src/HOL/Fields.thy
 author hoelzl Fri May 20 21:38:32 2011 +0200 (2011-05-20) changeset 42904 4aedcff42de6 parent 36774 9e444b09fbef child 44064 5bce8ff0d9ae permissions -rw-r--r--
```     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 header {* Fields *}
```
```    11
```
```    12 theory Fields
```
```    13 imports Rings
```
```    14 begin
```
```    15
```
```    16 class field = comm_ring_1 + inverse +
```
```    17   assumes field_inverse: "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
```
```    18   assumes field_divide_inverse: "a / b = a * inverse b"
```
```    19 begin
```
```    20
```
```    21 subclass division_ring
```
```    22 proof
```
```    23   fix a :: 'a
```
```    24   assume "a \<noteq> 0"
```
```    25   thus "inverse a * a = 1" by (rule field_inverse)
```
```    26   thus "a * inverse a = 1" by (simp only: mult_commute)
```
```    27 next
```
```    28   fix a b :: 'a
```
```    29   show "a / b = a * inverse b" by (rule field_divide_inverse)
```
```    30 qed
```
```    31
```
```    32 subclass idom ..
```
```    33
```
```    34 text{*There is no slick version using division by zero.*}
```
```    35 lemma inverse_add:
```
```    36   "[| a \<noteq> 0;  b \<noteq> 0 |]
```
```    37    ==> inverse a + inverse b = (a + b) * inverse a * inverse b"
```
```    38 by (simp add: division_ring_inverse_add mult_ac)
```
```    39
```
```    40 lemma nonzero_mult_divide_mult_cancel_left [simp, no_atp]:
```
```    41 assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" shows "(c*a)/(c*b) = a/b"
```
```    42 proof -
```
```    43   have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
```
```    44     by (simp add: divide_inverse nonzero_inverse_mult_distrib)
```
```    45   also have "... =  a * inverse b * (inverse c * c)"
```
```    46     by (simp only: mult_ac)
```
```    47   also have "... =  a * inverse b" by simp
```
```    48     finally show ?thesis by (simp add: divide_inverse)
```
```    49 qed
```
```    50
```
```    51 lemma nonzero_mult_divide_mult_cancel_right [simp, no_atp]:
```
```    52   "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (a * c) / (b * c) = a / b"
```
```    53 by (simp add: mult_commute [of _ c])
```
```    54
```
```    55 lemma times_divide_eq_left [simp]: "(b / c) * a = (b * a) / c"
```
```    56   by (simp add: divide_inverse mult_ac)
```
```    57
```
```    58 text {* These are later declared as simp rules. *}
```
```    59 lemmas times_divide_eq [no_atp] = times_divide_eq_right times_divide_eq_left
```
```    60
```
```    61 lemma add_frac_eq:
```
```    62   assumes "y \<noteq> 0" and "z \<noteq> 0"
```
```    63   shows "x / y + w / z = (x * z + w * y) / (y * z)"
```
```    64 proof -
```
```    65   have "x / y + w / z = (x * z) / (y * z) + (y * w) / (y * z)"
```
```    66     using assms by simp
```
```    67   also have "\<dots> = (x * z + y * w) / (y * z)"
```
```    68     by (simp only: add_divide_distrib)
```
```    69   finally show ?thesis
```
```    70     by (simp only: mult_commute)
```
```    71 qed
```
```    72
```
```    73 text{*Special Cancellation Simprules for Division*}
```
```    74
```
```    75 lemma nonzero_mult_divide_cancel_right [simp, no_atp]:
```
```    76   "b \<noteq> 0 \<Longrightarrow> a * b / b = a"
```
```    77   using nonzero_mult_divide_mult_cancel_right [of 1 b a] by simp
```
```    78
```
```    79 lemma nonzero_mult_divide_cancel_left [simp, no_atp]:
```
```    80   "a \<noteq> 0 \<Longrightarrow> a * b / a = b"
```
```    81 using nonzero_mult_divide_mult_cancel_left [of 1 a b] by simp
```
```    82
```
```    83 lemma nonzero_divide_mult_cancel_right [simp, no_atp]:
```
```    84   "\<lbrakk>a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> b / (a * b) = 1 / a"
```
```    85 using nonzero_mult_divide_mult_cancel_right [of a b 1] by simp
```
```    86
```
```    87 lemma nonzero_divide_mult_cancel_left [simp, no_atp]:
```
```    88   "\<lbrakk>a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / (a * b) = 1 / b"
```
```    89 using nonzero_mult_divide_mult_cancel_left [of b a 1] by simp
```
```    90
```
```    91 lemma nonzero_mult_divide_mult_cancel_left2 [simp, no_atp]:
```
```    92   "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (c * a) / (b * c) = a / b"
```
```    93 using nonzero_mult_divide_mult_cancel_left [of b c a] by (simp add: mult_ac)
```
```    94
```
```    95 lemma nonzero_mult_divide_mult_cancel_right2 [simp, no_atp]:
```
```    96   "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (a * c) / (c * b) = a / b"
```
```    97 using nonzero_mult_divide_mult_cancel_right [of b c a] by (simp add: mult_ac)
```
```    98
```
```    99 lemma add_divide_eq_iff [field_simps]:
```
```   100   "z \<noteq> 0 \<Longrightarrow> x + y / z = (z * x + y) / z"
```
```   101   by (simp add: add_divide_distrib)
```
```   102
```
```   103 lemma divide_add_eq_iff [field_simps]:
```
```   104   "z \<noteq> 0 \<Longrightarrow> x / z + y = (x + z * y) / z"
```
```   105   by (simp add: add_divide_distrib)
```
```   106
```
```   107 lemma diff_divide_eq_iff [field_simps]:
```
```   108   "z \<noteq> 0 \<Longrightarrow> x - y / z = (z * x - y) / z"
```
```   109   by (simp add: diff_divide_distrib)
```
```   110
```
```   111 lemma divide_diff_eq_iff [field_simps]:
```
```   112   "z \<noteq> 0 \<Longrightarrow> x / z - y = (x - z * y) / z"
```
```   113   by (simp add: diff_divide_distrib)
```
```   114
```
```   115 lemma diff_frac_eq:
```
```   116   "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y - w / z = (x * z - w * y) / (y * z)"
