src/HOL/Fields.thy
 author wenzelm Mon Mar 22 20:58:52 2010 +0100 (2010-03-22) changeset 35898 c890a3835d15 parent 35828 46cfc4b8112e child 36301 72f4d079ebf8 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 lemma right_inverse_eq: "b \<noteq> 0 \<Longrightarrow> a / b = 1 \<longleftrightarrow> a = b"
```
```    35 proof
```
```    36   assume neq: "b \<noteq> 0"
```
```    37   {
```
```    38     hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac)
```
```    39     also assume "a / b = 1"
```
```    40     finally show "a = b" by simp
```
```    41   next
```
```    42     assume "a = b"
```
```    43     with neq show "a / b = 1" by (simp add: divide_inverse)
```
```    44   }
```
```    45 qed
```
```    46
```
```    47 lemma nonzero_inverse_eq_divide: "a \<noteq> 0 \<Longrightarrow> inverse a = 1 / a"
```
```    48 by (simp add: divide_inverse)
```
```    49
```
```    50 lemma divide_self [simp]: "a \<noteq> 0 \<Longrightarrow> a / a = 1"
```
```    51 by (simp add: divide_inverse)
```
```    52
```
```    53 lemma divide_zero_left [simp]: "0 / a = 0"
```
```    54 by (simp add: divide_inverse)
```
```    55
```
```    56 lemma inverse_eq_divide: "inverse a = 1 / a"
```
```    57 by (simp add: divide_inverse)
```
```    58
```
```    59 lemma add_divide_distrib: "(a+b) / c = a/c + b/c"
```
```    60 by (simp add: divide_inverse algebra_simps)
```
```    61
```
```    62 text{*There is no slick version using division by zero.*}
```
```    63 lemma inverse_add:
```
```    64   "[| a \<noteq> 0;  b \<noteq> 0 |]
```
```    65    ==> inverse a + inverse b = (a + b) * inverse a * inverse b"
```
```    66 by (simp add: division_ring_inverse_add mult_ac)
```
```    67
```
```    68 lemma nonzero_mult_divide_mult_cancel_left [simp, no_atp]:
```
```    69 assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" shows "(c*a)/(c*b) = a/b"
```
```    70 proof -
```
```    71   have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
```
```    72     by (simp add: divide_inverse nonzero_inverse_mult_distrib)
```
```    73   also have "... =  a * inverse b * (inverse c * c)"
```
```    74     by (simp only: mult_ac)
```
```    75   also have "... =  a * inverse b" by simp
```
```    76     finally show ?thesis by (simp add: divide_inverse)
```
```    77 qed
```
```    78
```
```    79 lemma nonzero_mult_divide_mult_cancel_right [simp, no_atp]:
```
```    80   "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (a * c) / (b * c) = a / b"
```
```    81 by (simp add: mult_commute [of _ c])
```
```    82
```
```    83 lemma divide_1 [simp]: "a / 1 = a"
```
```    84 by (simp add: divide_inverse)
```
```    85
```
```    86 lemma times_divide_eq_right: "a * (b / c) = (a * b) / c"
```
```    87 by (simp add: divide_inverse mult_assoc)
```
```    88
```
```    89 lemma times_divide_eq_left: "(b / c) * a = (b * a) / c"
```
```    90 by (simp add: divide_inverse mult_ac)
```
```    91
```
```    92 text {* These are later declared as simp rules. *}
```
```    93 lemmas times_divide_eq [no_atp] = times_divide_eq_right times_divide_eq_left
```
```    94
```
```    95 lemma add_frac_eq:
```
```    96   assumes "y \<noteq> 0" and "z \<noteq> 0"
```
```    97   shows "x / y + w / z = (x * z + w * y) / (y * z)"
```
```    98 proof -
```
```    99   have "x / y + w / z = (x * z) / (y * z) + (y * w) / (y * z)"
```
```   100     using assms by simp
```
```   101   also have "\<dots> = (x * z + y * w) / (y * z)"
```
```   102     by (simp only: add_divide_distrib)
```
```   103   finally show ?thesis
```
```   104     by (simp only: mult_commute)
```
```   105 qed
```
```   106
```
```   107 text{*Special Cancellation Simprules for Division*}
```
```   108
```
```   109 lemma nonzero_mult_divide_cancel_right [simp, no_atp]:
```
```   110   "b \<noteq> 0 \<Longrightarrow> a * b / b = a"
```
```   111 using nonzero_mult_divide_mult_cancel_right [of 1 b a] by simp
```
```   112
```
```   113 lemma nonzero_mult_divide_cancel_left [simp, no_atp]:
```
```   114   "a \<noteq> 0 \<Longrightarrow> a * b / a = b"
```
```   115 using nonzero_mult_divide_mult_cancel_left [of 1 a b] by simp
```
```   116
```
```   117 lemma nonzero_divide_mult_cancel_right [simp, no_atp]:
```
```   118   "\<lbrakk>a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> b / (a * b) = 1 / a"
```
```   119 using nonzero_mult_divide_mult_cancel_right [of a b 1] by simp
```
```   120
```
```   121 lemma nonzero_divide_mult_cancel_left [simp, no_atp]:
```
```   122   "\<lbrakk>a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / (a * b) = 1 / b"
```
```   123 using nonzero_mult_divide_mult_cancel_left [of b a 1] by simp
```
```   124
```
```   125 lemma nonzero_mult_divide_mult_cancel_left2 [simp, no_atp]:
```
```   126   "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (c * a) / (b * c) = a / b"
```
```   127 using nonzero_mult_divide_mult_cancel_left [of b c a] by (simp add: mult_ac)
```
```   128
```
```   129 lemma nonzero_mult_divide_mult_cancel_right2 [simp, no_atp]:
```
```   130   "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (a * c) / (c * b) = a / b"
```
```   131 using nonzero_mult_divide_mult_cancel_right [of b c a] by (simp add: mult_ac)
```
```   132
```
```   133 lemma minus_divide_left: "- (a / b) = (-a) / b"
```
```   134 by (simp add: divide_inverse)
```
```   135
```
```   136 lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a / b) = a / (- b)"
```
```   137 by (simp add: divide_inverse nonzero_inverse_minus_eq)
```
```   138
```
```   139 lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a) / (-b) = a / b"
```
```   140 by (simp add: divide_inverse nonzero_inverse_minus_eq)
```
```   141
```
```   142 lemma divide_minus_left [simp, no_atp]: "(-a) / b = - (a / b)"
```
```   143 by (simp add: divide_inverse)
```
```   144
```
```   145 lemma diff_divide_distrib: "(a - b) / c = a / c - b / c"
```
```   146 by (simp add: diff_minus add_divide_distrib)
```
```   147
```
```   148 lemma add_divide_eq_iff:
```
```   149   "z \<noteq> 0 \<Longrightarrow> x + y / z = (z * x + y) / z"
```
```   150 by (simp add: add_divide_distrib)
```
```   151
```
```   152 lemma divide_add_eq_iff:
```
```   153   "z \<noteq> 0 \<Longrightarrow> x / z + y = (x + z * y) / z"
```
```   154 by (simp add: add_divide_distrib)
```
```   155
```
```   156 lemma diff_divide_eq_iff:
```
```   157   "z \<noteq> 0 \<Longrightarrow> x - y / z = (z * x - y) / z"
```
```   158 by (simp add: diff_divide_distrib)
```
```   159
```
```   160 lemma divide_diff_eq_iff:
```
```   161   "z \<noteq> 0 \<Longrightarrow> x / z - y = (x - z * y) / z"
```
```   162 by (simp add: diff_divide_distrib)
```
```   163
```
```   164 lemma nonzero_eq_divide_eq: "c \<noteq> 0 \<Longrightarrow> a = b / c \<longleftrightarrow> a * c = b"
```
```   165 proof -
```
```   166   assume [simp]: "c \<noteq> 0"
```
```   167   have "a = b / c \<longleftrightarrow> a * c = (b / c) * c" by simp
```
```   168   also have "... \<longleftrightarrow> a * c = b" by (simp add: divide_inverse mult_assoc)
```
```   169   finally show ?thesis .
