a < 0")

   384 apply (auto simp add: linorder_not_less order_le_less linorder_neq_iff)

   385 apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono)+

   386 done

   387

   388 lemma zero_le_mult_iff:

   389      "((0::'a::ordered_ring_strict) \<le> a*b) = (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"

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

   391                    zero_less_mult_iff)

   392

   393 lemma mult_less_0_iff:

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

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

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

   397 done

   398

   399 lemma mult_le_0_iff:

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

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

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

   403 done

   404

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

   406 by (auto simp add: mult_pos_le mult_neg_le)

   407

   408 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)"

   409 by (auto simp add: mult_pos_neg_le mult_pos_neg2_le)

   410

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

   412 by (simp add: zero_le_mult_iff linorder_linear)

   413

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

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

   416

   417 instance ordered_idom \<subseteq> ordered_semidom

   418 proof

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

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

   421 qed

   422

   423 instance ordered_ring_strict \<subseteq> axclass_no_zero_divisors

   424 by (intro_classes, simp)

   425

   426 instance ordered_idom \<subseteq> idom ..

   427

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

   429

   430 declare zero_neq_one [THEN not_sym, simp]

   431

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

   433   by (rule zero_less_one [THEN order_less_imp_le])

   434

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

   436 by (simp add: linorder_not_le)

   437

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

   439 by (simp add: linorder_not_less)

   440

   441 subsection{*More Monotonicity*}

   442

   443 text{*Strict monotonicity in both arguments*}

   444 lemma mult_strict_mono:

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

   446 apply (case_tac "c=0")

   447  apply (simp add: mult_pos)

   448 apply (erule mult_strict_right_mono [THEN order_less_trans])

   449  apply (force simp add: order_le_less)

   450 apply (erule mult_strict_left_mono, assumption)

   451 done

   452

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

   454 lemma mult_strict_mono':

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

   456 apply (rule mult_strict_mono)

   457 apply (blast intro: order_le_less_trans)+

   458 done

   459

   460 lemma mult_mono:

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

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

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

   464 apply (erule mult_left_mono, assumption)

   465 done

   466

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

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

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

   470 done

   471

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

   473

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

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

   476

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

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

   479

   480 lemma mult_less_cancel_right_disj:

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

   482 apply (case_tac "c = 0")

   483 apply (auto simp add: linorder_neq_iff mult_strict_right_mono

   484                       mult_strict_right_mono_neg)

   485 apply (auto simp add: linorder_not_less

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

   487                       linorder_not_le [symmetric, of a])

   488 apply (erule_tac [!] notE)

   489 apply (auto simp add: order_less_imp_le mult_right_mono

   490                       mult_right_mono_neg)

   491 done

   492

   493 lemma mult_less_cancel_left_disj:

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

   495 apply (case_tac "c = 0")

   496 apply (auto simp add: linorder_neq_iff mult_strict_left_mono

   497                       mult_strict_left_mono_neg)

   498 apply (auto simp add: linorder_not_less

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

   500                       linorder_not_le [symmetric, of a])

   501 apply (erule_tac [!] notE)

   502 apply (auto simp add: order_less_imp_le mult_left_mono

   503                       mult_left_mono_neg)

   504 done

   505

   506

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

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

   509

   510 lemma mult_less_cancel_right:

   511   fixes c :: "'a :: ordered_ring_strict"

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

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

   514

   515 lemma mult_less_cancel_left:

   516   fixes c :: "'a :: ordered_ring_strict"

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

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

   519

   520 lemma mult_le_cancel_right:

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

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

   523

   524 lemma mult_le_cancel_left:

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

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

   527

   528 lemma mult_less_imp_less_left:

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

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

   531 proof (rule ccontr)

   532   assume "~ a < b"

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

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

   535   with this and less show False

   536     by (simp add: linorder_not_less [symmetric])

   537 qed

   538

   539 lemma mult_less_imp_less_right:

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

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

   542 proof (rule ccontr)

   543   assume "~ a < b"

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

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

   546   with this and less show False

   547     by (simp add: linorder_not_less [symmetric])

   548 qed

   549

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

   551 lemma mult_cancel_right [simp]:

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

   553 apply (cut_tac linorder_less_linear [of 0 c])

   554 apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono

   555              simp add: linorder_neq_iff)

   556 done

   557

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

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

   560 lemma mult_cancel_left [simp]:

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

   562 apply (cut_tac linorder_less_linear [of 0 c])

   563 apply (force dest: mult_strict_left_mono_neg mult_strict_left_mono

   564              simp add: linorder_neq_iff)

   565 done

   566

   567

   568 subsubsection{*Special Cancellation Simprules for Multiplication*}

   569

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

   571

   572 lemma mult_le_cancel_right1:

   573   fixes c :: "'a :: ordered_idom"

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

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

   576

   577 lemma mult_le_cancel_right2:

   578   fixes c :: "'a :: ordered_idom"

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

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

   581

   582 lemma mult_le_cancel_left1:

   583   fixes c :: "'a :: ordered_idom"

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

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

   586

   587 lemma mult_le_cancel_left2:

   588   fixes c :: "'a :: ordered_idom"

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

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

   591

   592 lemma mult_less_cancel_right1:

   593   fixes c :: "'a :: ordered_idom"

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

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

   596

   597 lemma mult_less_cancel_right2:

   598   fixes c :: "'a :: ordered_idom"

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

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

   601

   602 lemma mult_less_cancel_left1:

   603   fixes c :: "'a :: ordered_idom"

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

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

   606

   607 lemma mult_less_cancel_left2:

   608   fixes c :: "'a :: ordered_idom"

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

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

   611

   612 lemma mult_cancel_right1 [simp]:

   613 fixes c :: "'a :: ordered_idom"

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

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

   616

   617 lemma mult_cancel_right2 [simp]:

   618 fixes c :: "'a :: ordered_idom"

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

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

   621

   622 lemma mult_cancel_left1 [simp]:

   623 fixes c :: "'a :: ordered_idom"

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

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

   626

   627 lemma mult_cancel_left2 [simp]:

   628 fixes c :: "'a :: ordered_idom"

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

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

   631

   632

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

   634 lemmas mult_compare_simps =

   635     mult_le_cancel_right mult_le_cancel_left

   636     mult_le_cancel_right1 mult_le_cancel_right2

   637     mult_le_cancel_left1 mult_le_cancel_left2

   638     mult_less_cancel_right mult_less_cancel_left

   639     mult_less_cancel_right1 mult_less_cancel_right2

   640     mult_less_cancel_left1 mult_less_cancel_left2

   641     mult_cancel_right mult_cancel_left

   642     mult_cancel_right1 mult_cancel_right2

   643     mult_cancel_left1 mult_cancel_left2

   644

   645

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

   647 lemmas ring_eq_simps =

   648 (*  mult_ac*)

   649   left_distrib right_distrib left_diff_distrib right_diff_distrib

   650   group_eq_simps

   651 (*  add_ac

   652   add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2

   653   diff_eq_eq eq_diff_eq *)

   654

   655 subsection {* Fields *}

   656

   657 lemma right_inverse [simp]:

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

   659 proof -

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

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

   662   finally show ?thesis .

