src/HOL/Fields.thy
 author paulson Mon May 23 15:33:24 2016 +0100 (2016-05-23) changeset 63114 27afe7af7379 parent 62481 b5d8e57826df child 63952 354808e9f44b permissions -rw-r--r--
Lots of new material for multivariate analysis
```     1 (*  Title:      HOL/Fields.thy
```
```     2     Author:     Gertrud Bauer
```
```     3     Author:     Steven Obua
```
```     4     Author:     Tobias Nipkow
```
```     5     Author:     Lawrence C Paulson
```
```     6     Author:     Markus Wenzel
```
```     7     Author:     Jeremy Avigad
```
```     8 *)
```
```     9
```
```    10 section \<open>Fields\<close>
```
```    11
```
```    12 theory Fields
```
```    13 imports Nat
```
```    14 begin
```
```    15
```
```    16 subsection \<open>Division rings\<close>
```
```    17
```
```    18 text \<open>
```
```    19   A division ring is like a field, but without the commutativity requirement.
```
```    20 \<close>
```
```    21
```
```    22 class inverse = divide +
```
```    23   fixes inverse :: "'a \<Rightarrow> 'a"
```
```    24 begin
```
```    25
```
```    26 abbreviation inverse_divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "'/" 70)
```
```    27 where
```
```    28   "inverse_divide \<equiv> divide"
```
```    29
```
```    30 end
```
```    31
```
```    32 text \<open>Setup for linear arithmetic prover\<close>
```
```    33
```
```    34 ML_file "~~/src/Provers/Arith/fast_lin_arith.ML"
```
```    35 ML_file "Tools/lin_arith.ML"
```
```    36 setup \<open>Lin_Arith.global_setup\<close>
```
```    37 declaration \<open>K Lin_Arith.setup\<close>
```
```    38
```
```    39 simproc_setup fast_arith_nat ("(m::nat) < n" | "(m::nat) \<le> n" | "(m::nat) = n") =
```
```    40   \<open>K Lin_Arith.simproc\<close>
```
```    41 (* Because of this simproc, the arithmetic solver is really only
```
```    42 useful to detect inconsistencies among the premises for subgoals which are
```
```    43 *not* themselves (in)equalities, because the latter activate
```
```    44 fast_nat_arith_simproc anyway. However, it seems cheaper to activate the
```
```    45 solver all the time rather than add the additional check. *)
```
```    46
```
```    47 lemmas [arith_split] = nat_diff_split split_min split_max
```
```    48
```
```    49
```
```    50 text\<open>Lemmas \<open>divide_simps\<close> move division to the outside and eliminates them on (in)equalities.\<close>
```
```    51
```
```    52 named_theorems divide_simps "rewrite rules to eliminate divisions"
```
```    53
```
```    54 class division_ring = ring_1 + inverse +
```
```    55   assumes left_inverse [simp]:  "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
```
```    56   assumes right_inverse [simp]: "a \<noteq> 0 \<Longrightarrow> a * inverse a = 1"
```
```    57   assumes divide_inverse: "a / b = a * inverse b"
```
```    58   assumes inverse_zero [simp]: "inverse 0 = 0"
```
```    59 begin
```
```    60
```
```    61 subclass ring_1_no_zero_divisors
```
```    62 proof
```
```    63   fix a b :: 'a
```
```    64   assume a: "a \<noteq> 0" and b: "b \<noteq> 0"
```
```    65   show "a * b \<noteq> 0"
```
```    66   proof
```
```    67     assume ab: "a * b = 0"
```
```    68     hence "0 = inverse a * (a * b) * inverse b" by simp
```
```    69     also have "\<dots> = (inverse a * a) * (b * inverse b)"
```
```    70       by (simp only: mult.assoc)
```
```    71     also have "\<dots> = 1" using a b by simp
```
```    72     finally show False by simp
```
```    73   qed
```
```    74 qed
```
```    75
```
```    76 lemma nonzero_imp_inverse_nonzero:
```
```    77   "a \<noteq> 0 \<Longrightarrow> inverse a \<noteq> 0"
```
```    78 proof
```
```    79   assume ianz: "inverse a = 0"
```
```    80   assume "a \<noteq> 0"
```
```    81   hence "1 = a * inverse a" by simp
```
```    82   also have "... = 0" by (simp add: ianz)
```
```    83   finally have "1 = 0" .
```
```    84   thus False by (simp add: eq_commute)
```
```    85 qed
```
```    86
```
```    87 lemma inverse_zero_imp_zero:
```
```    88   "inverse a = 0 \<Longrightarrow> a = 0"
```
```    89 apply (rule classical)
```
```    90 apply (drule nonzero_imp_inverse_nonzero)
```
```    91 apply auto
```
```    92 done
```
```    93
```
```    94 lemma inverse_unique:
```
```    95   assumes ab: "a * b = 1"
```
```    96   shows "inverse a = b"
```
```    97 proof -
```
```    98   have "a \<noteq> 0" using ab by (cases "a = 0") simp_all
```
```    99   moreover have "inverse a * (a * b) = inverse a" by (simp add: ab)
```
```   100   ultimately show ?thesis by (simp add: mult.assoc [symmetric])
```
```   101 qed
```
```   102
```
```   103 lemma nonzero_inverse_minus_eq:
```
```   104   "a \<noteq> 0 \<Longrightarrow> inverse (- a) = - inverse a"
```
```   105 by (rule inverse_unique) simp
```
```   106
```
```   107 lemma nonzero_inverse_inverse_eq:
```
```   108   "a \<noteq> 0 \<Longrightarrow> inverse (inverse a) = a"
```
```   109 by (rule inverse_unique) simp
```
```   110
```
```   111 lemma nonzero_inverse_eq_imp_eq:
```
```   112   assumes "inverse a = inverse b" and "a \<noteq> 0" and "b \<noteq> 0"
```
```   113   shows "a = b"
```
```   114 proof -
```
```   115   from \<open>inverse a = inverse b\<close>
```
```   116   have "inverse (inverse a) = inverse (inverse b)" by (rule arg_cong)
```
```   117   with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> show "a = b"
```
```   118     by (simp add: nonzero_inverse_inverse_eq)
```
```   119 qed
```
```   120
```
```   121 lemma inverse_1 [simp]: "inverse 1 = 1"
```
```   122 by (rule inverse_unique) simp
```
```   123
```
```   124 lemma nonzero_inverse_mult_distrib:
```
```   125   assumes "a \<noteq> 0" and "b \<noteq> 0"
```
```   126   shows "inverse (a * b) = inverse b * inverse a"
```
```   127 proof -
```
```   128   have "a * (b * inverse b) * inverse a = 1" using assms by simp
```
```   129   hence "a * b * (inverse b * inverse a) = 1" by (simp only: mult.assoc)
```
```   130   thus ?thesis by (rule inverse_unique)
```
```   131 qed
```
```   132
```
```   133 lemma division_ring_inverse_add:
```
```   134   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a + inverse b = inverse a * (a + b) * inverse b"
```
```   135 by (simp add: algebra_simps)
```
```   136
```
```   137 lemma division_ring_inverse_diff:
```
```   138   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a - inverse b = inverse a * (b - a) * inverse b"
```
```   139 by (simp add: algebra_simps)
```
```   140
```
```   141 lemma right_inverse_eq: "b \<noteq> 0 \<Longrightarrow> a / b = 1 \<longleftrightarrow> a = b"
```
```   142 proof
```
```   143   assume neq: "b \<noteq> 0"
```
```   144   {
```
```   145     hence "a = (a / b) * b" by (simp add: divide_inverse mult.assoc)
```
```   146     also assume "a / b = 1"
```
```   147     finally show "a = b" by simp
```
```   148   next
```
```   149     assume "a = b"
```
```   150     with neq show "a / b = 1" by (simp add: divide_inverse)
```
```   151   }
```
```   152 qed
```
```   153
```
```   154 lemma nonzero_inverse_eq_divide: "a \<noteq> 0 \<Longrightarrow> inverse a = 1 / a"
```
```   155 by (simp add: divide_inverse)
```
```   156
```
```   157 lemma divide_self [simp]: "a \<noteq> 0 \<Longrightarrow> a / a = 1"
```
```   158 by (simp add: divide_inverse)
```
```   159
```
```   160 lemma inverse_eq_divide [field_simps, divide_simps]: "inverse a = 1 / a"
```
```   161 by (simp add: divide_inverse)
```
```   162
```
```   163 lemma add_divide_distrib: "(a+b) / c = a/c + b/c"
```
```   164 by (simp add: divide_inverse algebra_simps)
```
```   165
```
```   166 lemma times_divide_eq_right [simp]: "a * (b / c) = (a * b) / c"
```
```   167   by (simp add: divide_inverse mult.assoc)
```
```   168
```
```   169 lemma minus_divide_left: "- (a / b) = (-a) / b"
```
```   170   by (simp add: divide_inverse)
```
```   171
```
```   172 lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a / b) = a / (- b)"
```
```   173   by (simp add: divide_inverse nonzero_inverse_minus_eq)
```
```   174
```
```   175 lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a) / (-b) = a / b"
```
```   176   by (simp add: divide_inverse nonzero_inverse_minus_eq)
```
```   177
```
```   178 lemma divide_minus_left [simp]: "(-a) / b = - (a / b)"
```
```   179   by (simp add: divide_inverse)
```
```   180
```
```   181 lemma diff_divide_distrib: "(a - b) / c = a / c - b / c"
```
```   182   using add_divide_distrib [of a "- b" c] by simp
```
```   183
```
```   184 lemma nonzero_eq_divide_eq [field_simps]: "c \<noteq> 0 \<Longrightarrow> a = b / c \<longleftrightarrow> a * c = b"
```
```   185 proof -
```
```   186   assume [simp]: "c \<noteq> 0"
```
```   187   have "a = b / c \<longleftrightarrow> a * c = (b / c) * c" by simp
```
```   188   also have "... \<longleftrightarrow> a * c = b" by (simp add: divide_inverse mult.assoc)
```
```   189   finally show ?thesis .
