src/HOL/Fields.thy
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explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
```     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 subsection {* Division rings *}
```
```    17
```
```    18 text {*
```
```    19   A division ring is like a field, but without the commutativity requirement.
```
```    20 *}
```
```    21
```
```    22 class inverse =
```
```    23   fixes inverse :: "'a \<Rightarrow> 'a"
```
```    24     and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "'/" 70)
```
```    25
```
```    26 text{* Lemmas @{text divide_simps} move division to the outside and eliminates them on (in)equalities. *}
```
```    27
```
```    28 named_theorems divide_simps "rewrite rules to eliminate divisions"
```
```    29
```
```    30
```
```    31 class division_ring = ring_1 + inverse +
```
```    32   assumes left_inverse [simp]:  "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
```
```    33   assumes right_inverse [simp]: "a \<noteq> 0 \<Longrightarrow> a * inverse a = 1"
```
```    34   assumes divide_inverse: "a / b = a * inverse b"
```
```    35 begin
```
```    36
```
```    37 subclass ring_1_no_zero_divisors
```
```    38 proof
```
```    39   fix a b :: 'a
```
```    40   assume a: "a \<noteq> 0" and b: "b \<noteq> 0"
```
```    41   show "a * b \<noteq> 0"
```
```    42   proof
```
```    43     assume ab: "a * b = 0"
```
```    44     hence "0 = inverse a * (a * b) * inverse b" by simp
```
```    45     also have "\<dots> = (inverse a * a) * (b * inverse b)"
```
```    46       by (simp only: mult.assoc)
```
```    47     also have "\<dots> = 1" using a b by simp
```
```    48     finally show False by simp
```
```    49   qed
```
```    50 qed
```
```    51
```
```    52 lemma nonzero_imp_inverse_nonzero:
```
```    53   "a \<noteq> 0 \<Longrightarrow> inverse a \<noteq> 0"
```
```    54 proof
```
```    55   assume ianz: "inverse a = 0"
```
```    56   assume "a \<noteq> 0"
```
```    57   hence "1 = a * inverse a" by simp
```
```    58   also have "... = 0" by (simp add: ianz)
```
```    59   finally have "1 = 0" .
```
```    60   thus False by (simp add: eq_commute)
```
```    61 qed
```
```    62
```
```    63 lemma inverse_zero_imp_zero:
```
```    64   "inverse a = 0 \<Longrightarrow> a = 0"
```
```    65 apply (rule classical)
```
```    66 apply (drule nonzero_imp_inverse_nonzero)
```
```    67 apply auto
```
```    68 done
```
```    69
```
```    70 lemma inverse_unique:
```
```    71   assumes ab: "a * b = 1"
```
```    72   shows "inverse a = b"
```
```    73 proof -
```
```    74   have "a \<noteq> 0" using ab by (cases "a = 0") simp_all
```
```    75   moreover have "inverse a * (a * b) = inverse a" by (simp add: ab)
```
```    76   ultimately show ?thesis by (simp add: mult.assoc [symmetric])
```
```    77 qed
```
```    78
```
```    79 lemma nonzero_inverse_minus_eq:
```
```    80   "a \<noteq> 0 \<Longrightarrow> inverse (- a) = - inverse a"
```
```    81 by (rule inverse_unique) simp
```
```    82
```
```    83 lemma nonzero_inverse_inverse_eq:
```
```    84   "a \<noteq> 0 \<Longrightarrow> inverse (inverse a) = a"
```
```    85 by (rule inverse_unique) simp
```
```    86
```
```    87 lemma nonzero_inverse_eq_imp_eq:
```
```    88   assumes "inverse a = inverse b" and "a \<noteq> 0" and "b \<noteq> 0"
```
```    89   shows "a = b"
```
```    90 proof -
```
```    91   from `inverse a = inverse b`
```
```    92   have "inverse (inverse a) = inverse (inverse b)" by (rule arg_cong)
```
```    93   with `a \<noteq> 0` and `b \<noteq> 0` show "a = b"
```
```    94     by (simp add: nonzero_inverse_inverse_eq)
```
```    95 qed
```
```    96
```
```    97 lemma inverse_1 [simp]: "inverse 1 = 1"
```
```    98 by (rule inverse_unique) simp
```
```    99
```
```   100 lemma nonzero_inverse_mult_distrib:
```
```   101   assumes "a \<noteq> 0" and "b \<noteq> 0"
```
```   102   shows "inverse (a * b) = inverse b * inverse a"
```
```   103 proof -
```
```   104   have "a * (b * inverse b) * inverse a = 1" using assms by simp
```
```   105   hence "a * b * (inverse b * inverse a) = 1" by (simp only: mult.assoc)
```
```   106   thus ?thesis by (rule inverse_unique)
```
```   107 qed
```
```   108
```
```   109 lemma division_ring_inverse_add:
```
```   110   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a + inverse b = inverse a * (a + b) * inverse b"
```
```   111 by (simp add: algebra_simps)
```
```   112
```
```   113 lemma division_ring_inverse_diff:
```
```   114   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a - inverse b = inverse a * (b - a) * inverse b"
```
```   115 by (simp add: algebra_simps)
```
```   116
```
```   117 lemma right_inverse_eq: "b \<noteq> 0 \<Longrightarrow> a / b = 1 \<longleftrightarrow> a = b"
```
```   118 proof
```
```   119   assume neq: "b \<noteq> 0"
```
```   120   {
```
```   121     hence "a = (a / b) * b" by (simp add: divide_inverse mult.assoc)
```
```   122     also assume "a / b = 1"
```
```   123     finally show "a = b" by simp
```
```   124   next
```
```   125     assume "a = b"
```
```   126     with neq show "a / b = 1" by (simp add: divide_inverse)
```
```   127   }
```
```   128 qed
```
```   129
```
```   130 lemma nonzero_inverse_eq_divide: "a \<noteq> 0 \<Longrightarrow> inverse a = 1 / a"
```
```   131 by (simp add: divide_inverse)
```
```   132
```
```   133 lemma divide_self [simp]: "a \<noteq> 0 \<Longrightarrow> a / a = 1"
```
```   134 by (simp add: divide_inverse)
```
```   135
```
```   136 lemma divide_zero_left [simp]: "0 / a = 0"
```
```   137 by (simp add: divide_inverse)
```
```   138
```
```   139 lemma inverse_eq_divide [field_simps, divide_simps]: "inverse a = 1 / a"
```
```   140 by (simp add: divide_inverse)
```
```   141
```
```   142 lemma add_divide_distrib: "(a+b) / c = a/c + b/c"
```
```   143 by (simp add: divide_inverse algebra_simps)
```
```   144
```
```   145 lemma divide_1 [simp]: "a / 1 = a"
```
```   146   by (simp add: divide_inverse)
```
```   147
```
```   148 lemma times_divide_eq_right [simp]: "a * (b / c) = (a * b) / c"
```
```   149   by (simp add: divide_inverse mult.assoc)
```
```   150
```
```   151 lemma minus_divide_left: "- (a / b) = (-a) / b"
```
```   152   by (simp add: divide_inverse)
```
```   153
```
```   154 lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a / b) = a / (- b)"
```
```   155   by (simp add: divide_inverse nonzero_inverse_minus_eq)
```
```   156
```
```   157 lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a) / (-b) = a / b"
```
```   158   by (simp add: divide_inverse nonzero_inverse_minus_eq)
```
```   159
```
```   160 lemma divide_minus_left [simp]: "(-a) / b = - (a / b)"
```
```   161   by (simp add: divide_inverse)
```
```   162
```
```   163 lemma diff_divide_distrib: "(a - b) / c = a / c - b / c"
```
```   164   using add_divide_distrib [of a "- b" c] by simp
```
```   165
```
```   166 lemma nonzero_eq_divide_eq [field_simps]: "c \<noteq> 0 \<Longrightarrow> a = b / c \<longleftrightarrow> a * c = b"
```
```   167 proof -
```
```   168   assume [simp]: "c \<noteq> 0"
```
```   169   have "a = b / c \<longleftrightarrow> a * c = (b / c) * c" by simp
```
```   170   also have "... \<longleftrightarrow> a * c = b" by (simp add: divide_inverse mult.assoc)
```
```   171   finally show ?thesis .