   663 qed

   664

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

   666 proof

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

   668   {

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

   670     also assume "a / b = 1"

   671     finally show "a = b" by simp

   672   next

   673     assume "a = b"

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

   675   }

   676 qed

   677

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

   679 by (simp add: divide_inverse)

   680

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

   682   by (simp add: divide_inverse)

   683

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

   685 by (simp add: divide_inverse)

   686

   687 lemma divide_self_if [simp]:

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

   689   by (simp add: divide_self)

   690

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

   692 by (simp add: divide_inverse)

   693

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

   695 by (simp add: divide_inverse)

   696

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

   698 by (simp add: divide_inverse left_distrib)

   699

   700

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

   702       of an ordering.*}

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

   704 proof cases

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

   706 next

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

   708   { assume "a * b = 0"

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

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

   711   thus ?thesis by force

   712 qed

   713

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

   715 lemma field_mult_cancel_right_lemma:

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

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

   718 	 shows "a=b"

   719 proof -

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

   721     by (simp add: eq)

   722   thus "a=b"

   723     by (simp add: mult_assoc cnz)

   724 qed

   725

   726 lemma field_mult_cancel_right [simp]:

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

   728 proof cases

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

   730 next

   731   assume "c\<noteq>0"

   732   thus ?thesis by (force dest: field_mult_cancel_right_lemma)

   733 qed

   734

   735 lemma field_mult_cancel_left [simp]:

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

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

   738

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

   740 proof

   741   assume ianz: "inverse a = 0"

   742   assume "a \<noteq> 0"

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

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

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

   746   thus False by (simp add: eq_commute)

   747 qed

   748

   749

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

   751

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

   753 apply (rule ccontr)

   754 apply (blast dest: nonzero_imp_inverse_nonzero)

   755 done

   756

   757 lemma inverse_nonzero_imp_nonzero:

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

   759 apply (rule ccontr)

   760 apply (blast dest: nonzero_imp_inverse_nonzero)

   761 done

   762

   763 lemma inverse_nonzero_iff_nonzero [simp]:

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

   765 by (force dest: inverse_nonzero_imp_nonzero)

   766

   767 lemma nonzero_inverse_minus_eq:

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

   769 proof -

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

   771     by simp

   772   thus ?thesis

   773     by (simp only: field_mult_cancel_left, simp)

   774 qed

   775

   776 lemma inverse_minus_eq [simp]:

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

   778 proof cases

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

   780 next

   781   assume "a\<noteq>0"

   782   thus ?thesis by (simp add: nonzero_inverse_minus_eq)

   783 qed

   784

   785 lemma nonzero_inverse_eq_imp_eq:

   786       assumes inveq: "inverse a = inverse b"

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

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

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

   790 proof -

   791   have "a * inverse b = a * inverse a"

   792     by (simp add: inveq)

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

   794     by simp

   795   thus "a = b"

   796     by (simp add: mult_assoc anz bnz)

   797 qed

   798

   799 lemma inverse_eq_imp_eq:

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

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

   802  apply (force dest!: inverse_zero_imp_zero

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

   804 apply (force dest!: nonzero_inverse_eq_imp_eq)

   805 done

   806

   807 lemma inverse_eq_iff_eq [simp]:

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

   809 by (force dest!: inverse_eq_imp_eq)

   810

   811 lemma nonzero_inverse_inverse_eq:

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

   813   proof -

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

   815     by (simp add: nonzero_imp_inverse_nonzero)

   816   thus ?thesis

   817     by (simp add: mult_assoc)

   818   qed

   819

   820 lemma inverse_inverse_eq [simp]:

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

   822   proof cases

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

   824   next

   825     assume "a\<noteq>0"

   826     thus ?thesis by (simp add: nonzero_inverse_inverse_eq)

   827   qed

   828

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

   830   proof -

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

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

   833   thus ?thesis  by simp

   834   qed

   835

   836 lemma inverse_unique:

   837   assumes ab: "a*b = 1"

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

   839 proof -

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

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

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

   843 qed

   844

   845 lemma nonzero_inverse_mult_distrib:

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

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

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

   849   proof -

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

   851     by (simp add: field_mult_eq_0_iff anz bnz)

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

   853     by (simp add: mult_assoc bnz)

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

   855     by simp

   856   thus ?thesis

   857     by (simp add: mult_assoc anz)

   858   qed

   859

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

   861       the right-hand side.*}

   862 lemma inverse_mult_distrib [simp]:

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

   864   proof cases

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

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

   867   next

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

   869     thus ?thesis  by force

   870   qed

   871

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

   873 lemma inverse_add:

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

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

   876 apply (simp add: left_distrib mult_assoc)

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

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

   879 done

   880

   881 lemma inverse_divide [simp]:

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

   883   by (simp add: divide_inverse mult_commute)

   884

   885 lemma nonzero_mult_divide_cancel_left:

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

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

   888 proof -

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

   890     by (simp add: field_mult_eq_0_iff divide_inverse

   891                   nonzero_inverse_mult_distrib)

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

   893     by (simp only: mult_ac)

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

   895     by simp

   896     finally show ?thesis

   897     by (simp add: divide_inverse)

   898 qed

   899

   900 lemma mult_divide_cancel_left:

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

   902 apply (case_tac "b = 0")

   903 apply (simp_all add: nonzero_mult_divide_cancel_left)

   904 done

   905

   906 lemma nonzero_mult_divide_cancel_right:

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

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

   909

   910 lemma mult_divide_cancel_right:

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

   912 apply (case_tac "b = 0")

   913 apply (simp_all add: nonzero_mult_divide_cancel_right)

   914 done

   915

   916 (*For ExtractCommonTerm*)

   917 lemma mult_divide_cancel_eq_if:

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

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

   920   by (simp add: mult_divide_cancel_left)

   921

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

   923   by (simp add: divide_inverse)

   924

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

   926 by (simp add: divide_inverse mult_assoc)

   927

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

   929 by (simp add: divide_inverse mult_ac)

   930

   931 lemma divide_divide_eq_right [simp]:

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

   933 by (simp add: divide_inverse mult_ac)

   934

   935 lemma divide_divide_eq_left [simp]:

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

   937 by (simp add: divide_inverse mult_assoc)

   938

   939

   940 subsubsection{*Special Cancellation Simprules for Division*}

   941

   942 lemma mult_divide_cancel_left_if [simp]:

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

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

   945 by (simp add: mult_divide_cancel_left)

   946

   947 lemma mult_divide_cancel_right_if [simp]:

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

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

   950 by (simp add: mult_divide_cancel_right)

   951

   952 lemma mult_divide_cancel_left_if1 [simp]:

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

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

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

   956 apply (simp del: mult_divide_cancel_left_if)

   957 done

   958

   959 lemma mult_divide_cancel_left_if2 [simp]:

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

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

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

   963 apply (simp del: mult_divide_cancel_left_if)

   964 done

   965

   966 lemma mult_divide_cancel_right_if1 [simp]:

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

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

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

   970 apply (simp del: mult_divide_cancel_right_if)

   971 done

   972

   973 lemma mult_divide_cancel_right_if2 [simp]:

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

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

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

   977 apply (simp del: mult_divide_cancel_right_if)

   978 done

   979

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

   981

   982 lemma times_divide_self_right [simp]:

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

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

   985 by (simp add: times_divide_eq_right)

   986

   987 lemma times_divide_self_left [simp]:

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

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

   990 by (simp add: times_divide_eq_left)

   991

   992

   993 subsection {* Division and Unary Minus *}

   994

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

   996 by (simp add: divide_inverse minus_mult_left)

   997

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

   999 by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right)

  1000

  1001 lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a)/(-b) = a / (b::'a::field)"

  1002 by (simp add: divide_inverse nonzero_inverse_minus_eq)

  1003

  1004 lemma minus_divide_left: "- (a/b) = (-a) / (b::'a::field)"

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

  1006

  1007 lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"

  1008 by (simp add: divide_inverse minus_mult_right [symmetric])

  1009

  1010

  1011 text{*The effect is to extract signs from divisions*}

  1012 declare minus_divide_left  [symmetric, simp]

  1013 declare minus_divide_right [symmetric, simp]

  1014

  1015 text{*Also, extract signs from products*}

  1016 declare minus_mult_left [symmetric, simp]

  1017 declare minus_mult_right [symmetric, simp]

  1018

  1019 lemma minus_divide_divide [simp]:

  1020      "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"

  1021 apply (case_tac "b=0", simp)

  1022 apply (simp add: nonzero_minus_divide_divide)

  1023 done

  1024

  1025 lemma diff_divide_distrib: "(a-b)/(c::'a::field) = a/c - b/c"

  1026 by (simp add: diff_minus add_divide_distrib)

  1027

  1028

  1029 subsection {* Ordered Fields *}

  1030

  1031 lemma positive_imp_inverse_positive:

  1032       assumes a_gt_0: "0 < a"  shows "0 < inverse (a::'a::ordered_field)"

  1033   proof -

  1034   have "0 < a * inverse a"

  1035     by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one)

  1036   thus "0 < inverse a"

  1037     by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff)

  1038   qed

  1039

  1040 lemma negative_imp_inverse_negative:

  1041      "a < 0 ==> inverse a < (0::'a::ordered_field)"

  1042   by (insert positive_imp_inverse_positive [of "-a"],

  1043       simp add: nonzero_inverse_minus_eq order_less_imp_not_eq)

  1044

  1045 lemma inverse_le_imp_le:

  1046       assumes invle: "inverse a \<le> inverse b"

  1047 	  and apos:  "0 < a"

  1048 	 shows "b \<le> (a::'a::ordered_field)"

  1049   proof (rule classical)

  1050   assume "~ b \<le> a"

  1051   hence "a < b"

  1052     by (simp add: linorder_not_le)

  1053   hence bpos: "0 < b"

  1054     by (blast intro: apos order_less_trans)

  1055   hence "a * inverse a \<le> a * inverse b"

  1056     by (simp add: apos invle order_less_imp_le mult_left_mono)

  1057   hence "(a * inverse a) * b \<le> (a * inverse b) * b"

  1058     by (simp add: bpos order_less_imp_le mult_right_mono)

  1059   thus "b \<le> a"

  1060     by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)

  1061   qed

  1062

  1063 lemma inverse_positive_imp_positive:

  1064       assumes inv_gt_0: "0 < inverse a"

  1065           and [simp]:   "a \<noteq> 0"

  1066         shows "0 < (a::'a::ordered_field)"

  1067   proof -

  1068   have "0 < inverse (inverse a)"

  1069     by (rule positive_imp_inverse_positive)

  1070   thus "0 < a"

  1071     by (simp add: nonzero_inverse_inverse_eq)

  1072   qed

  1073

  1074 lemma inverse_positive_iff_positive [simp]:

  1075       "(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))"

  1076 apply (case_tac "a = 0", simp)

  1077 apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)

  1078 done

  1079

  1080 lemma inverse_negative_imp_negative:

  1081       assumes inv_less_0: "inverse a < 0"

  1082           and [simp]:   "a \<noteq> 0"

  1083         shows "a < (0::'a::ordered_field)"

  1084   proof -

  1085   have "inverse (inverse a) < 0"

  1086     by (rule negative_imp_inverse_negative)

  1087   thus "a < 0"

  1088     by (simp add: nonzero_inverse_inverse_eq)

  1089   qed

  1090

  1091 lemma inverse_negative_iff_negative [simp]:

  1092       "(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))"

  1093 apply (case_tac "a = 0", simp)

  1094 apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)

  1095 done

  1096

  1097 lemma inverse_nonnegative_iff_nonnegative [simp]:

  1098       "(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))"

  1099 by (simp add: linorder_not_less [symmetric])

  1100

  1101 lemma inverse_nonpositive_iff_nonpositive [simp]:

  1102       "(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))"

  1103 by (simp add: linorder_not_less [symmetric])

  1104

  1105

  1106 subsection{*Anti-Monotonicity of @{term inverse}*}

  1107

  1108 lemma less_imp_inverse_less:

  1109       assumes less: "a < b"

  1110 	  and apos:  "0 < a"

  1111 	shows "inverse b < inverse (a::'a::ordered_field)"

  1112   proof (rule ccontr)

  1113   assume "~ inverse b < inverse a"

  1114   hence "inverse a \<le> inverse b"

  1115     by (simp add: linorder_not_less)

  1116   hence "~ (a < b)"

  1117     by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos])

  1118   thus False

  1119     by (rule notE [OF _ less])

  1120   qed

  1121

  1122 lemma inverse_less_imp_less:

  1123    "[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::ordered_field)"

  1124 apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"])

  1125 apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq)

  1126 done

  1127

  1128 text{*Both premises are essential. Consider -1 and 1.*}

  1129 lemma inverse_less_iff_less [simp]:

  1130      "[|0 < a; 0 < b|]

  1131       ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"

  1132 by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less)

  1133

  1134 lemma le_imp_inverse_le:

  1135    "[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::ordered_field)"

  1136   by (force simp add: order_le_less less_imp_inverse_less)

  1137

  1138 lemma inverse_le_iff_le [simp]:

  1139      "[|0 < a; 0 < b|]

  1140       ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"

  1141 by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le)

  1142

  1143

  1144 text{*These results refer to both operands being negative.  The opposite-sign

  1145 case is trivial, since inverse preserves signs.*}

  1146 lemma inverse_le_imp_le_neg:

  1147    "[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::ordered_field)"

  1148   apply (rule classical)

  1149   apply (subgoal_tac "a < 0")

  1150    prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans)

  1151   apply (insert inverse_le_imp_le [of "-b" "-a"])

  1152   apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq)

  1153   done

  1154

  1155 lemma less_imp_inverse_less_neg:

  1156    "[|a < b; b < 0|] ==> inverse b < inverse (a::'a::ordered_field)"

  1157   apply (subgoal_tac "a < 0")

  1158    prefer 2 apply (blast intro: order_less_trans)

  1159   apply (insert less_imp_inverse_less [of "-b" "-a"])

  1160   apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq)

  1161   done

  1162

  1163 lemma inverse_less_imp_less_neg:

  1164    "[|inverse a < inverse b; b < 0|] ==> b < (a::'a::ordered_field)"

  1165   apply (rule classical)

  1166   apply (subgoal_tac "a < 0")

  1167    prefer 2

  1168    apply (force simp add: linorder_not_less intro: order_le_less_trans)

  1169   apply (insert inverse_less_imp_less [of "-b" "-a"])

  1170   apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq)

  1171   done

  1172

  1173 lemma inverse_less_iff_less_neg [simp]:

  1174      "[|a < 0; b < 0|]

  1175       ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"

  1176   apply (insert inverse_less_iff_less [of "-b" "-a"])

  1177   apply (simp del: inverse_less_iff_less

  1178 	      add: order_less_imp_not_eq nonzero_inverse_minus_eq)

  1179   done

  1180

  1181 lemma le_imp_inverse_le_neg:

  1182    "[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)"

  1183   by (force simp add: order_le_less less_imp_inverse_less_neg)

  1184

  1185 lemma inverse_le_iff_le_neg [simp]:

  1186      "[|a < 0; b < 0|]

  1187       ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"

  1188 by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg)

  1189

  1190

  1191 subsection{*Inverses and the Number One*}

  1192

  1193 lemma one_less_inverse_iff:

  1194     "(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"proof cases

  1195   assume "0 < x"

  1196     with inverse_less_iff_less [OF zero_less_one, of x]

  1197     show ?thesis by simp

  1198 next

  1199   assume notless: "~ (0 < x)"

  1200   have "~ (1 < inverse x)"

  1201   proof

  1202     assume "1 < inverse x"

  1203     also with notless have "... \<le> 0" by (simp add: linorder_not_less)

  1204     also have "... < 1" by (rule zero_less_one)

  1205     finally show False by auto

  1206   qed

  1207   with notless show ?thesis by simp

  1208 qed

  1209

  1210 lemma inverse_eq_1_iff [simp]:

  1211     "(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))"

  1212 by (insert inverse_eq_iff_eq [of x 1], simp)

  1213

  1214 lemma one_le_inverse_iff:

  1215    "(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))"

  1216 by (force simp add: order_le_less one_less_inverse_iff zero_less_one

  1217                     eq_commute [of 1])

  1218

  1219 lemma inverse_less_1_iff:

  1220    "(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{ordered_field,division_by_zero}))"

  1221 by (simp add: linorder_not_le [symmetric] one_le_inverse_iff)

  1222

  1223 lemma inverse_le_1_iff:

  1224    "(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{ordered_field,division_by_zero}))"

  1225 by (simp add: linorder_not_less [symmetric] one_less_inverse_iff)

  1226

  1227

  1228 subsection{*Division and Signs*}

  1229

  1230 lemma zero_less_divide_iff:

  1231      "((0::'a::{ordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"

  1232 by (simp add: divide_inverse zero_less_mult_iff)

  1233

  1234 lemma divide_less_0_iff:

  1235      "(a/b < (0::'a::{ordered_field,division_by_zero})) =

  1236       (0 < a & b < 0 | a < 0 & 0 < b)"

  1237 by (simp add: divide_inverse mult_less_0_iff)

  1238

  1239 lemma zero_le_divide_iff:

  1240      "((0::'a::{ordered_field,division_by_zero}) \<le> a/b) =

  1241       (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"

  1242 by (simp add: divide_inverse zero_le_mult_iff)

  1243

  1244 lemma divide_le_0_iff:

  1245      "(a/b \<le> (0::'a::{ordered_field,division_by_zero})) =

  1246       (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"

  1247 by (simp add: divide_inverse mult_le_0_iff)

  1248

  1249 lemma divide_eq_0_iff [simp]:

  1250      "(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))"

  1251 by (simp add: divide_inverse field_mult_eq_0_iff)

  1252

  1253

  1254 subsection{*Simplification of Inequalities Involving Literal Divisors*}

  1255

  1256 lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"

  1257 proof -

  1258   assume less: "0<c"

  1259   hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"

  1260     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])

  1261   also have "... = (a*c \<le> b)"

  1262     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc)

  1263   finally show ?thesis .

  1264 qed

  1265

  1266

  1267 lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"

  1268 proof -

  1269   assume less: "c<0"

  1270   hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"

  1271     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])

  1272   also have "... = (b \<le> a*c)"

  1273     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc)

  1274   finally show ?thesis .

  1275 qed

  1276

  1277 lemma le_divide_eq:

  1278   "(a \<le> b/c) =

  1279    (if 0 < c then a*c \<le> b

  1280              else if c < 0 then b \<le> a*c

  1281              else  a \<le> (0::'a::{ordered_field,division_by_zero}))"

  1282 apply (case_tac "c=0", simp)

  1283 apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff)

  1284 done

  1285

  1286 lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)"

  1287 proof -

  1288   assume less: "0<c"

  1289   hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"

  1290     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])

  1291   also have "... = (b \<le> a*c)"

  1292     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc)

  1293   finally show ?thesis .

  1294 qed

  1295

  1296 lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)"

  1297 proof -

  1298   assume less: "c<0"

  1299   hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"

  1300     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])

  1301   also have "... = (a*c \<le> b)"

  1302     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc)

  1303   finally show ?thesis .

  1304 qed

  1305

  1306 lemma divide_le_eq:

  1307   "(b/c \<le> a) =

  1308    (if 0 < c then b \<le> a*c

  1309              else if c < 0 then a*c \<le> b

  1310              else 0 \<le> (a::'a::{ordered_field,division_by_zero}))"

  1311 apply (case_tac "c=0", simp)

  1312 apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff)

  1313 done

  1314

  1315

  1316 lemma pos_less_divide_eq:

  1317      "0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)"

  1318 proof -

  1319   assume less: "0<c"

  1320   hence "(a < b/c) = (a*c < (b/c)*c)"

  1321     by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])

  1322   also have "... = (a*c < b)"

  1323     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc)

  1324   finally show ?thesis .

  1325 qed

  1326

  1327 lemma neg_less_divide_eq:

  1328  "c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)"

  1329 proof -

  1330   assume less: "c<0"

  1331   hence "(a < b/c) = ((b/c)*c < a*c)"

  1332     by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])

  1333   also have "... = (b < a*c)"

  1334     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc)

  1335   finally show ?thesis .

  1336 qed

  1337

  1338 lemma less_divide_eq:

  1339   "(a < b/c) =

  1340    (if 0 < c then a*c < b

  1341              else if c < 0 then b < a*c

  1342              else  a < (0::'a::{ordered_field,division_by_zero}))"

  1343 apply (case_tac "c=0", simp)

  1344 apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff)

  1345 done

  1346

  1347 lemma pos_divide_less_eq:

  1348      "0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)"

  1349 proof -

  1350   assume less: "0<c"

  1351   hence "(b/c < a) = ((b/c)*c < a*c)"

  1352     by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])

  1353   also have "... = (b < a*c)"

  1354     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc)

  1355   finally show ?thesis .

  1356 qed

  1357

  1358 lemma neg_divide_less_eq:

  1359  "c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)"

  1360 proof -

  1361   assume less: "c<0"

  1362   hence "(b/c < a) = (a*c < (b/c)*c)"

  1363     by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])

  1364   also have "... = (a*c < b)"

  1365     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc)

  1366   finally show ?thesis .

  1367 qed

  1368

  1369 lemma divide_less_eq:

  1370   "(b/c < a) =

  1371    (if 0 < c then b < a*c

  1372              else if c < 0 then a*c < b

  1373              else 0 < (a::'a::{ordered_field,division_by_zero}))"

  1374 apply (case_tac "c=0", simp)

  1375 apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff)

  1376 done

  1377

  1378 lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)"

  1379 proof -

  1380   assume [simp]: "c\<noteq>0"

  1381   have "(a = b/c) = (a*c = (b/c)*c)"

  1382     by (simp add: field_mult_cancel_right)

  1383   also have "... = (a*c = b)"

  1384     by (simp add: divide_inverse mult_assoc)

  1385   finally show ?thesis .

  1386 qed

  1387

  1388 lemma eq_divide_eq:

  1389   "((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)"

  1390 by (simp add: nonzero_eq_divide_eq)

  1391

  1392 lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)"

  1393 proof -

  1394   assume [simp]: "c\<noteq>0"

  1395   have "(b/c = a) = ((b/c)*c = a*c)"

  1396     by (simp add: field_mult_cancel_right)

  1397   also have "... = (b = a*c)"

  1398     by (simp add: divide_inverse mult_assoc)

  1399   finally show ?thesis .

  1400 qed

  1401

  1402 lemma divide_eq_eq:

  1403   "(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)"

  1404 by (force simp add: nonzero_divide_eq_eq)

  1405

  1406

  1407 subsection{*Cancellation Laws for Division*}

  1408

  1409 lemma divide_cancel_right [simp]:

  1410      "(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))"

  1411 apply (case_tac "c=0", simp)

  1412 apply (simp add: divide_inverse field_mult_cancel_right)

  1413 done

  1414

  1415 lemma divide_cancel_left [simp]:

  1416      "(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))"

  1417 apply (case_tac "c=0", simp)

  1418 apply (simp add: divide_inverse field_mult_cancel_left)

  1419 done

  1420

  1421 subsection {* Division and the Number One *}

  1422

  1423 text{*Simplify expressions equated with 1*}

  1424 lemma divide_eq_1_iff [simp]:

  1425      "(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"

  1426 apply (case_tac "b=0", simp)

  1427 apply (simp add: right_inverse_eq)

  1428 done

  1429

  1430

  1431 lemma one_eq_divide_iff [simp]:

  1432      "(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"

  1433 by (simp add: eq_commute [of 1])

  1434

  1435 lemma zero_eq_1_divide_iff [simp]:

  1436      "((0::'a::{ordered_field,division_by_zero}) = 1/a) = (a = 0)"

  1437 apply (case_tac "a=0", simp)

  1438 apply (auto simp add: nonzero_eq_divide_eq)

  1439 done

  1440

  1441 lemma one_divide_eq_0_iff [simp]:

  1442      "(1/a = (0::'a::{ordered_field,division_by_zero})) = (a = 0)"

  1443 apply (case_tac "a=0", simp)

  1444 apply (insert zero_neq_one [THEN not_sym])

  1445 apply (auto simp add: nonzero_divide_eq_eq)

  1446 done

  1447

  1448 text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}

  1449 declare zero_less_divide_iff [of "1", simp]

  1450 declare divide_less_0_iff [of "1", simp]

  1451 declare zero_le_divide_iff [of "1", simp]

  1452 declare divide_le_0_iff [of "1", simp]

