src/HOL/Ring_and_Field.thy
author haftmann
Tue Nov 06 08:47:30 2007 +0100 (2007-11-06)
changeset 25304 7491c00f0915
parent 25267 1f745c599b5c
child 25450 c3b26e533b21
permissions -rw-r--r--
removed subclass edge ordered_ring < lordered_ring
     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 subclass pordered_comm_monoid_add by unfold_locales
   357 
   358 lemma mult_left_less_imp_less:
   359   "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
   360   by (force simp add: mult_left_mono not_le [symmetric])
   361  
   362 lemma mult_right_less_imp_less:
   363   "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
   364   by (force simp add: mult_right_mono not_le [symmetric])
   365 
   366 end
   367 
   368 class ordered_semiring_strict = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add +
   369   assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
   370   assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
   371 begin
   372 
   373 subclass semiring_0_cancel by unfold_locales
   374 
   375 subclass ordered_semiring
   376 proof unfold_locales
   377   fix a b c :: 'a
   378   assume A: "a \<le> b" "0 \<le> c"
   379   from A show "c * a \<le> c * b"
   380     unfolding le_less
   381     using mult_strict_left_mono by (cases "c = 0") auto
   382   from A show "a * c \<le> b * c"
   383     unfolding le_less
   384     using mult_strict_right_mono by (cases "c = 0") auto
   385 qed
   386 
   387 lemma mult_left_le_imp_le:
   388   "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
   389   by (force simp add: mult_strict_left_mono _not_less [symmetric])
   390  
   391 lemma mult_right_le_imp_le:
   392   "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
   393   by (force simp add: mult_strict_right_mono not_less [symmetric])
   394 
   395 lemma mult_pos_pos:
   396   "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
   397   by (drule mult_strict_left_mono [of zero b], auto)
   398 
   399 lemma mult_pos_neg:
   400   "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
   401   by (drule mult_strict_left_mono [of b zero], auto)
   402 
   403 lemma mult_pos_neg2:
   404   "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0" 
   405   by (drule mult_strict_right_mono [of b zero], auto)
   406 
   407 lemma zero_less_mult_pos:
   408   "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
   409 apply (cases "b\<le>0") 
   410  apply (auto simp add: le_less not_less)
   411 apply (drule_tac mult_pos_neg [of a b]) 
   412  apply (auto dest: less_not_sym)
   413 done
   414 
   415 lemma zero_less_mult_pos2:
   416   "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
   417 apply (cases "b\<le>0") 
   418  apply (auto simp add: le_less not_less)
   419 apply (drule_tac mult_pos_neg2 [of a b]) 
   420  apply (auto dest: less_not_sym)
   421 done
   422 
   423 end
   424 
   425 class mult_mono1 = times + zero + ord +
   426   assumes mult_mono1: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
   427 
   428 class pordered_comm_semiring = comm_semiring_0
   429   + pordered_ab_semigroup_add + mult_mono1
   430 begin
   431 
   432 subclass pordered_semiring
   433 proof unfold_locales
   434   fix a b c :: 'a
   435   assume "a \<le> b" "0 \<le> c"
   436   thus "c * a \<le> c * b" by (rule mult_mono1)
   437   thus "a * c \<le> b * c" by (simp only: mult_commute)
   438 qed
   439 
   440 end
   441 
   442 class pordered_cancel_comm_semiring = comm_semiring_0_cancel
   443   + pordered_ab_semigroup_add + mult_mono1
   444 begin
   445 
   446 subclass pordered_comm_semiring by unfold_locales
   447 subclass pordered_cancel_semiring by unfold_locales
   448 
   449 end
   450 
   451 class ordered_comm_semiring_strict = comm_semiring_0 + ordered_cancel_ab_semigroup_add +
   452   assumes mult_strict_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
   453 begin
   454 
   455 subclass ordered_semiring_strict
   456 proof unfold_locales
   457   fix a b c :: 'a
   458   assume "a < b" "0 < c"
   459   thus "c * a < c * b" by (rule mult_strict_mono)
   460   thus "a * c < b * c" by (simp only: mult_commute)
   461 qed
   462 
   463 subclass pordered_cancel_comm_semiring
   464 proof unfold_locales
   465   fix a b c :: 'a
   466   assume "a \<le> b" "0 \<le> c"
   467   thus "c * a \<le> c * b"
   468     unfolding le_less
   469     using mult_strict_mono by (cases "c = 0") auto
   470 qed
   471 
   472 end
   473 
   474 class pordered_ring = ring + pordered_cancel_semiring 
   475 begin
   476 
   477 subclass pordered_ab_group_add by unfold_locales
   478 
   479 lemmas ring_simps = ring_simps group_simps
   480 
   481 lemma less_add_iff1:
   482   "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
   483   by (simp add: ring_simps)
   484 
   485 lemma less_add_iff2:
   486   "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
   487   by (simp add: ring_simps)
   488 
   489 lemma le_add_iff1:
   490   "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
   491   by (simp add: ring_simps)
   492 
   493 lemma le_add_iff2:
   494   "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
   495   by (simp add: ring_simps)
   496 
   497 lemma mult_left_mono_neg:
   498   "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
   499   apply (drule mult_left_mono [of _ _ "uminus c"])
   500   apply (simp_all add: minus_mult_left [symmetric]) 
   501   done
   502 
   503 lemma mult_right_mono_neg:
   504   "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
   505   apply (drule mult_right_mono [of _ _ "uminus c"])
   506   apply (simp_all add: minus_mult_right [symmetric]) 
   507   done
   508 
   509 lemma mult_nonpos_nonpos:
   510   "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
   511   by (drule mult_right_mono_neg [of a zero b]) auto
   512 
   513 lemma split_mult_pos_le:
   514   "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
   515   by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos)
   516 
   517 end
   518 
   519 class abs_if = minus + ord + zero + abs +
   520   assumes abs_if: "\<bar>a\<bar> = (if a < 0 then (- a) else a)"
   521 
   522 class sgn_if = sgn + zero + one + minus + ord +
   523   assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
   524 
   525 class ordered_ring = ring + ordered_semiring
   526   + ordered_ab_group_add + abs_if
   527 begin
   528 
   529 subclass pordered_ring by unfold_locales
   530 
   531 subclass pordered_ab_group_add_abs
   532 proof unfold_locales
   533   fix a b
   534   show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
   535   by (auto simp add: abs_if not_less neg_less_eq_nonneg less_eq_neg_nonpos)
   536    (auto simp del: minus_add_distrib simp add: minus_add_distrib [symmetric]
   537      neg_less_eq_nonneg less_eq_neg_nonpos, auto intro: add_nonneg_nonneg,
   538       auto intro!: less_imp_le add_neg_neg)
   539 qed (auto simp add: abs_if less_eq_neg_nonpos neg_equal_zero)
   540 
   541 end
   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   + ordered_ab_group_add + abs_if
   548 begin
   549 
   550 subclass ordered_ring by unfold_locales
   551 
   552 lemma mult_strict_left_mono_neg:
   553   "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
   554   apply (drule mult_strict_left_mono [of _ _ "uminus c"])
   555   apply (simp_all add: minus_mult_left [symmetric]) 
   556   done
   557 
   558 lemma mult_strict_right_mono_neg:
   559   "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
   560   apply (drule mult_strict_right_mono [of _ _ "uminus c"])
   561   apply (simp_all add: minus_mult_right [symmetric]) 
   562   done
   563 
   564 lemma mult_neg_neg:
   565   "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
   566   by (drule mult_strict_right_mono_neg, auto)
   567 
   568 end
   569 
   570 instance ordered_ring_strict \<subseteq> ring_no_zero_divisors
   571 apply intro_classes
   572 apply (auto simp add: linorder_not_less order_le_less linorder_neq_iff)
   573 apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono)+
   574 done
   575 
   576 lemma zero_less_mult_iff:
   577   fixes a :: "'a::ordered_ring_strict"
   578   shows "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
   579   apply (auto simp add: le_less not_less mult_pos_pos mult_neg_neg)
   580   apply (blast dest: zero_less_mult_pos) 
   581   apply (blast dest: zero_less_mult_pos2)
   582   done
   583 
   584 lemma zero_le_mult_iff:
   585      "((0::'a::ordered_ring_strict) \<le> a*b) = (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
   586 by (auto simp add: eq_commute [of 0] order_le_less linorder_not_less
   587                    zero_less_mult_iff)
   588 
   589 lemma mult_less_0_iff:
   590      "(a*b < (0::'a::ordered_ring_strict)) = (0 < a & b < 0 | a < 0 & 0 < b)"
   591 apply (insert zero_less_mult_iff [of "-a" b]) 
   592 apply (force simp add: minus_mult_left[symmetric]) 
   593 done
   594 
   595 lemma mult_le_0_iff:
   596      "(a*b \<le> (0::'a::ordered_ring_strict)) = (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
   597 apply (insert zero_le_mult_iff [of "-a" b]) 
   598 apply (force simp add: minus_mult_left[symmetric]) 
   599 done
   600 
   601 lemma zero_le_square[simp]: "(0::'a::ordered_ring_strict) \<le> a*a"
   602 by (simp add: zero_le_mult_iff linorder_linear)
   603 
   604 lemma not_square_less_zero[simp]: "\<not> (a * a < (0::'a::ordered_ring_strict))"
   605 by (simp add: not_less)
   606 
   607 text{*This list of rewrites simplifies ring terms by multiplying
   608 everything out and bringing sums and products into a canonical form
   609 (by ordered rewriting). As a result it decides ring equalities but
   610 also helps with inequalities. *}
   611 lemmas ring_simps = group_simps ring_distribs
   612 
   613 
   614 class pordered_comm_ring = comm_ring + pordered_comm_semiring
   615 begin
   616 
   617 subclass pordered_ring by unfold_locales
   618 subclass pordered_cancel_comm_semiring by unfold_locales
   619 
   620 end
   621 
   622 class ordered_semidom = comm_semiring_1_cancel + ordered_comm_semiring_strict +
   623   (*previously ordered_semiring*)
   624   assumes zero_less_one [simp]: "0 < 1"
   625 begin
   626 
   627 lemma pos_add_strict:
   628   shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
   629   using add_strict_mono [of zero a b c] by simp
   630 
   631 end
   632 
   633 class ordered_idom =
   634   comm_ring_1 +
   635   ordered_comm_semiring_strict +
   636   ordered_ab_group_add +
   637   abs_if + sgn_if
   638   (*previously ordered_ring*)
   639 
   640 instance ordered_idom \<subseteq> ordered_ring_strict ..