```
```   172 qed
```
```   173
```
```   174 lemma nonzero_divide_eq_eq [field_simps]: "c \<noteq> 0 \<Longrightarrow> b / c = a \<longleftrightarrow> b = a * c"
```
```   175 proof -
```
```   176   assume [simp]: "c \<noteq> 0"
```
```   177   have "b / c = a \<longleftrightarrow> (b / c) * c = a * c" by simp
```
```   178   also have "... \<longleftrightarrow> b = a * c" by (simp add: divide_inverse mult.assoc)
```
```   179   finally show ?thesis .
```
```   180 qed
```
```   181
```
```   182 lemma nonzero_neg_divide_eq_eq [field_simps]: "b \<noteq> 0 \<Longrightarrow> - (a / b) = c \<longleftrightarrow> - a = c * b"
```
```   183   using nonzero_divide_eq_eq[of b "-a" c] by (simp add: divide_minus_left)
```
```   184
```
```   185 lemma nonzero_neg_divide_eq_eq2 [field_simps]: "b \<noteq> 0 \<Longrightarrow> c = - (a / b) \<longleftrightarrow> c * b = - a"
```
```   186   using nonzero_neg_divide_eq_eq[of b a c] by auto
```
```   187
```
```   188 lemma divide_eq_imp: "c \<noteq> 0 \<Longrightarrow> b = a * c \<Longrightarrow> b / c = a"
```
```   189   by (simp add: divide_inverse mult.assoc)
```
```   190
```
```   191 lemma eq_divide_imp: "c \<noteq> 0 \<Longrightarrow> a * c = b \<Longrightarrow> a = b / c"
```
```   192   by (drule sym) (simp add: divide_inverse mult.assoc)
```
```   193
```
```   194 lemma add_divide_eq_iff [field_simps]:
```
```   195   "z \<noteq> 0 \<Longrightarrow> x + y / z = (x * z + y) / z"
```
```   196   by (simp add: add_divide_distrib nonzero_eq_divide_eq)
```
```   197
```
```   198 lemma divide_add_eq_iff [field_simps]:
```
```   199   "z \<noteq> 0 \<Longrightarrow> x / z + y = (x + y * z) / z"
```
```   200   by (simp add: add_divide_distrib nonzero_eq_divide_eq)
```
```   201
```
```   202 lemma diff_divide_eq_iff [field_simps]:
```
```   203   "z \<noteq> 0 \<Longrightarrow> x - y / z = (x * z - y) / z"
```
```   204   by (simp add: diff_divide_distrib nonzero_eq_divide_eq eq_diff_eq)
```
```   205
```
```   206 lemma minus_divide_add_eq_iff [field_simps]:
```
```   207   "z \<noteq> 0 \<Longrightarrow> - (x / z) + y = (- x + y * z) / z"
```
```   208   by (simp add: add_divide_distrib diff_divide_eq_iff divide_minus_left)
```
```   209
```
```   210 lemma divide_diff_eq_iff [field_simps]:
```
```   211   "z \<noteq> 0 \<Longrightarrow> x / z - y = (x - y * z) / z"
```
```   212   by (simp add: field_simps)
```
```   213
```
```   214 lemma minus_divide_diff_eq_iff [field_simps]:
```
```   215   "z \<noteq> 0 \<Longrightarrow> - (x / z) - y = (- x - y * z) / z"
```
```   216   by (simp add: divide_diff_eq_iff[symmetric] divide_minus_left)
```
```   217
```
```   218 end
```
```   219
```
```   220 class division_ring_inverse_zero = division_ring +
```
```   221   assumes inverse_zero [simp]: "inverse 0 = 0"
```
```   222 begin
```
```   223
```
```   224 lemma divide_zero [simp]:
```
```   225   "a / 0 = 0"
```
```   226   by (simp add: divide_inverse)
```
```   227
```
```   228 lemma divide_self_if [simp]:
```
```   229   "a / a = (if a = 0 then 0 else 1)"
```
```   230   by simp
```
```   231
```
```   232 lemma inverse_nonzero_iff_nonzero [simp]:
```
```   233   "inverse a = 0 \<longleftrightarrow> a = 0"
```
```   234   by rule (fact inverse_zero_imp_zero, simp)
```
```   235
```
```   236 lemma inverse_minus_eq [simp]:
```
```   237   "inverse (- a) = - inverse a"
```
```   238 proof cases
```
```   239   assume "a=0" thus ?thesis by simp
```
```   240 next
```
```   241   assume "a\<noteq>0"
```
```   242   thus ?thesis by (simp add: nonzero_inverse_minus_eq)
```
```   243 qed
```
```   244
```
```   245 lemma inverse_inverse_eq [simp]:
```
```   246   "inverse (inverse a) = a"
```
```   247 proof cases
```
```   248   assume "a=0" thus ?thesis by simp
```
```   249 next
```
```   250   assume "a\<noteq>0"
```
```   251   thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
```
```   252 qed
```
```   253
```
```   254 lemma inverse_eq_imp_eq:
```
```   255   "inverse a = inverse b \<Longrightarrow> a = b"
```
```   256   by (drule arg_cong [where f="inverse"], simp)
```
```   257
```
```   258 lemma inverse_eq_iff_eq [simp]:
```
```   259   "inverse a = inverse b \<longleftrightarrow> a = b"
```
```   260   by (force dest!: inverse_eq_imp_eq)
```
```   261
```
```   262 lemma add_divide_eq_if_simps [divide_simps]:
```
```   263     "a + b / z = (if z = 0 then a else (a * z + b) / z)"
```
```   264     "a / z + b = (if z = 0 then b else (a + b * z) / z)"
```
```   265     "- (a / z) + b = (if z = 0 then b else (-a + b * z) / z)"
```
```   266     "a - b / z = (if z = 0 then a else (a * z - b) / z)"
```
```   267     "a / z - b = (if z = 0 then -b else (a - b * z) / z)"
```
```   268     "- (a / z) - b = (if z = 0 then -b else (- a - b * z) / z)"
```
```   269   by (simp_all add: add_divide_eq_iff divide_add_eq_iff diff_divide_eq_iff divide_diff_eq_iff
```
```   270       minus_divide_diff_eq_iff)
```
```   271
```
```   272 lemma [divide_simps]:
```
```   273   shows divide_eq_eq: "b / c = a \<longleftrightarrow> (if c \<noteq> 0 then b = a * c else a = 0)"
```
```   274     and eq_divide_eq: "a = b / c \<longleftrightarrow> (if c \<noteq> 0 then a * c = b else a = 0)"
```
```   275     and minus_divide_eq_eq: "- (b / c) = a \<longleftrightarrow> (if c \<noteq> 0 then - b = a * c else a = 0)"
```
```   276     and eq_minus_divide_eq: "a = - (b / c) \<longleftrightarrow> (if c \<noteq> 0 then a * c = - b else a = 0)"
```
```   277   by (auto simp add:  field_simps)
```
```   278
```
```   279 end
```
```   280
```
```   281 subsection {* Fields *}
```
```   282
```
```   283 class field = comm_ring_1 + inverse +
```
```   284   assumes field_inverse: "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
```
```   285   assumes field_divide_inverse: "a / b = a * inverse b"
```
```   286 begin
```
```   287
```
```   288 subclass division_ring
```
```   289 proof
```
```   290   fix a :: 'a
```
```   291   assume "a \<noteq> 0"
```
```   292   thus "inverse a * a = 1" by (rule field_inverse)
```
```   293   thus "a * inverse a = 1" by (simp only: mult.commute)
```
```   294 next
```
```   295   fix a b :: 'a
```
```   296   show "a / b = a * inverse b" by (rule field_divide_inverse)
```
```   297 qed
```
```   298
```
```   299 subclass idom ..