```
```   117   by (simp add: field_simps)
```
```   118
```
```   119 lemma frac_eq_eq:
```
```   120   "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> (x / y = w / z) = (x * z = w * y)"
```
```   121   by (simp add: field_simps)
```
```   122
```
```   123 end
```
```   124
```
```   125 class field_inverse_zero = field +
```
```   126   assumes field_inverse_zero: "inverse 0 = 0"
```
```   127 begin
```
```   128
```
```   129 subclass division_ring_inverse_zero proof
```
```   130 qed (fact field_inverse_zero)
```
```   131
```
```   132 text{*This version builds in division by zero while also re-orienting
```
```   133       the right-hand side.*}
```
```   134 lemma inverse_mult_distrib [simp]:
```
```   135   "inverse (a * b) = inverse a * inverse b"
```
```   136 proof cases
```
```   137   assume "a \<noteq> 0 & b \<noteq> 0"
```
```   138   thus ?thesis by (simp add: nonzero_inverse_mult_distrib mult_ac)
```
```   139 next
```
```   140   assume "~ (a \<noteq> 0 & b \<noteq> 0)"
```
```   141   thus ?thesis by force
```
```   142 qed
```
```   143
```
```   144 lemma inverse_divide [simp]:
```
```   145   "inverse (a / b) = b / a"
```
```   146   by (simp add: divide_inverse mult_commute)
```
```   147
```
```   148
```
```   149 text {* Calculations with fractions *}
```
```   150
```
```   151 text{* There is a whole bunch of simp-rules just for class @{text
```
```   152 field} but none for class @{text field} and @{text nonzero_divides}
```
```   153 because the latter are covered by a simproc. *}
```
```   154
```
```   155 lemma mult_divide_mult_cancel_left:
```
```   156   "c \<noteq> 0 \<Longrightarrow> (c * a) / (c * b) = a / b"
```
```   157 apply (cases "b = 0")
```
```   158 apply simp_all
```
```   159 done
```
```   160
```
```   161 lemma mult_divide_mult_cancel_right:
```
```   162   "c \<noteq> 0 \<Longrightarrow> (a * c) / (b * c) = a / b"
```
```   163 apply (cases "b = 0")
```
```   164 apply simp_all
```
```   165 done
```
```   166
```
```   167 lemma divide_divide_eq_right [simp, no_atp]:
```
```   168   "a / (b / c) = (a * c) / b"
```
```   169   by (simp add: divide_inverse mult_ac)
```
```   170
```
```   171 lemma divide_divide_eq_left [simp, no_atp]:
```
```   172   "(a / b) / c = a / (b * c)"
```
```   173   by (simp add: divide_inverse mult_assoc)
```
```   174
```
```   175
```
```   176 text {*Special Cancellation Simprules for Division*}
```
```   177
```
```   178 lemma mult_divide_mult_cancel_left_if [simp,no_atp]:
```
```   179   shows "(c * a) / (c * b) = (if c = 0 then 0 else a / b)"
```
```   180   by (simp add: mult_divide_mult_cancel_left)
```
```   181
```
```   182
```
```   183 text {* Division and Unary Minus *}
```
```   184
```
```   185 lemma minus_divide_right:
```
```   186   "- (a / b) = a / - b"
```
```   187   by (simp add: divide_inverse)
```
```   188
```
```   189 lemma divide_minus_right [simp, no_atp]:
```
```   190   "a / - b = - (a / b)"
```
```   191   by (simp add: divide_inverse)
```
```   192
```
```   193 lemma minus_divide_divide:
```
```   194   "(- a) / (- b) = a / b"
```
```   195 apply (cases "b=0", simp)
```
```   196 apply (simp add: nonzero_minus_divide_divide)
```
```   197 done
```
```   198
```
```   199 lemma eq_divide_eq:
```
```   200   "a = b / c \<longleftrightarrow> (if c \<noteq> 0 then a * c = b else a = 0)"
```
```   201   by (simp add: nonzero_eq_divide_eq)
```
```   202
```
```   203 lemma divide_eq_eq:
```
```   204   "b / c = a \<longleftrightarrow> (if c \<noteq> 0 then b = a * c else a = 0)"
```
```   205   by (force simp add: nonzero_divide_eq_eq)
```
```   206
```
```   207 lemma inverse_eq_1_iff [simp]:
```
```   208   "inverse x = 1 \<longleftrightarrow> x = 1"
```
```   209   by (insert inverse_eq_iff_eq [of x 1], simp)
```
```   210
```
```   211 lemma divide_eq_0_iff [simp, no_atp]:
```
```   212   "a / b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
```
```   213   by (simp add: divide_inverse)
```
```   214
```
```   215 lemma divide_cancel_right [simp, no_atp]:
```
```   216   "a / c = b / c \<longleftrightarrow> c = 0 \<or> a = b"
```
```   217   apply (cases "c=0", simp)
```
```   218   apply (simp add: divide_inverse)
```
```   219   done
```
```   220
```
```   221 lemma divide_cancel_left [simp, no_atp]:
```
```   222   "c / a = c / b \<longleftrightarrow> c = 0 \<or> a = b"
```
```   223   apply (cases "c=0", simp)
```
```   224   apply (simp add: divide_inverse)
```
```   225   done
```
```   226
```
```   227 lemma divide_eq_1_iff [simp, no_atp]:
```
```   228   "a / b = 1 \<longleftrightarrow> b \<noteq> 0 \<and> a = b"
```
```   229   apply (cases "b=0", simp)
```
```   230   apply (simp add: right_inverse_eq)
```
```   231   done
```
```   232
```
```   233 lemma one_eq_divide_iff [simp, no_atp]:
```
```   234   "1 = a / b \<longleftrightarrow> b \<noteq> 0 \<and> a = b"
```
```   235   by (simp add: eq_commute [of 1])
```
```   236
```
```   237 lemma times_divide_times_eq:
```
```   238   "(x / y) * (z / w) = (x * z) / (y * w)"
```
```   239   by simp
```
```   240
```
```   241 lemma add_frac_num:
```
```   242   "y \<noteq> 0 \<Longrightarrow> x / y + z = (x + z * y) / y"
```
```   243   by (simp add: add_divide_distrib)
```
```   244
```
```   245 lemma add_num_frac:
```
```   246   "y \<noteq> 0 \<Longrightarrow> z + x / y = (x + z * y) / y"
```
```   247   by (simp add: add_divide_distrib add.commute)
```
```   248
```
```   249 end
```
```   250
```