```
```   190 qed
```
```   191
```
```   192 lemma nonzero_divide_eq_eq [field_simps]: "c \<noteq> 0 \<Longrightarrow> b / c = a \<longleftrightarrow> b = a * c"
```
```   193 proof -
```
```   194   assume [simp]: "c \<noteq> 0"
```
```   195   have "b / c = a \<longleftrightarrow> (b / c) * c = a * c" by simp
```
```   196   also have "... \<longleftrightarrow> b = a * c" by (simp add: divide_inverse mult.assoc)
```
```   197   finally show ?thesis .
```
```   198 qed
```
```   199
```
```   200 lemma nonzero_neg_divide_eq_eq [field_simps]: "b \<noteq> 0 \<Longrightarrow> - (a / b) = c \<longleftrightarrow> - a = c * b"
```
```   201   using nonzero_divide_eq_eq[of b "-a" c] by simp
```
```   202
```
```   203 lemma nonzero_neg_divide_eq_eq2 [field_simps]: "b \<noteq> 0 \<Longrightarrow> c = - (a / b) \<longleftrightarrow> c * b = - a"
```
```   204   using nonzero_neg_divide_eq_eq[of b a c] by auto
```
```   205
```
```   206 lemma divide_eq_imp: "c \<noteq> 0 \<Longrightarrow> b = a * c \<Longrightarrow> b / c = a"
```
```   207   by (simp add: divide_inverse mult.assoc)
```
```   208
```
```   209 lemma eq_divide_imp: "c \<noteq> 0 \<Longrightarrow> a * c = b \<Longrightarrow> a = b / c"
```
```   210   by (drule sym) (simp add: divide_inverse mult.assoc)
```
```   211
```
```   212 lemma add_divide_eq_iff [field_simps]:
```
```   213   "z \<noteq> 0 \<Longrightarrow> x + y / z = (x * z + y) / z"
```
```   214   by (simp add: add_divide_distrib nonzero_eq_divide_eq)
```
```   215
```
```   216 lemma divide_add_eq_iff [field_simps]:
```
```   217   "z \<noteq> 0 \<Longrightarrow> x / z + y = (x + y * z) / z"
```
```   218   by (simp add: add_divide_distrib nonzero_eq_divide_eq)
```
```   219
```
```   220 lemma diff_divide_eq_iff [field_simps]:
```
```   221   "z \<noteq> 0 \<Longrightarrow> x - y / z = (x * z - y) / z"
```
```   222   by (simp add: diff_divide_distrib nonzero_eq_divide_eq eq_diff_eq)
```
```   223
```
```   224 lemma minus_divide_add_eq_iff [field_simps]:
```
```   225   "z \<noteq> 0 \<Longrightarrow> - (x / z) + y = (- x + y * z) / z"
```
```   226   by (simp add: add_divide_distrib diff_divide_eq_iff)
```
```   227
```
```   228 lemma divide_diff_eq_iff [field_simps]:
```
```   229   "z \<noteq> 0 \<Longrightarrow> x / z - y = (x - y * z) / z"
```
```   230   by (simp add: field_simps)
```
```   231
```
```   232 lemma minus_divide_diff_eq_iff [field_simps]:
```
```   233   "z \<noteq> 0 \<Longrightarrow> - (x / z) - y = (- x - y * z) / z"
```
```   234   by (simp add: divide_diff_eq_iff[symmetric])
```
```   235
```
```   236 lemma division_ring_divide_zero [simp]:
```
```   237   "a / 0 = 0"
```
```   238   by (simp add: divide_inverse)
```
```   239
```
```   240 lemma divide_self_if [simp]:
```
```   241   "a / a = (if a = 0 then 0 else 1)"
```
```   242   by simp
```
```   243
```
```   244 lemma inverse_nonzero_iff_nonzero [simp]:
```
```   245   "inverse a = 0 \<longleftrightarrow> a = 0"
```
```   246   by rule (fact inverse_zero_imp_zero, simp)
```
```   247
```
```   248 lemma inverse_minus_eq [simp]:
```
```   249   "inverse (- a) = - inverse a"
```
```   250 proof cases
```
```   251   assume "a=0" thus ?thesis by simp
```
```   252 next
```
```   253   assume "a\<noteq>0"
```
```   254   thus ?thesis by (simp add: nonzero_inverse_minus_eq)
```
```   255 qed
```
```   256
```
```   257 lemma inverse_inverse_eq [simp]:
```
```   258   "inverse (inverse a) = a"
```
```   259 proof cases
```
```   260   assume "a=0" thus ?thesis by simp
```
```   261 next
```
```   262   assume "a\<noteq>0"
```
```   263   thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
```
```   264 qed
```
```   265
```
```   266 lemma inverse_eq_imp_eq:
```
```   267   "inverse a = inverse b \<Longrightarrow> a = b"
```
```   268   by (drule arg_cong [where f="inverse"], simp)
```
```   269
```
```   270 lemma inverse_eq_iff_eq [simp]:
```
```   271   "inverse a = inverse b \<longleftrightarrow> a = b"
```
```   272   by (force dest!: inverse_eq_imp_eq)
```
```   273
```
```   274 lemma add_divide_eq_if_simps [divide_simps]:
```
```   275     "a + b / z = (if z = 0 then a else (a * z + b) / z)"
```
```   276     "a / z + b = (if z = 0 then b else (a + b * z) / z)"
```
```   277     "- (a / z) + b = (if z = 0 then b else (-a + b * z) / z)"
```
```   278     "a - b / z = (if z = 0 then a else (a * z - b) / z)"
```
```   279     "a / z - b = (if z = 0 then -b else (a - b * z) / z)"
```
```   280     "- (a / z) - b = (if z = 0 then -b else (- a - b * z) / z)"
```
```   281   by (simp_all add: add_divide_eq_iff divide_add_eq_iff diff_divide_eq_iff divide_diff_eq_iff
```
```   282       minus_divide_diff_eq_iff)
```
```   283
```
```   284 lemma [divide_simps]:
```
```   285   shows divide_eq_eq: "b / c = a \<longleftrightarrow> (if c \<noteq> 0 then b = a * c else a = 0)"
```
```   286     and eq_divide_eq: "a = b / c \<longleftrightarrow> (if c \<noteq> 0 then a * c = b else a = 0)"
```
```   287     and minus_divide_eq_eq: "- (b / c) = a \<longleftrightarrow> (if c \<noteq> 0 then - b = a * c else a = 0)"
```
```   288     and eq_minus_divide_eq: "a = - (b / c) \<longleftrightarrow> (if c \<noteq> 0 then a * c = - b else a = 0)"
```
```   289   by (auto simp add:  field_simps)
```
```   290
```
```   291 end
```
```   292
```
```   293 subsection \<open>Fields\<close>
```
```   294
```
```   295 class field = comm_ring_1 + inverse +
```
```   296   assumes field_inverse: "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
```
```   297   assumes field_divide_inverse: "a / b = a * inverse b"
```
```   298   assumes field_inverse_zero: "inverse 0 = 0"
```
```   299 begin
```
```   300
```
```   301 subclass division_ring
```
```   302 proof
```
```   303   fix a :: 'a
```
```   304   assume "a \<noteq> 0"
```
```   305   thus "inverse a * a = 1" by (rule field_inverse)
```
```   306   thus "a * inverse a = 1" by (simp only: mult.commute)
```
```   307 next
```
```   308   fix a b :: 'a
```
```   309   show "a / b = a * inverse b" by (rule field_divide_inverse)
```
```   310 next
```
```   311   show "inverse 0 = 0"
```
```   312     by (fact field_inverse_zero)
```
```   313 qed
```
```   314
```
```   315 subclass idom_divide
```
```   316 proof
```
```   317   fix b a
```
```   318   assume "b \<noteq> 0"
```
```   319   then show "a * b / b = a"
```
```   320     by (simp add: divide_inverse ac_simps)
```
```   321 next
```
```   322   fix a
```
```   323   show "a / 0 = 0"
```
```   324     by (simp add: divide_inverse)
```
```   325 qed
```
```   326
```
```   327 text\<open>There is no slick version using division by zero.\<close>
```
```   328 lemma inverse_add:
```
```   329   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a + inverse b = (a + b) * inverse a * inverse b"
```
```   330   by (simp add: division_ring_inverse_add ac_simps)
```
```   331
```
```   332 lemma nonzero_mult_divide_mult_cancel_left [simp]:
```
```   333   assumes [simp]: "c \<noteq> 0"
```
```   334   shows "(c * a) / (c * b) = a / b"
```
```   335 proof (cases "b = 0")
```
```   336   case True then show ?thesis by simp
```
```   337 next
```
```   338   case False
```
```   339   then have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
```
```   340     by (simp add: divide_inverse nonzero_inverse_mult_distrib)