```
```   300
```
```   301 text{*There is no slick version using division by zero.*}
```
```   302 lemma inverse_add:
```
```   303   "[| a \<noteq> 0;  b \<noteq> 0 |]
```
```   304    ==> inverse a + inverse b = (a + b) * inverse a * inverse b"
```
```   305 by (simp add: division_ring_inverse_add ac_simps)
```
```   306
```
```   307 lemma nonzero_mult_divide_mult_cancel_left [simp]:
```
```   308 assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" shows "(c*a)/(c*b) = a/b"
```
```   309 proof -
```
```   310   have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
```
```   311     by (simp add: divide_inverse nonzero_inverse_mult_distrib)
```
```   312   also have "... =  a * inverse b * (inverse c * c)"
```
```   313     by (simp only: ac_simps)
```
```   314   also have "... =  a * inverse b" by simp
```
```   315     finally show ?thesis by (simp add: divide_inverse)
```
```   316 qed
```
```   317
```
```   318 lemma nonzero_mult_divide_mult_cancel_right [simp]:
```
```   319   "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (a * c) / (b * c) = a / b"
```
```   320 by (simp add: mult.commute [of _ c])
```
```   321
```
```   322 lemma times_divide_eq_left [simp]: "(b / c) * a = (b * a) / c"
```
```   323   by (simp add: divide_inverse ac_simps)
```
```   324
```
```   325 text{*It's not obvious whether @{text times_divide_eq} should be
```
```   326   simprules or not. Their effect is to gather terms into one big
```
```   327   fraction, like a*b*c / x*y*z. The rationale for that is unclear, but
```
```   328   many proofs seem to need them.*}
```
```   329
```
```   330 lemmas times_divide_eq = times_divide_eq_right times_divide_eq_left
```
```   331
```
```   332 lemma add_frac_eq:
```
```   333   assumes "y \<noteq> 0" and "z \<noteq> 0"
```
```   334   shows "x / y + w / z = (x * z + w * y) / (y * z)"
```
```   335 proof -
```
```   336   have "x / y + w / z = (x * z) / (y * z) + (y * w) / (y * z)"
```
```   337     using assms by simp
```
```   338   also have "\<dots> = (x * z + y * w) / (y * z)"
```
```   339     by (simp only: add_divide_distrib)
```
```   340   finally show ?thesis
```
```   341     by (simp only: mult.commute)
```
```   342 qed
```
```   343
```
```   344 text{*Special Cancellation Simprules for Division*}
```
```   345
```
```   346 lemma nonzero_mult_divide_cancel_right [simp]:
```
```   347   "b \<noteq> 0 \<Longrightarrow> a * b / b = a"
```
```   348   using nonzero_mult_divide_mult_cancel_right [of 1 b a] by simp
```
```   349
```
```   350 lemma nonzero_mult_divide_cancel_left [simp]:
```
```   351   "a \<noteq> 0 \<Longrightarrow> a * b / a = b"
```
```   352 using nonzero_mult_divide_mult_cancel_left [of 1 a b] by simp
```
```   353
```
```   354 lemma nonzero_divide_mult_cancel_right [simp]:
```
```   355   "\<lbrakk>a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> b / (a * b) = 1 / a"
```
```   356 using nonzero_mult_divide_mult_cancel_right [of a b 1] by simp
```
```   357
```
```   358 lemma nonzero_divide_mult_cancel_left [simp]:
```
```   359   "\<lbrakk>a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / (a * b) = 1 / b"
```
```   360 using nonzero_mult_divide_mult_cancel_left [of b a 1] by simp
```
```   361
```
```   362 lemma nonzero_mult_divide_mult_cancel_left2 [simp]:
```
```   363   "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (c * a) / (b * c) = a / b"
```
```   364 using nonzero_mult_divide_mult_cancel_left [of b c a] by (simp add: ac_simps)
```
```   365
```
```   366 lemma nonzero_mult_divide_mult_cancel_right2 [simp]:
```
```   367   "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (a * c) / (c * b) = a / b"
```
```   368 using nonzero_mult_divide_mult_cancel_right [of b c a] by (simp add: ac_simps)
```
```   369
```
```   370 lemma diff_frac_eq:
```
```   371   "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y - w / z = (x * z - w * y) / (y * z)"
```
```   372   by (simp add: field_simps)
```
```   373
```
```   374 lemma frac_eq_eq:
```
```   375   "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> (x / y = w / z) = (x * z = w * y)"
```
```   376   by (simp add: field_simps)
```
```   377
```
```   378 end
```
```   379
```
```   380 class field_inverse_zero = field +
```
```   381   assumes field_inverse_zero: "inverse 0 = 0"
```
```   382 begin
```
```   383
```
```   384 subclass division_ring_inverse_zero proof
```
```   385 qed (fact field_inverse_zero)
```
```   386
```
```   387 text{*This version builds in division by zero while also re-orienting
```
```   388       the right-hand side.*}
```
```   389 lemma inverse_mult_distrib [simp]:
```
```   390   "inverse (a * b) = inverse a * inverse b"
```
```   391 proof cases
```
```   392   assume "a \<noteq> 0 & b \<noteq> 0"
```
```   393   thus ?thesis by (simp add: nonzero_inverse_mult_distrib ac_simps)
```
```   394 next
```
```   395   assume "~ (a \<noteq> 0 & b \<noteq> 0)"
```
```   396   thus ?thesis by force
```
```   397 qed
```
```   398
```
```   399 lemma inverse_divide [simp]:
```
```   400   "inverse (a / b) = b / a"
```
```   401   by (simp add: divide_inverse mult.commute)
```
```   402
```
```   403
```
```   404 text {* Calculations with fractions *}
```
```   405
```
```   406 text{* There is a whole bunch of simp-rules just for class @{text
```
```   407 field} but none for class @{text field} and @{text nonzero_divides}
```
```   408 because the latter are covered by a simproc. *}
```
```   409
```
```   410 lemma mult_divide_mult_cancel_left:
```
```   411   "c \<noteq> 0 \<Longrightarrow> (c * a) / (c * b) = a / b"
```
```   412 apply (cases "b = 0")
```
```   413 apply simp_all
```
```   414 done
```
```   415
```
```   416 lemma mult_divide_mult_cancel_right:
```
```   417   "c \<noteq> 0 \<Longrightarrow> (a * c) / (b * c) = a / b"
```
```   418 apply (cases "b = 0")
```
```   419 apply simp_all
```
```   420 done
```
```   421
```
```   422 lemma divide_divide_eq_right [simp]:
```
```   423   "a / (b / c) = (a * c) / b"
```
```   424   by (simp add: divide_inverse ac_simps)
```
```   425
```
```   426 lemma divide_divide_eq_left [simp]:
```
```   427   "(a / b) / c = a / (b * c)"
```
```   428   by (simp add: divide_inverse mult.assoc)
```
```   429
```
```   430 lemma divide_divide_times_eq:
```
```   431   "(x / y) / (z / w) = (x * w) / (y * z)"
```
```   432   by simp
```
```   433
```
```   434 text {*Special Cancellation Simprules for Division*}
```
```   435
```
```   436 lemma mult_divide_mult_cancel_left_if [simp]:
```
```   437   shows "(c * a) / (c * b) = (if c = 0 then 0 else a / b)"
```
```   438   by (simp add: mult_divide_mult_cancel_left)
```
```   439
```
```   440
```
```   441 text {* Division and Unary Minus *}
```
```   442
```
```   443 lemma minus_divide_right:
```
```   444   "- (a / b) = a / - b"
```
```   445   by (simp add: divide_inverse)
```
```   446
```
```   447 lemma divide_minus_right [simp]:
```
```   448   "a / - b = - (a / b)"
```
```   449   by (simp add: divide_inverse)
```
```   450
```
```   451 lemma minus_divide_divide:
```
```   452   "(- a) / (- b) = a / b"
```
```   453 apply (cases "b=0", simp)
```
```   454 apply (simp add: nonzero_minus_divide_divide)
```
```   455 done
```
```   456
```
```   457 lemma inverse_eq_1_iff [simp]:
```
```   458   "inverse x = 1 \<longleftrightarrow> x = 1"
```
```   459   by (insert inverse_eq_iff_eq [of x 1], simp)
```
```   460
```
```   461 lemma divide_eq_0_iff [simp]:
```
```   462   "a / b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
```
```   463   by (simp add: divide_inverse)
```
```   464
```
```   465 lemma divide_cancel_right [simp]:
```
```   466   "a / c = b / c \<longleftrightarrow> c = 0 \<or> a = b"
```
```   467   apply (cases "c=0", simp)
```
```   468   apply (simp add: divide_inverse)
```
```   469   done
```
```   470
```
```   471 lemma divide_cancel_left [simp]:
```
```   472   "c / a = c / b \<longleftrightarrow> c = 0 \<or> a = b"
```
```   473   apply (cases "c=0", simp)
```