   641 
   642 instance ordered_idom \<subseteq> pordered_comm_ring ..
   643 
   644 class ordered_field = field + ordered_idom
   645 
   646 lemma linorder_neqE_ordered_idom:
   647   fixes x y :: "'a :: ordered_idom"
   648   assumes "x \<noteq> y" obtains "x < y" | "y < x"
   649   using assms by (rule linorder_neqE)
   650 
   651 
   652 text{*Proving axiom @{text zero_less_one} makes all @{text ordered_semidom}
   653       theorems available to members of @{term ordered_idom} *}
   654 
   655 instance ordered_idom \<subseteq> ordered_semidom
   656 proof
   657   have "(0::'a) \<le> 1*1" by (rule zero_le_square)
   658   thus "(0::'a) < 1" by (simp add: order_le_less) 
   659 qed
   660 
   661 instance ordered_idom \<subseteq> idom ..
   662 
   663 text{*All three types of comparision involving 0 and 1 are covered.*}
   664 
   665 lemmas one_neq_zero = zero_neq_one [THEN not_sym]
   666 declare one_neq_zero [simp]
   667 
   668 lemma zero_le_one [simp]: "(0::'a::ordered_semidom) \<le> 1"
   669   by (rule zero_less_one [THEN order_less_imp_le]) 
   670 
   671 lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semidom) \<le> 0"
   672 by (simp add: linorder_not_le) 
   673 
   674 lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semidom) < 0"
   675 by (simp add: linorder_not_less) 
   676 
   677 
   678 subsection{*More Monotonicity*}
   679 
   680 text{*Strict monotonicity in both arguments*}
   681 lemma mult_strict_mono:
   682      "[|a<b; c<d; 0<b; 0\<le>c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
   683 apply (cases "c=0")
   684  apply (simp add: mult_pos_pos) 
   685 apply (erule mult_strict_right_mono [THEN order_less_trans])
   686  apply (force simp add: order_le_less) 
   687 apply (erule mult_strict_left_mono, assumption)
   688 done
   689 
   690 text{*This weaker variant has more natural premises*}
   691 lemma mult_strict_mono':
   692      "[| a<b; c<d; 0 \<le> a; 0 \<le> c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
   693 apply (rule mult_strict_mono)
   694 apply (blast intro: order_le_less_trans)+
   695 done
   696 
   697 lemma less_1_mult: "[| 1 < m; 1 < n |] ==> 1 < m*(n::'a::ordered_semidom)"
   698 apply (insert mult_strict_mono [of 1 m 1 n]) 
   699 apply (simp add:  order_less_trans [OF zero_less_one]) 
   700 done
   701 
   702 lemma mult_less_le_imp_less: "(a::'a::ordered_semiring_strict) < b ==>
   703     c <= d ==> 0 <= a ==> 0 < c ==> a * c < b * d"
   704   apply (subgoal_tac "a * c < b * c")
   705   apply (erule order_less_le_trans)
   706   apply (erule mult_left_mono)
   707   apply simp
   708   apply (erule mult_strict_right_mono)
   709   apply assumption
   710 done
   711 
   712 lemma mult_le_less_imp_less: "(a::'a::ordered_semiring_strict) <= b ==>
   713     c < d ==> 0 < a ==> 0 <= c ==> a * c < b * d"
   714   apply (subgoal_tac "a * c <= b * c")
   715   apply (erule order_le_less_trans)
   716   apply (erule mult_strict_left_mono)
   717   apply simp
   718   apply (erule mult_right_mono)
   719   apply simp
   720 done
   721 
   722 
   723 subsection{*Cancellation Laws for Relationships With a Common Factor*}
   724 
   725 text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
   726    also with the relations @{text "\<le>"} and equality.*}
   727 
   728 text{*These ``disjunction'' versions produce two cases when the comparison is
   729  an assumption, but effectively four when the comparison is a goal.*}
   730 
   731 lemma mult_less_cancel_right_disj:
   732     "(a*c < b*c) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
   733 apply (cases "c = 0")
   734 apply (auto simp add: linorder_neq_iff mult_strict_right_mono 
   735                       mult_strict_right_mono_neg)
   736 apply (auto simp add: linorder_not_less 
   737                       linorder_not_le [symmetric, of "a*c"]
   738                       linorder_not_le [symmetric, of a])
   739 apply (erule_tac [!] notE)
   740 apply (auto simp add: order_less_imp_le mult_right_mono 
   741                       mult_right_mono_neg)
   742 done
   743 
   744 lemma mult_less_cancel_left_disj:
   745     "(c*a < c*b) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
   746 apply (cases "c = 0")
   747 apply (auto simp add: linorder_neq_iff mult_strict_left_mono 
   748                       mult_strict_left_mono_neg)
   749 apply (auto simp add: linorder_not_less 
   750                       linorder_not_le [symmetric, of "c*a"]
   751                       linorder_not_le [symmetric, of a])
   752 apply (erule_tac [!] notE)
   753 apply (auto simp add: order_less_imp_le mult_left_mono 
   754                       mult_left_mono_neg)
   755 done
   756 
   757 
   758 text{*The ``conjunction of implication'' lemmas produce two cases when the
   759 comparison is a goal, but give four when the comparison is an assumption.*}
   760 
   761 lemma mult_less_cancel_right:
   762   fixes c :: "'a :: ordered_ring_strict"
   763   shows      "(a*c < b*c) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))"
   764 by (insert mult_less_cancel_right_disj [of a c b], auto)
   765 
   766 lemma mult_less_cancel_left:
   767   fixes c :: "'a :: ordered_ring_strict"
   768   shows      "(c*a < c*b) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))"
   769 by (insert mult_less_cancel_left_disj [of c a b], auto)
   770 
   771 lemma mult_le_cancel_right:
   772      "(a*c \<le> b*c) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
   773 by (simp add: linorder_not_less [symmetric] mult_less_cancel_right_disj)
   774 
   775 lemma mult_le_cancel_left:
   776      "(c*a \<le> c*b) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
   777 by (simp add: linorder_not_less [symmetric] mult_less_cancel_left_disj)
   778 
   779 lemma mult_less_imp_less_left:
   780       assumes less: "c*a < c*b" and nonneg: "0 \<le> c"
   781       shows "a < (b::'a::ordered_semiring_strict)"
   782 proof (rule ccontr)
   783   assume "~ a < b"
   784   hence "b \<le> a" by (simp add: linorder_not_less)
   785   hence "c*b \<le> c*a" using nonneg by (rule mult_left_mono)
   786   with this and less show False 
   787     by (simp add: linorder_not_less [symmetric])
   788 qed
   789 
   790 lemma mult_less_imp_less_right:
   791   assumes less: "a*c < b*c" and nonneg: "0 <= c"
   792   shows "a < (b::'a::ordered_semiring_strict)"
   793 proof (rule ccontr)
   794   assume "~ a < b"
   795   hence "b \<le> a" by (simp add: linorder_not_less)
   796   hence "b*c \<le> a*c" using nonneg by (rule mult_right_mono)
   797   with this and less show False 
   798     by (simp add: linorder_not_less [symmetric])
   799 qed  
   800 
   801 text{*Cancellation of equalities with a common factor*}
   802 lemma mult_cancel_right [simp,noatp]:
   803   fixes a b c :: "'a::ring_no_zero_divisors"
   804   shows "(a * c = b * c) = (c = 0 \<or> a = b)"
   805 proof -
   806   have "(a * c = b * c) = ((a - b) * c = 0)"
   807     by (simp add: ring_distribs)
   808   thus ?thesis
   809     by (simp add: disj_commute)
   810 qed
   811 
   812 lemma mult_cancel_left [simp,noatp]:
   813   fixes a b c :: "'a::ring_no_zero_divisors"
   814   shows "(c * a = c * b) = (c = 0 \<or> a = b)"
   815 proof -
   816   have "(c * a = c * b) = (c * (a - b) = 0)"
   817     by (simp add: ring_distribs)
   818   thus ?thesis
   819     by simp
   820 qed
   821 
   822 
   823 subsubsection{*Special Cancellation Simprules for Multiplication*}
   824 
   825 text{*These also produce two cases when the comparison is a goal.*}
   826 
   827 lemma mult_le_cancel_right1:
   828   fixes c :: "'a :: ordered_idom"
   829   shows "(c \<le> b*c) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))"
   830 by (insert mult_le_cancel_right [of 1 c b], simp)
   831 
   832 lemma mult_le_cancel_right2:
   833   fixes c :: "'a :: ordered_idom"
   834   shows "(a*c \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))"
   835 by (insert mult_le_cancel_right [of a c 1], simp)
   836 
   837 lemma mult_le_cancel_left1:
   838   fixes c :: "'a :: ordered_idom"
   839   shows "(c \<le> c*b) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))"
   840 by (insert mult_le_cancel_left [of c 1 b], simp)
   841 
   842 lemma mult_le_cancel_left2:
   843   fixes c :: "'a :: ordered_idom"
   844   shows "(c*a \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))"