```
```   170 qed
```
```   171
```
```   172 lemma nonzero_divide_eq_eq: "c \<noteq> 0 \<Longrightarrow> b / c = a \<longleftrightarrow> b = a * c"
```
```   173 proof -
```
```   174   assume [simp]: "c \<noteq> 0"
```
```   175   have "b / c = a \<longleftrightarrow> (b / c) * c = a * c" by simp
```
```   176   also have "... \<longleftrightarrow> b = a * c" by (simp add: divide_inverse mult_assoc)
```
```   177   finally show ?thesis .
```
```   178 qed
```
```   179
```
```   180 lemma divide_eq_imp: "c \<noteq> 0 \<Longrightarrow> b = a * c \<Longrightarrow> b / c = a"
```
```   181 by simp
```
```   182
```
```   183 lemma eq_divide_imp: "c \<noteq> 0 \<Longrightarrow> a * c = b \<Longrightarrow> a = b / c"
```
```   184 by (erule subst, simp)
```
```   185
```
```   186 lemmas field_eq_simps[no_atp] = algebra_simps
```
```   187   (* pull / out*)
```
```   188   add_divide_eq_iff divide_add_eq_iff
```
```   189   diff_divide_eq_iff divide_diff_eq_iff
```
```   190   (* multiply eqn *)
```
```   191   nonzero_eq_divide_eq nonzero_divide_eq_eq
```
```   192 (* is added later:
```
```   193   times_divide_eq_left times_divide_eq_right
```
```   194 *)
```
```   195
```
```   196 text{*An example:*}
```
```   197 lemma "\<lbrakk>a\<noteq>b; c\<noteq>d; e\<noteq>f\<rbrakk> \<Longrightarrow> ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) = 1"
```
```   198 apply(subgoal_tac "(c-d)*(e-f)*(a-b) \<noteq> 0")
```
```   199  apply(simp add:field_eq_simps)
```
```   200 apply(simp)
```
```   201 done
```
```   202
```
```   203 lemma diff_frac_eq:
```
```   204   "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y - w / z = (x * z - w * y) / (y * z)"
```
```   205 by (simp add: field_eq_simps times_divide_eq)
```
```   206
```
```   207 lemma frac_eq_eq:
```
```   208   "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> (x / y = w / z) = (x * z = w * y)"
```
```   209 by (simp add: field_eq_simps times_divide_eq)
```
```   210
```
```   211 end
```
```   212
```
```   213 class division_by_zero = zero + inverse +
```
```   214   assumes inverse_zero [simp]: "inverse 0 = 0"
```
```   215
```
```   216 lemma divide_zero [simp]:
```
```   217   "a / 0 = (0::'a::{field,division_by_zero})"
```
```   218 by (simp add: divide_inverse)
```
```   219
```
```   220 lemma divide_self_if [simp]:
```
```   221   "a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)"
```
```   222 by simp
```
```   223
```
```   224 class linordered_field = field + linordered_idom
```
```   225
```
```   226 lemma inverse_nonzero_iff_nonzero [simp]:
```
```   227    "(inverse a = 0) = (a = (0::'a::{division_ring,division_by_zero}))"
```
```   228 by (force dest: inverse_zero_imp_zero)
```
```   229
```
```   230 lemma inverse_minus_eq [simp]:
```
```   231    "inverse(-a) = -inverse(a::'a::{division_ring,division_by_zero})"
```
```   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_eq_imp_eq:
```
```   240   "inverse a = inverse b ==> a = (b::'a::{division_ring,division_by_zero})"
```
```   241 apply (cases "a=0 | b=0")
```
```   242  apply (force dest!: inverse_zero_imp_zero
```
```   243               simp add: eq_commute [of "0::'a"])
```
```   244 apply (force dest!: nonzero_inverse_eq_imp_eq)
```
```   245 done
```
```   246
```
```   247 lemma inverse_eq_iff_eq [simp]:
```
```   248   "(inverse a = inverse b) = (a = (b::'a::{division_ring,division_by_zero}))"
```
```   249 by (force dest!: inverse_eq_imp_eq)
```
```   250
```
```   251 lemma inverse_inverse_eq [simp]:
```
```   252      "inverse(inverse (a::'a::{division_ring,division_by_zero})) = a"
```
```   253   proof cases
```
```   254     assume "a=0" thus ?thesis by simp
```
```   255   next
```
```   256     assume "a\<noteq>0"
```
```   257     thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
```
```   258   qed
```
```   259
```
```   260 text{*This version builds in division by zero while also re-orienting
```
```   261       the right-hand side.*}
```
```   262 lemma inverse_mult_distrib [simp]:
```
```   263      "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"
```
```   264   proof cases
```
```   265     assume "a \<noteq> 0 & b \<noteq> 0"
```
```   266     thus ?thesis by (simp add: nonzero_inverse_mult_distrib mult_commute)
```
```   267   next
```
```   268     assume "~ (a \<noteq> 0 & b \<noteq> 0)"
```
```   269     thus ?thesis by force
```
```   270   qed
```
```   271
```
```   272 lemma inverse_divide [simp]:
```
```   273   "inverse (a/b) = b / (a::'a::{field,division_by_zero})"
```
```   274 by (simp add: divide_inverse mult_commute)
```
```   275
```
```   276
```
```   277 subsection {* Calculations with fractions *}
```
```   278
```
```   279 text{* There is a whole bunch of simp-rules just for class @{text
```
```   280 field} but none for class @{text field} and @{text nonzero_divides}
```
```   281 because the latter are covered by a simproc. *}
```
```   282
```
```   283 lemma mult_divide_mult_cancel_left:
```
```   284   "c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"
```
```   285 apply (cases "b = 0")
```
```   286 apply simp_all
```
```   287 done
```
```   288
```
```   289 lemma mult_divide_mult_cancel_right:
```
```   290   "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"
```
```   291 apply (cases "b = 0")
```
```   292 apply simp_all
```
```   293 done
```
```   294
```
```   295 lemma divide_divide_eq_right [simp,no_atp]:
```
```   296   "a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"
```
```   297 by (simp add: divide_inverse mult_ac)
```
```   298
```
```   299 lemma divide_divide_eq_left [simp,no_atp]:
```
```   300   "(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"
```
```   301 by (simp add: divide_inverse mult_assoc)
```
```   302
```
```   303
```
```   304 subsubsection{*Special Cancellation Simprules for Division*}
```
```   305
```
```   306 lemma mult_divide_mult_cancel_left_if[simp,no_atp]:
```
```   307 fixes c :: "'a :: {field,division_by_zero}"
```
```   308 shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)"
```
```   309 by (simp add: mult_divide_mult_cancel_left)
```
```   310
```
```   311
```
```   312 subsection {* Division and Unary Minus *}
```
```   313
```
```   314 lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"
```
```   315 by (simp add: divide_inverse)
```
```   316
```
```   317 lemma divide_minus_right [simp, no_atp]:
```
```   318   "a / -(b::'a::{field,division_by_zero}) = -(a / b)"
```
```   319 by (simp add: divide_inverse)
```
```   320
```
```   321 lemma minus_divide_divide:
```
```   322   "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"
```
```   323 apply (cases "b=0", simp)
```
```   324 apply (simp add: nonzero_minus_divide_divide)
```
```   325 done
```
```   326
```
```   327 lemma eq_divide_eq:
```
```   328   "((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)"
```
```   329 by (simp add: nonzero_eq_divide_eq)
```
```   330
```
```   331 lemma divide_eq_eq:
```
```   332   "(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)"
```
```   333 by (force simp add: nonzero_divide_eq_eq)
```
```   334
```
```   335
```
```   336 subsection {* Ordered Fields *}
```
```   337
```
```   338 lemma positive_imp_inverse_positive:
```
```   339 assumes a_gt_0: "0 < a"  shows "0 < inverse (a::'a::linordered_field)"
```
```   340 proof -
```
```   341   have "0 < a * inverse a"
```
```   342     by (simp add: a_gt_0 [THEN order_less_imp_not_eq2])
```
```   343   thus "0 < inverse a"
```
```   344     by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff)
```
```   345 qed
```
```   346
```
```   347 lemma negative_imp_inverse_negative:
```
```   348   "a < 0 ==> inverse a < (0::'a::linordered_field)"
```
```   349 by (insert positive_imp_inverse_positive [of "-a"],
```
```   350     simp add: nonzero_inverse_minus_eq order_less_imp_not_eq)
```
```   351
```
```   352 lemma inverse_le_imp_le:
```
```   353 assumes invle: "inverse a \<le> inverse b" and apos:  "0 < a"
```
```   354 shows "b \<le> (a::'a::linordered_field)"
```
```   355 proof (rule classical)
```
```   356   assume "~ b \<le> a"
```
```   357   hence "a < b"  by (simp add: linorder_not_le)
```
```   358   hence bpos: "0 < b"  by (blast intro: apos order_less_trans)
```
```   359   hence "a * inverse a \<le> a * inverse b"
```
```   360     by (simp add: apos invle order_less_imp_le mult_left_mono)
```
```   361   hence "(a * inverse a) * b \<le> (a * inverse b) * b"
```
```   362     by (simp add: bpos order_less_imp_le mult_right_mono)
```
```   363   thus "b \<le> a"  by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)
```
```   364 qed
```
```   365
```
```   366 lemma inverse_positive_imp_positive:
```
```   367 assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0"
```
```   368 shows "0 < (a::'a::linordered_field)"
```
```   369 proof -
```
```   370   have "0 < inverse (inverse a)"
```
```   371     using inv_gt_0 by (rule positive_imp_inverse_positive)
```
```   372   thus "0 < a"
```
```   373     using nz by (simp add: nonzero_inverse_inverse_eq)
```
```   374 qed
```
```   375
```
```   376 lemma inverse_positive_iff_positive [simp]:
```
```   377   "(0 < inverse a) = (0 < (a::'a::{linordered_field,division_by_zero}))"
```
```   378 apply (cases "a = 0", simp)
```
```   379 apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
```
```   380 done
```
```   381
```
```   382 lemma inverse_negative_imp_negative:
```
```   383 assumes inv_less_0: "inverse a < 0" and nz:  "a \<noteq> 0"
```
```   384 shows "a < (0::'a::linordered_field)"
```
```   385 proof -
```
```   386   have "inverse (inverse a) < 0"
```
```   387     using inv_less_0 by (rule negative_imp_inverse_negative)
```
```   388   thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq)
```
```   389 qed
```
```   390
```
```   391 lemma inverse_negative_iff_negative [simp]:
```
```   392   "(inverse a < 0) = (a < (0::'a::{linordered_field,division_by_zero}))"
```
```   393 apply (cases "a = 0", simp)
```
```   394 apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
```
```   395 done
```
```   396
```
```   397 lemma inverse_nonnegative_iff_nonnegative [simp]:
```
```   398   "(0 \<le> inverse a) = (0 \<le> (a::'a::{linordered_field,division_by_zero}))"
```
```   399 by (simp add: linorder_not_less [symmetric])
```
```   400
```
```   401 lemma inverse_nonpositive_iff_nonpositive [simp]:
```
```   402   "(inverse a \<le> 0) = (a \<le> (0::'a::{linordered_field,division_by_zero}))"
```
```   403 by (simp add: linorder_not_less [symmetric])
```
```   404
```
```   405 lemma linordered_field_no_lb: "\<forall> x. \<exists>y. y < (x::'a::linordered_field)"
```
```   406 proof
```
```   407   fix x::'a
```
```   408   have m1: "- (1::'a) < 0" by simp
```
```   409   from add_strict_right_mono[OF m1, where c=x]
```
```   410   have "(- 1) + x < x" by simp
```
```   411   thus "\<exists>y. y < x" by blast
```
```   412 qed
```
```   413
```
```   414 lemma linordered_field_no_ub: "\<forall> x. \<exists>y. y > (x::'a::linordered_field)"
```
```   415 proof
```
```   416   fix x::'a
```
```   417   have m1: " (1::'a) > 0" by simp
```
```   418   from add_strict_right_mono[OF m1, where c=x]
```
```   419   have "1 + x > x" by simp
```
```   420   thus "\<exists>y. y > x" by blast
```
```   421 qed
```
```   422
```
```   423 subsection{*Anti-Monotonicity of @{term inverse}*}
```
```   424
```
```   425 lemma less_imp_inverse_less:
```
```   426 assumes less: "a < b" and apos:  "0 < a"
```
```   427 shows "inverse b < inverse (a::'a::linordered_field)"
```
```   428 proof (rule ccontr)
```
```   429   assume "~ inverse b < inverse a"
```
```   430   hence "inverse a \<le> inverse b" by (simp add: linorder_not_less)
```
```   431   hence "~ (a < b)"
```
```   432     by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos])
