src/HOL/Ring_and_Field.thy
author paulson
Tue Jun 28 15:27:45 2005 +0200 (2005-06-28)
changeset 16587 b34c8aa657a5
parent 16568 e02fe7ae212b
child 16775 c1b87ef4a1c3
permissions -rw-r--r--
Constant "If" is now local
     1 (*  Title:   HOL/Ring_and_Field.thy
     2     ID:      $Id$
     3     Author:  Gertrud Bauer, Steven Obua, Lawrence C Paulson and Markus Wenzel
     4 *)
     5 
     6 header {* (Ordered) Rings and Fields *}
     7 
     8 theory Ring_and_Field
     9 imports OrderedGroup
    10 begin
    11 
    12 text {*
    13   The theory of partially ordered rings is taken from the books:
    14   \begin{itemize}
    15   \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
    16   \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
    17   \end{itemize}
    18   Most of the used notions can also be looked up in 
    19   \begin{itemize}
    20   \item \url{http://www.mathworld.com} by Eric Weisstein et. al.
    21   \item \emph{Algebra I} by van der Waerden, Springer.
    22   \end{itemize}
    23 *}
    24 
    25 axclass semiring \<subseteq> ab_semigroup_add, semigroup_mult
    26   left_distrib: "(a + b) * c = a * c + b * c"
    27   right_distrib: "a * (b + c) = a * b + a * c"
    28 
    29 axclass semiring_0 \<subseteq> semiring, comm_monoid_add
    30 
    31 axclass semiring_0_cancel \<subseteq> semiring_0, cancel_ab_semigroup_add
    32 
    33 axclass comm_semiring \<subseteq> ab_semigroup_add, ab_semigroup_mult  
    34   distrib: "(a + b) * c = a * c + b * c"
    35 
    36 instance comm_semiring \<subseteq> semiring
    37 proof
    38   fix a b c :: 'a
    39   show "(a + b) * c = a * c + b * c" by (simp add: distrib)
    40   have "a * (b + c) = (b + c) * a" by (simp add: mult_ac)
    41   also have "... = b * a + c * a" by (simp only: distrib)
    42   also have "... = a * b + a * c" by (simp add: mult_ac)
    43   finally show "a * (b + c) = a * b + a * c" by blast
    44 qed
    45 
    46 axclass comm_semiring_0 \<subseteq> comm_semiring, comm_monoid_add
    47 
    48 instance comm_semiring_0 \<subseteq> semiring_0 ..
    49 
    50 axclass comm_semiring_0_cancel \<subseteq> comm_semiring_0, cancel_ab_semigroup_add
    51 
    52 instance comm_semiring_0_cancel \<subseteq> semiring_0_cancel ..
    53 
    54 axclass axclass_0_neq_1 \<subseteq> zero, one
    55   zero_neq_one [simp]: "0 \<noteq> 1"
    56 
    57 axclass semiring_1 \<subseteq> axclass_0_neq_1, semiring_0, monoid_mult
    58 
    59 axclass comm_semiring_1 \<subseteq> axclass_0_neq_1, comm_semiring_0, comm_monoid_mult (* previously almost_semiring *)
    60 
    61 instance comm_semiring_1 \<subseteq> semiring_1 ..
    62 
    63 axclass axclass_no_zero_divisors \<subseteq> zero, times
    64   no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"
    65 
    66 axclass semiring_1_cancel \<subseteq> semiring_1, cancel_ab_semigroup_add
    67 
    68 instance semiring_1_cancel \<subseteq> semiring_0_cancel ..
    69 
    70 axclass comm_semiring_1_cancel \<subseteq> comm_semiring_1, cancel_ab_semigroup_add (* previously semiring *)
    71 
    72 instance comm_semiring_1_cancel \<subseteq> semiring_1_cancel ..
    73 
    74 instance comm_semiring_1_cancel \<subseteq> comm_semiring_0_cancel ..
    75 
    76 axclass ring \<subseteq> semiring, ab_group_add
    77 
    78 instance ring \<subseteq> semiring_0_cancel ..
    79 
    80 axclass comm_ring \<subseteq> comm_semiring_0, ab_group_add
    81 
    82 instance comm_ring \<subseteq> ring ..
    83 
    84 instance comm_ring \<subseteq> comm_semiring_0_cancel ..
    85 
    86 axclass ring_1 \<subseteq> ring, semiring_1
    87 
    88 instance ring_1 \<subseteq> semiring_1_cancel ..
    89 
    90 axclass comm_ring_1 \<subseteq> comm_ring, comm_semiring_1 (* previously ring *)
    91 
    92 instance comm_ring_1 \<subseteq> ring_1 ..
    93 
    94 instance comm_ring_1 \<subseteq> comm_semiring_1_cancel ..
    95 
    96 axclass idom \<subseteq> comm_ring_1, axclass_no_zero_divisors
    97 
    98 axclass field \<subseteq> comm_ring_1, inverse
    99   left_inverse [simp]: "a \<noteq> 0 ==> inverse a * a = 1"
   100   divide_inverse:      "a / b = a * inverse b"
   101 
   102 lemma mult_zero_left [simp]: "0 * a = (0::'a::semiring_0_cancel)"
   103 proof -
   104   have "0*a + 0*a = 0*a + 0"
   105     by (simp add: left_distrib [symmetric])
   106   thus ?thesis 
   107     by (simp only: add_left_cancel)
   108 qed
   109 
   110 lemma mult_zero_right [simp]: "a * 0 = (0::'a::semiring_0_cancel)"
   111 proof -
   112   have "a*0 + a*0 = a*0 + 0"
   113     by (simp add: right_distrib [symmetric])
   114   thus ?thesis 
   115     by (simp only: add_left_cancel)
   116 qed
   117 
   118 lemma field_mult_eq_0_iff [simp]: "(a*b = (0::'a::field)) = (a = 0 | b = 0)"
   119 proof cases
   120   assume "a=0" thus ?thesis by simp
   121 next
   122   assume anz [simp]: "a\<noteq>0"
   123   { assume "a * b = 0"
   124     hence "inverse a * (a * b) = 0" by simp
   125     hence "b = 0"  by (simp (no_asm_use) add: mult_assoc [symmetric])}
   126   thus ?thesis by force
   127 qed
   128 
   129 instance field \<subseteq> idom
   130 by (intro_classes, simp)
   131   
   132 axclass division_by_zero \<subseteq> zero, inverse
   133   inverse_zero [simp]: "inverse 0 = 0"
   134 
   135 subsection {* Distribution rules *}
   136 
   137 theorems ring_distrib = right_distrib left_distrib
   138 
   139 text{*For the @{text combine_numerals} simproc*}
   140 lemma combine_common_factor:
   141      "a*e + (b*e + c) = (a+b)*e + (c::'a::semiring)"
   142 by (simp add: left_distrib add_ac)
   143 
   144 lemma minus_mult_left: "- (a * b) = (-a) * (b::'a::ring)"
   145 apply (rule equals_zero_I)
   146 apply (simp add: left_distrib [symmetric]) 
   147 done
   148 
   149 lemma minus_mult_right: "- (a * b) = a * -(b::'a::ring)"
   150 apply (rule equals_zero_I)
   151 apply (simp add: right_distrib [symmetric]) 
   152 done
   153 
   154 lemma minus_mult_minus [simp]: "(- a) * (- b) = a * (b::'a::ring)"
   155   by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])
   156 
   157 lemma minus_mult_commute: "(- a) * b = a * (- b::'a::ring)"
   158   by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])
   159 
   160 lemma right_diff_distrib: "a * (b - c) = a * b - a * (c::'a::ring)"
   161 by (simp add: right_distrib diff_minus 
   162               minus_mult_left [symmetric] minus_mult_right [symmetric]) 
   163 
   164 lemma left_diff_distrib: "(a - b) * c = a * c - b * (c::'a::ring)"
   165 by (simp add: left_distrib diff_minus 
   166               minus_mult_left [symmetric] minus_mult_right [symmetric]) 
   167 
   168 axclass pordered_semiring \<subseteq> semiring_0, pordered_ab_semigroup_add 
   169   mult_left_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> c * a <= c * b"
   170   mult_right_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> a * c <= b * c"
   171 
   172 axclass pordered_cancel_semiring \<subseteq> pordered_semiring, cancel_ab_semigroup_add
   173 
   174 instance pordered_cancel_semiring \<subseteq> semiring_0_cancel ..
   175 
   176 axclass ordered_semiring_strict \<subseteq> semiring_0, ordered_cancel_ab_semigroup_add
   177   mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
   178   mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
   179 
   180 instance ordered_semiring_strict \<subseteq> semiring_0_cancel ..
   181 
   182 instance ordered_semiring_strict \<subseteq> pordered_cancel_semiring
   183 apply intro_classes
   184 apply (case_tac "a < b & 0 < c")
   185 apply (auto simp add: mult_strict_left_mono order_less_le)
   186 apply (auto simp add: mult_strict_left_mono order_le_less)
   187 apply (simp add: mult_strict_right_mono)
   188 done
   189 
   190 axclass pordered_comm_semiring \<subseteq> comm_semiring_0, pordered_ab_semigroup_add
   191   mult_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> c * a <= c * b"
   192 
   193 axclass pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring, cancel_ab_semigroup_add
   194 
   195 instance pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring ..
   196 
   197 axclass ordered_comm_semiring_strict \<subseteq> comm_semiring_0, ordered_cancel_ab_semigroup_add
   198   mult_strict_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
   199 
   200 instance pordered_comm_semiring \<subseteq> pordered_semiring
   201 by (intro_classes, insert mult_mono, simp_all add: mult_commute, blast+)
   202 
   203 instance pordered_cancel_comm_semiring \<subseteq> pordered_cancel_semiring ..
   204 
   205 instance ordered_comm_semiring_strict \<subseteq> ordered_semiring_strict
   206 by (intro_classes, insert mult_strict_mono, simp_all add: mult_commute, blast+)
   207 
   208 instance ordered_comm_semiring_strict \<subseteq> pordered_cancel_comm_semiring
   209 apply (intro_classes)
   210 apply (case_tac "a < b & 0 < c")
   211 apply (auto simp add: mult_strict_left_mono order_less_le)
   212 apply (auto simp add: mult_strict_left_mono order_le_less)
   213 done
   214 
   215 axclass pordered_ring \<subseteq> ring, pordered_semiring 
   216 
   217 instance pordered_ring \<subseteq> pordered_ab_group_add ..
   218 
   219 instance pordered_ring \<subseteq> pordered_cancel_semiring ..
   220 
   221 axclass lordered_ring \<subseteq> pordered_ring, lordered_ab_group_abs
   222 
   223 instance lordered_ring \<subseteq> lordered_ab_group_meet ..
   224 
   225 instance lordered_ring \<subseteq> lordered_ab_group_join ..
   226 
   227 axclass axclass_abs_if \<subseteq> minus, ord, zero
   228   abs_if: "abs a = (if (a < 0) then (-a) else a)"
   229 
   230 axclass ordered_ring_strict \<subseteq> ring, ordered_semiring_strict, axclass_abs_if
   231 
   232 instance ordered_ring_strict \<subseteq> lordered_ab_group ..
   233 
   234 instance ordered_ring_strict \<subseteq> lordered_ring
   235 by (intro_classes, simp add: abs_if join_eq_if)
   236 
   237 axclass pordered_comm_ring \<subseteq> comm_ring, pordered_comm_semiring
   238 
   239 axclass ordered_semidom \<subseteq> comm_semiring_1_cancel, ordered_comm_semiring_strict (* previously ordered_semiring *)
   240   zero_less_one [simp]: "0 < 1"
   241 
   242 axclass ordered_idom \<subseteq> comm_ring_1, ordered_comm_semiring_strict, axclass_abs_if (* previously ordered_ring *)
   243 
   244 instance ordered_idom \<subseteq> ordered_ring_strict ..