   845 by (insert mult_le_cancel_left [of c a 1], simp)
   846 
   847 lemma mult_less_cancel_right1:
   848   fixes c :: "'a :: ordered_idom"
   849   shows "(c < b*c) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))"
   850 by (insert mult_less_cancel_right [of 1 c b], simp)
   851 
   852 lemma mult_less_cancel_right2:
   853   fixes c :: "'a :: ordered_idom"
   854   shows "(a*c < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))"
   855 by (insert mult_less_cancel_right [of a c 1], simp)
   856 
   857 lemma mult_less_cancel_left1:
   858   fixes c :: "'a :: ordered_idom"
   859   shows "(c < c*b) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))"
   860 by (insert mult_less_cancel_left [of c 1 b], simp)
   861 
   862 lemma mult_less_cancel_left2:
   863   fixes c :: "'a :: ordered_idom"
   864   shows "(c*a < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))"
   865 by (insert mult_less_cancel_left [of c a 1], simp)
   866 
   867 lemma mult_cancel_right1 [simp]:
   868   fixes c :: "'a :: ring_1_no_zero_divisors"
   869   shows "(c = b*c) = (c = 0 | b=1)"
   870 by (insert mult_cancel_right [of 1 c b], force)
   871 
   872 lemma mult_cancel_right2 [simp]:
   873   fixes c :: "'a :: ring_1_no_zero_divisors"
   874   shows "(a*c = c) = (c = 0 | a=1)"
   875 by (insert mult_cancel_right [of a c 1], simp)
   876  
   877 lemma mult_cancel_left1 [simp]:
   878   fixes c :: "'a :: ring_1_no_zero_divisors"
   879   shows "(c = c*b) = (c = 0 | b=1)"
   880 by (insert mult_cancel_left [of c 1 b], force)
   881 
   882 lemma mult_cancel_left2 [simp]:
   883   fixes c :: "'a :: ring_1_no_zero_divisors"
   884   shows "(c*a = c) = (c = 0 | a=1)"
   885 by (insert mult_cancel_left [of c a 1], simp)
   886 
   887 
   888 text{*Simprules for comparisons where common factors can be cancelled.*}
   889 lemmas mult_compare_simps =
   890     mult_le_cancel_right mult_le_cancel_left
   891     mult_le_cancel_right1 mult_le_cancel_right2
   892     mult_le_cancel_left1 mult_le_cancel_left2
   893     mult_less_cancel_right mult_less_cancel_left
   894     mult_less_cancel_right1 mult_less_cancel_right2
   895     mult_less_cancel_left1 mult_less_cancel_left2
   896     mult_cancel_right mult_cancel_left
   897     mult_cancel_right1 mult_cancel_right2
   898     mult_cancel_left1 mult_cancel_left2
   899 
   900 
   901 (* what ordering?? this is a straight instance of mult_eq_0_iff
   902 text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement
   903       of an ordering.*}
   904 lemma field_mult_eq_0_iff [simp]:
   905   "(a*b = (0::'a::division_ring)) = (a = 0 | b = 0)"
   906 by simp
   907 *)
   908 (* subsumed by mult_cancel lemmas on ring_no_zero_divisors
   909 text{*Cancellation of equalities with a common factor*}
   910 lemma field_mult_cancel_right_lemma:
   911       assumes cnz: "c \<noteq> (0::'a::division_ring)"
   912          and eq:  "a*c = b*c"
   913         shows "a=b"
   914 proof -
   915   have "(a * c) * inverse c = (b * c) * inverse c"
   916     by (simp add: eq)
   917   thus "a=b"
   918     by (simp add: mult_assoc cnz)
   919 qed
   920 
   921 lemma field_mult_cancel_right [simp]:
   922      "(a*c = b*c) = (c = (0::'a::division_ring) | a=b)"
   923 by simp
   924 
   925 lemma field_mult_cancel_left [simp]:
   926      "(c*a = c*b) = (c = (0::'a::division_ring) | a=b)"
   927 by simp
   928 *)
   929 lemma nonzero_imp_inverse_nonzero:
   930   "a \<noteq> 0 ==> inverse a \<noteq> (0::'a::division_ring)"
   931 proof
   932   assume ianz: "inverse a = 0"
   933   assume "a \<noteq> 0"
   934   hence "1 = a * inverse a" by simp
   935   also have "... = 0" by (simp add: ianz)
   936   finally have "1 = (0::'a::division_ring)" .
   937   thus False by (simp add: eq_commute)
   938 qed
   939 
   940 
   941 subsection{*Basic Properties of @{term inverse}*}
   942 
   943 lemma inverse_zero_imp_zero: "inverse a = 0 ==> a = (0::'a::division_ring)"
   944 apply (rule ccontr) 
   945 apply (blast dest: nonzero_imp_inverse_nonzero) 
   946 done
   947 
   948 lemma inverse_nonzero_imp_nonzero:
   949    "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_iff_nonzero [simp]:
   955    "(inverse a = 0) = (a = (0::'a::{division_ring,division_by_zero}))"
   956 by (force dest: inverse_nonzero_imp_nonzero) 
   957 
   958 lemma nonzero_inverse_minus_eq:
   959       assumes [simp]: "a\<noteq>0"
   960       shows "inverse(-a) = -inverse(a::'a::division_ring)"
   961 proof -
   962   have "-a * inverse (- a) = -a * - inverse a"
   963     by simp
   964   thus ?thesis 
   965     by (simp only: mult_cancel_left, simp)
   966 qed
   967 
   968 lemma inverse_minus_eq [simp]:
   969    "inverse(-a) = -inverse(a::'a::{division_ring,division_by_zero})"
   970 proof cases
   971   assume "a=0" thus ?thesis by (simp add: inverse_zero)
   972 next
   973   assume "a\<noteq>0" 
   974   thus ?thesis by (simp add: nonzero_inverse_minus_eq)
   975 qed
   976 
   977 lemma nonzero_inverse_eq_imp_eq:
   978       assumes inveq: "inverse a = inverse b"
   979 	  and anz:  "a \<noteq> 0"
   980 	  and bnz:  "b \<noteq> 0"
   981 	 shows "a = (b::'a::division_ring)"
   982 proof -
   983   have "a * inverse b = a * inverse a"
   984     by (simp add: inveq)
   985   hence "(a * inverse b) * b = (a * inverse a) * b"
   986     by simp
   987   thus "a = b"
   988     by (simp add: mult_assoc anz bnz)
   989 qed
   990 
   991 lemma inverse_eq_imp_eq:
   992   "inverse a = inverse b ==> a = (b::'a::{division_ring,division_by_zero})"
   993 apply (cases "a=0 | b=0") 
   994  apply (force dest!: inverse_zero_imp_zero
   995               simp add: eq_commute [of "0::'a"])
   996 apply (force dest!: nonzero_inverse_eq_imp_eq) 
   997 done
   998 
   999 lemma inverse_eq_iff_eq [simp]:
  1000   "(inverse a = inverse b) = (a = (b::'a::{division_ring,division_by_zero}))"
  1001 by (force dest!: inverse_eq_imp_eq)
  1002 
  1003 lemma nonzero_inverse_inverse_eq:
  1004       assumes [simp]: "a \<noteq> 0"
  1005       shows "inverse(inverse (a::'a::division_ring)) = a"
  1006   proof -
  1007   have "(inverse (inverse a) * inverse a) * a = a" 
  1008     by (simp add: nonzero_imp_inverse_nonzero)
  1009   thus ?thesis
  1010     by (simp add: mult_assoc)
  1011   qed
  1012 
  1013 lemma inverse_inverse_eq [simp]:
  1014      "inverse(inverse (a::'a::{division_ring,division_by_zero})) = a"
  1015   proof cases
  1016     assume "a=0" thus ?thesis by simp
  1017   next
  1018     assume "a\<noteq>0" 
  1019     thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
  1020   qed
  1021 
  1022 lemma inverse_1 [simp]: "inverse 1 = (1::'a::division_ring)"
  1023   proof -
  1024   have "inverse 1 * 1 = (1::'a::division_ring)" 
  1025     by (rule left_inverse [OF zero_neq_one [symmetric]])
  1026   thus ?thesis  by simp
  1027   qed
  1028 
  1029 lemma inverse_unique: 
  1030   assumes ab: "a*b = 1"
  1031   shows "inverse a = (b::'a::division_ring)"
  1032 proof -
  1033   have "a \<noteq> 0" using ab by auto
  1034   moreover have "inverse a * (a * b) = inverse a" by (simp add: ab) 
  1035   ultimately show ?thesis by (simp add: mult_assoc [symmetric]) 
  1036 qed
  1037 
  1038 lemma nonzero_inverse_mult_distrib: 
  1039       assumes anz: "a \<noteq> 0"
  1040           and bnz: "b \<noteq> 0"
  1041       shows "inverse(a*b) = inverse(b) * inverse(a::'a::division_ring)"
  1042   proof -
  1043   have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" 
  1044     by (simp add: anz bnz)
  1045   hence "inverse(a*b) * a = inverse(b)" 
  1046     by (simp add: mult_assoc bnz)
  1047   hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" 
  1048     by simp
  1049   thus ?thesis
  1050     by (simp add: mult_assoc anz)
  1051   qed
  1052 
  1053 text{*This version builds in division by zero while also re-orienting
  1054       the right-hand side.*}
  1055 lemma inverse_mult_distrib [simp]:
  1056      "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"
  1057   proof cases
  1058     assume "a \<noteq> 0 & b \<noteq> 0" 
  1059     thus ?thesis
  1060       by (simp add: nonzero_inverse_mult_distrib mult_commute)
  1061   next
  1062     assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
  1063     thus ?thesis
  1064       by force
  1065   qed
  1066 