```   474   apply (simp add: divide_inverse)
```
```   475   done
```
```   476
```
```   477 lemma divide_eq_1_iff [simp]:
```
```   478   "a / b = 1 \<longleftrightarrow> b \<noteq> 0 \<and> a = b"
```
```   479   apply (cases "b=0", simp)
```
```   480   apply (simp add: right_inverse_eq)
```
```   481   done
```
```   482
```
```   483 lemma one_eq_divide_iff [simp]:
```
```   484   "1 = a / b \<longleftrightarrow> b \<noteq> 0 \<and> a = b"
```
```   485   by (simp add: eq_commute [of 1])
```
```   486
```
```   487 lemma times_divide_times_eq:
```
```   488   "(x / y) * (z / w) = (x * z) / (y * w)"
```
```   489   by simp
```
```   490
```
```   491 lemma add_frac_num:
```
```   492   "y \<noteq> 0 \<Longrightarrow> x / y + z = (x + z * y) / y"
```
```   493   by (simp add: add_divide_distrib)
```
```   494
```
```   495 lemma add_num_frac:
```
```   496   "y \<noteq> 0 \<Longrightarrow> z + x / y = (x + z * y) / y"
```
```   497   by (simp add: add_divide_distrib add.commute)
```
```   498
```
```   499 end
```
```   500
```
```   501
```
```   502 subsection {* Ordered fields *}
```
```   503
```
```   504 class linordered_field = field + linordered_idom
```
```   505 begin
```
```   506
```
```   507 lemma positive_imp_inverse_positive:
```
```   508   assumes a_gt_0: "0 < a"
```
```   509   shows "0 < inverse a"
```
```   510 proof -
```
```   511   have "0 < a * inverse a"
```
```   512     by (simp add: a_gt_0 [THEN less_imp_not_eq2])
```
```   513   thus "0 < inverse a"
```
```   514     by (simp add: a_gt_0 [THEN less_not_sym] zero_less_mult_iff)
```
```   515 qed
```
```   516
```
```   517 lemma negative_imp_inverse_negative:
```
```   518   "a < 0 \<Longrightarrow> inverse a < 0"
```
```   519   by (insert positive_imp_inverse_positive [of "-a"],
```
```   520     simp add: nonzero_inverse_minus_eq less_imp_not_eq)
```
```   521
```
```   522 lemma inverse_le_imp_le:
```
```   523   assumes invle: "inverse a \<le> inverse b" and apos: "0 < a"
```
```   524   shows "b \<le> a"
```
```   525 proof (rule classical)
```
```   526   assume "~ b \<le> a"
```
```   527   hence "a < b"  by (simp add: linorder_not_le)
```
```   528   hence bpos: "0 < b"  by (blast intro: apos less_trans)
```
```   529   hence "a * inverse a \<le> a * inverse b"
```
```   530     by (simp add: apos invle less_imp_le mult_left_mono)
```
```   531   hence "(a * inverse a) * b \<le> (a * inverse b) * b"
```
```   532     by (simp add: bpos less_imp_le mult_right_mono)
```
```   533   thus "b \<le> a"  by (simp add: mult.assoc apos bpos less_imp_not_eq2)
```
```   534 qed
```
```   535
```
```   536 lemma inverse_positive_imp_positive:
```
```   537   assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0"
```
```   538   shows "0 < a"
```
```   539 proof -
```
```   540   have "0 < inverse (inverse a)"
```
```   541     using inv_gt_0 by (rule positive_imp_inverse_positive)
```
```   542   thus "0 < a"
```
```   543     using nz by (simp add: nonzero_inverse_inverse_eq)
```
```   544 qed
```
```   545
```
```   546 lemma inverse_negative_imp_negative:
```
```   547   assumes inv_less_0: "inverse a < 0" and nz: "a \<noteq> 0"
```
```   548   shows "a < 0"
```
```   549 proof -
```
```   550   have "inverse (inverse a) < 0"
```
```   551     using inv_less_0 by (rule negative_imp_inverse_negative)
```
```   552   thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq)
```
```   553 qed
```
```   554
```
```   555 lemma linordered_field_no_lb:
```
```   556   "\<forall>x. \<exists>y. y < x"
```
```   557 proof
```
```   558   fix x::'a
```
```   559   have m1: "- (1::'a) < 0" by simp
```
```   560   from add_strict_right_mono[OF m1, where c=x]
```
```   561   have "(- 1) + x < x" by simp
```
```   562   thus "\<exists>y. y < x" by blast
```
```   563 qed
```
```   564
```
```   565 lemma linordered_field_no_ub:
```
```   566   "\<forall> x. \<exists>y. y > x"
```
```   567 proof
```
```   568   fix x::'a
```
```   569   have m1: " (1::'a) > 0" by simp
```
```   570   from add_strict_right_mono[OF m1, where c=x]
```
```   571   have "1 + x > x" by simp
```
```   572   thus "\<exists>y. y > x" by blast
```
```   573 qed
```
```   574
```
```   575 lemma less_imp_inverse_less:
```
```   576   assumes less: "a < b" and apos:  "0 < a"
```
```   577   shows "inverse b < inverse a"
```
```   578 proof (rule ccontr)
```
```   579   assume "~ inverse b < inverse a"
```
```   580   hence "inverse a \<le> inverse b" by simp
```
```   581   hence "~ (a < b)"
```
```   582     by (simp add: not_less inverse_le_imp_le [OF _ apos])
```
```   583   thus False by (rule notE [OF _ less])
```
```   584 qed
```
```   585
```
```   586 lemma inverse_less_imp_less:
```
```   587   "inverse a < inverse b \<Longrightarrow> 0 < a \<Longrightarrow> b < a"
```
```   588 apply (simp add: less_le [of "inverse a"] less_le [of "b"])
```
```   589 apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq)
```
```   590 done
```
```   591
```
```   592 text{*Both premises are essential. Consider -1 and 1.*}
```
```   593 lemma inverse_less_iff_less [simp]:
```
```   594   "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"
```
```   595   by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less)
```
```   596
```
```   597 lemma le_imp_inverse_le:
```
```   598   "a \<le> b \<Longrightarrow> 0 < a \<Longrightarrow> inverse b \<le> inverse a"
```
```   599   by (force simp add: le_less less_imp_inverse_less)
```
```   600
```
```   601 lemma inverse_le_iff_le [simp]:
```
```   602   "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a"
```
```   603   by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le)
```
```   604
```
```   605
```
```   606 text{*These results refer to both operands being negative.  The opposite-sign
```
```   607 case is trivial, since inverse preserves signs.*}
```
```   608 lemma inverse_le_imp_le_neg:
```
```   609   "inverse a \<le> inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b \<le> a"
```
```   610 apply (rule classical)
```
```   611 apply (subgoal_tac "a < 0")
```
```   612  prefer 2 apply force
```
```   613 apply (insert inverse_le_imp_le [of "-b" "-a"])
```
```   614 apply (simp add: nonzero_inverse_minus_eq)
```
```   615 done
```
```   616
```
```   617 lemma less_imp_inverse_less_neg:
```
```   618    "a < b \<Longrightarrow> b < 0 \<Longrightarrow> inverse b < inverse a"
```
```   619 apply (subgoal_tac "a < 0")
```
```   620  prefer 2 apply (blast intro: less_trans)
```
```   621 apply (insert less_imp_inverse_less [of "-b" "-a"])
```
```   622 apply (simp add: nonzero_inverse_minus_eq)
```
```   623 done
```
```   624
```
```   625 lemma inverse_less_imp_less_neg:
```
```   626    "inverse a < inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b < a"
```
```   627 apply (rule classical)
```
```   628 apply (subgoal_tac "a < 0")
```
```   629  prefer 2
```
```   630  apply force
```
```   631 apply (insert inverse_less_imp_less [of "-b" "-a"])
```
```   632 apply (simp add: nonzero_inverse_minus_eq)
```
```   633 done
```
```   634
```
```   635 lemma inverse_less_iff_less_neg [simp]:
```
```   636   "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"
```
```   637 apply (insert inverse_less_iff_less [of "-b" "-a"])
```
```   638 apply (simp del: inverse_less_iff_less
```
```   639             add: nonzero_inverse_minus_eq)
```
```   640 done
```
```   641
```
```   642 lemma le_imp_inverse_le_neg:
```
```   643   "a \<le> b \<Longrightarrow> b < 0 ==> inverse b \<le> inverse a"
```
```   644   by (force simp add: le_less less_imp_inverse_less_neg)
```
```   645
```
```   646 lemma inverse_le_iff_le_neg [simp]:
```
```   647   "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a"
```
```   648   by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg)
```
```   649
```
```   650 lemma one_less_inverse:
```
```   651   "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> 1 < inverse a"
```
```   652   using less_imp_inverse_less [of a 1, unfolded inverse_1] .
