misc tuning and modernization;
authorwenzelm
Mon Jun 20 21:40:48 2016 +0200 (2016-06-20)
changeset 633251086d56cde86
parent 63324 1e98146f3581
child 63326 9d2470571719
misc tuning and modernization;
src/HOL/Groups.thy
src/HOL/Rings.thy
     1.1 --- a/src/HOL/Groups.thy	Mon Jun 20 17:51:47 2016 +0200
     1.2 +++ b/src/HOL/Groups.thy	Mon Jun 20 21:40:48 2016 +0200
     1.3 @@ -13,22 +13,26 @@
     1.4  named_theorems ac_simps "associativity and commutativity simplification rules"
     1.5  
     1.6  
     1.7 -text\<open>The rewrites accumulated in \<open>algebra_simps\<close> deal with the
     1.8 -classical algebraic structures of groups, rings and family. They simplify
     1.9 -terms by multiplying everything out (in case of a ring) and bringing sums and
    1.10 -products into a canonical form (by ordered rewriting). As a result it decides
    1.11 -group and ring equalities but also helps with inequalities.
    1.12 +text \<open>
    1.13 +  The rewrites accumulated in \<open>algebra_simps\<close> deal with the
    1.14 +  classical algebraic structures of groups, rings and family. They simplify
    1.15 +  terms by multiplying everything out (in case of a ring) and bringing sums and
    1.16 +  products into a canonical form (by ordered rewriting). As a result it decides
    1.17 +  group and ring equalities but also helps with inequalities.
    1.18  
    1.19 -Of course it also works for fields, but it knows nothing about multiplicative
    1.20 -inverses or division. This is catered for by \<open>field_simps\<close>.\<close>
    1.21 +  Of course it also works for fields, but it knows nothing about multiplicative
    1.22 +  inverses or division. This is catered for by \<open>field_simps\<close>.
    1.23 +\<close>
    1.24  
    1.25  named_theorems algebra_simps "algebra simplification rules"
    1.26  
    1.27  
    1.28 -text\<open>Lemmas \<open>field_simps\<close> multiply with denominators in (in)equations
    1.29 -if they can be proved to be non-zero (for equations) or positive/negative
    1.30 -(for inequations). Can be too aggressive and is therefore separate from the
    1.31 -more benign \<open>algebra_simps\<close>.\<close>
    1.32 +text \<open>
    1.33 +  Lemmas \<open>field_simps\<close> multiply with denominators in (in)equations
    1.34 +  if they can be proved to be non-zero (for equations) or positive/negative
    1.35 +  (for inequations). Can be too aggressive and is therefore separate from the
    1.36 +  more benign \<open>algebra_simps\<close>.
    1.37 +\<close>
    1.38  
    1.39  named_theorems field_simps "algebra simplification rules for fields"
    1.40  
    1.41 @@ -42,15 +46,14 @@
    1.42  \<close>
    1.43  
    1.44  locale semigroup =
    1.45 -  fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<^bold>*" 70)
    1.46 +  fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "\<^bold>*" 70)
    1.47    assumes assoc [ac_simps]: "a \<^bold>* b \<^bold>* c = a \<^bold>* (b \<^bold>* c)"
    1.48  
    1.49  locale abel_semigroup = semigroup +
    1.50    assumes commute [ac_simps]: "a \<^bold>* b = b \<^bold>* a"
    1.51  begin
    1.52  
    1.53 -lemma left_commute [ac_simps]:
    1.54 -  "b \<^bold>* (a \<^bold>* c) = a \<^bold>* (b \<^bold>* c)"
    1.55 +lemma left_commute [ac_simps]: "b \<^bold>* (a \<^bold>* c) = a \<^bold>* (b \<^bold>* c)"
    1.56  proof -
    1.57    have "(b \<^bold>* a) \<^bold>* c = (a \<^bold>* b) \<^bold>* c"
    1.58      by (simp only: commute)
    1.59 @@ -238,13 +241,11 @@
    1.60    assumes add_right_imp_eq: "b + a = c + a \<Longrightarrow> b = c"
    1.61  begin
    1.62  
    1.63 -lemma add_left_cancel [simp]:
    1.64 -  "a + b = a + c \<longleftrightarrow> b = c"
    1.65 -by (blast dest: add_left_imp_eq)
    1.66 +lemma add_left_cancel [simp]: "a + b = a + c \<longleftrightarrow> b = c"
    1.67 +  by (blast dest: add_left_imp_eq)
    1.68  
    1.69 -lemma add_right_cancel [simp]:
    1.70 -  "b + a = c + a \<longleftrightarrow> b = c"
    1.71 -by (blast dest: add_right_imp_eq)
    1.72 +lemma add_right_cancel [simp]: "b + a = c + a \<longleftrightarrow> b = c"
    1.73 +  by (blast dest: add_right_imp_eq)
    1.74  
    1.75  end
    1.76  
    1.77 @@ -253,8 +254,7 @@
    1.78    assumes diff_diff_add [algebra_simps, field_simps]: "a - b - c = a - (b + c)"
    1.79  begin
    1.80  
    1.81 -lemma add_diff_cancel_right' [simp]:
    1.82 -  "(a + b) - b = a"
    1.83 +lemma add_diff_cancel_right' [simp]: "(a + b) - b = a"
    1.84    using add_diff_cancel_left' [of b a] by (simp add: ac_simps)
    1.85  
    1.86  subclass cancel_semigroup_add
    1.87 @@ -274,16 +274,13 @@
    1.88      by simp
    1.89  qed
    1.90  
    1.91 -lemma add_diff_cancel_left [simp]:
    1.92 -  "(c + a) - (c + b) = a - b"
    1.93 +lemma add_diff_cancel_left [simp]: "(c + a) - (c + b) = a - b"
    1.94    unfolding diff_diff_add [symmetric] by simp
    1.95  
    1.96 -lemma add_diff_cancel_right [simp]:
    1.97 -  "(a + c) - (b + c) = a - b"
    1.98 +lemma add_diff_cancel_right [simp]: "(a + c) - (b + c) = a - b"
    1.99    using add_diff_cancel_left [symmetric] by (simp add: ac_simps)
   1.100  
   1.101 -lemma diff_right_commute:
   1.102 -  "a - c - b = a - b - c"
   1.103 +lemma diff_right_commute: "a - c - b = a - b - c"
   1.104    by (simp add: diff_diff_add add.commute)
   1.105  
   1.106  end
   1.107 @@ -291,14 +288,13 @@
   1.108  class cancel_comm_monoid_add = cancel_ab_semigroup_add + comm_monoid_add
   1.109  begin
   1.110  
   1.111 -lemma diff_zero [simp]:
   1.112 -  "a - 0 = a"
   1.113 +lemma diff_zero [simp]: "a - 0 = a"
   1.114    using add_diff_cancel_right' [of a 0] by simp
   1.115  
   1.116 -lemma diff_cancel [simp]:
   1.117 -  "a - a = 0"
