src/HOL/Ring_and_Field.thy
 author obua Mon Jun 14 14:20:55 2004 +0200 (2004-06-14) changeset 14940 b9ab8babd8b3 parent 14770 fe9504ba63d5 child 15010 72fbe711e414 permissions -rw-r--r--
Further development of matrix theory
     1 (*  Title:   HOL/Ring_and_Field.thy

     2     ID:      $Id$

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

     4     License: GPL (GNU GENERAL PUBLIC LICENSE)

     5 *)

     6

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

     8

     9 theory Ring_and_Field = OrderedGroup:

    10

    11 text {*

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

    13   \begin{itemize}

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

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

    16   \end{itemize}

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

    18   \begin{itemize}

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

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

    21   \end{itemize}

    22 *}

    23

    24 axclass semiring \<subseteq> ab_semigroup_add, semigroup_mult

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

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

    27

    28 axclass semiring_0 \<subseteq> semiring, comm_monoid_add

    29

    30 axclass semiring_0_cancel \<subseteq> semiring_0, cancel_ab_semigroup_add

    31

    32 axclass comm_semiring \<subseteq> ab_semigroup_add, ab_semigroup_mult

    33   mult_commute: "a * b = b * a"

    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 lemma mult_less_cancel_right:

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

   476 apply (case_tac "c = 0")

   477 apply (auto simp add: linorder_neq_iff mult_strict_right_mono

   478                       mult_strict_right_mono_neg)

   479 apply (auto simp add: linorder_not_less

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

   481                       linorder_not_le [symmetric, of a])

   482 apply (erule_tac [!] notE)

   483 apply (auto simp add: order_less_imp_le mult_right_mono

   484                       mult_right_mono_neg)

   485 done

   486

   487 lemma mult_less_cancel_left:

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

   489 apply (case_tac "c = 0")

   490 apply (auto simp add: linorder_neq_iff mult_strict_left_mono

   491                       mult_strict_left_mono_neg)

   492 apply (auto simp add: linorder_not_less

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

   494                       linorder_not_le [symmetric, of a])

   495 apply (erule_tac [!] notE)

   496 apply (auto simp add: order_less_imp_le mult_left_mono

   497                       mult_left_mono_neg)

   498 done

   499

   500 lemma mult_le_cancel_right:

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

   502 by (simp add: linorder_not_less [symmetric] mult_less_cancel_right)

   503

   504 lemma mult_le_cancel_left:

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

   506 by (simp add: linorder_not_less [symmetric] mult_less_cancel_left)

   507

   508 lemma mult_less_imp_less_left:

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

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

   511 proof (rule ccontr)

   512   assume "~ a < b"

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

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

   515   with this and less show False

   516     by (simp add: linorder_not_less [symmetric])

   517 qed

   518

   519 lemma mult_less_imp_less_right:

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

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

   522 proof (rule ccontr)

   523   assume "~ a < b"

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

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

   526   with this and less show False

   527     by (simp add: linorder_not_less [symmetric])

   528 qed

   529

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

   531 lemma mult_cancel_right [simp]:

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

   533 apply (cut_tac linorder_less_linear [of 0 c])

   534 apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono

   535              simp add: linorder_neq_iff)

   536 done

   537

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

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

   540 lemma mult_cancel_left [simp]:

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

   542 apply (cut_tac linorder_less_linear [of 0 c])

   543 apply (force dest: mult_strict_left_mono_neg mult_strict_left_mono

   544              simp add: linorder_neq_iff)

   545 done

   546

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

   548 lemmas ring_eq_simps =

   549   mult_ac

   550   left_distrib right_distrib left_diff_distrib right_diff_distrib

   551   add_ac

   552   add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2

   553   diff_eq_eq eq_diff_eq

   554

   555

   556 subsection {* Fields *}

   557

   558 lemma right_inverse [simp]:

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

   560 proof -

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

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

   563   finally show ?thesis .

   564 qed

   565

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

   567 proof

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

   569   {

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

   571     also assume "a / b = 1"

   572     finally show "a = b" by simp

   573   next

   574     assume "a = b"

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

   576   }

   577 qed

   578

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

   580 by (simp add: divide_inverse)

   581

   582 lemma divide_self [simp]: "a \<noteq> 0 ==> a / (a::'a::field) = 1"

   583   by (simp add: divide_inverse)

   584

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

   586 by (simp add: divide_inverse)

   587

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

   589 by (simp add: divide_inverse)

   590

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

   592 by (simp add: divide_inverse)

   593

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

   595 by (simp add: divide_inverse left_distrib)

   596

   597

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

   599       of an ordering.*}

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

   601 proof cases

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

   603 next

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

   605   { assume "a * b = 0"

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

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

   608   thus ?thesis by force

   609 qed

   610

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

   612 lemma field_mult_cancel_right_lemma:

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

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

   615 	 shows "a=b"

   616 proof -

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

   618     by (simp add: eq)

   619   thus "a=b"

   620     by (simp add: mult_assoc cnz)

   621 qed

   622

   623 lemma field_mult_cancel_right [simp]:

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

   625 proof cases

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

   627 next

   628   assume "c\<noteq>0"

   629   thus ?thesis by (force dest: field_mult_cancel_right_lemma)

   630 qed

   631

   632 lemma field_mult_cancel_left [simp]:

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

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

   635

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

   637 proof

   638   assume ianz: "inverse a = 0"

   639   assume "a \<noteq> 0"

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

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

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

   643   thus False by (simp add: eq_commute)

   644 qed

   645

   646

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

   648

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

   650 apply (rule ccontr)

   651 apply (blast dest: nonzero_imp_inverse_nonzero)

   652 done

   653

   654 lemma inverse_nonzero_imp_nonzero:

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

   656 apply (rule ccontr)

   657 apply (blast dest: nonzero_imp_inverse_nonzero)

   658 done

   659

   660 lemma inverse_nonzero_iff_nonzero [simp]:

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

   662 by (force dest: inverse_nonzero_imp_nonzero)

   663

   664 lemma nonzero_inverse_minus_eq:

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

   666 proof -

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

   668     by simp

   669   thus ?thesis

   670     by (simp only: field_mult_cancel_left, simp)

   671 qed

   672

   673 lemma inverse_minus_eq [simp]:

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

   675 proof cases

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

   677 next

   678   assume "a\<noteq>0"

   679   thus ?thesis by (simp add: nonzero_inverse_minus_eq)

   680 qed

   681

   682 lemma nonzero_inverse_eq_imp_eq:

   683       assumes inveq: "inverse a = inverse b"

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

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

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

   687 proof -

   688   have "a * inverse b = a * inverse a"

   689     by (simp add: inveq)

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

   691     by simp

   692   thus "a = b"

   693     by (simp add: mult_assoc anz bnz)

   694 qed

   695

   696 lemma inverse_eq_imp_eq:

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

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

   699  apply (force dest!: inverse_zero_imp_zero

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

   701 apply (force dest!: nonzero_inverse_eq_imp_eq)

   702 done

   703

   704 lemma inverse_eq_iff_eq [simp]:

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

   706 by (force dest!: inverse_eq_imp_eq)

   707

   708 lemma nonzero_inverse_inverse_eq:

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

   710   proof -

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

   712     by (simp add: nonzero_imp_inverse_nonzero)

   713   thus ?thesis

   714     by (simp add: mult_assoc)

   715   qed

   716

   717 lemma inverse_inverse_eq [simp]:

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

   719   proof cases

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

   721   next

   722     assume "a\<noteq>0"

   723     thus ?thesis by (simp add: nonzero_inverse_inverse_eq)

   724   qed

   725

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

   727   proof -

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

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

   730   thus ?thesis  by simp

   731   qed

   732

   733 lemma nonzero_inverse_mult_distrib:

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

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

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

   737   proof -

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

   739     by (simp add: field_mult_eq_0_iff anz bnz)

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

   741     by (simp add: mult_assoc bnz)

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

   743     by simp

   744   thus ?thesis

   745     by (simp add: mult_assoc anz)

   746   qed

   747

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

   749       the right-hand side.*}

   750 lemma inverse_mult_distrib [simp]:

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

   752   proof cases

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

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

   755   next

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

   757     thus ?thesis  by force

   758   qed

   759

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

   761 lemma inverse_add:

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

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

   764 apply (simp add: left_distrib mult_assoc)

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

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

   767 done

   768

   769 lemma inverse_divide [simp]:

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

   771   by (simp add: divide_inverse mult_commute)

   772

   773 lemma nonzero_mult_divide_cancel_left:

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

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

   776 proof -

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

   778     by (simp add: field_mult_eq_0_iff divide_inverse

   779                   nonzero_inverse_mult_distrib)

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

   781     by (simp only: mult_ac)

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

   783     by simp

   784     finally show ?thesis

   785     by (simp add: divide_inverse)

   786 qed

   787

   788 lemma mult_divide_cancel_left:

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

   790 apply (case_tac "b = 0")

   791 apply (simp_all add: nonzero_mult_divide_cancel_left)

   792 done

   793

   794 lemma nonzero_mult_divide_cancel_right:

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

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

   797

   798 lemma mult_divide_cancel_right:

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

   800 apply (case_tac "b = 0")

   801 apply (simp_all add: nonzero_mult_divide_cancel_right)

   802 done

   803

   804 (*For ExtractCommonTerm*)

   805 lemma mult_divide_cancel_eq_if:

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

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

   808   by (simp add: mult_divide_cancel_left)

   809

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

   811   by (simp add: divide_inverse)

   812

   813 lemma times_divide_eq_right [simp]: "a * (b/c) = (a*b) / (c::'a::field)"

   814 by (simp add: divide_inverse mult_assoc)

   815

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

   817 by (simp add: divide_inverse mult_ac)

   818

   819 lemma divide_divide_eq_right [simp]:

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

   821 by (simp add: divide_inverse mult_ac)

   822

   823 lemma divide_divide_eq_left [simp]:

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

   825 by (simp add: divide_inverse mult_assoc)

   826

   827

   828 subsection {* Division and Unary Minus *}

   829

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

   831 by (simp add: divide_inverse minus_mult_left)

   832

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

   834 by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right)

   835

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

   837 by (simp add: divide_inverse nonzero_inverse_minus_eq)

   838

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

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

   841

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

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

   844

   845

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

   847 declare minus_divide_left  [symmetric, simp]

   848 declare minus_divide_right [symmetric, simp]

   849

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

   851 declare minus_mult_left [symmetric, simp]

   852 declare minus_mult_right [symmetric, simp]

   853

   854 lemma minus_divide_divide [simp]:

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

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

   857 apply (simp add: nonzero_minus_divide_divide)

   858 done

   859

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

   861 by (simp add: diff_minus add_divide_distrib)

   862

   863

   864 subsection {* Ordered Fields *}

   865

   866 lemma positive_imp_inverse_positive:

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

   868   proof -

   869   have "0 < a * inverse a"

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

   871   thus "0 < inverse a"

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

   873   qed

   874

   875 lemma negative_imp_inverse_negative:

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

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

   878       simp add: nonzero_inverse_minus_eq order_less_imp_not_eq)

   879

   880 lemma inverse_le_imp_le:

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

   882 	  and apos:  "0 < a"

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

   884   proof (rule classical)

   885   assume "~ b \<le> a"

   886   hence "a < b"

   887     by (simp add: linorder_not_le)

   888   hence bpos: "0 < b"

   889     by (blast intro: apos order_less_trans)

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

   891     by (simp add: apos invle order_less_imp_le mult_left_mono)

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

   893     by (simp add: bpos order_less_imp_le mult_right_mono)

   894   thus "b \<le> a"

   895     by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)

   896   qed

   897

   898 lemma inverse_positive_imp_positive:

   899       assumes inv_gt_0: "0 < inverse a"

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

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

   902   proof -

   903   have "0 < inverse (inverse a)"

   904     by (rule positive_imp_inverse_positive)

   905   thus "0 < a"

   906     by (simp add: nonzero_inverse_inverse_eq)

   907   qed

   908

   909 lemma inverse_positive_iff_positive [simp]:

   910       "(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))"

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

   912 apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)

   913 done

   914

   915 lemma inverse_negative_imp_negative:

   916       assumes inv_less_0: "inverse a < 0"

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

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

   919   proof -

   920   have "inverse (inverse a) < 0"

   921     by (rule negative_imp_inverse_negative)

   922   thus "a < 0"