```
```   341   also have "... =  a * inverse b * (inverse c * c)"
```
```   342     by (simp only: ac_simps)
```
```   343   also have "... =  a * inverse b" by simp
```
```   344     finally show ?thesis by (simp add: divide_inverse)
```
```   345 qed
```
```   346
```
```   347 lemma nonzero_mult_divide_mult_cancel_right [simp]:
```
```   348   "c \<noteq> 0 \<Longrightarrow> (a * c) / (b * c) = a / b"
```
```   349   using nonzero_mult_divide_mult_cancel_left [of c a b] by (simp add: ac_simps)
```
```   350
```
```   351 lemma times_divide_eq_left [simp]: "(b / c) * a = (b * a) / c"
```
```   352   by (simp add: divide_inverse ac_simps)
```
```   353
```
```   354 lemma divide_inverse_commute: "a / b = inverse b * a"
```
```   355   by (simp add: divide_inverse mult.commute)
```
```   356
```
```   357 lemma add_frac_eq:
```
```   358   assumes "y \<noteq> 0" and "z \<noteq> 0"
```
```   359   shows "x / y + w / z = (x * z + w * y) / (y * z)"
```
```   360 proof -
```
```   361   have "x / y + w / z = (x * z) / (y * z) + (y * w) / (y * z)"
```
```   362     using assms by simp
```
```   363   also have "\<dots> = (x * z + y * w) / (y * z)"
```
```   364     by (simp only: add_divide_distrib)
```
```   365   finally show ?thesis
```
```   366     by (simp only: mult.commute)
```
```   367 qed
```
```   368
```
```   369 text\<open>Special Cancellation Simprules for Division\<close>
```
```   370
```
```   371 lemma nonzero_divide_mult_cancel_right [simp]:
```
```   372   "b \<noteq> 0 \<Longrightarrow> b / (a * b) = 1 / a"
```
```   373   using nonzero_mult_divide_mult_cancel_right [of b 1 a] by simp
```
```   374
```
```   375 lemma nonzero_divide_mult_cancel_left [simp]:
```
```   376   "a \<noteq> 0 \<Longrightarrow> a / (a * b) = 1 / b"
```
```   377   using nonzero_mult_divide_mult_cancel_left [of a 1 b] by simp
```
```   378
```
```   379 lemma nonzero_mult_divide_mult_cancel_left2 [simp]:
```
```   380   "c \<noteq> 0 \<Longrightarrow> (c * a) / (b * c) = a / b"
```
```   381   using nonzero_mult_divide_mult_cancel_left [of c a b] by (simp add: ac_simps)
```
```   382
```
```   383 lemma nonzero_mult_divide_mult_cancel_right2 [simp]:
```
```   384   "c \<noteq> 0 \<Longrightarrow> (a * c) / (c * b) = a / b"
```
```   385   using nonzero_mult_divide_mult_cancel_right [of b c a] by (simp add: ac_simps)
```
```   386
```
```   387 lemma diff_frac_eq:
```
```   388   "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y - w / z = (x * z - w * y) / (y * z)"
```
```   389   by (simp add: field_simps)
```
```   390
```
```   391 lemma frac_eq_eq:
```
```   392   "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> (x / y = w / z) = (x * z = w * y)"
```
```   393   by (simp add: field_simps)
```
```   394
```
```   395 lemma divide_minus1 [simp]: "x / - 1 = - x"
```
```   396   using nonzero_minus_divide_right [of "1" x] by simp
```
```   397
```
```   398 text\<open>This version builds in division by zero while also re-orienting
```
```   399       the right-hand side.\<close>
```
```   400 lemma inverse_mult_distrib [simp]:
```
```   401   "inverse (a * b) = inverse a * inverse b"
```
```   402 proof cases
```
```   403   assume "a \<noteq> 0 & b \<noteq> 0"
```
```   404   thus ?thesis by (simp add: nonzero_inverse_mult_distrib ac_simps)
```
```   405 next
```
```   406   assume "~ (a \<noteq> 0 & b \<noteq> 0)"
```
```   407   thus ?thesis by force
```
```   408 qed
```
```   409
```
```   410 lemma inverse_divide [simp]:
```
```   411   "inverse (a / b) = b / a"
```
```   412   by (simp add: divide_inverse mult.commute)
```
```   413
```
```   414
```
```   415 text \<open>Calculations with fractions\<close>
```
```   416
```
```   417 text\<open>There is a whole bunch of simp-rules just for class \<open>field\<close> but none for class \<open>field\<close> and \<open>nonzero_divides\<close>
```
```   418 because the latter are covered by a simproc.\<close>
```
```   419
```
```   420 lemma mult_divide_mult_cancel_left:
```
```   421   "c \<noteq> 0 \<Longrightarrow> (c * a) / (c * b) = a / b"
```
```   422 apply (cases "b = 0")
```
```   423 apply simp_all
```
```   424 done
```
```   425
```
```   426 lemma mult_divide_mult_cancel_right:
```
```   427   "c \<noteq> 0 \<Longrightarrow> (a * c) / (b * c) = a / b"
```
```   428 apply (cases "b = 0")
```
```   429 apply simp_all
```
```   430 done
```
```   431
```
```   432 lemma divide_divide_eq_right [simp]:
```
```   433   "a / (b / c) = (a * c) / b"
```
```   434   by (simp add: divide_inverse ac_simps)
```
```   435
```
```   436 lemma divide_divide_eq_left [simp]:
```
```   437   "(a / b) / c = a / (b * c)"
```
```   438   by (simp add: divide_inverse mult.assoc)
```
```   439
```
```   440 lemma divide_divide_times_eq:
```
```   441   "(x / y) / (z / w) = (x * w) / (y * z)"
```
```   442   by simp
```
```   443
```
```   444 text \<open>Special Cancellation Simprules for Division\<close>
```
```   445
```
```   446 lemma mult_divide_mult_cancel_left_if [simp]:
```
```   447   shows "(c * a) / (c * b) = (if c = 0 then 0 else a / b)"
```
```   448   by simp
```
```   449
```
```   450
```
```   451 text \<open>Division and Unary Minus\<close>
```
```   452
```
```   453 lemma minus_divide_right:
```
```   454   "- (a / b) = a / - b"
```
```   455   by (simp add: divide_inverse)
```
```   456
```
```   457 lemma divide_minus_right [simp]:
```
```   458   "a / - b = - (a / b)"
```
```   459   by (simp add: divide_inverse)
```
```   460
```
```   461 lemma minus_divide_divide:
```
```   462   "(- a) / (- b) = a / b"
```
```   463 apply (cases "b=0", simp)
```
```   464 apply (simp add: nonzero_minus_divide_divide)
```
```   465 done
```
```   466
```
```   467 lemma inverse_eq_1_iff [simp]:
```
```   468   "inverse x = 1 \<longleftrightarrow> x = 1"
```
```   469   by (insert inverse_eq_iff_eq [of x 1], simp)
```
```   470
```
```   471 lemma divide_eq_0_iff [simp]:
```
```   472   "a / b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
```
```   473   by (simp add: divide_inverse)
```
```   474
```
```   475 lemma divide_cancel_right [simp]:
```
```   476   "a / c = b / c \<longleftrightarrow> c = 0 \<or> a = b"
```
```   477   apply (cases "c=0", simp)
```
```   478   apply (simp add: divide_inverse)
```
```   479   done
```
```   480
```
```   481 lemma divide_cancel_left [simp]:
```
```   482   "c / a = c / b \<longleftrightarrow> c = 0 \<or> a = b"
```
```   483   apply (cases "c=0", simp)
```
```   484   apply (simp add: divide_inverse)
```
```   485   done
```
```   486
```
```   487 lemma divide_eq_1_iff [simp]:
```
```   488   "a / b = 1 \<longleftrightarrow> b \<noteq> 0 \<and> a = b"
```
```   489   apply (cases "b=0", simp)
```
```   490   apply (simp add: right_inverse_eq)
```
```   491   done
```
```   492
```
```   493 lemma one_eq_divide_iff [simp]:
```
```   494   "1 = a / b \<longleftrightarrow> b \<noteq> 0 \<and> a = b"
```
```   495   by (simp add: eq_commute [of 1])
```
```   496
```
```   497 lemma times_divide_times_eq:
```
```   498   "(x / y) * (z / w) = (x * z) / (y * w)"
```
```   499   by simp
```
```   500
```
```   501 lemma add_frac_num:
```
```   502   "y \<noteq> 0 \<Longrightarrow> x / y + z = (x + z * y) / y"
```
```   503   by (simp add: add_divide_distrib)
```