  1067 lemma division_ring_inverse_add:
  1068   "[|(a::'a::division_ring) \<noteq> 0; b \<noteq> 0|]
  1069    ==> inverse a + inverse b = inverse a * (a+b) * inverse b"
  1070 by (simp add: ring_simps)
  1071 
  1072 lemma division_ring_inverse_diff:
  1073   "[|(a::'a::division_ring) \<noteq> 0; b \<noteq> 0|]
  1074    ==> inverse a - inverse b = inverse a * (b-a) * inverse b"
  1075 by (simp add: ring_simps)
  1076 
  1077 text{*There is no slick version using division by zero.*}
  1078 lemma inverse_add:
  1079   "[|a \<noteq> 0;  b \<noteq> 0|]
  1080    ==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)"
  1081 by (simp add: division_ring_inverse_add mult_ac)
  1082 
  1083 lemma inverse_divide [simp]:
  1084   "inverse (a/b) = b / (a::'a::{field,division_by_zero})"
  1085 by (simp add: divide_inverse mult_commute)
  1086 
  1087 
  1088 subsection {* Calculations with fractions *}
  1089 
  1090 text{* There is a whole bunch of simp-rules just for class @{text
  1091 field} but none for class @{text field} and @{text nonzero_divides}
  1092 because the latter are covered by a simproc. *}
  1093 
  1094 lemma nonzero_mult_divide_mult_cancel_left[simp,noatp]:
  1095 assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" shows "(c*a)/(c*b) = a/(b::'a::field)"
  1096 proof -
  1097   have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
  1098     by (simp add: divide_inverse nonzero_inverse_mult_distrib)
  1099   also have "... =  a * inverse b * (inverse c * c)"
  1100     by (simp only: mult_ac)
  1101   also have "... =  a * inverse b"
  1102     by simp
  1103     finally show ?thesis 
  1104     by (simp add: divide_inverse)
  1105 qed
  1106 
  1107 lemma mult_divide_mult_cancel_left:
  1108   "c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"
  1109 apply (cases "b = 0")
  1110 apply (simp_all add: nonzero_mult_divide_mult_cancel_left)
  1111 done
  1112 
  1113 lemma nonzero_mult_divide_mult_cancel_right [noatp]:
  1114   "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)"
  1115 by (simp add: mult_commute [of _ c] nonzero_mult_divide_mult_cancel_left) 
  1116 
  1117 lemma mult_divide_mult_cancel_right:
  1118   "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"
  1119 apply (cases "b = 0")
  1120 apply (simp_all add: nonzero_mult_divide_mult_cancel_right)
  1121 done
  1122 
  1123 lemma divide_1 [simp]: "a/1 = (a::'a::field)"
  1124 by (simp add: divide_inverse)
  1125 
  1126 lemma times_divide_eq_right: "a * (b/c) = (a*b) / (c::'a::field)"
  1127 by (simp add: divide_inverse mult_assoc)
  1128 
  1129 lemma times_divide_eq_left: "(b/c) * a = (b*a) / (c::'a::field)"
  1130 by (simp add: divide_inverse mult_ac)
  1131 
  1132 lemmas times_divide_eq = times_divide_eq_right times_divide_eq_left
  1133 
  1134 lemma divide_divide_eq_right [simp,noatp]:
  1135   "a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"
  1136 by (simp add: divide_inverse mult_ac)
  1137 
  1138 lemma divide_divide_eq_left [simp,noatp]:
  1139   "(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"
  1140 by (simp add: divide_inverse mult_assoc)
  1141 
  1142 lemma add_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
  1143     x / y + w / z = (x * z + w * y) / (y * z)"
  1144 apply (subgoal_tac "x / y = (x * z) / (y * z)")
  1145 apply (erule ssubst)
  1146 apply (subgoal_tac "w / z = (w * y) / (y * z)")
  1147 apply (erule ssubst)
  1148 apply (rule add_divide_distrib [THEN sym])
  1149 apply (subst mult_commute)
  1150 apply (erule nonzero_mult_divide_mult_cancel_left [THEN sym])
  1151 apply assumption
  1152 apply (erule nonzero_mult_divide_mult_cancel_right [THEN sym])
  1153 apply assumption
  1154 done
  1155 
  1156 
  1157 subsubsection{*Special Cancellation Simprules for Division*}
  1158 
  1159 lemma mult_divide_mult_cancel_left_if[simp,noatp]:
  1160 fixes c :: "'a :: {field,division_by_zero}"
  1161 shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)"
  1162 by (simp add: mult_divide_mult_cancel_left)
  1163 
  1164 lemma nonzero_mult_divide_cancel_right[simp,noatp]:
  1165   "b \<noteq> 0 \<Longrightarrow> a * b / b = (a::'a::field)"
  1166 using nonzero_mult_divide_mult_cancel_right[of 1 b a] by simp
  1167 
  1168 lemma nonzero_mult_divide_cancel_left[simp,noatp]:
  1169   "a \<noteq> 0 \<Longrightarrow> a * b / a = (b::'a::field)"
  1170 using nonzero_mult_divide_mult_cancel_left[of 1 a b] by simp
  1171 
  1172 
  1173 lemma nonzero_divide_mult_cancel_right[simp,noatp]:
  1174   "\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> b / (a * b) = 1/(a::'a::field)"
  1175 using nonzero_mult_divide_mult_cancel_right[of a b 1] by simp
  1176 
  1177 lemma nonzero_divide_mult_cancel_left[simp,noatp]:
  1178   "\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> a / (a * b) = 1/(b::'a::field)"
  1179 using nonzero_mult_divide_mult_cancel_left[of b a 1] by simp
  1180 
  1181 
  1182 lemma nonzero_mult_divide_mult_cancel_left2[simp,noatp]:
  1183   "[|b\<noteq>0; c\<noteq>0|] ==> (c*a) / (b*c) = a/(b::'a::field)"
  1184 using nonzero_mult_divide_mult_cancel_left[of b c a] by(simp add:mult_ac)
  1185 
  1186 lemma nonzero_mult_divide_mult_cancel_right2[simp,noatp]:
  1187   "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (c*b) = a/(b::'a::field)"
  1188 using nonzero_mult_divide_mult_cancel_right[of b c a] by(simp add:mult_ac)
  1189 
  1190 
  1191 subsection {* Division and Unary Minus *}
  1192 
  1193 lemma nonzero_minus_divide_left: "b \<noteq> 0 ==> - (a/b) = (-a) / (b::'a::field)"
  1194 by (simp add: divide_inverse minus_mult_left)
  1195 
  1196 lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a/b) = a / -(b::'a::field)"
  1197 by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right)
  1198 
  1199 lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a)/(-b) = a / (b::'a::field)"
  1200 by (simp add: divide_inverse nonzero_inverse_minus_eq)
  1201 
  1202 lemma minus_divide_left: "- (a/b) = (-a) / (b::'a::field)"
  1203 by (simp add: divide_inverse minus_mult_left [symmetric])
  1204 
  1205 lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"
  1206 by (simp add: divide_inverse minus_mult_right [symmetric])
  1207 
  1208 
  1209 text{*The effect is to extract signs from divisions*}
  1210 lemmas divide_minus_left = minus_divide_left [symmetric]
  1211 lemmas divide_minus_right = minus_divide_right [symmetric]
  1212 declare divide_minus_left [simp]   divide_minus_right [simp]
  1213 
  1214 text{*Also, extract signs from products*}
  1215 lemmas mult_minus_left = minus_mult_left [symmetric]
  1216 lemmas mult_minus_right = minus_mult_right [symmetric]
  1217 declare mult_minus_left [simp]   mult_minus_right [simp]
  1218 
  1219 lemma minus_divide_divide [simp]:
  1220   "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"
  1221 apply (cases "b=0", simp) 
  1222 apply (simp add: nonzero_minus_divide_divide) 
  1223 done
  1224 
  1225 lemma diff_divide_distrib: "(a-b)/(c::'a::field) = a/c - b/c"
  1226 by (simp add: diff_minus add_divide_distrib) 
  1227 
  1228 lemma add_divide_eq_iff:
  1229   "(z::'a::field) \<noteq> 0 \<Longrightarrow> x + y/z = (z*x + y)/z"
  1230 by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left)
  1231 
  1232 lemma divide_add_eq_iff:
  1233   "(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z + y = (x + z*y)/z"
  1234 by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left)
  1235 
  1236 lemma diff_divide_eq_iff:
  1237   "(z::'a::field) \<noteq> 0 \<Longrightarrow> x - y/z = (z*x - y)/z"
  1238 by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left)
  1239 
  1240 lemma divide_diff_eq_iff:
  1241   "(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z - y = (x - z*y)/z"
  1242 by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left)
  1243 
  1244 lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)"
  1245 proof -
  1246   assume [simp]: "c\<noteq>0"
  1247   have "(a = b/c) = (a*c = (b/c)*c)" by simp
  1248   also have "... = (a*c = b)" by (simp add: divide_inverse mult_assoc)
  1249   finally show ?thesis .
  1250 qed
  1251 
  1252 lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)"
  1253 proof -
  1254   assume [simp]: "c\<noteq>0"
  1255   have "(b/c = a) = ((b/c)*c = a*c)"  by simp
  1256   also have "... = (b = a*c)"  by (simp add: divide_inverse mult_assoc) 
  1257   finally show ?thesis .
