src/HOL/Fields.thy
author paulson <lp15@cam.ac.uk>
Mon Feb 22 14:37:56 2016 +0000 (2016-02-22)
changeset 62379 340738057c8c
parent 62347 2230b7047376
child 62481 b5d8e57826df
permissions -rw-r--r--
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
     1 (*  Title:      HOL/Fields.thy
     2     Author:     Gertrud Bauer
     3     Author:     Steven Obua
     4     Author:     Tobias Nipkow
     5     Author:     Lawrence C Paulson
     6     Author:     Markus Wenzel
     7     Author:     Jeremy Avigad
     8 *)
     9 
    10 section \<open>Fields\<close>
    11 
    12 theory Fields
    13 imports Rings
    14 begin
    15 
    16 subsection \<open>Division rings\<close>
    17 
    18 text \<open>
    19   A division ring is like a field, but without the commutativity requirement.
    20 \<close>
    21 
    22 class inverse = divide +
    23   fixes inverse :: "'a \<Rightarrow> 'a"
    24 begin
    25   
    26 abbreviation inverse_divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "'/" 70)
    27 where
    28   "inverse_divide \<equiv> divide"
    29 
    30 end
    31 
    32 text\<open>Lemmas \<open>divide_simps\<close> move division to the outside and eliminates them on (in)equalities.\<close>
    33 
    34 named_theorems divide_simps "rewrite rules to eliminate divisions"
    35 
    36 class division_ring = ring_1 + inverse +
    37   assumes left_inverse [simp]:  "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
    38   assumes right_inverse [simp]: "a \<noteq> 0 \<Longrightarrow> a * inverse a = 1"
    39   assumes divide_inverse: "a / b = a * inverse b"
    40   assumes inverse_zero [simp]: "inverse 0 = 0"
    41 begin
    42 
    43 subclass ring_1_no_zero_divisors
    44 proof
    45   fix a b :: 'a
    46   assume a: "a \<noteq> 0" and b: "b \<noteq> 0"
    47   show "a * b \<noteq> 0"
    48   proof
    49     assume ab: "a * b = 0"
    50     hence "0 = inverse a * (a * b) * inverse b" by simp
    51     also have "\<dots> = (inverse a * a) * (b * inverse b)"
    52       by (simp only: mult.assoc)
    53     also have "\<dots> = 1" using a b by simp
    54     finally show False by simp
    55   qed
    56 qed
    57 
    58 lemma nonzero_imp_inverse_nonzero:
    59   "a \<noteq> 0 \<Longrightarrow> inverse a \<noteq> 0"
    60 proof
    61   assume ianz: "inverse a = 0"
    62   assume "a \<noteq> 0"
    63   hence "1 = a * inverse a" by simp
    64   also have "... = 0" by (simp add: ianz)
    65   finally have "1 = 0" .
    66   thus False by (simp add: eq_commute)
    67 qed
    68 
    69 lemma inverse_zero_imp_zero:
    70   "inverse a = 0 \<Longrightarrow> a = 0"
    71 apply (rule classical)
    72 apply (drule nonzero_imp_inverse_nonzero)
    73 apply auto
    74 done
    75 
    76 lemma inverse_unique:
    77   assumes ab: "a * b = 1"
    78   shows "inverse a = b"
    79 proof -
    80   have "a \<noteq> 0" using ab by (cases "a = 0") simp_all
    81   moreover have "inverse a * (a * b) = inverse a" by (simp add: ab)
    82   ultimately show ?thesis by (simp add: mult.assoc [symmetric])
    83 qed
    84 
    85 lemma nonzero_inverse_minus_eq:
    86   "a \<noteq> 0 \<Longrightarrow> inverse (- a) = - inverse a"
    87 by (rule inverse_unique) simp
    88 
    89 lemma nonzero_inverse_inverse_eq:
    90   "a \<noteq> 0 \<Longrightarrow> inverse (inverse a) = a"
    91 by (rule inverse_unique) simp
    92 
    93 lemma nonzero_inverse_eq_imp_eq:
    94   assumes "inverse a = inverse b" and "a \<noteq> 0" and "b \<noteq> 0"
    95   shows "a = b"
    96 proof -
    97   from \<open>inverse a = inverse b\<close>
    98   have "inverse (inverse a) = inverse (inverse b)" by (rule arg_cong)
    99   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> show "a = b"
   100     by (simp add: nonzero_inverse_inverse_eq)
   101 qed
   102 
   103 lemma inverse_1 [simp]: "inverse 1 = 1"
   104 by (rule inverse_unique) simp
   105 
   106 lemma nonzero_inverse_mult_distrib:
   107   assumes "a \<noteq> 0" and "b \<noteq> 0"
   108   shows "inverse (a * b) = inverse b * inverse a"
   109 proof -
   110   have "a * (b * inverse b) * inverse a = 1" using assms by simp
   111   hence "a * b * (inverse b * inverse a) = 1" by (simp only: mult.assoc)
   112   thus ?thesis by (rule inverse_unique)
   113 qed
   114 
   115 lemma division_ring_inverse_add:
   116   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a + inverse b = inverse a * (a + b) * inverse b"
   117 by (simp add: algebra_simps)
   118 
   119 lemma division_ring_inverse_diff:
   120   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a - inverse b = inverse a * (b - a) * inverse b"
   121 by (simp add: algebra_simps)
   122 
   123 lemma right_inverse_eq: "b \<noteq> 0 \<Longrightarrow> a / b = 1 \<longleftrightarrow> a = b"
   124 proof
   125   assume neq: "b \<noteq> 0"
   126   {
   127     hence "a = (a / b) * b" by (simp add: divide_inverse mult.assoc)
   128     also assume "a / b = 1"
   129     finally show "a = b" by simp
   130   next
   131     assume "a = b"
   132     with neq show "a / b = 1" by (simp add: divide_inverse)
   133   }
   134 qed
   135 
   136 lemma nonzero_inverse_eq_divide: "a \<noteq> 0 \<Longrightarrow> inverse a = 1 / a"
   137 by (simp add: divide_inverse)
   138 
   139 lemma divide_self [simp]: "a \<noteq> 0 \<Longrightarrow> a / a = 1"
   140 by (simp add: divide_inverse)
   141 
   142 lemma inverse_eq_divide [field_simps, divide_simps]: "inverse a = 1 / a"
   143 by (simp add: divide_inverse)
   144 
   145 lemma add_divide_distrib: "(a+b) / c = a/c + b/c"
   146 by (simp add: divide_inverse algebra_simps)
   147 
   148 lemma times_divide_eq_right [simp]: "a * (b / c) = (a * b) / c"
   149   by (simp add: divide_inverse mult.assoc)
   150 
   151 lemma minus_divide_left: "- (a / b) = (-a) / b"
   152   by (simp add: divide_inverse)
   153 
   154 lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a / b) = a / (- b)"
   155   by (simp add: divide_inverse nonzero_inverse_minus_eq)
   156 
   157 lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a) / (-b) = a / b"
   158   by (simp add: divide_inverse nonzero_inverse_minus_eq)
   159 
   160 lemma divide_minus_left [simp]: "(-a) / b = - (a / b)"
   161   by (simp add: divide_inverse)
   162 
   163 lemma diff_divide_distrib: "(a - b) / c = a / c - b / c"
   164   using add_divide_distrib [of a "- b" c] by simp
   165 
   166 lemma nonzero_eq_divide_eq [field_simps]: "c \<noteq> 0 \<Longrightarrow> a = b / c \<longleftrightarrow> a * c = b"
   167 proof -
   168   assume [simp]: "c \<noteq> 0"
   169   have "a = b / c \<longleftrightarrow> a * c = (b / c) * c" by simp
   170   also have "... \<longleftrightarrow> a * c = b" by (simp add: divide_inverse mult.assoc)
   171   finally show ?thesis .
