src/HOL/Ring_and_Field.thy
author haftmann
Fri Nov 02 18:52:58 2007 +0100 (2007-11-02)
changeset 25267 1f745c599b5c
parent 25238 ee73d4c33a88
child 25304 7491c00f0915
permissions -rw-r--r--
proper reinitialisation after subclass
     1 (*  Title:   HOL/Ring_and_Field.thy
     2     ID:      $Id$
     3     Author:  Gertrud Bauer, Steven Obua, Tobias Nipkow, Lawrence C Paulson, and Markus Wenzel,
     4              with contributions by Jeremy Avigad
     5 *)
     6 
     7 header {* (Ordered) Rings and Fields *}
     8 
     9 theory Ring_and_Field
    10 imports OrderedGroup
    11 begin
    12 
    13 text {*
    14   The theory of partially ordered rings is taken from the books:
    15   \begin{itemize}
    16   \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
    17   \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
    18   \end{itemize}
    19   Most of the used notions can also be looked up in 
    20   \begin{itemize}
    21   \item \url{http://www.mathworld.com} by Eric Weisstein et. al.
    22   \item \emph{Algebra I} by van der Waerden, Springer.
    23   \end{itemize}
    24 *}
    25 
    26 class semiring = ab_semigroup_add + semigroup_mult +
    27   assumes left_distrib: "(a + b) * c = a * c + b * c"
    28   assumes right_distrib: "a * (b + c) = a * b + a * c"
    29 begin
    30 
    31 text{*For the @{text combine_numerals} simproc*}
    32 lemma combine_common_factor:
    33   "a * e + (b * e + c) = (a + b) * e + c"
    34   by (simp add: left_distrib add_ac)
    35 
    36 end
    37 
    38 class mult_zero = times + zero +
    39   assumes mult_zero_left [simp]: "0 * a = 0"
    40   assumes mult_zero_right [simp]: "a * 0 = 0"
    41 
    42 class semiring_0 = semiring + comm_monoid_add + mult_zero
    43 
    44 class semiring_0_cancel = semiring + comm_monoid_add + cancel_ab_semigroup_add
    45 begin
    46 
    47 subclass semiring_0
    48 proof unfold_locales
    49   fix a :: 'a
    50   have "0 * a + 0 * a = 0 * a + 0"
    51     by (simp add: left_distrib [symmetric])
    52   thus "0 * a = 0"
    53     by (simp only: add_left_cancel)
    54 next
    55   fix a :: 'a
    56   have "a * 0 + a * 0 = a * 0 + 0"
    57     by (simp add: right_distrib [symmetric])
    58   thus "a * 0 = 0"
    59     by (simp only: add_left_cancel)
    60 qed
    61 
    62 end
    63 
    64 class comm_semiring = ab_semigroup_add + ab_semigroup_mult +
    65   assumes distrib: "(a + b) * c = a * c + b * c"
    66 begin
    67 
    68 subclass semiring
    69 proof unfold_locales
    70   fix a b c :: 'a
    71   show "(a + b) * c = a * c + b * c" by (simp add: distrib)
    72   have "a * (b + c) = (b + c) * a" by (simp add: mult_ac)
    73   also have "... = b * a + c * a" by (simp only: distrib)
    74   also have "... = a * b + a * c" by (simp add: mult_ac)
    75   finally show "a * (b + c) = a * b + a * c" by blast
    76 qed
    77 
    78 end
    79 
    80 class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero
    81 begin
    82 
    83 subclass semiring_0 by unfold_locales
    84 
    85 end
    86 
    87 class comm_semiring_0_cancel = comm_semiring + comm_monoid_add + cancel_ab_semigroup_add
    88 begin
    89 
    90 subclass semiring_0_cancel by unfold_locales
    91 
    92 end
    93 
    94 class zero_neq_one = zero + one +
    95   assumes zero_neq_one [simp]: "0 \<noteq> 1"
    96 
    97 class semiring_1 = zero_neq_one + semiring_0 + monoid_mult
    98 
    99 class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult
   100   (*previously almost_semiring*)
   101 begin
   102 
   103 subclass semiring_1 by unfold_locales
   104 
   105 end
   106 
   107 class no_zero_divisors = zero + times +
   108   assumes no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"
   109 
   110 class semiring_1_cancel = semiring + comm_monoid_add + zero_neq_one
   111   + cancel_ab_semigroup_add + monoid_mult
   112 begin
   113 
   114 subclass semiring_0_cancel by unfold_locales
   115 
   116 subclass semiring_1 by unfold_locales
   117 
   118 end
   119 
   120 class comm_semiring_1_cancel = comm_semiring + comm_monoid_add + comm_monoid_mult
   121   + zero_neq_one + cancel_ab_semigroup_add
   122 begin
   123 
   124 subclass semiring_1_cancel by unfold_locales
   125 subclass comm_semiring_0_cancel by unfold_locales
   126 subclass comm_semiring_1 by unfold_locales
   127 
   128 end
   129 
   130 class ring = semiring + ab_group_add
   131 begin
   132 
   133 subclass semiring_0_cancel by unfold_locales
   134 
   135 text {* Distribution rules *}
   136 
   137 lemma minus_mult_left: "- (a * b) = - a * b"
   138   by (rule equals_zero_I) (simp add: left_distrib [symmetric]) 
   139 
   140 lemma minus_mult_right: "- (a * b) = a * - b"
   141   by (rule equals_zero_I) (simp add: right_distrib [symmetric]) 
   142 
   143 lemma minus_mult_minus [simp]: "- a * - b = a * b"
   144   by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])
   145 
   146 lemma minus_mult_commute: "- a * b = a * - b"
   147   by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])
   148 
   149 lemma right_diff_distrib: "a * (b - c) = a * b - a * c"
   150   by (simp add: right_distrib diff_minus 
   151     minus_mult_left [symmetric] minus_mult_right [symmetric]) 
   152 
   153 lemma left_diff_distrib: "(a - b) * c = a * c - b * c"
   154   by (simp add: left_distrib diff_minus 
   155     minus_mult_left [symmetric] minus_mult_right [symmetric]) 
   156 
   157 lemmas ring_distribs =
   158   right_distrib left_distrib left_diff_distrib right_diff_distrib
   159 
   160 lemmas ring_simps =
   161   add_ac
   162   add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
   163   diff_eq_eq eq_diff_eq diff_minus [symmetric] uminus_add_conv_diff
   164   ring_distribs
   165 
   166 lemma eq_add_iff1:
   167   "a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d"
   168   by (simp add: ring_simps)
   169 
   170 lemma eq_add_iff2:
   171   "a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d"
   172   by (simp add: ring_simps)
   173 
   174 end
   175 
   176 lemmas ring_distribs =
   177   right_distrib left_distrib left_diff_distrib right_diff_distrib
   178 
   179 class comm_ring = comm_semiring + ab_group_add
   180 begin
   181 
   182 subclass ring by unfold_locales
   183 subclass comm_semiring_0 by unfold_locales
   184 
   185 end
   186 
   187 class ring_1 = ring + zero_neq_one + monoid_mult
   188 begin
   189 
   190 subclass semiring_1_cancel by unfold_locales
   191 
   192 end
   193 
   194 class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult
   195   (*previously ring*)
   196 begin
   197 
   198 subclass ring_1 by unfold_locales
   199 subclass comm_semiring_1_cancel by unfold_locales
   200 
   201 end
   202 
   203 class ring_no_zero_divisors = ring + no_zero_divisors
   204 begin
   205 
   206 lemma mult_eq_0_iff [simp]:
   207   shows "a * b = 0 \<longleftrightarrow> (a = 0 \<or> b = 0)"
   208 proof (cases "a = 0 \<or> b = 0")
   209   case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
   210     then show ?thesis using no_zero_divisors by simp
   211 next
   212   case True then show ?thesis by auto
   213 qed
   214 
   215 end
   216 
   217 class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
   218 
   219 class idom = comm_ring_1 + no_zero_divisors
   220 begin
   221 
   222 subclass ring_1_no_zero_divisors by unfold_locales
   223 
   224 end
   225 
   226 class division_ring = ring_1 + inverse +
   227   assumes left_inverse [simp]:  "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
   228   assumes right_inverse [simp]: "a \<noteq> 0 \<Longrightarrow> a * inverse a = 1"
   229 begin
   230 
   231 subclass ring_1_no_zero_divisors
   232 proof unfold_locales
   233   fix a b :: 'a
   234   assume a: "a \<noteq> 0" and b: "b \<noteq> 0"
   235   show "a * b \<noteq> 0"
   236   proof
   237     assume ab: "a * b = 0"
   238     hence "0 = inverse a * (a * b) * inverse b"
   239       by simp
   240     also have "\<dots> = (inverse a * a) * (b * inverse b)"
   241       by (simp only: mult_assoc)
   242     also have "\<dots> = 1"
   243       using a b by simp
   244     finally show False
   245       by simp
   246   qed
   247 qed
   248 
   249 end
   250 
   251 class field = comm_ring_1 + inverse +
   252   assumes field_inverse:  "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
   253   assumes divide_inverse: "a / b = a * inverse b"
   254 begin
   255 
   256 subclass division_ring
   257 proof unfold_locales
   258   fix a :: 'a
   259   assume "a \<noteq> 0"
   260   thus "inverse a * a = 1" by (rule field_inverse)
   261   thus "a * inverse a = 1" by (simp only: mult_commute)
   262 qed
   263 
   264 subclass idom by unfold_locales
   265 
   266 lemma right_inverse_eq: "b \<noteq> 0 \<Longrightarrow> a / b = 1 \<longleftrightarrow> a = b"
   267 proof
   268   assume neq: "b \<noteq> 0"
   269   {
   270     hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac)
   271     also assume "a / b = 1"
   272     finally show "a = b" by simp
   273   next
   274     assume "a = b"
   275     with neq show "a / b = 1" by (simp add: divide_inverse)
   276   }
   277 qed
   278 
   279 lemma nonzero_inverse_eq_divide: "a \<noteq> 0 \<Longrightarrow> inverse a = 1 / a"
   280   by (simp add: divide_inverse)
   281 
   282 lemma divide_self [simp]: "a \<noteq> 0 \<Longrightarrow> a / a = 1"
   283   by (simp add: divide_inverse)
   284 
   285 lemma divide_zero_left [simp]: "0 / a = 0"
   286   by (simp add: divide_inverse)
   287 
   288 lemma inverse_eq_divide: "inverse a = 1 / a"
   289   by (simp add: divide_inverse)
   290 
   291 lemma add_divide_distrib: "(a+b) / c = a/c + b/c"
   292   by (simp add: divide_inverse ring_distribs) 
   293 
   294 end
   295 
   296 class division_by_zero = zero + inverse +
   297   assumes inverse_zero [simp]: "inverse 0 = 0"
   298 
   299 lemma divide_zero [simp]:
   300   "a / 0 = (0::'a::{field,division_by_zero})"
   301   by (simp add: divide_inverse)
   302 
   303 lemma divide_self_if [simp]:
   304   "a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)"
   305   by (simp add: divide_self)
   306 
   307 class mult_mono = times + zero + ord +
   308   assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
   309   assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
   310 
   311 class pordered_semiring = mult_mono + semiring_0 + pordered_ab_semigroup_add 
   312 begin
   313 
   314 lemma mult_mono:
   315   "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c
   316      \<Longrightarrow> a * c \<le> b * d"
   317 apply (erule mult_right_mono [THEN order_trans], assumption)
   318 apply (erule mult_left_mono, assumption)
   319 done
   320 
   321 lemma mult_mono':
   322   "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c
   323      \<Longrightarrow> a * c \<le> b * d"
   324 apply (rule mult_mono)
   325 apply (fast intro: order_trans)+
   326 done
   327 
   328 end
   329 
   330 class pordered_cancel_semiring = mult_mono + pordered_ab_semigroup_add
   331   + semiring + comm_monoid_add + cancel_ab_semigroup_add
   332 begin
   333 
   334 subclass semiring_0_cancel by unfold_locales
   335 subclass pordered_semiring by unfold_locales
   336 
   337 lemma mult_nonneg_nonneg: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