  1453

  1454

  1455 subsection {* Ordering Rules for Division *}

  1456

  1457 lemma divide_strict_right_mono:

  1458      "[|a < b; 0 < c|] ==> a / c < b / (c::'a::ordered_field)"

  1459 by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono

  1460               positive_imp_inverse_positive)

  1461

  1462 lemma divide_right_mono:

  1463      "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{ordered_field,division_by_zero})"

  1464   by (force simp add: divide_strict_right_mono order_le_less)

  1465

  1466

  1467 text{*The last premise ensures that @{term a} and @{term b}

  1468       have the same sign*}

  1469 lemma divide_strict_left_mono:

  1470        "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"

  1471 by (force simp add: zero_less_mult_iff divide_inverse mult_strict_left_mono

  1472       order_less_imp_not_eq order_less_imp_not_eq2

  1473       less_imp_inverse_less less_imp_inverse_less_neg)

  1474

  1475 lemma divide_left_mono:

  1476      "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::ordered_field)"

  1477   apply (subgoal_tac "a \<noteq> 0 & b \<noteq> 0")

  1478    prefer 2

  1479    apply (force simp add: zero_less_mult_iff order_less_imp_not_eq)

  1480   apply (case_tac "c=0", simp add: divide_inverse)

  1481   apply (force simp add: divide_strict_left_mono order_le_less)

  1482   done

  1483

  1484 lemma divide_strict_left_mono_neg:

  1485      "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"

  1486   apply (subgoal_tac "a \<noteq> 0 & b \<noteq> 0")

  1487    prefer 2

  1488    apply (force simp add: zero_less_mult_iff order_less_imp_not_eq)

  1489   apply (drule divide_strict_left_mono [of _ _ "-c"])

  1490    apply (simp_all add: mult_commute nonzero_minus_divide_left [symmetric])

  1491   done

  1492

  1493 lemma divide_strict_right_mono_neg:

  1494      "[|b < a; c < 0|] ==> a / c < b / (c::'a::ordered_field)"

  1495 apply (drule divide_strict_right_mono [of _ _ "-c"], simp)

  1496 apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric])

  1497 done

  1498

  1499

  1500 subsection {* Ordered Fields are Dense *}

  1501

  1502 lemma less_add_one: "a < (a+1::'a::ordered_semidom)"

  1503 proof -

  1504   have "a+0 < (a+1::'a::ordered_semidom)"

  1505     by (blast intro: zero_less_one add_strict_left_mono)

  1506   thus ?thesis by simp

  1507 qed

  1508

  1509 lemma zero_less_two: "0 < (1+1::'a::ordered_semidom)"

  1510   by (blast intro: order_less_trans zero_less_one less_add_one)

  1511

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

  1513 by (simp add: zero_less_two pos_less_divide_eq right_distrib)

  1514

  1515 lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::ordered_field) < b"

  1516 by (simp add: zero_less_two pos_divide_less_eq right_distrib)

  1517

  1518 lemma dense: "a < b ==> \<exists>r::'a::ordered_field. a < r & r < b"

  1519 by (blast intro!: less_half_sum gt_half_sum)

  1520

  1521

  1522 lemmas times_divide_eq = times_divide_eq_right times_divide_eq_left

  1523

  1524 text{*It's not obvious whether these should be simprules or not.

  1525   Their effect is to gather terms into one big fraction, like

  1526   a*b*c / x*y*z. The rationale for that is unclear, but many proofs

  1527   seem to need them.*}

  1528

  1529 declare times_divide_eq [simp]

  1530

  1531

  1532 subsection {* Absolute Value *}

  1533

  1534 lemma abs_one [simp]: "abs 1 = (1::'a::ordered_idom)"

  1535   by (simp add: abs_if zero_less_one [THEN order_less_not_sym])

  1536

  1537 lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lordered_ring))"

  1538 proof -

  1539   let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"

  1540   let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"

  1541   have a: "(abs a) * (abs b) = ?x"

  1542     by (simp only: abs_prts[of a] abs_prts[of b] ring_eq_simps)

  1543   {

  1544     fix u v :: 'a

  1545     have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow>

  1546               u * v = pprt a * pprt b + pprt a * nprt b +

  1547                       nprt a * pprt b + nprt a * nprt b"

  1548       apply (subst prts[of u], subst prts[of v])

  1549       apply (simp add: left_distrib right_distrib add_ac)

  1550       done

  1551   }

  1552   note b = this[OF refl[of a] refl[of b]]

  1553   note addm = add_mono[of "0::'a" _ "0::'a", simplified]

  1554   note addm2 = add_mono[of _ "0::'a" _ "0::'a", simplified]

  1555   have xy: "- ?x <= ?y"

  1556     apply (simp)

  1557     apply (rule_tac y="0::'a" in order_trans)

  1558     apply (rule addm2)

  1559     apply (simp_all add: mult_pos_le mult_neg_le)

  1560     apply (rule addm)

  1561     apply (simp_all add: mult_pos_le mult_neg_le)

  1562     done

  1563   have yx: "?y <= ?x"

  1564     apply (simp add:diff_def)

  1565     apply (rule_tac y=0 in order_trans)

  1566     apply (rule addm2, (simp add: mult_pos_neg_le mult_pos_neg2_le)+)

  1567     apply (rule addm, (simp add: mult_pos_neg_le mult_pos_neg2_le)+)

  1568     done

  1569   have i1: "a*b <= abs a * abs b" by (simp only: a b yx)

  1570   have i2: "- (abs a * abs b) <= a*b" by (simp only: a b xy)

  1571   show ?thesis

  1572     apply (rule abs_leI)

  1573     apply (simp add: i1)

  1574     apply (simp add: i2[simplified minus_le_iff])

  1575     done

  1576 qed

  1577

  1578 lemma abs_eq_mult:

  1579   assumes "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)"

  1580   shows "abs (a*b) = abs a * abs (b::'a::lordered_ring)"

  1581 proof -

  1582   have s: "(0 <= a*b) | (a*b <= 0)"

  1583     apply (auto)

  1584     apply (rule_tac split_mult_pos_le)

  1585     apply (rule_tac contrapos_np[of "a*b <= 0"])

  1586     apply (simp)

  1587     apply (rule_tac split_mult_neg_le)

  1588     apply (insert prems)

  1589     apply (blast)

  1590     done

  1591   have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)"

  1592     by (simp add: prts[symmetric])

  1593   show ?thesis

  1594   proof cases

  1595     assume "0 <= a * b"

  1596     then show ?thesis

  1597       apply (simp_all add: mulprts abs_prts)

  1598       apply (insert prems)

  1599       apply (auto simp add:

  1600 	ring_eq_simps

  1601 	iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_zero_pprt]

  1602 	iff2imp[OF le_zero_iff_pprt_id] iff2imp[OF zero_le_iff_nprt_id])

  1603 	apply(drule (1) mult_pos_neg_le[of a b], simp)

  1604 	apply(drule (1) mult_pos_neg2_le[of b a], simp)

  1605       done

  1606   next

  1607     assume "~(0 <= a*b)"

  1608     with s have "a*b <= 0" by simp

  1609     then show ?thesis

  1610       apply (simp_all add: mulprts abs_prts)

  1611       apply (insert prems)

  1612       apply (auto simp add: ring_eq_simps)

  1613       apply(drule (1) mult_pos_le[of a b],simp)

  1614       apply(drule (1) mult_neg_le[of a b],simp)

  1615       done

  1616   qed

  1617 qed

  1618

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

  1620 by (simp add: abs_eq_mult linorder_linear)

  1621

  1622 lemma abs_mult_self: "abs a * abs a = a * (a::'a::ordered_idom)"

  1623 by (simp add: abs_if)

  1624

  1625 lemma nonzero_abs_inverse:

  1626      "a \<noteq> 0 ==> abs (inverse (a::'a::ordered_field)) = inverse (abs a)"

  1627 apply (auto simp add: linorder_neq_iff abs_if nonzero_inverse_minus_eq

  1628                       negative_imp_inverse_negative)