```   251
```
```   252 text {* Ordered Fields *}
```
```   253
```
```   254 class linordered_field = field + linordered_idom
```
```   255 begin
```
```   256
```
```   257 lemma positive_imp_inverse_positive:
```
```   258   assumes a_gt_0: "0 < a"
```
```   259   shows "0 < inverse a"
```
```   260 proof -
```
```   261   have "0 < a * inverse a"
```
```   262     by (simp add: a_gt_0 [THEN less_imp_not_eq2])
```
```   263   thus "0 < inverse a"
```
```   264     by (simp add: a_gt_0 [THEN less_not_sym] zero_less_mult_iff)
```
```   265 qed
```
```   266
```
```   267 lemma negative_imp_inverse_negative:
```
```   268   "a < 0 \<Longrightarrow> inverse a < 0"
```
```   269   by (insert positive_imp_inverse_positive [of "-a"],
```
```   270     simp add: nonzero_inverse_minus_eq less_imp_not_eq)
```
```   271
```
```   272 lemma inverse_le_imp_le:
```
```   273   assumes invle: "inverse a \<le> inverse b" and apos: "0 < a"
```
```   274   shows "b \<le> a"
```
```   275 proof (rule classical)
```
```   276   assume "~ b \<le> a"
```
```   277   hence "a < b"  by (simp add: linorder_not_le)
```
```   278   hence bpos: "0 < b"  by (blast intro: apos less_trans)
```
```   279   hence "a * inverse a \<le> a * inverse b"
```
```   280     by (simp add: apos invle less_imp_le mult_left_mono)
```
```   281   hence "(a * inverse a) * b \<le> (a * inverse b) * b"
```
```   282     by (simp add: bpos less_imp_le mult_right_mono)
```
```   283   thus "b \<le> a"  by (simp add: mult_assoc apos bpos less_imp_not_eq2)
```
```   284 qed
```
```   285
```
```   286 lemma inverse_positive_imp_positive:
```
```   287   assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0"
```
```   288   shows "0 < a"
```
```   289 proof -
```
```   290   have "0 < inverse (inverse a)"
```
```   291     using inv_gt_0 by (rule positive_imp_inverse_positive)
```
```   292   thus "0 < a"
```
```   293     using nz by (simp add: nonzero_inverse_inverse_eq)
```
```   294 qed
```
```   295
```
```   296 lemma inverse_negative_imp_negative:
```
```   297   assumes inv_less_0: "inverse a < 0" and nz: "a \<noteq> 0"
```
```   298   shows "a < 0"
```
```   299 proof -
```
```   300   have "inverse (inverse a) < 0"
```
```   301     using inv_less_0 by (rule negative_imp_inverse_negative)
```
```   302   thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq)
```
```   303 qed
```
```   304
```
```   305 lemma linordered_field_no_lb:
```
```   306   "\<forall>x. \<exists>y. y < x"
```
```   307 proof
```
```   308   fix x::'a
```
```   309   have m1: "- (1::'a) < 0" by simp
```
```   310   from add_strict_right_mono[OF m1, where c=x]
```
```   311   have "(- 1) + x < x" by simp
```
```   312   thus "\<exists>y. y < x" by blast
```
```   313 qed
```
```   314
```
```   315 lemma linordered_field_no_ub:
```
```   316   "\<forall> x. \<exists>y. y > x"
```
```   317 proof
```
```   318   fix x::'a
```
```   319   have m1: " (1::'a) > 0" by simp
```
```   320   from add_strict_right_mono[OF m1, where c=x]
```
```   321   have "1 + x > x" by simp
```
```   322   thus "\<exists>y. y > x" by blast
```
```   323 qed
```
```   324
```
```   325 lemma less_imp_inverse_less:
```
```   326   assumes less: "a < b" and apos:  "0 < a"
```
```   327   shows "inverse b < inverse a"
```
```   328 proof (rule ccontr)
```
```   329   assume "~ inverse b < inverse a"
```
```   330   hence "inverse a \<le> inverse b" by simp
```
```   331   hence "~ (a < b)"
```
```   332     by (simp add: not_less inverse_le_imp_le [OF _ apos])
```
```   333   thus False by (rule notE [OF _ less])
```
```   334 qed
```
```   335
```
```   336 lemma inverse_less_imp_less:
```
```   337   "inverse a < inverse b \<Longrightarrow> 0 < a \<Longrightarrow> b < a"
```
```   338 apply (simp add: less_le [of "inverse a"] less_le [of "b"])
```
```   339 apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq)
```
```   340 done
```
```   341
```
```   342 text{*Both premises are essential. Consider -1 and 1.*}
```
```   343 lemma inverse_less_iff_less [simp,no_atp]:
```
```   344   "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"
```
```   345   by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less)
```
```   346
```
```   347 lemma le_imp_inverse_le:
```
```   348   "a \<le> b \<Longrightarrow> 0 < a \<Longrightarrow> inverse b \<le> inverse a"
```
```   349   by (force simp add: le_less less_imp_inverse_less)
```
```   350
```
```   351 lemma inverse_le_iff_le [simp,no_atp]:
```
```   352   "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a"
```
```   353   by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le)
```
```   354
```
```   355
```
```   356 text{*These results refer to both operands being negative.  The opposite-sign
```
```   357 case is trivial, since inverse preserves signs.*}
```
```   358 lemma inverse_le_imp_le_neg:
```
```   359   "inverse a \<le> inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b \<le> a"
```
```   360 apply (rule classical)
```
```   361 apply (subgoal_tac "a < 0")
```
```   362  prefer 2 apply force
```
```   363 apply (insert inverse_le_imp_le [of "-b" "-a"])
```
```   364 apply (simp add: nonzero_inverse_minus_eq)
```
```   365 done
```
```   366
```
```   367 lemma less_imp_inverse_less_neg:
```
```   368    "a < b \<Longrightarrow> b < 0 \<Longrightarrow> inverse b < inverse a"
```
```   369 apply (subgoal_tac "a < 0")
```
```   370  prefer 2 apply (blast intro: less_trans)
```
```   371 apply (insert less_imp_inverse_less [of "-b" "-a"])
```