   1.118 +lemma diff_cancel [simp]: "a - a = 0"
   1.119  proof -
   1.120 -  have "(a + 0) - (a + 0) = 0" by (simp only: add_diff_cancel_left diff_zero)
   1.121 +  have "(a + 0) - (a + 0) = 0"
   1.122 +    by (simp only: add_diff_cancel_left diff_zero)
   1.123    then show ?thesis by simp
   1.124  qed
   1.125  
   1.126 @@ -306,29 +302,29 @@
   1.127    assumes "c + b = a"
   1.128    shows "c = a - b"
   1.129  proof -
   1.130 -  from assms have "(b + c) - (b + 0) = a - b" by (simp add: add.commute)
   1.131 +  from assms have "(b + c) - (b + 0) = a - b"
   1.132 +    by (simp add: add.commute)
   1.133    then show "c = a - b" by simp
   1.134  qed
   1.135  
   1.136 -lemma add_cancel_right_right [simp]:
   1.137 -  "a = a + b \<longleftrightarrow> b = 0" (is "?P \<longleftrightarrow> ?Q")
   1.138 +lemma add_cancel_right_right [simp]: "a = a + b \<longleftrightarrow> b = 0"
   1.139 +  (is "?P \<longleftrightarrow> ?Q")
   1.140  proof
   1.141 -  assume ?Q then show ?P by simp
   1.142 +  assume ?Q
   1.143 +  then show ?P by simp
   1.144  next
   1.145 -  assume ?P then have "a - a = a + b - a" by simp
   1.146 +  assume ?P
   1.147 +  then have "a - a = a + b - a" by simp
   1.148    then show ?Q by simp
   1.149  qed
   1.150  
   1.151 -lemma add_cancel_right_left [simp]:
   1.152 -  "a = b + a \<longleftrightarrow> b = 0"
   1.153 +lemma add_cancel_right_left [simp]: "a = b + a \<longleftrightarrow> b = 0"
   1.154    using add_cancel_right_right [of a b] by (simp add: ac_simps)
   1.155  
   1.156 -lemma add_cancel_left_right [simp]:
   1.157 -  "a + b = a \<longleftrightarrow> b = 0"
   1.158 +lemma add_cancel_left_right [simp]: "a + b = a \<longleftrightarrow> b = 0"
   1.159    by (auto dest: sym)
   1.160  
   1.161 -lemma add_cancel_left_left [simp]:
   1.162 -  "b + a = a \<longleftrightarrow> b = 0"
   1.163 +lemma add_cancel_left_left [simp]: "b + a = a \<longleftrightarrow> b = 0"
   1.164    by (auto dest: sym)
   1.165  
   1.166  end
   1.167 @@ -337,11 +333,12 @@
   1.168    assumes zero_diff [simp]: "0 - a = 0"
   1.169  begin
   1.170  
   1.171 -lemma diff_add_zero [simp]:
   1.172 -  "a - (a + b) = 0"
   1.173 +lemma diff_add_zero [simp]: "a - (a + b) = 0"
   1.174  proof -
   1.175 -  have "a - (a + b) = (a + 0) - (a + b)" by simp
   1.176 -  also have "\<dots> = 0" by (simp only: add_diff_cancel_left zero_diff)
   1.177 +  have "a - (a + b) = (a + 0) - (a + b)"
   1.178 +    by simp
   1.179 +  also have "\<dots> = 0"
   1.180 +    by (simp only: add_diff_cancel_left zero_diff)
   1.181    finally show ?thesis .
   1.182  qed
   1.183  
   1.184 @@ -355,14 +352,14 @@
   1.185    assumes add_uminus_conv_diff [simp]: "a + (- b) = a - b"
   1.186  begin
   1.187  
   1.188 -lemma diff_conv_add_uminus:
   1.189 -  "a - b = a + (- b)"
   1.190 +lemma diff_conv_add_uminus: "a - b = a + (- b)"
   1.191    by simp
   1.192  
   1.193  lemma minus_unique:
   1.194 -  assumes "a + b = 0" shows "- a = b"
   1.195 +  assumes "a + b = 0"
   1.196 +  shows "- a = b"
   1.197  proof -
   1.198 -  have "- a = - a + (a + b)" using assms by simp
   1.199 +  from assms have "- a = - a + (a + b)" by simp
   1.200    also have "\<dots> = b" by (simp add: add.assoc [symmetric])
   1.201    finally show ?thesis .
   1.202  qed
   1.203 @@ -370,13 +367,13 @@
   1.204  lemma minus_zero [simp]: "- 0 = 0"
   1.205  proof -
   1.206    have "0 + 0 = 0" by (rule add_0_right)
   1.207 -  thus "- 0 = 0" by (rule minus_unique)
   1.208 +  then show "- 0 = 0" by (rule minus_unique)
   1.209  qed
   1.210  
   1.211  lemma minus_minus [simp]: "- (- a) = a"
   1.212  proof -
   1.213    have "- a + a = 0" by (rule left_minus)
   1.214 -  thus "- (- a) = a" by (rule minus_unique)
   1.215 +  then show "- (- a) = a" by (rule minus_unique)
   1.216  qed
   1.217  
   1.218  lemma right_minus: "a + - a = 0"
   1.219 @@ -386,8 +383,7 @@
   1.220    finally show ?thesis .
   1.221  qed
   1.222  
   1.223 -lemma diff_self [simp]:
   1.224 -  "a - a = 0"
   1.225 +lemma diff_self [simp]: "a - a = 0"
   1.226    using right_minus [of a] by simp
   1.227  
   1.228  subclass cancel_semigroup_add
   1.229 @@ -404,24 +400,19 @@
   1.230    then show "b = c" unfolding add.assoc by simp
   1.231  qed
   1.232  
   1.233 -lemma minus_add_cancel [simp]:
   1.234 -  "- a + (a + b) = b"
   1.235 +lemma minus_add_cancel [simp]: "- a + (a + b) = b"
   1.236    by (simp add: add.assoc [symmetric])
   1.237  
   1.238 -lemma add_minus_cancel [simp]:
   1.239 -  "a + (- a + b) = b"
   1.240 +lemma add_minus_cancel [simp]: "a + (- a + b) = b"
   1.241    by (simp add: add.assoc [symmetric])
   1.242  
   1.243 -lemma diff_add_cancel [simp]:
   1.244 -  "a - b + b = a"
   1.245 +lemma diff_add_cancel [simp]: "a - b + b = a"
   1.246    by (simp only: diff_conv_add_uminus add.assoc) simp
   1.247  
   1.248 -lemma add_diff_cancel [simp]:
   1.249 -  "a + b - b = a"
   1.250 +lemma add_diff_cancel [simp]: "a + b - b = a"
   1.251    by (simp only: diff_conv_add_uminus add.assoc) simp
   1.252  
   1.253 -lemma minus_add:
   1.254 -  "- (a + b) = - b + - a"
   1.255 +lemma minus_add: "- (a + b) = - b + - a"
   1.256  proof -
   1.257    have "(a + b) + (- b + - a) = 0"
   1.258      by (simp only: add.assoc add_minus_cancel) simp
   1.259 @@ -429,117 +420,103 @@
   1.260      by (rule minus_unique)
   1.261  qed
   1.262  
   1.263 -lemma right_minus_eq [simp]:
   1.264 -  "a - b = 0 \<longleftrightarrow> a = b"
   1.265 +lemma right_minus_eq [simp]: "a - b = 0 \<longleftrightarrow> a = b"
   1.266  proof
   1.267    assume "a - b = 0"
   1.268    have "a = (a - b) + b" by (simp add: add.assoc)
   1.269    also have "\<dots> = b" using \<open>a - b = 0\<close> by simp
   1.270    finally show "a = b" .