```
```   433   thus False by (rule notE [OF _ less])
```
```   434 qed
```
```   435
```
```   436 lemma inverse_less_imp_less:
```
```   437   "[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::linordered_field)"
```
```   438 apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"])
```
```   439 apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq)
```
```   440 done
```
```   441
```
```   442 text{*Both premises are essential. Consider -1 and 1.*}
```
```   443 lemma inverse_less_iff_less [simp,no_atp]:
```
```   444   "[|0 < a; 0 < b|] ==> (inverse a < inverse b) = (b < (a::'a::linordered_field))"
```
```   445 by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less)
```
```   446
```
```   447 lemma le_imp_inverse_le:
```
```   448   "[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::linordered_field)"
```
```   449 by (force simp add: order_le_less less_imp_inverse_less)
```
```   450
```
```   451 lemma inverse_le_iff_le [simp,no_atp]:
```
```   452  "[|0 < a; 0 < b|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::linordered_field))"
```
```   453 by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le)
```
```   454
```
```   455
```
```   456 text{*These results refer to both operands being negative.  The opposite-sign
```
```   457 case is trivial, since inverse preserves signs.*}
```
```   458 lemma inverse_le_imp_le_neg:
```
```   459   "[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::linordered_field)"
```
```   460 apply (rule classical)
```
```   461 apply (subgoal_tac "a < 0")
```
```   462  prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans)
```
```   463 apply (insert inverse_le_imp_le [of "-b" "-a"])
```
```   464 apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq)
```
```   465 done
```
```   466
```
```   467 lemma less_imp_inverse_less_neg:
```
```   468    "[|a < b; b < 0|] ==> inverse b < inverse (a::'a::linordered_field)"
```
```   469 apply (subgoal_tac "a < 0")
```
```   470  prefer 2 apply (blast intro: order_less_trans)
```
```   471 apply (insert less_imp_inverse_less [of "-b" "-a"])
```
```   472 apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq)
```
```   473 done
```
```   474
```
```   475 lemma inverse_less_imp_less_neg:
```
```   476    "[|inverse a < inverse b; b < 0|] ==> b < (a::'a::linordered_field)"
```
```   477 apply (rule classical)
```
```   478 apply (subgoal_tac "a < 0")
```
```   479  prefer 2
```
```   480  apply (force simp add: linorder_not_less intro: order_le_less_trans)
```
```   481 apply (insert inverse_less_imp_less [of "-b" "-a"])
```
```   482 apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq)
```
```   483 done
```
```   484
```
```   485 lemma inverse_less_iff_less_neg [simp,no_atp]:
```
```   486   "[|a < 0; b < 0|] ==> (inverse a < inverse b) = (b < (a::'a::linordered_field))"
```
```   487 apply (insert inverse_less_iff_less [of "-b" "-a"])
```
```   488 apply (simp del: inverse_less_iff_less
```
```   489             add: order_less_imp_not_eq nonzero_inverse_minus_eq)
```
```   490 done
```
```   491
```
```   492 lemma le_imp_inverse_le_neg:
```
```   493   "[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::linordered_field)"
```
```   494 by (force simp add: order_le_less less_imp_inverse_less_neg)
```
```   495
```
```   496 lemma inverse_le_iff_le_neg [simp,no_atp]:
```
```   497  "[|a < 0; b < 0|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::linordered_field))"
```
```   498 by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg)
```
```   499
```
```   500
```
```   501 subsection{*Inverses and the Number One*}
```
```   502
```
```   503 lemma one_less_inverse_iff:
```
```   504   "(1 < inverse x) = (0 < x & x < (1::'a::{linordered_field,division_by_zero}))"
```
```   505 proof cases
```
```   506   assume "0 < x"
```
```   507     with inverse_less_iff_less [OF zero_less_one, of x]
```
```   508     show ?thesis by simp
```
```   509 next
```
```   510   assume notless: "~ (0 < x)"
```
```   511   have "~ (1 < inverse x)"
```
```   512   proof
```
```   513     assume "1 < inverse x"
```
```   514     also with notless have "... \<le> 0" by (simp add: linorder_not_less)
```
```   515     also have "... < 1" by (rule zero_less_one)
```
```   516     finally show False by auto
```
```   517   qed
```
```   518   with notless show ?thesis by simp
```
```   519 qed
```
```   520
```
```   521 lemma inverse_eq_1_iff [simp]:
```
```   522   "(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))"
```
```   523 by (insert inverse_eq_iff_eq [of x 1], simp)
```
```   524
```
```   525 lemma one_le_inverse_iff:
```
```   526   "(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{linordered_field,division_by_zero}))"
```
```   527 by (force simp add: order_le_less one_less_inverse_iff)
```
```   528
```
```   529 lemma inverse_less_1_iff:
```
```   530   "(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{linordered_field,division_by_zero}))"
```
```   531 by (simp add: linorder_not_le [symmetric] one_le_inverse_iff)
```
```   532
```
```   533 lemma inverse_le_1_iff:
```
```   534   "(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{linordered_field,division_by_zero}))"
```
```   535 by (simp add: linorder_not_less [symmetric] one_less_inverse_iff)
```
```   536
```
```   537
```
```   538 subsection{*Simplification of Inequalities Involving Literal Divisors*}
```
```   539
```
```   540 lemma pos_le_divide_eq: "0 < (c::'a::linordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"
```
```   541 proof -
```
```   542   assume less: "0<c"
```
```   543   hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
```
```   544     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
```
```   545   also have "... = (a*c \<le> b)"
```
```   546     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc)
```
```   547   finally show ?thesis .