```   504
```
```   505 lemma add_num_frac:
```
```   506   "y \<noteq> 0 \<Longrightarrow> z + x / y = (x + z * y) / y"
```
```   507   by (simp add: add_divide_distrib add.commute)
```
```   508
```
```   509 end
```
```   510
```
```   511 class field_char_0 = field + ring_char_0
```
```   512
```
```   513
```
```   514 subsection \<open>Ordered fields\<close>
```
```   515
```
```   516 class linordered_field = field + linordered_idom
```
```   517 begin
```
```   518
```
```   519 lemma positive_imp_inverse_positive:
```
```   520   assumes a_gt_0: "0 < a"
```
```   521   shows "0 < inverse a"
```
```   522 proof -
```
```   523   have "0 < a * inverse a"
```
```   524     by (simp add: a_gt_0 [THEN less_imp_not_eq2])
```
```   525   thus "0 < inverse a"
```
```   526     by (simp add: a_gt_0 [THEN less_not_sym] zero_less_mult_iff)
```
```   527 qed
```
```   528
```
```   529 lemma negative_imp_inverse_negative:
```
```   530   "a < 0 \<Longrightarrow> inverse a < 0"
```
```   531   by (insert positive_imp_inverse_positive [of "-a"],
```
```   532     simp add: nonzero_inverse_minus_eq less_imp_not_eq)
```
```   533
```
```   534 lemma inverse_le_imp_le:
```
```   535   assumes invle: "inverse a \<le> inverse b" and apos: "0 < a"
```
```   536   shows "b \<le> a"
```
```   537 proof (rule classical)
```
```   538   assume "~ b \<le> a"
```
```   539   hence "a < b"  by (simp add: linorder_not_le)
```
```   540   hence bpos: "0 < b"  by (blast intro: apos less_trans)
```
```   541   hence "a * inverse a \<le> a * inverse b"
```
```   542     by (simp add: apos invle less_imp_le mult_left_mono)
```
```   543   hence "(a * inverse a) * b \<le> (a * inverse b) * b"
```
```   544     by (simp add: bpos less_imp_le mult_right_mono)
```
```   545   thus "b \<le> a"  by (simp add: mult.assoc apos bpos less_imp_not_eq2)
```
```   546 qed
```
```   547
```
```   548 lemma inverse_positive_imp_positive:
```
```   549   assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0"
```
```   550   shows "0 < a"
```
```   551 proof -
```
```   552   have "0 < inverse (inverse a)"
```
```   553     using inv_gt_0 by (rule positive_imp_inverse_positive)
```
```   554   thus "0 < a"
```
```   555     using nz by (simp add: nonzero_inverse_inverse_eq)
```
```   556 qed
```
```   557
```
```   558 lemma inverse_negative_imp_negative:
```
```   559   assumes inv_less_0: "inverse a < 0" and nz: "a \<noteq> 0"
```
```   560   shows "a < 0"
```
```   561 proof -
```
```   562   have "inverse (inverse a) < 0"
```
```   563     using inv_less_0 by (rule negative_imp_inverse_negative)
```
```   564   thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq)
```
```   565 qed
```
```   566
```
```   567 lemma linordered_field_no_lb:
```
```   568   "\<forall>x. \<exists>y. y < x"
```
```   569 proof
```
```   570   fix x::'a
```
```   571   have m1: "- (1::'a) < 0" by simp
```
```   572   from add_strict_right_mono[OF m1, where c=x]
```
```   573   have "(- 1) + x < x" by simp
```
```   574   thus "\<exists>y. y < x" by blast
```
```   575 qed
```
```   576
```
```   577 lemma linordered_field_no_ub:
```
```   578   "\<forall> x. \<exists>y. y > x"
```
```   579 proof
```
```   580   fix x::'a
```
```   581   have m1: " (1::'a) > 0" by simp
```
```   582   from add_strict_right_mono[OF m1, where c=x]
```
```   583   have "1 + x > x" by simp
```
```   584   thus "\<exists>y. y > x" by blast
```
```   585 qed
```
```   586
```
```   587 lemma less_imp_inverse_less:
```
```   588   assumes less: "a < b" and apos:  "0 < a"
```
```   589   shows "inverse b < inverse a"
```
```   590 proof (rule ccontr)
```
```   591   assume "~ inverse b < inverse a"
```
```   592   hence "inverse a \<le> inverse b" by simp
```
```   593   hence "~ (a < b)"
```
```   594     by (simp add: not_less inverse_le_imp_le [OF _ apos])
```
```   595   thus False by (rule notE [OF _ less])
```
```   596 qed
```
```   597
```
```   598 lemma inverse_less_imp_less:
```
```   599   "inverse a < inverse b \<Longrightarrow> 0 < a \<Longrightarrow> b < a"
```
```   600 apply (simp add: less_le [of "inverse a"] less_le [of "b"])
```
```   601 apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq)
```
```   602 done
```
```   603
```
```   604 text\<open>Both premises are essential. Consider -1 and 1.\<close>
```
```   605 lemma inverse_less_iff_less [simp]:
```
```   606   "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"
```
```   607   by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less)
```
```   608
```
```   609 lemma le_imp_inverse_le:
```
```   610   "a \<le> b \<Longrightarrow> 0 < a \<Longrightarrow> inverse b \<le> inverse a"
```
```   611   by (force simp add: le_less less_imp_inverse_less)
```
```   612
```
```   613 lemma inverse_le_iff_le [simp]:
```
```   614   "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a"
```
```   615   by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le)
```
```   616
```
```   617
```
```   618 text\<open>These results refer to both operands being negative.  The opposite-sign
```
```   619 case is trivial, since inverse preserves signs.\<close>
```
```   620 lemma inverse_le_imp_le_neg:
```
```   621   "inverse a \<le> inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b \<le> a"
```
```   622 apply (rule classical)
```
```   623 apply (subgoal_tac "a < 0")
```
```   624  prefer 2 apply force
```
```   625 apply (insert inverse_le_imp_le [of "-b" "-a"])
```
```   626 apply (simp add: nonzero_inverse_minus_eq)
```
```   627 done
```
```   628
```
```   629 lemma less_imp_inverse_less_neg:
```
```   630    "a < b \<Longrightarrow> b < 0 \<Longrightarrow> inverse b < inverse a"
```
```   631 apply (subgoal_tac "a < 0")
```
```   632  prefer 2 apply (blast intro: less_trans)
```
```   633 apply (insert less_imp_inverse_less [of "-b" "-a"])
```
```   634 apply (simp add: nonzero_inverse_minus_eq)
```
```   635 done
```
```   636
```
```   637 lemma inverse_less_imp_less_neg:
```
```   638    "inverse a < inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b < a"
```
```   639 apply (rule classical)
```
```   640 apply (subgoal_tac "a < 0")
```
```   641  prefer 2
```
```   642  apply force
```
```   643 apply (insert inverse_less_imp_less [of "-b" "-a"])
```
```   644 apply (simp add: nonzero_inverse_minus_eq)
```
```   645 done
```
```   646
```
```   647 lemma inverse_less_iff_less_neg [simp]:
```
```   648   "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"
```
```   649 apply (insert inverse_less_iff_less [of "-b" "-a"])
```
```   650 apply (simp del: inverse_less_iff_less
```
```   651             add: nonzero_inverse_minus_eq)
```
```   652 done
```
```   653
```
```   654 lemma le_imp_inverse_le_neg:
```
```   655   "a \<le> b \<Longrightarrow> b < 0 ==> inverse b \<le> inverse a"
```
```   656   by (force simp add: le_less less_imp_inverse_less_neg)
```
```   657
```
```   658 lemma inverse_le_iff_le_neg [simp]:
```
```   659   "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a"
```
```   660   by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg)
```
```   661
```
```   662 lemma one_less_inverse:
```
```   663   "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> 1 < inverse a"
```
```   664   using less_imp_inverse_less [of a 1, unfolded inverse_1] .