   1.271  next
   1.272 -  assume "a = b" thus "a - b = 0" by simp
   1.273 +  assume "a = b"
   1.274 +  then show "a - b = 0" by simp
   1.275  qed
   1.276  
   1.277 -lemma eq_iff_diff_eq_0:
   1.278 -  "a = b \<longleftrightarrow> a - b = 0"
   1.279 +lemma eq_iff_diff_eq_0: "a = b \<longleftrightarrow> a - b = 0"
   1.280    by (fact right_minus_eq [symmetric])
   1.281  
   1.282 -lemma diff_0 [simp]:
   1.283 -  "0 - a = - a"
   1.284 +lemma diff_0 [simp]: "0 - a = - a"
   1.285    by (simp only: diff_conv_add_uminus add_0_left)
   1.286  
   1.287 -lemma diff_0_right [simp]:
   1.288 -  "a - 0 = a"
   1.289 +lemma diff_0_right [simp]: "a - 0 = a"
   1.290    by (simp only: diff_conv_add_uminus minus_zero add_0_right)
   1.291  
   1.292 -lemma diff_minus_eq_add [simp]:
   1.293 -  "a - - b = a + b"
   1.294 +lemma diff_minus_eq_add [simp]: "a - - b = a + b"
   1.295    by (simp only: diff_conv_add_uminus minus_minus)
   1.296  
   1.297 -lemma neg_equal_iff_equal [simp]:
   1.298 -  "- a = - b \<longleftrightarrow> a = b"
   1.299 +lemma neg_equal_iff_equal [simp]: "- a = - b \<longleftrightarrow> a = b"
   1.300  proof
   1.301    assume "- a = - b"
   1.302 -  hence "- (- a) = - (- b)" by simp
   1.303 -  thus "a = b" by simp
   1.304 +  then have "- (- a) = - (- b)" by simp
   1.305 +  then show "a = b" by simp
   1.306  next
   1.307    assume "a = b"
   1.308 -  thus "- a = - b" by simp
   1.309 +  then show "- a = - b" by simp
   1.310  qed
   1.311  
   1.312 -lemma neg_equal_0_iff_equal [simp]:
   1.313 -  "- a = 0 \<longleftrightarrow> a = 0"
   1.314 +lemma neg_equal_0_iff_equal [simp]: "- a = 0 \<longleftrightarrow> a = 0"
   1.315    by (subst neg_equal_iff_equal [symmetric]) simp
   1.316  
   1.317 -lemma neg_0_equal_iff_equal [simp]:
   1.318 -  "0 = - a \<longleftrightarrow> 0 = a"
   1.319 +lemma neg_0_equal_iff_equal [simp]: "0 = - a \<longleftrightarrow> 0 = a"
   1.320    by (subst neg_equal_iff_equal [symmetric]) simp
   1.321  
   1.322 -text\<open>The next two equations can make the simplifier loop!\<close>
   1.323 +text \<open>The next two equations can make the simplifier loop!\<close>
   1.324  
   1.325 -lemma equation_minus_iff:
   1.326 -  "a = - b \<longleftrightarrow> b = - a"
   1.327 +lemma equation_minus_iff: "a = - b \<longleftrightarrow> b = - a"
   1.328  proof -
   1.329 -  have "- (- a) = - b \<longleftrightarrow> - a = b" by (rule neg_equal_iff_equal)
   1.330 -  thus ?thesis by (simp add: eq_commute)
   1.331 +  have "- (- a) = - b \<longleftrightarrow> - a = b"
   1.332 +    by (rule neg_equal_iff_equal)
   1.333 +  then show ?thesis
   1.334 +    by (simp add: eq_commute)
   1.335  qed
   1.336  
   1.337 -lemma minus_equation_iff:
   1.338 -  "- a = b \<longleftrightarrow> - b = a"
   1.339 +lemma minus_equation_iff: "- a = b \<longleftrightarrow> - b = a"
   1.340  proof -
   1.341 -  have "- a = - (- b) \<longleftrightarrow> a = -b" by (rule neg_equal_iff_equal)
   1.342 -  thus ?thesis by (simp add: eq_commute)
   1.343 +  have "- a = - (- b) \<longleftrightarrow> a = -b"
   1.344 +    by (rule neg_equal_iff_equal)
   1.345 +  then show ?thesis
   1.346 +    by (simp add: eq_commute)
   1.347  qed
   1.348  
   1.349 -lemma eq_neg_iff_add_eq_0:
   1.350 -  "a = - b \<longleftrightarrow> a + b = 0"
   1.351 +lemma eq_neg_iff_add_eq_0: "a = - b \<longleftrightarrow> a + b = 0"
   1.352  proof
   1.353 -  assume "a = - b" then show "a + b = 0" by simp
   1.354 +  assume "a = - b"
   1.355 +  then show "a + b = 0" by simp
   1.356  next
   1.357    assume "a + b = 0"
   1.358    moreover have "a + (b + - b) = (a + b) + - b"
   1.359      by (simp only: add.assoc)
   1.360 -  ultimately show "a = - b" by simp
   1.361 +  ultimately show "a = - b"
   1.362 +    by simp
   1.363  qed
   1.364  
   1.365 -lemma add_eq_0_iff2:
   1.366 -  "a + b = 0 \<longleftrightarrow> a = - b"
   1.367 +lemma add_eq_0_iff2: "a + b = 0 \<longleftrightarrow> a = - b"
   1.368    by (fact eq_neg_iff_add_eq_0 [symmetric])
   1.369  
   1.370 -lemma neg_eq_iff_add_eq_0:
   1.371 -  "- a = b \<longleftrightarrow> a + b = 0"
   1.372 +lemma neg_eq_iff_add_eq_0: "- a = b \<longleftrightarrow> a + b = 0"
   1.373    by (auto simp add: add_eq_0_iff2)
   1.374  
   1.375 -lemma add_eq_0_iff:
   1.376 -  "a + b = 0 \<longleftrightarrow> b = - a"
   1.377 +lemma add_eq_0_iff: "a + b = 0 \<longleftrightarrow> b = - a"
   1.378    by (auto simp add: neg_eq_iff_add_eq_0 [symmetric])
   1.379  
   1.380 -lemma minus_diff_eq [simp]:
   1.381 -  "- (a - b) = b - a"
   1.382 +lemma minus_diff_eq [simp]: "- (a - b) = b - a"
   1.383    by (simp only: neg_eq_iff_add_eq_0 diff_conv_add_uminus add.assoc minus_add_cancel) simp
   1.384  
   1.385 -lemma add_diff_eq [algebra_simps, field_simps]:
   1.386 -  "a + (b - c) = (a + b) - c"
   1.387 +lemma add_diff_eq [algebra_simps, field_simps]: "a + (b - c) = (a + b) - c"
   1.388    by (simp only: diff_conv_add_uminus add.assoc)
   1.389  
   1.390 -lemma diff_add_eq_diff_diff_swap:
   1.391 -  "a - (b + c) = a - c - b"
   1.392 +lemma diff_add_eq_diff_diff_swap: "a - (b + c) = a - c - b"
   1.393    by (simp only: diff_conv_add_uminus add.assoc minus_add)
   1.394  
   1.395 -lemma diff_eq_eq [algebra_simps, field_simps]:
   1.396 -  "a - b = c \<longleftrightarrow> a = c + b"
   1.397 +lemma diff_eq_eq [algebra_simps, field_simps]: "a - b = c \<longleftrightarrow> a = c + b"
   1.398    by auto
   1.399  
   1.400 -lemma eq_diff_eq [algebra_simps, field_simps]:
   1.401 -  "a = c - b \<longleftrightarrow> a + b = c"
   1.402 +lemma eq_diff_eq [algebra_simps, field_simps]: "a = c - b \<longleftrightarrow> a + b = c"
   1.403    by auto
   1.404  
   1.405 -lemma diff_diff_eq2 [algebra_simps, field_simps]:
   1.406 -  "a - (b - c) = (a + c) - b"
   1.407 +lemma diff_diff_eq2 [algebra_simps, field_simps]: "a - (b - c) = (a + c) - b"
   1.408    by (simp only: diff_conv_add_uminus add.assoc) simp
   1.409  
   1.410 -lemma diff_eq_diff_eq:
   1.411 -  "a - b = c - d \<Longrightarrow> a = b \<longleftrightarrow> c = d"
   1.412 +lemma diff_eq_diff_eq: "a - b = c - d \<Longrightarrow> a = b \<longleftrightarrow> c = d"
   1.413    by (simp only: eq_iff_diff_eq_0 [of a b] eq_iff_diff_eq_0 [of c d])