   338   by (drule mult_left_mono [of zero b], auto)
   339 
   340 lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
   341   by (drule mult_left_mono [of b zero], auto)
   342 
   343 lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0" 
   344   by (drule mult_right_mono [of b zero], auto)
   345 
   346 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)" 
   347   by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
   348 
   349 end
   350 
   351 class ordered_semiring = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add + mult_mono
   352 begin
   353 
   354 subclass pordered_cancel_semiring by unfold_locales
   355 
   356 lemma mult_left_less_imp_less:
   357   "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
   358   by (force simp add: mult_left_mono not_le [symmetric])
   359  
   360 lemma mult_right_less_imp_less:
   361   "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
   362   by (force simp add: mult_right_mono not_le [symmetric])
   363 
   364 end
   365 
   366 class ordered_semiring_strict = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add +
   367   assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
   368   assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
   369 begin
   370 
   371 subclass semiring_0_cancel by unfold_locales
   372 
   373 subclass ordered_semiring
   374 proof unfold_locales
   375   fix a b c :: 'a
   376   assume A: "a \<le> b" "0 \<le> c"
   377   from A show "c * a \<le> c * b"
   378     unfolding le_less
   379     using mult_strict_left_mono by (cases "c = 0") auto
   380   from A show "a * c \<le> b * c"
   381     unfolding le_less
   382     using mult_strict_right_mono by (cases "c = 0") auto
   383 qed
   384 
   385 lemma mult_left_le_imp_le:
   386   "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
   387   by (force simp add: mult_strict_left_mono _not_less [symmetric])
   388  
   389 lemma mult_right_le_imp_le:
   390   "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
   391   by (force simp add: mult_strict_right_mono not_less [symmetric])
   392 
   393 lemma mult_pos_pos:
   394   "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
   395   by (drule mult_strict_left_mono [of zero b], auto)
   396 
   397 lemma mult_pos_neg:
   398   "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
   399   by (drule mult_strict_left_mono [of b zero], auto)
   400 
   401 lemma mult_pos_neg2:
   402   "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0" 
   403   by (drule mult_strict_right_mono [of b zero], auto)
   404 
   405 lemma zero_less_mult_pos:
   406   "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
   407 apply (cases "b\<le>0") 
   408  apply (auto simp add: le_less not_less)
   409 apply (drule_tac mult_pos_neg [of a b]) 
   410  apply (auto dest: less_not_sym)
   411 done
   412 
   413 lemma zero_less_mult_pos2:
   414   "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
   415 apply (cases "b\<le>0") 
   416  apply (auto simp add: le_less not_less)
   417 apply (drule_tac mult_pos_neg2 [of a b]) 
   418  apply (auto dest: less_not_sym)
   419 done
   420 
   421 end
   422 
   423 class mult_mono1 = times + zero + ord +
   424   assumes mult_mono1: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
   425 
   426 class pordered_comm_semiring = comm_semiring_0
   427   + pordered_ab_semigroup_add + mult_mono1
   428 begin
   429 
   430 subclass pordered_semiring
   431 proof unfold_locales
   432   fix a b c :: 'a
   433   assume "a \<le> b" "0 \<le> c"
   434   thus "c * a \<le> c * b" by (rule mult_mono1)
   435   thus "a * c \<le> b * c" by (simp only: mult_commute)
   436 qed
   437 
   438 end
   439 
   440 class pordered_cancel_comm_semiring = comm_semiring_0_cancel
   441   + pordered_ab_semigroup_add + mult_mono1
   442 begin
   443 
   444 subclass pordered_comm_semiring by unfold_locales
   445 subclass pordered_cancel_semiring by unfold_locales
   446 
   447 end
   448 
   449 class ordered_comm_semiring_strict = comm_semiring_0 + ordered_cancel_ab_semigroup_add +
   450   assumes mult_strict_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
   451 begin
   452 
   453 subclass ordered_semiring_strict
   454 proof unfold_locales
   455   fix a b c :: 'a
   456   assume "a < b" "0 < c"
   457   thus "c * a < c * b" by (rule mult_strict_mono)
   458   thus "a * c < b * c" by (simp only: mult_commute)
   459 qed
   460 
   461 subclass pordered_cancel_comm_semiring
   462 proof unfold_locales
   463   fix a b c :: 'a
   464   assume "a \<le> b" "0 \<le> c"
   465   thus "c * a \<le> c * b"
   466     unfolding le_less
   467     using mult_strict_mono by (cases "c = 0") auto
   468 qed
   469 
   470 end
   471 
   472 class pordered_ring = ring + pordered_cancel_semiring 
   473 begin
   474 
   475 subclass pordered_ab_group_add by unfold_locales
   476 
   477 lemmas ring_simps = ring_simps group_simps
   478 
   479 lemma less_add_iff1:
   480   "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
   481   by (simp add: ring_simps)
   482 
   483 lemma less_add_iff2:
   484   "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
   485   by (simp add: ring_simps)
   486 
   487 lemma le_add_iff1:
   488   "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
   489   by (simp add: ring_simps)
   490 
   491 lemma le_add_iff2:
   492   "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
   493   by (simp add: ring_simps)
   494 
   495 lemma mult_left_mono_neg:
   496   "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
   497   apply (drule mult_left_mono [of _ _ "uminus c"])
   498   apply (simp_all add: minus_mult_left [symmetric]) 
   499   done
   500 
   501 lemma mult_right_mono_neg:
   502   "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
   503   apply (drule mult_right_mono [of _ _ "uminus c"])
   504   apply (simp_all add: minus_mult_right [symmetric]) 
   505   done
   506 
   507 lemma mult_nonpos_nonpos:
   508   "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
   509   by (drule mult_right_mono_neg [of a zero b]) auto
   510 
   511 lemma split_mult_pos_le:
   512   "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
   513   by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos)
   514 
   515 end
   516 
   517 class lordered_ring = pordered_ring + lordered_ab_group_abs
   518 begin
   519 
   520 subclass lordered_ab_group_meet by unfold_locales
   521 subclass lordered_ab_group_join by unfold_locales
   522 
   523 end
   524 
   525 class abs_if = minus + ord + zero + abs +
   526   assumes abs_if: "\<bar>a\<bar> = (if a < 0 then (- a) else a)"
   527 
   528 class sgn_if = sgn + zero + one + minus + ord +
   529   assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
   530 
   531 class ordered_ring = ring + ordered_semiring
   532   + lordered_ab_group + abs_if
   533   -- {*FIXME: should inherit from @{text ordered_ab_group_add} rather than
   534          @{text lordered_ab_group}*}
   535 
   536 instance ordered_ring \<subseteq> lordered_ring
   537 proof 
   538   fix x :: 'a
   539   show "\<bar>x\<bar> = sup x (- x)"
   540     by (simp only: abs_if sup_eq_if)
   541 qed
   542 
   543 (* The "strict" suffix can be seen as describing the combination of ordered_ring and no_zero_divisors.
   544    Basically, ordered_ring + no_zero_divisors = ordered_ring_strict.
   545  *)
   546 class ordered_ring_strict = ring + ordered_semiring_strict
   547   + lordered_ab_group + abs_if
   548   -- {*FIXME: should inherit from @{text ordered_ab_group_add} rather than
   549          @{text lordered_ab_group}*}
   550 
   551 instance ordered_ring_strict \<subseteq> ordered_ring by intro_classes
   552 
   553 context ordered_ring_strict
   554 begin
   555 
   556 lemma mult_strict_left_mono_neg:
   557   "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
   558   apply (drule mult_strict_left_mono [of _ _ "uminus c"])
   559   apply (simp_all add: minus_mult_left [symmetric]) 
   560   done
   561 
   562 lemma mult_strict_right_mono_neg:
   563   "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
   564   apply (drule mult_strict_right_mono [of _ _ "uminus c"])
   565   apply (simp_all add: minus_mult_right [symmetric]) 
   566   done
   567 
   568 lemma mult_neg_neg:
   569   "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
   570   by (drule mult_strict_right_mono_neg, auto)
   571 
   572 end
   573 
   574 lemma zero_less_mult_iff:
   575   fixes a :: "'a::ordered_ring_strict"
   576   shows "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
   577   apply (auto simp add: le_less not_less mult_pos_pos mult_neg_neg)
   578   apply (blast dest: zero_less_mult_pos) 
   579   apply (blast dest: zero_less_mult_pos2)
   580   done
   581 
   582 instance ordered_ring_strict \<subseteq> ring_no_zero_divisors
   583 apply intro_classes
   584 apply (auto simp add: linorder_not_less order_le_less linorder_neq_iff)
   585 apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono)+
   586 done
   587 
   588 lemma zero_le_mult_iff:
   589      "((0::'a::ordered_ring_strict) \<le> a*b) = (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
   590 by (auto simp add: eq_commute [of 0] order_le_less linorder_not_less
   591                    zero_less_mult_iff)
   592 
   593 lemma mult_less_0_iff:
   594      "(a*b < (0::'a::ordered_ring_strict)) = (0 < a & b < 0 | a < 0 & 0 < b)"
   595 apply (insert zero_less_mult_iff [of "-a" b]) 
   596 apply (force simp add: minus_mult_left[symmetric]) 
   597 done
   598 
   599 lemma mult_le_0_iff:
   600      "(a*b \<le> (0::'a::ordered_ring_strict)) = (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
   601 apply (insert zero_le_mult_iff [of "-a" b]) 
   602 apply (force simp add: minus_mult_left[symmetric]) 
   603 done
   604 
   605 lemma zero_le_square[simp]: "(0::'a::ordered_ring_strict) \<le> a*a"
   606 by (simp add: zero_le_mult_iff linorder_linear)
   607 
   608 lemma not_square_less_zero[simp]: "\<not> (a * a < (0::'a::ordered_ring_strict))"
   609 by (simp add: not_less)
   610 
   611 text{*This list of rewrites simplifies ring terms by multiplying
   612 everything out and bringing sums and products into a canonical form
   613 (by ordered rewriting). As a result it decides ring equalities but
   614 also helps with inequalities. *}
   615 lemmas ring_simps = group_simps ring_distribs
   616 
   617 
   618 class pordered_comm_ring = comm_ring + pordered_comm_semiring
   619 begin
   620 
   621 subclass pordered_ring by unfold_locales
   622 subclass pordered_cancel_comm_semiring by unfold_locales
   623 
   624 end
   625 
   626 class ordered_semidom = comm_semiring_1_cancel + ordered_comm_semiring_strict +
   627   (*previously ordered_semiring*)
   628   assumes zero_less_one [simp]: "0 < 1"
   629 begin
   630 
   631 lemma pos_add_strict:
   632   shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
   633   using add_strict_mono [of zero a b c] by simp
   634 
   635 end
   636 
   637 class ordered_idom =
   638   comm_ring_1 +
   639   ordered_comm_semiring_strict +
   640   lordered_ab_group +
   641   abs_if + sgn_if
   642   (*previously ordered_ring*)
   643 
   644 instance ordered_idom \<subseteq> ordered_ring_strict ..