```
```   665
```
```   666 lemma one_le_inverse:
```
```   667   "0 < a \<Longrightarrow> a \<le> 1 \<Longrightarrow> 1 \<le> inverse a"
```
```   668   using le_imp_inverse_le [of a 1, unfolded inverse_1] .
```
```   669
```
```   670 lemma pos_le_divide_eq [field_simps]:
```
```   671   assumes "0 < c"
```
```   672   shows "a \<le> b / c \<longleftrightarrow> a * c \<le> b"
```
```   673 proof -
```
```   674   from assms have "a \<le> b / c \<longleftrightarrow> a * c \<le> (b / c) * c"
```
```   675     using mult_le_cancel_right [of a c "b * inverse c"] by (auto simp add: field_simps)
```
```   676   also have "... \<longleftrightarrow> a * c \<le> b"
```
```   677     by (simp add: less_imp_not_eq2 [OF assms] divide_inverse mult.assoc)
```
```   678   finally show ?thesis .
```
```   679 qed
```
```   680
```
```   681 lemma pos_less_divide_eq [field_simps]:
```
```   682   assumes "0 < c"
```
```   683   shows "a < b / c \<longleftrightarrow> a * c < b"
```
```   684 proof -
```
```   685   from assms have "a < b / c \<longleftrightarrow> a * c < (b / c) * c"
```
```   686     using mult_less_cancel_right [of a c "b / c"] by auto
```
```   687   also have "... = (a*c < b)"
```
```   688     by (simp add: less_imp_not_eq2 [OF assms] divide_inverse mult.assoc)
```
```   689   finally show ?thesis .
```
```   690 qed
```
```   691
```
```   692 lemma neg_less_divide_eq [field_simps]:
```
```   693   assumes "c < 0"
```
```   694   shows "a < b / c \<longleftrightarrow> b < a * c"
```
```   695 proof -
```
```   696   from assms have "a < b / c \<longleftrightarrow> (b / c) * c < a * c"
```
```   697     using mult_less_cancel_right [of "b / c" c a] by auto
```
```   698   also have "... \<longleftrightarrow> b < a * c"
```
```   699     by (simp add: less_imp_not_eq [OF assms] divide_inverse mult.assoc)
```
```   700   finally show ?thesis .
```
```   701 qed
```
```   702
```
```   703 lemma neg_le_divide_eq [field_simps]:
```
```   704   assumes "c < 0"
```
```   705   shows "a \<le> b / c \<longleftrightarrow> b \<le> a * c"
```
```   706 proof -
```
```   707   from assms have "a \<le> b / c \<longleftrightarrow> (b / c) * c \<le> a * c"
```
```   708     using mult_le_cancel_right [of "b * inverse c" c a] by (auto simp add: field_simps)
```
```   709   also have "... \<longleftrightarrow> b \<le> a * c"
```
```   710     by (simp add: less_imp_not_eq [OF assms] divide_inverse mult.assoc)
```
```   711   finally show ?thesis .
```
```   712 qed
```
```   713
```
```   714 lemma pos_divide_le_eq [field_simps]:
```
```   715   assumes "0 < c"
```
```   716   shows "b / c \<le> a \<longleftrightarrow> b \<le> a * c"
```
```   717 proof -
```
```   718   from assms have "b / c \<le> a \<longleftrightarrow> (b / c) * c \<le> a * c"
```
```   719     using mult_le_cancel_right [of "b / c" c a] by auto
```
```   720   also have "... \<longleftrightarrow> b \<le> a * c"
```
```   721     by (simp add: less_imp_not_eq2 [OF assms] divide_inverse mult.assoc)
```
```   722   finally show ?thesis .
```
```   723 qed
```
```   724
```
```   725 lemma pos_divide_less_eq [field_simps]:
```
```   726   assumes "0 < c"
```
```   727   shows "b / c < a \<longleftrightarrow> b < a * c"
```
```   728 proof -
```
```   729   from assms have "b / c < a \<longleftrightarrow> (b / c) * c < a * c"
```
```   730     using mult_less_cancel_right [of "b / c" c a] by auto
```
```   731   also have "... \<longleftrightarrow> b < a * c"
```
```   732     by (simp add: less_imp_not_eq2 [OF assms] divide_inverse mult.assoc)
```
```   733   finally show ?thesis .
```
```   734 qed
```
```   735
```
```   736 lemma neg_divide_le_eq [field_simps]:
```
```   737   assumes "c < 0"
```
```   738   shows "b / c \<le> a \<longleftrightarrow> a * c \<le> b"
```
```   739 proof -
```
```   740   from assms have "b / c \<le> a \<longleftrightarrow> a * c \<le> (b / c) * c"
```
```   741     using mult_le_cancel_right [of a c "b / c"] by auto
```
```   742   also have "... \<longleftrightarrow> a * c \<le> b"
```
```   743     by (simp add: less_imp_not_eq [OF assms] divide_inverse mult.assoc)
```
```   744   finally show ?thesis .
```
```   745 qed
```
```   746
```
```   747 lemma neg_divide_less_eq [field_simps]:
```
```   748   assumes "c < 0"
```
```   749   shows "b / c < a \<longleftrightarrow> a * c < b"
```
```   750 proof -
```
```   751   from assms have "b / c < a \<longleftrightarrow> a * c < b / c * c"
```
```   752     using mult_less_cancel_right [of a c "b / c"] by auto
```
```   753   also have "... \<longleftrightarrow> a * c < b"
```
```   754     by (simp add: less_imp_not_eq [OF assms] divide_inverse mult.assoc)
```
```   755   finally show ?thesis .