   245 
   246 axclass ordered_field \<subseteq> field, ordered_idom
   247 
   248 lemmas linorder_neqE_ordered_idom =
   249  linorder_neqE[where 'a = "?'b::ordered_idom"]
   250 
   251 lemma eq_add_iff1:
   252      "(a*e + c = b*e + d) = ((a-b)*e + c = (d::'a::ring))"
   253 apply (simp add: diff_minus left_distrib)
   254 apply (simp add: diff_minus left_distrib add_ac)
   255 apply (simp add: compare_rls minus_mult_left [symmetric])
   256 done
   257 
   258 lemma eq_add_iff2:
   259      "(a*e + c = b*e + d) = (c = (b-a)*e + (d::'a::ring))"
   260 apply (simp add: diff_minus left_distrib add_ac)
   261 apply (simp add: compare_rls minus_mult_left [symmetric]) 
   262 done
   263 
   264 lemma less_add_iff1:
   265      "(a*e + c < b*e + d) = ((a-b)*e + c < (d::'a::pordered_ring))"
   266 apply (simp add: diff_minus left_distrib add_ac)
   267 apply (simp add: compare_rls minus_mult_left [symmetric]) 
   268 done
   269 
   270 lemma less_add_iff2:
   271      "(a*e + c < b*e + d) = (c < (b-a)*e + (d::'a::pordered_ring))"
   272 apply (simp add: diff_minus left_distrib add_ac)
   273 apply (simp add: compare_rls minus_mult_left [symmetric]) 
   274 done
   275 
   276 lemma le_add_iff1:
   277      "(a*e + c \<le> b*e + d) = ((a-b)*e + c \<le> (d::'a::pordered_ring))"
   278 apply (simp add: diff_minus left_distrib add_ac)
   279 apply (simp add: compare_rls minus_mult_left [symmetric]) 
   280 done
   281 
   282 lemma le_add_iff2:
   283      "(a*e + c \<le> b*e + d) = (c \<le> (b-a)*e + (d::'a::pordered_ring))"
   284 apply (simp add: diff_minus left_distrib add_ac)
   285 apply (simp add: compare_rls minus_mult_left [symmetric]) 
   286 done
   287 
   288 subsection {* Ordering Rules for Multiplication *}
   289 
   290 lemma mult_left_le_imp_le:
   291      "[|c*a \<le> c*b; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)"
   292   by (force simp add: mult_strict_left_mono linorder_not_less [symmetric])
   293  
   294 lemma mult_right_le_imp_le:
   295      "[|a*c \<le> b*c; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)"
   296   by (force simp add: mult_strict_right_mono linorder_not_less [symmetric])
   297 
   298 lemma mult_left_less_imp_less:
   299      "[|c*a < c*b; 0 \<le> c|] ==> a < (b::'a::ordered_semiring_strict)"
   300   by (force simp add: mult_left_mono linorder_not_le [symmetric])
   301  
   302 lemma mult_right_less_imp_less:
   303      "[|a*c < b*c; 0 \<le> c|] ==> a < (b::'a::ordered_semiring_strict)"
   304   by (force simp add: mult_right_mono linorder_not_le [symmetric])
   305 
   306 lemma mult_strict_left_mono_neg:
   307      "[|b < a; c < 0|] ==> c * a < c * (b::'a::ordered_ring_strict)"
   308 apply (drule mult_strict_left_mono [of _ _ "-c"])
   309 apply (simp_all add: minus_mult_left [symmetric]) 
   310 done
   311 
   312 lemma mult_left_mono_neg:
   313      "[|b \<le> a; c \<le> 0|] ==> c * a \<le>  c * (b::'a::pordered_ring)"
   314 apply (drule mult_left_mono [of _ _ "-c"])
   315 apply (simp_all add: minus_mult_left [symmetric]) 
   316 done
   317 
   318 lemma mult_strict_right_mono_neg:
   319      "[|b < a; c < 0|] ==> a * c < b * (c::'a::ordered_ring_strict)"
   320 apply (drule mult_strict_right_mono [of _ _ "-c"])
   321 apply (simp_all add: minus_mult_right [symmetric]) 
   322 done
   323 
   324 lemma mult_right_mono_neg:
   325      "[|b \<le> a; c \<le> 0|] ==> a * c \<le>  (b::'a::pordered_ring) * c"
   326 apply (drule mult_right_mono [of _ _ "-c"])
   327 apply (simp)
   328 apply (simp_all add: minus_mult_right [symmetric]) 
   329 done
   330 
   331 subsection{* Products of Signs *}
   332 
   333 lemma mult_pos: "[| (0::'a::ordered_semiring_strict) < a; 0 < b |] ==> 0 < a*b"
   334 by (drule mult_strict_left_mono [of 0 b], auto)
   335 
   336 lemma mult_pos_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; 0 \<le> b |] ==> 0 \<le> a*b"
   337 by (drule mult_left_mono [of 0 b], auto)
   338 
   339 lemma mult_pos_neg: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> a*b < 0"
   340 by (drule mult_strict_left_mono [of b 0], auto)
   341 
   342 lemma mult_pos_neg_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> a*b \<le> 0"
   343 by (drule mult_left_mono [of b 0], auto)
   344 
   345 lemma mult_pos_neg2: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> b*a < 0" 
   346 by (drule mult_strict_right_mono[of b 0], auto)
   347 
   348 lemma mult_pos_neg2_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> b*a \<le> 0" 
   349 by (drule mult_right_mono[of b 0], auto)
   350 
   351 lemma mult_neg: "[| a < (0::'a::ordered_ring_strict); b < 0 |] ==> 0 < a*b"
   352 by (drule mult_strict_right_mono_neg, auto)
   353 
   354 lemma mult_neg_le: "[| a \<le> (0::'a::pordered_ring); b \<le> 0 |] ==> 0 \<le> a*b"
   355 by (drule mult_right_mono_neg[of a 0 b ], auto)
   356 
   357 lemma zero_less_mult_pos:
   358      "[| 0 < a*b; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"
   359 apply (case_tac "b\<le>0") 
   360  apply (auto simp add: order_le_less linorder_not_less)
   361 apply (drule_tac mult_pos_neg [of a b]) 
   362  apply (auto dest: order_less_not_sym)
   363 done
   364 
   365 lemma zero_less_mult_pos2:
   366      "[| 0 < b*a; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"
   367 apply (case_tac "b\<le>0") 
   368  apply (auto simp add: order_le_less linorder_not_less)
   369 apply (drule_tac mult_pos_neg2 [of a b]) 
   370  apply (auto dest: order_less_not_sym)
   371 done
   372 
   373 lemma zero_less_mult_iff:
   374      "((0::'a::ordered_ring_strict) < a*b) = (0 < a & 0 < b | a < 0 & b < 0)"
   375 apply (auto simp add: order_le_less linorder_not_less mult_pos mult_neg)
   376 apply (blast dest: zero_less_mult_pos) 
   377 apply (blast dest: zero_less_mult_pos2)
   378 done
   379 
   380 text{*A field has no "zero divisors", and this theorem holds without the
   381       assumption of an ordering.  See @{text field_mult_eq_0_iff} below.*}
   382 lemma mult_eq_0_iff [simp]: "(a*b = (0::'a::ordered_ring_strict)) = (a = 0 | b = 0)"
   383 apply (case_tac "a < 0")
   384 apply (auto simp add: linorder_not_less order_le_less linorder_neq_iff)
   385 apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono)+
   386 done
   387 
   388 lemma zero_le_mult_iff:
   389      "((0::'a::ordered_ring_strict) \<le> a*b) = (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
   390 by (auto simp add: eq_commute [of 0] order_le_less linorder_not_less
   391                    zero_less_mult_iff)
   392 
   393 lemma mult_less_0_iff:
   394      "(a*b < (0::'a::ordered_ring_strict)) = (0 < a & b < 0 | a < 0 & 0 < b)"
   395 apply (insert zero_less_mult_iff [of "-a" b]) 
   396 apply (force simp add: minus_mult_left[symmetric]) 
   397 done
   398 
   399 lemma mult_le_0_iff:
   400      "(a*b \<le> (0::'a::ordered_ring_strict)) = (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
   401 apply (insert zero_le_mult_iff [of "-a" b]) 
   402 apply (force simp add: minus_mult_left[symmetric]) 
   403 done
   404 
   405 lemma split_mult_pos_le: "(0 \<le> a & 0 \<le> b) | (a \<le> 0 & b \<le> 0) \<Longrightarrow> 0 \<le> a * (b::_::pordered_ring)"
   406 by (auto simp add: mult_pos_le mult_neg_le)
   407 
   408 lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> (0::_::pordered_cancel_semiring)" 
   409 by (auto simp add: mult_pos_neg_le mult_pos_neg2_le)
   410 
   411 lemma zero_le_square: "(0::'a::ordered_ring_strict) \<le> a*a"
   412 by (simp add: zero_le_mult_iff linorder_linear) 
   413 
   414 text{*Proving axiom @{text zero_less_one} makes all @{text ordered_semidom}
   415       theorems available to members of @{term ordered_idom} *}
   416 
   417 instance ordered_idom \<subseteq> ordered_semidom
   418 proof
   419   have "(0::'a) \<le> 1*1" by (rule zero_le_square)
   420   thus "(0::'a) < 1" by (simp add: order_le_less) 
   421 qed
   422 
   423 instance ordered_ring_strict \<subseteq> axclass_no_zero_divisors 
   424 by (intro_classes, simp)
   425 
   426 instance ordered_idom \<subseteq> idom ..