  1629 apply (blast intro: positive_imp_inverse_positive elim: order_less_asym)

  1630 done

  1631

  1632 lemma abs_inverse [simp]:

  1633      "abs (inverse (a::'a::{ordered_field,division_by_zero})) =

  1634       inverse (abs a)"

  1635 apply (case_tac "a=0", simp)

  1636 apply (simp add: nonzero_abs_inverse)

  1637 done

  1638

  1639 lemma nonzero_abs_divide:

  1640      "b \<noteq> 0 ==> abs (a / (b::'a::ordered_field)) = abs a / abs b"

  1641 by (simp add: divide_inverse abs_mult nonzero_abs_inverse)

  1642

  1643 lemma abs_divide [simp]:

  1644      "abs (a / (b::'a::{ordered_field,division_by_zero})) = abs a / abs b"

  1645 apply (case_tac "b=0", simp)

  1646 apply (simp add: nonzero_abs_divide)

  1647 done

  1648

  1649 lemma abs_mult_less:

  1650      "[| abs a < c; abs b < d |] ==> abs a * abs b < c*(d::'a::ordered_idom)"

  1651 proof -

  1652   assume ac: "abs a < c"

  1653   hence cpos: "0<c" by (blast intro: order_le_less_trans abs_ge_zero)

  1654   assume "abs b < d"

  1655   thus ?thesis by (simp add: ac cpos mult_strict_mono)

  1656 qed

  1657

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

  1659 by (force simp add: order_eq_iff le_minus_self_iff minus_le_self_iff)

  1660

  1661 lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::ordered_idom))"

  1662 by (simp add: order_less_le le_minus_self_iff eq_minus_self_iff)

  1663

  1664 lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::ordered_idom))"

  1665 apply (simp add: order_less_le abs_le_iff)

  1666 apply (auto simp add: abs_if minus_le_self_iff eq_minus_self_iff)

  1667 apply (simp add: le_minus_self_iff linorder_neq_iff)

  1668 done

  1669

  1670 lemma linprog_dual_estimate:

  1671   assumes

  1672   "A * x \<le> (b::'a::lordered_ring)"

  1673   "0 \<le> y"

  1674   "abs (A - A') \<le> \<delta>A"

  1675   "b \<le> b'"

  1676   "abs (c - c') \<le> \<delta>c"

  1677   "abs x \<le> r"

  1678   shows

  1679   "c * x \<le> y * b' + (y * \<delta>A + abs (y * A' - c') + \<delta>c) * r"

  1680 proof -

  1681   from prems have 1: "y * b <= y * b'" by (simp add: mult_left_mono)

  1682   from prems have 2: "y * (A * x) <= y * b" by (simp add: mult_left_mono)

  1683   have 3: "y * (A * x) = c * x + (y * (A - A') + (y * A' - c') + (c'-c)) * x" by (simp add: ring_eq_simps)

  1684   from 1 2 3 have 4: "c * x + (y * (A - A') + (y * A' - c') + (c'-c)) * x <= y * b'" by simp

  1685   have 5: "c * x <= y * b' + abs((y * (A - A') + (y * A' - c') + (c'-c)) * x)"

  1686     by (simp only: 4 estimate_by_abs)

  1687   have 6: "abs((y * (A - A') + (y * A' - c') + (c'-c)) * x) <= abs (y * (A - A') + (y * A' - c') + (c'-c)) * abs x"

  1688     by (simp add: abs_le_mult)

  1689   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"

  1690     by(rule abs_triangle_ineq [THEN mult_right_mono]) simp

  1691   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"

  1692     by (simp add: abs_triangle_ineq mult_right_mono)

  1693   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"

  1694     by (simp add: abs_le_mult mult_right_mono)

  1695   have 10: "c'-c = -(c-c')" by (simp add: ring_eq_simps)

  1696   have 11: "abs (c'-c) = abs (c-c')"

  1697     by (subst 10, subst abs_minus_cancel, simp)

  1698   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"

  1699     by (simp add: 11 prems mult_right_mono)

  1700   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"

  1701     by (simp add: prems mult_right_mono mult_left_mono)

  1702   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"

  1703     apply (rule mult_left_mono)

  1704     apply (simp add: prems)

  1705     apply (rule_tac add_mono[of "0::'a" _ "0", simplified])+

  1706     apply (rule mult_left_mono[of "0" "\<delta>A", simplified])

  1707     apply (simp_all)

  1708     apply (rule order_trans[where y="abs (A-A')"], simp_all add: prems)

  1709     apply (rule order_trans[where y="abs (c-c')"], simp_all add: prems)

  1710     done

  1711   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"

  1712     by (simp)

  1713   show ?thesis

  1714     apply (rule_tac le_add_right_mono[of _ _ "abs((y * (A - A') + (y * A' - c') + (c'-c)) * x)"])

  1715     apply (simp_all only: 5 14[simplified abs_of_ge_0[of y, simplified prems]])

  1716     done

  1717 qed

  1718

  1719 lemma le_ge_imp_abs_diff_1:

  1720   assumes

  1721   "A1 <= (A::'a::lordered_ring)"

  1722   "A <= A2"

  1723   shows "abs (A-A1) <= A2-A1"

  1724 proof -

  1725   have "0 <= A - A1"

  1726   proof -

  1727     have 1: "A - A1 = A + (- A1)" by simp

  1728     show ?thesis by (simp only: 1 add_right_mono[of A1 A "-A1", simplified, simplified prems])

  1729   qed

  1730   then have "abs (A-A1) = A-A1" by (rule abs_of_ge_0)

  1731   with prems show "abs (A-A1) <= (A2-A1)" by simp

  1732 qed

  1733

  1734 lemma mult_le_prts:

  1735   assumes

  1736   "a1 <= (a::'a::lordered_ring)"

  1737   "a <= a2"

  1738   "b1 <= b"

  1739   "b <= b2"

  1740   shows

  1741   "a * b <= pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"

  1742 proof -

  1743   have "a * b = (pprt a + nprt a) * (pprt b + nprt b)"

  1744     apply (subst prts[symmetric])+

  1745     apply simp

  1746     done

  1747   then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"

  1748     by (simp add: ring_eq_simps)

  1749   moreover have "pprt a * pprt b <= pprt a2 * pprt b2"

  1750     by (simp_all add: prems mult_mono)

  1751   moreover have "pprt a * nprt b <= pprt a1 * nprt b2"

  1752   proof -

  1753     have "pprt a * nprt b <= pprt a * nprt b2"

  1754       by (simp add: mult_left_mono prems)

  1755     moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2"

  1756       by (simp add: mult_right_mono_neg prems)

  1757     ultimately show ?thesis

  1758       by simp

  1759   qed

  1760   moreover have "nprt a * pprt b <= nprt a2 * pprt b1"

  1761   proof -

  1762     have "nprt a * pprt b <= nprt a2 * pprt b"

  1763       by (simp add: mult_right_mono prems)

  1764     moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"

  1765       by (simp add: mult_left_mono_neg prems)

  1766     ultimately show ?thesis

  1767       by simp

  1768   qed

  1769   moreover have "nprt a * nprt b <= nprt a1 * nprt b1"

  1770   proof -

  1771     have "nprt a * nprt b <= nprt a * nprt b1"

  1772       by (simp add: mult_left_mono_neg prems)

  1773     moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1"

  1774       by (simp add: mult_right_mono_neg prems)

  1775     ultimately show ?thesis

  1776       by simp

  1777   qed

  1778   ultimately show ?thesis

  1779     by - (rule add_mono | simp)+

  1780 qed

  1781

  1782 lemma mult_le_dual_prts:

  1783   assumes

  1784   "A * x \<le> (b::'a::lordered_ring)"

  1785   "0 \<le> y"

  1786   "A1 \<le> A"

  1787   "A \<le> A2"

  1788   "c1 \<le> c"

  1789   "c \<le> c2"

  1790   "r1 \<le> x"

  1791   "x \<le> r2"

  1792   shows

  1793   "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)"

  1794   (is "_ <= _ + ?C")