   1.414  
   1.415  end
   1.416 @@ -550,7 +527,7 @@
   1.417  begin
   1.418  
   1.419  subclass group_add
   1.420 -  proof qed (simp_all add: ab_left_minus ab_diff_conv_add_uminus)
   1.421 +  by standard (simp_all add: ab_left_minus ab_diff_conv_add_uminus)
   1.422  
   1.423  subclass cancel_comm_monoid_add
   1.424  proof
   1.425 @@ -563,16 +540,13 @@
   1.426      by (simp add: algebra_simps)
   1.427  qed
   1.428  
   1.429 -lemma uminus_add_conv_diff [simp]:
   1.430 -  "- a + b = b - a"
   1.431 +lemma uminus_add_conv_diff [simp]: "- a + b = b - a"
   1.432    by (simp add: add.commute)
   1.433  
   1.434 -lemma minus_add_distrib [simp]:
   1.435 -  "- (a + b) = - a + - b"
   1.436 +lemma minus_add_distrib [simp]: "- (a + b) = - a + - b"
   1.437    by (simp add: algebra_simps)
   1.438  
   1.439 -lemma diff_add_eq [algebra_simps, field_simps]:
   1.440 -  "(a - b) + c = (a + c) - b"
   1.441 +lemma diff_add_eq [algebra_simps, field_simps]: "(a - b) + c = (a + c) - b"
   1.442    by (simp add: algebra_simps)
   1.443  
   1.444  end
   1.445 @@ -582,35 +556,31 @@
   1.446  
   1.447  text \<open>
   1.448    The theory of partially ordered groups is taken from the books:
   1.449 -  \begin{itemize}
   1.450 -  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979
   1.451 -  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
   1.452 -  \end{itemize}
   1.453 +
   1.454 +    \<^item> \<^emph>\<open>Lattice Theory\<close> by Garret Birkhoff, American Mathematical Society, 1979
   1.455 +    \<^item> \<^emph>\<open>Partially Ordered Algebraic Systems\<close>, Pergamon Press, 1963
   1.456 +
   1.457    Most of the used notions can also be looked up in
   1.458 -  \begin{itemize}
   1.459 -  \item @{url "http://www.mathworld.com"} by Eric Weisstein et. al.
   1.460 -  \item \emph{Algebra I} by van der Waerden, Springer.
   1.461 -  \end{itemize}
   1.462 +    \<^item> @{url "http://www.mathworld.com"} by Eric Weisstein et. al.
   1.463 +    \<^item> \<^emph>\<open>Algebra I\<close> by van der Waerden, Springer
   1.464  \<close>
   1.465  
   1.466  class ordered_ab_semigroup_add = order + ab_semigroup_add +
   1.467    assumes add_left_mono: "a \<le> b \<Longrightarrow> c + a \<le> c + b"
   1.468  begin
   1.469  
   1.470 -lemma add_right_mono:
   1.471 -  "a \<le> b \<Longrightarrow> a + c \<le> b + c"
   1.472 -by (simp add: add.commute [of _ c] add_left_mono)
   1.473 +lemma add_right_mono: "a \<le> b \<Longrightarrow> a + c \<le> b + c"
   1.474 +  by (simp add: add.commute [of _ c] add_left_mono)
   1.475  
   1.476  text \<open>non-strict, in both arguments\<close>
   1.477 -lemma add_mono:
   1.478 -  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c \<le> b + d"
   1.479 +lemma add_mono: "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c \<le> b + d"
   1.480    apply (erule add_right_mono [THEN order_trans])
   1.481    apply (simp add: add.commute add_left_mono)
   1.482    done
   1.483  
   1.484  end
   1.485  
   1.486 -text\<open>Strict monotonicity in both arguments\<close>
   1.487 +text \<open>Strict monotonicity in both arguments\<close>
   1.488  class strict_ordered_ab_semigroup_add = ordered_ab_semigroup_add +
   1.489    assumes add_strict_mono: "a < b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
   1.490  
   1.491 @@ -618,13 +588,11 @@
   1.492    ordered_ab_semigroup_add + cancel_ab_semigroup_add
   1.493  begin
   1.494  
   1.495 -lemma add_strict_left_mono:
   1.496 -  "a < b \<Longrightarrow> c + a < c + b"
   1.497 -by (auto simp add: less_le add_left_mono)
   1.498 +lemma add_strict_left_mono: "a < b \<Longrightarrow> c + a < c + b"
   1.499 +  by (auto simp add: less_le add_left_mono)
   1.500  
   1.501 -lemma add_strict_right_mono:
   1.502 -  "a < b \<Longrightarrow> a + c < b + c"
   1.503 -by (simp add: add.commute [of _ c] add_strict_left_mono)
   1.504 +lemma add_strict_right_mono: "a < b \<Longrightarrow> a + c < b + c"
   1.505 +  by (simp add: add.commute [of _ c] add_strict_left_mono)
   1.506  
   1.507  subclass strict_ordered_ab_semigroup_add
   1.508    apply standard
   1.509 @@ -632,17 +600,15 @@
   1.510    apply (erule add_strict_left_mono)
   1.511    done
   1.512  
   1.513 -lemma add_less_le_mono:
   1.514 -  "a < b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c < b + d"
   1.515 -apply (erule add_strict_right_mono [THEN less_le_trans])
   1.516 -apply (erule add_left_mono)
   1.517 -done
   1.518 +lemma add_less_le_mono: "a < b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c < b + d"
   1.519 +  apply (erule add_strict_right_mono [THEN less_le_trans])
   1.520 +  apply (erule add_left_mono)
   1.521 +  done
   1.522  
   1.523 -lemma add_le_less_mono:
   1.524 -  "a \<le> b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
   1.525 -apply (erule add_right_mono [THEN le_less_trans])
   1.526 -apply (erule add_strict_left_mono)
   1.527 -done
   1.528 +lemma add_le_less_mono: "a \<le> b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
   1.529 +  apply (erule add_right_mono [THEN le_less_trans])
   1.530 +  apply (erule add_strict_left_mono)
   1.531 +  done
   1.532  
   1.533  end
   1.534  
   1.535 @@ -651,63 +617,60 @@
   1.536  begin
   1.537  
   1.538  lemma add_less_imp_less_left:
   1.539 -  assumes less: "c + a < c + b" shows "a < b"
   1.540 +  assumes less: "c + a < c + b"
   1.541 +  shows "a < b"
   1.542  proof -
   1.543 -  from less have le: "c + a <= c + b" by (simp add: order_le_less)
   1.544 -  have "a <= b"
   1.545 +  from less have le: "c + a \<le> c + b"
   1.546 +    by (simp add: order_le_less)
   1.547 +  have "a \<le> b"
   1.548      apply (insert le)
   1.549      apply (drule add_le_imp_le_left)
   1.550 -    by (insert le, drule add_le_imp_le_left, assumption)
   1.551 +    apply (insert le)
   1.552 +    apply (drule add_le_imp_le_left)
   1.553 +    apply assumption
   1.554 +    done
   1.555    moreover have "a \<noteq> b"
   1.556    proof (rule ccontr)
   1.557 -    assume "~(a \<noteq> b)"
   1.558 +    assume "\<not> ?thesis"
   1.559      then have "a = b" by simp
   1.560      then have "c + a = c + b" by simp
   1.561 -    with less show "False"by simp