   645 
   646 instance ordered_idom \<subseteq> pordered_comm_ring ..
   647 
   648 class ordered_field = field + ordered_idom
   649 
   650 lemma linorder_neqE_ordered_idom:
   651   fixes x y :: "'a :: ordered_idom"
   652   assumes "x \<noteq> y" obtains "x < y" | "y < x"
   653   using assms by (rule linorder_neqE)
   654 
   655 -- {* FIXME continue localization here *}
   656 
   657 
   658 text{*Proving axiom @{text zero_less_one} makes all @{text ordered_semidom}
   659       theorems available to members of @{term ordered_idom} *}
   660 
   661 instance ordered_idom \<subseteq> ordered_semidom
   662 proof
   663   have "(0::'a) \<le> 1*1" by (rule zero_le_square)
   664   thus "(0::'a) < 1" by (simp add: order_le_less) 
   665 qed
   666 
   667 instance ordered_idom \<subseteq> idom ..
   668 
   669 text{*All three types of comparision involving 0 and 1 are covered.*}
   670 
   671 lemmas one_neq_zero = zero_neq_one [THEN not_sym]
   672 declare one_neq_zero [simp]
   673 
   674 lemma zero_le_one [simp]: "(0::'a::ordered_semidom) \<le> 1"
   675   by (rule zero_less_one [THEN order_less_imp_le]) 
   676 
   677 lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semidom) \<le> 0"
   678 by (simp add: linorder_not_le) 
   679 
   680 lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semidom) < 0"
   681 by (simp add: linorder_not_less) 
   682 
   683 
   684 subsection{*More Monotonicity*}
   685 
   686 text{*Strict monotonicity in both arguments*}
   687 lemma mult_strict_mono:
   688      "[|a<b; c<d; 0<b; 0\<le>c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
   689 apply (cases "c=0")
   690  apply (simp add: mult_pos_pos) 
   691 apply (erule mult_strict_right_mono [THEN order_less_trans])
   692  apply (force simp add: order_le_less) 
   693 apply (erule mult_strict_left_mono, assumption)
   694 done
   695 
   696 text{*This weaker variant has more natural premises*}
   697 lemma mult_strict_mono':
   698      "[| a<b; c<d; 0 \<le> a; 0 \<le> c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
   699 apply (rule mult_strict_mono)
   700 apply (blast intro: order_le_less_trans)+
   701 done
   702 
   703 lemma less_1_mult: "[| 1 < m; 1 < n |] ==> 1 < m*(n::'a::ordered_semidom)"
   704 apply (insert mult_strict_mono [of 1 m 1 n]) 
   705 apply (simp add:  order_less_trans [OF zero_less_one]) 
   706 done
   707 
   708 lemma mult_less_le_imp_less: "(a::'a::ordered_semiring_strict) < b ==>
   709     c <= d ==> 0 <= a ==> 0 < c ==> a * c < b * d"
   710   apply (subgoal_tac "a * c < b * c")
   711   apply (erule order_less_le_trans)
   712   apply (erule mult_left_mono)
   713   apply simp
   714   apply (erule mult_strict_right_mono)
   715   apply assumption
   716 done
   717 
   718 lemma mult_le_less_imp_less: "(a::'a::ordered_semiring_strict) <= b ==>
   719     c < d ==> 0 < a ==> 0 <= c ==> a * c < b * d"
   720   apply (subgoal_tac "a * c <= b * c")
   721   apply (erule order_le_less_trans)
   722   apply (erule mult_strict_left_mono)
   723   apply simp
   724   apply (erule mult_right_mono)
   725   apply simp
   726 done
   727 
   728 
   729 subsection{*Cancellation Laws for Relationships With a Common Factor*}
   730 
   731 text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
   732    also with the relations @{text "\<le>"} and equality.*}
   733 
   734 text{*These ``disjunction'' versions produce two cases when the comparison is
   735  an assumption, but effectively four when the comparison is a goal.*}
   736 
   737 lemma mult_less_cancel_right_disj:
   738     "(a*c < b*c) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
   739 apply (cases "c = 0")
   740 apply (auto simp add: linorder_neq_iff mult_strict_right_mono 
   741                       mult_strict_right_mono_neg)
   742 apply (auto simp add: linorder_not_less 
   743                       linorder_not_le [symmetric, of "a*c"]
   744                       linorder_not_le [symmetric, of a])
   745 apply (erule_tac [!] notE)
   746 apply (auto simp add: order_less_imp_le mult_right_mono 
   747                       mult_right_mono_neg)
   748 done
   749 
   750 lemma mult_less_cancel_left_disj:
   751     "(c*a < c*b) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
   752 apply (cases "c = 0")
   753 apply (auto simp add: linorder_neq_iff mult_strict_left_mono 
   754                       mult_strict_left_mono_neg)
   755 apply (auto simp add: linorder_not_less 
   756                       linorder_not_le [symmetric, of "c*a"]
   757                       linorder_not_le [symmetric, of a])
   758 apply (erule_tac [!] notE)
   759 apply (auto simp add: order_less_imp_le mult_left_mono 
   760                       mult_left_mono_neg)
   761 done
   762 
   763 
   764 text{*The ``conjunction of implication'' lemmas produce two cases when the
   765 comparison is a goal, but give four when the comparison is an assumption.*}
   766 
   767 lemma mult_less_cancel_right:
   768   fixes c :: "'a :: ordered_ring_strict"
   769   shows      "(a*c < b*c) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))"
   770 by (insert mult_less_cancel_right_disj [of a c b], auto)
   771 
   772 lemma mult_less_cancel_left:
   773   fixes c :: "'a :: ordered_ring_strict"
   774   shows      "(c*a < c*b) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))"
   775 by (insert mult_less_cancel_left_disj [of c a b], auto)
   776 
   777 lemma mult_le_cancel_right:
   778      "(a*c \<le> b*c) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
   779 by (simp add: linorder_not_less [symmetric] mult_less_cancel_right_disj)
   780 
   781 lemma mult_le_cancel_left:
   782      "(c*a \<le> c*b) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
   783 by (simp add: linorder_not_less [symmetric] mult_less_cancel_left_disj)
   784 
   785 lemma mult_less_imp_less_left:
   786       assumes less: "c*a < c*b" and nonneg: "0 \<le> c"
   787       shows "a < (b::'a::ordered_semiring_strict)"
   788 proof (rule ccontr)
   789   assume "~ a < b"
   790   hence "b \<le> a" by (simp add: linorder_not_less)
   791   hence "c*b \<le> c*a" using nonneg by (rule mult_left_mono)
   792   with this and less show False 
   793     by (simp add: linorder_not_less [symmetric])
   794 qed
   795 
   796 lemma mult_less_imp_less_right:
   797   assumes less: "a*c < b*c" and nonneg: "0 <= c"
   798   shows "a < (b::'a::ordered_semiring_strict)"
   799 proof (rule ccontr)
   800   assume "~ a < b"
   801   hence "b \<le> a" by (simp add: linorder_not_less)
   802   hence "b*c \<le> a*c" using nonneg by (rule mult_right_mono)
   803   with this and less show False 
   804     by (simp add: linorder_not_less [symmetric])
   805 qed  
   806 
   807 text{*Cancellation of equalities with a common factor*}
   808 lemma mult_cancel_right [simp,noatp]:
   809   fixes a b c :: "'a::ring_no_zero_divisors"
   810   shows "(a * c = b * c) = (c = 0 \<or> a = b)"
   811 proof -
   812   have "(a * c = b * c) = ((a - b) * c = 0)"
   813     by (simp add: ring_distribs)
   814   thus ?thesis
   815     by (simp add: disj_commute)
   816 qed
   817 
   818 lemma mult_cancel_left [simp,noatp]:
   819   fixes a b c :: "'a::ring_no_zero_divisors"
   820   shows "(c * a = c * b) = (c = 0 \<or> a = b)"
   821 proof -
   822   have "(c * a = c * b) = (c * (a - b) = 0)"
   823     by (simp add: ring_distribs)
   824   thus ?thesis
   825     by simp
   826 qed
   827 
   828 
   829 subsubsection{*Special Cancellation Simprules for Multiplication*}
   830 
   831 text{*These also produce two cases when the comparison is a goal.*}
   832 
   833 lemma mult_le_cancel_right1:
   834   fixes c :: "'a :: ordered_idom"
   835   shows "(c \<le> b*c) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))"
   836 by (insert mult_le_cancel_right [of 1 c b], simp)
   837 
   838 lemma mult_le_cancel_right2:
   839   fixes c :: "'a :: ordered_idom"
   840   shows "(a*c \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))"
   841 by (insert mult_le_cancel_right [of a c 1], simp)
   842 
   843 lemma mult_le_cancel_left1:
   844   fixes c :: "'a :: ordered_idom"
   845   shows "(c \<le> c*b) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))"
   846 by (insert mult_le_cancel_left [of c 1 b], simp)
   847 
   848 lemma mult_le_cancel_left2:
   849   fixes c :: "'a :: ordered_idom"
   850   shows "(c*a \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))"
   851 by (insert mult_le_cancel_left [of c a 1], simp)
   852 
   853 lemma mult_less_cancel_right1:
   854   fixes c :: "'a :: ordered_idom"
   855   shows "(c < b*c) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))"
   856 by (insert mult_less_cancel_right [of 1 c b], simp)
   857 
   858 lemma mult_less_cancel_right2:
   859   fixes c :: "'a :: ordered_idom"
   860   shows "(a*c < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))"
   861 by (insert mult_less_cancel_right [of a c 1], simp)
   862 
   863 lemma mult_less_cancel_left1:
   864   fixes c :: "'a :: ordered_idom"
   865   shows "(c < c*b) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))"
   866 by (insert mult_less_cancel_left [of c 1 b], simp)
   867 
   868 lemma mult_less_cancel_left2:
   869   fixes c :: "'a :: ordered_idom"
   870   shows "(c*a < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))"
   871 by (insert mult_less_cancel_left [of c a 1], simp)
   872 
   873 lemma mult_cancel_right1 [simp]:
   874   fixes c :: "'a :: ring_1_no_zero_divisors"
   875   shows "(c = b*c) = (c = 0 | b=1)"
   876 by (insert mult_cancel_right [of 1 c b], force)
   877 
   878 lemma mult_cancel_right2 [simp]:
   879   fixes c :: "'a :: ring_1_no_zero_divisors"
   880   shows "(a*c = c) = (c = 0 | a=1)"
   881 by (insert mult_cancel_right [of a c 1], simp)
   882  
   883 lemma mult_cancel_left1 [simp]:
   884   fixes c :: "'a :: ring_1_no_zero_divisors"
   885   shows "(c = c*b) = (c = 0 | b=1)"
   886 by (insert mult_cancel_left [of c 1 b], force)
   887 
   888 lemma mult_cancel_left2 [simp]:
   889   fixes c :: "'a :: ring_1_no_zero_divisors"
   890   shows "(c*a = c) = (c = 0 | a=1)"
   891 by (insert mult_cancel_left [of c a 1], simp)
   892 
   893 
   894 text{*Simprules for comparisons where common factors can be cancelled.*}
   895 lemmas mult_compare_simps =
   896     mult_le_cancel_right mult_le_cancel_left
   897     mult_le_cancel_right1 mult_le_cancel_right2
   898     mult_le_cancel_left1 mult_le_cancel_left2
   899     mult_less_cancel_right mult_less_cancel_left
   900     mult_less_cancel_right1 mult_less_cancel_right2
   901     mult_less_cancel_left1 mult_less_cancel_left2
   902     mult_cancel_right mult_cancel_left
   903     mult_cancel_right1 mult_cancel_right2
   904     mult_cancel_left1 mult_cancel_left2
   905 
   906 
   907 (* what ordering?? this is a straight instance of mult_eq_0_iff
   908 text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement
   909       of an ordering.*}
   910 lemma field_mult_eq_0_iff [simp]:
   911   "(a*b = (0::'a::division_ring)) = (a = 0 | b = 0)"
   912 by simp
   913 *)
   914 (* subsumed by mult_cancel lemmas on ring_no_zero_divisors
   915 text{*Cancellation of equalities with a common factor*}
   916 lemma field_mult_cancel_right_lemma:
   917       assumes cnz: "c \<noteq> (0::'a::division_ring)"
   918          and eq:  "a*c = b*c"
   919         shows "a=b"
   920 proof -
   921   have "(a * c) * inverse c = (b * c) * inverse c"
   922     by (simp add: eq)
   923   thus "a=b"
   924     by (simp add: mult_assoc cnz)
   925 qed
   926 
   927 lemma field_mult_cancel_right [simp]:
   928      "(a*c = b*c) = (c = (0::'a::division_ring) | a=b)"
   929 by simp
   930 
   931 lemma field_mult_cancel_left [simp]:
   932      "(c*a = c*b) = (c = (0::'a::division_ring) | a=b)"
   933 by simp
   934 *)
   935 lemma nonzero_imp_inverse_nonzero:
   936   "a \<noteq> 0 ==> inverse a \<noteq> (0::'a::division_ring)"
   937 proof
   938   assume ianz: "inverse a = 0"
   939   assume "a \<noteq> 0"
   940   hence "1 = a * inverse a" by simp
   941   also have "... = 0" by (simp add: ianz)
   942   finally have "1 = (0::'a::division_ring)" .