```
```   756 qed
```
```   757
```
```   758 text\<open>The following \<open>field_simps\<close> rules are necessary, as minus is always moved atop of
```
```   759 division but we want to get rid of division.\<close>
```
```   760
```
```   761 lemma pos_le_minus_divide_eq [field_simps]: "0 < c \<Longrightarrow> a \<le> - (b / c) \<longleftrightarrow> a * c \<le> - b"
```
```   762   unfolding minus_divide_left by (rule pos_le_divide_eq)
```
```   763
```
```   764 lemma neg_le_minus_divide_eq [field_simps]: "c < 0 \<Longrightarrow> a \<le> - (b / c) \<longleftrightarrow> - b \<le> a * c"
```
```   765   unfolding minus_divide_left by (rule neg_le_divide_eq)
```
```   766
```
```   767 lemma pos_less_minus_divide_eq [field_simps]: "0 < c \<Longrightarrow> a < - (b / c) \<longleftrightarrow> a * c < - b"
```
```   768   unfolding minus_divide_left by (rule pos_less_divide_eq)
```
```   769
```
```   770 lemma neg_less_minus_divide_eq [field_simps]: "c < 0 \<Longrightarrow> a < - (b / c) \<longleftrightarrow> - b < a * c"
```
```   771   unfolding minus_divide_left by (rule neg_less_divide_eq)
```
```   772
```
```   773 lemma pos_minus_divide_less_eq [field_simps]: "0 < c \<Longrightarrow> - (b / c) < a \<longleftrightarrow> - b < a * c"
```
```   774   unfolding minus_divide_left by (rule pos_divide_less_eq)
```
```   775
```
```   776 lemma neg_minus_divide_less_eq [field_simps]: "c < 0 \<Longrightarrow> - (b / c) < a \<longleftrightarrow> a * c < - b"
```
```   777   unfolding minus_divide_left by (rule neg_divide_less_eq)
```
```   778
```
```   779 lemma pos_minus_divide_le_eq [field_simps]: "0 < c \<Longrightarrow> - (b / c) \<le> a \<longleftrightarrow> - b \<le> a * c"
```
```   780   unfolding minus_divide_left by (rule pos_divide_le_eq)
```
```   781
```
```   782 lemma neg_minus_divide_le_eq [field_simps]: "c < 0 \<Longrightarrow> - (b / c) \<le> a \<longleftrightarrow> a * c \<le> - b"
```
```   783   unfolding minus_divide_left by (rule neg_divide_le_eq)
```
```   784
```
```   785 lemma frac_less_eq:
```
```   786   "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y < w / z \<longleftrightarrow> (x * z - w * y) / (y * z) < 0"
```
```   787   by (subst less_iff_diff_less_0) (simp add: diff_frac_eq )
```
```   788
```
```   789 lemma frac_le_eq:
```
```   790   "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y \<le> w / z \<longleftrightarrow> (x * z - w * y) / (y * z) \<le> 0"
```
```   791   by (subst le_iff_diff_le_0) (simp add: diff_frac_eq )
```
```   792
```
```   793 text\<open>Lemmas \<open>sign_simps\<close> is a first attempt to automate proofs
```
```   794 of positivity/negativity needed for \<open>field_simps\<close>. Have not added \<open>sign_simps\<close> to \<open>field_simps\<close> because the former can lead to case
```
```   795 explosions.\<close>
```
```   796
```
```   797 lemmas sign_simps = algebra_simps zero_less_mult_iff mult_less_0_iff
```
```   798
```
```   799 lemmas (in -) sign_simps = algebra_simps zero_less_mult_iff mult_less_0_iff
```
```   800
```
```   801 (* Only works once linear arithmetic is installed:
```
```   802 text{*An example:*}
```
```   803 lemma fixes a b c d e f :: "'a::linordered_field"
```
```   804 shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow>
```
```   805  ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) <
```
```   806  ((e-f)*(a-b)*(c-d))/((e-f)*(a-b)*(c-d)) + u"
```
```   807 apply(subgoal_tac "(c-d)*(e-f)*(a-b) > 0")
```
```   808  prefer 2 apply(simp add:sign_simps)
```
```   809 apply(subgoal_tac "(c-d)*(e-f)*(a-b)*u > 0")
```
```   810  prefer 2 apply(simp add:sign_simps)
```
```   811 apply(simp add:field_simps)
```
```   812 done
```
```   813 *)
```
```   814
```
```   815 lemma divide_pos_pos[simp]:
```
```   816   "0 < x ==> 0 < y ==> 0 < x / y"
```
```   817 by(simp add:field_simps)
```
```   818
```
```   819 lemma divide_nonneg_pos:
```
```   820   "0 <= x ==> 0 < y ==> 0 <= x / y"
```
```   821 by(simp add:field_simps)
```
```   822
```
```   823 lemma divide_neg_pos:
```
```   824   "x < 0 ==> 0 < y ==> x / y < 0"
```
```   825 by(simp add:field_simps)
```
```   826
```
```   827 lemma divide_nonpos_pos:
```
```   828   "x <= 0 ==> 0 < y ==> x / y <= 0"
```
```   829 by(simp add:field_simps)
```
```   830
```
```   831 lemma divide_pos_neg:
```
```   832   "0 < x ==> y < 0 ==> x / y < 0"
```
```   833 by(simp add:field_simps)
```
```   834
```
```   835 lemma divide_nonneg_neg:
```
```   836   "0 <= x ==> y < 0 ==> x / y <= 0"
```
```   837 by(simp add:field_simps)
```
```   838
```
```   839 lemma divide_neg_neg:
```
```   840   "x < 0 ==> y < 0 ==> 0 < x / y"
```
```   841 by(simp add:field_simps)
```
```   842
```
```   843 lemma divide_nonpos_neg:
```
```   844   "x <= 0 ==> y < 0 ==> 0 <= x / y"
```
```   845 by(simp add:field_simps)
```
```   846
```
```   847 lemma divide_strict_right_mono:
```
```   848      "[|a < b; 0 < c|] ==> a / c < b / c"
```
```   849 by (simp add: less_imp_not_eq2 divide_inverse mult_strict_right_mono
```
```   850               positive_imp_inverse_positive)
```
```   851
```
```   852
```
```   853 lemma divide_strict_right_mono_neg:
```
```   854      "[|b < a; c < 0|] ==> a / c < b / c"
```
```   855 apply (drule divide_strict_right_mono [of _ _ "-c"], simp)
```
```   856 apply (simp add: less_imp_not_eq nonzero_minus_divide_right [symmetric])
```
```   857 done
```
```   858
```
```   859 text\<open>The last premise ensures that @{term a} and @{term b}
```
```   860       have the same sign\<close>
```
```   861 lemma divide_strict_left_mono:
```
```   862   "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / b"
```
```   863   by (auto simp: field_simps zero_less_mult_iff mult_strict_right_mono)
```
```   864
```
```   865 lemma divide_left_mono:
```
```   866   "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / b"
```
```   867   by (auto simp: field_simps zero_less_mult_iff mult_right_mono)
```
```   868
```
```   869 lemma divide_strict_left_mono_neg:
```
```   870   "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / b"
```
```   871   by (auto simp: field_simps zero_less_mult_iff mult_strict_right_mono_neg)
```
```   872
```
```   873 lemma mult_imp_div_pos_le: "0 < y ==> x <= z * y ==>
```
```   874     x / y <= z"
```
```   875 by (subst pos_divide_le_eq, assumption+)
```
```   876
```
```   877 lemma mult_imp_le_div_pos: "0 < y ==> z * y <= x ==>
```
```   878     z <= x / y"
```
```   879 by(simp add:field_simps)
```
```   880
```
```   881 lemma mult_imp_div_pos_less: "0 < y ==> x < z * y ==>
```
```   882     x / y < z"
```
```   883 by(simp add:field_simps)
```
```   884
```
```   885 lemma mult_imp_less_div_pos: "0 < y ==> z * y < x ==>
```
```   886     z < x / y"
```
```   887 by(simp add:field_simps)
```
```   888
```
```   889 lemma frac_le: "0 <= x ==>
```
```   890     x <= y ==> 0 < w ==> w <= z  ==> x / z <= y / w"
```
```   891   apply (rule mult_imp_div_pos_le)
```
```   892   apply simp
```
```   893   apply (subst times_divide_eq_left)
```
```   894   apply (rule mult_imp_le_div_pos, assumption)
```
```   895   apply (rule mult_mono)
```
```   896   apply simp_all
```
```   897 done
```
```   898
```
```   899 lemma frac_less: "0 <= x ==>
```
```   900     x < y ==> 0 < w ==> w <= z  ==> x / z < y / w"
```
```   901   apply (rule mult_imp_div_pos_less)
```
```   902   apply simp
```
```   903   apply (subst times_divide_eq_left)
```
```   904   apply (rule mult_imp_less_div_pos, assumption)
```
```   905   apply (erule mult_less_le_imp_less)
```
```   906   apply simp_all
```
```   907 done
```
```   908
```
```   909 lemma frac_less2: "0 < x ==>
```
```   910     x <= y ==> 0 < w ==> w < z  ==> x / z < y / w"
```
```   911   apply (rule mult_imp_div_pos_less)
```
```   912   apply simp_all
```
```   913   apply (rule mult_imp_less_div_pos, assumption)
```
```   914   apply (erule mult_le_less_imp_less)
```
```   915   apply simp_all
```
```   916 done
```
```   917
```
```   918 lemma less_half_sum: "a < b ==> a < (a+b) / (1+1)"
```
```   919 by (simp add: field_simps zero_less_two)
```
```   920
```
```   921 lemma gt_half_sum: "a < b ==> (a+b)/(1+1) < b"
```
```   922 by (simp add: field_simps zero_less_two)
```
```   923
```
```   924 subclass unbounded_dense_linorder
```
```   925 proof
```
```   926   fix x y :: 'a
```
```   927   from less_add_one show "\<exists>y. x < y" ..