```
```   653
```
```   654 lemma one_le_inverse:
```
```   655   "0 < a \<Longrightarrow> a \<le> 1 \<Longrightarrow> 1 \<le> inverse a"
```
```   656   using le_imp_inverse_le [of a 1, unfolded inverse_1] .
```
```   657
```
```   658 lemma pos_le_divide_eq [field_simps]: "0 < c \<Longrightarrow> a \<le> b / c \<longleftrightarrow> a * c \<le> b"
```
```   659 proof -
```
```   660   assume less: "0<c"
```
```   661   hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
```
```   662     by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
```
```   663   also have "... = (a*c \<le> b)"
```
```   664     by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult.assoc)
```
```   665   finally show ?thesis .
```
```   666 qed
```
```   667
```
```   668 lemma neg_le_divide_eq [field_simps]: "c < 0 \<Longrightarrow> a \<le> b / c \<longleftrightarrow> b \<le> a * c"
```
```   669 proof -
```
```   670   assume less: "c<0"
```
```   671   hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
```
```   672     by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
```
```   673   also have "... = (b \<le> a*c)"
```
```   674     by (simp add: less_imp_not_eq [OF less] divide_inverse mult.assoc)
```
```   675   finally show ?thesis .
```
```   676 qed
```
```   677
```
```   678 lemma pos_less_divide_eq [field_simps]: "0 < c \<Longrightarrow> a < b / c \<longleftrightarrow> a * c < b"
```
```   679 proof -
```
```   680   assume less: "0<c"
```
```   681   hence "(a < b/c) = (a*c < (b/c)*c)"
```
```   682     by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq)
```
```   683   also have "... = (a*c < b)"
```
```   684     by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult.assoc)
```
```   685   finally show ?thesis .
```
```   686 qed
```
```   687
```
```   688 lemma neg_less_divide_eq [field_simps]: "c < 0 \<Longrightarrow> a < b / c \<longleftrightarrow> b < a * c"
```
```   689 proof -
```
```   690   assume less: "c<0"
```
```   691   hence "(a < b/c) = ((b/c)*c < a*c)"
```
```   692     by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq)
```
```   693   also have "... = (b < a*c)"
```
```   694     by (simp add: less_imp_not_eq [OF less] divide_inverse mult.assoc)
```
```   695   finally show ?thesis .
```
```   696 qed
```
```   697
```
```   698 lemma pos_divide_less_eq [field_simps]: "0 < c \<Longrightarrow> b / c < a \<longleftrightarrow> b < a * c"
```
```   699 proof -
```
```   700   assume less: "0<c"
```
```   701   hence "(b/c < a) = ((b/c)*c < a*c)"
```
```   702     by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq)
```
```   703   also have "... = (b < a*c)"
```
```   704     by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult.assoc)
```
```   705   finally show ?thesis .
```
```   706 qed
```
```   707
```
```   708 lemma neg_divide_less_eq [field_simps]: "c < 0 \<Longrightarrow> b / c < a \<longleftrightarrow> a * c < b"
```
```   709 proof -
```
```   710   assume less: "c<0"
```
```   711   hence "(b/c < a) = (a*c < (b/c)*c)"
```
```   712     by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq)
```
```   713   also have "... = (a*c < b)"
```
```   714     by (simp add: less_imp_not_eq [OF less] divide_inverse mult.assoc)
```
```   715   finally show ?thesis .
```
```   716 qed
```
```   717
```
```   718 lemma pos_divide_le_eq [field_simps]: "0 < c \<Longrightarrow> b / c \<le> a \<longleftrightarrow> b \<le> a * c"
```
```   719 proof -
```
```   720   assume less: "0<c"
```
```   721   hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
```
```   722     by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
```
```   723   also have "... = (b \<le> a*c)"
```
```   724     by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult.assoc)
```
```   725   finally show ?thesis .
```
```   726 qed
```
```   727
```
```   728 lemma neg_divide_le_eq [field_simps]: "c < 0 \<Longrightarrow> b / c \<le> a \<longleftrightarrow> a * c \<le> b"
```
```   729 proof -
```
```   730   assume less: "c<0"
```
```   731   hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
```
```   732     by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
```
```   733   also have "... = (a*c \<le> b)"
```
```   734     by (simp add: less_imp_not_eq [OF less] divide_inverse mult.assoc)
```
```   735   finally show ?thesis .