```
```   548 qed
```
```   549
```
```   550 lemma neg_le_divide_eq: "c < (0::'a::linordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"
```
```   551 proof -
```
```   552   assume less: "c<0"
```
```   553   hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
```
```   554     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
```
```   555   also have "... = (b \<le> a*c)"
```
```   556     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc)
```
```   557   finally show ?thesis .
```
```   558 qed
```
```   559
```
```   560 lemma le_divide_eq:
```
```   561   "(a \<le> b/c) =
```
```   562    (if 0 < c then a*c \<le> b
```
```   563              else if c < 0 then b \<le> a*c
```
```   564              else  a \<le> (0::'a::{linordered_field,division_by_zero}))"
```
```   565 apply (cases "c=0", simp)
```
```   566 apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff)
```
```   567 done
```
```   568
```
```   569 lemma pos_divide_le_eq: "0 < (c::'a::linordered_field) ==> (b/c \<le> a) = (b \<le> a*c)"
```
```   570 proof -
```
```   571   assume less: "0<c"
```
```   572   hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
```
```   573     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
```
```   574   also have "... = (b \<le> a*c)"
```
```   575     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc)
```
```   576   finally show ?thesis .
```
```   577 qed
```
```   578
```
```   579 lemma neg_divide_le_eq: "c < (0::'a::linordered_field) ==> (b/c \<le> a) = (a*c \<le> b)"
```
```   580 proof -
```
```   581   assume less: "c<0"
```
```   582   hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
```
```   583     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
```
```   584   also have "... = (a*c \<le> b)"
```
```   585     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc)
```
```   586   finally show ?thesis .
```
```   587 qed
```
```   588
```
```   589 lemma divide_le_eq:
```
```   590   "(b/c \<le> a) =
```
```   591    (if 0 < c then b \<le> a*c
```
```   592              else if c < 0 then a*c \<le> b
```
```   593              else 0 \<le> (a::'a::{linordered_field,division_by_zero}))"
```
```   594 apply (cases "c=0", simp)
```
```   595 apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff)
```
```   596 done
```
```   597
```
```   598 lemma pos_less_divide_eq:
```
```   599      "0 < (c::'a::linordered_field) ==> (a < b/c) = (a*c < b)"
```
```   600 proof -
```
```   601   assume less: "0<c"
```
```   602   hence "(a < b/c) = (a*c < (b/c)*c)"
```
```   603     by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
```
```   604   also have "... = (a*c < b)"
```
```   605     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc)
```
```   606   finally show ?thesis .
```
```   607 qed
```
```   608
```
```   609 lemma neg_less_divide_eq:
```
```   610  "c < (0::'a::linordered_field) ==> (a < b/c) = (b < a*c)"
```
```   611 proof -
```
```   612   assume less: "c<0"
```
```   613   hence "(a < b/c) = ((b/c)*c < a*c)"
```
```   614     by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
```
```   615   also have "... = (b < a*c)"
```
```   616     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc)
```
```   617   finally show ?thesis .
```
```   618 qed
```
```   619
```
```   620 lemma less_divide_eq:
```
```   621   "(a < b/c) =
```
```   622    (if 0 < c then a*c < b
```
```   623              else if c < 0 then b < a*c
```
```   624              else  a < (0::'a::{linordered_field,division_by_zero}))"
```
```   625 apply (cases "c=0", simp)
```
```   626 apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff)
```
```   627 done
```
```   628
```
```   629 lemma pos_divide_less_eq:
```
```   630      "0 < (c::'a::linordered_field) ==> (b/c < a) = (b < a*c)"
```
```   631 proof -
```
```   632   assume less: "0<c"
```
```   633   hence "(b/c < a) = ((b/c)*c < a*c)"
```
```   634     by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
```
```   635   also have "... = (b < a*c)"
```
```   636     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc)
```
```   637   finally show ?thesis .
```
```   638 qed
```
```   639
```
```   640 lemma neg_divide_less_eq:
```
```   641  "c < (0::'a::linordered_field) ==> (b/c < a) = (a*c < b)"
```
```   642 proof -
```
```   643   assume less: "c<0"
```
```   644   hence "(b/c < a) = (a*c < (b/c)*c)"
```
```   645     by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
```
```   646   also have "... = (a*c < b)"
```
```   647     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc)
```
```   648   finally show ?thesis .