  1795 proof -

  1796   from prems have "y * (A * x) <= y * b" by (simp add: mult_left_mono)

  1797   moreover have "y * (A * x) = c * x + (y * A - c) * x" by (simp add: ring_eq_simps)

  1798   ultimately have "c * x + (y * A - c) * x <= y * b" by simp

  1799   then have "c * x <= y * b - (y * A - c) * x" by (simp add: le_diff_eq)

  1800   then have cx: "c * x <= y * b + (c - y * A) * x" by (simp add: ring_eq_simps)

  1801   have s2: "c - y * A <= c2 - y * A1"

  1802     by (simp add: diff_def prems add_mono mult_left_mono)

  1803   have s1: "c1 - y * A2 <= c - y * A"

  1804     by (simp add: diff_def prems add_mono mult_left_mono)

  1805   have prts: "(c - y * A) * x <= ?C"

  1806     apply (simp add: Let_def)

  1807     apply (rule mult_le_prts)

  1808     apply (simp_all add: prems s1 s2)

  1809     done

  1810   then have "y * b + (c - y * A) * x <= y * b + ?C"

  1811     by simp

  1812   with cx show ?thesis

  1813     by(simp only:)

  1814 qed

  1815

  1816 ML {*

  1817 val left_distrib = thm "left_distrib";

  1818 val right_distrib = thm "right_distrib";

  1819 val mult_commute = thm "mult_commute";

  1820 val distrib = thm "distrib";

  1821 val zero_neq_one = thm "zero_neq_one";

  1822 val no_zero_divisors = thm "no_zero_divisors";

  1823 val left_inverse = thm "left_inverse";

  1824 val divide_inverse = thm "divide_inverse";

  1825 val mult_zero_left = thm "mult_zero_left";

  1826 val mult_zero_right = thm "mult_zero_right";

  1827 val field_mult_eq_0_iff = thm "field_mult_eq_0_iff";

  1828 val inverse_zero = thm "inverse_zero";

  1829 val ring_distrib = thms "ring_distrib";

  1830 val combine_common_factor = thm "combine_common_factor";

  1831 val minus_mult_left = thm "minus_mult_left";

  1832 val minus_mult_right = thm "minus_mult_right";

  1833 val minus_mult_minus = thm "minus_mult_minus";

  1834 val minus_mult_commute = thm "minus_mult_commute";

  1835 val right_diff_distrib = thm "right_diff_distrib";

  1836 val left_diff_distrib = thm "left_diff_distrib";

  1837 val mult_left_mono = thm "mult_left_mono";

  1838 val mult_right_mono = thm "mult_right_mono";

  1839 val mult_strict_left_mono = thm "mult_strict_left_mono";

  1840 val mult_strict_right_mono = thm "mult_strict_right_mono";

  1841 val mult_mono = thm "mult_mono";

  1842 val mult_strict_mono = thm "mult_strict_mono";

  1843 val abs_if = thm "abs_if";

  1844 val zero_less_one = thm "zero_less_one";

  1845 val eq_add_iff1 = thm "eq_add_iff1";

  1846 val eq_add_iff2 = thm "eq_add_iff2";

  1847 val less_add_iff1 = thm "less_add_iff1";

  1848 val less_add_iff2 = thm "less_add_iff2";

  1849 val le_add_iff1 = thm "le_add_iff1";

  1850 val le_add_iff2 = thm "le_add_iff2";

  1851 val mult_left_le_imp_le = thm "mult_left_le_imp_le";

  1852 val mult_right_le_imp_le = thm "mult_right_le_imp_le";

  1853 val mult_left_less_imp_less = thm "mult_left_less_imp_less";

  1854 val mult_right_less_imp_less = thm "mult_right_less_imp_less";

  1855 val mult_strict_left_mono_neg = thm "mult_strict_left_mono_neg";

  1856 val mult_left_mono_neg = thm "mult_left_mono_neg";

  1857 val mult_strict_right_mono_neg = thm "mult_strict_right_mono_neg";

  1858 val mult_right_mono_neg = thm "mult_right_mono_neg";

  1859 val mult_pos = thm "mult_pos";

  1860 val mult_pos_le = thm "mult_pos_le";

  1861 val mult_pos_neg = thm "mult_pos_neg";

  1862 val mult_pos_neg_le = thm "mult_pos_neg_le";

  1863 val mult_pos_neg2 = thm "mult_pos_neg2";

  1864 val mult_pos_neg2_le = thm "mult_pos_neg2_le";

  1865 val mult_neg = thm "mult_neg";

  1866 val mult_neg_le = thm "mult_neg_le";

  1867 val zero_less_mult_pos = thm "zero_less_mult_pos";

  1868 val zero_less_mult_pos2 = thm "zero_less_mult_pos2";

  1869 val zero_less_mult_iff = thm "zero_less_mult_iff";

  1870 val mult_eq_0_iff = thm "mult_eq_0_iff";

  1871 val zero_le_mult_iff = thm "zero_le_mult_iff";

  1872 val mult_less_0_iff = thm "mult_less_0_iff";

  1873 val mult_le_0_iff = thm "mult_le_0_iff";

  1874 val split_mult_pos_le = thm "split_mult_pos_le";

  1875 val split_mult_neg_le = thm "split_mult_neg_le";

  1876 val zero_le_square = thm "zero_le_square";

  1877 val zero_le_one = thm "zero_le_one";

  1878 val not_one_le_zero = thm "not_one_le_zero";

  1879 val not_one_less_zero = thm "not_one_less_zero";

  1880 val mult_left_mono_neg = thm "mult_left_mono_neg";

  1881 val mult_right_mono_neg = thm "mult_right_mono_neg";

  1882 val mult_strict_mono = thm "mult_strict_mono";

  1883 val mult_strict_mono' = thm "mult_strict_mono'";

  1884 val mult_mono = thm "mult_mono";

  1885 val less_1_mult = thm "less_1_mult";

  1886 val mult_less_cancel_right_disj = thm "mult_less_cancel_right_disj";

  1887 val mult_less_cancel_left_disj = thm "mult_less_cancel_left_disj";

  1888 val mult_less_cancel_right = thm "mult_less_cancel_right";

  1889 val mult_less_cancel_left = thm "mult_less_cancel_left";

  1890 val mult_le_cancel_right = thm "mult_le_cancel_right";

  1891 val mult_le_cancel_left = thm "mult_le_cancel_left";

  1892 val mult_less_imp_less_left = thm "mult_less_imp_less_left";

  1893 val mult_less_imp_less_right = thm "mult_less_imp_less_right";

  1894 val mult_cancel_right = thm "mult_cancel_right";

  1895 val mult_cancel_left = thm "mult_cancel_left";

  1896 val ring_eq_simps = thms "ring_eq_simps";

  1897 val right_inverse = thm "right_inverse";

  1898 val right_inverse_eq = thm "right_inverse_eq";

  1899 val nonzero_inverse_eq_divide = thm "nonzero_inverse_eq_divide";

  1900 val divide_self = thm "divide_self";

  1901 val divide_zero = thm "divide_zero";

  1902 val divide_zero_left = thm "divide_zero_left";

  1903 val inverse_eq_divide = thm "inverse_eq_divide";

  1904 val add_divide_distrib = thm "add_divide_distrib";

  1905 val field_mult_eq_0_iff = thm "field_mult_eq_0_iff";

  1906 val field_mult_cancel_right_lemma = thm "field_mult_cancel_right_lemma";

  1907 val field_mult_cancel_right = thm "field_mult_cancel_right";

  1908 val field_mult_cancel_left = thm "field_mult_cancel_left";

  1909 val nonzero_imp_inverse_nonzero = thm "nonzero_imp_inverse_nonzero";

  1910 val inverse_zero_imp_zero = thm "inverse_zero_imp_zero";

  1911 val inverse_nonzero_imp_nonzero = thm "inverse_nonzero_imp_nonzero";

  1912 val inverse_nonzero_iff_nonzero = thm "inverse_nonzero_iff_nonzero";

  1913 val nonzero_inverse_minus_eq = thm "nonzero_inverse_minus_eq";