   427 
   428 text{*All three types of comparision involving 0 and 1 are covered.*}
   429 
   430 declare zero_neq_one [THEN not_sym, simp]
   431 
   432 lemma zero_le_one [simp]: "(0::'a::ordered_semidom) \<le> 1"
   433   by (rule zero_less_one [THEN order_less_imp_le]) 
   434 
   435 lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semidom) \<le> 0"
   436 by (simp add: linorder_not_le) 
   437 
   438 lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semidom) < 0"
   439 by (simp add: linorder_not_less) 
   440 
   441 subsection{*More Monotonicity*}
   442 
   443 text{*Strict monotonicity in both arguments*}
   444 lemma mult_strict_mono:
   445      "[|a<b; c<d; 0<b; 0\<le>c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
   446 apply (case_tac "c=0")
   447  apply (simp add: mult_pos) 
   448 apply (erule mult_strict_right_mono [THEN order_less_trans])
   449  apply (force simp add: order_le_less) 
   450 apply (erule mult_strict_left_mono, assumption)
   451 done
   452 
   453 text{*This weaker variant has more natural premises*}
   454 lemma mult_strict_mono':
   455      "[| a<b; c<d; 0 \<le> a; 0 \<le> c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
   456 apply (rule mult_strict_mono)
   457 apply (blast intro: order_le_less_trans)+
   458 done
   459 
   460 lemma mult_mono:
   461      "[|a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c|] 
   462       ==> a * c  \<le>  b * (d::'a::pordered_semiring)"
   463 apply (erule mult_right_mono [THEN order_trans], assumption)
   464 apply (erule mult_left_mono, assumption)
   465 done
   466 
   467 lemma less_1_mult: "[| 1 < m; 1 < n |] ==> 1 < m*(n::'a::ordered_semidom)"
   468 apply (insert mult_strict_mono [of 1 m 1 n]) 
   469 apply (simp add:  order_less_trans [OF zero_less_one]) 
   470 done
   471 
   472 subsection{*Cancellation Laws for Relationships With a Common Factor*}
   473 
   474 text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
   475    also with the relations @{text "\<le>"} and equality.*}
   476 
   477 text{*These ``disjunction'' versions produce two cases when the comparison is
   478  an assumption, but effectively four when the comparison is a goal.*}
   479 
   480 lemma mult_less_cancel_right_disj:
   481     "(a*c < b*c) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
   482 apply (case_tac "c = 0")
   483 apply (auto simp add: linorder_neq_iff mult_strict_right_mono 
   484                       mult_strict_right_mono_neg)
   485 apply (auto simp add: linorder_not_less 
   486                       linorder_not_le [symmetric, of "a*c"]
   487                       linorder_not_le [symmetric, of a])
   488 apply (erule_tac [!] notE)
   489 apply (auto simp add: order_less_imp_le mult_right_mono 
   490                       mult_right_mono_neg)
   491 done
   492 
   493 lemma mult_less_cancel_left_disj:
   494     "(c*a < c*b) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
   495 apply (case_tac "c = 0")
   496 apply (auto simp add: linorder_neq_iff mult_strict_left_mono 
   497                       mult_strict_left_mono_neg)
   498 apply (auto simp add: linorder_not_less 
   499                       linorder_not_le [symmetric, of "c*a"]
   500                       linorder_not_le [symmetric, of a])
   501 apply (erule_tac [!] notE)
   502 apply (auto simp add: order_less_imp_le mult_left_mono 
   503                       mult_left_mono_neg)
   504 done
   505 
   506 
   507 text{*The ``conjunction of implication'' lemmas produce two cases when the
   508 comparison is a goal, but give four when the comparison is an assumption.*}
   509 
   510 lemma mult_less_cancel_right:
   511   fixes c :: "'a :: ordered_ring_strict"
   512   shows      "(a*c < b*c) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))"
   513 by (insert mult_less_cancel_right_disj [of a c b], auto)
   514 
   515 lemma mult_less_cancel_left:
   516   fixes c :: "'a :: ordered_ring_strict"
   517   shows      "(c*a < c*b) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))"
   518 by (insert mult_less_cancel_left_disj [of c a b], auto)
   519 
   520 lemma mult_le_cancel_right:
   521      "(a*c \<le> b*c) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
   522 by (simp add: linorder_not_less [symmetric] mult_less_cancel_right_disj)
   523 
   524 lemma mult_le_cancel_left:
   525      "(c*a \<le> c*b) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
   526 by (simp add: linorder_not_less [symmetric] mult_less_cancel_left_disj)
   527 
   528 lemma mult_less_imp_less_left:
   529       assumes less: "c*a < c*b" and nonneg: "0 \<le> c"
   530       shows "a < (b::'a::ordered_semiring_strict)"
   531 proof (rule ccontr)
   532   assume "~ a < b"
   533   hence "b \<le> a" by (simp add: linorder_not_less)
   534   hence "c*b \<le> c*a" by (rule mult_left_mono)
   535   with this and less show False 
   536     by (simp add: linorder_not_less [symmetric])
   537 qed
   538 
   539 lemma mult_less_imp_less_right:
   540   assumes less: "a*c < b*c" and nonneg: "0 <= c"
   541   shows "a < (b::'a::ordered_semiring_strict)"
   542 proof (rule ccontr)
   543   assume "~ a < b"
   544   hence "b \<le> a" by (simp add: linorder_not_less)
   545   hence "b*c \<le> a*c" by (rule mult_right_mono)
   546   with this and less show False 
   547     by (simp add: linorder_not_less [symmetric])
   548 qed  
   549 
   550 text{*Cancellation of equalities with a common factor*}
   551 lemma mult_cancel_right [simp]:
   552      "(a*c = b*c) = (c = (0::'a::ordered_ring_strict) | a=b)"
   553 apply (cut_tac linorder_less_linear [of 0 c])
   554 apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono
   555              simp add: linorder_neq_iff)
   556 done
   557 
   558 text{*These cancellation theorems require an ordering. Versions are proved
   559       below that work for fields without an ordering.*}
   560 lemma mult_cancel_left [simp]:
   561      "(c*a = c*b) = (c = (0::'a::ordered_ring_strict) | a=b)"
   562 apply (cut_tac linorder_less_linear [of 0 c])
   563 apply (force dest: mult_strict_left_mono_neg mult_strict_left_mono
   564              simp add: linorder_neq_iff)
   565 done
   566 
   567 
   568 subsubsection{*Special Cancellation Simprules for Multiplication*}
   569 
   570 text{*These also produce two cases when the comparison is a goal.*}
   571 
   572 lemma mult_le_cancel_right1:
   573   fixes c :: "'a :: ordered_idom"
   574   shows "(c \<le> b*c) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))"
   575 by (insert mult_le_cancel_right [of 1 c b], simp)
   576 
   577 lemma mult_le_cancel_right2:
   578   fixes c :: "'a :: ordered_idom"
   579   shows "(a*c \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))"
   580 by (insert mult_le_cancel_right [of a c 1], simp)
   581 
   582 lemma mult_le_cancel_left1:
   583   fixes c :: "'a :: ordered_idom"
   584   shows "(c \<le> c*b) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))"
   585 by (insert mult_le_cancel_left [of c 1 b], simp)
   586 
   587 lemma mult_le_cancel_left2:
   588   fixes c :: "'a :: ordered_idom"
   589   shows "(c*a \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))"
   590 by (insert mult_le_cancel_left [of c a 1], simp)
   591 
   592 lemma mult_less_cancel_right1:
   593   fixes c :: "'a :: ordered_idom"
   594   shows "(c < b*c) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))"
   595 by (insert mult_less_cancel_right [of 1 c b], simp)
   596 
   597 lemma mult_less_cancel_right2:
   598   fixes c :: "'a :: ordered_idom"
   599   shows "(a*c < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))"
   600 by (insert mult_less_cancel_right [of a c 1], simp)
   601 
   602 lemma mult_less_cancel_left1:
   603   fixes c :: "'a :: ordered_idom"
   604   shows "(c < c*b) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))"
   605 by (insert mult_less_cancel_left [of c 1 b], simp)
   606 
   607 lemma mult_less_cancel_left2:
   608   fixes c :: "'a :: ordered_idom"
   609   shows "(c*a < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))"
   610 by (insert mult_less_cancel_left [of c a 1], simp)
   611 
   612 lemma mult_cancel_right1 [simp]:
   613 fixes c :: "'a :: ordered_idom"
   614   shows "(c = b*c) = (c = 0 | b=1)"
   615 by (insert mult_cancel_right [of 1 c b], force)
   616 
   617 lemma mult_cancel_right2 [simp]:
   618 fixes c :: "'a :: ordered_idom"
   619   shows "(a*c = c) = (c = 0 | a=1)"
   620 by (insert mult_cancel_right [of a c 1], simp)
   621  
   622 lemma mult_cancel_left1 [simp]:
   623 fixes c :: "'a :: ordered_idom"
   624   shows "(c = c*b) = (c = 0 | b=1)"
   625 by (insert mult_cancel_left [of c 1 b], force)
   626 
   627 lemma mult_cancel_left2 [simp]:
   628 fixes c :: "'a :: ordered_idom"
   629   shows "(c*a = c) = (c = 0 | a=1)"
   630 by (insert mult_cancel_left [of c a 1], simp)
   631 
   632 
   633 text{*Simprules for comparisons where common factors can be cancelled.*}
   634 lemmas mult_compare_simps =
   635     mult_le_cancel_right mult_le_cancel_left
   636     mult_le_cancel_right1 mult_le_cancel_right2
   637     mult_le_cancel_left1 mult_le_cancel_left2
   638     mult_less_cancel_right mult_less_cancel_left
   639     mult_less_cancel_right1 mult_less_cancel_right2
   640     mult_less_cancel_left1 mult_less_cancel_left2
   641     mult_cancel_right mult_cancel_left
   642     mult_cancel_right1 mult_cancel_right2
   643     mult_cancel_left1 mult_cancel_left2
   644 
   645 
   646 text{*This list of rewrites decides ring equalities by ordered rewriting.*}
   647 lemmas ring_eq_simps =  
   648 (*  mult_ac*)
   649   left_distrib right_distrib left_diff_distrib right_diff_distrib
   650   group_eq_simps
   651 (*  add_ac
   652   add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
   653   diff_eq_eq eq_diff_eq *)
   654     
   655 subsection {* Fields *}
   656 
   657 lemma right_inverse [simp]:
   658       assumes not0: "a \<noteq> 0" shows "a * inverse (a::'a::field) = 1"
   659 proof -
   660   have "a * inverse a = inverse a * a" by (simp add: mult_ac)
   661   also have "... = 1" using not0 by simp
   662   finally show ?thesis .
   663 qed
   664 
   665 lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))"
   666 proof
   667   assume neq: "b \<noteq> 0"
   668   {
   669     hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac)
   670     also assume "a / b = 1"
   671     finally show "a = b" by simp
   672   next
   673     assume "a = b"
   674     with neq show "a / b = 1" by (simp add: divide_inverse)
   675   }
   676 qed
   677 
   678 lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a"
   679 by (simp add: divide_inverse)
   680 
   681 lemma divide_self: "a \<noteq> 0 ==> a / (a::'a::field) = 1"
   682   by (simp add: divide_inverse)
   683 
   684 lemma divide_zero [simp]: "a / 0 = (0::'a::{field,division_by_zero})"
   685 by (simp add: divide_inverse)
   686 
   687 lemma divide_self_if [simp]:
   688      "a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)"
   689   by (simp add: divide_self)
   690 
   691 lemma divide_zero_left [simp]: "0/a = (0::'a::field)"
   692 by (simp add: divide_inverse)
   693 
   694 lemma inverse_eq_divide: "inverse (a::'a::field) = 1/a"
   695 by (simp add: divide_inverse)
   696 
   697 lemma add_divide_distrib: "(a+b)/(c::'a::field) = a/c + b/c"
   698 by (simp add: divide_inverse left_distrib) 
   699 
   700 
   701 text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement
   702       of an ordering.*}
   703 lemma field_mult_eq_0_iff [simp]: "(a*b = (0::'a::field)) = (a = 0 | b = 0)"
   704 proof cases
   705   assume "a=0" thus ?thesis by simp
   706 next
   707   assume anz [simp]: "a\<noteq>0"
   708   { assume "a * b = 0"
   709     hence "inverse a * (a * b) = 0" by simp
   710     hence "b = 0"  by (simp (no_asm_use) add: mult_assoc [symmetric])}
   711   thus ?thesis by force
   712 qed
   713 
   714 text{*Cancellation of equalities with a common factor*}
   715 lemma field_mult_cancel_right_lemma:
   716       assumes cnz: "c \<noteq> (0::'a::field)"
   717 	  and eq:  "a*c = b*c"
   718 	 shows "a=b"
   719 proof -
   720   have "(a * c) * inverse c = (b * c) * inverse c"
   721     by (simp add: eq)
   722   thus "a=b"
   723     by (simp add: mult_assoc cnz)
   724 qed
   725 
   726 lemma field_mult_cancel_right [simp]:
   727      "(a*c = b*c) = (c = (0::'a::field) | a=b)"
   728 proof cases
   729   assume "c=0" thus ?thesis by simp
   730 next
   731   assume "c\<noteq>0" 
   732   thus ?thesis by (force dest: field_mult_cancel_right_lemma)
   733 qed
   734 
   735 lemma field_mult_cancel_left [simp]:
   736      "(c*a = c*b) = (c = (0::'a::field) | a=b)"
   737   by (simp add: mult_commute [of c] field_mult_cancel_right) 
   738 
   739 lemma nonzero_imp_inverse_nonzero: "a \<noteq> 0 ==> inverse a \<noteq> (0::'a::field)"
   740 proof
   741   assume ianz: "inverse a = 0"
   742   assume "a \<noteq> 0"
   743   hence "1 = a * inverse a" by simp
   744   also have "... = 0" by (simp add: ianz)
   745   finally have "1 = (0::'a::field)" .