   923     by (simp add: nonzero_inverse_inverse_eq)

   924   qed

   925

   926 lemma inverse_negative_iff_negative [simp]:

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

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

   929 apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)

   930 done

   931

   932 lemma inverse_nonnegative_iff_nonnegative [simp]:

   933       "(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))"

   934 by (simp add: linorder_not_less [symmetric])

   935

   936 lemma inverse_nonpositive_iff_nonpositive [simp]:

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

   938 by (simp add: linorder_not_less [symmetric])

   939

   940

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

   942

   943 lemma less_imp_inverse_less:

   944       assumes less: "a < b"

   945 	  and apos:  "0 < a"

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

   947   proof (rule ccontr)

   948   assume "~ inverse b < inverse a"

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

   950     by (simp add: linorder_not_less)

   951   hence "~ (a < b)"

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

   953   thus False

   954     by (rule notE [OF _ less])

   955   qed

   956

   957 lemma inverse_less_imp_less:

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

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

   960 apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq)

   961 done

   962

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

   964 lemma inverse_less_iff_less [simp]:

   965      "[|0 < a; 0 < b|]

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

   967 by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less)

   968

   969 lemma le_imp_inverse_le:

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

   971   by (force simp add: order_le_less less_imp_inverse_less)

   972

   973 lemma inverse_le_iff_le [simp]:

   974      "[|0 < a; 0 < b|]

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

   976 by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le)

   977

   978

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

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

   981 lemma inverse_le_imp_le_neg:

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

   983   apply (rule classical)

   984   apply (subgoal_tac "a < 0")

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

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

   987   apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq)

   988   done

   989

   990 lemma less_imp_inverse_less_neg:

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

   992   apply (subgoal_tac "a < 0")

   993    prefer 2 apply (blast intro: order_less_trans)

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

   995   apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq)

   996   done

   997

   998 lemma inverse_less_imp_less_neg:

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

  1000   apply (rule classical)

  1001   apply (subgoal_tac "a < 0")

  1002    prefer 2

  1003    apply (force simp add: linorder_not_less intro: order_le_less_trans)

  1004   apply (insert inverse_less_imp_less [of "-b" "-a"])

  1005   apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq)

  1006   done

  1007

  1008 lemma inverse_less_iff_less_neg [simp]:

  1009      "[|a < 0; b < 0|]

  1010       ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"

  1011   apply (insert inverse_less_iff_less [of "-b" "-a"])

  1012   apply (simp del: inverse_less_iff_less

  1013 	      add: order_less_imp_not_eq nonzero_inverse_minus_eq)

  1014   done

  1015

  1016 lemma le_imp_inverse_le_neg:

  1017    "[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)"

  1018   by (force simp add: order_le_less less_imp_inverse_less_neg)

  1019

  1020 lemma inverse_le_iff_le_neg [simp]:

  1021      "[|a < 0; b < 0|]

  1022       ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"

  1023 by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg)

  1024

  1025

  1026 subsection{*Inverses and the Number One*}

  1027

  1028 lemma one_less_inverse_iff:

  1029     "(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"proof cases

  1030   assume "0 < x"

  1031     with inverse_less_iff_less [OF zero_less_one, of x]

  1032     show ?thesis by simp

  1033 next

  1034   assume notless: "~ (0 < x)"

  1035   have "~ (1 < inverse x)"

  1036   proof

  1037     assume "1 < inverse x"

  1038     also with notless have "... \<le> 0" by (simp add: linorder_not_less)

  1039     also have "... < 1" by (rule zero_less_one)

  1040     finally show False by auto

  1041   qed

  1042   with notless show ?thesis by simp

  1043 qed

  1044

  1045 lemma inverse_eq_1_iff [simp]:

  1046     "(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))"

  1047 by (insert inverse_eq_iff_eq [of x 1], simp)

  1048

  1049 lemma one_le_inverse_iff:

  1050    "(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))"

  1051 by (force simp add: order_le_less one_less_inverse_iff zero_less_one

  1052                     eq_commute [of 1])

  1053

  1054 lemma inverse_less_1_iff:

  1055    "(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{ordered_field,division_by_zero}))"

  1056 by (simp add: linorder_not_le [symmetric] one_le_inverse_iff)

  1057

  1058 lemma inverse_le_1_iff:

  1059    "(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{ordered_field,division_by_zero}))"

  1060 by (simp add: linorder_not_less [symmetric] one_less_inverse_iff)

  1061

  1062

  1063 subsection{*Division and Signs*}

  1064

  1065 lemma zero_less_divide_iff:

  1066      "((0::'a::{ordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"

  1067 by (simp add: divide_inverse zero_less_mult_iff)

  1068

  1069 lemma divide_less_0_iff:

  1070      "(a/b < (0::'a::{ordered_field,division_by_zero})) =

  1071       (0 < a & b < 0 | a < 0 & 0 < b)"

  1072 by (simp add: divide_inverse mult_less_0_iff)

  1073

  1074 lemma zero_le_divide_iff:

  1075      "((0::'a::{ordered_field,division_by_zero}) \<le> a/b) =

  1076       (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"

  1077 by (simp add: divide_inverse zero_le_mult_iff)

  1078

  1079 lemma divide_le_0_iff:

  1080      "(a/b \<le> (0::'a::{ordered_field,division_by_zero})) =

  1081       (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"

  1082 by (simp add: divide_inverse mult_le_0_iff)

  1083

  1084 lemma divide_eq_0_iff [simp]:

  1085      "(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))"

  1086 by (simp add: divide_inverse field_mult_eq_0_iff)

  1087

  1088

  1089 subsection{*Simplification of Inequalities Involving Literal Divisors*}

  1090

  1091 lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"

  1092 proof -

  1093   assume less: "0<c"

  1094   hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"

  1095     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])

  1096   also have "... = (a*c \<le> b)"

  1097     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc)

  1098   finally show ?thesis .

  1099 qed

  1100

  1101

  1102 lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"

  1103 proof -

  1104   assume less: "c<0"

  1105   hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"

  1106     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])

  1107   also have "... = (b \<le> a*c)"

  1108     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc)

  1109   finally show ?thesis .