```
```   649 qed
```
```   650
```
```   651 lemma divide_less_eq:
```
```   652   "(b/c < a) =
```
```   653    (if 0 < c then b < a*c
```
```   654              else if c < 0 then a*c < b
```
```   655              else 0 < (a::'a::{linordered_field,division_by_zero}))"
```
```   656 apply (cases "c=0", simp)
```
```   657 apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff)
```
```   658 done
```
```   659
```
```   660
```
```   661 subsection{*Field simplification*}
```
```   662
```
```   663 text{* Lemmas @{text field_simps} multiply with denominators in in(equations)
```
```   664 if they can be proved to be non-zero (for equations) or positive/negative
```
```   665 (for inequations). Can be too aggressive and is therefore separate from the
```
```   666 more benign @{text algebra_simps}. *}
```
```   667
```
```   668 lemmas field_simps[no_atp] = field_eq_simps
```
```   669   (* multiply ineqn *)
```
```   670   pos_divide_less_eq neg_divide_less_eq
```
```   671   pos_less_divide_eq neg_less_divide_eq
```
```   672   pos_divide_le_eq neg_divide_le_eq
```
```   673   pos_le_divide_eq neg_le_divide_eq
```
```   674
```
```   675 text{* Lemmas @{text sign_simps} is a first attempt to automate proofs
```
```   676 of positivity/negativity needed for @{text field_simps}. Have not added @{text
```
```   677 sign_simps} to @{text field_simps} because the former can lead to case
```
```   678 explosions. *}
```
```   679
```
```   680 lemmas sign_simps[no_atp] = group_simps
```
```   681   zero_less_mult_iff  mult_less_0_iff
```
```   682
```
```   683 (* Only works once linear arithmetic is installed:
```
```   684 text{*An example:*}
```
```   685 lemma fixes a b c d e f :: "'a::linordered_field"
```
```   686 shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow>
```
```   687  ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) <
```
```   688  ((e-f)*(a-b)*(c-d))/((e-f)*(a-b)*(c-d)) + u"
```
```   689 apply(subgoal_tac "(c-d)*(e-f)*(a-b) > 0")
```
```   690  prefer 2 apply(simp add:sign_simps)
```
```   691 apply(subgoal_tac "(c-d)*(e-f)*(a-b)*u > 0")
```
```   692  prefer 2 apply(simp add:sign_simps)
```
```   693 apply(simp add:field_simps)
```
```   694 done
```
```   695 *)
```
```   696
```
```   697
```
```   698 subsection{*Division and Signs*}
```
```   699
```
```   700 lemma zero_less_divide_iff:
```
```   701      "((0::'a::{linordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
```
```   702 by (simp add: divide_inverse zero_less_mult_iff)
```
```   703
```
```   704 lemma divide_less_0_iff:
```
```   705      "(a/b < (0::'a::{linordered_field,division_by_zero})) =
```
```   706       (0 < a & b < 0 | a < 0 & 0 < b)"
```
```   707 by (simp add: divide_inverse mult_less_0_iff)
```
```   708
```
```   709 lemma zero_le_divide_iff:
```
```   710      "((0::'a::{linordered_field,division_by_zero}) \<le> a/b) =
```
```   711       (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
```
```   712 by (simp add: divide_inverse zero_le_mult_iff)
```
```   713
```
```   714 lemma divide_le_0_iff:
```
```   715      "(a/b \<le> (0::'a::{linordered_field,division_by_zero})) =
```
```   716       (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
```
```   717 by (simp add: divide_inverse mult_le_0_iff)
```
```   718
```
```   719 lemma divide_eq_0_iff [simp,no_atp]:
```
```   720      "(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))"
```
```   721 by (simp add: divide_inverse)
```
```   722
```
```   723 lemma divide_pos_pos:
```
```   724   "0 < (x::'a::linordered_field) ==> 0 < y ==> 0 < x / y"
```
```   725 by(simp add:field_simps)
```
```   726
```
```   727
```
```   728 lemma divide_nonneg_pos:
```
```   729   "0 <= (x::'a::linordered_field) ==> 0 < y ==> 0 <= x / y"
```
```   730 by(simp add:field_simps)
```
```   731
```
```   732 lemma divide_neg_pos:
```
```   733   "(x::'a::linordered_field) < 0 ==> 0 < y ==> x / y < 0"
```
```   734 by(simp add:field_simps)
```
```   735
```
```   736 lemma divide_nonpos_pos:
```
```   737   "(x::'a::linordered_field) <= 0 ==> 0 < y ==> x / y <= 0"
```
```   738 by(simp add:field_simps)
```
```   739
```
```   740 lemma divide_pos_neg:
```
```   741   "0 < (x::'a::linordered_field) ==> y < 0 ==> x / y < 0"
```
```   742 by(simp add:field_simps)
```
```   743
```
```   744 lemma divide_nonneg_neg:
```
```   745   "0 <= (x::'a::linordered_field) ==> y < 0 ==> x / y <= 0"
```
```   746 by(simp add:field_simps)
```
```   747
```
```   748 lemma divide_neg_neg:
```
```   749   "(x::'a::linordered_field) < 0 ==> y < 0 ==> 0 < x / y"
```
```   750 by(simp add:field_simps)
```
```   751
```
```   752 lemma divide_nonpos_neg:
```
```   753   "(x::'a::linordered_field) <= 0 ==> y < 0 ==> 0 <= x / y"
```
```   754 by(simp add:field_simps)
```
```   755
```
```   756
```
```   757 subsection{*Cancellation Laws for Division*}
```
```   758
```
```   759 lemma divide_cancel_right [simp,no_atp]:
```
```   760      "(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))"
```
```   761 apply (cases "c=0", simp)
```
```   762 apply (simp add: divide_inverse)
```
```   763 done
```
```   764
```
```   765 lemma divide_cancel_left [simp,no_atp]:
```
```   766      "(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))"
```
```   767 apply (cases "c=0", simp)
```
```   768 apply (simp add: divide_inverse)
```
```   769 done
```
```   770
```
```   771
```
```   772 subsection {* Division and the Number One *}
```
```   773
```
```   774 text{*Simplify expressions equated with 1*}
```
```   775 lemma divide_eq_1_iff [simp,no_atp]:
```
```   776      "(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
```
```   777 apply (cases "b=0", simp)
```
```   778 apply (simp add: right_inverse_eq)
```
```   779 done
```
```   780
```
```   781 lemma one_eq_divide_iff [simp,no_atp]:
```
```   782      "(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
```
```   783 by (simp add: eq_commute [of 1])
```
```   784
```
```   785 lemma zero_eq_1_divide_iff [simp,no_atp]:
```
```   786      "((0::'a::{linordered_field,division_by_zero}) = 1/a) = (a = 0)"
```
```   787 apply (cases "a=0", simp)
```
```   788 apply (auto simp add: nonzero_eq_divide_eq)
```
```   789 done
```
```   790
```
```   791 lemma one_divide_eq_0_iff [simp,no_atp]:
```
```   792      "(1/a = (0::'a::{linordered_field,division_by_zero})) = (a = 0)"
```
```   793 apply (cases "a=0", simp)
```
```   794 apply (insert zero_neq_one [THEN not_sym])
```
```   795 apply (auto simp add: nonzero_divide_eq_eq)
```
```   796 done
```
```   797
```
```   798 text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}
```
```   799 lemmas zero_less_divide_1_iff = zero_less_divide_iff [of 1, simplified]
```
```   800 lemmas divide_less_0_1_iff = divide_less_0_iff [of 1, simplified]
```
```   801 lemmas zero_le_divide_1_iff = zero_le_divide_iff [of 1, simplified]
```
```   802 lemmas divide_le_0_1_iff = divide_le_0_iff [of 1, simplified]
```
```   803
```
```   804 declare zero_less_divide_1_iff [simp,no_atp]
```
```   805 declare divide_less_0_1_iff [simp,no_atp]
```
```   806 declare zero_le_divide_1_iff [simp,no_atp]
```
```   807 declare divide_le_0_1_iff [simp,no_atp]
```
```   808
```
```   809
```
```   810 subsection {* Ordering Rules for Division *}
```
```   811
```
```   812 lemma divide_strict_right_mono:
```
```   813      "[|a < b; 0 < c|] ==> a / c < b / (c::'a::linordered_field)"
```
```   814 by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono
```
```   815               positive_imp_inverse_positive)
```
```   816
```
```   817 lemma divide_right_mono:
```
```   818      "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{linordered_field,division_by_zero})"