   746   thus False by (simp add: eq_commute)
   747 qed
   748 
   749 
   750 subsection{*Basic Properties of @{term inverse}*}
   751 
   752 lemma inverse_zero_imp_zero: "inverse a = 0 ==> a = (0::'a::field)"
   753 apply (rule ccontr) 
   754 apply (blast dest: nonzero_imp_inverse_nonzero) 
   755 done
   756 
   757 lemma inverse_nonzero_imp_nonzero:
   758    "inverse a = 0 ==> a = (0::'a::field)"
   759 apply (rule ccontr) 
   760 apply (blast dest: nonzero_imp_inverse_nonzero) 
   761 done
   762 
   763 lemma inverse_nonzero_iff_nonzero [simp]:
   764    "(inverse a = 0) = (a = (0::'a::{field,division_by_zero}))"
   765 by (force dest: inverse_nonzero_imp_nonzero) 
   766 
   767 lemma nonzero_inverse_minus_eq:
   768       assumes [simp]: "a\<noteq>0"  shows "inverse(-a) = -inverse(a::'a::field)"
   769 proof -
   770   have "-a * inverse (- a) = -a * - inverse a"
   771     by simp
   772   thus ?thesis 
   773     by (simp only: field_mult_cancel_left, simp)
   774 qed
   775 
   776 lemma inverse_minus_eq [simp]:
   777    "inverse(-a) = -inverse(a::'a::{field,division_by_zero})";
   778 proof cases
   779   assume "a=0" thus ?thesis by (simp add: inverse_zero)
   780 next
   781   assume "a\<noteq>0" 
   782   thus ?thesis by (simp add: nonzero_inverse_minus_eq)
   783 qed
   784 
   785 lemma nonzero_inverse_eq_imp_eq:
   786       assumes inveq: "inverse a = inverse b"
   787 	  and anz:  "a \<noteq> 0"
   788 	  and bnz:  "b \<noteq> 0"
   789 	 shows "a = (b::'a::field)"
   790 proof -
   791   have "a * inverse b = a * inverse a"
   792     by (simp add: inveq)
   793   hence "(a * inverse b) * b = (a * inverse a) * b"
   794     by simp
   795   thus "a = b"
   796     by (simp add: mult_assoc anz bnz)
   797 qed
   798 
   799 lemma inverse_eq_imp_eq:
   800      "inverse a = inverse b ==> a = (b::'a::{field,division_by_zero})"
   801 apply (case_tac "a=0 | b=0") 
   802  apply (force dest!: inverse_zero_imp_zero
   803               simp add: eq_commute [of "0::'a"])
   804 apply (force dest!: nonzero_inverse_eq_imp_eq) 
   805 done
   806 
   807 lemma inverse_eq_iff_eq [simp]:
   808      "(inverse a = inverse b) = (a = (b::'a::{field,division_by_zero}))"
   809 by (force dest!: inverse_eq_imp_eq) 
   810 
   811 lemma nonzero_inverse_inverse_eq:
   812       assumes [simp]: "a \<noteq> 0"  shows "inverse(inverse (a::'a::field)) = a"
   813   proof -
   814   have "(inverse (inverse a) * inverse a) * a = a" 
   815     by (simp add: nonzero_imp_inverse_nonzero)
   816   thus ?thesis
   817     by (simp add: mult_assoc)
   818   qed
   819 
   820 lemma inverse_inverse_eq [simp]:
   821      "inverse(inverse (a::'a::{field,division_by_zero})) = a"
   822   proof cases
   823     assume "a=0" thus ?thesis by simp
   824   next
   825     assume "a\<noteq>0" 
   826     thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
   827   qed
   828 
   829 lemma inverse_1 [simp]: "inverse 1 = (1::'a::field)"
   830   proof -
   831   have "inverse 1 * 1 = (1::'a::field)" 
   832     by (rule left_inverse [OF zero_neq_one [symmetric]])
   833   thus ?thesis  by simp
   834   qed
   835 
   836 lemma inverse_unique: 
   837   assumes ab: "a*b = 1"
   838   shows "inverse a = (b::'a::field)"
   839 proof -
   840   have "a \<noteq> 0" using ab by auto
   841   moreover have "inverse a * (a * b) = inverse a" by (simp add: ab) 
   842   ultimately show ?thesis by (simp add: mult_assoc [symmetric]) 
   843 qed
   844 
   845 lemma nonzero_inverse_mult_distrib: 
   846       assumes anz: "a \<noteq> 0"
   847           and bnz: "b \<noteq> 0"
   848       shows "inverse(a*b) = inverse(b) * inverse(a::'a::field)"
   849   proof -
   850   have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" 
   851     by (simp add: field_mult_eq_0_iff anz bnz)
   852   hence "inverse(a*b) * a = inverse(b)" 
   853     by (simp add: mult_assoc bnz)
   854   hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" 
   855     by simp
   856   thus ?thesis
   857     by (simp add: mult_assoc anz)
   858   qed
   859 
   860 text{*This version builds in division by zero while also re-orienting
   861       the right-hand side.*}
   862 lemma inverse_mult_distrib [simp]:
   863      "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"
   864   proof cases
   865     assume "a \<noteq> 0 & b \<noteq> 0" 
   866     thus ?thesis  by (simp add: nonzero_inverse_mult_distrib mult_commute)
   867   next
   868     assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
   869     thus ?thesis  by force
   870   qed
   871 
   872 text{*There is no slick version using division by zero.*}
   873 lemma inverse_add:
   874      "[|a \<noteq> 0;  b \<noteq> 0|]
   875       ==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)"
   876 apply (simp add: left_distrib mult_assoc)
   877 apply (simp add: mult_commute [of "inverse a"]) 
   878 apply (simp add: mult_assoc [symmetric] add_commute)
   879 done
   880 
   881 lemma inverse_divide [simp]:
   882       "inverse (a/b) = b / (a::'a::{field,division_by_zero})"
   883   by (simp add: divide_inverse mult_commute)
   884 
   885 lemma nonzero_mult_divide_cancel_left:
   886   assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" 
   887     shows "(c*a)/(c*b) = a/(b::'a::field)"
   888 proof -
   889   have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
   890     by (simp add: field_mult_eq_0_iff divide_inverse 
   891                   nonzero_inverse_mult_distrib)
   892   also have "... =  a * inverse b * (inverse c * c)"
   893     by (simp only: mult_ac)
   894   also have "... =  a * inverse b"
   895     by simp
   896     finally show ?thesis 
   897     by (simp add: divide_inverse)
   898 qed
   899 
   900 lemma mult_divide_cancel_left:
   901      "c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"
   902 apply (case_tac "b = 0")
   903 apply (simp_all add: nonzero_mult_divide_cancel_left)
   904 done
   905 
   906 lemma nonzero_mult_divide_cancel_right:
   907      "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)"
   908 by (simp add: mult_commute [of _ c] nonzero_mult_divide_cancel_left) 
   909 
   910 lemma mult_divide_cancel_right:
   911      "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"
   912 apply (case_tac "b = 0")
   913 apply (simp_all add: nonzero_mult_divide_cancel_right)
   914 done
   915 
   916 (*For ExtractCommonTerm*)
   917 lemma mult_divide_cancel_eq_if:
   918      "(c*a) / (c*b) = 
   919       (if c=0 then 0 else a / (b::'a::{field,division_by_zero}))"
   920   by (simp add: mult_divide_cancel_left)
   921 
   922 lemma divide_1 [simp]: "a/1 = (a::'a::field)"
   923   by (simp add: divide_inverse)
   924 
   925 lemma times_divide_eq_right: "a * (b/c) = (a*b) / (c::'a::field)"
   926 by (simp add: divide_inverse mult_assoc)
   927 
   928 lemma times_divide_eq_left: "(b/c) * a = (b*a) / (c::'a::field)"
   929 by (simp add: divide_inverse mult_ac)
   930 
   931 lemma divide_divide_eq_right [simp]:
   932      "a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"
   933 by (simp add: divide_inverse mult_ac)
   934 
   935 lemma divide_divide_eq_left [simp]:
   936      "(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"
   937 by (simp add: divide_inverse mult_assoc)
   938 
   939 
   940 subsubsection{*Special Cancellation Simprules for Division*}
   941 
   942 lemma mult_divide_cancel_left_if [simp]:
   943   fixes c :: "'a :: {field,division_by_zero}"
   944   shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)"
   945 by (simp add: mult_divide_cancel_left)
   946 
   947 lemma mult_divide_cancel_right_if [simp]:
   948   fixes c :: "'a :: {field,division_by_zero}"
   949   shows "(a*c) / (b*c) = (if c=0 then 0 else a/b)"
   950 by (simp add: mult_divide_cancel_right)
   951 
   952 lemma mult_divide_cancel_left_if1 [simp]:
   953   fixes c :: "'a :: {field,division_by_zero}"
   954   shows "c / (c*b) = (if c=0 then 0 else 1/b)"
   955 apply (insert mult_divide_cancel_left_if [of c 1 b]) 
   956 apply (simp del: mult_divide_cancel_left_if)
   957 done
   958 
   959 lemma mult_divide_cancel_left_if2 [simp]:
   960   fixes c :: "'a :: {field,division_by_zero}"
   961   shows "(c*a) / c = (if c=0 then 0 else a)" 
   962 apply (insert mult_divide_cancel_left_if [of c a 1]) 
   963 apply (simp del: mult_divide_cancel_left_if)
   964 done
   965 
   966 lemma mult_divide_cancel_right_if1 [simp]:
   967   fixes c :: "'a :: {field,division_by_zero}"
   968   shows "c / (b*c) = (if c=0 then 0 else 1/b)"
   969 apply (insert mult_divide_cancel_right_if [of 1 c b]) 
   970 apply (simp del: mult_divide_cancel_right_if)
   971 done
   972 
   973 lemma mult_divide_cancel_right_if2 [simp]:
   974   fixes c :: "'a :: {field,division_by_zero}"
   975   shows "(a*c) / c = (if c=0 then 0 else a)" 
   976 apply (insert mult_divide_cancel_right_if [of a c 1]) 
   977 apply (simp del: mult_divide_cancel_right_if)
   978 done
   979 
   980 text{*Two lemmas for cancelling the denominator*}
   981 
   982 lemma times_divide_self_right [simp]: 
   983   fixes a :: "'a :: {field,division_by_zero}"
   984   shows "a * (b/a) = (if a=0 then 0 else b)"
   985 by (simp add: times_divide_eq_right)
   986 
   987 lemma times_divide_self_left [simp]: 
   988   fixes a :: "'a :: {field,division_by_zero}"
   989   shows "(b/a) * a = (if a=0 then 0 else b)"
   990 by (simp add: times_divide_eq_left)
   991 
   992 
   993 subsection {* Division and Unary Minus *}
   994 
   995 lemma nonzero_minus_divide_left: "b \<noteq> 0 ==> - (a/b) = (-a) / (b::'a::field)"
   996 by (simp add: divide_inverse minus_mult_left)
   997 
   998 lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a/b) = a / -(b::'a::field)"
   999 by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right)
  1000 
  1001 lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a)/(-b) = a / (b::'a::field)"
  1002 by (simp add: divide_inverse nonzero_inverse_minus_eq)
  1003 
  1004 lemma minus_divide_left: "- (a/b) = (-a) / (b::'a::field)"
  1005 by (simp add: divide_inverse minus_mult_left [symmetric])
  1006 
  1007 lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"
  1008 by (simp add: divide_inverse minus_mult_right [symmetric])
  1009 
  1010 
  1011 text{*The effect is to extract signs from divisions*}
  1012 declare minus_divide_left  [symmetric, simp]
  1013 declare minus_divide_right [symmetric, simp]
  1014 
  1015 text{*Also, extract signs from products*}
  1016 declare minus_mult_left [symmetric, simp]
  1017 declare minus_mult_right [symmetric, simp]
  1018 
  1019 lemma minus_divide_divide [simp]:
  1020      "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"
  1021 apply (case_tac "b=0", simp) 
  1022 apply (simp add: nonzero_minus_divide_divide) 
  1023 done
  1024 
  1025 lemma diff_divide_distrib: "(a-b)/(c::'a::field) = a/c - b/c"
  1026 by (simp add: diff_minus add_divide_distrib) 
  1027 
  1028 
  1029 subsection {* Ordered Fields *}
  1030 
  1031 lemma positive_imp_inverse_positive: 
  1032       assumes a_gt_0: "0 < a"  shows "0 < inverse (a::'a::ordered_field)"
  1033   proof -
  1034   have "0 < a * inverse a" 
  1035     by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one)
  1036   thus "0 < inverse a" 
  1037     by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff)
  1038   qed
  1039 
  1040 lemma negative_imp_inverse_negative:
  1041      "a < 0 ==> inverse a < (0::'a::ordered_field)"
  1042   by (insert positive_imp_inverse_positive [of "-a"], 
  1043       simp add: nonzero_inverse_minus_eq order_less_imp_not_eq) 
  1044 
  1045 lemma inverse_le_imp_le:
  1046       assumes invle: "inverse a \<le> inverse b"
  1047 	  and apos:  "0 < a"
  1048 	 shows "b \<le> (a::'a::ordered_field)"
  1049   proof (rule classical)
  1050   assume "~ b \<le> a"
  1051   hence "a < b"
  1052     by (simp add: linorder_not_le)
  1053   hence bpos: "0 < b"
  1054     by (blast intro: apos order_less_trans)
  1055   hence "a * inverse a \<le> a * inverse b"
  1056     by (simp add: apos invle order_less_imp_le mult_left_mono)
  1057   hence "(a * inverse a) * b \<le> (a * inverse b) * b"
  1058     by (simp add: bpos order_less_imp_le mult_right_mono)
  1059   thus "b \<le> a"
  1060     by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)
  1061   qed
  1062 
  1063 lemma inverse_positive_imp_positive:
  1064       assumes inv_gt_0: "0 < inverse a"
  1065           and [simp]:   "a \<noteq> 0"
  1066         shows "0 < (a::'a::ordered_field)"
  1067   proof -