```   372 apply (simp add: nonzero_inverse_minus_eq)
```
```   373 done
```
```   374
```
```   375 lemma inverse_less_imp_less_neg:
```
```   376    "inverse a < inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b < a"
```
```   377 apply (rule classical)
```
```   378 apply (subgoal_tac "a < 0")
```
```   379  prefer 2
```
```   380  apply force
```
```   381 apply (insert inverse_less_imp_less [of "-b" "-a"])
```
```   382 apply (simp add: nonzero_inverse_minus_eq)
```
```   383 done
```
```   384
```
```   385 lemma inverse_less_iff_less_neg [simp,no_atp]:
```
```   386   "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"
```
```   387 apply (insert inverse_less_iff_less [of "-b" "-a"])
```
```   388 apply (simp del: inverse_less_iff_less
```
```   389             add: nonzero_inverse_minus_eq)
```
```   390 done
```
```   391
```
```   392 lemma le_imp_inverse_le_neg:
```
```   393   "a \<le> b \<Longrightarrow> b < 0 ==> inverse b \<le> inverse a"
```
```   394   by (force simp add: le_less less_imp_inverse_less_neg)
```
```   395
```
```   396 lemma inverse_le_iff_le_neg [simp,no_atp]:
```
```   397   "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a"
```
```   398   by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg)
```
```   399
```
```   400 lemma one_less_inverse:
```
```   401   "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> 1 < inverse a"
```
```   402   using less_imp_inverse_less [of a 1, unfolded inverse_1] .
```
```   403
```
```   404 lemma one_le_inverse:
```
```   405   "0 < a \<Longrightarrow> a \<le> 1 \<Longrightarrow> 1 \<le> inverse a"
```
```   406   using le_imp_inverse_le [of a 1, unfolded inverse_1] .
```
```   407
```
```   408 lemma pos_le_divide_eq [field_simps]: "0 < c ==> (a \<le> b/c) = (a*c \<le> b)"
```
```   409 proof -
```
```   410   assume less: "0<c"
```
```   411   hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
```
```   412     by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
```
```   413   also have "... = (a*c \<le> b)"
```
```   414     by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc)
```
```   415   finally show ?thesis .
```
```   416 qed
```
```   417
```
```   418 lemma neg_le_divide_eq [field_simps]: "c < 0 ==> (a \<le> b/c) = (b \<le> a*c)"
```
```   419 proof -
```
```   420   assume less: "c<0"
```
```   421   hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
```
```   422     by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
```
```   423   also have "... = (b \<le> a*c)"
```
```   424     by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc)
```
```   425   finally show ?thesis .
```
```   426 qed
```
```   427
```
```   428 lemma pos_less_divide_eq [field_simps]:
```
```   429      "0 < c ==> (a < b/c) = (a*c < b)"
```
```   430 proof -
```
```   431   assume less: "0<c"
```
```   432   hence "(a < b/c) = (a*c < (b/c)*c)"
```
```   433     by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq)
```
```   434   also have "... = (a*c < b)"
```
```   435     by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc)
```
```   436   finally show ?thesis .
```
```   437 qed
```
```   438
```
```   439 lemma neg_less_divide_eq [field_simps]:
```
```   440  "c < 0 ==> (a < b/c) = (b < a*c)"
```
```   441 proof -
```
```   442   assume less: "c<0"
```
```   443   hence "(a < b/c) = ((b/c)*c < a*c)"
```
```   444     by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq)
```
```   445   also have "... = (b < a*c)"
```
```   446     by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc)
```
```   447   finally show ?thesis .
```
```   448 qed
```
```   449
```
```   450 lemma pos_divide_less_eq [field_simps]:
```
```   451      "0 < c ==> (b/c < a) = (b < a*c)"
```
```   452 proof -
```
```   453   assume less: "0<c"
```
```   454   hence "(b/c < a) = ((b/c)*c < a*c)"
```
```   455     by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq)
```
```   456   also have "... = (b < a*c)"
```
```   457     by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc)
```
```   458   finally show ?thesis .
```
```   459 qed
```
```   460
```
```   461 lemma neg_divide_less_eq [field_simps]:
```
```   462  "c < 0 ==> (b/c < a) = (a*c < b)"
```
```   463 proof -
```
```   464   assume less: "c<0"
```
```   465   hence "(b/c < a) = (a*c < (b/c)*c)"
```
```   466     by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq)
```
```   467   also have "... = (a*c < b)"
```
```   468     by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc)
```
```   469   finally show ?thesis .
```
```   470 qed
```
```   471
```
```   472 lemma pos_divide_le_eq [field_simps]: "0 < c ==> (b/c \<le> a) = (b \<le> a*c)"
```
```   473 proof -
```
```   474   assume less: "0<c"
```
```   475   hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
```
```   476     by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
```
```   477   also have "... = (b \<le> a*c)"
```
```   478     by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc)
```
```   479   finally show ?thesis .
```
```   480 qed
```
```   481
```
```   482 lemma neg_divide_le_eq [field_simps]: "c < 0 ==> (b/c \<le> a) = (a*c \<le> b)"
```
```   483 proof -
```
```   484   assume less: "c<0"
```
```   485   hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
```
```   486     by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
```
```   487   also have "... = (a*c \<le> b)"
```
```   488     by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc)
```
```   489   finally show ?thesis .