   943   thus False by (simp add: eq_commute)
   944 qed
   945 
   946 
   947 subsection{*Basic Properties of @{term inverse}*}
   948 
   949 lemma inverse_zero_imp_zero: "inverse a = 0 ==> a = (0::'a::division_ring)"
   950 apply (rule ccontr) 
   951 apply (blast dest: nonzero_imp_inverse_nonzero) 
   952 done
   953 
   954 lemma inverse_nonzero_imp_nonzero:
   955    "inverse a = 0 ==> a = (0::'a::division_ring)"
   956 apply (rule ccontr) 
   957 apply (blast dest: nonzero_imp_inverse_nonzero) 
   958 done
   959 
   960 lemma inverse_nonzero_iff_nonzero [simp]:
   961    "(inverse a = 0) = (a = (0::'a::{division_ring,division_by_zero}))"
   962 by (force dest: inverse_nonzero_imp_nonzero) 
   963 
   964 lemma nonzero_inverse_minus_eq:
   965       assumes [simp]: "a\<noteq>0"
   966       shows "inverse(-a) = -inverse(a::'a::division_ring)"
   967 proof -
   968   have "-a * inverse (- a) = -a * - inverse a"
   969     by simp
   970   thus ?thesis 
   971     by (simp only: mult_cancel_left, simp)
   972 qed
   973 
   974 lemma inverse_minus_eq [simp]:
   975    "inverse(-a) = -inverse(a::'a::{division_ring,division_by_zero})"
   976 proof cases
   977   assume "a=0" thus ?thesis by (simp add: inverse_zero)
   978 next
   979   assume "a\<noteq>0" 
   980   thus ?thesis by (simp add: nonzero_inverse_minus_eq)
   981 qed
   982 
   983 lemma nonzero_inverse_eq_imp_eq:
   984       assumes inveq: "inverse a = inverse b"
   985 	  and anz:  "a \<noteq> 0"
   986 	  and bnz:  "b \<noteq> 0"
   987 	 shows "a = (b::'a::division_ring)"
   988 proof -
   989   have "a * inverse b = a * inverse a"
   990     by (simp add: inveq)
   991   hence "(a * inverse b) * b = (a * inverse a) * b"
   992     by simp
   993   thus "a = b"
   994     by (simp add: mult_assoc anz bnz)
   995 qed
   996 
   997 lemma inverse_eq_imp_eq:
   998   "inverse a = inverse b ==> a = (b::'a::{division_ring,division_by_zero})"
   999 apply (cases "a=0 | b=0") 
  1000  apply (force dest!: inverse_zero_imp_zero
  1001               simp add: eq_commute [of "0::'a"])
  1002 apply (force dest!: nonzero_inverse_eq_imp_eq) 
  1003 done
  1004 
  1005 lemma inverse_eq_iff_eq [simp]:
  1006   "(inverse a = inverse b) = (a = (b::'a::{division_ring,division_by_zero}))"
  1007 by (force dest!: inverse_eq_imp_eq)
  1008 
  1009 lemma nonzero_inverse_inverse_eq:
  1010       assumes [simp]: "a \<noteq> 0"
  1011       shows "inverse(inverse (a::'a::division_ring)) = a"
  1012   proof -
  1013   have "(inverse (inverse a) * inverse a) * a = a" 
  1014     by (simp add: nonzero_imp_inverse_nonzero)
  1015   thus ?thesis
  1016     by (simp add: mult_assoc)
  1017   qed
  1018 
  1019 lemma inverse_inverse_eq [simp]:
  1020      "inverse(inverse (a::'a::{division_ring,division_by_zero})) = a"
  1021   proof cases
  1022     assume "a=0" thus ?thesis by simp
  1023   next
  1024     assume "a\<noteq>0" 
  1025     thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
  1026   qed
  1027 
  1028 lemma inverse_1 [simp]: "inverse 1 = (1::'a::division_ring)"
  1029   proof -
  1030   have "inverse 1 * 1 = (1::'a::division_ring)" 
  1031     by (rule left_inverse [OF zero_neq_one [symmetric]])
  1032   thus ?thesis  by simp
  1033   qed
  1034 
  1035 lemma inverse_unique: 
  1036   assumes ab: "a*b = 1"
  1037   shows "inverse a = (b::'a::division_ring)"
  1038 proof -
  1039   have "a \<noteq> 0" using ab by auto
  1040   moreover have "inverse a * (a * b) = inverse a" by (simp add: ab) 
  1041   ultimately show ?thesis by (simp add: mult_assoc [symmetric]) 
  1042 qed
  1043 
  1044 lemma nonzero_inverse_mult_distrib: 
  1045       assumes anz: "a \<noteq> 0"
  1046           and bnz: "b \<noteq> 0"
  1047       shows "inverse(a*b) = inverse(b) * inverse(a::'a::division_ring)"
  1048   proof -
  1049   have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" 
  1050     by (simp add: anz bnz)
  1051   hence "inverse(a*b) * a = inverse(b)" 
  1052     by (simp add: mult_assoc bnz)
  1053   hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" 
  1054     by simp
  1055   thus ?thesis
  1056     by (simp add: mult_assoc anz)
  1057   qed
  1058 
  1059 text{*This version builds in division by zero while also re-orienting
  1060       the right-hand side.*}
  1061 lemma inverse_mult_distrib [simp]:
  1062      "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"
  1063   proof cases
  1064     assume "a \<noteq> 0 & b \<noteq> 0" 
  1065     thus ?thesis
  1066       by (simp add: nonzero_inverse_mult_distrib mult_commute)
  1067   next
  1068     assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
  1069     thus ?thesis
  1070       by force
  1071   qed
  1072 
  1073 lemma division_ring_inverse_add:
  1074   "[|(a::'a::division_ring) \<noteq> 0; b \<noteq> 0|]
  1075    ==> inverse a + inverse b = inverse a * (a+b) * inverse b"
  1076 by (simp add: ring_simps)
  1077 
  1078 lemma division_ring_inverse_diff:
  1079   "[|(a::'a::division_ring) \<noteq> 0; b \<noteq> 0|]
  1080    ==> inverse a - inverse b = inverse a * (b-a) * inverse b"
  1081 by (simp add: ring_simps)
  1082 
  1083 text{*There is no slick version using division by zero.*}
  1084 lemma inverse_add:
  1085   "[|a \<noteq> 0;  b \<noteq> 0|]
  1086    ==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)"
  1087 by (simp add: division_ring_inverse_add mult_ac)
  1088 
  1089 lemma inverse_divide [simp]:
  1090   "inverse (a/b) = b / (a::'a::{field,division_by_zero})"
  1091 by (simp add: divide_inverse mult_commute)
  1092 
  1093 
  1094 subsection {* Calculations with fractions *}
  1095 
  1096 text{* There is a whole bunch of simp-rules just for class @{text
  1097 field} but none for class @{text field} and @{text nonzero_divides}
  1098 because the latter are covered by a simproc. *}
  1099 
  1100 lemma nonzero_mult_divide_mult_cancel_left[simp,noatp]:
  1101 assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" shows "(c*a)/(c*b) = a/(b::'a::field)"
  1102 proof -
  1103   have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
  1104     by (simp add: divide_inverse nonzero_inverse_mult_distrib)
  1105   also have "... =  a * inverse b * (inverse c * c)"
  1106     by (simp only: mult_ac)
  1107   also have "... =  a * inverse b"
  1108     by simp
  1109     finally show ?thesis 
  1110     by (simp add: divide_inverse)
  1111 qed
  1112 
  1113 lemma mult_divide_mult_cancel_left:
  1114   "c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"
  1115 apply (cases "b = 0")
  1116 apply (simp_all add: nonzero_mult_divide_mult_cancel_left)
  1117 done
  1118 
  1119 lemma nonzero_mult_divide_mult_cancel_right [noatp]:
  1120   "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)"
  1121 by (simp add: mult_commute [of _ c] nonzero_mult_divide_mult_cancel_left) 
  1122 
  1123 lemma mult_divide_mult_cancel_right:
  1124   "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"
  1125 apply (cases "b = 0")
  1126 apply (simp_all add: nonzero_mult_divide_mult_cancel_right)
  1127 done
  1128 
  1129 lemma divide_1 [simp]: "a/1 = (a::'a::field)"
  1130 by (simp add: divide_inverse)
  1131 
  1132 lemma times_divide_eq_right: "a * (b/c) = (a*b) / (c::'a::field)"
  1133 by (simp add: divide_inverse mult_assoc)
  1134 
  1135 lemma times_divide_eq_left: "(b/c) * a = (b*a) / (c::'a::field)"
  1136 by (simp add: divide_inverse mult_ac)
  1137 
  1138 lemmas times_divide_eq = times_divide_eq_right times_divide_eq_left
  1139 
  1140 lemma divide_divide_eq_right [simp,noatp]:
  1141   "a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"
  1142 by (simp add: divide_inverse mult_ac)
  1143 
  1144 lemma divide_divide_eq_left [simp,noatp]:
  1145   "(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"
  1146 by (simp add: divide_inverse mult_assoc)
  1147 
  1148 lemma add_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
  1149     x / y + w / z = (x * z + w * y) / (y * z)"
  1150 apply (subgoal_tac "x / y = (x * z) / (y * z)")
  1151 apply (erule ssubst)
  1152 apply (subgoal_tac "w / z = (w * y) / (y * z)")
  1153 apply (erule ssubst)
  1154 apply (rule add_divide_distrib [THEN sym])
  1155 apply (subst mult_commute)
  1156 apply (erule nonzero_mult_divide_mult_cancel_left [THEN sym])
  1157 apply assumption
  1158 apply (erule nonzero_mult_divide_mult_cancel_right [THEN sym])
  1159 apply assumption
  1160 done
  1161 
  1162 
  1163 subsubsection{*Special Cancellation Simprules for Division*}
  1164 
  1165 lemma mult_divide_mult_cancel_left_if[simp,noatp]:
  1166 fixes c :: "'a :: {field,division_by_zero}"
  1167 shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)"
  1168 by (simp add: mult_divide_mult_cancel_left)
  1169 
  1170 lemma nonzero_mult_divide_cancel_right[simp,noatp]:
  1171   "b \<noteq> 0 \<Longrightarrow> a * b / b = (a::'a::field)"
  1172 using nonzero_mult_divide_mult_cancel_right[of 1 b a] by simp
  1173 
  1174 lemma nonzero_mult_divide_cancel_left[simp,noatp]:
  1175   "a \<noteq> 0 \<Longrightarrow> a * b / a = (b::'a::field)"
  1176 using nonzero_mult_divide_mult_cancel_left[of 1 a b] by simp
  1177 
  1178 
  1179 lemma nonzero_divide_mult_cancel_right[simp,noatp]:
  1180   "\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> b / (a * b) = 1/(a::'a::field)"
  1181 using nonzero_mult_divide_mult_cancel_right[of a b 1] by simp
  1182 
  1183 lemma nonzero_divide_mult_cancel_left[simp,noatp]:
  1184   "\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> a / (a * b) = 1/(b::'a::field)"
  1185 using nonzero_mult_divide_mult_cancel_left[of b a 1] by simp
  1186 
  1187 
  1188 lemma nonzero_mult_divide_mult_cancel_left2[simp,noatp]:
  1189   "[|b\<noteq>0; c\<noteq>0|] ==> (c*a) / (b*c) = a/(b::'a::field)"
  1190 using nonzero_mult_divide_mult_cancel_left[of b c a] by(simp add:mult_ac)
  1191 
  1192 lemma nonzero_mult_divide_mult_cancel_right2[simp,noatp]:
  1193   "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (c*b) = a/(b::'a::field)"
  1194 using nonzero_mult_divide_mult_cancel_right[of b c a] by(simp add:mult_ac)
  1195 
  1196 
  1197 subsection {* Division and Unary Minus *}
  1198 
  1199 lemma nonzero_minus_divide_left: "b \<noteq> 0 ==> - (a/b) = (-a) / (b::'a::field)"