   1.562 +    with less show "False" by simp
   1.563    qed
   1.564 -  ultimately show "a < b" by (simp add: order_le_less)
   1.565 +  ultimately show "a < b"
   1.566 +    by (simp add: order_le_less)
   1.567  qed
   1.568  
   1.569 -lemma add_less_imp_less_right:
   1.570 -  "a + c < b + c \<Longrightarrow> a < b"
   1.571 -apply (rule add_less_imp_less_left [of c])
   1.572 -apply (simp add: add.commute)
   1.573 -done
   1.574 +lemma add_less_imp_less_right: "a + c < b + c \<Longrightarrow> a < b"
   1.575 +  by (rule add_less_imp_less_left [of c]) (simp add: add.commute)
   1.576  
   1.577 -lemma add_less_cancel_left [simp]:
   1.578 -  "c + a < c + b \<longleftrightarrow> a < b"
   1.579 +lemma add_less_cancel_left [simp]: "c + a < c + b \<longleftrightarrow> a < b"
   1.580    by (blast intro: add_less_imp_less_left add_strict_left_mono)
   1.581  
   1.582 -lemma add_less_cancel_right [simp]:
   1.583 -  "a + c < b + c \<longleftrightarrow> a < b"
   1.584 +lemma add_less_cancel_right [simp]: "a + c < b + c \<longleftrightarrow> a < b"
   1.585    by (blast intro: add_less_imp_less_right add_strict_right_mono)
   1.586  
   1.587 -lemma add_le_cancel_left [simp]:
   1.588 -  "c + a \<le> c + b \<longleftrightarrow> a \<le> b"
   1.589 -  by (auto, drule add_le_imp_le_left, simp_all add: add_left_mono)
   1.590 +lemma add_le_cancel_left [simp]: "c + a \<le> c + b \<longleftrightarrow> a \<le> b"
   1.591 +  apply auto
   1.592 +  apply (drule add_le_imp_le_left)
   1.593 +  apply (simp_all add: add_left_mono)
   1.594 +  done
   1.595  
   1.596 -lemma add_le_cancel_right [simp]:
   1.597 -  "a + c \<le> b + c \<longleftrightarrow> a \<le> b"
   1.598 +lemma add_le_cancel_right [simp]: "a + c \<le> b + c \<longleftrightarrow> a \<le> b"
   1.599    by (simp add: add.commute [of a c] add.commute [of b c])
   1.600  
   1.601 -lemma add_le_imp_le_right:
   1.602 -  "a + c \<le> b + c \<Longrightarrow> a \<le> b"
   1.603 -by simp
   1.604 +lemma add_le_imp_le_right: "a + c \<le> b + c \<Longrightarrow> a \<le> b"
   1.605 +  by simp
   1.606  
   1.607 -lemma max_add_distrib_left:
   1.608 -  "max x y + z = max (x + z) (y + z)"
   1.609 +lemma max_add_distrib_left: "max x y + z = max (x + z) (y + z)"
   1.610    unfolding max_def by auto
   1.611  
   1.612 -lemma min_add_distrib_left:
   1.613 -  "min x y + z = min (x + z) (y + z)"
   1.614 +lemma min_add_distrib_left: "min x y + z = min (x + z) (y + z)"
   1.615    unfolding min_def by auto
   1.616  
   1.617 -lemma max_add_distrib_right:
   1.618 -  "x + max y z = max (x + y) (x + z)"
   1.619 +lemma max_add_distrib_right: "x + max y z = max (x + y) (x + z)"
   1.620    unfolding max_def by auto
   1.621  
   1.622 -lemma min_add_distrib_right:
   1.623 -  "x + min y z = min (x + y) (x + z)"
   1.624 +lemma min_add_distrib_right: "x + min y z = min (x + y) (x + z)"
   1.625    unfolding min_def by auto
   1.626  
   1.627  end
   1.628 @@ -717,36 +680,28 @@
   1.629  class ordered_comm_monoid_add = comm_monoid_add + ordered_ab_semigroup_add
   1.630  begin
   1.631  
   1.632 -lemma add_nonneg_nonneg [simp]:
   1.633 -  "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a + b"
   1.634 +lemma add_nonneg_nonneg [simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a + b"
   1.635    using add_mono[of 0 a 0 b] by simp
   1.636  
   1.637 -lemma add_nonpos_nonpos:
   1.638 -  "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> a + b \<le> 0"
   1.639 +lemma add_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> a + b \<le> 0"
   1.640    using add_mono[of a 0 b 0] by simp
   1.641  
   1.642 -lemma add_nonneg_eq_0_iff:
   1.643 -  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x + y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
   1.644 +lemma add_nonneg_eq_0_iff: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x + y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
   1.645    using add_left_mono[of 0 y x] add_right_mono[of 0 x y] by auto
   1.646  
   1.647 -lemma add_nonpos_eq_0_iff:
   1.648 -  "x \<le> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> x + y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
   1.649 +lemma add_nonpos_eq_0_iff: "x \<le> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> x + y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
   1.650    using add_left_mono[of y 0 x] add_right_mono[of x 0 y] by auto
   1.651  
   1.652 -lemma add_increasing:
   1.653 -  "0 \<le> a \<Longrightarrow> b \<le> c \<Longrightarrow> b \<le> a + c"
   1.654 -  by (insert add_mono [of 0 a b c], simp)
   1.655 +lemma add_increasing: "0 \<le> a \<Longrightarrow> b \<le> c \<Longrightarrow> b \<le> a + c"
   1.656 +  using add_mono [of 0 a b c] by simp
   1.657  
   1.658 -lemma add_increasing2:
   1.659 -  "0 \<le> c \<Longrightarrow> b \<le> a \<Longrightarrow> b \<le> a + c"
   1.660 +lemma add_increasing2: "0 \<le> c \<Longrightarrow> b \<le> a \<Longrightarrow> b \<le> a + c"
   1.661    by (simp add: add_increasing add.commute [of a])
   1.662  
   1.663 -lemma add_decreasing:
   1.664 -  "a \<le> 0 \<Longrightarrow> c \<le> b \<Longrightarrow> a + c \<le> b"
   1.665 -  using add_mono[of a 0 c b] by simp
   1.666 +lemma add_decreasing: "a \<le> 0 \<Longrightarrow> c \<le> b \<Longrightarrow> a + c \<le> b"
   1.667 +  using add_mono [of a 0 c b] by simp
   1.668  
   1.669 -lemma add_decreasing2:
   1.670 -  "c \<le> 0 \<Longrightarrow> a \<le> b \<Longrightarrow> a + c \<le> b"
   1.671 +lemma add_decreasing2: "c \<le> 0 \<Longrightarrow> a \<le> b \<Longrightarrow> a + c \<le> b"
   1.672    using add_mono[of a b c 0] by simp
   1.673  
   1.674  lemma add_pos_nonneg: "0 < a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < a + b"
   1.675 @@ -776,8 +731,7 @@
   1.676  class strict_ordered_comm_monoid_add = comm_monoid_add + strict_ordered_ab_semigroup_add
   1.677  begin
   1.678  
   1.679 -lemma pos_add_strict:
   1.680 -  shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
   1.681 +lemma pos_add_strict: "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
   1.682    using add_strict_mono [of 0 a b c] by simp
   1.683  
   1.684  end
   1.685 @@ -788,13 +742,11 @@
   1.686  subclass ordered_cancel_ab_semigroup_add ..