```
```   736 qed
```
```   737
```
```   738 text{* The following @{text field_simps} rules are necessary, as minus is always moved atop of
```
```   739 division but we want to get rid of division. *}
```
```   740
```
```   741 lemma pos_le_minus_divide_eq [field_simps]: "0 < c \<Longrightarrow> a \<le> - (b / c) \<longleftrightarrow> a * c \<le> - b"
```
```   742   unfolding minus_divide_left by (rule pos_le_divide_eq)
```
```   743
```
```   744 lemma neg_le_minus_divide_eq [field_simps]: "c < 0 \<Longrightarrow> a \<le> - (b / c) \<longleftrightarrow> - b \<le> a * c"
```
```   745   unfolding minus_divide_left by (rule neg_le_divide_eq)
```
```   746
```
```   747 lemma pos_less_minus_divide_eq [field_simps]: "0 < c \<Longrightarrow> a < - (b / c) \<longleftrightarrow> a * c < - b"
```
```   748   unfolding minus_divide_left by (rule pos_less_divide_eq)
```
```   749
```
```   750 lemma neg_less_minus_divide_eq [field_simps]: "c < 0 \<Longrightarrow> a < - (b / c) \<longleftrightarrow> - b < a * c"
```
```   751   unfolding minus_divide_left by (rule neg_less_divide_eq)
```
```   752
```
```   753 lemma pos_minus_divide_less_eq [field_simps]: "0 < c \<Longrightarrow> - (b / c) < a \<longleftrightarrow> - b < a * c"
```
```   754   unfolding minus_divide_left by (rule pos_divide_less_eq)
```
```   755
```
```   756 lemma neg_minus_divide_less_eq [field_simps]: "c < 0 \<Longrightarrow> - (b / c) < a \<longleftrightarrow> a * c < - b"
```
```   757   unfolding minus_divide_left by (rule neg_divide_less_eq)
```
```   758
```
```   759 lemma pos_minus_divide_le_eq [field_simps]: "0 < c \<Longrightarrow> - (b / c) \<le> a \<longleftrightarrow> - b \<le> a * c"
```
```   760   unfolding minus_divide_left by (rule pos_divide_le_eq)
```
```   761
```
```   762 lemma neg_minus_divide_le_eq [field_simps]: "c < 0 \<Longrightarrow> - (b / c) \<le> a \<longleftrightarrow> a * c \<le> - b"
```
```   763   unfolding minus_divide_left by (rule neg_divide_le_eq)
```
```   764
```
```   765 lemma frac_less_eq:
```
```   766   "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y < w / z \<longleftrightarrow> (x * z - w * y) / (y * z) < 0"
```
```   767   by (subst less_iff_diff_less_0) (simp add: diff_frac_eq )
```
```   768
```
```   769 lemma frac_le_eq:
```
```   770   "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y \<le> w / z \<longleftrightarrow> (x * z - w * y) / (y * z) \<le> 0"
```
```   771   by (subst le_iff_diff_le_0) (simp add: diff_frac_eq )
```
```   772
```
```   773 text{* Lemmas @{text sign_simps} is a first attempt to automate proofs
```
```   774 of positivity/negativity needed for @{text field_simps}. Have not added @{text
```
```   775 sign_simps} to @{text field_simps} because the former can lead to case
```
```   776 explosions. *}
```
```   777
```
```   778 lemmas sign_simps = algebra_simps zero_less_mult_iff mult_less_0_iff
```
```   779
```
```   780 lemmas (in -) sign_simps = algebra_simps zero_less_mult_iff mult_less_0_iff
```
```   781
```
```   782 (* Only works once linear arithmetic is installed:
```
```   783 text{*An example:*}
```
```   784 lemma fixes a b c d e f :: "'a::linordered_field"
```
```   785 shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow>
```
```   786  ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) <
```
```   787  ((e-f)*(a-b)*(c-d))/((e-f)*(a-b)*(c-d)) + u"
```
```   788 apply(subgoal_tac "(c-d)*(e-f)*(a-b) > 0")
```
```   789  prefer 2 apply(simp add:sign_simps)
```
```   790 apply(subgoal_tac "(c-d)*(e-f)*(a-b)*u > 0")
```
```   791  prefer 2 apply(simp add:sign_simps)
```
```   792 apply(simp add:field_simps)
```
```   793 done
```
```   794 *)
```
```   795
```
```   796 lemma divide_pos_pos[simp]:
```
```   797   "0 < x ==> 0 < y ==> 0 < x / y"
```
```   798 by(simp add:field_simps)
```
```   799
```
```   800 lemma divide_nonneg_pos:
```
```   801   "0 <= x ==> 0 < y ==> 0 <= x / y"
```
```   802 by(simp add:field_simps)
```
```   803
```
```   804 lemma divide_neg_pos:
```
```   805   "x < 0 ==> 0 < y ==> x / y < 0"
```
```   806 by(simp add:field_simps)
```
```   807
```
```   808 lemma divide_nonpos_pos:
```
```   809   "x <= 0 ==> 0 < y ==> x / y <= 0"
```
```   810 by(simp add:field_simps)
```
```   811
```
```   812 lemma divide_pos_neg:
```
```   813   "0 < x ==> y < 0 ==> x / y < 0"
```
```   814 by(simp add:field_simps)
```
```   815
```
```   816 lemma divide_nonneg_neg:
```
```   817   "0 <= x ==> y < 0 ==> x / y <= 0"
```
```   818 by(simp add:field_simps)
```
```   819
```
```   820 lemma divide_neg_neg:
```
```   821   "x < 0 ==> y < 0 ==> 0 < x / y"
```
```   822 by(simp add:field_simps)
```
```   823
```
```   824 lemma divide_nonpos_neg:
```
```   825   "x <= 0 ==> y < 0 ==> 0 <= x / y"
```
```   826 by(simp add:field_simps)
```
```   827
```
```   828 lemma divide_strict_right_mono:
```
```   829      "[|a < b; 0 < c|] ==> a / c < b / c"
```
```   830 by (simp add: less_imp_not_eq2 divide_inverse mult_strict_right_mono
```
```   831               positive_imp_inverse_positive)
```
```   832
```
```   833
```
```   834 lemma divide_strict_right_mono_neg:
```
```   835      "[|b < a; c < 0|] ==> a / c < b / c"
```
```   836 apply (drule divide_strict_right_mono [of _ _ "-c"], simp)
```
```   837 apply (simp add: less_imp_not_eq nonzero_minus_divide_right [symmetric])
```
```   838 done
```
```   839
```
```   840 text{*The last premise ensures that @{term a} and @{term b}
```
```   841       have the same sign*}
```
```   842 lemma divide_strict_left_mono:
```
```   843   "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / b"
```
```   844   by (auto simp: field_simps zero_less_mult_iff mult_strict_right_mono)
```
```   845
```
```   846 lemma divide_left_mono:
```
```   847   "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / b"
```
```   848   by (auto simp: field_simps zero_less_mult_iff mult_right_mono)
```
```   849
```
```   850 lemma divide_strict_left_mono_neg:
```
```   851   "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / b"
```
```   852   by (auto simp: field_simps zero_less_mult_iff mult_strict_right_mono_neg)
```
```   853
```
```   854 lemma mult_imp_div_pos_le: "0 < y ==> x <= z * y ==>
```
```   855     x / y <= z"
```
```   856 by (subst pos_divide_le_eq, assumption+)
```
```   857
```
```   858 lemma mult_imp_le_div_pos: "0 < y ==> z * y <= x ==>
```
```   859     z <= x / y"
```
```   860 by(simp add:field_simps)
```
```   861
```
```   862 lemma mult_imp_div_pos_less: "0 < y ==> x < z * y ==>
```
```   863     x / y < z"
```
```   864 by(simp add:field_simps)
```
```   865
```
```   866 lemma mult_imp_less_div_pos: "0 < y ==> z * y < x ==>
```
```   867     z < x / y"
```
```   868 by(simp add:field_simps)
```
```   869
```
```   870 lemma frac_le: "0 <= x ==>
```
```   871     x <= y ==> 0 < w ==> w <= z  ==> x / z <= y / w"
```
```   872   apply (rule mult_imp_div_pos_le)
```
```   873   apply simp
```
```   874   apply (subst times_divide_eq_left)
```
```   875   apply (rule mult_imp_le_div_pos, assumption)
```
```   876   apply (rule mult_mono)
```
```   877   apply simp_all
```
```   878 done
```
```   879
```
```   880 lemma frac_less: "0 <= x ==>
```
```   881     x < y ==> 0 < w ==> w <= z  ==> x / z < y / w"
```
```   882   apply (rule mult_imp_div_pos_less)
```
```   883   apply simp
```
```   884   apply (subst times_divide_eq_left)
```
```   885   apply (rule mult_imp_less_div_pos, assumption)
```
```   886   apply (erule mult_less_le_imp_less)
```
```   887   apply simp_all
```
```   888 done
```
```   889
```
```   890 lemma frac_less2: "0 < x ==>
```
```   891     x <= y ==> 0 < w ==> w < z  ==> x / z < y / w"
```
```   892   apply (rule mult_imp_div_pos_less)
```
```   893   apply simp_all
```
```   894   apply (rule mult_imp_less_div_pos, assumption)
```
```   895   apply (erule mult_le_less_imp_less)
```
```   896   apply simp_all
```
```   897 done
```
```   898
```
```   899 lemma less_half_sum: "a < b ==> a < (a+b) / (1+1)"
```
```   900 by (simp add: field_simps zero_less_two)
```
```   901
```
```   902 lemma gt_half_sum: "a < b ==> (a+b)/(1+1) < b"
```
```   903 by (simp add: field_simps zero_less_two)
```
```   904
```
```   905 subclass unbounded_dense_linorder
```
```   906 proof
```
```   907   fix x y :: 'a
```
```   908   from less_add_one show "\<exists>y. x < y" ..