  1200 by (simp add: divide_inverse minus_mult_left)
  1201 
  1202 lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a/b) = a / -(b::'a::field)"
  1203 by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right)
  1204 
  1205 lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a)/(-b) = a / (b::'a::field)"
  1206 by (simp add: divide_inverse nonzero_inverse_minus_eq)
  1207 
  1208 lemma minus_divide_left: "- (a/b) = (-a) / (b::'a::field)"
  1209 by (simp add: divide_inverse minus_mult_left [symmetric])
  1210 
  1211 lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"
  1212 by (simp add: divide_inverse minus_mult_right [symmetric])
  1213 
  1214 
  1215 text{*The effect is to extract signs from divisions*}
  1216 lemmas divide_minus_left = minus_divide_left [symmetric]
  1217 lemmas divide_minus_right = minus_divide_right [symmetric]
  1218 declare divide_minus_left [simp]   divide_minus_right [simp]
  1219 
  1220 text{*Also, extract signs from products*}
  1221 lemmas mult_minus_left = minus_mult_left [symmetric]
  1222 lemmas mult_minus_right = minus_mult_right [symmetric]
  1223 declare mult_minus_left [simp]   mult_minus_right [simp]
  1224 
  1225 lemma minus_divide_divide [simp]:
  1226   "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"
  1227 apply (cases "b=0", simp) 
  1228 apply (simp add: nonzero_minus_divide_divide) 
  1229 done
  1230 
  1231 lemma diff_divide_distrib: "(a-b)/(c::'a::field) = a/c - b/c"
  1232 by (simp add: diff_minus add_divide_distrib) 
  1233 
  1234 lemma add_divide_eq_iff:
  1235   "(z::'a::field) \<noteq> 0 \<Longrightarrow> x + y/z = (z*x + y)/z"
  1236 by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left)
  1237 
  1238 lemma divide_add_eq_iff:
  1239   "(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z + y = (x + z*y)/z"
  1240 by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left)
  1241 
  1242 lemma diff_divide_eq_iff:
  1243   "(z::'a::field) \<noteq> 0 \<Longrightarrow> x - y/z = (z*x - y)/z"
  1244 by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left)
  1245 
  1246 lemma divide_diff_eq_iff:
  1247   "(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z - y = (x - z*y)/z"
  1248 by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left)
  1249 
  1250 lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)"
  1251 proof -
  1252   assume [simp]: "c\<noteq>0"
  1253   have "(a = b/c) = (a*c = (b/c)*c)" by simp
  1254   also have "... = (a*c = b)" by (simp add: divide_inverse mult_assoc)
  1255   finally show ?thesis .
  1256 qed
  1257 
  1258 lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)"
  1259 proof -
  1260   assume [simp]: "c\<noteq>0"
  1261   have "(b/c = a) = ((b/c)*c = a*c)"  by simp
  1262   also have "... = (b = a*c)"  by (simp add: divide_inverse mult_assoc) 
  1263   finally show ?thesis .
  1264 qed
  1265 
  1266 lemma eq_divide_eq:
  1267   "((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)"
  1268 by (simp add: nonzero_eq_divide_eq) 
  1269 
  1270 lemma divide_eq_eq:
  1271   "(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)"
  1272 by (force simp add: nonzero_divide_eq_eq) 
  1273 
  1274 lemma divide_eq_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
  1275     b = a * c ==> b / c = a"
  1276   by (subst divide_eq_eq, simp)
  1277 
  1278 lemma eq_divide_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
  1279     a * c = b ==> a = b / c"
  1280   by (subst eq_divide_eq, simp)
  1281 
  1282 
  1283 lemmas field_eq_simps = ring_simps
  1284   (* pull / out*)
  1285   add_divide_eq_iff divide_add_eq_iff
  1286   diff_divide_eq_iff divide_diff_eq_iff
  1287   (* multiply eqn *)
  1288   nonzero_eq_divide_eq nonzero_divide_eq_eq
  1289 (* is added later:
  1290   times_divide_eq_left times_divide_eq_right
  1291 *)
  1292 
  1293 text{*An example:*}
  1294 lemma fixes a b c d e f :: "'a::field"
  1295 shows "\<lbrakk>a\<noteq>b; c\<noteq>d; e\<noteq>f \<rbrakk> \<Longrightarrow> ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) = 1"
  1296 apply(subgoal_tac "(c-d)*(e-f)*(a-b) \<noteq> 0")
  1297  apply(simp add:field_eq_simps)
  1298 apply(simp)
  1299 done
  1300 
  1301 
  1302 lemma diff_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
  1303     x / y - w / z = (x * z - w * y) / (y * z)"
  1304 by (simp add:field_eq_simps times_divide_eq)
  1305 
  1306 lemma frac_eq_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
  1307     (x / y = w / z) = (x * z = w * y)"
  1308 by (simp add:field_eq_simps times_divide_eq)
  1309 
  1310 
  1311 subsection {* Ordered Fields *}
  1312 
  1313 lemma positive_imp_inverse_positive: 
  1314 assumes a_gt_0: "0 < a"  shows "0 < inverse (a::'a::ordered_field)"
  1315 proof -
  1316   have "0 < a * inverse a" 
  1317     by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one)
  1318   thus "0 < inverse a" 
  1319     by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff)
  1320 qed
  1321 
  1322 lemma negative_imp_inverse_negative:
  1323   "a < 0 ==> inverse a < (0::'a::ordered_field)"
  1324 by (insert positive_imp_inverse_positive [of "-a"], 
  1325     simp add: nonzero_inverse_minus_eq order_less_imp_not_eq)
  1326 
  1327 lemma inverse_le_imp_le:
  1328 assumes invle: "inverse a \<le> inverse b" and apos:  "0 < a"
  1329 shows "b \<le> (a::'a::ordered_field)"
  1330 proof (rule classical)
  1331   assume "~ b \<le> a"
  1332   hence "a < b"  by (simp add: linorder_not_le)
  1333   hence bpos: "0 < b"  by (blast intro: apos order_less_trans)
  1334   hence "a * inverse a \<le> a * inverse b"
  1335     by (simp add: apos invle order_less_imp_le mult_left_mono)
  1336   hence "(a * inverse a) * b \<le> (a * inverse b) * b"
  1337     by (simp add: bpos order_less_imp_le mult_right_mono)
  1338   thus "b \<le> a"  by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)
  1339 qed
  1340 
  1341 lemma inverse_positive_imp_positive:
  1342 assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0"
  1343 shows "0 < (a::'a::ordered_field)"
  1344 proof -
  1345   have "0 < inverse (inverse a)"
  1346     using inv_gt_0 by (rule positive_imp_inverse_positive)
  1347   thus "0 < a"
  1348     using nz by (simp add: nonzero_inverse_inverse_eq)
  1349 qed
  1350 
  1351 lemma inverse_positive_iff_positive [simp]:
  1352   "(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))"
  1353 apply (cases "a = 0", simp)
  1354 apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
  1355 done
  1356 
  1357 lemma inverse_negative_imp_negative:
  1358 assumes inv_less_0: "inverse a < 0" and nz:  "a \<noteq> 0"
  1359 shows "a < (0::'a::ordered_field)"
  1360 proof -
  1361   have "inverse (inverse a) < 0"
  1362     using inv_less_0 by (rule negative_imp_inverse_negative)
  1363   thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq)
  1364 qed
  1365 
  1366 lemma inverse_negative_iff_negative [simp]:
  1367   "(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))"
  1368 apply (cases "a = 0", simp)
  1369 apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
  1370 done
  1371 
  1372 lemma inverse_nonnegative_iff_nonnegative [simp]:
  1373   "(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))"
  1374 by (simp add: linorder_not_less [symmetric])
  1375 
  1376 lemma inverse_nonpositive_iff_nonpositive [simp]:
  1377   "(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))"
  1378 by (simp add: linorder_not_less [symmetric])
  1379 
  1380 lemma ordered_field_no_lb: "\<forall> x. \<exists>y. y < (x::'a::ordered_field)"
  1381 proof
  1382   fix x::'a
  1383   have m1: "- (1::'a) < 0" by simp
  1384   from add_strict_right_mono[OF m1, where c=x] 
  1385   have "(- 1) + x < x" by simp
  1386   thus "\<exists>y. y < x" by blast
  1387 qed
  1388 
  1389 lemma ordered_field_no_ub: "\<forall> x. \<exists>y. y > (x::'a::ordered_field)"
  1390 proof
  1391   fix x::'a
  1392   have m1: " (1::'a) > 0" by simp
  1393   from add_strict_right_mono[OF m1, where c=x] 
  1394   have "1 + x > x" by simp
  1395   thus "\<exists>y. y > x" by blast
  1396 qed
  1397 
  1398 subsection{*Anti-Monotonicity of @{term inverse}*}
  1399 
  1400 lemma less_imp_inverse_less:
  1401 assumes less: "a < b" and apos:  "0 < a"
  1402 shows "inverse b < inverse (a::'a::ordered_field)"
  1403 proof (rule ccontr)
  1404   assume "~ inverse b < inverse a"
  1405   hence "inverse a \<le> inverse b"
  1406     by (simp add: linorder_not_less)
  1407   hence "~ (a < b)"
  1408     by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos])
  1409   thus False
  1410     by (rule notE [OF _ less])
  1411 qed
  1412 
  1413 lemma inverse_less_imp_less:
  1414   "[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::ordered_field)"
  1415 apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"])
  1416 apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
  1417 done
  1418 
  1419 text{*Both premises are essential. Consider -1 and 1.*}
  1420 lemma inverse_less_iff_less [simp,noatp]:
  1421   "[|0 < a; 0 < b|] ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
  1422 by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
  1423 
  1424 lemma le_imp_inverse_le:
  1425   "[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
  1426 by (force simp add: order_le_less less_imp_inverse_less)
  1427 
  1428 lemma inverse_le_iff_le [simp,noatp]:
  1429  "[|0 < a; 0 < b|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
  1430 by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
  1431 
  1432 
  1433 text{*These results refer to both operands being negative.  The opposite-sign
  1434 case is trivial, since inverse preserves signs.*}
  1435 lemma inverse_le_imp_le_neg:
  1436   "[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::ordered_field)"
  1437 apply (rule classical) 
  1438 apply (subgoal_tac "a < 0") 
  1439  prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans) 
  1440 apply (insert inverse_le_imp_le [of "-b" "-a"])
  1441 apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
  1442 done
  1443 
  1444 lemma less_imp_inverse_less_neg:
  1445    "[|a < b; b < 0|] ==> inverse b < inverse (a::'a::ordered_field)"