```
```   490 qed
```
```   491
```
```   492 text{* Lemmas @{text sign_simps} is a first attempt to automate proofs
```
```   493 of positivity/negativity needed for @{text field_simps}. Have not added @{text
```
```   494 sign_simps} to @{text field_simps} because the former can lead to case
```
```   495 explosions. *}
```
```   496
```
```   497 lemmas sign_simps [no_atp] = algebra_simps
```
```   498   zero_less_mult_iff mult_less_0_iff
```
```   499
```
```   500 lemmas (in -) sign_simps [no_atp] = algebra_simps
```
```   501   zero_less_mult_iff mult_less_0_iff
```
```   502
```
```   503 (* Only works once linear arithmetic is installed:
```
```   504 text{*An example:*}
```
```   505 lemma fixes a b c d e f :: "'a::linordered_field"
```
```   506 shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow>
```
```   507  ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) <
```
```   508  ((e-f)*(a-b)*(c-d))/((e-f)*(a-b)*(c-d)) + u"
```
```   509 apply(subgoal_tac "(c-d)*(e-f)*(a-b) > 0")
```
```   510  prefer 2 apply(simp add:sign_simps)
```
```   511 apply(subgoal_tac "(c-d)*(e-f)*(a-b)*u > 0")
```
```   512  prefer 2 apply(simp add:sign_simps)
```
```   513 apply(simp add:field_simps)
```
```   514 done
```
```   515 *)
```
```   516
```
```   517 lemma divide_pos_pos:
```
```   518   "0 < x ==> 0 < y ==> 0 < x / y"
```
```   519 by(simp add:field_simps)
```
```   520
```
```   521 lemma divide_nonneg_pos:
```
```   522   "0 <= x ==> 0 < y ==> 0 <= x / y"
```
```   523 by(simp add:field_simps)
```
```   524
```
```   525 lemma divide_neg_pos:
```
```   526   "x < 0 ==> 0 < y ==> x / y < 0"
```
```   527 by(simp add:field_simps)
```
```   528
```
```   529 lemma divide_nonpos_pos:
```
```   530   "x <= 0 ==> 0 < y ==> x / y <= 0"
```
```   531 by(simp add:field_simps)
```
```   532
```
```   533 lemma divide_pos_neg:
```
```   534   "0 < x ==> y < 0 ==> x / y < 0"
```
```   535 by(simp add:field_simps)
```
```   536
```
```   537 lemma divide_nonneg_neg:
```
```   538   "0 <= x ==> y < 0 ==> x / y <= 0"
```
```   539 by(simp add:field_simps)
```
```   540
```
```   541 lemma divide_neg_neg:
```
```   542   "x < 0 ==> y < 0 ==> 0 < x / y"
```
```   543 by(simp add:field_simps)
```
```   544
```
```   545 lemma divide_nonpos_neg:
```
```   546   "x <= 0 ==> y < 0 ==> 0 <= x / y"
```
```   547 by(simp add:field_simps)
```
```   548
```
```   549 lemma divide_strict_right_mono:
```
```   550      "[|a < b; 0 < c|] ==> a / c < b / c"
```
```   551 by (simp add: less_imp_not_eq2 divide_inverse mult_strict_right_mono
```
```   552               positive_imp_inverse_positive)
```
```   553
```
```   554
```
```   555 lemma divide_strict_right_mono_neg:
```
```   556      "[|b < a; c < 0|] ==> a / c < b / c"
```
```   557 apply (drule divide_strict_right_mono [of _ _ "-c"], simp)
```
```   558 apply (simp add: less_imp_not_eq nonzero_minus_divide_right [symmetric])
```
```   559 done
```
```   560
```
```   561 text{*The last premise ensures that @{term a} and @{term b}
```
```   562       have the same sign*}
```
```   563 lemma divide_strict_left_mono:
```
```   564   "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / b"
```
```   565 by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono)
```
```   566
```
```   567 lemma divide_left_mono:
```
```   568   "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / b"
```
```   569 by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_right_mono)
```
```   570
```
```   571 lemma divide_strict_left_mono_neg:
```
```   572   "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / b"
```
```   573 by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono_neg)
```
```   574
```
```   575 lemma mult_imp_div_pos_le: "0 < y ==> x <= z * y ==>
```
```   576     x / y <= z"
```
```   577 by (subst pos_divide_le_eq, assumption+)
```
```   578
```
```   579 lemma mult_imp_le_div_pos: "0 < y ==> z * y <= x ==>
```
```   580     z <= x / y"
```
```   581 by(simp add:field_simps)
```
```   582
```
```   583 lemma mult_imp_div_pos_less: "0 < y ==> x < z * y ==>
```
```   584     x / y < z"
```
```   585 by(simp add:field_simps)
```
```   586
```
```   587 lemma mult_imp_less_div_pos: "0 < y ==> z * y < x ==>
```
```   588     z < x / y"
```
```   589 by(simp add:field_simps)
```
```   590
```
```   591 lemma frac_le: "0 <= x ==>
```
```   592     x <= y ==> 0 < w ==> w <= z  ==> x / z <= y / w"
```
```   593   apply (rule mult_imp_div_pos_le)
```
```   594   apply simp
```
```   595   apply (subst times_divide_eq_left)
```
```   596   apply (rule mult_imp_le_div_pos, assumption)
```
```   597   apply (rule mult_mono)
```
```   598   apply simp_all
```
```   599 done
```
```   600
```
```   601 lemma frac_less: "0 <= x ==>
```
```   602     x < y ==> 0 < w ==> w <= z  ==> x / z < y / w"
```
```   603   apply (rule mult_imp_div_pos_less)
```
```   604   apply simp
```
```   605   apply (subst times_divide_eq_left)
```
```   606   apply (rule mult_imp_less_div_pos, assumption)
```
```   607   apply (erule mult_less_le_imp_less)
```
```   608   apply simp_all
```
```   609 done
```
```   610
```
```   611 lemma frac_less2: "0 < x ==>
```
```   612     x <= y ==> 0 < w ==> w < z  ==> x / z < y / w"
```
```   613   apply (rule mult_imp_div_pos_less)
```
```   614   apply simp_all
```
```   615   apply (rule mult_imp_less_div_pos, assumption)
```
```   616   apply (erule mult_le_less_imp_less)
```
```   617   apply simp_all
```
```   618 done
```
```   619
```
```   620 text{*It's not obvious whether these should be simprules or not.