  1068   have "0 < inverse (inverse a)"
  1069     by (rule positive_imp_inverse_positive)
  1070   thus "0 < a"
  1071     by (simp add: nonzero_inverse_inverse_eq)
  1072   qed
  1073 
  1074 lemma inverse_positive_iff_positive [simp]:
  1075       "(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))"
  1076 apply (case_tac "a = 0", simp)
  1077 apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
  1078 done
  1079 
  1080 lemma inverse_negative_imp_negative:
  1081       assumes inv_less_0: "inverse a < 0"
  1082           and [simp]:   "a \<noteq> 0"
  1083         shows "a < (0::'a::ordered_field)"
  1084   proof -
  1085   have "inverse (inverse a) < 0"
  1086     by (rule negative_imp_inverse_negative)
  1087   thus "a < 0"
  1088     by (simp add: nonzero_inverse_inverse_eq)
  1089   qed
  1090 
  1091 lemma inverse_negative_iff_negative [simp]:
  1092       "(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))"
  1093 apply (case_tac "a = 0", simp)
  1094 apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
  1095 done
  1096 
  1097 lemma inverse_nonnegative_iff_nonnegative [simp]:
  1098       "(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))"
  1099 by (simp add: linorder_not_less [symmetric])
  1100 
  1101 lemma inverse_nonpositive_iff_nonpositive [simp]:
  1102       "(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))"
  1103 by (simp add: linorder_not_less [symmetric])
  1104 
  1105 
  1106 subsection{*Anti-Monotonicity of @{term inverse}*}
  1107 
  1108 lemma less_imp_inverse_less:
  1109       assumes less: "a < b"
  1110 	  and apos:  "0 < a"
  1111 	shows "inverse b < inverse (a::'a::ordered_field)"
  1112   proof (rule ccontr)
  1113   assume "~ inverse b < inverse a"
  1114   hence "inverse a \<le> inverse b"
  1115     by (simp add: linorder_not_less)
  1116   hence "~ (a < b)"
  1117     by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos])
  1118   thus False
  1119     by (rule notE [OF _ less])
  1120   qed
  1121 
  1122 lemma inverse_less_imp_less:
  1123    "[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::ordered_field)"
  1124 apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"])
  1125 apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
  1126 done
  1127 
  1128 text{*Both premises are essential. Consider -1 and 1.*}
  1129 lemma inverse_less_iff_less [simp]:
  1130      "[|0 < a; 0 < b|] 
  1131       ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
  1132 by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
  1133 
  1134 lemma le_imp_inverse_le:
  1135    "[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
  1136   by (force simp add: order_le_less less_imp_inverse_less)
  1137 
  1138 lemma inverse_le_iff_le [simp]:
  1139      "[|0 < a; 0 < b|] 
  1140       ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
  1141 by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
  1142 
  1143 
  1144 text{*These results refer to both operands being negative.  The opposite-sign
  1145 case is trivial, since inverse preserves signs.*}
  1146 lemma inverse_le_imp_le_neg:
  1147    "[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::ordered_field)"
  1148   apply (rule classical) 
  1149   apply (subgoal_tac "a < 0") 
  1150    prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans) 
  1151   apply (insert inverse_le_imp_le [of "-b" "-a"])
  1152   apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
  1153   done
  1154 
  1155 lemma less_imp_inverse_less_neg:
  1156    "[|a < b; b < 0|] ==> inverse b < inverse (a::'a::ordered_field)"
  1157   apply (subgoal_tac "a < 0") 
  1158    prefer 2 apply (blast intro: order_less_trans) 
  1159   apply (insert less_imp_inverse_less [of "-b" "-a"])
  1160   apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
  1161   done
  1162 
  1163 lemma inverse_less_imp_less_neg:
  1164    "[|inverse a < inverse b; b < 0|] ==> b < (a::'a::ordered_field)"
  1165   apply (rule classical) 
  1166   apply (subgoal_tac "a < 0") 
  1167    prefer 2
  1168    apply (force simp add: linorder_not_less intro: order_le_less_trans) 
  1169   apply (insert inverse_less_imp_less [of "-b" "-a"])
  1170   apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
  1171   done
  1172 
  1173 lemma inverse_less_iff_less_neg [simp]:
  1174      "[|a < 0; b < 0|] 
  1175       ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
  1176   apply (insert inverse_less_iff_less [of "-b" "-a"])
  1177   apply (simp del: inverse_less_iff_less 
  1178 	      add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
  1179   done
  1180 
  1181 lemma le_imp_inverse_le_neg:
  1182    "[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
  1183   by (force simp add: order_le_less less_imp_inverse_less_neg)
  1184 
  1185 lemma inverse_le_iff_le_neg [simp]:
  1186      "[|a < 0; b < 0|] 
  1187       ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
  1188 by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
  1189 
  1190 
  1191 subsection{*Inverses and the Number One*}
  1192 
  1193 lemma one_less_inverse_iff:
  1194     "(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"proof cases
  1195   assume "0 < x"
  1196     with inverse_less_iff_less [OF zero_less_one, of x]
  1197     show ?thesis by simp
  1198 next
  1199   assume notless: "~ (0 < x)"
  1200   have "~ (1 < inverse x)"
  1201   proof
  1202     assume "1 < inverse x"
  1203     also with notless have "... \<le> 0" by (simp add: linorder_not_less)
  1204     also have "... < 1" by (rule zero_less_one) 
  1205     finally show False by auto
  1206   qed
  1207   with notless show ?thesis by simp
  1208 qed
  1209 
  1210 lemma inverse_eq_1_iff [simp]:
  1211     "(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))"
  1212 by (insert inverse_eq_iff_eq [of x 1], simp) 
  1213 
  1214 lemma one_le_inverse_iff:
  1215    "(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))"
  1216 by (force simp add: order_le_less one_less_inverse_iff zero_less_one 
  1217                     eq_commute [of 1]) 
  1218 
  1219 lemma inverse_less_1_iff:
  1220    "(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{ordered_field,division_by_zero}))"
  1221 by (simp add: linorder_not_le [symmetric] one_le_inverse_iff) 
  1222 
  1223 lemma inverse_le_1_iff:
  1224    "(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{ordered_field,division_by_zero}))"
  1225 by (simp add: linorder_not_less [symmetric] one_less_inverse_iff) 
  1226 
  1227 
  1228 subsection{*Division and Signs*}
  1229 
  1230 lemma zero_less_divide_iff:
  1231      "((0::'a::{ordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
  1232 by (simp add: divide_inverse zero_less_mult_iff)
  1233 
  1234 lemma divide_less_0_iff:
  1235      "(a/b < (0::'a::{ordered_field,division_by_zero})) = 
  1236       (0 < a & b < 0 | a < 0 & 0 < b)"
  1237 by (simp add: divide_inverse mult_less_0_iff)
  1238 
  1239 lemma zero_le_divide_iff:
  1240      "((0::'a::{ordered_field,division_by_zero}) \<le> a/b) =
  1241       (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
  1242 by (simp add: divide_inverse zero_le_mult_iff)
  1243 
  1244 lemma divide_le_0_iff:
  1245      "(a/b \<le> (0::'a::{ordered_field,division_by_zero})) =
  1246       (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
  1247 by (simp add: divide_inverse mult_le_0_iff)
  1248 
  1249 lemma divide_eq_0_iff [simp]:
  1250      "(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))"
  1251 by (simp add: divide_inverse field_mult_eq_0_iff)
  1252 
  1253 
  1254 subsection{*Simplification of Inequalities Involving Literal Divisors*}
  1255 
  1256 lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"
  1257 proof -
  1258   assume less: "0<c"
  1259   hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
  1260     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
  1261   also have "... = (a*c \<le> b)"
  1262     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
  1263   finally show ?thesis .
  1264 qed
  1265 
  1266 
  1267 lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"
  1268 proof -
  1269   assume less: "c<0"
  1270   hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
  1271     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
  1272   also have "... = (b \<le> a*c)"
  1273     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
  1274   finally show ?thesis .
  1275 qed
  1276 
  1277 lemma le_divide_eq:
  1278   "(a \<le> b/c) = 
  1279    (if 0 < c then a*c \<le> b
  1280              else if c < 0 then b \<le> a*c
  1281              else  a \<le> (0::'a::{ordered_field,division_by_zero}))"
  1282 apply (case_tac "c=0", simp) 
  1283 apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
  1284 done
  1285 
  1286 lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)"
  1287 proof -
  1288   assume less: "0<c"
  1289   hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
  1290     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
  1291   also have "... = (b \<le> a*c)"
  1292     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
  1293   finally show ?thesis .
  1294 qed
  1295 
  1296 lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)"
  1297 proof -
  1298   assume less: "c<0"
  1299   hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
  1300     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
  1301   also have "... = (a*c \<le> b)"
  1302     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
  1303   finally show ?thesis .
  1304 qed
  1305 
  1306 lemma divide_le_eq:
  1307   "(b/c \<le> a) = 
  1308    (if 0 < c then b \<le> a*c
  1309              else if c < 0 then a*c \<le> b
  1310              else 0 \<le> (a::'a::{ordered_field,division_by_zero}))"
  1311 apply (case_tac "c=0", simp) 
  1312 apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) 
  1313 done
  1314 
  1315 
  1316 lemma pos_less_divide_eq:
  1317      "0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)"
  1318 proof -
  1319   assume less: "0<c"
  1320   hence "(a < b/c) = (a*c < (b/c)*c)"
  1321     by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
  1322   also have "... = (a*c < b)"
  1323     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
  1324   finally show ?thesis .
  1325 qed
  1326 
  1327 lemma neg_less_divide_eq:
  1328  "c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)"
  1329 proof -
  1330   assume less: "c<0"
  1331   hence "(a < b/c) = ((b/c)*c < a*c)"
  1332     by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
  1333   also have "... = (b < a*c)"
  1334     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
  1335   finally show ?thesis .
  1336 qed
  1337 
  1338 lemma less_divide_eq:
  1339   "(a < b/c) = 
  1340    (if 0 < c then a*c < b
  1341              else if c < 0 then b < a*c
  1342              else  a < (0::'a::{ordered_field,division_by_zero}))"
  1343 apply (case_tac "c=0", simp) 
  1344 apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) 
  1345 done
  1346 
  1347 lemma pos_divide_less_eq:
  1348      "0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)"
  1349 proof -
  1350   assume less: "0<c"
  1351   hence "(b/c < a) = ((b/c)*c < a*c)"
  1352     by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
  1353   also have "... = (b < a*c)"
  1354     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
  1355   finally show ?thesis .
  1356 qed
  1357 
  1358 lemma neg_divide_less_eq:
  1359  "c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)"
  1360 proof -
  1361   assume less: "c<0"
  1362   hence "(b/c < a) = (a*c < (b/c)*c)"
  1363     by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
  1364   also have "... = (a*c < b)"
  1365     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
  1366   finally show ?thesis .
  1367 qed
  1368 
  1369 lemma divide_less_eq:
  1370   "(b/c < a) = 
  1371    (if 0 < c then b < a*c
  1372              else if c < 0 then a*c < b
  1373              else 0 < (a::'a::{ordered_field,division_by_zero}))"
  1374 apply (case_tac "c=0", simp) 
  1375 apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) 
  1376 done
  1377 
  1378 lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)"
  1379 proof -
  1380   assume [simp]: "c\<noteq>0"
  1381   have "(a = b/c) = (a*c = (b/c)*c)"
  1382     by (simp add: field_mult_cancel_right)
  1383   also have "... = (a*c = b)"
  1384     by (simp add: divide_inverse mult_assoc) 
  1385   finally show ?thesis .