   172 qed
   173 
   174 lemma nonzero_divide_eq_eq [field_simps]: "c \<noteq> 0 \<Longrightarrow> b / c = a \<longleftrightarrow> b = a * c"
   175 proof -
   176   assume [simp]: "c \<noteq> 0"
   177   have "b / c = a \<longleftrightarrow> (b / c) * c = a * c" by simp
   178   also have "... \<longleftrightarrow> b = a * c" by (simp add: divide_inverse mult.assoc)
   179   finally show ?thesis .
   180 qed
   181 
   182 lemma nonzero_neg_divide_eq_eq [field_simps]: "b \<noteq> 0 \<Longrightarrow> - (a / b) = c \<longleftrightarrow> - a = c * b"
   183   using nonzero_divide_eq_eq[of b "-a" c] by simp
   184 
   185 lemma nonzero_neg_divide_eq_eq2 [field_simps]: "b \<noteq> 0 \<Longrightarrow> c = - (a / b) \<longleftrightarrow> c * b = - a"
   186   using nonzero_neg_divide_eq_eq[of b a c] by auto
   187 
   188 lemma divide_eq_imp: "c \<noteq> 0 \<Longrightarrow> b = a * c \<Longrightarrow> b / c = a"
   189   by (simp add: divide_inverse mult.assoc)
   190 
   191 lemma eq_divide_imp: "c \<noteq> 0 \<Longrightarrow> a * c = b \<Longrightarrow> a = b / c"
   192   by (drule sym) (simp add: divide_inverse mult.assoc)
   193 
   194 lemma add_divide_eq_iff [field_simps]:
   195   "z \<noteq> 0 \<Longrightarrow> x + y / z = (x * z + y) / z"
   196   by (simp add: add_divide_distrib nonzero_eq_divide_eq)
   197 
   198 lemma divide_add_eq_iff [field_simps]:
   199   "z \<noteq> 0 \<Longrightarrow> x / z + y = (x + y * z) / z"
   200   by (simp add: add_divide_distrib nonzero_eq_divide_eq)
   201 
   202 lemma diff_divide_eq_iff [field_simps]:
   203   "z \<noteq> 0 \<Longrightarrow> x - y / z = (x * z - y) / z"
   204   by (simp add: diff_divide_distrib nonzero_eq_divide_eq eq_diff_eq)
   205 
   206 lemma minus_divide_add_eq_iff [field_simps]:
   207   "z \<noteq> 0 \<Longrightarrow> - (x / z) + y = (- x + y * z) / z"
   208   by (simp add: add_divide_distrib diff_divide_eq_iff)
   209 
   210 lemma divide_diff_eq_iff [field_simps]:
   211   "z \<noteq> 0 \<Longrightarrow> x / z - y = (x - y * z) / z"
   212   by (simp add: field_simps)
   213 
   214 lemma minus_divide_diff_eq_iff [field_simps]:
   215   "z \<noteq> 0 \<Longrightarrow> - (x / z) - y = (- x - y * z) / z"
   216   by (simp add: divide_diff_eq_iff[symmetric])
   217 
   218 lemma division_ring_divide_zero [simp]:
   219   "a / 0 = 0"
   220   by (simp add: divide_inverse)
   221 
   222 lemma divide_self_if [simp]:
   223   "a / a = (if a = 0 then 0 else 1)"
   224   by simp
   225 
   226 lemma inverse_nonzero_iff_nonzero [simp]:
   227   "inverse a = 0 \<longleftrightarrow> a = 0"
   228   by rule (fact inverse_zero_imp_zero, simp)
   229 
   230 lemma inverse_minus_eq [simp]:
   231   "inverse (- a) = - inverse a"
   232 proof cases
   233   assume "a=0" thus ?thesis by simp
   234 next
   235   assume "a\<noteq>0"
   236   thus ?thesis by (simp add: nonzero_inverse_minus_eq)
   237 qed
   238 
   239 lemma inverse_inverse_eq [simp]:
   240   "inverse (inverse a) = a"
   241 proof cases
   242   assume "a=0" thus ?thesis by simp
   243 next
   244   assume "a\<noteq>0"
   245   thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
   246 qed
   247 
   248 lemma inverse_eq_imp_eq:
   249   "inverse a = inverse b \<Longrightarrow> a = b"
   250   by (drule arg_cong [where f="inverse"], simp)
   251 
   252 lemma inverse_eq_iff_eq [simp]:
   253   "inverse a = inverse b \<longleftrightarrow> a = b"
   254   by (force dest!: inverse_eq_imp_eq)
   255 
   256 lemma add_divide_eq_if_simps [divide_simps]:
   257     "a + b / z = (if z = 0 then a else (a * z + b) / z)"
   258     "a / z + b = (if z = 0 then b else (a + b * z) / z)"
   259     "- (a / z) + b = (if z = 0 then b else (-a + b * z) / z)"
   260     "a - b / z = (if z = 0 then a else (a * z - b) / z)"
   261     "a / z - b = (if z = 0 then -b else (a - b * z) / z)"
   262     "- (a / z) - b = (if z = 0 then -b else (- a - b * z) / z)"
   263   by (simp_all add: add_divide_eq_iff divide_add_eq_iff diff_divide_eq_iff divide_diff_eq_iff
   264       minus_divide_diff_eq_iff)
   265 
   266 lemma [divide_simps]:
   267   shows divide_eq_eq: "b / c = a \<longleftrightarrow> (if c \<noteq> 0 then b = a * c else a = 0)"
   268     and eq_divide_eq: "a = b / c \<longleftrightarrow> (if c \<noteq> 0 then a * c = b else a = 0)"
   269     and minus_divide_eq_eq: "- (b / c) = a \<longleftrightarrow> (if c \<noteq> 0 then - b = a * c else a = 0)"
   270     and eq_minus_divide_eq: "a = - (b / c) \<longleftrightarrow> (if c \<noteq> 0 then a * c = - b else a = 0)"
   271   by (auto simp add:  field_simps)
   272 
   273 end
   274 
   275 subsection \<open>Fields\<close>
   276 
   277 class field = comm_ring_1 + inverse +
   278   assumes field_inverse: "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
   279   assumes field_divide_inverse: "a / b = a * inverse b"
   280   assumes field_inverse_zero: "inverse 0 = 0"
   281 begin
   282 
   283 subclass division_ring
   284 proof
   285   fix a :: 'a
   286   assume "a \<noteq> 0"
   287   thus "inverse a * a = 1" by (rule field_inverse)
   288   thus "a * inverse a = 1" by (simp only: mult.commute)
   289 next
   290   fix a b :: 'a
   291   show "a / b = a * inverse b" by (rule field_divide_inverse)
   292 next
   293   show "inverse 0 = 0"
   294     by (fact field_inverse_zero) 
   295 qed
   296 
   297 subclass idom_divide
   298 proof
   299   fix b a
   300   assume "b \<noteq> 0"
   301   then show "a * b / b = a"
   302     by (simp add: divide_inverse ac_simps)
   303 next
   304   fix a
   305   show "a / 0 = 0"
   306     by (simp add: divide_inverse)
   307 qed
   308 
   309 text\<open>There is no slick version using division by zero.\<close>
   310 lemma inverse_add:
   311   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a + inverse b = (a + b) * inverse a * inverse b"
   312   by (simp add: division_ring_inverse_add ac_simps)
   313 
   314 lemma nonzero_mult_divide_mult_cancel_left [simp]:
   315   assumes [simp]: "c \<noteq> 0"
   316   shows "(c * a) / (c * b) = a / b"
   317 proof (cases "b = 0")
   318   case True then show ?thesis by simp
   319 next
   320   case False
   321   then have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
   322     by (simp add: divide_inverse nonzero_inverse_mult_distrib)