  1110 qed

  1111

  1112 lemma le_divide_eq:

  1113   "(a \<le> b/c) =

  1114    (if 0 < c then a*c \<le> b

  1115              else if c < 0 then b \<le> a*c

  1116              else  a \<le> (0::'a::{ordered_field,division_by_zero}))"

  1117 apply (case_tac "c=0", simp)

  1118 apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff)

  1119 done

  1120

  1121 lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)"

  1122 proof -

  1123   assume less: "0<c"

  1124   hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"

  1125     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])

  1126   also have "... = (b \<le> a*c)"

  1127     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc)

  1128   finally show ?thesis .

  1129 qed

  1130

  1131 lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)"

  1132 proof -

  1133   assume less: "c<0"

  1134   hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"

  1135     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])

  1136   also have "... = (a*c \<le> b)"

  1137     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc)

  1138   finally show ?thesis .

  1139 qed

  1140

  1141 lemma divide_le_eq:

  1142   "(b/c \<le> a) =

  1143    (if 0 < c then b \<le> a*c

  1144              else if c < 0 then a*c \<le> b

  1145              else 0 \<le> (a::'a::{ordered_field,division_by_zero}))"

  1146 apply (case_tac "c=0", simp)

  1147 apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff)

  1148 done

  1149

  1150

  1151 lemma pos_less_divide_eq:

  1152      "0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)"

  1153 proof -

  1154   assume less: "0<c"

  1155   hence "(a < b/c) = (a*c < (b/c)*c)"

  1156     by (simp add: mult_less_cancel_right order_less_not_sym [OF less])

  1157   also have "... = (a*c < b)"

  1158     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc)

  1159   finally show ?thesis .

  1160 qed

  1161

  1162 lemma neg_less_divide_eq:

  1163  "c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)"

  1164 proof -

  1165   assume less: "c<0"

  1166   hence "(a < b/c) = ((b/c)*c < a*c)"

  1167     by (simp add: mult_less_cancel_right order_less_not_sym [OF less])

  1168   also have "... = (b < a*c)"

  1169     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc)

  1170   finally show ?thesis .

  1171 qed

  1172

  1173 lemma less_divide_eq:

  1174   "(a < b/c) =

  1175    (if 0 < c then a*c < b

  1176              else if c < 0 then b < a*c

  1177              else  a < (0::'a::{ordered_field,division_by_zero}))"

  1178 apply (case_tac "c=0", simp)

  1179 apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff)

  1180 done

  1181

  1182 lemma pos_divide_less_eq:

  1183      "0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)"

  1184 proof -

  1185   assume less: "0<c"

  1186   hence "(b/c < a) = ((b/c)*c < a*c)"

  1187     by (simp add: mult_less_cancel_right order_less_not_sym [OF less])

  1188   also have "... = (b < a*c)"

  1189     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc)

  1190   finally show ?thesis .

  1191 qed

  1192

  1193 lemma neg_divide_less_eq:

  1194  "c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)"

  1195 proof -

  1196   assume less: "c<0"

  1197   hence "(b/c < a) = (a*c < (b/c)*c)"

  1198     by (simp add: mult_less_cancel_right order_less_not_sym [OF less])

  1199   also have "... = (a*c < b)"

  1200     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc)

  1201   finally show ?thesis .

  1202 qed

  1203

  1204 lemma divide_less_eq:

  1205   "(b/c < a) =

  1206    (if 0 < c then b < a*c

  1207              else if c < 0 then a*c < b

  1208              else 0 < (a::'a::{ordered_field,division_by_zero}))"

  1209 apply (case_tac "c=0", simp)

  1210 apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff)

  1211 done

  1212

  1213 lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)"

  1214 proof -

  1215   assume [simp]: "c\<noteq>0"

  1216   have "(a = b/c) = (a*c = (b/c)*c)"

  1217     by (simp add: field_mult_cancel_right)

  1218   also have "... = (a*c = b)"

  1219     by (simp add: divide_inverse mult_assoc)

  1220   finally show ?thesis .

  1221 qed

  1222

  1223 lemma eq_divide_eq:

  1224   "((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)"

  1225 by (simp add: nonzero_eq_divide_eq)

  1226

  1227 lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)"

  1228 proof -

  1229   assume [simp]: "c\<noteq>0"

  1230   have "(b/c = a) = ((b/c)*c = a*c)"

  1231     by (simp add: field_mult_cancel_right)

  1232   also have "... = (b = a*c)"

  1233     by (simp add: divide_inverse mult_assoc)

  1234   finally show ?thesis .

  1235 qed

  1236

  1237 lemma divide_eq_eq:

  1238   "(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)"

  1239 by (force simp add: nonzero_divide_eq_eq)

  1240

  1241 subsection{*Cancellation Laws for Division*}

  1242

  1243 lemma divide_cancel_right [simp]:

  1244      "(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))"

  1245 apply (case_tac "c=0", simp)

  1246 apply (simp add: divide_inverse field_mult_cancel_right)

  1247 done

  1248

  1249 lemma divide_cancel_left [simp]:

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

  1251 apply (case_tac "c=0", simp)

  1252 apply (simp add: divide_inverse field_mult_cancel_left)

  1253 done

  1254

  1255 subsection {* Division and the Number One *}

  1256

  1257 text{*Simplify expressions equated with 1*}

  1258 lemma divide_eq_1_iff [simp]:

  1259      "(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"

  1260 apply (case_tac "b=0", simp)

  1261 apply (simp add: right_inverse_eq)

  1262 done

  1263

  1264

  1265 lemma one_eq_divide_iff [simp]:

  1266      "(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"

  1267 by (simp add: eq_commute [of 1])

  1268

  1269 lemma zero_eq_1_divide_iff [simp]:

  1270      "((0::'a::{ordered_field,division_by_zero}) = 1/a) = (a = 0)"

  1271 apply (case_tac "a=0", simp)

  1272 apply (auto simp add: nonzero_eq_divide_eq)

  1273 done

  1274

  1275 lemma one_divide_eq_0_iff [simp]:

  1276      "(1/a = (0::'a::{ordered_field,division_by_zero})) = (a = 0)"

  1277 apply (case_tac "a=0", simp)

  1278 apply (insert zero_neq_one [THEN not_sym])

  1279 apply (auto simp add: nonzero_divide_eq_eq)

  1280 done

  1281

  1282 text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}

  1283 declare zero_less_divide_iff [of "1", simp]