  1258 qed
  1259 
  1260 lemma eq_divide_eq:
  1261   "((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)"
  1262 by (simp add: nonzero_eq_divide_eq) 
  1263 
  1264 lemma divide_eq_eq:
  1265   "(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)"
  1266 by (force simp add: nonzero_divide_eq_eq) 
  1267 
  1268 lemma divide_eq_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
  1269     b = a * c ==> b / c = a"
  1270   by (subst divide_eq_eq, simp)
  1271 
  1272 lemma eq_divide_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
  1273     a * c = b ==> a = b / c"
  1274   by (subst eq_divide_eq, simp)
  1275 
  1276 
  1277 lemmas field_eq_simps = ring_simps
  1278   (* pull / out*)
  1279   add_divide_eq_iff divide_add_eq_iff
  1280   diff_divide_eq_iff divide_diff_eq_iff
  1281   (* multiply eqn *)
  1282   nonzero_eq_divide_eq nonzero_divide_eq_eq
  1283 (* is added later:
  1284   times_divide_eq_left times_divide_eq_right
  1285 *)
  1286 
  1287 text{*An example:*}
  1288 lemma fixes a b c d e f :: "'a::field"
  1289 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"
  1290 apply(subgoal_tac "(c-d)*(e-f)*(a-b) \<noteq> 0")
  1291  apply(simp add:field_eq_simps)
  1292 apply(simp)
  1293 done
  1294 
  1295 
  1296 lemma diff_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
  1297     x / y - w / z = (x * z - w * y) / (y * z)"
  1298 by (simp add:field_eq_simps times_divide_eq)
  1299 
  1300 lemma frac_eq_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
  1301     (x / y = w / z) = (x * z = w * y)"
  1302 by (simp add:field_eq_simps times_divide_eq)
  1303 
  1304 
  1305 subsection {* Ordered Fields *}
  1306 
  1307 lemma positive_imp_inverse_positive: 
  1308 assumes a_gt_0: "0 < a"  shows "0 < inverse (a::'a::ordered_field)"
  1309 proof -
  1310   have "0 < a * inverse a" 
  1311     by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one)
  1312   thus "0 < inverse a" 
  1313     by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff)
  1314 qed
  1315 
  1316 lemma negative_imp_inverse_negative:
  1317   "a < 0 ==> inverse a < (0::'a::ordered_field)"
  1318 by (insert positive_imp_inverse_positive [of "-a"], 
  1319     simp add: nonzero_inverse_minus_eq order_less_imp_not_eq)
  1320 
  1321 lemma inverse_le_imp_le:
  1322 assumes invle: "inverse a \<le> inverse b" and apos:  "0 < a"
  1323 shows "b \<le> (a::'a::ordered_field)"
  1324 proof (rule classical)
  1325   assume "~ b \<le> a"
  1326   hence "a < b"  by (simp add: linorder_not_le)
  1327   hence bpos: "0 < b"  by (blast intro: apos order_less_trans)
  1328   hence "a * inverse a \<le> a * inverse b"
  1329     by (simp add: apos invle order_less_imp_le mult_left_mono)
  1330   hence "(a * inverse a) * b \<le> (a * inverse b) * b"
  1331     by (simp add: bpos order_less_imp_le mult_right_mono)
  1332   thus "b \<le> a"  by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)
  1333 qed
  1334 
  1335 lemma inverse_positive_imp_positive:
  1336 assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0"
  1337 shows "0 < (a::'a::ordered_field)"
  1338 proof -
  1339   have "0 < inverse (inverse a)"
  1340     using inv_gt_0 by (rule positive_imp_inverse_positive)
  1341   thus "0 < a"
  1342     using nz by (simp add: nonzero_inverse_inverse_eq)
  1343 qed
  1344 
  1345 lemma inverse_positive_iff_positive [simp]:
  1346   "(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))"
  1347 apply (cases "a = 0", simp)
  1348 apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
  1349 done
  1350 
  1351 lemma inverse_negative_imp_negative:
  1352 assumes inv_less_0: "inverse a < 0" and nz:  "a \<noteq> 0"
  1353 shows "a < (0::'a::ordered_field)"
  1354 proof -
  1355   have "inverse (inverse a) < 0"
  1356     using inv_less_0 by (rule negative_imp_inverse_negative)
  1357   thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq)
  1358 qed
  1359 
  1360 lemma inverse_negative_iff_negative [simp]:
  1361   "(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))"
  1362 apply (cases "a = 0", simp)
  1363 apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
  1364 done
  1365 
  1366 lemma inverse_nonnegative_iff_nonnegative [simp]:
  1367   "(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))"
  1368 by (simp add: linorder_not_less [symmetric])
  1369 
  1370 lemma inverse_nonpositive_iff_nonpositive [simp]:
  1371   "(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))"
  1372 by (simp add: linorder_not_less [symmetric])
  1373 
  1374 lemma ordered_field_no_lb: "\<forall> x. \<exists>y. y < (x::'a::ordered_field)"
  1375 proof
  1376   fix x::'a
  1377   have m1: "- (1::'a) < 0" by simp
  1378   from add_strict_right_mono[OF m1, where c=x] 
  1379   have "(- 1) + x < x" by simp
  1380   thus "\<exists>y. y < x" by blast
  1381 qed
  1382 
  1383 lemma ordered_field_no_ub: "\<forall> x. \<exists>y. y > (x::'a::ordered_field)"
  1384 proof
  1385   fix x::'a
  1386   have m1: " (1::'a) > 0" by simp
  1387   from add_strict_right_mono[OF m1, where c=x] 
  1388   have "1 + x > x" by simp
  1389   thus "\<exists>y. y > x" by blast
  1390 qed
  1391 
  1392 subsection{*Anti-Monotonicity of @{term inverse}*}
  1393 
  1394 lemma less_imp_inverse_less:
  1395 assumes less: "a < b" and apos:  "0 < a"
  1396 shows "inverse b < inverse (a::'a::ordered_field)"
  1397 proof (rule ccontr)
  1398   assume "~ inverse b < inverse a"
  1399   hence "inverse a \<le> inverse b"
  1400     by (simp add: linorder_not_less)
  1401   hence "~ (a < b)"
  1402     by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos])
  1403   thus False
  1404     by (rule notE [OF _ less])
  1405 qed
  1406 
  1407 lemma inverse_less_imp_less:
  1408   "[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::ordered_field)"
  1409 apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"])
  1410 apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
  1411 done
  1412 
  1413 text{*Both premises are essential. Consider -1 and 1.*}
  1414 lemma inverse_less_iff_less [simp,noatp]:
  1415   "[|0 < a; 0 < b|] ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
  1416 by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
  1417 
  1418 lemma le_imp_inverse_le:
  1419   "[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
  1420 by (force simp add: order_le_less less_imp_inverse_less)
  1421 
  1422 lemma inverse_le_iff_le [simp,noatp]:
  1423  "[|0 < a; 0 < b|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
  1424 by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
  1425 
  1426 
  1427 text{*These results refer to both operands being negative.  The opposite-sign
  1428 case is trivial, since inverse preserves signs.*}
  1429 lemma inverse_le_imp_le_neg:
  1430   "[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::ordered_field)"
  1431 apply (rule classical) 
  1432 apply (subgoal_tac "a < 0") 
  1433  prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans) 
  1434 apply (insert inverse_le_imp_le [of "-b" "-a"])
  1435 apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
  1436 done
  1437 
  1438 lemma less_imp_inverse_less_neg:
  1439    "[|a < b; b < 0|] ==> inverse b < inverse (a::'a::ordered_field)"
  1440 apply (subgoal_tac "a < 0") 
  1441  prefer 2 apply (blast intro: order_less_trans) 
  1442 apply (insert less_imp_inverse_less [of "-b" "-a"])
  1443 apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
  1444 done
  1445 
  1446 lemma inverse_less_imp_less_neg:
  1447    "[|inverse a < inverse b; b < 0|] ==> b < (a::'a::ordered_field)"
  1448 apply (rule classical) 
  1449 apply (subgoal_tac "a < 0") 
  1450  prefer 2
  1451  apply (force simp add: linorder_not_less intro: order_le_less_trans) 
  1452 apply (insert inverse_less_imp_less [of "-b" "-a"])
  1453 apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
  1454 done
  1455 
  1456 lemma inverse_less_iff_less_neg [simp,noatp]:
  1457   "[|a < 0; b < 0|] ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
  1458 apply (insert inverse_less_iff_less [of "-b" "-a"])
  1459 apply (simp del: inverse_less_iff_less 
  1460             add: order_less_imp_not_eq nonzero_inverse_minus_eq)
  1461 done
  1462 
  1463 lemma le_imp_inverse_le_neg:
  1464   "[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
  1465 by (force simp add: order_le_less less_imp_inverse_less_neg)
  1466 
  1467 lemma inverse_le_iff_le_neg [simp,noatp]:
  1468  "[|a < 0; b < 0|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
  1469 by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
  1470 
  1471 
  1472 subsection{*Inverses and the Number One*}
  1473 
  1474 lemma one_less_inverse_iff:
  1475   "(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"
  1476 proof cases
  1477   assume "0 < x"
  1478     with inverse_less_iff_less [OF zero_less_one, of x]
  1479     show ?thesis by simp
  1480 next
  1481   assume notless: "~ (0 < x)"
  1482   have "~ (1 < inverse x)"
  1483   proof
  1484     assume "1 < inverse x"
  1485     also with notless have "... \<le> 0" by (simp add: linorder_not_less)
  1486     also have "... < 1" by (rule zero_less_one) 
  1487     finally show False by auto
  1488   qed
  1489   with notless show ?thesis by simp
  1490 qed
  1491 
  1492 lemma inverse_eq_1_iff [simp]:
  1493   "(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))"
  1494 by (insert inverse_eq_iff_eq [of x 1], simp) 
  1495 
  1496 lemma one_le_inverse_iff:
  1497   "(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))"
  1498 by (force simp add: order_le_less one_less_inverse_iff zero_less_one 
  1499                     eq_commute [of 1]) 
  1500 
  1501 lemma inverse_less_1_iff:
  1502   "(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{ordered_field,division_by_zero}))"
  1503 by (simp add: linorder_not_le [symmetric] one_le_inverse_iff) 
  1504 
  1505 lemma inverse_le_1_iff:
  1506   "(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{ordered_field,division_by_zero}))"
  1507 by (simp add: linorder_not_less [symmetric] one_less_inverse_iff) 
  1508 
  1509 
  1510 subsection{*Simplification of Inequalities Involving Literal Divisors*}
  1511 
  1512 lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"
  1513 proof -
  1514   assume less: "0<c"
  1515   hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
  1516     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
  1517   also have "... = (a*c \<le> b)"
  1518     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
  1519   finally show ?thesis .
  1520 qed
  1521 
  1522 lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"
  1523 proof -
  1524   assume less: "c<0"
  1525   hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
  1526     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
  1527   also have "... = (b \<le> a*c)"
  1528     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
  1529   finally show ?thesis .
  1530 qed
  1531 
  1532 lemma le_divide_eq:
  1533   "(a \<le> b/c) = 
  1534    (if 0 < c then a*c \<le> b
  1535              else if c < 0 then b \<le> a*c
  1536              else  a \<le> (0::'a::{ordered_field,division_by_zero}))"
  1537 apply (cases "c=0", simp) 
  1538 apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
  1539 done
  1540 
  1541 lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)"
  1542 proof -
  1543   assume less: "0<c"
  1544   hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
  1545     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
  1546   also have "... = (b \<le> a*c)"
  1547     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
  1548   finally show ?thesis .