  1386 qed
  1387 
  1388 lemma eq_divide_eq:
  1389   "((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)"
  1390 by (simp add: nonzero_eq_divide_eq) 
  1391 
  1392 lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)"
  1393 proof -
  1394   assume [simp]: "c\<noteq>0"
  1395   have "(b/c = a) = ((b/c)*c = a*c)"
  1396     by (simp add: field_mult_cancel_right)
  1397   also have "... = (b = a*c)"
  1398     by (simp add: divide_inverse mult_assoc) 
  1399   finally show ?thesis .
  1400 qed
  1401 
  1402 lemma divide_eq_eq:
  1403   "(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)"
  1404 by (force simp add: nonzero_divide_eq_eq) 
  1405 
  1406 
  1407 subsection{*Cancellation Laws for Division*}
  1408 
  1409 lemma divide_cancel_right [simp]:
  1410      "(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))"
  1411 apply (case_tac "c=0", simp) 
  1412 apply (simp add: divide_inverse field_mult_cancel_right) 
  1413 done
  1414 
  1415 lemma divide_cancel_left [simp]:
  1416      "(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))" 
  1417 apply (case_tac "c=0", simp) 
  1418 apply (simp add: divide_inverse field_mult_cancel_left) 
  1419 done
  1420 
  1421 subsection {* Division and the Number One *}
  1422 
  1423 text{*Simplify expressions equated with 1*}
  1424 lemma divide_eq_1_iff [simp]:
  1425      "(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
  1426 apply (case_tac "b=0", simp) 
  1427 apply (simp add: right_inverse_eq) 
  1428 done
  1429 
  1430 
  1431 lemma one_eq_divide_iff [simp]:
  1432      "(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
  1433 by (simp add: eq_commute [of 1])  
  1434 
  1435 lemma zero_eq_1_divide_iff [simp]:
  1436      "((0::'a::{ordered_field,division_by_zero}) = 1/a) = (a = 0)"
  1437 apply (case_tac "a=0", simp) 
  1438 apply (auto simp add: nonzero_eq_divide_eq) 
  1439 done
  1440 
  1441 lemma one_divide_eq_0_iff [simp]:
  1442      "(1/a = (0::'a::{ordered_field,division_by_zero})) = (a = 0)"
  1443 apply (case_tac "a=0", simp) 
  1444 apply (insert zero_neq_one [THEN not_sym]) 
  1445 apply (auto simp add: nonzero_divide_eq_eq) 
  1446 done
  1447 
  1448 text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}
  1449 declare zero_less_divide_iff [of "1", simp]
  1450 declare divide_less_0_iff [of "1", simp]
  1451 declare zero_le_divide_iff [of "1", simp]
  1452 declare divide_le_0_iff [of "1", simp]
  1453 
  1454 
  1455 subsection {* Ordering Rules for Division *}
  1456 
  1457 lemma divide_strict_right_mono:
  1458      "[|a < b; 0 < c|] ==> a / c < b / (c::'a::ordered_field)"
  1459 by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono 
  1460               positive_imp_inverse_positive) 
  1461 
  1462 lemma divide_right_mono:
  1463      "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{ordered_field,division_by_zero})"
  1464   by (force simp add: divide_strict_right_mono order_le_less) 
  1465 
  1466 
  1467 text{*The last premise ensures that @{term a} and @{term b} 
  1468       have the same sign*}
  1469 lemma divide_strict_left_mono:
  1470        "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
  1471 by (force simp add: zero_less_mult_iff divide_inverse mult_strict_left_mono 
  1472       order_less_imp_not_eq order_less_imp_not_eq2  
  1473       less_imp_inverse_less less_imp_inverse_less_neg) 
  1474 
  1475 lemma divide_left_mono:
  1476      "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::ordered_field)"
  1477   apply (subgoal_tac "a \<noteq> 0 & b \<noteq> 0") 
  1478    prefer 2 
  1479    apply (force simp add: zero_less_mult_iff order_less_imp_not_eq) 
  1480   apply (case_tac "c=0", simp add: divide_inverse)
  1481   apply (force simp add: divide_strict_left_mono order_le_less) 
  1482   done
  1483 
  1484 lemma divide_strict_left_mono_neg:
  1485      "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
  1486   apply (subgoal_tac "a \<noteq> 0 & b \<noteq> 0") 
  1487    prefer 2 
  1488    apply (force simp add: zero_less_mult_iff order_less_imp_not_eq) 
  1489   apply (drule divide_strict_left_mono [of _ _ "-c"]) 
  1490    apply (simp_all add: mult_commute nonzero_minus_divide_left [symmetric]) 
  1491   done
  1492 
  1493 lemma divide_strict_right_mono_neg:
  1494      "[|b < a; c < 0|] ==> a / c < b / (c::'a::ordered_field)"
  1495 apply (drule divide_strict_right_mono [of _ _ "-c"], simp) 
  1496 apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric]) 
  1497 done
  1498 
  1499 
  1500 subsection {* Ordered Fields are Dense *}
  1501 
  1502 lemma less_add_one: "a < (a+1::'a::ordered_semidom)"
  1503 proof -
  1504   have "a+0 < (a+1::'a::ordered_semidom)"
  1505     by (blast intro: zero_less_one add_strict_left_mono) 
  1506   thus ?thesis by simp
  1507 qed
  1508 
  1509 lemma zero_less_two: "0 < (1+1::'a::ordered_semidom)"
  1510   by (blast intro: order_less_trans zero_less_one less_add_one) 
  1511 
  1512 lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::ordered_field)"
  1513 by (simp add: zero_less_two pos_less_divide_eq right_distrib) 
  1514 
  1515 lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::ordered_field) < b"
  1516 by (simp add: zero_less_two pos_divide_less_eq right_distrib) 
  1517 
  1518 lemma dense: "a < b ==> \<exists>r::'a::ordered_field. a < r & r < b"
  1519 by (blast intro!: less_half_sum gt_half_sum)
  1520 
  1521 
  1522 lemmas times_divide_eq = times_divide_eq_right times_divide_eq_left
  1523 
  1524 text{*It's not obvious whether these should be simprules or not. 
  1525   Their effect is to gather terms into one big fraction, like
  1526   a*b*c / x*y*z. The rationale for that is unclear, but many proofs 
  1527   seem to need them.*}
  1528 
  1529 declare times_divide_eq [simp]
  1530 
  1531 
  1532 subsection {* Absolute Value *}
  1533 
  1534 lemma abs_one [simp]: "abs 1 = (1::'a::ordered_idom)"
  1535   by (simp add: abs_if zero_less_one [THEN order_less_not_sym]) 
  1536 
  1537 lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lordered_ring))" 
  1538 proof -
  1539   let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"
  1540   let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
  1541   have a: "(abs a) * (abs b) = ?x"
  1542     by (simp only: abs_prts[of a] abs_prts[of b] ring_eq_simps)
  1543   {
  1544     fix u v :: 'a
  1545     have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow> 
  1546               u * v = pprt a * pprt b + pprt a * nprt b + 
  1547                       nprt a * pprt b + nprt a * nprt b"
  1548       apply (subst prts[of u], subst prts[of v])
  1549       apply (simp add: left_distrib right_distrib add_ac) 
  1550       done
  1551   }
  1552   note b = this[OF refl[of a] refl[of b]]
  1553   note addm = add_mono[of "0::'a" _ "0::'a", simplified]
  1554   note addm2 = add_mono[of _ "0::'a" _ "0::'a", simplified]
  1555   have xy: "- ?x <= ?y"
  1556     apply (simp)
  1557     apply (rule_tac y="0::'a" in order_trans)
  1558     apply (rule addm2)
  1559     apply (simp_all add: mult_pos_le mult_neg_le)
  1560     apply (rule addm)
  1561     apply (simp_all add: mult_pos_le mult_neg_le)
  1562     done
  1563   have yx: "?y <= ?x"
  1564     apply (simp add:diff_def)
  1565     apply (rule_tac y=0 in order_trans)
  1566     apply (rule addm2, (simp add: mult_pos_neg_le mult_pos_neg2_le)+)
  1567     apply (rule addm, (simp add: mult_pos_neg_le mult_pos_neg2_le)+)
  1568     done
  1569   have i1: "a*b <= abs a * abs b" by (simp only: a b yx)
  1570   have i2: "- (abs a * abs b) <= a*b" by (simp only: a b xy)
  1571   show ?thesis
  1572     apply (rule abs_leI)
  1573     apply (simp add: i1)
  1574     apply (simp add: i2[simplified minus_le_iff])
  1575     done
  1576 qed
  1577 
  1578 lemma abs_eq_mult: 
  1579   assumes "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)"
  1580   shows "abs (a*b) = abs a * abs (b::'a::lordered_ring)"
  1581 proof -
  1582   have s: "(0 <= a*b) | (a*b <= 0)"
  1583     apply (auto)    
  1584     apply (rule_tac split_mult_pos_le)
  1585     apply (rule_tac contrapos_np[of "a*b <= 0"])
  1586     apply (simp)
  1587     apply (rule_tac split_mult_neg_le)
  1588     apply (insert prems)
  1589     apply (blast)
  1590     done
  1591   have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
  1592     by (simp add: prts[symmetric])
  1593   show ?thesis
  1594   proof cases
  1595     assume "0 <= a * b"
  1596     then show ?thesis
  1597       apply (simp_all add: mulprts abs_prts)
  1598       apply (insert prems)
  1599       apply (auto simp add: 
  1600 	ring_eq_simps 
  1601 	iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_zero_pprt]
  1602 	iff2imp[OF le_zero_iff_pprt_id] iff2imp[OF zero_le_iff_nprt_id])
  1603 	apply(drule (1) mult_pos_neg_le[of a b], simp)
  1604 	apply(drule (1) mult_pos_neg2_le[of b a], simp)
  1605       done
  1606   next
  1607     assume "~(0 <= a*b)"
  1608     with s have "a*b <= 0" by simp
  1609     then show ?thesis
  1610       apply (simp_all add: mulprts abs_prts)
  1611       apply (insert prems)
  1612       apply (auto simp add: ring_eq_simps)
  1613       apply(drule (1) mult_pos_le[of a b],simp)
  1614       apply(drule (1) mult_neg_le[of a b],simp)
  1615       done
  1616   qed
  1617 qed
  1618 
  1619 lemma abs_mult: "abs (a * b) = abs a * abs (b::'a::ordered_idom)" 
  1620 by (simp add: abs_eq_mult linorder_linear)
  1621 
  1622 lemma abs_mult_self: "abs a * abs a = a * (a::'a::ordered_idom)"
  1623 by (simp add: abs_if) 
  1624 
  1625 lemma nonzero_abs_inverse:
  1626      "a \<noteq> 0 ==> abs (inverse (a::'a::ordered_field)) = inverse (abs a)"
  1627 apply (auto simp add: linorder_neq_iff abs_if nonzero_inverse_minus_eq 
  1628                       negative_imp_inverse_negative)
  1629 apply (blast intro: positive_imp_inverse_positive elim: order_less_asym) 
  1630 done
  1631 
  1632 lemma abs_inverse [simp]:
  1633      "abs (inverse (a::'a::{ordered_field,division_by_zero})) = 