   1.687  subclass strict_ordered_comm_monoid_add ..
   1.688  
   1.689 -lemma add_strict_increasing:
   1.690 -  "0 < a \<Longrightarrow> b \<le> c \<Longrightarrow> b < a + c"
   1.691 -  by (insert add_less_le_mono [of 0 a b c], simp)
   1.692 +lemma add_strict_increasing: "0 < a \<Longrightarrow> b \<le> c \<Longrightarrow> b < a + c"
   1.693 +  using add_less_le_mono [of 0 a b c] by simp
   1.694  
   1.695 -lemma add_strict_increasing2:
   1.696 -  "0 \<le> a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
   1.697 -  by (insert add_le_less_mono [of 0 a b c], simp)
   1.698 +lemma add_strict_increasing2: "0 \<le> a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
   1.699 +  using add_le_less_mono [of 0 a b c] by simp
   1.700  
   1.701  end
   1.702  
   1.703 @@ -807,105 +759,108 @@
   1.704  proof
   1.705    fix a b c :: 'a
   1.706    assume "c + a \<le> c + b"
   1.707 -  hence "(-c) + (c + a) \<le> (-c) + (c + b)" by (rule add_left_mono)
   1.708 -  hence "((-c) + c) + a \<le> ((-c) + c) + b" by (simp only: add.assoc)
   1.709 -  thus "a \<le> b" by simp
   1.710 +  then have "(-c) + (c + a) \<le> (-c) + (c + b)"
   1.711 +    by (rule add_left_mono)
   1.712 +  then have "((-c) + c) + a \<le> ((-c) + c) + b"
   1.713 +    by (simp only: add.assoc)
   1.714 +  then show "a \<le> b" by simp
   1.715  qed
   1.716  
   1.717  subclass ordered_cancel_comm_monoid_add ..
   1.718  
   1.719 -lemma add_less_same_cancel1 [simp]:
   1.720 -  "b + a < b \<longleftrightarrow> a < 0"
   1.721 +lemma add_less_same_cancel1 [simp]: "b + a < b \<longleftrightarrow> a < 0"
   1.722    using add_less_cancel_left [of _ _ 0] by simp
   1.723  
   1.724 -lemma add_less_same_cancel2 [simp]:
   1.725 -  "a + b < b \<longleftrightarrow> a < 0"
   1.726 +lemma add_less_same_cancel2 [simp]: "a + b < b \<longleftrightarrow> a < 0"
   1.727    using add_less_cancel_right [of _ _ 0] by simp
   1.728  
   1.729 -lemma less_add_same_cancel1 [simp]:
   1.730 -  "a < a + b \<longleftrightarrow> 0 < b"
   1.731 +lemma less_add_same_cancel1 [simp]: "a < a + b \<longleftrightarrow> 0 < b"
   1.732    using add_less_cancel_left [of _ 0] by simp
   1.733  
   1.734 -lemma less_add_same_cancel2 [simp]:
   1.735 -  "a < b + a \<longleftrightarrow> 0 < b"
   1.736 +lemma less_add_same_cancel2 [simp]: "a < b + a \<longleftrightarrow> 0 < b"
   1.737    using add_less_cancel_right [of 0] by simp
   1.738  
   1.739 -lemma add_le_same_cancel1 [simp]:
   1.740 -  "b + a \<le> b \<longleftrightarrow> a \<le> 0"
   1.741 +lemma add_le_same_cancel1 [simp]: "b + a \<le> b \<longleftrightarrow> a \<le> 0"
   1.742    using add_le_cancel_left [of _ _ 0] by simp
   1.743  
   1.744 -lemma add_le_same_cancel2 [simp]:
   1.745 -  "a + b \<le> b \<longleftrightarrow> a \<le> 0"
   1.746 +lemma add_le_same_cancel2 [simp]: "a + b \<le> b \<longleftrightarrow> a \<le> 0"
   1.747    using add_le_cancel_right [of _ _ 0] by simp
   1.748  
   1.749 -lemma le_add_same_cancel1 [simp]:
   1.750 -  "a \<le> a + b \<longleftrightarrow> 0 \<le> b"
   1.751 +lemma le_add_same_cancel1 [simp]: "a \<le> a + b \<longleftrightarrow> 0 \<le> b"
   1.752    using add_le_cancel_left [of _ 0] by simp
   1.753  
   1.754 -lemma le_add_same_cancel2 [simp]:
   1.755 -  "a \<le> b + a \<longleftrightarrow> 0 \<le> b"
   1.756 +lemma le_add_same_cancel2 [simp]: "a \<le> b + a \<longleftrightarrow> 0 \<le> b"
   1.757    using add_le_cancel_right [of 0] by simp
   1.758  
   1.759 -lemma max_diff_distrib_left:
   1.760 -  shows "max x y - z = max (x - z) (y - z)"
   1.761 +lemma max_diff_distrib_left: "max x y - z = max (x - z) (y - z)"
   1.762    using max_add_distrib_left [of x y "- z"] by simp
   1.763  
   1.764 -lemma min_diff_distrib_left:
   1.765 -  shows "min x y - z = min (x - z) (y - z)"
   1.766 +lemma min_diff_distrib_left: "min x y - z = min (x - z) (y - z)"
   1.767    using min_add_distrib_left [of x y "- z"] by simp
   1.768  
   1.769  lemma le_imp_neg_le:
   1.770 -  assumes "a \<le> b" shows "-b \<le> -a"
   1.771 +  assumes "a \<le> b"
   1.772 +  shows "- b \<le> - a"
   1.773  proof -
   1.774 -  have "-a+a \<le> -a+b" using \<open>a \<le> b\<close> by (rule add_left_mono)
   1.775 -  then have "0 \<le> -a+b" by simp
   1.776 -  then have "0 + (-b) \<le> (-a + b) + (-b)" by (rule add_right_mono)
   1.777 -  then show ?thesis by (simp add: algebra_simps)
   1.778 +  from assms have "- a + a \<le> - a + b"
   1.779 +    by (rule add_left_mono)
   1.780 +  then have "0 \<le> - a + b"
   1.781 +    by simp
   1.782 +  then have "0 + (- b) \<le> (- a + b) + (- b)"
   1.783 +    by (rule add_right_mono)
   1.784 +  then show ?thesis
   1.785 +    by (simp add: algebra_simps)
   1.786  qed
   1.787  
   1.788  lemma neg_le_iff_le [simp]: "- b \<le> - a \<longleftrightarrow> a \<le> b"
   1.789  proof
   1.790    assume "- b \<le> - a"
   1.791 -  hence "- (- a) \<le> - (- b)" by (rule le_imp_neg_le)
   1.792 -  thus "a\<le>b" by simp
   1.793 +  then have "- (- a) \<le> - (- b)"
   1.794 +    by (rule le_imp_neg_le)
   1.795 +  then show "a \<le> b"
   1.796 +    by simp
   1.797  next
   1.798 -  assume "a\<le>b"
   1.799 -  thus "-b \<le> -a" by (rule le_imp_neg_le)
   1.800 +  assume "a \<le> b"
   1.801 +  then show "- b \<le> - a"
   1.802 +    by (rule le_imp_neg_le)
   1.803  qed