  1446 apply (subgoal_tac "a < 0") 
  1447  prefer 2 apply (blast intro: order_less_trans) 
  1448 apply (insert less_imp_inverse_less [of "-b" "-a"])
  1449 apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
  1450 done
  1451 
  1452 lemma inverse_less_imp_less_neg:
  1453    "[|inverse a < inverse b; b < 0|] ==> b < (a::'a::ordered_field)"
  1454 apply (rule classical) 
  1455 apply (subgoal_tac "a < 0") 
  1456  prefer 2
  1457  apply (force simp add: linorder_not_less intro: order_le_less_trans) 
  1458 apply (insert inverse_less_imp_less [of "-b" "-a"])
  1459 apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
  1460 done
  1461 
  1462 lemma inverse_less_iff_less_neg [simp,noatp]:
  1463   "[|a < 0; b < 0|] ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
  1464 apply (insert inverse_less_iff_less [of "-b" "-a"])
  1465 apply (simp del: inverse_less_iff_less 
  1466             add: order_less_imp_not_eq nonzero_inverse_minus_eq)
  1467 done
  1468 
  1469 lemma le_imp_inverse_le_neg:
  1470   "[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
  1471 by (force simp add: order_le_less less_imp_inverse_less_neg)
  1472 
  1473 lemma inverse_le_iff_le_neg [simp,noatp]:
  1474  "[|a < 0; b < 0|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
  1475 by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
  1476 
  1477 
  1478 subsection{*Inverses and the Number One*}
  1479 
  1480 lemma one_less_inverse_iff:
  1481   "(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"
  1482 proof cases
  1483   assume "0 < x"
  1484     with inverse_less_iff_less [OF zero_less_one, of x]
  1485     show ?thesis by simp
  1486 next
  1487   assume notless: "~ (0 < x)"
  1488   have "~ (1 < inverse x)"
  1489   proof
  1490     assume "1 < inverse x"
  1491     also with notless have "... \<le> 0" by (simp add: linorder_not_less)
  1492     also have "... < 1" by (rule zero_less_one) 
  1493     finally show False by auto
  1494   qed
  1495   with notless show ?thesis by simp
  1496 qed
  1497 
  1498 lemma inverse_eq_1_iff [simp]:
  1499   "(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))"
  1500 by (insert inverse_eq_iff_eq [of x 1], simp) 
  1501 
  1502 lemma one_le_inverse_iff:
  1503   "(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))"
  1504 by (force simp add: order_le_less one_less_inverse_iff zero_less_one 
  1505                     eq_commute [of 1]) 
  1506 
  1507 lemma inverse_less_1_iff:
  1508   "(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{ordered_field,division_by_zero}))"
  1509 by (simp add: linorder_not_le [symmetric] one_le_inverse_iff) 
  1510 
  1511 lemma inverse_le_1_iff:
  1512   "(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{ordered_field,division_by_zero}))"
  1513 by (simp add: linorder_not_less [symmetric] one_less_inverse_iff) 
  1514 
  1515 
  1516 subsection{*Simplification of Inequalities Involving Literal Divisors*}
  1517 
  1518 lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"
  1519 proof -
  1520   assume less: "0<c"
  1521   hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
  1522     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
  1523   also have "... = (a*c \<le> b)"
  1524     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
  1525   finally show ?thesis .
  1526 qed
  1527 
  1528 lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"
  1529 proof -
  1530   assume less: "c<0"
  1531   hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
  1532     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
  1533   also have "... = (b \<le> a*c)"
  1534     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
  1535   finally show ?thesis .
  1536 qed
  1537 
  1538 lemma le_divide_eq:
  1539   "(a \<le> b/c) = 
  1540    (if 0 < c then a*c \<le> b
  1541              else if c < 0 then b \<le> a*c
  1542              else  a \<le> (0::'a::{ordered_field,division_by_zero}))"
  1543 apply (cases "c=0", simp) 
  1544 apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
  1545 done
  1546 
  1547 lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)"
  1548 proof -
  1549   assume less: "0<c"
  1550   hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
  1551     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
  1552   also have "... = (b \<le> a*c)"
  1553     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
  1554   finally show ?thesis .
  1555 qed
  1556 
  1557 lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)"
  1558 proof -
  1559   assume less: "c<0"
  1560   hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
  1561     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
  1562   also have "... = (a*c \<le> b)"
  1563     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
  1564   finally show ?thesis .
  1565 qed
  1566 
  1567 lemma divide_le_eq:
  1568   "(b/c \<le> a) = 
  1569    (if 0 < c then b \<le> a*c
  1570              else if c < 0 then a*c \<le> b
  1571              else 0 \<le> (a::'a::{ordered_field,division_by_zero}))"
  1572 apply (cases "c=0", simp) 
  1573 apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) 
  1574 done
  1575 
  1576 lemma pos_less_divide_eq:
  1577      "0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)"
  1578 proof -
  1579   assume less: "0<c"
  1580   hence "(a < b/c) = (a*c < (b/c)*c)"
  1581     by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
  1582   also have "... = (a*c < b)"
  1583     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
  1584   finally show ?thesis .
  1585 qed
  1586 
  1587 lemma neg_less_divide_eq:
  1588  "c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)"
  1589 proof -
  1590   assume less: "c<0"
  1591   hence "(a < b/c) = ((b/c)*c < a*c)"
  1592     by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
  1593   also have "... = (b < a*c)"
  1594     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
  1595   finally show ?thesis .
  1596 qed
  1597 
  1598 lemma less_divide_eq:
  1599   "(a < b/c) = 
  1600    (if 0 < c then a*c < b
  1601              else if c < 0 then b < a*c
  1602              else  a < (0::'a::{ordered_field,division_by_zero}))"
  1603 apply (cases "c=0", simp) 
  1604 apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) 
  1605 done
  1606 
  1607 lemma pos_divide_less_eq:
  1608      "0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)"
  1609 proof -
  1610   assume less: "0<c"
  1611   hence "(b/c < a) = ((b/c)*c < a*c)"
  1612     by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
  1613   also have "... = (b < a*c)"
  1614     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
  1615   finally show ?thesis .
  1616 qed
  1617 
  1618 lemma neg_divide_less_eq:
  1619  "c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)"
  1620 proof -
  1621   assume less: "c<0"
  1622   hence "(b/c < a) = (a*c < (b/c)*c)"
  1623     by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
  1624   also have "... = (a*c < b)"
  1625     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
  1626   finally show ?thesis .
  1627 qed
  1628 
  1629 lemma divide_less_eq:
  1630   "(b/c < a) = 
  1631    (if 0 < c then b < a*c
  1632              else if c < 0 then a*c < b
  1633              else 0 < (a::'a::{ordered_field,division_by_zero}))"
  1634 apply (cases "c=0", simp) 
  1635 apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) 
  1636 done
  1637 
  1638 
  1639 subsection{*Field simplification*}
  1640 
  1641 text{* Lemmas @{text field_simps} multiply with denominators in
  1642 in(equations) if they can be proved to be non-zero (for equations) or
  1643 positive/negative (for inequations). *}
  1644 
  1645 lemmas field_simps = field_eq_simps
  1646   (* multiply ineqn *)
  1647   pos_divide_less_eq neg_divide_less_eq
  1648   pos_less_divide_eq neg_less_divide_eq
  1649   pos_divide_le_eq neg_divide_le_eq
  1650   pos_le_divide_eq neg_le_divide_eq
  1651 
  1652 text{* Lemmas @{text sign_simps} is a first attempt to automate proofs
  1653 of positivity/negativity needed for @{text field_simps}. Have not added @{text
  1654 sign_simps} to @{text field_simps} because the former can lead to case
  1655 explosions. *}
  1656 
  1657 lemmas sign_simps = group_simps
  1658   zero_less_mult_iff  mult_less_0_iff
  1659 
  1660 (* Only works once linear arithmetic is installed:
  1661 text{*An example:*}
  1662 lemma fixes a b c d e f :: "'a::ordered_field"
  1663 shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow>
  1664  ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) <
  1665  ((e-f)*(a-b)*(c-d))/((e-f)*(a-b)*(c-d)) + u"
  1666 apply(subgoal_tac "(c-d)*(e-f)*(a-b) > 0")
  1667  prefer 2 apply(simp add:sign_simps)
  1668 apply(subgoal_tac "(c-d)*(e-f)*(a-b)*u > 0")
  1669  prefer 2 apply(simp add:sign_simps)
  1670 apply(simp add:field_simps)
  1671 done
  1672 *)
  1673 
  1674 
  1675 subsection{*Division and Signs*}
  1676 
  1677 lemma zero_less_divide_iff:
  1678      "((0::'a::{ordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
  1679 by (simp add: divide_inverse zero_less_mult_iff)
  1680 
  1681 lemma divide_less_0_iff:
  1682      "(a/b < (0::'a::{ordered_field,division_by_zero})) = 
  1683       (0 < a & b < 0 | a < 0 & 0 < b)"
  1684 by (simp add: divide_inverse mult_less_0_iff)
  1685 
  1686 lemma zero_le_divide_iff:
  1687      "((0::'a::{ordered_field,division_by_zero}) \<le> a/b) =
  1688       (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
  1689 by (simp add: divide_inverse zero_le_mult_iff)
  1690 
  1691 lemma divide_le_0_iff:
  1692      "(a/b \<le> (0::'a::{ordered_field,division_by_zero})) =
  1693       (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
  1694 by (simp add: divide_inverse mult_le_0_iff)
  1695 
  1696 lemma divide_eq_0_iff [simp,noatp]:
  1697      "(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))"
  1698 by (simp add: divide_inverse)
  1699 
  1700 lemma divide_pos_pos:
  1701   "0 < (x::'a::ordered_field) ==> 0 < y ==> 0 < x / y"
  1702 by(simp add:field_simps)
  1703 
  1704 
  1705 lemma divide_nonneg_pos:
  1706   "0 <= (x::'a::ordered_field) ==> 0 < y ==> 0 <= x / y"