  1284 declare divide_less_0_iff [of "1", simp]

  1285 declare zero_le_divide_iff [of "1", simp]

  1286 declare divide_le_0_iff [of "1", simp]

  1287

  1288

  1289 subsection {* Ordering Rules for Division *}

  1290

  1291 lemma divide_strict_right_mono:

  1292      "[|a < b; 0 < c|] ==> a / c < b / (c::'a::ordered_field)"

  1293 by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono

  1294               positive_imp_inverse_positive)

  1295

  1296 lemma divide_right_mono:

  1297      "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{ordered_field,division_by_zero})"

  1298   by (force simp add: divide_strict_right_mono order_le_less)

  1299

  1300

  1301 text{*The last premise ensures that @{term a} and @{term b}

  1302       have the same sign*}

  1303 lemma divide_strict_left_mono:

  1304        "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"

  1305 by (force simp add: zero_less_mult_iff divide_inverse mult_strict_left_mono

  1306       order_less_imp_not_eq order_less_imp_not_eq2

  1307       less_imp_inverse_less less_imp_inverse_less_neg)

  1308

  1309 lemma divide_left_mono:

  1310      "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::ordered_field)"

  1311   apply (subgoal_tac "a \<noteq> 0 & b \<noteq> 0")

  1312    prefer 2

  1313    apply (force simp add: zero_less_mult_iff order_less_imp_not_eq)

  1314   apply (case_tac "c=0", simp add: divide_inverse)

  1315   apply (force simp add: divide_strict_left_mono order_le_less)

  1316   done

  1317

  1318 lemma divide_strict_left_mono_neg:

  1319      "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"

  1320   apply (subgoal_tac "a \<noteq> 0 & b \<noteq> 0")

  1321    prefer 2

  1322    apply (force simp add: zero_less_mult_iff order_less_imp_not_eq)

  1323   apply (drule divide_strict_left_mono [of _ _ "-c"])

  1324    apply (simp_all add: mult_commute nonzero_minus_divide_left [symmetric])

  1325   done

  1326

  1327 lemma divide_strict_right_mono_neg:

  1328      "[|b < a; c < 0|] ==> a / c < b / (c::'a::ordered_field)"

  1329 apply (drule divide_strict_right_mono [of _ _ "-c"], simp)

  1330 apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric])

  1331 done

  1332

  1333

  1334 subsection {* Ordered Fields are Dense *}

  1335

  1336 lemma less_add_one: "a < (a+1::'a::ordered_semidom)"

  1337 proof -

  1338   have "a+0 < (a+1::'a::ordered_semidom)"

  1339     by (blast intro: zero_less_one add_strict_left_mono)

  1340   thus ?thesis by simp

  1341 qed

  1342

  1343 lemma zero_less_two: "0 < (1+1::'a::ordered_semidom)"

  1344   by (blast intro: order_less_trans zero_less_one less_add_one)

  1345

  1346 lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::ordered_field)"

  1347 by (simp add: zero_less_two pos_less_divide_eq right_distrib)

  1348

  1349 lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::ordered_field) < b"

  1350 by (simp add: zero_less_two pos_divide_less_eq right_distrib)

  1351

  1352 lemma dense: "a < b ==> \<exists>r::'a::ordered_field. a < r & r < b"

  1353 by (blast intro!: less_half_sum gt_half_sum)

  1354

  1355 subsection {* Absolute Value *}

  1356

  1357 lemma abs_one [simp]: "abs 1 = (1::'a::ordered_idom)"

  1358   by (simp add: abs_if zero_less_one [THEN order_less_not_sym])

  1359

  1360 lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lordered_ring))"

  1361 proof -

  1362   let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"

  1363   let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"

  1364   have a: "(abs a) * (abs b) = ?x"

  1365     by (simp only: abs_prts[of a] abs_prts[of b] ring_eq_simps)

  1366   {

  1367     fix u v :: 'a

  1368     have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow> u * v = ?y"

  1369       apply (subst prts[of u], subst prts[of v])

  1370       apply (simp add: left_distrib right_distrib add_ac)

  1371       done

  1372   }

  1373   note b = this[OF refl[of a] refl[of b]]

  1374   note addm = add_mono[of "0::'a" _ "0::'a", simplified]

  1375   note addm2 = add_mono[of _ "0::'a" _ "0::'a", simplified]

  1376   have xy: "- ?x <= ?y"

  1377     apply (simp)

  1378     apply (rule_tac y="0::'a" in order_trans)

  1379     apply (rule addm2)+

  1380     apply (simp_all add: mult_pos_le mult_neg_le)

  1381     apply (rule addm)+

  1382     apply (simp_all add: mult_pos_le mult_neg_le)

  1383     done

  1384   have yx: "?y <= ?x"

  1385     apply (simp add: add_ac)

  1386     apply (rule_tac y=0 in order_trans)

  1387     apply (rule addm2, (simp add: mult_pos_neg_le mult_pos_neg2_le)+)

  1388     apply (rule addm, (simp add: mult_pos_neg_le mult_pos_neg2_le)+)

  1389     done

  1390   have i1: "a*b <= abs a * abs b" by (simp only: a b yx)

  1391   have i2: "- (abs a * abs b) <= a*b" by (simp only: a b xy)

  1392   show ?thesis

  1393     apply (rule abs_leI)

  1394     apply (simp add: i1)

  1395     apply (simp add: i2[simplified minus_le_iff])

  1396     done

  1397 qed

  1398

  1399 lemma abs_eq_mult:

  1400   assumes "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)"

  1401   shows "abs (a*b) = abs a * abs (b::'a::lordered_ring)"

  1402 proof -

  1403   have s: "(0 <= a*b) | (a*b <= 0)"

  1404     apply (auto)

  1405     apply (rule_tac split_mult_pos_le)

  1406     apply (rule_tac contrapos_np[of "a*b <= 0"])

  1407     apply (simp)

  1408     apply (rule_tac split_mult_neg_le)

  1409     apply (insert prems)

  1410     apply (blast)

  1411     done

  1412   have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)"

  1413     by (simp add: prts[symmetric])

  1414   show ?thesis

  1415   proof cases

  1416     assume "0 <= a * b"

  1417     then show ?thesis

  1418       apply (simp_all add: mulprts abs_prts)