```
```   819 by (force simp add: divide_strict_right_mono order_le_less)
```
```   820
```
```   821 lemma divide_right_mono_neg: "(a::'a::{division_by_zero,linordered_field}) <= b
```
```   822     ==> c <= 0 ==> b / c <= a / c"
```
```   823 apply (drule divide_right_mono [of _ _ "- c"])
```
```   824 apply auto
```
```   825 done
```
```   826
```
```   827 lemma divide_strict_right_mono_neg:
```
```   828      "[|b < a; c < 0|] ==> a / c < b / (c::'a::linordered_field)"
```
```   829 apply (drule divide_strict_right_mono [of _ _ "-c"], simp)
```
```   830 apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric])
```
```   831 done
```
```   832
```
```   833 text{*The last premise ensures that @{term a} and @{term b}
```
```   834       have the same sign*}
```
```   835 lemma divide_strict_left_mono:
```
```   836   "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::linordered_field)"
```
```   837 by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono)
```
```   838
```
```   839 lemma divide_left_mono:
```
```   840   "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::linordered_field)"
```
```   841 by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_right_mono)
```
```   842
```
```   843 lemma divide_left_mono_neg: "(a::'a::{division_by_zero,linordered_field}) <= b
```
```   844     ==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"
```
```   845   apply (drule divide_left_mono [of _ _ "- c"])
```
```   846   apply (auto simp add: mult_commute)
```
```   847 done
```
```   848
```
```   849 lemma divide_strict_left_mono_neg:
```
```   850   "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::linordered_field)"
```
```   851 by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono_neg)
```
```   852
```
```   853
```
```   854 text{*Simplify quotients that are compared with the value 1.*}
```
```   855
```
```   856 lemma le_divide_eq_1 [no_atp]:
```
```   857   fixes a :: "'a :: {linordered_field,division_by_zero}"
```
```   858   shows "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"
```
```   859 by (auto simp add: le_divide_eq)
```
```   860
```
```   861 lemma divide_le_eq_1 [no_atp]:
```
```   862   fixes a :: "'a :: {linordered_field,division_by_zero}"
```
```   863   shows "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"
```
```   864 by (auto simp add: divide_le_eq)
```
```   865
```
```   866 lemma less_divide_eq_1 [no_atp]:
```
```   867   fixes a :: "'a :: {linordered_field,division_by_zero}"
```
```   868   shows "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
```
```   869 by (auto simp add: less_divide_eq)
```
```   870
```
```   871 lemma divide_less_eq_1 [no_atp]:
```
```   872   fixes a :: "'a :: {linordered_field,division_by_zero}"
```
```   873   shows "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
```
```   874 by (auto simp add: divide_less_eq)
```
```   875
```
```   876
```
```   877 subsection{*Conditional Simplification Rules: No Case Splits*}
```
```   878
```
```   879 lemma le_divide_eq_1_pos [simp,no_atp]:
```
```   880   fixes a :: "'a :: {linordered_field,division_by_zero}"
```
```   881   shows "0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)"
```
```   882 by (auto simp add: le_divide_eq)
```
```   883
```
```   884 lemma le_divide_eq_1_neg [simp,no_atp]:
```
```   885   fixes a :: "'a :: {linordered_field,division_by_zero}"
```
```   886   shows "a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)"
```
```   887 by (auto simp add: le_divide_eq)
```
```   888
```
```   889 lemma divide_le_eq_1_pos [simp,no_atp]:
```
```   890   fixes a :: "'a :: {linordered_field,division_by_zero}"
```
```   891   shows "0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)"
```
```   892 by (auto simp add: divide_le_eq)
```
```   893
```
```   894 lemma divide_le_eq_1_neg [simp,no_atp]:
```
```   895   fixes a :: "'a :: {linordered_field,division_by_zero}"
```
```   896   shows "a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)"
```
```   897 by (auto simp add: divide_le_eq)
```
```   898
```
```   899 lemma less_divide_eq_1_pos [simp,no_atp]:
```
```   900   fixes a :: "'a :: {linordered_field,division_by_zero}"
```
```   901   shows "0 < a \<Longrightarrow> (1 < b/a) = (a < b)"
```
```   902 by (auto simp add: less_divide_eq)
```
```   903
```
```   904 lemma less_divide_eq_1_neg [simp,no_atp]:
```
```   905   fixes a :: "'a :: {linordered_field,division_by_zero}"
```
```   906   shows "a < 0 \<Longrightarrow> (1 < b/a) = (b < a)"
```
```   907 by (auto simp add: less_divide_eq)
```
```   908
```
```   909 lemma divide_less_eq_1_pos [simp,no_atp]:
```
```   910   fixes a :: "'a :: {linordered_field,division_by_zero}"
```
```   911   shows "0 < a \<Longrightarrow> (b/a < 1) = (b < a)"
```
```   912 by (auto simp add: divide_less_eq)
```
```   913
```
```   914 lemma divide_less_eq_1_neg [simp,no_atp]:
```
```   915   fixes a :: "'a :: {linordered_field,division_by_zero}"
```
```   916   shows "a < 0 \<Longrightarrow> b/a < 1 <-> a < b"
```
```   917 by (auto simp add: divide_less_eq)
```
```   918
```
```   919 lemma eq_divide_eq_1 [simp,no_atp]:
```
```   920   fixes a :: "'a :: {linordered_field,division_by_zero}"
```
```   921   shows "(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   fixes a :: "'a :: {linordered_field,division_by_zero}"
```
```   926   shows "(b/a = 1) = ((a \<noteq> 0 & a = b))"
```
```   927 by (auto simp add: divide_eq_eq)
```
```   928
```
```   929
```
```   930 subsection {* Reasoning about inequalities with division *}
```
```   931
```
```   932 lemma mult_imp_div_pos_le: "0 < (y::'a::linordered_field) ==> x <= z * y ==>
```
```   933     x / y <= z"
```
```   934 by (subst pos_divide_le_eq, assumption+)
```
```   935
```
```   936 lemma mult_imp_le_div_pos: "0 < (y::'a::linordered_field) ==> z * y <= x ==>
```
```   937     z <= x / y"
```
```   938 by(simp add:field_simps)
```
```   939
```
```   940 lemma mult_imp_div_pos_less: "0 < (y::'a::linordered_field) ==> x < z * y ==>
```
```   941     x / y < z"
```
```   942 by(simp add:field_simps)
```
```   943
```
```   944 lemma mult_imp_less_div_pos: "0 < (y::'a::linordered_field) ==> z * y < x ==>
```
```   945     z < x / y"
```
```   946 by(simp add:field_simps)
```
```   947
```
```   948 lemma frac_le: "(0::'a::linordered_field) <= x ==>
```
```   949     x <= y ==> 0 < w ==> w <= z  ==> x / z <= y / w"
```
```   950   apply (rule mult_imp_div_pos_le)
```
```   951   apply simp
```
```   952   apply (subst times_divide_eq_left)
```
```   953   apply (rule mult_imp_le_div_pos, assumption)
```
```   954   apply (rule mult_mono)
```
```   955   apply simp_all
```
```   956 done
```
```   957
```
```   958 lemma frac_less: "(0::'a::linordered_field) <= x ==>
```
```   959     x < y ==> 0 < w ==> w <= z  ==> x / z < y / w"
```
```   960   apply (rule mult_imp_div_pos_less)
```
```   961   apply simp
```
```   962   apply (subst times_divide_eq_left)
```
```   963   apply (rule mult_imp_less_div_pos, assumption)
```
```   964   apply (erule mult_less_le_imp_less)
```
```   965   apply simp_all
```
```   966 done
```
```   967
```
```   968 lemma frac_less2: "(0::'a::linordered_field) < x ==>
```
```   969     x <= y ==> 0 < w ==> w < z  ==> x / z < y / w"
```
```   970   apply (rule mult_imp_div_pos_less)
```
```   971   apply simp_all
```
```   972   apply (subst times_divide_eq_left)
```
```   973   apply (rule mult_imp_less_div_pos, assumption)
```
```   974   apply (erule mult_le_less_imp_less)
```
```   975   apply simp_all
```
```   976 done
```
```   977
```
```   978 text{*It's not obvious whether these should be simprules or not.