```
```   621   Their effect is to gather terms into one big fraction, like
```
```   622   a*b*c / x*y*z. The rationale for that is unclear, but many proofs
```
```   623   seem to need them.*}
```
```   624
```
```   625 lemma less_half_sum: "a < b ==> a < (a+b) / (1+1)"
```
```   626 by (simp add: field_simps zero_less_two)
```
```   627
```
```   628 lemma gt_half_sum: "a < b ==> (a+b)/(1+1) < b"
```
```   629 by (simp add: field_simps zero_less_two)
```
```   630
```
```   631 subclass dense_linorder
```
```   632 proof
```
```   633   fix x y :: 'a
```
```   634   from less_add_one show "\<exists>y. x < y" ..
```
```   635   from less_add_one have "x + (- 1) < (x + 1) + (- 1)" by (rule add_strict_right_mono)
```
```   636   then have "x - 1 < x + 1 - 1" by (simp only: diff_minus [symmetric])
```
```   637   then have "x - 1 < x" by (simp add: algebra_simps)
```
```   638   then show "\<exists>y. y < x" ..
```
```   639   show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum)
```
```   640 qed
```
```   641
```
```   642 lemma nonzero_abs_inverse:
```
```   643      "a \<noteq> 0 ==> \<bar>inverse a\<bar> = inverse \<bar>a\<bar>"
```
```   644 apply (auto simp add: neq_iff abs_if nonzero_inverse_minus_eq
```
```   645                       negative_imp_inverse_negative)
```
```   646 apply (blast intro: positive_imp_inverse_positive elim: less_asym)
```
```   647 done
```
```   648
```
```   649 lemma nonzero_abs_divide:
```
```   650      "b \<noteq> 0 ==> \<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
```
```   651   by (simp add: divide_inverse abs_mult nonzero_abs_inverse)
```
```   652
```
```   653 lemma field_le_epsilon:
```
```   654   assumes e: "\<And>e. 0 < e \<Longrightarrow> x \<le> y + e"
```
```   655   shows "x \<le> y"
```
```   656 proof (rule dense_le)
```
```   657   fix t assume "t < x"
```
```   658   hence "0 < x - t" by (simp add: less_diff_eq)
```
```   659   from e [OF this] have "x + 0 \<le> x + (y - t)" by (simp add: algebra_simps)
```
```   660   then have "0 \<le> y - t" by (simp only: add_le_cancel_left)
```
```   661   then show "t \<le> y" by (simp add: algebra_simps)
```
```   662 qed
```
```   663
```
```   664 end
```
```   665
```
```   666 class linordered_field_inverse_zero = linordered_field + field_inverse_zero
```
```   667 begin
```
```   668
```
```   669 lemma le_divide_eq:
```
```   670   "(a \<le> b/c) =
```
```   671    (if 0 < c then a*c \<le> b
```
```   672              else if c < 0 then b \<le> a*c
```
```   673              else  a \<le> 0)"
```
```   674 apply (cases "c=0", simp)
```
```   675 apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff)
```
```   676 done
```
```   677
```
```   678 lemma inverse_positive_iff_positive [simp]:
```
```   679   "(0 < inverse a) = (0 < a)"
```
```   680 apply (cases "a = 0", simp)
```
```   681 apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
```
```   682 done
```
```   683
```
```   684 lemma inverse_negative_iff_negative [simp]:
```
```   685   "(inverse a < 0) = (a < 0)"
```
```   686 apply (cases "a = 0", simp)
```
```   687 apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
```
```   688 done
```
```   689
```
```   690 lemma inverse_nonnegative_iff_nonnegative [simp]:
```
```   691   "0 \<le> inverse a \<longleftrightarrow> 0 \<le> a"
```
```   692   by (simp add: not_less [symmetric])
```
```   693
```
```   694 lemma inverse_nonpositive_iff_nonpositive [simp]:
```
```   695   "inverse a \<le> 0 \<longleftrightarrow> a \<le> 0"
```
```   696   by (simp add: not_less [symmetric])
```
```   697
```
```   698 lemma one_less_inverse_iff:
```
```   699   "1 < inverse x \<longleftrightarrow> 0 < x \<and> x < 1"
```
```   700 proof cases
```
```   701   assume "0 < x"
```
```   702     with inverse_less_iff_less [OF zero_less_one, of x]
```
```   703     show ?thesis by simp
```
```   704 next
```
```   705   assume notless: "~ (0 < x)"
```
```   706   have "~ (1 < inverse x)"
```
```   707   proof
```
```   708     assume "1 < inverse x"
```
```   709     also with notless have "... \<le> 0" by simp
```
```   710     also have "... < 1" by (rule zero_less_one)
```
```   711     finally show False by auto
```
```   712   qed
```
```   713   with notless show ?thesis by simp
```
```   714 qed
```
```   715
```
```   716 lemma one_le_inverse_iff:
```
```   717   "1 \<le> inverse x \<longleftrightarrow> 0 < x \<and> x \<le> 1"
```
```   718 proof (cases "x = 1")
```
```   719   case True then show ?thesis by simp
```
```   720 next
```
```   721   case False then have "inverse x \<noteq> 1" by simp
```
```   722   then have "1 \<noteq> inverse x" by blast
```
```   723   then have "1 \<le> inverse x \<longleftrightarrow> 1 < inverse x" by (simp add: le_less)
```
```   724   with False show ?thesis by (auto simp add: one_less_inverse_iff)
```
```   725 qed
```
```   726
```
```   727 lemma inverse_less_1_iff:
```
```   728   "inverse x < 1 \<longleftrightarrow> x \<le> 0 \<or> 1 < x"
```
```   729   by (simp add: not_le [symmetric] one_le_inverse_iff)
```
```   730
```
```   731 lemma inverse_le_1_iff:
```
```   732   "inverse x \<le> 1 \<longleftrightarrow> x \<le> 0 \<or> 1 \<le> x"
```