   323   also have "... =  a * inverse b * (inverse c * c)"
   324     by (simp only: ac_simps)
   325   also have "... =  a * inverse b" by simp
   326     finally show ?thesis by (simp add: divide_inverse)
   327 qed
   328 
   329 lemma nonzero_mult_divide_mult_cancel_right [simp]:
   330   "c \<noteq> 0 \<Longrightarrow> (a * c) / (b * c) = a / b"
   331   using nonzero_mult_divide_mult_cancel_left [of c a b] by (simp add: ac_simps)
   332 
   333 lemma times_divide_eq_left [simp]: "(b / c) * a = (b * a) / c"
   334   by (simp add: divide_inverse ac_simps)
   335 
   336 lemma divide_inverse_commute: "a / b = inverse b * a"
   337   by (simp add: divide_inverse mult.commute)
   338 
   339 lemma add_frac_eq:
   340   assumes "y \<noteq> 0" and "z \<noteq> 0"
   341   shows "x / y + w / z = (x * z + w * y) / (y * z)"
   342 proof -
   343   have "x / y + w / z = (x * z) / (y * z) + (y * w) / (y * z)"
   344     using assms by simp
   345   also have "\<dots> = (x * z + y * w) / (y * z)"
   346     by (simp only: add_divide_distrib)
   347   finally show ?thesis
   348     by (simp only: mult.commute)
   349 qed
   350 
   351 text\<open>Special Cancellation Simprules for Division\<close>
   352 
   353 lemma nonzero_divide_mult_cancel_right [simp]:
   354   "b \<noteq> 0 \<Longrightarrow> b / (a * b) = 1 / a"
   355   using nonzero_mult_divide_mult_cancel_right [of b 1 a] by simp
   356 
   357 lemma nonzero_divide_mult_cancel_left [simp]:
   358   "a \<noteq> 0 \<Longrightarrow> a / (a * b) = 1 / b"
   359   using nonzero_mult_divide_mult_cancel_left [of a 1 b] by simp
   360 
   361 lemma nonzero_mult_divide_mult_cancel_left2 [simp]:
   362   "c \<noteq> 0 \<Longrightarrow> (c * a) / (b * c) = a / b"
   363   using nonzero_mult_divide_mult_cancel_left [of c a b] by (simp add: ac_simps)
   364 
   365 lemma nonzero_mult_divide_mult_cancel_right2 [simp]:
   366   "c \<noteq> 0 \<Longrightarrow> (a * c) / (c * b) = a / b"
   367   using nonzero_mult_divide_mult_cancel_right [of b c a] by (simp add: ac_simps)
   368 
   369 lemma diff_frac_eq:
   370   "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y - w / z = (x * z - w * y) / (y * z)"
   371   by (simp add: field_simps)
   372 
   373 lemma frac_eq_eq:
   374   "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> (x / y = w / z) = (x * z = w * y)"
   375   by (simp add: field_simps)
   376 
   377 lemma divide_minus1 [simp]: "x / - 1 = - x"
   378   using nonzero_minus_divide_right [of "1" x] by simp
   379 
   380 text\<open>This version builds in division by zero while also re-orienting
   381       the right-hand side.\<close>
   382 lemma inverse_mult_distrib [simp]:
   383   "inverse (a * b) = inverse a * inverse b"
   384 proof cases
   385   assume "a \<noteq> 0 & b \<noteq> 0"
   386   thus ?thesis by (simp add: nonzero_inverse_mult_distrib ac_simps)
   387 next
   388   assume "~ (a \<noteq> 0 & b \<noteq> 0)"
   389   thus ?thesis by force
   390 qed
   391 
   392 lemma inverse_divide [simp]:
   393   "inverse (a / b) = b / a"
   394   by (simp add: divide_inverse mult.commute)
   395 
   396 
   397 text \<open>Calculations with fractions\<close>
   398 
   399 text\<open>There is a whole bunch of simp-rules just for class \<open>field\<close> but none for class \<open>field\<close> and \<open>nonzero_divides\<close>
   400 because the latter are covered by a simproc.\<close>
   401 
   402 lemma mult_divide_mult_cancel_left:
   403   "c \<noteq> 0 \<Longrightarrow> (c * a) / (c * b) = a / b"
   404 apply (cases "b = 0")
   405 apply simp_all
   406 done
   407 
   408 lemma mult_divide_mult_cancel_right:
   409   "c \<noteq> 0 \<Longrightarrow> (a * c) / (b * c) = a / b"
   410 apply (cases "b = 0")
   411 apply simp_all
   412 done
   413 
   414 lemma divide_divide_eq_right [simp]:
   415   "a / (b / c) = (a * c) / b"
   416   by (simp add: divide_inverse ac_simps)
   417 
   418 lemma divide_divide_eq_left [simp]:
   419   "(a / b) / c = a / (b * c)"
   420   by (simp add: divide_inverse mult.assoc)
   421 
   422 lemma divide_divide_times_eq:
   423   "(x / y) / (z / w) = (x * w) / (y * z)"
   424   by simp
   425 
   426 text \<open>Special Cancellation Simprules for Division\<close>
   427 
   428 lemma mult_divide_mult_cancel_left_if [simp]:
   429   shows "(c * a) / (c * b) = (if c = 0 then 0 else a / b)"
   430   by simp
   431 
   432 
   433 text \<open>Division and Unary Minus\<close>
   434 
   435 lemma minus_divide_right:
   436   "- (a / b) = a / - b"
   437   by (simp add: divide_inverse)
   438 
   439 lemma divide_minus_right [simp]:
   440   "a / - b = - (a / b)"
   441   by (simp add: divide_inverse)
   442 
   443 lemma minus_divide_divide:
   444   "(- a) / (- b) = a / b"
   445 apply (cases "b=0", simp)
   446 apply (simp add: nonzero_minus_divide_divide)
   447 done
   448 
   449 lemma inverse_eq_1_iff [simp]:
   450   "inverse x = 1 \<longleftrightarrow> x = 1"
   451   by (insert inverse_eq_iff_eq [of x 1], simp)
   452 
   453 lemma divide_eq_0_iff [simp]:
   454   "a / b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
   455   by (simp add: divide_inverse)
   456 
   457 lemma divide_cancel_right [simp]:
   458   "a / c = b / c \<longleftrightarrow> c = 0 \<or> a = b"
   459   apply (cases "c=0", simp)
   460   apply (simp add: divide_inverse)
   461   done
   462 
   463 lemma divide_cancel_left [simp]:
   464   "c / a = c / b \<longleftrightarrow> c = 0 \<or> a = b"
   465   apply (cases "c=0", simp)
   466   apply (simp add: divide_inverse)
   467   done
   468 
   469 lemma divide_eq_1_iff [simp]:
   470   "a / b = 1 \<longleftrightarrow> b \<noteq> 0 \<and> a = b"
   471   apply (cases "b=0", simp)
   472   apply (simp add: right_inverse_eq)
   473   done
   474 
   475 lemma one_eq_divide_iff [simp]:
   476   "1 = a / b \<longleftrightarrow> b \<noteq> 0 \<and> a = b"
   477   by (simp add: eq_commute [of 1])
   478 
   479 lemma times_divide_times_eq:
   480   "(x / y) * (z / w) = (x * z) / (y * w)"
   481   by simp
   482 
   483 lemma add_frac_num:
   484   "y \<noteq> 0 \<Longrightarrow> x / y + z = (x + z * y) / y"
   485   by (simp add: add_divide_distrib)
   486 
   487 lemma add_num_frac:
   488   "y \<noteq> 0 \<Longrightarrow> z + x / y = (x + z * y) / y"
   489   by (simp add: add_divide_distrib add.commute)
   490 
   491 end
   492 
   493 
   494 subsection \<open>Ordered fields\<close>
   495 
   496 class linordered_field = field + linordered_idom
   497 begin
   498 
   499 lemma positive_imp_inverse_positive:
   500   assumes a_gt_0: "0 < a"
   501   shows "0 < inverse a"
   502 proof -
   503   have "0 < a * inverse a"
   504     by (simp add: a_gt_0 [THEN less_imp_not_eq2])
   505   thus "0 < inverse a"
   506     by (simp add: a_gt_0 [THEN less_not_sym] zero_less_mult_iff)
   507 qed
   508 
   509 lemma negative_imp_inverse_negative:
   510   "a < 0 \<Longrightarrow> inverse a < 0"
   511   by (insert positive_imp_inverse_positive [of "-a"],
   512     simp add: nonzero_inverse_minus_eq less_imp_not_eq)
   513 
   514 lemma inverse_le_imp_le:
   515   assumes invle: "inverse a \<le> inverse b" and apos: "0 < a"
   516   shows "b \<le> a"
   517 proof (rule classical)
   518   assume "~ b \<le> a"
   519   hence "a < b"  by (simp add: linorder_not_le)
   520   hence bpos: "0 < b"  by (blast intro: apos less_trans)
   521   hence "a * inverse a \<le> a * inverse b"
   522     by (simp add: apos invle less_imp_le mult_left_mono)
   523   hence "(a * inverse a) * b \<le> (a * inverse b) * b"
   524     by (simp add: bpos less_imp_le mult_right_mono)
   525   thus "b \<le> a"  by (simp add: mult.assoc apos bpos less_imp_not_eq2)
   526 qed
   527 
   528 lemma inverse_positive_imp_positive:
   529   assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0"
   530   shows "0 < a"
   531 proof -
   532   have "0 < inverse (inverse a)"
   533     using inv_gt_0 by (rule positive_imp_inverse_positive)
   534   thus "0 < a"
   535     using nz by (simp add: nonzero_inverse_inverse_eq)
   536 qed
   537 
   538 lemma inverse_negative_imp_negative:
   539   assumes inv_less_0: "inverse a < 0" and nz: "a \<noteq> 0"
   540   shows "a < 0"
   541 proof -
   542   have "inverse (inverse a) < 0"
   543     using inv_less_0 by (rule negative_imp_inverse_negative)
   544   thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq)
   545 qed
   546 
   547 lemma linordered_field_no_lb:
   548   "\<forall>x. \<exists>y. y < x"
   549 proof
   550   fix x::'a
   551   have m1: "- (1::'a) < 0" by simp
   552   from add_strict_right_mono[OF m1, where c=x]
   553   have "(- 1) + x < x" by simp
   554   thus "\<exists>y. y < x" by blast
   555 qed
   556 
   557 lemma linordered_field_no_ub:
   558   "\<forall> x. \<exists>y. y > x"
   559 proof
   560   fix x::'a
   561   have m1: " (1::'a) > 0" by simp
   562   from add_strict_right_mono[OF m1, where c=x]
   563   have "1 + x > x" by simp
   564   thus "\<exists>y. y > x" by blast
   565 qed
   566 
   567 lemma less_imp_inverse_less:
   568   assumes less: "a < b" and apos:  "0 < a"
   569   shows "inverse b < inverse a"
   570 proof (rule ccontr)
   571   assume "~ inverse b < inverse a"
   572   hence "inverse a \<le> inverse b" by simp
   573   hence "~ (a < b)"
   574     by (simp add: not_less inverse_le_imp_le [OF _ apos])
   575   thus False by (rule notE [OF _ less])
   576 qed
   577 
   578 lemma inverse_less_imp_less:
   579   "inverse a < inverse b \<Longrightarrow> 0 < a \<Longrightarrow> b < a"
   580 apply (simp add: less_le [of "inverse a"] less_le [of "b"])
   581 apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq)
   582 done
   583 
   584 text\<open>Both premises are essential. Consider -1 and 1.\<close>
   585 lemma inverse_less_iff_less [simp]:
   586   "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"
   587   by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less)
   588 
   589 lemma le_imp_inverse_le:
   590   "a \<le> b \<Longrightarrow> 0 < a \<Longrightarrow> inverse b \<le> inverse a"
   591   by (force simp add: le_less less_imp_inverse_less)
   592 
   593 lemma inverse_le_iff_le [simp]:
   594   "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a"
   595   by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le)
   596 
   597 
   598 text\<open>These results refer to both operands being negative.  The opposite-sign
   599 case is trivial, since inverse preserves signs.\<close>
   600 lemma inverse_le_imp_le_neg:
   601   "inverse a \<le> inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b \<le> a"
   602 apply (rule classical)
   603 apply (subgoal_tac "a < 0")
   604  prefer 2 apply force
   605 apply (insert inverse_le_imp_le [of "-b" "-a"])
   606 apply (simp add: nonzero_inverse_minus_eq)
   607 done
   608 
   609 lemma less_imp_inverse_less_neg:
   610    "a < b \<Longrightarrow> b < 0 \<Longrightarrow> inverse b < inverse a"
   611 apply (subgoal_tac "a < 0")
   612  prefer 2 apply (blast intro: less_trans)
   613 apply (insert less_imp_inverse_less [of "-b" "-a"])
   614 apply (simp add: nonzero_inverse_minus_eq)
   615 done
   616 
   617 lemma inverse_less_imp_less_neg:
   618    "inverse a < inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b < a"
   619 apply (rule classical)
   620 apply (subgoal_tac "a < 0")
   621  prefer 2
   622  apply force
   623 apply (insert inverse_less_imp_less [of "-b" "-a"])
   624 apply (simp add: nonzero_inverse_minus_eq)
   625 done
   626 
   627 lemma inverse_less_iff_less_neg [simp]:
   628   "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"
   629 apply (insert inverse_less_iff_less [of "-b" "-a"])
   630 apply (simp del: inverse_less_iff_less
   631             add: nonzero_inverse_minus_eq)
   632 done
   633 
   634 lemma le_imp_inverse_le_neg:
   635   "a \<le> b \<Longrightarrow> b < 0 ==> inverse b \<le> inverse a"
   636   by (force simp add: le_less less_imp_inverse_less_neg)
   637 
   638 lemma inverse_le_iff_le_neg [simp]:
   639   "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a"
   640   by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg)
   641 
   642 lemma one_less_inverse:
   643   "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> 1 < inverse a"
   644   using less_imp_inverse_less [of a 1, unfolded inverse_1] .
   645 
   646 lemma one_le_inverse:
   647   "0 < a \<Longrightarrow> a \<le> 1 \<Longrightarrow> 1 \<le> inverse a"
   648   using le_imp_inverse_le [of a 1, unfolded inverse_1] .