   1.804  
   1.805  lemma neg_le_0_iff_le [simp]: "- a \<le> 0 \<longleftrightarrow> 0 \<le> a"
   1.806 -by (subst neg_le_iff_le [symmetric], simp)
   1.807 +  by (subst neg_le_iff_le [symmetric]) simp
   1.808  
   1.809  lemma neg_0_le_iff_le [simp]: "0 \<le> - a \<longleftrightarrow> a \<le> 0"
   1.810 -by (subst neg_le_iff_le [symmetric], simp)
   1.811 +  by (subst neg_le_iff_le [symmetric]) simp
   1.812  
   1.813  lemma neg_less_iff_less [simp]: "- b < - a \<longleftrightarrow> a < b"
   1.814 -by (force simp add: less_le)
   1.815 +  by (auto simp add: less_le)
   1.816  
   1.817  lemma neg_less_0_iff_less [simp]: "- a < 0 \<longleftrightarrow> 0 < a"
   1.818 -by (subst neg_less_iff_less [symmetric], simp)
   1.819 +  by (subst neg_less_iff_less [symmetric]) simp
   1.820  
   1.821  lemma neg_0_less_iff_less [simp]: "0 < - a \<longleftrightarrow> a < 0"
   1.822 -by (subst neg_less_iff_less [symmetric], simp)
   1.823 +  by (subst neg_less_iff_less [symmetric]) simp
   1.824  
   1.825 -text\<open>The next several equations can make the simplifier loop!\<close>
   1.826 +text \<open>The next several equations can make the simplifier loop!\<close>
   1.827  
   1.828  lemma less_minus_iff: "a < - b \<longleftrightarrow> b < - a"
   1.829  proof -
   1.830 -  have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less)
   1.831 -  thus ?thesis by simp
   1.832 +  have "- (-a) < - b \<longleftrightarrow> b < - a"
   1.833 +    by (rule neg_less_iff_less)
   1.834 +  then show ?thesis by simp
   1.835  qed
   1.836  
   1.837  lemma minus_less_iff: "- a < b \<longleftrightarrow> - b < a"
   1.838  proof -
   1.839 -  have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less)
   1.840 -  thus ?thesis by simp
   1.841 +  have "- a < - (- b) \<longleftrightarrow> - b < a"
   1.842 +    by (rule neg_less_iff_less)
   1.843 +  then show ?thesis by simp
   1.844  qed
   1.845  
   1.846  lemma le_minus_iff: "a \<le> - b \<longleftrightarrow> b \<le> - a"
   1.847  proof -
   1.848 -  have mm: "!! a (b::'a). (-(-a)) < -b \<Longrightarrow> -(-b) < -a" by (simp only: minus_less_iff)
   1.849 -  have "(- (- a) <= -b) = (b <= - a)"
   1.850 +  have mm: "- (- a) < -b \<Longrightarrow> - (- b) < -a" for a b :: 'a
   1.851 +    by (simp only: minus_less_iff)
   1.852 +  have "- (- a) \<le> -b \<longleftrightarrow> b \<le> - a"
   1.853      apply (auto simp only: le_less)
   1.854      apply (drule mm)
   1.855      apply (simp_all)
   1.856 @@ -915,60 +870,52 @@
   1.857  qed
   1.858  
   1.859  lemma minus_le_iff: "- a \<le> b \<longleftrightarrow> - b \<le> a"
   1.860 -by (auto simp add: le_less minus_less_iff)
   1.861 +  by (auto simp add: le_less minus_less_iff)
   1.862  
   1.863 -lemma diff_less_0_iff_less [simp]:
   1.864 -  "a - b < 0 \<longleftrightarrow> a < b"
   1.865 +lemma diff_less_0_iff_less [simp]: "a - b < 0 \<longleftrightarrow> a < b"
   1.866  proof -
   1.867 -  have "a - b < 0 \<longleftrightarrow> a + (- b) < b + (- b)" by simp
   1.868 -  also have "... \<longleftrightarrow> a < b" by (simp only: add_less_cancel_right)
   1.869 +  have "a - b < 0 \<longleftrightarrow> a + (- b) < b + (- b)"
   1.870 +    by simp
   1.871 +  also have "\<dots> \<longleftrightarrow> a < b"
   1.872 +    by (simp only: add_less_cancel_right)
   1.873    finally show ?thesis .
   1.874  qed
   1.875  
   1.876  lemmas less_iff_diff_less_0 = diff_less_0_iff_less [symmetric]
   1.877  
   1.878 -lemma diff_less_eq [algebra_simps, field_simps]:
   1.879 -  "a - b < c \<longleftrightarrow> a < c + b"
   1.880 -apply (subst less_iff_diff_less_0 [of a])
   1.881 -apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst])
   1.882 -apply (simp add: algebra_simps)
   1.883 -done
   1.884 +lemma diff_less_eq [algebra_simps, field_simps]: "a - b < c \<longleftrightarrow> a < c + b"
   1.885 +  apply (subst less_iff_diff_less_0 [of a])
   1.886 +  apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst])
   1.887 +  apply (simp add: algebra_simps)
   1.888 +  done
   1.889  
   1.890 -lemma less_diff_eq[algebra_simps, field_simps]:
   1.891 -  "a < c - b \<longleftrightarrow> a + b < c"
   1.892 -apply (subst less_iff_diff_less_0 [of "a + b"])
   1.893 -apply (subst less_iff_diff_less_0 [of a])
   1.894 -apply (simp add: algebra_simps)
   1.895 -done
   1.896 +lemma less_diff_eq[algebra_simps, field_simps]: "a < c - b \<longleftrightarrow> a + b < c"
   1.897 +  apply (subst less_iff_diff_less_0 [of "a + b"])
   1.898 +  apply (subst less_iff_diff_less_0 [of a])
   1.899 +  apply (simp add: algebra_simps)
   1.900 +  done
   1.901  
   1.902 -lemma diff_gt_0_iff_gt [simp]:
   1.903 -  "a - b > 0 \<longleftrightarrow> a > b"
   1.904 +lemma diff_gt_0_iff_gt [simp]: "a - b > 0 \<longleftrightarrow> a > b"
   1.905    by (simp add: less_diff_eq)
   1.906  
   1.907 -lemma diff_le_eq [algebra_simps, field_simps]:
   1.908 -  "a - b \<le> c \<longleftrightarrow> a \<le> c + b"
   1.909 +lemma diff_le_eq [algebra_simps, field_simps]: "a - b \<le> c \<longleftrightarrow> a \<le> c + b"
   1.910    by (auto simp add: le_less diff_less_eq )
   1.911  
   1.912 -lemma le_diff_eq [algebra_simps, field_simps]:
   1.913 -  "a \<le> c - b \<longleftrightarrow> a + b \<le> c"
   1.914 +lemma le_diff_eq [algebra_simps, field_simps]: "a \<le> c - b \<longleftrightarrow> a + b \<le> c"