```
```   909   from less_add_one have "x + (- 1) < (x + 1) + (- 1)" by (rule add_strict_right_mono)
```
```   910   then have "x - 1 < x + 1 - 1" by simp
```
```   911   then have "x - 1 < x" by (simp add: algebra_simps)
```
```   912   then show "\<exists>y. y < x" ..
```
```   913   show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum)
```
```   914 qed
```
```   915
```
```   916 lemma nonzero_abs_inverse:
```
```   917      "a \<noteq> 0 ==> \<bar>inverse a\<bar> = inverse \<bar>a\<bar>"
```
```   918 apply (auto simp add: neq_iff abs_if nonzero_inverse_minus_eq
```
```   919                       negative_imp_inverse_negative)
```
```   920 apply (blast intro: positive_imp_inverse_positive elim: less_asym)
```
```   921 done
```
```   922
```
```   923 lemma nonzero_abs_divide:
```
```   924      "b \<noteq> 0 ==> \<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
```
```   925   by (simp add: divide_inverse abs_mult nonzero_abs_inverse)
```
```   926
```
```   927 lemma field_le_epsilon:
```
```   928   assumes e: "\<And>e. 0 < e \<Longrightarrow> x \<le> y + e"
```
```   929   shows "x \<le> y"
```
```   930 proof (rule dense_le)
```
```   931   fix t assume "t < x"
```
```   932   hence "0 < x - t" by (simp add: less_diff_eq)
```
```   933   from e [OF this] have "x + 0 \<le> x + (y - t)" by (simp add: algebra_simps)
```
```   934   then have "0 \<le> y - t" by (simp only: add_le_cancel_left)
```
```   935   then show "t \<le> y" by (simp add: algebra_simps)
```
```   936 qed
```
```   937
```
```   938 end
```
```   939
```
```   940 class linordered_field_inverse_zero = linordered_field + field_inverse_zero
```
```   941 begin
```
```   942
```
```   943 lemma inverse_positive_iff_positive [simp]:
```
```   944   "(0 < inverse a) = (0 < a)"
```
```   945 apply (cases "a = 0", simp)
```
```   946 apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
```
```   947 done
```
```   948
```
```   949 lemma inverse_negative_iff_negative [simp]:
```
```   950   "(inverse a < 0) = (a < 0)"
```
```   951 apply (cases "a = 0", simp)
```
```   952 apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
```
```   953 done
```
```   954
```
```   955 lemma inverse_nonnegative_iff_nonnegative [simp]:
```
```   956   "0 \<le> inverse a \<longleftrightarrow> 0 \<le> a"
```
```   957   by (simp add: not_less [symmetric])
```
```   958
```
```   959 lemma inverse_nonpositive_iff_nonpositive [simp]:
```
```   960   "inverse a \<le> 0 \<longleftrightarrow> a \<le> 0"
```
```   961   by (simp add: not_less [symmetric])
```
```   962
```
```   963 lemma one_less_inverse_iff: "1 < inverse x \<longleftrightarrow> 0 < x \<and> x < 1"
```
```   964   using less_trans[of 1 x 0 for x]
```
```   965   by (cases x 0 rule: linorder_cases) (auto simp add: field_simps)
```
```   966
```
```   967 lemma one_le_inverse_iff: "1 \<le> inverse x \<longleftrightarrow> 0 < x \<and> x \<le> 1"
```
```   968 proof (cases "x = 1")
```
```   969   case True then show ?thesis by simp
```
```   970 next
```
```   971   case False then have "inverse x \<noteq> 1" by simp
```
```   972   then have "1 \<noteq> inverse x" by blast
```
```   973   then have "1 \<le> inverse x \<longleftrightarrow> 1 < inverse x" by (simp add: le_less)
```
```   974   with False show ?thesis by (auto simp add: one_less_inverse_iff)
```
```   975 qed
```
```   976
```
```   977 lemma inverse_less_1_iff: "inverse x < 1 \<longleftrightarrow> x \<le> 0 \<or> 1 < x"
```
```   978   by (simp add: not_le [symmetric] one_le_inverse_iff)
```
```   979
```
```   980 lemma inverse_le_1_iff: "inverse x \<le> 1 \<longleftrightarrow> x \<le> 0 \<or> 1 \<le> x"
```
```   981   by (simp add: not_less [symmetric] one_less_inverse_iff)
```
```   982
```
```   983 lemma [divide_simps]:
```
```   984   shows le_divide_eq: "a \<le> b / c \<longleftrightarrow> (if 0 < c then a * c \<le> b else if c < 0 then b \<le> a * c else a \<le> 0)"
```
```   985     and divide_le_eq: "b / c \<le> a \<longleftrightarrow> (if 0 < c then b \<le> a * c else if c < 0 then a * c \<le> b else 0 \<le> a)"
```
```   986     and less_divide_eq: "a < b / c \<longleftrightarrow> (if 0 < c then a * c < b else if c < 0 then b < a * c else a < 0)"
```
```   987     and divide_less_eq: "b / c < a \<longleftrightarrow> (if 0 < c then b < a * c else if c < 0 then a * c < b else 0 < a)"
```
```   988     and le_minus_divide_eq: "a \<le> - (b / c) \<longleftrightarrow> (if 0 < c then a * c \<le> - b else if c < 0 then - b \<le> a * c else a \<le> 0)"
```
```   989     and minus_divide_le_eq: "- (b / c) \<le> a \<longleftrightarrow> (if 0 < c then - b \<le> a * c else if c < 0 then a * c \<le> - b else 0 \<le> a)"
```
```   990     and less_minus_divide_eq: "a < - (b / c) \<longleftrightarrow> (if 0 < c then a * c < - b else if c < 0 then - b < a * c else  a < 0)"
```
```   991     and minus_divide_less_eq: "- (b / c) < a \<longleftrightarrow> (if 0 < c then - b < a * c else if c < 0 then a * c < - b else 0 < a)"
```
```   992   by (auto simp: field_simps not_less dest: antisym)
```
```   993
```
```   994 text {*Division and Signs*}
```
```   995
```
```   996 lemma
```
```   997   shows zero_less_divide_iff: "0 < a / b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
```
```   998     and divide_less_0_iff: "a / b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
```
```   999     and zero_le_divide_iff: "0 \<le> a / b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0"
```
```  1000     and divide_le_0_iff: "a / b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
```
```  1001   by (auto simp add: divide_simps)
```
```  1002
```
```  1003 text {* Division and the Number One *}
```
```  1004
```
```  1005 text{*Simplify expressions equated with 1*}
```
```  1006
```
```  1007 lemma zero_eq_1_divide_iff [simp]: "0 = 1 / a \<longleftrightarrow> a = 0"
```
```  1008   by (cases "a = 0") (auto simp: field_simps)
```
```  1009
```
```  1010 lemma one_divide_eq_0_iff [simp]: "1 / a = 0 \<longleftrightarrow> a = 0"
```
```  1011   using zero_eq_1_divide_iff[of a] by simp
```
```  1012
```
```  1013 text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}
```
```  1014
```
```  1015 lemma zero_le_divide_1_iff [simp]:
```
```  1016   "0 \<le> 1 / a \<longleftrightarrow> 0 \<le> a"
```
```  1017   by (simp add: zero_le_divide_iff)
```
```  1018
```
```  1019 lemma zero_less_divide_1_iff [simp]:
```
```  1020   "0 < 1 / a \<longleftrightarrow> 0 < a"
```
```  1021   by (simp add: zero_less_divide_iff)
```
```  1022
```
```  1023 lemma divide_le_0_1_iff [simp]:
```
```  1024   "1 / a \<le> 0 \<longleftrightarrow> a \<le> 0"
```
```  1025   by (simp add: divide_le_0_iff)
```
```  1026
```
```  1027 lemma divide_less_0_1_iff [simp]:
```
```  1028   "1 / a < 0 \<longleftrightarrow> a < 0"
```
```  1029   by (simp add: divide_less_0_iff)
```
```  1030
```
```  1031 lemma divide_right_mono:
```
```  1032      "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/c"