   649 
   650 lemma pos_le_divide_eq [field_simps]:
   651   assumes "0 < c"
   652   shows "a \<le> b / c \<longleftrightarrow> a * c \<le> b"
   653 proof -
   654   from assms have "a \<le> b / c \<longleftrightarrow> a * c \<le> (b / c) * c"
   655     using mult_le_cancel_right [of a c "b * inverse c"] by (auto simp add: field_simps)
   656   also have "... \<longleftrightarrow> a * c \<le> b"
   657     by (simp add: less_imp_not_eq2 [OF assms] divide_inverse mult.assoc)
   658   finally show ?thesis .
   659 qed
   660 
   661 lemma pos_less_divide_eq [field_simps]:
   662   assumes "0 < c"
   663   shows "a < b / c \<longleftrightarrow> a * c < b"
   664 proof -
   665   from assms have "a < b / c \<longleftrightarrow> a * c < (b / c) * c"
   666     using mult_less_cancel_right [of a c "b / c"] by auto
   667   also have "... = (a*c < b)"
   668     by (simp add: less_imp_not_eq2 [OF assms] divide_inverse mult.assoc)
   669   finally show ?thesis .
   670 qed
   671 
   672 lemma neg_less_divide_eq [field_simps]:
   673   assumes "c < 0"
   674   shows "a < b / c \<longleftrightarrow> b < a * c"
   675 proof -
   676   from assms have "a < b / c \<longleftrightarrow> (b / c) * c < a * c"
   677     using mult_less_cancel_right [of "b / c" c a] by auto
   678   also have "... \<longleftrightarrow> b < a * c"
   679     by (simp add: less_imp_not_eq [OF assms] divide_inverse mult.assoc)
   680   finally show ?thesis .
   681 qed
   682 
   683 lemma neg_le_divide_eq [field_simps]:
   684   assumes "c < 0"
   685   shows "a \<le> b / c \<longleftrightarrow> b \<le> a * c"
   686 proof -
   687   from assms have "a \<le> b / c \<longleftrightarrow> (b / c) * c \<le> a * c"
   688     using mult_le_cancel_right [of "b * inverse c" c a] by (auto simp add: field_simps)
   689   also have "... \<longleftrightarrow> b \<le> a * c"
   690     by (simp add: less_imp_not_eq [OF assms] divide_inverse mult.assoc)
   691   finally show ?thesis .
   692 qed
   693 
   694 lemma pos_divide_le_eq [field_simps]:
   695   assumes "0 < c"
   696   shows "b / c \<le> a \<longleftrightarrow> b \<le> a * c"
   697 proof -
   698   from assms have "b / c \<le> a \<longleftrightarrow> (b / c) * c \<le> a * c"
   699     using mult_le_cancel_right [of "b / c" c a] by auto
   700   also have "... \<longleftrightarrow> b \<le> a * c"
   701     by (simp add: less_imp_not_eq2 [OF assms] divide_inverse mult.assoc)
   702   finally show ?thesis .
   703 qed
   704 
   705 lemma pos_divide_less_eq [field_simps]:
   706   assumes "0 < c"
   707   shows "b / c < a \<longleftrightarrow> b < a * c"
   708 proof -
   709   from assms have "b / c < a \<longleftrightarrow> (b / c) * c < a * c"
   710     using mult_less_cancel_right [of "b / c" c a] by auto
   711   also have "... \<longleftrightarrow> b < a * c"
   712     by (simp add: less_imp_not_eq2 [OF assms] divide_inverse mult.assoc)
   713   finally show ?thesis .
   714 qed
   715 
   716 lemma neg_divide_le_eq [field_simps]:
   717   assumes "c < 0"
   718   shows "b / c \<le> a \<longleftrightarrow> a * c \<le> b"
   719 proof -
   720   from assms have "b / c \<le> a \<longleftrightarrow> a * c \<le> (b / c) * c"
   721     using mult_le_cancel_right [of a c "b / c"] by auto
   722   also have "... \<longleftrightarrow> a * c \<le> b"
   723     by (simp add: less_imp_not_eq [OF assms] divide_inverse mult.assoc)
   724   finally show ?thesis .
   725 qed
   726 
   727 lemma neg_divide_less_eq [field_simps]:
   728   assumes "c < 0"
   729   shows "b / c < a \<longleftrightarrow> a * c < b"
   730 proof -
   731   from assms have "b / c < a \<longleftrightarrow> a * c < b / c * c"
   732     using mult_less_cancel_right [of a c "b / c"] by auto
   733   also have "... \<longleftrightarrow> a * c < b"
   734     by (simp add: less_imp_not_eq [OF assms] divide_inverse mult.assoc)
   735   finally show ?thesis .
   736 qed
   737 
   738 text\<open>The following \<open>field_simps\<close> rules are necessary, as minus is always moved atop of
   739 division but we want to get rid of division.\<close>
   740 
   741 lemma pos_le_minus_divide_eq [field_simps]: "0 < c \<Longrightarrow> a \<le> - (b / c) \<longleftrightarrow> a * c \<le> - b"
   742   unfolding minus_divide_left by (rule pos_le_divide_eq)
   743 
   744 lemma neg_le_minus_divide_eq [field_simps]: "c < 0 \<Longrightarrow> a \<le> - (b / c) \<longleftrightarrow> - b \<le> a * c"
   745   unfolding minus_divide_left by (rule neg_le_divide_eq)
   746 
   747 lemma pos_less_minus_divide_eq [field_simps]: "0 < c \<Longrightarrow> a < - (b / c) \<longleftrightarrow> a * c < - b"
   748   unfolding minus_divide_left by (rule pos_less_divide_eq)
   749 
   750 lemma neg_less_minus_divide_eq [field_simps]: "c < 0 \<Longrightarrow> a < - (b / c) \<longleftrightarrow> - b < a * c"
   751   unfolding minus_divide_left by (rule neg_less_divide_eq)
   752 
   753 lemma pos_minus_divide_less_eq [field_simps]: "0 < c \<Longrightarrow> - (b / c) < a \<longleftrightarrow> - b < a * c"
   754   unfolding minus_divide_left by (rule pos_divide_less_eq)
   755 
   756 lemma neg_minus_divide_less_eq [field_simps]: "c < 0 \<Longrightarrow> - (b / c) < a \<longleftrightarrow> a * c < - b"
   757   unfolding minus_divide_left by (rule neg_divide_less_eq)
   758 
   759 lemma pos_minus_divide_le_eq [field_simps]: "0 < c \<Longrightarrow> - (b / c) \<le> a \<longleftrightarrow> - b \<le> a * c"
   760   unfolding minus_divide_left by (rule pos_divide_le_eq)
   761 
   762 lemma neg_minus_divide_le_eq [field_simps]: "c < 0 \<Longrightarrow> - (b / c) \<le> a \<longleftrightarrow> a * c \<le> - b"
   763   unfolding minus_divide_left by (rule neg_divide_le_eq)