  1707 by(simp add:field_simps)
  1708 
  1709 lemma divide_neg_pos:
  1710   "(x::'a::ordered_field) < 0 ==> 0 < y ==> x / y < 0"
  1711 by(simp add:field_simps)
  1712 
  1713 lemma divide_nonpos_pos:
  1714   "(x::'a::ordered_field) <= 0 ==> 0 < y ==> x / y <= 0"
  1715 by(simp add:field_simps)
  1716 
  1717 lemma divide_pos_neg:
  1718   "0 < (x::'a::ordered_field) ==> y < 0 ==> x / y < 0"
  1719 by(simp add:field_simps)
  1720 
  1721 lemma divide_nonneg_neg:
  1722   "0 <= (x::'a::ordered_field) ==> y < 0 ==> x / y <= 0" 
  1723 by(simp add:field_simps)
  1724 
  1725 lemma divide_neg_neg:
  1726   "(x::'a::ordered_field) < 0 ==> y < 0 ==> 0 < x / y"
  1727 by(simp add:field_simps)
  1728 
  1729 lemma divide_nonpos_neg:
  1730   "(x::'a::ordered_field) <= 0 ==> y < 0 ==> 0 <= x / y"
  1731 by(simp add:field_simps)
  1732 
  1733 
  1734 subsection{*Cancellation Laws for Division*}
  1735 
  1736 lemma divide_cancel_right [simp,noatp]:
  1737      "(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))"
  1738 apply (cases "c=0", simp)
  1739 apply (simp add: divide_inverse)
  1740 done
  1741 
  1742 lemma divide_cancel_left [simp,noatp]:
  1743      "(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))" 
  1744 apply (cases "c=0", simp)
  1745 apply (simp add: divide_inverse)
  1746 done
  1747 
  1748 
  1749 subsection {* Division and the Number One *}
  1750 
  1751 text{*Simplify expressions equated with 1*}
  1752 lemma divide_eq_1_iff [simp,noatp]:
  1753      "(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
  1754 apply (cases "b=0", simp)
  1755 apply (simp add: right_inverse_eq)
  1756 done
  1757 
  1758 lemma one_eq_divide_iff [simp,noatp]:
  1759      "(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
  1760 by (simp add: eq_commute [of 1])
  1761 
  1762 lemma zero_eq_1_divide_iff [simp,noatp]:
  1763      "((0::'a::{ordered_field,division_by_zero}) = 1/a) = (a = 0)"
  1764 apply (cases "a=0", simp)
  1765 apply (auto simp add: nonzero_eq_divide_eq)
  1766 done
  1767 
  1768 lemma one_divide_eq_0_iff [simp,noatp]:
  1769      "(1/a = (0::'a::{ordered_field,division_by_zero})) = (a = 0)"
  1770 apply (cases "a=0", simp)
  1771 apply (insert zero_neq_one [THEN not_sym])
  1772 apply (auto simp add: nonzero_divide_eq_eq)
  1773 done
  1774 
  1775 text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}
  1776 lemmas zero_less_divide_1_iff = zero_less_divide_iff [of 1, simplified]
  1777 lemmas divide_less_0_1_iff = divide_less_0_iff [of 1, simplified]
  1778 lemmas zero_le_divide_1_iff = zero_le_divide_iff [of 1, simplified]
  1779 lemmas divide_le_0_1_iff = divide_le_0_iff [of 1, simplified]
  1780 
  1781 declare zero_less_divide_1_iff [simp]
  1782 declare divide_less_0_1_iff [simp,noatp]
  1783 declare zero_le_divide_1_iff [simp]
  1784 declare divide_le_0_1_iff [simp,noatp]
  1785 
  1786 
  1787 subsection {* Ordering Rules for Division *}
  1788 
  1789 lemma divide_strict_right_mono:
  1790      "[|a < b; 0 < c|] ==> a / c < b / (c::'a::ordered_field)"
  1791 by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono 
  1792               positive_imp_inverse_positive)
  1793 
  1794 lemma divide_right_mono:
  1795      "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{ordered_field,division_by_zero})"
  1796 by (force simp add: divide_strict_right_mono order_le_less)
  1797 
  1798 lemma divide_right_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b 
  1799     ==> c <= 0 ==> b / c <= a / c"
  1800 apply (drule divide_right_mono [of _ _ "- c"])
  1801 apply auto
  1802 done
  1803 
  1804 lemma divide_strict_right_mono_neg:
  1805      "[|b < a; c < 0|] ==> a / c < b / (c::'a::ordered_field)"
  1806 apply (drule divide_strict_right_mono [of _ _ "-c"], simp)
  1807 apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric])
  1808 done
  1809 
  1810 text{*The last premise ensures that @{term a} and @{term b} 
  1811       have the same sign*}
  1812 lemma divide_strict_left_mono:
  1813   "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
  1814 by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono)
  1815 
  1816 lemma divide_left_mono:
  1817   "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::ordered_field)"
  1818 by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_right_mono)
  1819 
  1820 lemma divide_left_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b 
  1821     ==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"
  1822   apply (drule divide_left_mono [of _ _ "- c"])
  1823   apply (auto simp add: mult_commute)
  1824 done
  1825 
  1826 lemma divide_strict_left_mono_neg:
  1827   "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
  1828 by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono_neg)
  1829 
  1830 
  1831 text{*Simplify quotients that are compared with the value 1.*}
  1832 
  1833 lemma le_divide_eq_1 [noatp]:
  1834   fixes a :: "'a :: {ordered_field,division_by_zero}"
  1835   shows "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"
  1836 by (auto simp add: le_divide_eq)
  1837 
  1838 lemma divide_le_eq_1 [noatp]:
  1839   fixes a :: "'a :: {ordered_field,division_by_zero}"
  1840   shows "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"
  1841 by (auto simp add: divide_le_eq)
  1842 
  1843 lemma less_divide_eq_1 [noatp]:
  1844   fixes a :: "'a :: {ordered_field,division_by_zero}"
  1845   shows "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
  1846 by (auto simp add: less_divide_eq)
  1847 
  1848 lemma divide_less_eq_1 [noatp]:
  1849   fixes a :: "'a :: {ordered_field,division_by_zero}"
  1850   shows "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
  1851 by (auto simp add: divide_less_eq)
  1852 
  1853 
  1854 subsection{*Conditional Simplification Rules: No Case Splits*}
  1855 
  1856 lemma le_divide_eq_1_pos [simp,noatp]:
  1857   fixes a :: "'a :: {ordered_field,division_by_zero}"
  1858   shows "0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)"
  1859 by (auto simp add: le_divide_eq)
  1860 
  1861 lemma le_divide_eq_1_neg [simp,noatp]:
  1862   fixes a :: "'a :: {ordered_field,division_by_zero}"
  1863   shows "a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)"
  1864 by (auto simp add: le_divide_eq)
  1865 
  1866 lemma divide_le_eq_1_pos [simp,noatp]:
  1867   fixes a :: "'a :: {ordered_field,division_by_zero}"
  1868   shows "0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)"
  1869 by (auto simp add: divide_le_eq)
  1870 
  1871 lemma divide_le_eq_1_neg [simp,noatp]:
  1872   fixes a :: "'a :: {ordered_field,division_by_zero}"
  1873   shows "a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)"
  1874 by (auto simp add: divide_le_eq)
  1875 
  1876 lemma less_divide_eq_1_pos [simp,noatp]:
  1877   fixes a :: "'a :: {ordered_field,division_by_zero}"
  1878   shows "0 < a \<Longrightarrow> (1 < b/a) = (a < b)"
  1879 by (auto simp add: less_divide_eq)
  1880 
  1881 lemma less_divide_eq_1_neg [simp,noatp]:
  1882   fixes a :: "'a :: {ordered_field,division_by_zero}"
  1883   shows "a < 0 \<Longrightarrow> (1 < b/a) = (b < a)"
  1884 by (auto simp add: less_divide_eq)
  1885 
  1886 lemma divide_less_eq_1_pos [simp,noatp]:
  1887   fixes a :: "'a :: {ordered_field,division_by_zero}"
  1888   shows "0 < a \<Longrightarrow> (b/a < 1) = (b < a)"
  1889 by (auto simp add: divide_less_eq)
  1890 
  1891 lemma divide_less_eq_1_neg [simp,noatp]:
  1892   fixes a :: "'a :: {ordered_field,division_by_zero}"
  1893   shows "a < 0 \<Longrightarrow> b/a < 1 <-> a < b"
  1894 by (auto simp add: divide_less_eq)
  1895 
  1896 lemma eq_divide_eq_1 [simp,noatp]:
  1897   fixes a :: "'a :: {ordered_field,division_by_zero}"
  1898   shows "(1 = b/a) = ((a \<noteq> 0 & a = b))"
  1899 by (auto simp add: eq_divide_eq)
  1900 
  1901 lemma divide_eq_eq_1 [simp,noatp]:
  1902   fixes a :: "'a :: {ordered_field,division_by_zero}"
  1903   shows "(b/a = 1) = ((a \<noteq> 0 & a = b))"
  1904 by (auto simp add: divide_eq_eq)
  1905 
  1906 
  1907 subsection {* Reasoning about inequalities with division *}
  1908 
  1909 lemma mult_right_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
  1910     ==> x * y <= x"
  1911   by (auto simp add: mult_compare_simps);
  1912 
  1913 lemma mult_left_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
  1914     ==> y * x <= x"
  1915   by (auto simp add: mult_compare_simps);
  1916 
  1917 lemma mult_imp_div_pos_le: "0 < (y::'a::ordered_field) ==> x <= z * y ==>
  1918     x / y <= z";
  1919   by (subst pos_divide_le_eq, assumption+);
  1920 
  1921 lemma mult_imp_le_div_pos: "0 < (y::'a::ordered_field) ==> z * y <= x ==>
  1922     z <= x / y"
  1923 by(simp add:field_simps)
  1924 
  1925 lemma mult_imp_div_pos_less: "0 < (y::'a::ordered_field) ==> x < z * y ==>
  1926     x / y < z"
  1927 by(simp add:field_simps)
  1928 
  1929 lemma mult_imp_less_div_pos: "0 < (y::'a::ordered_field) ==> z * y < x ==>
  1930     z < x / y"
  1931 by(simp add:field_simps)
  1932 
  1933 lemma frac_le: "(0::'a::ordered_field) <= x ==> 
  1934     x <= y ==> 0 < w ==> w <= z  ==> x / z <= y / w"
  1935   apply (rule mult_imp_div_pos_le)
  1936   apply simp
  1937   apply (subst times_divide_eq_left)
  1938   apply (rule mult_imp_le_div_pos, assumption)
  1939   thm mult_mono
  1940   thm mult_mono'
  1941   apply (rule mult_mono)
  1942   apply simp_all
  1943 done
  1944 
  1945 lemma frac_less: "(0::'a::ordered_field) <= x ==> 
  1946     x < y ==> 0 < w ==> w <= z  ==> x / z < y / w"
  1947   apply (rule mult_imp_div_pos_less)
  1948   apply simp;
  1949   apply (subst times_divide_eq_left);
  1950   apply (rule mult_imp_less_div_pos, assumption)
  1951   apply (erule mult_less_le_imp_less)
  1952   apply simp_all
  1953 done
  1954 
  1955 lemma frac_less2: "(0::'a::ordered_field) < x ==> 
  1956     x <= y ==> 0 < w ==> w < z  ==> x / z < y / w"
  1957   apply (rule mult_imp_div_pos_less)
  1958   apply simp_all
  1959   apply (subst times_divide_eq_left);
  1960   apply (rule mult_imp_less_div_pos, assumption)
  1961   apply (erule mult_le_less_imp_less)
  1962   apply simp_all
  1963 done
  1964 
  1965 text{*It's not obvious whether these should be simprules or not. 