```
```  1033 by (force simp add: divide_strict_right_mono le_less)
```
```  1034
```
```  1035 lemma divide_right_mono_neg: "a <= b
```
```  1036     ==> c <= 0 ==> b / c <= a / c"
```
```  1037 apply (drule divide_right_mono [of _ _ "- c"])
```
```  1038 apply auto
```
```  1039 done
```
```  1040
```
```  1041 lemma divide_left_mono_neg: "a <= b
```
```  1042     ==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"
```
```  1043   apply (drule divide_left_mono [of _ _ "- c"])
```
```  1044   apply (auto simp add: mult.commute)
```
```  1045 done
```
```  1046
```
```  1047 lemma inverse_le_iff: "inverse a \<le> inverse b \<longleftrightarrow> (0 < a * b \<longrightarrow> b \<le> a) \<and> (a * b \<le> 0 \<longrightarrow> a \<le> b)"
```
```  1048   by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])
```
```  1049      (auto simp add: field_simps zero_less_mult_iff mult_le_0_iff)
```
```  1050
```
```  1051 lemma inverse_less_iff: "inverse a < inverse b \<longleftrightarrow> (0 < a * b \<longrightarrow> b < a) \<and> (a * b \<le> 0 \<longrightarrow> a < b)"
```
```  1052   by (subst less_le) (auto simp: inverse_le_iff)
```
```  1053
```
```  1054 lemma divide_le_cancel: "a / c \<le> b / c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
```
```  1055   by (simp add: divide_inverse mult_le_cancel_right)
```
```  1056
```
```  1057 lemma divide_less_cancel: "a / c < b / c \<longleftrightarrow> (0 < c \<longrightarrow> a < b) \<and> (c < 0 \<longrightarrow> b < a) \<and> c \<noteq> 0"
```
```  1058   by (auto simp add: divide_inverse mult_less_cancel_right)
```
```  1059
```
```  1060 text{*Simplify quotients that are compared with the value 1.*}
```
```  1061
```
```  1062 lemma le_divide_eq_1:
```
```  1063   "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"
```
```  1064 by (auto simp add: le_divide_eq)
```
```  1065
```
```  1066 lemma divide_le_eq_1:
```
```  1067   "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"
```
```  1068 by (auto simp add: divide_le_eq)
```
```  1069
```
```  1070 lemma less_divide_eq_1:
```
```  1071   "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
```
```  1072 by (auto simp add: less_divide_eq)
```
```  1073
```
```  1074 lemma divide_less_eq_1:
```
```  1075   "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
```
```  1076 by (auto simp add: divide_less_eq)
```
```  1077
```
```  1078 lemma divide_nonneg_nonneg [simp]:
```
```  1079   "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> x / y"
```
```  1080   by (auto simp add: divide_simps)
```
```  1081
```
```  1082 lemma divide_nonpos_nonpos:
```
```  1083   "x \<le> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> 0 \<le> x / y"
```
```  1084   by (auto simp add: divide_simps)
```
```  1085
```
```  1086 lemma divide_nonneg_nonpos:
```
```  1087   "0 \<le> x \<Longrightarrow> y \<le> 0 \<Longrightarrow> x / y \<le> 0"
```
```  1088   by (auto simp add: divide_simps)
```
```  1089
```
```  1090 lemma divide_nonpos_nonneg:
```
```  1091   "x \<le> 0 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x / y \<le> 0"
```
```  1092   by (auto simp add: divide_simps)
```
```  1093
```
```  1094 text {*Conditional Simplification Rules: No Case Splits*}
```
```  1095
```
```  1096 lemma le_divide_eq_1_pos [simp]:
```
```  1097   "0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)"
```
```  1098 by (auto simp add: le_divide_eq)
```
```  1099
```
```  1100 lemma le_divide_eq_1_neg [simp]:
```
```  1101   "a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)"
```
```  1102 by (auto simp add: le_divide_eq)
```
```  1103
```
```  1104 lemma divide_le_eq_1_pos [simp]:
```
```  1105   "0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)"
```
```  1106 by (auto simp add: divide_le_eq)
```
```  1107
```
```  1108 lemma divide_le_eq_1_neg [simp]:
```
```  1109   "a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)"
```
```  1110 by (auto simp add: divide_le_eq)
```
```  1111
```
```  1112 lemma less_divide_eq_1_pos [simp]:
```
```  1113   "0 < a \<Longrightarrow> (1 < b/a) = (a < b)"
```
```  1114 by (auto simp add: less_divide_eq)
```
```  1115
```
```  1116 lemma less_divide_eq_1_neg [simp]:
```
```  1117   "a < 0 \<Longrightarrow> (1 < b/a) = (b < a)"
```
```  1118 by (auto simp add: less_divide_eq)
```
```  1119
```
```  1120 lemma divide_less_eq_1_pos [simp]:
```
```  1121   "0 < a \<Longrightarrow> (b/a < 1) = (b < a)"
```
```  1122 by (auto simp add: divide_less_eq)
```
```  1123
```
```  1124 lemma divide_less_eq_1_neg [simp]:
```
```  1125   "a < 0 \<Longrightarrow> b/a < 1 <-> a < b"
```
```  1126 by (auto simp add: divide_less_eq)
```
```  1127
```
```  1128 lemma eq_divide_eq_1 [simp]:
```
```  1129   "(1 = b/a) = ((a \<noteq> 0 & a = b))"
```
```  1130 by (auto simp add: eq_divide_eq)
```
```  1131
```
```  1132 lemma divide_eq_eq_1 [simp]:
```
```  1133   "(b/a = 1) = ((a \<noteq> 0 & a = b))"
```
```  1134 by (auto simp add: divide_eq_eq)
```
```  1135
```
```  1136 lemma abs_inverse [simp]:
```
```  1137      "\<bar>inverse a\<bar> =
```
```  1138       inverse \<bar>a\<bar>"
```
```  1139 apply (cases "a=0", simp)
```
```  1140 apply (simp add: nonzero_abs_inverse)
```
```  1141 done
```
```  1142
```
```  1143 lemma abs_divide [simp]:
```
```  1144      "\<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
```
```  1145 apply (cases "b=0", simp)
```
```  1146 apply (simp add: nonzero_abs_divide)
```
```  1147 done
```
```  1148
```
```  1149 lemma abs_div_pos: "0 < y ==>
```
```  1150     \<bar>x\<bar> / y = \<bar>x / y\<bar>"
```
```  1151   apply (subst abs_divide)
```
```  1152   apply (simp add: order_less_imp_le)
```
```  1153 done
```
```  1154
```
```  1155 lemma zero_le_divide_abs_iff [simp]: "(0 \<le> a / abs b) = (0 \<le> a | b = 0)"
```
```  1156 by (auto simp: zero_le_divide_iff)
```
```  1157
```
```  1158 lemma divide_le_0_abs_iff [simp]: "(a / abs b \<le> 0) = (a \<le> 0 | b = 0)"
```
```  1159 by (auto simp: divide_le_0_iff)
```
```  1160
```
```  1161 lemma field_le_mult_one_interval:
```
```  1162   assumes *: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y"
```
```  1163   shows "x \<le> y"
```
```  1164 proof (cases "0 < x")
```
```  1165   assume "0 < x"
```
```  1166   thus ?thesis
```
```  1167     using dense_le_bounded[of 0 1 "y/x"] *
```
```  1168     unfolding le_divide_eq if_P[OF `0 < x`] by simp
```
```  1169 next
```
```  1170   assume "\<not>0 < x" hence "x \<le> 0" by simp
```
```  1171   obtain s::'a where s: "0 < s" "s < 1" using dense[of 0 "1\<Colon>'a"] by auto
```
```  1172   hence "x \<le> s * x" using mult_le_cancel_right[of 1 x s] `x \<le> 0` by auto
```
```  1173   also note *[OF s]
```
```  1174   finally show ?thesis .
```
```  1175 qed
```
```  1176
```
```  1177 end
```
```  1178
```
```  1179 code_identifier
```
```  1180   code_module Fields \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
```
```  1181
```
```  1182 end
```