   764 
   765 lemma frac_less_eq:
   766   "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y < w / z \<longleftrightarrow> (x * z - w * y) / (y * z) < 0"
   767   by (subst less_iff_diff_less_0) (simp add: diff_frac_eq )
   768 
   769 lemma frac_le_eq:
   770   "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y \<le> w / z \<longleftrightarrow> (x * z - w * y) / (y * z) \<le> 0"
   771   by (subst le_iff_diff_le_0) (simp add: diff_frac_eq )
   772 
   773 text\<open>Lemmas \<open>sign_simps\<close> is a first attempt to automate proofs
   774 of positivity/negativity needed for \<open>field_simps\<close>. Have not added \<open>sign_simps\<close> to \<open>field_simps\<close> because the former can lead to case
   775 explosions.\<close>
   776 
   777 lemmas sign_simps = algebra_simps zero_less_mult_iff mult_less_0_iff
   778 
   779 lemmas (in -) sign_simps = algebra_simps zero_less_mult_iff mult_less_0_iff
   780 
   781 (* Only works once linear arithmetic is installed:
   782 text{*An example:*}
   783 lemma fixes a b c d e f :: "'a::linordered_field"
   784 shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow>
   785  ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) <
   786  ((e-f)*(a-b)*(c-d))/((e-f)*(a-b)*(c-d)) + u"
   787 apply(subgoal_tac "(c-d)*(e-f)*(a-b) > 0")
   788  prefer 2 apply(simp add:sign_simps)
   789 apply(subgoal_tac "(c-d)*(e-f)*(a-b)*u > 0")
   790  prefer 2 apply(simp add:sign_simps)
   791 apply(simp add:field_simps)
   792 done
   793 *)
   794 
   795 lemma divide_pos_pos[simp]:
   796   "0 < x ==> 0 < y ==> 0 < x / y"
   797 by(simp add:field_simps)
   798 
   799 lemma divide_nonneg_pos:
   800   "0 <= x ==> 0 < y ==> 0 <= x / y"
   801 by(simp add:field_simps)
   802 
   803 lemma divide_neg_pos:
   804   "x < 0 ==> 0 < y ==> x / y < 0"
   805 by(simp add:field_simps)
   806 
   807 lemma divide_nonpos_pos:
   808   "x <= 0 ==> 0 < y ==> x / y <= 0"
   809 by(simp add:field_simps)
   810 
   811 lemma divide_pos_neg:
   812   "0 < x ==> y < 0 ==> x / y < 0"
   813 by(simp add:field_simps)
   814 
   815 lemma divide_nonneg_neg:
   816   "0 <= x ==> y < 0 ==> x / y <= 0"
   817 by(simp add:field_simps)
   818 
   819 lemma divide_neg_neg:
   820   "x < 0 ==> y < 0 ==> 0 < x / y"
   821 by(simp add:field_simps)
   822 
   823 lemma divide_nonpos_neg:
   824   "x <= 0 ==> y < 0 ==> 0 <= x / y"
   825 by(simp add:field_simps)
   826 
   827 lemma divide_strict_right_mono:
   828      "[|a < b; 0 < c|] ==> a / c < b / c"
   829 by (simp add: less_imp_not_eq2 divide_inverse mult_strict_right_mono
   830               positive_imp_inverse_positive)
   831 
   832 
   833 lemma divide_strict_right_mono_neg:
   834      "[|b < a; c < 0|] ==> a / c < b / c"
   835 apply (drule divide_strict_right_mono [of _ _ "-c"], simp)
   836 apply (simp add: less_imp_not_eq nonzero_minus_divide_right [symmetric])
   837 done
   838 
   839 text\<open>The last premise ensures that @{term a} and @{term b}
   840       have the same sign\<close>
   841 lemma divide_strict_left_mono:
   842   "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / b"
   843   by (auto simp: field_simps zero_less_mult_iff mult_strict_right_mono)
   844 
   845 lemma divide_left_mono:
   846   "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / b"
   847   by (auto simp: field_simps zero_less_mult_iff mult_right_mono)
   848 
   849 lemma divide_strict_left_mono_neg:
   850   "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / b"
   851   by (auto simp: field_simps zero_less_mult_iff mult_strict_right_mono_neg)
   852 
   853 lemma mult_imp_div_pos_le: "0 < y ==> x <= z * y ==>
   854     x / y <= z"
   855 by (subst pos_divide_le_eq, assumption+)
   856 
   857 lemma mult_imp_le_div_pos: "0 < y ==> z * y <= x ==>
   858     z <= x / y"
   859 by(simp add:field_simps)
   860 
   861 lemma mult_imp_div_pos_less: "0 < y ==> x < z * y ==>
   862     x / y < z"
   863 by(simp add:field_simps)
   864 
   865 lemma mult_imp_less_div_pos: "0 < y ==> z * y < x ==>
   866     z < x / y"
   867 by(simp add:field_simps)
   868 
   869 lemma frac_le: "0 <= x ==>
   870     x <= y ==> 0 < w ==> w <= z  ==> x / z <= y / w"
   871   apply (rule mult_imp_div_pos_le)
   872   apply simp
   873   apply (subst times_divide_eq_left)
   874   apply (rule mult_imp_le_div_pos, assumption)
   875   apply (rule mult_mono)
   876   apply simp_all
   877 done
   878 
   879 lemma frac_less: "0 <= x ==>
   880     x < y ==> 0 < w ==> w <= z  ==> x / z < y / w"
   881   apply (rule mult_imp_div_pos_less)
   882   apply simp
   883   apply (subst times_divide_eq_left)
   884   apply (rule mult_imp_less_div_pos, assumption)
   885   apply (erule mult_less_le_imp_less)
   886   apply simp_all
   887 done
   888 
   889 lemma frac_less2: "0 < x ==>
   890     x <= y ==> 0 < w ==> w < z  ==> x / z < y / w"
   891   apply (rule mult_imp_div_pos_less)
   892   apply simp_all
   893   apply (rule mult_imp_less_div_pos, assumption)
   894   apply (erule mult_le_less_imp_less)
   895   apply simp_all
   896 done
   897 
   898 lemma less_half_sum: "a < b ==> a < (a+b) / (1+1)"
   899 by (simp add: field_simps zero_less_two)
   900 
   901 lemma gt_half_sum: "a < b ==> (a+b)/(1+1) < b"
   902 by (simp add: field_simps zero_less_two)
   903 
   904 subclass unbounded_dense_linorder
   905 proof
   906   fix x y :: 'a
   907   from less_add_one show "\<exists>y. x < y" ..
   908   from less_add_one have "x + (- 1) < (x + 1) + (- 1)" by (rule add_strict_right_mono)
   909   then have "x - 1 < x + 1 - 1" by simp
   910   then have "x - 1 < x" by (simp add: algebra_simps)
   911   then show "\<exists>y. y < x" ..