```
```   928   from less_add_one have "x + (- 1) < (x + 1) + (- 1)" by (rule add_strict_right_mono)
```
```   929   then have "x - 1 < x + 1 - 1" by simp
```
```   930   then have "x - 1 < x" by (simp add: algebra_simps)
```
```   931   then show "\<exists>y. y < x" ..
```
```   932   show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum)
```
```   933 qed
```
```   934
```
```   935 lemma nonzero_abs_inverse:
```
```   936      "a \<noteq> 0 ==> \<bar>inverse a\<bar> = inverse \<bar>a\<bar>"
```
```   937 apply (auto simp add: neq_iff abs_if nonzero_inverse_minus_eq
```
```   938                       negative_imp_inverse_negative)
```
```   939 apply (blast intro: positive_imp_inverse_positive elim: less_asym)
```
```   940 done
```
```   941
```
```   942 lemma nonzero_abs_divide:
```
```   943      "b \<noteq> 0 ==> \<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
```
```   944   by (simp add: divide_inverse abs_mult nonzero_abs_inverse)
```
```   945
```
```   946 lemma field_le_epsilon:
```
```   947   assumes e: "\<And>e. 0 < e \<Longrightarrow> x \<le> y + e"
```
```   948   shows "x \<le> y"
```
```   949 proof (rule dense_le)
```
```   950   fix t assume "t < x"
```
```   951   hence "0 < x - t" by (simp add: less_diff_eq)
```
```   952   from e [OF this] have "x + 0 \<le> x + (y - t)" by (simp add: algebra_simps)
```
```   953   then have "0 \<le> y - t" by (simp only: add_le_cancel_left)
```
```   954   then show "t \<le> y" by (simp add: algebra_simps)
```
```   955 qed
```
```   956
```
```   957 lemma inverse_positive_iff_positive [simp]:
```
```   958   "(0 < inverse a) = (0 < a)"
```
```   959 apply (cases "a = 0", simp)
```
```   960 apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
```
```   961 done
```
```   962
```
```   963 lemma inverse_negative_iff_negative [simp]:
```
```   964   "(inverse a < 0) = (a < 0)"
```
```   965 apply (cases "a = 0", simp)
```
```   966 apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
```
```   967 done
```
```   968
```
```   969 lemma inverse_nonnegative_iff_nonnegative [simp]:
```
```   970   "0 \<le> inverse a \<longleftrightarrow> 0 \<le> a"
```
```   971   by (simp add: not_less [symmetric])
```
```   972
```
```   973 lemma inverse_nonpositive_iff_nonpositive [simp]:
```
```   974   "inverse a \<le> 0 \<longleftrightarrow> a \<le> 0"
```
```   975   by (simp add: not_less [symmetric])
```
```   976
```
```   977 lemma one_less_inverse_iff: "1 < inverse x \<longleftrightarrow> 0 < x \<and> x < 1"
```
```   978   using less_trans[of 1 x 0 for x]
```
```   979   by (cases x 0 rule: linorder_cases) (auto simp add: field_simps)
```
```   980
```
```   981 lemma one_le_inverse_iff: "1 \<le> inverse x \<longleftrightarrow> 0 < x \<and> x \<le> 1"
```
```   982 proof (cases "x = 1")
```
```   983   case True then show ?thesis by simp
```
```   984 next
```
```   985   case False then have "inverse x \<noteq> 1" by simp
```
```   986   then have "1 \<noteq> inverse x" by blast
```
```   987   then have "1 \<le> inverse x \<longleftrightarrow> 1 < inverse x" by (simp add: le_less)
```
```   988   with False show ?thesis by (auto simp add: one_less_inverse_iff)
```
```   989 qed
```
```   990
```
```   991 lemma inverse_less_1_iff: "inverse x < 1 \<longleftrightarrow> x \<le> 0 \<or> 1 < x"
```
```   992   by (simp add: not_le [symmetric] one_le_inverse_iff)
```
```   993
```
```   994 lemma inverse_le_1_iff: "inverse x \<le> 1 \<longleftrightarrow> x \<le> 0 \<or> 1 \<le> x"
```
```   995   by (simp add: not_less [symmetric] one_less_inverse_iff)
```
```   996
```
```   997 lemma [divide_simps]:
```
```   998   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)"
```
```   999     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)"
```
```  1000     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)"
```
```  1001     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)"
```
```  1002     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)"
```
```  1003     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)"
```
```  1004     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)"
```
```  1005     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)"
```
```  1006   by (auto simp: field_simps not_less dest: antisym)
```
```  1007
```
```  1008 text \<open>Division and Signs\<close>
```
```  1009
```
```  1010 lemma
```
```  1011   shows zero_less_divide_iff: "0 < a / b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
```
```  1012     and divide_less_0_iff: "a / b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
```
```  1013     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"
```
```  1014     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"
```
```  1015   by (auto simp add: divide_simps)
```
```  1016
```
```  1017 text \<open>Division and the Number One\<close>
```
```  1018
```
```  1019 text\<open>Simplify expressions equated with 1\<close>
```
```  1020
```
```  1021 lemma zero_eq_1_divide_iff [simp]: "0 = 1 / a \<longleftrightarrow> a = 0"
```
```  1022   by (cases "a = 0") (auto simp: field_simps)
```
```  1023
```
```  1024 lemma one_divide_eq_0_iff [simp]: "1 / a = 0 \<longleftrightarrow> a = 0"
```
```  1025   using zero_eq_1_divide_iff[of a] by simp
```
```  1026
```
```  1027 text\<open>Simplify expressions such as \<open>0 < 1/x\<close> to \<open>0 < x\<close>\<close>
```
```  1028
```
```  1029 lemma zero_le_divide_1_iff [simp]:
```
```  1030   "0 \<le> 1 / a \<longleftrightarrow> 0 \<le> a"
```
```  1031   by (simp add: zero_le_divide_iff)
```
```  1032
```
```  1033 lemma zero_less_divide_1_iff [simp]:
```
```  1034   "0 < 1 / a \<longleftrightarrow> 0 < a"
```
```  1035   by (simp add: zero_less_divide_iff)
```
```  1036
```
```  1037 lemma divide_le_0_1_iff [simp]:
```
```  1038   "1 / a \<le> 0 \<longleftrightarrow> a \<le> 0"
```
```  1039   by (simp add: divide_le_0_iff)
```
```  1040
```
```  1041 lemma divide_less_0_1_iff [simp]:
```
```  1042   "1 / a < 0 \<longleftrightarrow> a < 0"
```
```  1043   by (simp add: divide_less_0_iff)
```
```  1044
```
```  1045 lemma divide_right_mono:
```
```  1046      "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/c"
```
```  1047 by (force simp add: divide_strict_right_mono le_less)
```
```  1048
```
```  1049 lemma divide_right_mono_neg: "a <= b
```
```  1050     ==> c <= 0 ==> b / c <= a / c"
```
```  1051 apply (drule divide_right_mono [of _ _ "- c"])
```
```  1052 apply auto
```
```  1053 done
```
```  1054
```
```  1055 lemma divide_left_mono_neg: "a <= b
```
```  1056     ==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"
```
```  1057   apply (drule divide_left_mono [of _ _ "- c"])
```
```  1058   apply (auto simp add: mult.commute)
```
```  1059 done
```
```  1060
```
```  1061 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)"
```
```  1062   by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])
```
```  1063      (auto simp add: field_simps zero_less_mult_iff mult_le_0_iff)
```
```  1064
```
```  1065 lemma inverse_less_iff: "inverse a < inverse b \<longleftrightarrow> (0 < a * b \<longrightarrow> b < a) \<and> (a * b \<le> 0 \<longrightarrow> a < b)"
```
```  1066   by (subst less_le) (auto simp: inverse_le_iff)
```
```  1067
```
```  1068 lemma divide_le_cancel: "a / c \<le> b / c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
```
```  1069   by (simp add: divide_inverse mult_le_cancel_right)
```
```  1070
```