  1549 qed
  1550 
  1551 lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)"
  1552 proof -
  1553   assume less: "c<0"
  1554   hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
  1555     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
  1556   also have "... = (a*c \<le> b)"
  1557     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
  1558   finally show ?thesis .
  1559 qed
  1560 
  1561 lemma divide_le_eq:
  1562   "(b/c \<le> a) = 
  1563    (if 0 < c then b \<le> a*c
  1564              else if c < 0 then a*c \<le> b
  1565              else 0 \<le> (a::'a::{ordered_field,division_by_zero}))"
  1566 apply (cases "c=0", simp) 
  1567 apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) 
  1568 done
  1569 
  1570 lemma pos_less_divide_eq:
  1571      "0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)"
  1572 proof -
  1573   assume less: "0<c"
  1574   hence "(a < b/c) = (a*c < (b/c)*c)"
  1575     by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
  1576   also have "... = (a*c < b)"
  1577     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
  1578   finally show ?thesis .
  1579 qed
  1580 
  1581 lemma neg_less_divide_eq:
  1582  "c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)"
  1583 proof -
  1584   assume less: "c<0"
  1585   hence "(a < b/c) = ((b/c)*c < a*c)"
  1586     by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
  1587   also have "... = (b < a*c)"
  1588     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
  1589   finally show ?thesis .
  1590 qed
  1591 
  1592 lemma less_divide_eq:
  1593   "(a < b/c) = 
  1594    (if 0 < c then a*c < b
  1595              else if c < 0 then b < a*c
  1596              else  a < (0::'a::{ordered_field,division_by_zero}))"
  1597 apply (cases "c=0", simp) 
  1598 apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) 
  1599 done
  1600 
  1601 lemma pos_divide_less_eq:
  1602      "0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)"
  1603 proof -
  1604   assume less: "0<c"
  1605   hence "(b/c < a) = ((b/c)*c < a*c)"
  1606     by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
  1607   also have "... = (b < a*c)"
  1608     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
  1609   finally show ?thesis .
  1610 qed
  1611 
  1612 lemma neg_divide_less_eq:
  1613  "c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)"
  1614 proof -
  1615   assume less: "c<0"
  1616   hence "(b/c < a) = (a*c < (b/c)*c)"
  1617     by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
  1618   also have "... = (a*c < b)"
  1619     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
  1620   finally show ?thesis .
  1621 qed
  1622 
  1623 lemma divide_less_eq:
  1624   "(b/c < a) = 
  1625    (if 0 < c then b < a*c
  1626              else if c < 0 then a*c < b
  1627              else 0 < (a::'a::{ordered_field,division_by_zero}))"
  1628 apply (cases "c=0", simp) 
  1629 apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) 
  1630 done
  1631 
  1632 
  1633 subsection{*Field simplification*}
  1634 
  1635 text{* Lemmas @{text field_simps} multiply with denominators in
  1636 in(equations) if they can be proved to be non-zero (for equations) or
  1637 positive/negative (for inequations). *}
  1638 
  1639 lemmas field_simps = field_eq_simps
  1640   (* multiply ineqn *)
  1641   pos_divide_less_eq neg_divide_less_eq
  1642   pos_less_divide_eq neg_less_divide_eq
  1643   pos_divide_le_eq neg_divide_le_eq
  1644   pos_le_divide_eq neg_le_divide_eq
  1645 
  1646 text{* Lemmas @{text sign_simps} is a first attempt to automate proofs
  1647 of positivity/negativity needed for @{text field_simps}. Have not added @{text
  1648 sign_simps} to @{text field_simps} because the former can lead to case
  1649 explosions. *}
  1650 
  1651 lemmas sign_simps = group_simps
  1652   zero_less_mult_iff  mult_less_0_iff
  1653 
  1654 (* Only works once linear arithmetic is installed:
  1655 text{*An example:*}
  1656 lemma fixes a b c d e f :: "'a::ordered_field"
  1657 shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow>
  1658  ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) <
  1659  ((e-f)*(a-b)*(c-d))/((e-f)*(a-b)*(c-d)) + u"
  1660 apply(subgoal_tac "(c-d)*(e-f)*(a-b) > 0")
  1661  prefer 2 apply(simp add:sign_simps)
  1662 apply(subgoal_tac "(c-d)*(e-f)*(a-b)*u > 0")
  1663  prefer 2 apply(simp add:sign_simps)
  1664 apply(simp add:field_simps)
  1665 done
  1666 *)
  1667 
  1668 
  1669 subsection{*Division and Signs*}
  1670 
  1671 lemma zero_less_divide_iff:
  1672      "((0::'a::{ordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
  1673 by (simp add: divide_inverse zero_less_mult_iff)
  1674 
  1675 lemma divide_less_0_iff:
  1676      "(a/b < (0::'a::{ordered_field,division_by_zero})) = 
  1677       (0 < a & b < 0 | a < 0 & 0 < b)"
  1678 by (simp add: divide_inverse mult_less_0_iff)
  1679 
  1680 lemma zero_le_divide_iff:
  1681      "((0::'a::{ordered_field,division_by_zero}) \<le> a/b) =
  1682       (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
  1683 by (simp add: divide_inverse zero_le_mult_iff)
  1684 
  1685 lemma divide_le_0_iff:
  1686      "(a/b \<le> (0::'a::{ordered_field,division_by_zero})) =
  1687       (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
  1688 by (simp add: divide_inverse mult_le_0_iff)
  1689 
  1690 lemma divide_eq_0_iff [simp,noatp]:
  1691      "(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))"
  1692 by (simp add: divide_inverse)
  1693 
  1694 lemma divide_pos_pos:
  1695   "0 < (x::'a::ordered_field) ==> 0 < y ==> 0 < x / y"
  1696 by(simp add:field_simps)
  1697 
  1698 
  1699 lemma divide_nonneg_pos:
  1700   "0 <= (x::'a::ordered_field) ==> 0 < y ==> 0 <= x / y"
  1701 by(simp add:field_simps)
  1702 
  1703 lemma divide_neg_pos:
  1704   "(x::'a::ordered_field) < 0 ==> 0 < y ==> x / y < 0"
  1705 by(simp add:field_simps)
  1706 
  1707 lemma divide_nonpos_pos:
  1708   "(x::'a::ordered_field) <= 0 ==> 0 < y ==> x / y <= 0"
  1709 by(simp add:field_simps)
  1710 
  1711 lemma divide_pos_neg:
  1712   "0 < (x::'a::ordered_field) ==> y < 0 ==> x / y < 0"
  1713 by(simp add:field_simps)
  1714 
  1715 lemma divide_nonneg_neg:
  1716   "0 <= (x::'a::ordered_field) ==> y < 0 ==> x / y <= 0" 
  1717 by(simp add:field_simps)
  1718 
  1719 lemma divide_neg_neg:
  1720   "(x::'a::ordered_field) < 0 ==> y < 0 ==> 0 < x / y"
  1721 by(simp add:field_simps)
  1722 
  1723 lemma divide_nonpos_neg:
  1724   "(x::'a::ordered_field) <= 0 ==> y < 0 ==> 0 <= x / y"
  1725 by(simp add:field_simps)
  1726 
  1727 
  1728 subsection{*Cancellation Laws for Division*}
  1729 
  1730 lemma divide_cancel_right [simp,noatp]:
  1731      "(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))"
  1732 apply (cases "c=0", simp)
  1733 apply (simp add: divide_inverse)
  1734 done
  1735 
  1736 lemma divide_cancel_left [simp,noatp]:
  1737      "(c/a = c/b) = (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 
  1743 subsection {* Division and the Number One *}
  1744 
  1745 text{*Simplify expressions equated with 1*}
  1746 lemma divide_eq_1_iff [simp,noatp]:
  1747      "(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
  1748 apply (cases "b=0", simp)
  1749 apply (simp add: right_inverse_eq)
  1750 done
  1751 
  1752 lemma one_eq_divide_iff [simp,noatp]:
  1753      "(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
  1754 by (simp add: eq_commute [of 1])
  1755 
  1756 lemma zero_eq_1_divide_iff [simp,noatp]:
  1757      "((0::'a::{ordered_field,division_by_zero}) = 1/a) = (a = 0)"
  1758 apply (cases "a=0", simp)
  1759 apply (auto simp add: nonzero_eq_divide_eq)
  1760 done
  1761 
  1762 lemma one_divide_eq_0_iff [simp,noatp]:
  1763      "(1/a = (0::'a::{ordered_field,division_by_zero})) = (a = 0)"
  1764 apply (cases "a=0", simp)
  1765 apply (insert zero_neq_one [THEN not_sym])
  1766 apply (auto simp add: nonzero_divide_eq_eq)
  1767 done
  1768 
  1769 text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}
  1770 lemmas zero_less_divide_1_iff = zero_less_divide_iff [of 1, simplified]
  1771 lemmas divide_less_0_1_iff = divide_less_0_iff [of 1, simplified]
  1772 lemmas zero_le_divide_1_iff = zero_le_divide_iff [of 1, simplified]
  1773 lemmas divide_le_0_1_iff = divide_le_0_iff [of 1, simplified]
  1774 
  1775 declare zero_less_divide_1_iff [simp]
  1776 declare divide_less_0_1_iff [simp,noatp]
  1777 declare zero_le_divide_1_iff [simp]
  1778 declare divide_le_0_1_iff [simp,noatp]
  1779 