```   733   by (simp add: not_less [symmetric] one_less_inverse_iff)
```
```   734
```
```   735 lemma divide_le_eq:
```
```   736   "(b/c \<le> a) =
```
```   737    (if 0 < c then b \<le> a*c
```
```   738              else if c < 0 then a*c \<le> b
```
```   739              else 0 \<le> a)"
```
```   740 apply (cases "c=0", simp)
```
```   741 apply (force simp add: pos_divide_le_eq neg_divide_le_eq)
```
```   742 done
```
```   743
```
```   744 lemma less_divide_eq:
```
```   745   "(a < b/c) =
```
```   746    (if 0 < c then a*c < b
```
```   747              else if c < 0 then b < a*c
```
```   748              else  a < 0)"
```
```   749 apply (cases "c=0", simp)
```
```   750 apply (force simp add: pos_less_divide_eq neg_less_divide_eq)
```
```   751 done
```
```   752
```
```   753 lemma divide_less_eq:
```
```   754   "(b/c < a) =
```
```   755    (if 0 < c then b < a*c
```
```   756              else if c < 0 then a*c < b
```
```   757              else 0 < a)"
```
```   758 apply (cases "c=0", simp)
```
```   759 apply (force simp add: pos_divide_less_eq neg_divide_less_eq)
```
```   760 done
```
```   761
```
```   762 text {*Division and Signs*}
```
```   763
```
```   764 lemma zero_less_divide_iff:
```
```   765      "(0 < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
```
```   766 by (simp add: divide_inverse zero_less_mult_iff)
```
```   767
```
```   768 lemma divide_less_0_iff:
```
```   769      "(a/b < 0) =
```
```   770       (0 < a & b < 0 | a < 0 & 0 < b)"
```
```   771 by (simp add: divide_inverse mult_less_0_iff)
```
```   772
```
```   773 lemma zero_le_divide_iff:
```
```   774      "(0 \<le> a/b) =
```
```   775       (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
```
```   776 by (simp add: divide_inverse zero_le_mult_iff)
```
```   777
```
```   778 lemma divide_le_0_iff:
```
```   779      "(a/b \<le> 0) =
```
```   780       (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
```
```   781 by (simp add: divide_inverse mult_le_0_iff)
```
```   782
```
```   783 text {* Division and the Number One *}
```
```   784
```
```   785 text{*Simplify expressions equated with 1*}
```
```   786
```
```   787 lemma zero_eq_1_divide_iff [simp,no_atp]:
```
```   788      "(0 = 1/a) = (a = 0)"
```
```   789 apply (cases "a=0", simp)
```
```   790 apply (auto simp add: nonzero_eq_divide_eq)
```
```   791 done
```
```   792
```
```   793 lemma one_divide_eq_0_iff [simp,no_atp]:
```
```   794      "(1/a = 0) = (a = 0)"
```
```   795 apply (cases "a=0", simp)
```
```   796 apply (insert zero_neq_one [THEN not_sym])
```
```   797 apply (auto simp add: nonzero_divide_eq_eq)
```
```   798 done
```
```   799
```
```   800 text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}
```
```   801
```
```   802 lemma zero_le_divide_1_iff [simp, no_atp]:
```
```   803   "0 \<le> 1 / a \<longleftrightarrow> 0 \<le> a"
```
```   804   by (simp add: zero_le_divide_iff)
```
```   805
```
```   806 lemma zero_less_divide_1_iff [simp, no_atp]:
```
```   807   "0 < 1 / a \<longleftrightarrow> 0 < a"
```
```   808   by (simp add: zero_less_divide_iff)
```
```   809
```
```   810 lemma divide_le_0_1_iff [simp, no_atp]:
```
```   811   "1 / a \<le> 0 \<longleftrightarrow> a \<le> 0"
```
```   812   by (simp add: divide_le_0_iff)
```
```   813
```
```   814 lemma divide_less_0_1_iff [simp, no_atp]:
```
```   815   "1 / a < 0 \<longleftrightarrow> a < 0"
```
```   816   by (simp add: divide_less_0_iff)
```
```   817
```
```   818 lemma divide_right_mono:
```
```   819      "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/c"
```
```   820 by (force simp add: divide_strict_right_mono le_less)
```
```   821
```
```   822 lemma divide_right_mono_neg: "a <= b
```
```   823     ==> c <= 0 ==> b / c <= a / c"
```
```   824 apply (drule divide_right_mono [of _ _ "- c"])
```
```   825 apply auto
```
```   826 done
```
```   827
```
```   828 lemma divide_left_mono_neg: "a <= b
```
```   829     ==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"
```
```   830   apply (drule divide_left_mono [of _ _ "- c"])
```
```   831   apply (auto simp add: mult_commute)
```
```   832 done
```
```   833
```
```   834 lemma inverse_le_iff:
```
```   835   "inverse a \<le> inverse b \<longleftrightarrow> (0 < a * b \<longrightarrow> b \<le> a) \<and> (a * b \<le> 0 \<longrightarrow> a \<le> b)"
```
```   836 proof -
```
```   837   { assume "a < 0"
```
```   838     then have "inverse a < 0" by simp
```
```   839     moreover assume "0 < b"
```
```   840     then have "0 < inverse b" by simp
```
```   841     ultimately have "inverse a < inverse b" by (rule less_trans)
```
```   842     then have "inverse a \<le> inverse b" by simp }
```
```   843   moreover
```
```   844   { assume "b < 0"
```
```   845     then have "inverse b < 0" by simp
```
```   846     moreover assume "0 < a"
```
```   847     then have "0 < inverse a" by simp
```
```   848     ultimately have "inverse b < inverse a" by (rule less_trans)
```
```   849     then have "\<not> inverse a \<le> inverse b" by simp }
```
```   850   ultimately show ?thesis
```
```   851     by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])
```
```   852        (auto simp: not_less zero_less_mult_iff mult_le_0_iff)
```
```   853 qed