```
```   979   Their effect is to gather terms into one big fraction, like
```
```   980   a*b*c / x*y*z. The rationale for that is unclear, but many proofs
```
```   981   seem to need them.*}
```
```   982
```
```   983 declare times_divide_eq [simp]
```
```   984
```
```   985
```
```   986 subsection {* Ordered Fields are Dense *}
```
```   987
```
```   988 lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::linordered_field)"
```
```   989 by (simp add: field_simps zero_less_two)
```
```   990
```
```   991 lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::linordered_field) < b"
```
```   992 by (simp add: field_simps zero_less_two)
```
```   993
```
```   994 instance linordered_field < dense_linorder
```
```   995 proof
```
```   996   fix x y :: 'a
```
```   997   have "x < x + 1" by simp
```
```   998   then show "\<exists>y. x < y" ..
```
```   999   have "x - 1 < x" by simp
```
```  1000   then show "\<exists>y. y < x" ..
```
```  1001   show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum)
```
```  1002 qed
```
```  1003
```
```  1004
```
```  1005 subsection {* Absolute Value *}
```
```  1006
```
```  1007 lemma nonzero_abs_inverse:
```
```  1008      "a \<noteq> 0 ==> abs (inverse (a::'a::linordered_field)) = inverse (abs a)"
```
```  1009 apply (auto simp add: linorder_neq_iff abs_if nonzero_inverse_minus_eq
```
```  1010                       negative_imp_inverse_negative)
```
```  1011 apply (blast intro: positive_imp_inverse_positive elim: order_less_asym)
```
```  1012 done
```
```  1013
```
```  1014 lemma abs_inverse [simp]:
```
```  1015      "abs (inverse (a::'a::{linordered_field,division_by_zero})) =
```
```  1016       inverse (abs a)"
```
```  1017 apply (cases "a=0", simp)
```
```  1018 apply (simp add: nonzero_abs_inverse)
```
```  1019 done
```
```  1020
```
```  1021 lemma nonzero_abs_divide:
```
```  1022      "b \<noteq> 0 ==> abs (a / (b::'a::linordered_field)) = abs a / abs b"
```
```  1023 by (simp add: divide_inverse abs_mult nonzero_abs_inverse)
```
```  1024
```
```  1025 lemma abs_divide [simp]:
```
```  1026      "abs (a / (b::'a::{linordered_field,division_by_zero})) = abs a / abs b"
```
```  1027 apply (cases "b=0", simp)
```
```  1028 apply (simp add: nonzero_abs_divide)
```
```  1029 done
```
```  1030
```
```  1031 lemma abs_div_pos: "(0::'a::{division_by_zero,linordered_field}) < y ==>
```
```  1032     abs x / y = abs (x / y)"
```
```  1033   apply (subst abs_divide)
```
```  1034   apply (simp add: order_less_imp_le)
```
```  1035 done
```
```  1036
```
```  1037 lemma field_le_epsilon:
```
```  1038   fixes x y :: "'a\<Colon>linordered_field"
```
```  1039   assumes e: "\<And>e. 0 < e \<Longrightarrow> x \<le> y + e"
```
```  1040   shows "x \<le> y"
```
```  1041 proof (rule dense_le)
```
```  1042   fix t assume "t < x"
```
```  1043   hence "0 < x - t" by (simp add: less_diff_eq)
```
```  1044   from e[OF this]
```
```  1045   show "t \<le> y" by (simp add: field_simps)
```
```  1046 qed
```
```  1047
```
```  1048 lemma field_le_mult_one_interval:
```
```  1049   fixes x :: "'a\<Colon>{linordered_field,division_by_zero}"
```
```  1050   assumes *: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y"
```
```  1051   shows "x \<le> y"
```
```  1052 proof (cases "0 < x")
```
```  1053   assume "0 < x"
```
```  1054   thus ?thesis
```
```  1055     using dense_le_bounded[of 0 1 "y/x"] *
```
```  1056     unfolding le_divide_eq if_P[OF `0 < x`] by simp
```
```  1057 next
```
```  1058   assume "\<not>0 < x" hence "x \<le> 0" by simp
```
```  1059   obtain s::'a where s: "0 < s" "s < 1" using dense[of 0 "1\<Colon>'a"] by auto
```
```  1060   hence "x \<le> s * x" using mult_le_cancel_right[of 1 x s] `x \<le> 0` by auto
```
```  1061   also note *[OF s]
```
```  1062   finally show ?thesis .
```
```  1063 qed
```
```  1064
```
```  1065 code_modulename SML
```
```  1066   Fields Arith
```
```  1067
```
```  1068 code_modulename OCaml
```
```  1069   Fields Arith
```
```  1070
```
```  1071 code_modulename Haskell
```
```  1072   Fields Arith
```
```  1073
```
```  1074 end
```