  1419       apply (simp add:

  1420 	iff2imp[OF zero_le_iff_zero_nprt]

  1421 	iff2imp[OF le_zero_iff_pprt_id]

  1422       )

  1423       apply (insert prems)

  1424       apply (auto simp add:

  1425 	ring_eq_simps

  1426 	iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_zero_pprt]

  1427 	iff2imp[OF le_zero_iff_pprt_id] iff2imp[OF zero_le_iff_nprt_id]

  1428 	order_antisym mult_pos_neg_le[of a b] mult_pos_neg2_le[of b a])

  1429       done

  1430   next

  1431     assume "~(0 <= a*b)"

  1432     with s have "a*b <= 0" by simp

  1433     then show ?thesis

  1434       apply (simp_all add: mulprts abs_prts)

  1435       apply (insert prems)

  1436       apply (auto simp add: ring_eq_simps iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_zero_pprt]

  1437 	iff2imp[OF le_zero_iff_pprt_id] iff2imp[OF zero_le_iff_nprt_id] order_antisym mult_pos_le[of a b] mult_neg_le[of a b])

  1438       done

  1439   qed

  1440 qed

  1441

  1442 lemma abs_mult: "abs (a * b) = abs a * abs (b::'a::ordered_idom)"

  1443 by (simp add: abs_eq_mult linorder_linear)

  1444

  1445 lemma abs_mult_self: "abs a * abs a = a * (a::'a::ordered_idom)"

  1446 by (simp add: abs_if)

  1447

  1448 lemma nonzero_abs_inverse:

  1449      "a \<noteq> 0 ==> abs (inverse (a::'a::ordered_field)) = inverse (abs a)"

  1450 apply (auto simp add: linorder_neq_iff abs_if nonzero_inverse_minus_eq

  1451                       negative_imp_inverse_negative)

  1452 apply (blast intro: positive_imp_inverse_positive elim: order_less_asym)

  1453 done

  1454

  1455 lemma abs_inverse [simp]:

  1456      "abs (inverse (a::'a::{ordered_field,division_by_zero})) =

  1457       inverse (abs a)"

  1458 apply (case_tac "a=0", simp)

  1459 apply (simp add: nonzero_abs_inverse)

  1460 done

  1461

  1462 lemma nonzero_abs_divide:

  1463      "b \<noteq> 0 ==> abs (a / (b::'a::ordered_field)) = abs a / abs b"

  1464 by (simp add: divide_inverse abs_mult nonzero_abs_inverse)

  1465

  1466 lemma abs_divide:

  1467      "abs (a / (b::'a::{ordered_field,division_by_zero})) = abs a / abs b"

  1468 apply (case_tac "b=0", simp)

  1469 apply (simp add: nonzero_abs_divide)

  1470 done

  1471

  1472 lemma abs_mult_less:

  1473      "[| abs a < c; abs b < d |] ==> abs a * abs b < c*(d::'a::ordered_idom)"

  1474 proof -

  1475   assume ac: "abs a < c"

  1476   hence cpos: "0<c" by (blast intro: order_le_less_trans abs_ge_zero)

  1477   assume "abs b < d"

  1478   thus ?thesis by (simp add: ac cpos mult_strict_mono)

  1479 qed

  1480

  1481 lemma eq_minus_self_iff: "(a = -a) = (a = (0::'a::ordered_idom))"

  1482 by (force simp add: order_eq_iff le_minus_self_iff minus_le_self_iff)

  1483

  1484 lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::ordered_idom))"

  1485 by (simp add: order_less_le le_minus_self_iff eq_minus_self_iff)

  1486

  1487 lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::ordered_idom))"

  1488 apply (simp add: order_less_le abs_le_iff)

  1489 apply (auto simp add: abs_if minus_le_self_iff eq_minus_self_iff)

  1490 apply (simp add: le_minus_self_iff linorder_neq_iff)

  1491 done

  1492

  1493 text{*Moving this up spoils many proofs using @{text mult_le_cancel_right}*}

  1494 declare times_divide_eq_left [simp]

  1495

  1496 ML {*

  1497 val left_distrib = thm "left_distrib";

  1498 val right_distrib = thm "right_distrib";

  1499 val mult_commute = thm "mult_commute";

  1500 val distrib = thm "distrib";

  1501 val zero_neq_one = thm "zero_neq_one";

  1502 val no_zero_divisors = thm "no_zero_divisors";

  1503 val left_inverse = thm "left_inverse";

  1504 val divide_inverse = thm "divide_inverse";

  1505 val mult_zero_left = thm "mult_zero_left";

  1506 val mult_zero_right = thm "mult_zero_right";

  1507 val field_mult_eq_0_iff = thm "field_mult_eq_0_iff";

  1508 val inverse_zero = thm "inverse_zero";

  1509 val ring_distrib = thms "ring_distrib";

  1510 val combine_common_factor = thm "combine_common_factor";

  1511 val minus_mult_left = thm "minus_mult_left";

  1512 val minus_mult_right = thm "minus_mult_right";

  1513 val minus_mult_minus = thm "minus_mult_minus";

  1514 val minus_mult_commute = thm "minus_mult_commute";

  1515 val right_diff_distrib = thm "right_diff_distrib";

  1516 val left_diff_distrib = thm "left_diff_distrib";

  1517 val mult_left_mono = thm "mult_left_mono";

  1518 val mult_right_mono = thm "mult_right_mono";

  1519 val mult_strict_left_mono = thm "mult_strict_left_mono";

  1520 val mult_strict_right_mono = thm "mult_strict_right_mono";

  1521 val mult_mono = thm "mult_mono";

  1522 val mult_strict_mono = thm "mult_strict_mono";

  1523 val abs_if = thm "abs_if";

  1524 val zero_less_one = thm "zero_less_one";

  1525 val eq_add_iff1 = thm "eq_add_iff1";

  1526 val eq_add_iff2 = thm "eq_add_iff2";

  1527 val less_add_iff1 = thm "less_add_iff1";

  1528 val less_add_iff2 = thm "less_add_iff2";

  1529 val le_add_iff1 = thm "le_add_iff1";

  1530 val le_add_iff2 = thm "le_add_iff2";

  1531 val mult_left_le_imp_le = thm "mult_left_le_imp_le";