   912   show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum)
   913 qed
   914 
   915 lemma nonzero_abs_inverse:
   916      "a \<noteq> 0 ==> \<bar>inverse a\<bar> = inverse \<bar>a\<bar>"
   917 apply (auto simp add: neq_iff abs_if nonzero_inverse_minus_eq
   918                       negative_imp_inverse_negative)
   919 apply (blast intro: positive_imp_inverse_positive elim: less_asym)
   920 done
   921 
   922 lemma nonzero_abs_divide:
   923      "b \<noteq> 0 ==> \<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
   924   by (simp add: divide_inverse abs_mult nonzero_abs_inverse)
   925 
   926 lemma field_le_epsilon:
   927   assumes e: "\<And>e. 0 < e \<Longrightarrow> x \<le> y + e"
   928   shows "x \<le> y"
   929 proof (rule dense_le)
   930   fix t assume "t < x"
   931   hence "0 < x - t" by (simp add: less_diff_eq)
   932   from e [OF this] have "x + 0 \<le> x + (y - t)" by (simp add: algebra_simps)
   933   then have "0 \<le> y - t" by (simp only: add_le_cancel_left)
   934   then show "t \<le> y" by (simp add: algebra_simps)
   935 qed
   936 
   937 lemma inverse_positive_iff_positive [simp]:
   938   "(0 < inverse a) = (0 < a)"
   939 apply (cases "a = 0", simp)
   940 apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
   941 done
   942 
   943 lemma inverse_negative_iff_negative [simp]:
   944   "(inverse a < 0) = (a < 0)"
   945 apply (cases "a = 0", simp)
   946 apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
   947 done
   948 
   949 lemma inverse_nonnegative_iff_nonnegative [simp]:
   950   "0 \<le> inverse a \<longleftrightarrow> 0 \<le> a"
   951   by (simp add: not_less [symmetric])
   952 
   953 lemma inverse_nonpositive_iff_nonpositive [simp]:
   954   "inverse a \<le> 0 \<longleftrightarrow> a \<le> 0"
   955   by (simp add: not_less [symmetric])
   956 
   957 lemma one_less_inverse_iff: "1 < inverse x \<longleftrightarrow> 0 < x \<and> x < 1"
   958   using less_trans[of 1 x 0 for x]
   959   by (cases x 0 rule: linorder_cases) (auto simp add: field_simps)
   960 
   961 lemma one_le_inverse_iff: "1 \<le> inverse x \<longleftrightarrow> 0 < x \<and> x \<le> 1"
   962 proof (cases "x = 1")
   963   case True then show ?thesis by simp
   964 next
   965   case False then have "inverse x \<noteq> 1" by simp
   966   then have "1 \<noteq> inverse x" by blast
   967   then have "1 \<le> inverse x \<longleftrightarrow> 1 < inverse x" by (simp add: le_less)
   968   with False show ?thesis by (auto simp add: one_less_inverse_iff)
   969 qed
   970 
   971 lemma inverse_less_1_iff: "inverse x < 1 \<longleftrightarrow> x \<le> 0 \<or> 1 < x"
   972   by (simp add: not_le [symmetric] one_le_inverse_iff)
   973 
   974 lemma inverse_le_1_iff: "inverse x \<le> 1 \<longleftrightarrow> x \<le> 0 \<or> 1 \<le> x"
   975   by (simp add: not_less [symmetric] one_less_inverse_iff)
   976 
   977 lemma [divide_simps]:
   978   shows le_divide_eq: "a \<le> b / c \<longleftrightarrow> (if 0 < c then a * c \<le> b else if c < 0 then b \<le> a * c else a \<le> 0)"
   979     and divide_le_eq: "b / c \<le> a \<longleftrightarrow> (if 0 < c then b \<le> a * c else if c < 0 then a * c \<le> b else 0 \<le> a)"
   980     and less_divide_eq: "a < b / c \<longleftrightarrow> (if 0 < c then a * c < b else if c < 0 then b < a * c else a < 0)"
   981     and divide_less_eq: "b / c < a \<longleftrightarrow> (if 0 < c then b < a * c else if c < 0 then a * c < b else 0 < a)"
   982     and le_minus_divide_eq: "a \<le> - (b / c) \<longleftrightarrow> (if 0 < c then a * c \<le> - b else if c < 0 then - b \<le> a * c else a \<le> 0)"
   983     and minus_divide_le_eq: "- (b / c) \<le> a \<longleftrightarrow> (if 0 < c then - b \<le> a * c else if c < 0 then a * c \<le> - b else 0 \<le> a)"
   984     and less_minus_divide_eq: "a < - (b / c) \<longleftrightarrow> (if 0 < c then a * c < - b else if c < 0 then - b < a * c else  a < 0)"
   985     and minus_divide_less_eq: "- (b / c) < a \<longleftrightarrow> (if 0 < c then - b < a * c else if c < 0 then a * c < - b else 0 < a)"
   986   by (auto simp: field_simps not_less dest: antisym)
   987 
   988 text \<open>Division and Signs\<close>
   989 
   990 lemma
   991   shows zero_less_divide_iff: "0 < a / b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
   992     and divide_less_0_iff: "a / b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
   993     and zero_le_divide_iff: "0 \<le> a / b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0"
   994     and divide_le_0_iff: "a / b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
   995   by (auto simp add: divide_simps)
   996 
   997 text \<open>Division and the Number One\<close>
   998 
   999 text\<open>Simplify expressions equated with 1\<close>
  1000 
  1001 lemma zero_eq_1_divide_iff [simp]: "0 = 1 / a \<longleftrightarrow> a = 0"
  1002   by (cases "a = 0") (auto simp: field_simps)
  1003 
  1004 lemma one_divide_eq_0_iff [simp]: "1 / a = 0 \<longleftrightarrow> a = 0"
  1005   using zero_eq_1_divide_iff[of a] by simp
  1006 
  1007 text\<open>Simplify expressions such as \<open>0 < 1/x\<close> to \<open>0 < x\<close>\<close>
  1008 
  1009 lemma zero_le_divide_1_iff [simp]:
  1010   "0 \<le> 1 / a \<longleftrightarrow> 0 \<le> a"
  1011   by (simp add: zero_le_divide_iff)
  1012 
  1013 lemma zero_less_divide_1_iff [simp]:
  1014   "0 < 1 / a \<longleftrightarrow> 0 < a"
  1015   by (simp add: zero_less_divide_iff)
  1016 
  1017 lemma divide_le_0_1_iff [simp]:
  1018   "1 / a \<le> 0 \<longleftrightarrow> a \<le> 0"
  1019   by (simp add: divide_le_0_iff)
  1020 
  1021 lemma divide_less_0_1_iff [simp]:
  1022   "1 / a < 0 \<longleftrightarrow> a < 0"
  1023   by (simp add: divide_less_0_iff)
  1024 
  1025 lemma divide_right_mono:
  1026      "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/c"
  1027 by (force simp add: divide_strict_right_mono le_less)
  1028 
  1029 lemma divide_right_mono_neg: "a <= b
  1030     ==> c <= 0 ==> b / c <= a / c"
  1031 apply (drule divide_right_mono [of _ _ "- c"])
  1032 apply auto
  1033 done
  1034 
  1035 lemma divide_left_mono_neg: "a <= b
  1036     ==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"
  1037   apply (drule divide_left_mono [of _ _ "- c"])
  1038   apply (auto simp add: mult.commute)
  1039 done
  1040 
  1041 lemma inverse_le_iff: "inverse a \<le> inverse b \<longleftrightarrow> (0 < a * b \<longrightarrow> b \<le> a) \<and> (a * b \<le> 0 \<longrightarrow> a \<le> b)"
  1042   by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])
  1043      (auto simp add: field_simps zero_less_mult_iff mult_le_0_iff)
  1044 
  1045 lemma inverse_less_iff: "inverse a < inverse b \<longleftrightarrow> (0 < a * b \<longrightarrow> b < a) \<and> (a * b \<le> 0 \<longrightarrow> a < b)"