  1966   Their effect is to gather terms into one big fraction, like
  1967   a*b*c / x*y*z. The rationale for that is unclear, but many proofs 
  1968   seem to need them.*}
  1969 
  1970 declare times_divide_eq [simp]
  1971 
  1972 
  1973 subsection {* Ordered Fields are Dense *}
  1974 
  1975 context ordered_semidom
  1976 begin
  1977 
  1978 lemma less_add_one: "a < a + 1"
  1979 proof -
  1980   have "a + 0 < a + 1"
  1981     by (blast intro: zero_less_one add_strict_left_mono)
  1982   thus ?thesis by simp
  1983 qed
  1984 
  1985 lemma zero_less_two: "0 < 1 + 1"
  1986   by (blast intro: less_trans zero_less_one less_add_one)
  1987 
  1988 end
  1989 
  1990 lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::ordered_field)"
  1991 by (simp add: field_simps zero_less_two)
  1992 
  1993 lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::ordered_field) < b"
  1994 by (simp add: field_simps zero_less_two)
  1995 
  1996 instance ordered_field < dense_linear_order
  1997 proof
  1998   fix x y :: 'a
  1999   have "x < x + 1" by simp
  2000   then show "\<exists>y. x < y" .. 
  2001   have "x - 1 < x" by simp
  2002   then show "\<exists>y. y < x" ..
  2003   show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum)
  2004 qed
  2005 
  2006 
  2007 subsection {* Absolute Value *}
  2008 
  2009 lemma mult_sgn_abs: "sgn x * abs x = (x::'a::{ordered_idom,linorder})"
  2010 using less_linear[of x 0]
  2011 by(auto simp: sgn_if abs_if)
  2012 
  2013 lemma abs_one [simp]: "abs 1 = (1::'a::ordered_idom)"
  2014 by (simp add: abs_if zero_less_one [THEN order_less_not_sym])
  2015 
  2016 lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lordered_ring))" 
  2017 proof -
  2018   let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"
  2019   let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
  2020   have a: "(abs a) * (abs b) = ?x"
  2021     by (simp only: abs_prts[of a] abs_prts[of b] ring_simps)
  2022   {
  2023     fix u v :: 'a
  2024     have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow> 
  2025               u * v = pprt a * pprt b + pprt a * nprt b + 
  2026                       nprt a * pprt b + nprt a * nprt b"
  2027       apply (subst prts[of u], subst prts[of v])
  2028       apply (simp add: ring_simps) 
  2029       done
  2030   }
  2031   note b = this[OF refl[of a] refl[of b]]
  2032   note addm = add_mono[of "0::'a" _ "0::'a", simplified]
  2033   note addm2 = add_mono[of _ "0::'a" _ "0::'a", simplified]
  2034   have xy: "- ?x <= ?y"
  2035     apply (simp)
  2036     apply (rule_tac y="0::'a" in order_trans)
  2037     apply (rule addm2)
  2038     apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos)
  2039     apply (rule addm)
  2040     apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos)
  2041     done
  2042   have yx: "?y <= ?x"
  2043     apply (simp add:diff_def)
  2044     apply (rule_tac y=0 in order_trans)
  2045     apply (rule addm2, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+)
  2046     apply (rule addm, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+)
  2047     done
  2048   have i1: "a*b <= abs a * abs b" by (simp only: a b yx)
  2049   have i2: "- (abs a * abs b) <= a*b" by (simp only: a b xy)
  2050   show ?thesis
  2051     apply (rule abs_leI)
  2052     apply (simp add: i1)
  2053     apply (simp add: i2[simplified minus_le_iff])
  2054     done
  2055 qed
  2056 
  2057 lemma abs_eq_mult: 
  2058   assumes "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)"
  2059   shows "abs (a*b) = abs a * abs (b::'a::lordered_ring)"
  2060 proof -
  2061   have s: "(0 <= a*b) | (a*b <= 0)"
  2062     apply (auto)    
  2063     apply (rule_tac split_mult_pos_le)
  2064     apply (rule_tac contrapos_np[of "a*b <= 0"])
  2065     apply (simp)
  2066     apply (rule_tac split_mult_neg_le)
  2067     apply (insert prems)
  2068     apply (blast)
  2069     done
  2070   have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
  2071     by (simp add: prts[symmetric])
  2072   show ?thesis
  2073   proof cases
  2074     assume "0 <= a * b"
  2075     then show ?thesis
  2076       apply (simp_all add: mulprts abs_prts)
  2077       apply (insert prems)
  2078       apply (auto simp add: 
  2079 	ring_simps 
  2080 	iffD1[OF zero_le_iff_zero_nprt] iffD1[OF le_zero_iff_zero_pprt]
  2081 	iffD1[OF le_zero_iff_pprt_id] iffD1[OF zero_le_iff_nprt_id])
  2082 	apply(drule (1) mult_nonneg_nonpos[of a b], simp)
  2083 	apply(drule (1) mult_nonneg_nonpos2[of b a], simp)
  2084       done
  2085   next
  2086     assume "~(0 <= a*b)"
  2087     with s have "a*b <= 0" by simp
  2088     then show ?thesis
  2089       apply (simp_all add: mulprts abs_prts)
  2090       apply (insert prems)
  2091       apply (auto simp add: ring_simps)
  2092       apply(drule (1) mult_nonneg_nonneg[of a b],simp)
  2093       apply(drule (1) mult_nonpos_nonpos[of a b],simp)
  2094       done
  2095   qed
  2096 qed
  2097 
  2098 lemma abs_mult: "abs (a * b) = abs a * abs (b::'a::ordered_idom)" 
  2099 by (simp add: abs_eq_mult linorder_linear)
  2100 
  2101 lemma abs_mult_self: "abs a * abs a = a * (a::'a::ordered_idom)"
  2102 by (simp add: abs_if) 
  2103 
  2104 lemma nonzero_abs_inverse:
  2105      "a \<noteq> 0 ==> abs (inverse (a::'a::ordered_field)) = inverse (abs a)"
  2106 apply (auto simp add: linorder_neq_iff abs_if nonzero_inverse_minus_eq 
  2107                       negative_imp_inverse_negative)
  2108 apply (blast intro: positive_imp_inverse_positive elim: order_less_asym) 
  2109 done
  2110 
  2111 lemma abs_inverse [simp]:
  2112      "abs (inverse (a::'a::{ordered_field,division_by_zero})) = 
  2113       inverse (abs a)"
  2114 apply (cases "a=0", simp) 
  2115 apply (simp add: nonzero_abs_inverse) 
  2116 done
  2117 
  2118 lemma nonzero_abs_divide:
  2119      "b \<noteq> 0 ==> abs (a / (b::'a::ordered_field)) = abs a / abs b"
  2120 by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 
  2121 
  2122 lemma abs_divide [simp]:
  2123      "abs (a / (b::'a::{ordered_field,division_by_zero})) = abs a / abs b"
  2124 apply (cases "b=0", simp) 
  2125 apply (simp add: nonzero_abs_divide) 
  2126 done
  2127 
  2128 lemma abs_mult_less:
  2129      "[| abs a < c; abs b < d |] ==> abs a * abs b < c*(d::'a::ordered_idom)"
  2130 proof -
  2131   assume ac: "abs a < c"
  2132   hence cpos: "0<c" by (blast intro: order_le_less_trans abs_ge_zero)
  2133   assume "abs b < d"
  2134   thus ?thesis by (simp add: ac cpos mult_strict_mono) 
  2135 qed
  2136 
  2137 lemma eq_minus_self_iff: "(a = -a) = (a = (0::'a::ordered_idom))"
  2138 by (force simp add: order_eq_iff le_minus_self_iff minus_le_self_iff)
  2139 
  2140 lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::ordered_idom))"
  2141 by (simp add: order_less_le le_minus_self_iff eq_minus_self_iff)
  2142 
  2143 lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::ordered_idom))" 
  2144 apply (simp add: order_less_le abs_le_iff)  
  2145 apply (auto simp add: abs_if minus_le_self_iff eq_minus_self_iff)
  2146 apply (simp add: le_minus_self_iff linorder_neq_iff) 
  2147 done
  2148 
  2149 lemma abs_mult_pos: "(0::'a::ordered_idom) <= x ==> 
  2150     (abs y) * x = abs (y * x)";
  2151   apply (subst abs_mult);
  2152   apply simp;
  2153 done;
  2154 
  2155 lemma abs_div_pos: "(0::'a::{division_by_zero,ordered_field}) < y ==> 
  2156     abs x / y = abs (x / y)";
  2157   apply (subst abs_divide);
  2158   apply (simp add: order_less_imp_le);
  2159 done;
  2160 
  2161 
  2162 subsection {* Bounds of products via negative and positive Part *}
  2163 
  2164 lemma mult_le_prts:
  2165   assumes
  2166   "a1 <= (a::'a::lordered_ring)"
  2167   "a <= a2"
  2168   "b1 <= b"
  2169   "b <= b2"
  2170   shows
  2171   "a * b <= pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
  2172 proof - 
  2173   have "a * b = (pprt a + nprt a) * (pprt b + nprt b)" 
  2174     apply (subst prts[symmetric])+
  2175     apply simp
  2176     done
  2177   then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
  2178     by (simp add: ring_simps)
  2179   moreover have "pprt a * pprt b <= pprt a2 * pprt b2"
  2180     by (simp_all add: prems mult_mono)
  2181   moreover have "pprt a * nprt b <= pprt a1 * nprt b2"
  2182   proof -
  2183     have "pprt a * nprt b <= pprt a * nprt b2"
  2184       by (simp add: mult_left_mono prems)
  2185     moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2"
  2186       by (simp add: mult_right_mono_neg prems)
  2187     ultimately show ?thesis
  2188       by simp
  2189   qed
  2190   moreover have "nprt a * pprt b <= nprt a2 * pprt b1"
  2191   proof - 
  2192     have "nprt a * pprt b <= nprt a2 * pprt b"
  2193       by (simp add: mult_right_mono prems)
  2194     moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"
  2195       by (simp add: mult_left_mono_neg prems)
  2196     ultimately show ?thesis
  2197       by simp
  2198   qed
  2199   moreover have "nprt a * nprt b <= nprt a1 * nprt b1"
  2200   proof -
  2201     have "nprt a * nprt b <= nprt a * nprt b1"
  2202       by (simp add: mult_left_mono_neg prems)
  2203     moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1"
  2204       by (simp add: mult_right_mono_neg prems)
  2205     ultimately show ?thesis
  2206       by simp
  2207   qed
  2208   ultimately show ?thesis
  2209     by - (rule add_mono | simp)+
  2210 qed
  2211 
  2212 lemma mult_ge_prts:
  2213   assumes
  2214   "a1 <= (a::'a::lordered_ring)"
  2215   "a <= a2"
  2216   "b1 <= b"
  2217   "b <= b2"
  2218   shows
  2219   "a * b >= nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"
  2220 proof - 
  2221   from prems have a1:"- a2 <= -a" by auto
  2222   from prems have a2: "-a <= -a1" by auto
  2223   from mult_le_prts[of "-a2" "-a" "-a1" "b1" b "b2", OF a1 a2 prems(3) prems(4), simplified nprt_neg pprt_neg] 
  2224   have le: "- (a * b) <= - nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1" by simp  
  2225   then have "-(- nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1) <= a * b"
  2226     by (simp only: minus_le_iff)
  2227   then show ?thesis by simp
  2228 qed
  2229 
  2230 end