```
```   854
```
```   855 lemma inverse_less_iff:
```
```   856   "inverse a < inverse b \<longleftrightarrow> (0 < a * b \<longrightarrow> b < a) \<and> (a * b \<le> 0 \<longrightarrow> a < b)"
```
```   857   by (subst less_le) (auto simp: inverse_le_iff)
```
```   858
```
```   859 lemma divide_le_cancel:
```
```   860   "a / c \<le> b / c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
```
```   861   by (simp add: divide_inverse mult_le_cancel_right)
```
```   862
```
```   863 lemma divide_less_cancel:
```
```   864   "a / c < b / c \<longleftrightarrow> (0 < c \<longrightarrow> a < b) \<and> (c < 0 \<longrightarrow> b < a) \<and> c \<noteq> 0"
```
```   865   by (auto simp add: divide_inverse mult_less_cancel_right)
```
```   866
```
```   867 text{*Simplify quotients that are compared with the value 1.*}
```
```   868
```
```   869 lemma le_divide_eq_1 [no_atp]:
```
```   870   "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"
```
```   871 by (auto simp add: le_divide_eq)
```
```   872
```
```   873 lemma divide_le_eq_1 [no_atp]:
```
```   874   "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"
```
```   875 by (auto simp add: divide_le_eq)
```
```   876
```
```   877 lemma less_divide_eq_1 [no_atp]:
```
```   878   "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
```
```   879 by (auto simp add: less_divide_eq)
```
```   880
```
```   881 lemma divide_less_eq_1 [no_atp]:
```
```   882   "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
```
```   883 by (auto simp add: divide_less_eq)
```
```   884
```
```   885
```
```   886 text {*Conditional Simplification Rules: No Case Splits*}
```
```   887
```
```   888 lemma le_divide_eq_1_pos [simp,no_atp]:
```
```   889   "0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)"
```
```   890 by (auto simp add: le_divide_eq)
```
```   891
```
```   892 lemma le_divide_eq_1_neg [simp,no_atp]:
```
```   893   "a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)"
```
```   894 by (auto simp add: le_divide_eq)
```
```   895
```
```   896 lemma divide_le_eq_1_pos [simp,no_atp]:
```
```   897   "0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)"
```
```   898 by (auto simp add: divide_le_eq)
```
```   899
```
```   900 lemma divide_le_eq_1_neg [simp,no_atp]:
```
```   901   "a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)"
```
```   902 by (auto simp add: divide_le_eq)
```
```   903
```
```   904 lemma less_divide_eq_1_pos [simp,no_atp]:
```
```   905   "0 < a \<Longrightarrow> (1 < b/a) = (a < b)"
```
```   906 by (auto simp add: less_divide_eq)
```
```   907
```
```   908 lemma less_divide_eq_1_neg [simp,no_atp]:
```
```   909   "a < 0 \<Longrightarrow> (1 < b/a) = (b < a)"
```
```   910 by (auto simp add: less_divide_eq)
```
```   911
```
```   912 lemma divide_less_eq_1_pos [simp,no_atp]:
```
```   913   "0 < a \<Longrightarrow> (b/a < 1) = (b < a)"
```
```   914 by (auto simp add: divide_less_eq)
```
```   915
```
```   916 lemma divide_less_eq_1_neg [simp,no_atp]:
```
```   917   "a < 0 \<Longrightarrow> b/a < 1 <-> a < b"
```
```   918 by (auto simp add: divide_less_eq)
```
```   919
```
```   920 lemma eq_divide_eq_1 [simp,no_atp]:
```
```   921   "(1 = b/a) = ((a \<noteq> 0 & a = b))"
```
```   922 by (auto simp add: eq_divide_eq)
```
```   923
```
```   924 lemma divide_eq_eq_1 [simp,no_atp]:
```
```   925   "(b/a = 1) = ((a \<noteq> 0 & a = b))"
```
```   926 by (auto simp add: divide_eq_eq)
```
```   927
```
```   928 lemma abs_inverse [simp]:
```
```   929      "\<bar>inverse a\<bar> =
```
```   930       inverse \<bar>a\<bar>"
```
```   931 apply (cases "a=0", simp)
```
```   932 apply (simp add: nonzero_abs_inverse)
```
```   933 done
```
```   934
```
```   935 lemma abs_divide [simp]:
```
```   936      "\<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
```
```   937 apply (cases "b=0", simp)
```
```   938 apply (simp add: nonzero_abs_divide)
```
```   939 done
```
```   940
```
```   941 lemma abs_div_pos: "0 < y ==>
```
```   942     \<bar>x\<bar> / y = \<bar>x / y\<bar>"
```
```   943   apply (subst abs_divide)
```
```   944   apply (simp add: order_less_imp_le)
```
```   945 done
```
```   946
```
```   947 lemma field_le_mult_one_interval:
```
```   948   assumes *: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y"
```
```   949   shows "x \<le> y"
```
```   950 proof (cases "0 < x")
```
```   951   assume "0 < x"
```
```   952   thus ?thesis
```
```   953     using dense_le_bounded[of 0 1 "y/x"] *
```
```   954     unfolding le_divide_eq if_P[OF `0 < x`] by simp
```
```   955 next
```
```   956   assume "\<not>0 < x" hence "x \<le> 0" by simp
```
```   957   obtain s::'a where s: "0 < s" "s < 1" using dense[of 0 "1\<Colon>'a"] by auto
```
```   958   hence "x \<le> s * x" using mult_le_cancel_right[of 1 x s] `x \<le> 0` by auto
```
```   959   also note *[OF s]
```
```   960   finally show ?thesis .
```
```   961 qed
```
```   962
```
```   963 end
```
```   964
```
```   965 code_modulename SML
```
```   966   Fields Arith
```
```   967
```
```   968 code_modulename OCaml
```
```   969   Fields Arith
```
```   970
```
```   971 code_modulename Haskell
```
```   972   Fields Arith
```
```   973
```
```   974 end
```