  1780 
  1781 subsection {* Ordering Rules for Division *}
  1782 
  1783 lemma divide_strict_right_mono:
  1784      "[|a < b; 0 < c|] ==> a / c < b / (c::'a::ordered_field)"
  1785 by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono 
  1786               positive_imp_inverse_positive)
  1787 
  1788 lemma divide_right_mono:
  1789      "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{ordered_field,division_by_zero})"
  1790 by (force simp add: divide_strict_right_mono order_le_less)
  1791 
  1792 lemma divide_right_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b 
  1793     ==> c <= 0 ==> b / c <= a / c"
  1794 apply (drule divide_right_mono [of _ _ "- c"])
  1795 apply auto
  1796 done
  1797 
  1798 lemma divide_strict_right_mono_neg:
  1799      "[|b < a; c < 0|] ==> a / c < b / (c::'a::ordered_field)"
  1800 apply (drule divide_strict_right_mono [of _ _ "-c"], simp)
  1801 apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric])
  1802 done
  1803 
  1804 text{*The last premise ensures that @{term a} and @{term b} 
  1805       have the same sign*}
  1806 lemma divide_strict_left_mono:
  1807   "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
  1808 by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono)
  1809 
  1810 lemma divide_left_mono:
  1811   "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::ordered_field)"
  1812 by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_right_mono)
  1813 
  1814 lemma divide_left_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b 
  1815     ==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"
  1816   apply (drule divide_left_mono [of _ _ "- c"])
  1817   apply (auto simp add: mult_commute)
  1818 done
  1819 
  1820 lemma divide_strict_left_mono_neg:
  1821   "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
  1822 by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono_neg)
  1823 
  1824 
  1825 text{*Simplify quotients that are compared with the value 1.*}
  1826 
  1827 lemma le_divide_eq_1 [noatp]:
  1828   fixes a :: "'a :: {ordered_field,division_by_zero}"
  1829   shows "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"
  1830 by (auto simp add: le_divide_eq)
  1831 
  1832 lemma divide_le_eq_1 [noatp]:
  1833   fixes a :: "'a :: {ordered_field,division_by_zero}"
  1834   shows "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"
  1835 by (auto simp add: divide_le_eq)
  1836 
  1837 lemma less_divide_eq_1 [noatp]:
  1838   fixes a :: "'a :: {ordered_field,division_by_zero}"
  1839   shows "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
  1840 by (auto simp add: less_divide_eq)
  1841 
  1842 lemma divide_less_eq_1 [noatp]:
  1843   fixes a :: "'a :: {ordered_field,division_by_zero}"
  1844   shows "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
  1845 by (auto simp add: divide_less_eq)
  1846 
  1847 
  1848 subsection{*Conditional Simplification Rules: No Case Splits*}
  1849 
  1850 lemma le_divide_eq_1_pos [simp,noatp]:
  1851   fixes a :: "'a :: {ordered_field,division_by_zero}"
  1852   shows "0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)"
  1853 by (auto simp add: le_divide_eq)
  1854 
  1855 lemma le_divide_eq_1_neg [simp,noatp]:
  1856   fixes a :: "'a :: {ordered_field,division_by_zero}"
  1857   shows "a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)"
  1858 by (auto simp add: le_divide_eq)
  1859 
  1860 lemma divide_le_eq_1_pos [simp,noatp]:
  1861   fixes a :: "'a :: {ordered_field,division_by_zero}"
  1862   shows "0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)"
  1863 by (auto simp add: divide_le_eq)
  1864 
  1865 lemma divide_le_eq_1_neg [simp,noatp]:
  1866   fixes a :: "'a :: {ordered_field,division_by_zero}"
  1867   shows "a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)"
  1868 by (auto simp add: divide_le_eq)
  1869 
  1870 lemma less_divide_eq_1_pos [simp,noatp]:
  1871   fixes a :: "'a :: {ordered_field,division_by_zero}"
  1872   shows "0 < a \<Longrightarrow> (1 < b/a) = (a < b)"
  1873 by (auto simp add: less_divide_eq)
  1874 
  1875 lemma less_divide_eq_1_neg [simp,noatp]:
  1876   fixes a :: "'a :: {ordered_field,division_by_zero}"
  1877   shows "a < 0 \<Longrightarrow> (1 < b/a) = (b < a)"
  1878 by (auto simp add: less_divide_eq)
  1879 
  1880 lemma divide_less_eq_1_pos [simp,noatp]:
  1881   fixes a :: "'a :: {ordered_field,division_by_zero}"
  1882   shows "0 < a \<Longrightarrow> (b/a < 1) = (b < a)"
  1883 by (auto simp add: divide_less_eq)
  1884 
  1885 lemma divide_less_eq_1_neg [simp,noatp]:
  1886   fixes a :: "'a :: {ordered_field,division_by_zero}"
  1887   shows "a < 0 \<Longrightarrow> b/a < 1 <-> a < b"
  1888 by (auto simp add: divide_less_eq)
  1889 
  1890 lemma eq_divide_eq_1 [simp,noatp]:
  1891   fixes a :: "'a :: {ordered_field,division_by_zero}"
  1892   shows "(1 = b/a) = ((a \<noteq> 0 & a = b))"
  1893 by (auto simp add: eq_divide_eq)
  1894 
  1895 lemma divide_eq_eq_1 [simp,noatp]:
  1896   fixes a :: "'a :: {ordered_field,division_by_zero}"
  1897   shows "(b/a = 1) = ((a \<noteq> 0 & a = b))"
  1898 by (auto simp add: divide_eq_eq)
  1899 
  1900 
  1901 subsection {* Reasoning about inequalities with division *}
  1902 
  1903 lemma mult_right_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
  1904     ==> x * y <= x"
  1905   by (auto simp add: mult_compare_simps);
  1906 
  1907 lemma mult_left_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
  1908     ==> y * x <= x"
  1909   by (auto simp add: mult_compare_simps);
  1910 
  1911 lemma mult_imp_div_pos_le: "0 < (y::'a::ordered_field) ==> x <= z * y ==>
  1912     x / y <= z";
  1913   by (subst pos_divide_le_eq, assumption+);
  1914 
  1915 lemma mult_imp_le_div_pos: "0 < (y::'a::ordered_field) ==> z * y <= x ==>
  1916     z <= x / y"
  1917 by(simp add:field_simps)
  1918 
  1919 lemma mult_imp_div_pos_less: "0 < (y::'a::ordered_field) ==> x < z * y ==>
  1920     x / y < z"
  1921 by(simp add:field_simps)
  1922 
  1923 lemma mult_imp_less_div_pos: "0 < (y::'a::ordered_field) ==> z * y < x ==>
  1924     z < x / y"
  1925 by(simp add:field_simps)
  1926 
  1927 lemma frac_le: "(0::'a::ordered_field) <= x ==> 
  1928     x <= y ==> 0 < w ==> w <= z  ==> x / z <= y / w"
  1929   apply (rule mult_imp_div_pos_le)
  1930   apply simp
  1931   apply (subst times_divide_eq_left)
  1932   apply (rule mult_imp_le_div_pos, assumption)
  1933   thm mult_mono
  1934   thm mult_mono'
  1935   apply (rule mult_mono)
  1936   apply simp_all
  1937 done
  1938 
  1939 lemma frac_less: "(0::'a::ordered_field) <= x ==> 
  1940     x < y ==> 0 < w ==> w <= z  ==> x / z < y / w"
  1941   apply (rule mult_imp_div_pos_less)
  1942   apply simp;
  1943   apply (subst times_divide_eq_left);
  1944   apply (rule mult_imp_less_div_pos, assumption)
  1945   apply (erule mult_less_le_imp_less)
  1946   apply simp_all
  1947 done
  1948 
  1949 lemma frac_less2: "(0::'a::ordered_field) < x ==> 
  1950     x <= y ==> 0 < w ==> w < z  ==> x / z < y / w"
  1951   apply (rule mult_imp_div_pos_less)
  1952   apply simp_all
  1953   apply (subst times_divide_eq_left);
  1954   apply (rule mult_imp_less_div_pos, assumption)
  1955   apply (erule mult_le_less_imp_less)
  1956   apply simp_all
  1957 done
  1958 
  1959 text{*It's not obvious whether these should be simprules or not. 
  1960   Their effect is to gather terms into one big fraction, like
  1961   a*b*c / x*y*z. The rationale for that is unclear, but many proofs 
  1962   seem to need them.*}
  1963 
  1964 declare times_divide_eq [simp]
  1965 
  1966 
  1967 subsection {* Ordered Fields are Dense *}
  1968 
  1969 context ordered_semidom
  1970 begin
  1971 
  1972 lemma less_add_one: "a < a + 1"
  1973 proof -
  1974   have "a + 0 < a + 1"
  1975     by (blast intro: zero_less_one add_strict_left_mono)
  1976   thus ?thesis by simp
  1977 qed
  1978 
  1979 lemma zero_less_two: "0 < 1 + 1"
  1980   by (blast intro: less_trans zero_less_one less_add_one)
  1981 
  1982 end
  1983 
  1984 lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::ordered_field)"
  1985 by (simp add: field_simps zero_less_two)
  1986 
  1987 lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::ordered_field) < b"
  1988 by (simp add: field_simps zero_less_two)
  1989 
  1990 instance ordered_field < dense_linear_order
  1991 proof
  1992   fix x y :: 'a
  1993   have "x < x + 1" by simp
  1994   then show "\<exists>y. x < y" .. 
  1995   have "x - 1 < x" by simp
  1996   then show "\<exists>y. y < x" ..