  1634       inverse (abs a)"
  1635 apply (case_tac "a=0", simp) 
  1636 apply (simp add: nonzero_abs_inverse) 
  1637 done
  1638 
  1639 lemma nonzero_abs_divide:
  1640      "b \<noteq> 0 ==> abs (a / (b::'a::ordered_field)) = abs a / abs b"
  1641 by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 
  1642 
  1643 lemma abs_divide [simp]:
  1644      "abs (a / (b::'a::{ordered_field,division_by_zero})) = abs a / abs b"
  1645 apply (case_tac "b=0", simp) 
  1646 apply (simp add: nonzero_abs_divide) 
  1647 done
  1648 
  1649 lemma abs_mult_less:
  1650      "[| abs a < c; abs b < d |] ==> abs a * abs b < c*(d::'a::ordered_idom)"
  1651 proof -
  1652   assume ac: "abs a < c"
  1653   hence cpos: "0<c" by (blast intro: order_le_less_trans abs_ge_zero)
  1654   assume "abs b < d"
  1655   thus ?thesis by (simp add: ac cpos mult_strict_mono) 
  1656 qed
  1657 
  1658 lemma eq_minus_self_iff: "(a = -a) = (a = (0::'a::ordered_idom))"
  1659 by (force simp add: order_eq_iff le_minus_self_iff minus_le_self_iff)
  1660 
  1661 lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::ordered_idom))"
  1662 by (simp add: order_less_le le_minus_self_iff eq_minus_self_iff)
  1663 
  1664 lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::ordered_idom))" 
  1665 apply (simp add: order_less_le abs_le_iff)  
  1666 apply (auto simp add: abs_if minus_le_self_iff eq_minus_self_iff)
  1667 apply (simp add: le_minus_self_iff linorder_neq_iff) 
  1668 done
  1669 
  1670 lemma linprog_dual_estimate:
  1671   assumes
  1672   "A * x \<le> (b::'a::lordered_ring)"
  1673   "0 \<le> y"
  1674   "abs (A - A') \<le> \<delta>A"
  1675   "b \<le> b'"
  1676   "abs (c - c') \<le> \<delta>c"
  1677   "abs x \<le> r"
  1678   shows
  1679   "c * x \<le> y * b' + (y * \<delta>A + abs (y * A' - c') + \<delta>c) * r"
  1680 proof -
  1681   from prems have 1: "y * b <= y * b'" by (simp add: mult_left_mono)
  1682   from prems have 2: "y * (A * x) <= y * b" by (simp add: mult_left_mono) 
  1683   have 3: "y * (A * x) = c * x + (y * (A - A') + (y * A' - c') + (c'-c)) * x" by (simp add: ring_eq_simps)  
  1684   from 1 2 3 have 4: "c * x + (y * (A - A') + (y * A' - c') + (c'-c)) * x <= y * b'" by simp
  1685   have 5: "c * x <= y * b' + abs((y * (A - A') + (y * A' - c') + (c'-c)) * x)"
  1686     by (simp only: 4 estimate_by_abs)  
  1687   have 6: "abs((y * (A - A') + (y * A' - c') + (c'-c)) * x) <= abs (y * (A - A') + (y * A' - c') + (c'-c)) * abs x"
  1688     by (simp add: abs_le_mult)
  1689   have 7: "(abs (y * (A - A') + (y * A' - c') + (c'-c))) * abs x <= (abs (y * (A-A') + (y*A'-c')) + abs(c'-c)) * abs x"
  1690     by(rule abs_triangle_ineq [THEN mult_right_mono]) simp
  1691   have 8: " (abs (y * (A-A') + (y*A'-c')) + abs(c'-c)) * abs x <=  (abs (y * (A-A')) + abs (y*A'-c') + abs(c'-c)) * abs x"
  1692     by (simp add: abs_triangle_ineq mult_right_mono)    
  1693   have 9: "(abs (y * (A-A')) + abs (y*A'-c') + abs(c'-c)) * abs x <= (abs y * abs (A-A') + abs (y*A'-c') + abs (c'-c)) * abs x"
  1694     by (simp add: abs_le_mult mult_right_mono)  
  1695   have 10: "c'-c = -(c-c')" by (simp add: ring_eq_simps)
  1696   have 11: "abs (c'-c) = abs (c-c')" 
  1697     by (subst 10, subst abs_minus_cancel, simp)
  1698   have 12: "(abs y * abs (A-A') + abs (y*A'-c') + abs (c'-c)) * abs x <= (abs y * abs (A-A') + abs (y*A'-c') + \<delta>c) * abs x"
  1699     by (simp add: 11 prems mult_right_mono)
  1700   have 13: "(abs y * abs (A-A') + abs (y*A'-c') + \<delta>c) * abs x <= (abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * abs x"
  1701     by (simp add: prems mult_right_mono mult_left_mono)  
  1702   have r: "(abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * abs x <=  (abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * r"
  1703     apply (rule mult_left_mono)
  1704     apply (simp add: prems)
  1705     apply (rule_tac add_mono[of "0::'a" _ "0", simplified])+
  1706     apply (rule mult_left_mono[of "0" "\<delta>A", simplified])
  1707     apply (simp_all)
  1708     apply (rule order_trans[where y="abs (A-A')"], simp_all add: prems)
  1709     apply (rule order_trans[where y="abs (c-c')"], simp_all add: prems)
  1710     done    
  1711   from 6 7 8 9 12 13 r have 14:" abs((y * (A - A') + (y * A' - c') + (c'-c)) * x) <=(abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * r"     
  1712     by (simp)
  1713   show ?thesis 
  1714     apply (rule_tac le_add_right_mono[of _ _ "abs((y * (A - A') + (y * A' - c') + (c'-c)) * x)"])
  1715     apply (simp_all only: 5 14[simplified abs_of_ge_0[of y, simplified prems]])
  1716     done
  1717 qed
  1718 
  1719 lemma le_ge_imp_abs_diff_1:
  1720   assumes
  1721   "A1 <= (A::'a::lordered_ring)"
  1722   "A <= A2" 
  1723   shows "abs (A-A1) <= A2-A1"
  1724 proof -
  1725   have "0 <= A - A1"    
  1726   proof -
  1727     have 1: "A - A1 = A + (- A1)" by simp
  1728     show ?thesis by (simp only: 1 add_right_mono[of A1 A "-A1", simplified, simplified prems])
  1729   qed
  1730   then have "abs (A-A1) = A-A1" by (rule abs_of_ge_0)
  1731   with prems show "abs (A-A1) <= (A2-A1)" by simp
  1732 qed
  1733 
  1734 lemma mult_le_prts:
  1735   assumes
  1736   "a1 <= (a::'a::lordered_ring)"
  1737   "a <= a2"
  1738   "b1 <= b"
  1739   "b <= b2"
  1740   shows
  1741   "a * b <= pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
  1742 proof - 
  1743   have "a * b = (pprt a + nprt a) * (pprt b + nprt b)" 
  1744     apply (subst prts[symmetric])+
  1745     apply simp
  1746     done
  1747   then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
  1748     by (simp add: ring_eq_simps)
  1749   moreover have "pprt a * pprt b <= pprt a2 * pprt b2"
  1750     by (simp_all add: prems mult_mono)
  1751   moreover have "pprt a * nprt b <= pprt a1 * nprt b2"
  1752   proof -
  1753     have "pprt a * nprt b <= pprt a * nprt b2"
  1754       by (simp add: mult_left_mono prems)
  1755     moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2"
  1756       by (simp add: mult_right_mono_neg prems)
  1757     ultimately show ?thesis
  1758       by simp
  1759   qed
  1760   moreover have "nprt a * pprt b <= nprt a2 * pprt b1"
  1761   proof - 
  1762     have "nprt a * pprt b <= nprt a2 * pprt b"
  1763       by (simp add: mult_right_mono prems)
  1764     moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"
  1765       by (simp add: mult_left_mono_neg prems)
  1766     ultimately show ?thesis
  1767       by simp
  1768   qed
  1769   moreover have "nprt a * nprt b <= nprt a1 * nprt b1"
  1770   proof -
  1771     have "nprt a * nprt b <= nprt a * nprt b1"
  1772       by (simp add: mult_left_mono_neg prems)
  1773     moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1"
  1774       by (simp add: mult_right_mono_neg prems)
  1775     ultimately show ?thesis
  1776       by simp
  1777   qed
  1778   ultimately show ?thesis
  1779     by - (rule add_mono | simp)+
  1780 qed
  1781     
  1782 lemma mult_le_dual_prts: 
  1783   assumes
  1784   "A * x \<le> (b::'a::lordered_ring)"
  1785   "0 \<le> y"
  1786   "A1 \<le> A"
  1787   "A \<le> A2"
  1788   "c1 \<le> c"
  1789   "c \<le> c2"
  1790   "r1 \<le> x"
  1791   "x \<le> r2"
  1792   shows
  1793   "c * x \<le> y * b + (let s1 = c1 - y * A2; s2 = c2 - y * A1 in pprt s2 * pprt r2 + pprt s1 * nprt r2 + nprt s2 * pprt r1 + nprt s1 * nprt r1)"
  1794   (is "_ <= _ + ?C")
  1795 proof -
  1796   from prems have "y * (A * x) <= y * b" by (simp add: mult_left_mono) 
  1797   moreover have "y * (A * x) = c * x + (y * A - c) * x" by (simp add: ring_eq_simps)  
  1798   ultimately have "c * x + (y * A - c) * x <= y * b" by simp
  1799   then have "c * x <= y * b - (y * A - c) * x" by (simp add: le_diff_eq)
  1800   then have cx: "c * x <= y * b + (c - y * A) * x" by (simp add: ring_eq_simps)
  1801   have s2: "c - y * A <= c2 - y * A1"
  1802     by (simp add: diff_def prems add_mono mult_left_mono)
  1803   have s1: "c1 - y * A2 <= c - y * A"
  1804     by (simp add: diff_def prems add_mono mult_left_mono)
  1805   have prts: "(c - y * A) * x <= ?C"
  1806     apply (simp add: Let_def)
  1807     apply (rule mult_le_prts)
  1808     apply (simp_all add: prems s1 s2)
  1809     done
  1810   then have "y * b + (c - y * A) * x <= y * b + ?C"
  1811     by simp
  1812   with cx show ?thesis
  1813     by(simp only:)
  1814 qed
  1815 
  1816 ML {*
  1817 val left_distrib = thm "left_distrib";
  1818 val right_distrib = thm "right_distrib";
  1819 val mult_commute = thm "mult_commute";
  1820 val distrib = thm "distrib";
  1821 val zero_neq_one = thm "zero_neq_one";
  1822 val no_zero_divisors = thm "no_zero_divisors";
  1823 val left_inverse = thm "left_inverse";
  1824 val divide_inverse = thm "divide_inverse";
  1825 val mult_zero_left = thm "mult_zero_left";
  1826 val mult_zero_right = thm "mult_zero_right";
  1827 val field_mult_eq_0_iff = thm "field_mult_eq_0_iff";
  1828 val inverse_zero = thm "inverse_zero";
  1829 val ring_distrib = thms "ring_distrib";
  1830 val combine_common_factor = thm "combine_common_factor";
  1831 val minus_mult_left = thm "minus_mult_left";
  1832 val minus_mult_right = thm "minus_mult_right";
  1833 val minus_mult_minus = thm "minus_mult_minus";
  1834 val minus_mult_commute = thm "minus_mult_commute";
  1835 val right_diff_distrib = thm "right_diff_distrib";
  1836 val left_diff_distrib = thm "left_diff_distrib";
  1837 val mult_left_mono = thm "mult_left_mono";
  1838 val mult_right_mono = thm "mult_right_mono";
  1839 val mult_strict_left_mono = thm "mult_strict_left_mono";
  1840 val mult_strict_right_mono = thm "mult_strict_right_mono";
  1841 val mult_mono = thm "mult_mono";
  1842 val mult_strict_mono = thm "mult_strict_mono";
  1843 val abs_if = thm "abs_if";
  1844 val zero_less_one = thm "zero_less_one";
  1845 val eq_add_iff1 = thm "eq_add_iff1";
  1846 val eq_add_iff2 = thm "eq_add_iff2";
  1847 val less_add_iff1 = thm "less_add_iff1";