  1046   by (subst less_le) (auto simp: inverse_le_iff)
  1047 
  1048 lemma divide_le_cancel: "a / c \<le> b / c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
  1049   by (simp add: divide_inverse mult_le_cancel_right)
  1050 
  1051 lemma divide_less_cancel: "a / c < b / c \<longleftrightarrow> (0 < c \<longrightarrow> a < b) \<and> (c < 0 \<longrightarrow> b < a) \<and> c \<noteq> 0"
  1052   by (auto simp add: divide_inverse mult_less_cancel_right)
  1053 
  1054 text\<open>Simplify quotients that are compared with the value 1.\<close>
  1055 
  1056 lemma le_divide_eq_1:
  1057   "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"
  1058 by (auto simp add: le_divide_eq)
  1059 
  1060 lemma divide_le_eq_1:
  1061   "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"
  1062 by (auto simp add: divide_le_eq)
  1063 
  1064 lemma less_divide_eq_1:
  1065   "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
  1066 by (auto simp add: less_divide_eq)
  1067 
  1068 lemma divide_less_eq_1:
  1069   "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
  1070 by (auto simp add: divide_less_eq)
  1071 
  1072 lemma divide_nonneg_nonneg [simp]:
  1073   "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> x / y"
  1074   by (auto simp add: divide_simps)
  1075 
  1076 lemma divide_nonpos_nonpos:
  1077   "x \<le> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> 0 \<le> x / y"
  1078   by (auto simp add: divide_simps)
  1079 
  1080 lemma divide_nonneg_nonpos:
  1081   "0 \<le> x \<Longrightarrow> y \<le> 0 \<Longrightarrow> x / y \<le> 0"
  1082   by (auto simp add: divide_simps)
  1083 
  1084 lemma divide_nonpos_nonneg:
  1085   "x \<le> 0 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x / y \<le> 0"
  1086   by (auto simp add: divide_simps)
  1087 
  1088 text \<open>Conditional Simplification Rules: No Case Splits\<close>
  1089 
  1090 lemma le_divide_eq_1_pos [simp]:
  1091   "0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)"
  1092 by (auto simp add: le_divide_eq)
  1093 
  1094 lemma le_divide_eq_1_neg [simp]:
  1095   "a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)"
  1096 by (auto simp add: le_divide_eq)
  1097 
  1098 lemma divide_le_eq_1_pos [simp]:
  1099   "0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)"
  1100 by (auto simp add: divide_le_eq)
  1101 
  1102 lemma divide_le_eq_1_neg [simp]:
  1103   "a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)"
  1104 by (auto simp add: divide_le_eq)
  1105 
  1106 lemma less_divide_eq_1_pos [simp]:
  1107   "0 < a \<Longrightarrow> (1 < b/a) = (a < b)"
  1108 by (auto simp add: less_divide_eq)
  1109 
  1110 lemma less_divide_eq_1_neg [simp]:
  1111   "a < 0 \<Longrightarrow> (1 < b/a) = (b < a)"
  1112 by (auto simp add: less_divide_eq)
  1113 
  1114 lemma divide_less_eq_1_pos [simp]:
  1115   "0 < a \<Longrightarrow> (b/a < 1) = (b < a)"
  1116 by (auto simp add: divide_less_eq)
  1117 
  1118 lemma divide_less_eq_1_neg [simp]:
  1119   "a < 0 \<Longrightarrow> b/a < 1 \<longleftrightarrow> a < b"
  1120 by (auto simp add: divide_less_eq)
  1121 
  1122 lemma eq_divide_eq_1 [simp]:
  1123   "(1 = b/a) = ((a \<noteq> 0 & a = b))"
  1124 by (auto simp add: eq_divide_eq)
  1125 
  1126 lemma divide_eq_eq_1 [simp]:
  1127   "(b/a = 1) = ((a \<noteq> 0 & a = b))"
  1128 by (auto simp add: divide_eq_eq)
  1129 
  1130 lemma abs_inverse [simp]:
  1131      "\<bar>inverse a\<bar> =
  1132       inverse \<bar>a\<bar>"
  1133 apply (cases "a=0", simp)
  1134 apply (simp add: nonzero_abs_inverse)
  1135 done
  1136 
  1137 lemma abs_divide [simp]:
  1138      "\<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
  1139 apply (cases "b=0", simp)
  1140 apply (simp add: nonzero_abs_divide)
  1141 done
  1142 
  1143 lemma abs_div_pos: "0 < y ==>
  1144     \<bar>x\<bar> / y = \<bar>x / y\<bar>"
  1145   apply (subst abs_divide)
  1146   apply (simp add: order_less_imp_le)
  1147 done
  1148 
  1149 lemma zero_le_divide_abs_iff [simp]: "(0 \<le> a / \<bar>b\<bar>) = (0 \<le> a | b = 0)"
  1150 by (auto simp: zero_le_divide_iff)
  1151 
  1152 lemma divide_le_0_abs_iff [simp]: "(a / \<bar>b\<bar> \<le> 0) = (a \<le> 0 | b = 0)"
  1153 by (auto simp: divide_le_0_iff)
  1154 
  1155 lemma inverse_sgn:
  1156   "sgn (inverse a) = inverse (sgn a)"
  1157   by (simp add: sgn_if)
  1158 
  1159 lemma field_le_mult_one_interval:
  1160   assumes *: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y"
  1161   shows "x \<le> y"
  1162 proof (cases "0 < x")
  1163   assume "0 < x"
  1164   thus ?thesis
  1165     using dense_le_bounded[of 0 1 "y/x"] *
  1166     unfolding le_divide_eq if_P[OF \<open>0 < x\<close>] by simp
  1167 next
  1168   assume "\<not>0 < x" hence "x \<le> 0" by simp
  1169   obtain s::'a where s: "0 < s" "s < 1" using dense[of 0 "1::'a"] by auto
  1170   hence "x \<le> s * x" using mult_le_cancel_right[of 1 x s] \<open>x \<le> 0\<close> by auto
  1171   also note *[OF s]
  1172   finally show ?thesis .
  1173 qed
  1174 
  1175 end
  1176 
  1177 text \<open>Min/max Simplification Rules\<close>
  1178 
  1179 lemma min_mult_distrib_left:
  1180   fixes x::"'a::linordered_idom" 
  1181   shows "p * min x y = (if 0 \<le> p then min (p*x) (p*y) else max (p*x) (p*y))"
  1182 by (auto simp add: min_def max_def mult_le_cancel_left)
  1183 
  1184 lemma min_mult_distrib_right:
  1185   fixes x::"'a::linordered_idom" 
  1186   shows "min x y * p = (if 0 \<le> p then min (x*p) (y*p) else max (x*p) (y*p))"
  1187 by (auto simp add: min_def max_def mult_le_cancel_right)
  1188 
  1189 lemma min_divide_distrib_right:
  1190   fixes x::"'a::linordered_field" 
  1191   shows "min x y / p = (if 0 \<le> p then min (x/p) (y/p) else max (x/p) (y/p))"
  1192 by (simp add: min_mult_distrib_right divide_inverse)
  1193 
  1194 lemma max_mult_distrib_left:
  1195   fixes x::"'a::linordered_idom" 
  1196   shows "p * max x y = (if 0 \<le> p then max (p*x) (p*y) else min (p*x) (p*y))"
  1197 by (auto simp add: min_def max_def mult_le_cancel_left)
  1198 
  1199 lemma max_mult_distrib_right:
  1200   fixes x::"'a::linordered_idom" 
  1201   shows "max x y * p = (if 0 \<le> p then max (x*p) (y*p) else min (x*p) (y*p))"
  1202 by (auto simp add: min_def max_def mult_le_cancel_right)
  1203 
  1204 lemma max_divide_distrib_right:
  1205   fixes x::"'a::linordered_field" 
  1206   shows "max x y / p = (if 0 \<le> p then max (x/p) (y/p) else min (x/p) (y/p))"
  1207 by (simp add: max_mult_distrib_right divide_inverse)
  1208 
  1209 hide_fact (open) field_inverse field_divide_inverse field_inverse_zero
  1210 
  1211 code_identifier
  1212   code_module Fields \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
  1213 
  1214 end