  1997   show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum)
  1998 qed
  1999 
  2000 
  2001 subsection {* Absolute Value *}
  2002 
  2003 context ordered_idom
  2004 begin
  2005 
  2006 lemma mult_sgn_abs: "sgn x * abs x = x"
  2007   unfolding abs_if sgn_if by auto
  2008 
  2009 end
  2010 
  2011 lemma abs_one [simp]: "abs 1 = (1::'a::ordered_idom)"
  2012   by (simp add: abs_if zero_less_one [THEN order_less_not_sym])
  2013 
  2014 class pordered_ring_abs = pordered_ring + pordered_ab_group_add_abs +
  2015   assumes abs_eq_mult:
  2016     "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0) \<Longrightarrow> \<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
  2017 
  2018 
  2019 class lordered_ring = pordered_ring + lordered_ab_group_add_abs
  2020 begin
  2021 
  2022 subclass lordered_ab_group_add_meet by unfold_locales
  2023 subclass lordered_ab_group_add_join by unfold_locales
  2024 
  2025 end
  2026 
  2027 lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lordered_ring))" 
  2028 proof -
  2029   let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"
  2030   let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
  2031   have a: "(abs a) * (abs b) = ?x"
  2032     by (simp only: abs_prts[of a] abs_prts[of b] ring_simps)
  2033   {
  2034     fix u v :: 'a
  2035     have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow> 
  2036               u * v = pprt a * pprt b + pprt a * nprt b + 
  2037                       nprt a * pprt b + nprt a * nprt b"
  2038       apply (subst prts[of u], subst prts[of v])
  2039       apply (simp add: ring_simps) 
  2040       done
  2041   }
  2042   note b = this[OF refl[of a] refl[of b]]
  2043   note addm = add_mono[of "0::'a" _ "0::'a", simplified]
  2044   note addm2 = add_mono[of _ "0::'a" _ "0::'a", simplified]
  2045   have xy: "- ?x <= ?y"
  2046     apply (simp)
  2047     apply (rule_tac y="0::'a" in order_trans)
  2048     apply (rule addm2)
  2049     apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos)
  2050     apply (rule addm)
  2051     apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos)
  2052     done
  2053   have yx: "?y <= ?x"
  2054     apply (simp add:diff_def)
  2055     apply (rule_tac y=0 in order_trans)
  2056     apply (rule addm2, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+)
  2057     apply (rule addm, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+)
  2058     done
  2059   have i1: "a*b <= abs a * abs b" by (simp only: a b yx)
  2060   have i2: "- (abs a * abs b) <= a*b" by (simp only: a b xy)
  2061   show ?thesis
  2062     apply (rule abs_leI)
  2063     apply (simp add: i1)
  2064     apply (simp add: i2[simplified minus_le_iff])
  2065     done
  2066 qed
  2067 
  2068 instance lordered_ring \<subseteq> pordered_ring_abs
  2069 proof
  2070   fix a b :: "'a\<Colon> lordered_ring"
  2071   assume "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)"
  2072   show "abs (a*b) = abs a * abs b"
  2073 proof -
  2074   have s: "(0 <= a*b) | (a*b <= 0)"
  2075     apply (auto)    
  2076     apply (rule_tac split_mult_pos_le)
  2077     apply (rule_tac contrapos_np[of "a*b <= 0"])
  2078     apply (simp)
  2079     apply (rule_tac split_mult_neg_le)
  2080     apply (insert prems)
  2081     apply (blast)
  2082     done
  2083   have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
  2084     by (simp add: prts[symmetric])
  2085   show ?thesis
  2086   proof cases
  2087     assume "0 <= a * b"
  2088     then show ?thesis
  2089       apply (simp_all add: mulprts abs_prts)
  2090       apply (insert prems)
  2091       apply (auto simp add: 
  2092 	ring_simps 
  2093 	iffD1[OF zero_le_iff_zero_nprt] iffD1[OF le_zero_iff_zero_pprt]
  2094 	iffD1[OF le_zero_iff_pprt_id] iffD1[OF zero_le_iff_nprt_id])
  2095 	apply(drule (1) mult_nonneg_nonpos[of a b], simp)
  2096 	apply(drule (1) mult_nonneg_nonpos2[of b a], simp)
  2097       done
  2098   next
  2099     assume "~(0 <= a*b)"
  2100     with s have "a*b <= 0" by simp
  2101     then show ?thesis
  2102       apply (simp_all add: mulprts abs_prts)
  2103       apply (insert prems)
  2104       apply (auto simp add: ring_simps)
  2105       apply(drule (1) mult_nonneg_nonneg[of a b],simp)
  2106       apply(drule (1) mult_nonpos_nonpos[of a b],simp)
  2107       done
  2108   qed
  2109 qed
  2110 qed
  2111 
  2112 instance ordered_idom \<subseteq> pordered_ring_abs
  2113 by default (auto simp add: abs_if not_less
  2114   equal_neg_zero neg_equal_zero mult_less_0_iff)
  2115 
  2116 lemma abs_mult: "abs (a * b) = abs a * abs (b::'a::ordered_idom)" 
  2117   by (simp add: abs_eq_mult linorder_linear)
  2118 
  2119 lemma abs_mult_self: "abs a * abs a = a * (a::'a::ordered_idom)"
  2120   by (simp add: abs_if) 
  2121 
  2122 lemma nonzero_abs_inverse:
  2123      "a \<noteq> 0 ==> abs (inverse (a::'a::ordered_field)) = inverse (abs a)"
  2124 apply (auto simp add: linorder_neq_iff abs_if nonzero_inverse_minus_eq 
  2125                       negative_imp_inverse_negative)
  2126 apply (blast intro: positive_imp_inverse_positive elim: order_less_asym) 
  2127 done
  2128 
  2129 lemma abs_inverse [simp]:
  2130      "abs (inverse (a::'a::{ordered_field,division_by_zero})) = 
  2131       inverse (abs a)"
  2132 apply (cases "a=0", simp) 
  2133 apply (simp add: nonzero_abs_inverse) 
  2134 done
  2135 
  2136 lemma nonzero_abs_divide:
  2137      "b \<noteq> 0 ==> abs (a / (b::'a::ordered_field)) = abs a / abs b"
  2138 by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 
  2139 
  2140 lemma abs_divide [simp]:
  2141      "abs (a / (b::'a::{ordered_field,division_by_zero})) = abs a / abs b"
  2142 apply (cases "b=0", simp) 
  2143 apply (simp add: nonzero_abs_divide) 
  2144 done
  2145 
  2146 lemma abs_mult_less:
  2147      "[| abs a < c; abs b < d |] ==> abs a * abs b < c*(d::'a::ordered_idom)"
  2148 proof -
  2149   assume ac: "abs a < c"
  2150   hence cpos: "0<c" by (blast intro: order_le_less_trans abs_ge_zero)
  2151   assume "abs b < d"
  2152   thus ?thesis by (simp add: ac cpos mult_strict_mono) 
  2153 qed
  2154 
  2155 lemmas eq_minus_self_iff = equal_neg_zero
  2156 
  2157 lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::ordered_idom))"
  2158   unfolding order_less_le less_eq_neg_nonpos equal_neg_zero ..
  2159 
  2160 lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::ordered_idom))" 
  2161 apply (simp add: order_less_le abs_le_iff)  
  2162 apply (auto simp add: abs_if neg_less_eq_nonneg less_eq_neg_nonpos)
  2163 done
  2164 
  2165 lemma abs_mult_pos: "(0::'a::ordered_idom) <= x ==> 
  2166     (abs y) * x = abs (y * x)"
  2167   apply (subst abs_mult)
  2168   apply simp
  2169 done
  2170 
  2171 lemma abs_div_pos: "(0::'a::{division_by_zero,ordered_field}) < y ==> 
  2172     abs x / y = abs (x / y)"
  2173   apply (subst abs_divide)
  2174   apply (simp add: order_less_imp_le)
  2175 done
  2176 
  2177 
  2178 subsection {* Bounds of products via negative and positive Part *}
  2179 
  2180 lemma mult_le_prts:
  2181   assumes
  2182   "a1 <= (a::'a::lordered_ring)"
  2183   "a <= a2"
  2184   "b1 <= b"
  2185   "b <= b2"
  2186   shows
  2187   "a * b <= pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
  2188 proof - 
  2189   have "a * b = (pprt a + nprt a) * (pprt b + nprt b)" 
  2190     apply (subst prts[symmetric])+
  2191     apply simp
  2192     done
  2193   then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
  2194     by (simp add: ring_simps)
  2195   moreover have "pprt a * pprt b <= pprt a2 * pprt b2"
  2196     by (simp_all add: prems mult_mono)
  2197   moreover have "pprt a * nprt b <= pprt a1 * nprt b2"
  2198   proof -
  2199     have "pprt a * nprt b <= pprt a * nprt b2"
  2200       by (simp add: mult_left_mono prems)
  2201     moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2"
  2202       by (simp add: mult_right_mono_neg prems)
  2203     ultimately show ?thesis
  2204       by simp
  2205   qed
  2206   moreover have "nprt a * pprt b <= nprt a2 * pprt b1"
  2207   proof - 
  2208     have "nprt a * pprt b <= nprt a2 * pprt b"
  2209       by (simp add: mult_right_mono prems)
  2210     moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"
  2211       by (simp add: mult_left_mono_neg prems)
  2212     ultimately show ?thesis
  2213       by simp
  2214   qed
  2215   moreover have "nprt a * nprt b <= nprt a1 * nprt b1"
  2216   proof -
  2217     have "nprt a * nprt b <= nprt a * nprt b1"
  2218       by (simp add: mult_left_mono_neg prems)
  2219     moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1"
  2220       by (simp add: mult_right_mono_neg prems)
  2221     ultimately show ?thesis
  2222       by simp
  2223   qed
  2224   ultimately show ?thesis
  2225     by - (rule add_mono | simp)+
  2226 qed
  2227 
  2228 lemma mult_ge_prts:
  2229   assumes
  2230   "a1 <= (a::'a::lordered_ring)"
  2231   "a <= a2"
  2232   "b1 <= b"
  2233   "b <= b2"
  2234   shows
  2235   "a * b >= nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"
  2236 proof - 
  2237   from prems have a1:"- a2 <= -a" by auto
  2238   from prems have a2: "-a <= -a1" by auto
  2239   from mult_le_prts[of "-a2" "-a" "-a1" "b1" b "b2", OF a1 a2 prems(3) prems(4), simplified nprt_neg pprt_neg] 
  2240   have le: "- (a * b) <= - nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1" by simp  
  2241   then have "-(- nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1) <= a * b"
  2242     by (simp only: minus_le_iff)
  2243   then show ?thesis by simp
  2244 qed
  2245 
  2246 end