  1532 val mult_right_le_imp_le = thm "mult_right_le_imp_le";

  1533 val mult_left_less_imp_less = thm "mult_left_less_imp_less";

  1534 val mult_right_less_imp_less = thm "mult_right_less_imp_less";

  1535 val mult_strict_left_mono_neg = thm "mult_strict_left_mono_neg";

  1536 val mult_left_mono_neg = thm "mult_left_mono_neg";

  1537 val mult_strict_right_mono_neg = thm "mult_strict_right_mono_neg";

  1538 val mult_right_mono_neg = thm "mult_right_mono_neg";

  1539 val mult_pos = thm "mult_pos";

  1540 val mult_pos_le = thm "mult_pos_le";

  1541 val mult_pos_neg = thm "mult_pos_neg";

  1542 val mult_pos_neg_le = thm "mult_pos_neg_le";

  1543 val mult_pos_neg2 = thm "mult_pos_neg2";

  1544 val mult_pos_neg2_le = thm "mult_pos_neg2_le";

  1545 val mult_neg = thm "mult_neg";

  1546 val mult_neg_le = thm "mult_neg_le";

  1547 val zero_less_mult_pos = thm "zero_less_mult_pos";

  1548 val zero_less_mult_pos2 = thm "zero_less_mult_pos2";

  1549 val zero_less_mult_iff = thm "zero_less_mult_iff";

  1550 val mult_eq_0_iff = thm "mult_eq_0_iff";

  1551 val zero_le_mult_iff = thm "zero_le_mult_iff";

  1552 val mult_less_0_iff = thm "mult_less_0_iff";

  1553 val mult_le_0_iff = thm "mult_le_0_iff";

  1554 val split_mult_pos_le = thm "split_mult_pos_le";

  1555 val split_mult_neg_le = thm "split_mult_neg_le";

  1556 val zero_le_square = thm "zero_le_square";

  1557 val zero_le_one = thm "zero_le_one";

  1558 val not_one_le_zero = thm "not_one_le_zero";

  1559 val not_one_less_zero = thm "not_one_less_zero";

  1560 val mult_left_mono_neg = thm "mult_left_mono_neg";

  1561 val mult_right_mono_neg = thm "mult_right_mono_neg";

  1562 val mult_strict_mono = thm "mult_strict_mono";

  1563 val mult_strict_mono' = thm "mult_strict_mono'";

  1564 val mult_mono = thm "mult_mono";

  1565 val less_1_mult = thm "less_1_mult";

  1566 val mult_less_cancel_right = thm "mult_less_cancel_right";

  1567 val mult_less_cancel_left = thm "mult_less_cancel_left";

  1568 val mult_le_cancel_right = thm "mult_le_cancel_right";

  1569 val mult_le_cancel_left = thm "mult_le_cancel_left";

  1570 val mult_less_imp_less_left = thm "mult_less_imp_less_left";

  1571 val mult_less_imp_less_right = thm "mult_less_imp_less_right";

  1572 val mult_cancel_right = thm "mult_cancel_right";

  1573 val mult_cancel_left = thm "mult_cancel_left";

  1574 val ring_eq_simps = thms "ring_eq_simps";

  1575 val right_inverse = thm "right_inverse";

  1576 val right_inverse_eq = thm "right_inverse_eq";

  1577 val nonzero_inverse_eq_divide = thm "nonzero_inverse_eq_divide";

  1578 val divide_self = thm "divide_self";

  1579 val divide_zero = thm "divide_zero";

  1580 val divide_zero_left = thm "divide_zero_left";

  1581 val inverse_eq_divide = thm "inverse_eq_divide";

  1582 val add_divide_distrib = thm "add_divide_distrib";

  1583 val field_mult_eq_0_iff = thm "field_mult_eq_0_iff";

  1584 val field_mult_cancel_right_lemma = thm "field_mult_cancel_right_lemma";

  1585 val field_mult_cancel_right = thm "field_mult_cancel_right";

  1586 val field_mult_cancel_left = thm "field_mult_cancel_left";

  1587 val nonzero_imp_inverse_nonzero = thm "nonzero_imp_inverse_nonzero";

  1588 val inverse_zero_imp_zero = thm "inverse_zero_imp_zero";

  1589 val inverse_nonzero_imp_nonzero = thm "inverse_nonzero_imp_nonzero";

  1590 val inverse_nonzero_iff_nonzero = thm "inverse_nonzero_iff_nonzero";

  1591 val nonzero_inverse_minus_eq = thm "nonzero_inverse_minus_eq";

  1592 val inverse_minus_eq = thm "inverse_minus_eq";

  1593 val nonzero_inverse_eq_imp_eq = thm "nonzero_inverse_eq_imp_eq";

  1594 val inverse_eq_imp_eq = thm "inverse_eq_imp_eq";

  1595 val inverse_eq_iff_eq = thm "inverse_eq_iff_eq";

  1596 val nonzero_inverse_inverse_eq = thm "nonzero_inverse_inverse_eq";

  1597 val inverse_inverse_eq = thm "inverse_inverse_eq";

  1598 val inverse_1 = thm "inverse_1";

  1599 val nonzero_inverse_mult_distrib = thm "nonzero_inverse_mult_distrib";

  1600 val inverse_mult_distrib = thm "inverse_mult_distrib";

  1601 val inverse_add = thm "inverse_add";

  1602 val inverse_divide = thm "inverse_divide";

  1603 val nonzero_mult_divide_cancel_left = thm "nonzero_mult_divide_cancel_left";

  1604 val mult_divide_cancel_left = thm "mult_divide_cancel_left";

  1605 val nonzero_mult_divide_cancel_right = thm "nonzero_mult_divide_cancel_right";

  1606 val mult_divide_cancel_right = thm "mult_divide_cancel_right";

  1607 val mult_divide_cancel_eq_if = thm "mult_divide_cancel_eq_if";

  1608 val divide_1 = thm "divide_1";

  1609 val times_divide_eq_right = thm "times_divide_eq_right";

  1610 val times_divide_eq_left = thm "times_divide_eq_left";

  1611 val divide_divide_eq_right = thm "divide_divide_eq_right";

  1612 val divide_divide_eq_left = thm "divide_divide_eq_left";

  1613 val nonzero_minus_divide_left = thm "nonzero_minus_divide_left";