```  1071 lemma divide_less_cancel: "a / c < b / c \<longleftrightarrow> (0 < c \<longrightarrow> a < b) \<and> (c < 0 \<longrightarrow> b < a) \<and> c \<noteq> 0"
```
```  1072   by (auto simp add: divide_inverse mult_less_cancel_right)
```
```  1073
```
```  1074 text\<open>Simplify quotients that are compared with the value 1.\<close>
```
```  1075
```
```  1076 lemma le_divide_eq_1:
```
```  1077   "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"
```
```  1078 by (auto simp add: le_divide_eq)
```
```  1079
```
```  1080 lemma divide_le_eq_1:
```
```  1081   "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"
```
```  1082 by (auto simp add: divide_le_eq)
```
```  1083
```
```  1084 lemma less_divide_eq_1:
```
```  1085   "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
```
```  1086 by (auto simp add: less_divide_eq)
```
```  1087
```
```  1088 lemma divide_less_eq_1:
```
```  1089   "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
```
```  1090 by (auto simp add: divide_less_eq)
```
```  1091
```
```  1092 lemma divide_nonneg_nonneg [simp]:
```
```  1093   "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> x / y"
```
```  1094   by (auto simp add: divide_simps)
```
```  1095
```
```  1096 lemma divide_nonpos_nonpos:
```
```  1097   "x \<le> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> 0 \<le> x / y"
```
```  1098   by (auto simp add: divide_simps)
```
```  1099
```
```  1100 lemma divide_nonneg_nonpos:
```
```  1101   "0 \<le> x \<Longrightarrow> y \<le> 0 \<Longrightarrow> x / y \<le> 0"
```
```  1102   by (auto simp add: divide_simps)
```
```  1103
```
```  1104 lemma divide_nonpos_nonneg:
```
```  1105   "x \<le> 0 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x / y \<le> 0"
```
```  1106   by (auto simp add: divide_simps)
```
```  1107
```
```  1108 text \<open>Conditional Simplification Rules: No Case Splits\<close>
```
```  1109
```
```  1110 lemma le_divide_eq_1_pos [simp]:
```
```  1111   "0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)"
```
```  1112 by (auto simp add: le_divide_eq)
```
```  1113
```
```  1114 lemma le_divide_eq_1_neg [simp]:
```
```  1115   "a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)"
```
```  1116 by (auto simp add: le_divide_eq)
```
```  1117
```
```  1118 lemma divide_le_eq_1_pos [simp]:
```
```  1119   "0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)"
```
```  1120 by (auto simp add: divide_le_eq)
```
```  1121
```
```  1122 lemma divide_le_eq_1_neg [simp]:
```
```  1123   "a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)"
```
```  1124 by (auto simp add: divide_le_eq)
```
```  1125
```
```  1126 lemma less_divide_eq_1_pos [simp]:
```
```  1127   "0 < a \<Longrightarrow> (1 < b/a) = (a < b)"
```
```  1128 by (auto simp add: less_divide_eq)
```
```  1129
```
```  1130 lemma less_divide_eq_1_neg [simp]:
```
```  1131   "a < 0 \<Longrightarrow> (1 < b/a) = (b < a)"
```
```  1132 by (auto simp add: less_divide_eq)
```
```  1133
```
```  1134 lemma divide_less_eq_1_pos [simp]:
```
```  1135   "0 < a \<Longrightarrow> (b/a < 1) = (b < a)"
```
```  1136 by (auto simp add: divide_less_eq)
```
```  1137
```
```  1138 lemma divide_less_eq_1_neg [simp]:
```
```  1139   "a < 0 \<Longrightarrow> b/a < 1 \<longleftrightarrow> a < b"
```
```  1140 by (auto simp add: divide_less_eq)
```
```  1141
```
```  1142 lemma eq_divide_eq_1 [simp]:
```
```  1143   "(1 = b/a) = ((a \<noteq> 0 & a = b))"
```
```  1144 by (auto simp add: eq_divide_eq)
```
```  1145
```
```  1146 lemma divide_eq_eq_1 [simp]:
```
```  1147   "(b/a = 1) = ((a \<noteq> 0 & a = b))"
```
```  1148 by (auto simp add: divide_eq_eq)
```
```  1149
```
```  1150 lemma abs_inverse [simp]:
```
```  1151      "\<bar>inverse a\<bar> =
```
```  1152       inverse \<bar>a\<bar>"
```
```  1153 apply (cases "a=0", simp)
```
```  1154 apply (simp add: nonzero_abs_inverse)
```
```  1155 done
```
```  1156
```
```  1157 lemma abs_divide [simp]:
```
```  1158      "\<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
```
```  1159 apply (cases "b=0", simp)
```
```  1160 apply (simp add: nonzero_abs_divide)
```
```  1161 done
```
```  1162
```
```  1163 lemma abs_div_pos: "0 < y ==>
```
```  1164     \<bar>x\<bar> / y = \<bar>x / y\<bar>"
```
```  1165   apply (subst abs_divide)
```
```  1166   apply (simp add: order_less_imp_le)
```
```  1167 done
```
```  1168
```
```  1169 lemma zero_le_divide_abs_iff [simp]: "(0 \<le> a / \<bar>b\<bar>) = (0 \<le> a | b = 0)"
```
```  1170 by (auto simp: zero_le_divide_iff)
```
```  1171
```
```  1172 lemma divide_le_0_abs_iff [simp]: "(a / \<bar>b\<bar> \<le> 0) = (a \<le> 0 | b = 0)"
```
```  1173 by (auto simp: divide_le_0_iff)
```
```  1174
```
```  1175 lemma inverse_sgn:
```
```  1176   "sgn (inverse a) = inverse (sgn a)"
```
```  1177   by (simp add: sgn_if)
```
```  1178
```
```  1179 lemma field_le_mult_one_interval:
```
```  1180   assumes *: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y"
```
```  1181   shows "x \<le> y"
```
```  1182 proof (cases "0 < x")
```
```  1183   assume "0 < x"
```
```  1184   thus ?thesis
```
```  1185     using dense_le_bounded[of 0 1 "y/x"] *
```
```  1186     unfolding le_divide_eq if_P[OF \<open>0 < x\<close>] by simp
```
```  1187 next
```
```  1188   assume "\<not>0 < x" hence "x \<le> 0" by simp
```
```  1189   obtain s::'a where s: "0 < s" "s < 1" using dense[of 0 "1::'a"] by auto
```
```  1190   hence "x \<le> s * x" using mult_le_cancel_right[of 1 x s] \<open>x \<le> 0\<close> by auto
```
```  1191   also note *[OF s]
```
```  1192   finally show ?thesis .
```
```  1193 qed
```
```  1194
```
```  1195 end
```
```  1196
```
```  1197 text \<open>Min/max Simplification Rules\<close>
```
```  1198
```
```  1199 lemma min_mult_distrib_left:
```
```  1200   fixes x::"'a::linordered_idom"
```
```  1201   shows "p * min x y = (if 0 \<le> p then min (p*x) (p*y) else max (p*x) (p*y))"
```
```  1202 by (auto simp add: min_def max_def mult_le_cancel_left)
```
```  1203
```
```  1204 lemma min_mult_distrib_right:
```
```  1205   fixes x::"'a::linordered_idom"
```
```  1206   shows "min x y * p = (if 0 \<le> p then min (x*p) (y*p) else max (x*p) (y*p))"
```
```  1207 by (auto simp add: min_def max_def mult_le_cancel_right)
```
```  1208
```
```  1209 lemma min_divide_distrib_right:
```
```  1210   fixes x::"'a::linordered_field"
```
```  1211   shows "min x y / p = (if 0 \<le> p then min (x/p) (y/p) else max (x/p) (y/p))"
```
```  1212 by (simp add: min_mult_distrib_right divide_inverse)
```
```  1213
```
```  1214 lemma max_mult_distrib_left:
```
```  1215   fixes x::"'a::linordered_idom"
```
```  1216   shows "p * max x y = (if 0 \<le> p then max (p*x) (p*y) else min (p*x) (p*y))"
```
```  1217 by (auto simp add: min_def max_def mult_le_cancel_left)
```
```  1218
```
```  1219 lemma max_mult_distrib_right:
```
```  1220   fixes x::"'a::linordered_idom"
```
```  1221   shows "max x y * p = (if 0 \<le> p then max (x*p) (y*p) else min (x*p) (y*p))"
```
```  1222 by (auto simp add: min_def max_def mult_le_cancel_right)
```
```  1223
```
```  1224 lemma max_divide_distrib_right:
```
```  1225   fixes x::"'a::linordered_field"
```
```  1226   shows "max x y / p = (if 0 \<le> p then max (x/p) (y/p) else min (x/p) (y/p))"
```
```  1227 by (simp add: max_mult_distrib_right divide_inverse)
```
```  1228
```
```  1229 hide_fact (open) field_inverse field_divide_inverse field_inverse_zero
```
```  1230
```
```  1231 code_identifier
```
```  1232   code_module Fields \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
```
```  1233
```
```  1234 end
```