  1848 val less_add_iff2 = thm "less_add_iff2";
  1849 val le_add_iff1 = thm "le_add_iff1";
  1850 val le_add_iff2 = thm "le_add_iff2";
  1851 val mult_left_le_imp_le = thm "mult_left_le_imp_le";
  1852 val mult_right_le_imp_le = thm "mult_right_le_imp_le";
  1853 val mult_left_less_imp_less = thm "mult_left_less_imp_less";
  1854 val mult_right_less_imp_less = thm "mult_right_less_imp_less";
  1855 val mult_strict_left_mono_neg = thm "mult_strict_left_mono_neg";
  1856 val mult_left_mono_neg = thm "mult_left_mono_neg";
  1857 val mult_strict_right_mono_neg = thm "mult_strict_right_mono_neg";
  1858 val mult_right_mono_neg = thm "mult_right_mono_neg";
  1859 val mult_pos = thm "mult_pos";
  1860 val mult_pos_le = thm "mult_pos_le";
  1861 val mult_pos_neg = thm "mult_pos_neg";
  1862 val mult_pos_neg_le = thm "mult_pos_neg_le";
  1863 val mult_pos_neg2 = thm "mult_pos_neg2";
  1864 val mult_pos_neg2_le = thm "mult_pos_neg2_le";
  1865 val mult_neg = thm "mult_neg";
  1866 val mult_neg_le = thm "mult_neg_le";
  1867 val zero_less_mult_pos = thm "zero_less_mult_pos";
  1868 val zero_less_mult_pos2 = thm "zero_less_mult_pos2";
  1869 val zero_less_mult_iff = thm "zero_less_mult_iff";
  1870 val mult_eq_0_iff = thm "mult_eq_0_iff";
  1871 val zero_le_mult_iff = thm "zero_le_mult_iff";
  1872 val mult_less_0_iff = thm "mult_less_0_iff";
  1873 val mult_le_0_iff = thm "mult_le_0_iff";
  1874 val split_mult_pos_le = thm "split_mult_pos_le";
  1875 val split_mult_neg_le = thm "split_mult_neg_le";
  1876 val zero_le_square = thm "zero_le_square";
  1877 val zero_le_one = thm "zero_le_one";
  1878 val not_one_le_zero = thm "not_one_le_zero";
  1879 val not_one_less_zero = thm "not_one_less_zero";
  1880 val mult_left_mono_neg = thm "mult_left_mono_neg";
  1881 val mult_right_mono_neg = thm "mult_right_mono_neg";
  1882 val mult_strict_mono = thm "mult_strict_mono";
  1883 val mult_strict_mono' = thm "mult_strict_mono'";
  1884 val mult_mono = thm "mult_mono";
  1885 val less_1_mult = thm "less_1_mult";
  1886 val mult_less_cancel_right_disj = thm "mult_less_cancel_right_disj";
  1887 val mult_less_cancel_left_disj = thm "mult_less_cancel_left_disj";
  1888 val mult_less_cancel_right = thm "mult_less_cancel_right";
  1889 val mult_less_cancel_left = thm "mult_less_cancel_left";
  1890 val mult_le_cancel_right = thm "mult_le_cancel_right";
  1891 val mult_le_cancel_left = thm "mult_le_cancel_left";
  1892 val mult_less_imp_less_left = thm "mult_less_imp_less_left";
  1893 val mult_less_imp_less_right = thm "mult_less_imp_less_right";
  1894 val mult_cancel_right = thm "mult_cancel_right";
  1895 val mult_cancel_left = thm "mult_cancel_left";
  1896 val ring_eq_simps = thms "ring_eq_simps";
  1897 val right_inverse = thm "right_inverse";
  1898 val right_inverse_eq = thm "right_inverse_eq";
  1899 val nonzero_inverse_eq_divide = thm "nonzero_inverse_eq_divide";
  1900 val divide_self = thm "divide_self";
  1901 val divide_zero = thm "divide_zero";
  1902 val divide_zero_left = thm "divide_zero_left";
  1903 val inverse_eq_divide = thm "inverse_eq_divide";
  1904 val add_divide_distrib = thm "add_divide_distrib";
  1905 val field_mult_eq_0_iff = thm "field_mult_eq_0_iff";
  1906 val field_mult_cancel_right_lemma = thm "field_mult_cancel_right_lemma";
  1907 val field_mult_cancel_right = thm "field_mult_cancel_right";
  1908 val field_mult_cancel_left = thm "field_mult_cancel_left";
  1909 val nonzero_imp_inverse_nonzero = thm "nonzero_imp_inverse_nonzero";
  1910 val inverse_zero_imp_zero = thm "inverse_zero_imp_zero";
  1911 val inverse_nonzero_imp_nonzero = thm "inverse_nonzero_imp_nonzero";
  1912 val inverse_nonzero_iff_nonzero = thm "inverse_nonzero_iff_nonzero";
  1913 val nonzero_inverse_minus_eq = thm "nonzero_inverse_minus_eq";
  1914 val inverse_minus_eq = thm "inverse_minus_eq";
  1915 val nonzero_inverse_eq_imp_eq = thm "nonzero_inverse_eq_imp_eq";
  1916 val inverse_eq_imp_eq = thm "inverse_eq_imp_eq";
  1917 val inverse_eq_iff_eq = thm "inverse_eq_iff_eq";
  1918 val nonzero_inverse_inverse_eq = thm "nonzero_inverse_inverse_eq";
  1919 val inverse_inverse_eq = thm "inverse_inverse_eq";
  1920 val inverse_1 = thm "inverse_1";
  1921 val nonzero_inverse_mult_distrib = thm "nonzero_inverse_mult_distrib";
  1922 val inverse_mult_distrib = thm "inverse_mult_distrib";
  1923 val inverse_add = thm "inverse_add";
  1924 val inverse_divide = thm "inverse_divide";
  1925 val nonzero_mult_divide_cancel_left = thm "nonzero_mult_divide_cancel_left";
  1926 val mult_divide_cancel_left = thm "mult_divide_cancel_left";
  1927 val nonzero_mult_divide_cancel_right = thm "nonzero_mult_divide_cancel_right";
  1928 val mult_divide_cancel_right = thm "mult_divide_cancel_right";
  1929 val mult_divide_cancel_eq_if = thm "mult_divide_cancel_eq_if";
  1930 val divide_1 = thm "divide_1";
  1931 val times_divide_eq_right = thm "times_divide_eq_right";
  1932 val times_divide_eq_left = thm "times_divide_eq_left";
  1933 val divide_divide_eq_right = thm "divide_divide_eq_right";
  1934 val divide_divide_eq_left = thm "divide_divide_eq_left";
  1935 val nonzero_minus_divide_left = thm "nonzero_minus_divide_left";
  1936 val nonzero_minus_divide_right = thm "nonzero_minus_divide_right";
  1937 val nonzero_minus_divide_divide = thm "nonzero_minus_divide_divide";
  1938 val minus_divide_left = thm "minus_divide_left";
  1939 val minus_divide_right = thm "minus_divide_right";
  1940 val minus_divide_divide = thm "minus_divide_divide";
  1941 val diff_divide_distrib = thm "diff_divide_distrib";
  1942 val positive_imp_inverse_positive = thm "positive_imp_inverse_positive";
  1943 val negative_imp_inverse_negative = thm "negative_imp_inverse_negative";
  1944 val inverse_le_imp_le = thm "inverse_le_imp_le";
  1945 val inverse_positive_imp_positive = thm "inverse_positive_imp_positive";
  1946 val inverse_positive_iff_positive = thm "inverse_positive_iff_positive";
  1947 val inverse_negative_imp_negative = thm "inverse_negative_imp_negative";
  1948 val inverse_negative_iff_negative = thm "inverse_negative_iff_negative";
  1949 val inverse_nonnegative_iff_nonnegative = thm "inverse_nonnegative_iff_nonnegative";
  1950 val inverse_nonpositive_iff_nonpositive = thm "inverse_nonpositive_iff_nonpositive";
  1951 val less_imp_inverse_less = thm "less_imp_inverse_less";
  1952 val inverse_less_imp_less = thm "inverse_less_imp_less";
  1953 val inverse_less_iff_less = thm "inverse_less_iff_less";
  1954 val le_imp_inverse_le = thm "le_imp_inverse_le";
  1955 val inverse_le_iff_le = thm "inverse_le_iff_le";
  1956 val inverse_le_imp_le_neg = thm "inverse_le_imp_le_neg";
  1957 val less_imp_inverse_less_neg = thm "less_imp_inverse_less_neg";
  1958 val inverse_less_imp_less_neg = thm "inverse_less_imp_less_neg";
  1959 val inverse_less_iff_less_neg = thm "inverse_less_iff_less_neg";
  1960 val le_imp_inverse_le_neg = thm "le_imp_inverse_le_neg";
  1961 val inverse_le_iff_le_neg = thm "inverse_le_iff_le_neg";
  1962 val one_less_inverse_iff = thm "one_less_inverse_iff";
  1963 val inverse_eq_1_iff = thm "inverse_eq_1_iff";
  1964 val one_le_inverse_iff = thm "one_le_inverse_iff";
  1965 val inverse_less_1_iff = thm "inverse_less_1_iff";
  1966 val inverse_le_1_iff = thm "inverse_le_1_iff";
  1967 val zero_less_divide_iff = thm "zero_less_divide_iff";
  1968 val divide_less_0_iff = thm "divide_less_0_iff";
  1969 val zero_le_divide_iff = thm "zero_le_divide_iff";
  1970 val divide_le_0_iff = thm "divide_le_0_iff";
  1971 val divide_eq_0_iff = thm "divide_eq_0_iff";
  1972 val pos_le_divide_eq = thm "pos_le_divide_eq";
  1973 val neg_le_divide_eq = thm "neg_le_divide_eq";
  1974 val le_divide_eq = thm "le_divide_eq";
  1975 val pos_divide_le_eq = thm "pos_divide_le_eq";
  1976 val neg_divide_le_eq = thm "neg_divide_le_eq";
  1977 val divide_le_eq = thm "divide_le_eq";
  1978 val pos_less_divide_eq = thm "pos_less_divide_eq";
  1979 val neg_less_divide_eq = thm "neg_less_divide_eq";
  1980 val less_divide_eq = thm "less_divide_eq";
  1981 val pos_divide_less_eq = thm "pos_divide_less_eq";
  1982 val neg_divide_less_eq = thm "neg_divide_less_eq";
  1983 val divide_less_eq = thm "divide_less_eq";
  1984 val nonzero_eq_divide_eq = thm "nonzero_eq_divide_eq";
  1985 val eq_divide_eq = thm "eq_divide_eq";
  1986 val nonzero_divide_eq_eq = thm "nonzero_divide_eq_eq";
  1987 val divide_eq_eq = thm "divide_eq_eq";
  1988 val divide_cancel_right = thm "divide_cancel_right";
  1989 val divide_cancel_left = thm "divide_cancel_left";
  1990 val divide_eq_1_iff = thm "divide_eq_1_iff";
  1991 val one_eq_divide_iff = thm "one_eq_divide_iff";
  1992 val zero_eq_1_divide_iff = thm "zero_eq_1_divide_iff";
  1993 val one_divide_eq_0_iff = thm "one_divide_eq_0_iff";
  1994 val divide_strict_right_mono = thm "divide_strict_right_mono";
  1995 val divide_right_mono = thm "divide_right_mono";
  1996 val divide_strict_left_mono = thm "divide_strict_left_mono";
  1997 val divide_left_mono = thm "divide_left_mono";
  1998 val divide_strict_left_mono_neg = thm "divide_strict_left_mono_neg";
  1999 val divide_strict_right_mono_neg = thm "divide_strict_right_mono_neg";
  2000 val less_add_one = thm "less_add_one";
  2001 val zero_less_two = thm "zero_less_two";
  2002 val less_half_sum = thm "less_half_sum";
  2003 val gt_half_sum = thm "gt_half_sum";
  2004 val dense = thm "dense";
  2005 val abs_one = thm "abs_one";
  2006 val abs_le_mult = thm "abs_le_mult";
  2007 val abs_eq_mult = thm "abs_eq_mult";
  2008 val abs_mult = thm "abs_mult";
  2009 val abs_mult_self = thm "abs_mult_self";
  2010 val nonzero_abs_inverse = thm "nonzero_abs_inverse";
  2011 val abs_inverse = thm "abs_inverse";
  2012 val nonzero_abs_divide = thm "nonzero_abs_divide";
  2013 val abs_divide = thm "abs_divide";
  2014 val abs_mult_less = thm "abs_mult_less";
  2015 val eq_minus_self_iff = thm "eq_minus_self_iff";
  2016 val less_minus_self_iff = thm "less_minus_self_iff";
  2017 val abs_less_iff = thm "abs_less_iff";
  2018 *}
  2019 
  2020 end