  1614 val nonzero_minus_divide_right = thm "nonzero_minus_divide_right";

  1615 val nonzero_minus_divide_divide = thm "nonzero_minus_divide_divide";

  1616 val minus_divide_left = thm "minus_divide_left";

  1617 val minus_divide_right = thm "minus_divide_right";

  1618 val minus_divide_divide = thm "minus_divide_divide";

  1619 val diff_divide_distrib = thm "diff_divide_distrib";

  1620 val positive_imp_inverse_positive = thm "positive_imp_inverse_positive";

  1621 val negative_imp_inverse_negative = thm "negative_imp_inverse_negative";

  1622 val inverse_le_imp_le = thm "inverse_le_imp_le";

  1623 val inverse_positive_imp_positive = thm "inverse_positive_imp_positive";

  1624 val inverse_positive_iff_positive = thm "inverse_positive_iff_positive";

  1625 val inverse_negative_imp_negative = thm "inverse_negative_imp_negative";

  1626 val inverse_negative_iff_negative = thm "inverse_negative_iff_negative";

  1627 val inverse_nonnegative_iff_nonnegative = thm "inverse_nonnegative_iff_nonnegative";

  1628 val inverse_nonpositive_iff_nonpositive = thm "inverse_nonpositive_iff_nonpositive";

  1629 val less_imp_inverse_less = thm "less_imp_inverse_less";

  1630 val inverse_less_imp_less = thm "inverse_less_imp_less";

  1631 val inverse_less_iff_less = thm "inverse_less_iff_less";

  1632 val le_imp_inverse_le = thm "le_imp_inverse_le";

  1633 val inverse_le_iff_le = thm "inverse_le_iff_le";

  1634 val inverse_le_imp_le_neg = thm "inverse_le_imp_le_neg";

  1635 val less_imp_inverse_less_neg = thm "less_imp_inverse_less_neg";

  1636 val inverse_less_imp_less_neg = thm "inverse_less_imp_less_neg";

  1637 val inverse_less_iff_less_neg = thm "inverse_less_iff_less_neg";

  1638 val le_imp_inverse_le_neg = thm "le_imp_inverse_le_neg";

  1639 val inverse_le_iff_le_neg = thm "inverse_le_iff_le_neg";

  1640 val one_less_inverse_iff = thm "one_less_inverse_iff";

  1641 val inverse_eq_1_iff = thm "inverse_eq_1_iff";

  1642 val one_le_inverse_iff = thm "one_le_inverse_iff";

  1643 val inverse_less_1_iff = thm "inverse_less_1_iff";

  1644 val inverse_le_1_iff = thm "inverse_le_1_iff";

  1645 val zero_less_divide_iff = thm "zero_less_divide_iff";

  1646 val divide_less_0_iff = thm "divide_less_0_iff";

  1647 val zero_le_divide_iff = thm "zero_le_divide_iff";

  1648 val divide_le_0_iff = thm "divide_le_0_iff";

  1649 val divide_eq_0_iff = thm "divide_eq_0_iff";

  1650 val pos_le_divide_eq = thm "pos_le_divide_eq";

  1651 val neg_le_divide_eq = thm "neg_le_divide_eq";

  1652 val le_divide_eq = thm "le_divide_eq";

  1653 val pos_divide_le_eq = thm "pos_divide_le_eq";

  1654 val neg_divide_le_eq = thm "neg_divide_le_eq";

  1655 val divide_le_eq = thm "divide_le_eq";

  1656 val pos_less_divide_eq = thm "pos_less_divide_eq";

  1657 val neg_less_divide_eq = thm "neg_less_divide_eq";

  1658 val less_divide_eq = thm "less_divide_eq";

  1659 val pos_divide_less_eq = thm "pos_divide_less_eq";

  1660 val neg_divide_less_eq = thm "neg_divide_less_eq";

  1661 val divide_less_eq = thm "divide_less_eq";

  1662 val nonzero_eq_divide_eq = thm "nonzero_eq_divide_eq";

  1663 val eq_divide_eq = thm "eq_divide_eq";

  1664 val nonzero_divide_eq_eq = thm "nonzero_divide_eq_eq";

  1665 val divide_eq_eq = thm "divide_eq_eq";

  1666 val divide_cancel_right = thm "divide_cancel_right";

  1667 val divide_cancel_left = thm "divide_cancel_left";

  1668 val divide_eq_1_iff = thm "divide_eq_1_iff";

  1669 val one_eq_divide_iff = thm "one_eq_divide_iff";

  1670 val zero_eq_1_divide_iff = thm "zero_eq_1_divide_iff";

  1671 val one_divide_eq_0_iff = thm "one_divide_eq_0_iff";

  1672 val divide_strict_right_mono = thm "divide_strict_right_mono";

  1673 val divide_right_mono = thm "divide_right_mono";

  1674 val divide_strict_left_mono = thm "divide_strict_left_mono";

  1675 val divide_left_mono = thm "divide_left_mono";

  1676 val divide_strict_left_mono_neg = thm "divide_strict_left_mono_neg";

  1677 val divide_strict_right_mono_neg = thm "divide_strict_right_mono_neg";

  1678 val less_add_one = thm "less_add_one";

  1679 val zero_less_two = thm "zero_less_two";

  1680 val less_half_sum = thm "less_half_sum";

  1681 val gt_half_sum = thm "gt_half_sum";

  1682 val dense = thm "dense";

  1683 val abs_one = thm "abs_one";

  1684 val abs_le_mult = thm "abs_le_mult";

  1685 val abs_eq_mult = thm "abs_eq_mult";

  1686 val abs_mult = thm "abs_mult";

  1687 val abs_mult_self = thm "abs_mult_self";

  1688 val nonzero_abs_inverse = thm "nonzero_abs_inverse";

  1689 val abs_inverse = thm "abs_inverse";

  1690 val nonzero_abs_divide = thm "nonzero_abs_divide";

  1691 val abs_divide = thm "abs_divide";

  1692 val abs_mult_less = thm "abs_mult_less";

  1693 val eq_minus_self_iff = thm "eq_minus_self_iff";

  1694 val less_minus_self_iff = thm "less_minus_self_iff";

  1695 val abs_less_iff = thm "abs_less_iff";

  1696 *}

  1697

  1698 end