  1914 val inverse_minus_eq = thm "inverse_minus_eq";

  1915 val nonzero_inverse_eq_imp_eq = thm "nonzero_inverse_eq_imp_eq";

  1916 val inverse_eq_imp_eq = thm "inverse_eq_imp_eq";

  1917 val inverse_eq_iff_eq = thm "inverse_eq_iff_eq";

  1918 val nonzero_inverse_inverse_eq = thm "nonzero_inverse_inverse_eq";

  1919 val inverse_inverse_eq = thm "inverse_inverse_eq";

  1920 val inverse_1 = thm "inverse_1";

  1921 val nonzero_inverse_mult_distrib = thm "nonzero_inverse_mult_distrib";

  1922 val inverse_mult_distrib = thm "inverse_mult_distrib";

  1923 val inverse_add = thm "inverse_add";

  1924 val inverse_divide = thm "inverse_divide";

  1925 val nonzero_mult_divide_cancel_left = thm "nonzero_mult_divide_cancel_left";

  1926 val mult_divide_cancel_left = thm "mult_divide_cancel_left";

  1927 val nonzero_mult_divide_cancel_right = thm "nonzero_mult_divide_cancel_right";

  1928 val mult_divide_cancel_right = thm "mult_divide_cancel_right";

  1929 val mult_divide_cancel_eq_if = thm "mult_divide_cancel_eq_if";

  1930 val divide_1 = thm "divide_1";

  1931 val times_divide_eq_right = thm "times_divide_eq_right";

  1932 val times_divide_eq_left = thm "times_divide_eq_left";

  1933 val divide_divide_eq_right = thm "divide_divide_eq_right";

  1934 val divide_divide_eq_left = thm "divide_divide_eq_left";

  1935 val nonzero_minus_divide_left = thm "nonzero_minus_divide_left";

  1936 val nonzero_minus_divide_right = thm "nonzero_minus_divide_right";

  1937 val nonzero_minus_divide_divide = thm "nonzero_minus_divide_divide";

  1938 val minus_divide_left = thm "minus_divide_left";

  1939 val minus_divide_right = thm "minus_divide_right";

  1940 val minus_divide_divide = thm "minus_divide_divide";

  1941 val diff_divide_distrib = thm "diff_divide_distrib";

  1942 val positive_imp_inverse_positive = thm "positive_imp_inverse_positive";

  1943 val negative_imp_inverse_negative = thm "negative_imp_inverse_negative";

  1944 val inverse_le_imp_le = thm "inverse_le_imp_le";

  1945 val inverse_positive_imp_positive = thm "inverse_positive_imp_positive";

  1946 val inverse_positive_iff_positive = thm "inverse_positive_iff_positive";

  1947 val inverse_negative_imp_negative = thm "inverse_negative_imp_negative";

  1948 val inverse_negative_iff_negative = thm "inverse_negative_iff_negative";

  1949 val inverse_nonnegative_iff_nonnegative = thm "inverse_nonnegative_iff_nonnegative";

  1950 val inverse_nonpositive_iff_nonpositive = thm "inverse_nonpositive_iff_nonpositive";

  1951 val less_imp_inverse_less = thm "less_imp_inverse_less";

  1952 val inverse_less_imp_less = thm "inverse_less_imp_less";

  1953 val inverse_less_iff_less = thm "inverse_less_iff_less";

  1954 val le_imp_inverse_le = thm "le_imp_inverse_le";

  1955 val inverse_le_iff_le = thm "inverse_le_iff_le";

  1956 val inverse_le_imp_le_neg = thm "inverse_le_imp_le_neg";

  1957 val less_imp_inverse_less_neg = thm "less_imp_inverse_less_neg";

  1958 val inverse_less_imp_less_neg = thm "inverse_less_imp_less_neg";

  1959 val inverse_less_iff_less_neg = thm "inverse_less_iff_less_neg";

  1960 val le_imp_inverse_le_neg = thm "le_imp_inverse_le_neg";

  1961 val inverse_le_iff_le_neg = thm "inverse_le_iff_le_neg";

  1962 val one_less_inverse_iff = thm "one_less_inverse_iff";

  1963 val inverse_eq_1_iff = thm "inverse_eq_1_iff";

  1964 val one_le_inverse_iff = thm "one_le_inverse_iff";

  1965 val inverse_less_1_iff = thm "inverse_less_1_iff";

  1966 val inverse_le_1_iff = thm "inverse_le_1_iff";

  1967 val zero_less_divide_iff = thm "zero_less_divide_iff";

  1968 val divide_less_0_iff = thm "divide_less_0_iff";

  1969 val zero_le_divide_iff = thm "zero_le_divide_iff";

  1970 val divide_le_0_iff = thm "divide_le_0_iff";

  1971 val divide_eq_0_iff = thm "divide_eq_0_iff";

  1972 val pos_le_divide_eq = thm "pos_le_divide_eq";

  1973 val neg_le_divide_eq = thm "neg_le_divide_eq";

  1974 val le_divide_eq = thm "le_divide_eq";

  1975 val pos_divide_le_eq = thm "pos_divide_le_eq";

  1976 val neg_divide_le_eq = thm "neg_divide_le_eq";

  1977 val divide_le_eq = thm "divide_le_eq";

  1978 val pos_less_divide_eq = thm "pos_less_divide_eq";

  1979 val neg_less_divide_eq = thm "neg_less_divide_eq";

  1980 val less_divide_eq = thm "less_divide_eq";

  1981 val pos_divide_less_eq = thm "pos_divide_less_eq";

  1982 val neg_divide_less_eq = thm "neg_divide_less_eq";

  1983 val divide_less_eq = thm "divide_less_eq";

  1984 val nonzero_eq_divide_eq = thm "nonzero_eq_divide_eq";

  1985 val eq_divide_eq = thm "eq_divide_eq";

  1986 val nonzero_divide_eq_eq = thm "nonzero_divide_eq_eq";

  1987 val divide_eq_eq = thm "divide_eq_eq";

  1988 val divide_cancel_right = thm "divide_cancel_right";

  1989 val divide_cancel_left = thm "divide_cancel_left";

  1990 val divide_eq_1_iff = thm "divide_eq_1_iff";

  1991 val one_eq_divide_iff = thm "one_eq_divide_iff";

  1992 val zero_eq_1_divide_iff = thm "zero_eq_1_divide_iff";

  1993 val one_divide_eq_0_iff = thm "one_divide_eq_0_iff";

  1994 val divide_strict_right_mono = thm "divide_strict_right_mono";

  1995 val divide_right_mono = thm "divide_right_mono";

  1996 val divide_strict_left_mono = thm "divide_strict_left_mono";

  1997 val divide_left_mono = thm "divide_left_mono";

  1998 val divide_strict_left_mono_neg = thm "divide_strict_left_mono_neg";

  1999 val divide_strict_right_mono_neg = thm "divide_strict_right_mono_neg";

  2000 val less_add_one = thm "less_add_one";

  2001 val zero_less_two = thm "zero_less_two";

  2002 val less_half_sum = thm "less_half_sum";

  2003 val gt_half_sum = thm "gt_half_sum";

  2004 val dense = thm "dense";

  2005 val abs_one = thm "abs_one";

  2006 val abs_le_mult = thm "abs_le_mult";

  2007 val abs_eq_mult = thm "abs_eq_mult";

  2008 val abs_mult = thm "abs_mult";

  2009 val abs_mult_self = thm "abs_mult_self";

  2010 val nonzero_abs_inverse = thm "nonzero_abs_inverse";

  2011 val abs_inverse = thm "abs_inverse";

  2012 val nonzero_abs_divide = thm "nonzero_abs_divide";

  2013 val abs_divide = thm "abs_divide";

  2014 val abs_mult_less = thm "abs_mult_less";

  2015 val eq_minus_self_iff = thm "eq_minus_self_iff";

  2016 val less_minus_self_iff = thm "less_minus_self_iff";

  2017 val abs_less_iff = thm "abs_less_iff";

  2018 *}

  2019

  2020 end