   1.915    by (auto simp add: le_less less_diff_eq)
   1.916  
   1.917 -lemma diff_le_0_iff_le [simp]:
   1.918 -  "a - b \<le> 0 \<longleftrightarrow> a \<le> b"
   1.919 +lemma diff_le_0_iff_le [simp]: "a - b \<le> 0 \<longleftrightarrow> a \<le> b"
   1.920    by (simp add: algebra_simps)
   1.921  
   1.922  lemmas le_iff_diff_le_0 = diff_le_0_iff_le [symmetric]
   1.923  
   1.924 -lemma diff_ge_0_iff_ge [simp]:
   1.925 -  "a - b \<ge> 0 \<longleftrightarrow> a \<ge> b"
   1.926 +lemma diff_ge_0_iff_ge [simp]: "a - b \<ge> 0 \<longleftrightarrow> a \<ge> b"
   1.927    by (simp add: le_diff_eq)
   1.928  
   1.929 -lemma diff_eq_diff_less:
   1.930 -  "a - b = c - d \<Longrightarrow> a < b \<longleftrightarrow> c < d"
   1.931 +lemma diff_eq_diff_less: "a - b = c - d \<Longrightarrow> a < b \<longleftrightarrow> c < d"
   1.932    by (auto simp only: less_iff_diff_less_0 [of a b] less_iff_diff_less_0 [of c d])
   1.933  
   1.934 -lemma diff_eq_diff_less_eq:
   1.935 -  "a - b = c - d \<Longrightarrow> a \<le> b \<longleftrightarrow> c \<le> d"
   1.936 +lemma diff_eq_diff_less_eq: "a - b = c - d \<Longrightarrow> a \<le> b \<longleftrightarrow> c \<le> d"
   1.937    by (auto simp only: le_iff_diff_le_0 [of a b] le_iff_diff_le_0 [of c d])
   1.938  
   1.939  lemma diff_mono: "a \<le> b \<Longrightarrow> d \<le> c \<Longrightarrow> a - c \<le> b - d"
   1.940 @@ -1020,18 +967,18 @@
   1.941  subclass ordered_ab_semigroup_add_imp_le
   1.942  proof
   1.943    fix a b c :: 'a
   1.944 -  assume le: "c + a <= c + b"
   1.945 -  show "a <= b"
   1.946 +  assume le1: "c + a \<le> c + b"
   1.947 +  show "a \<le> b"
   1.948    proof (rule ccontr)
   1.949 -    assume w: "~ a \<le> b"
   1.950 -    hence "b <= a" by (simp add: linorder_not_le)
   1.951 -    hence le2: "c + b <= c + a" by (rule add_left_mono)
   1.952 +    assume *: "\<not> ?thesis"
   1.953 +    then have "b \<le> a" by (simp add: linorder_not_le)
   1.954 +    then have le2: "c + b \<le> c + a" by (rule add_left_mono)
   1.955      have "a = b"
   1.956 -      apply (insert le)
   1.957 -      apply (insert le2)
   1.958 -      apply (drule antisym, simp_all)
   1.959 +      apply (insert le1 le2)
   1.960 +      apply (drule antisym)
   1.961 +      apply simp_all
   1.962        done
   1.963 -    with w show False
   1.964 +    with * show False
   1.965        by (simp add: linorder_not_le [symmetric])
   1.966    qed
   1.967  qed
   1.968 @@ -1043,72 +990,71 @@
   1.969  
   1.970  subclass linordered_cancel_ab_semigroup_add ..
   1.971  
   1.972 -lemma equal_neg_zero [simp]:
   1.973 -  "a = - a \<longleftrightarrow> a = 0"
   1.974 +lemma equal_neg_zero [simp]: "a = - a \<longleftrightarrow> a = 0"
   1.975  proof
   1.976 -  assume "a = 0" then show "a = - a" by simp
   1.977 +  assume "a = 0"
   1.978 +  then show "a = - a" by simp
   1.979  next
   1.980 -  assume A: "a = - a" show "a = 0"
   1.981 +  assume A: "a = - a"
   1.982 +  show "a = 0"
   1.983    proof (cases "0 \<le> a")
   1.984 -    case True with A have "0 \<le> - a" by auto
   1.985 +    case True
   1.986 +    with A have "0 \<le> - a" by auto
   1.987      with le_minus_iff have "a \<le> 0" by simp
   1.988      with True show ?thesis by (auto intro: order_trans)
   1.989    next
   1.990 -    case False then have B: "a \<le> 0" by auto
   1.991 +    case False
   1.992 +    then have B: "a \<le> 0" by auto
   1.993      with A have "- a \<le> 0" by auto
   1.994      with B show ?thesis by (auto intro: order_trans)
   1.995    qed
   1.996  qed
   1.997  
   1.998 -lemma neg_equal_zero [simp]:
   1.999 -  "- a = a \<longleftrightarrow> a = 0"
  1.1000 +lemma neg_equal_zero [simp]: "- a = a \<longleftrightarrow> a = 0"
  1.1001    by (auto dest: sym)
  1.1002  
  1.1003 -lemma neg_less_eq_nonneg [simp]:
  1.1004 -  "- a \<le> a \<longleftrightarrow> 0 \<le> a"
  1.1005 +lemma neg_less_eq_nonneg [simp]: "- a \<le> a \<longleftrightarrow> 0 \<le> a"
  1.1006  proof
  1.1007 -  assume A: "- a \<le> a" show "0 \<le> a"
  1.1008 +  assume *: "- a \<le> a"
  1.1009 +  show "0 \<le> a"
  1.1010    proof (rule classical)
  1.1011 -    assume "\<not> 0 \<le> a"
  1.1012 +    assume "\<not> ?thesis"
  1.1013      then have "a < 0" by auto
  1.1014 -    with A have "- a < 0" by (rule le_less_trans)
  1.1015 +    with * have "- a < 0" by (rule le_less_trans)
  1.1016      then show ?thesis by auto
  1.1017    qed
  1.1018  next
  1.1019 -  assume A: "0 \<le> a" show "- a \<le> a"
  1.1020 -  proof (rule order_trans)
  1.1021 -    show "- a \<le> 0" using A by (simp add: minus_le_iff)
  1.1022 -  next
  1.1023 -    show "0 \<le> a" using A .
  1.1024 -  qed
  1.1025 +  assume *: "0 \<le> a"
  1.1026 +  then have "- a \<le> 0" by (simp add: minus_le_iff)
  1.1027 +  from this * show "- a \<le> a" by (rule order_trans)
  1.1028  qed
  1.1029  
  1.1030 -lemma neg_less_pos [simp]:
  1.1031 -  "- a < a \<longleftrightarrow> 0 < a"
  1.1032 +lemma neg_less_pos [simp]: "- a < a \<longleftrightarrow> 0 < a"
  1.1033    by (auto simp add: less_le)
  1.1034  
  1.1035 -lemma less_eq_neg_nonpos [simp]:
  1.1036 -  "a \<le> - a \<longleftrightarrow> a \<le> 0"
  1.1037 +lemma less_eq_neg_nonpos [simp]: "a \<le> - a \<longleftrightarrow> a \<le> 0"
  1.1038    using neg_less_eq_nonneg [of "- a"] by simp
  1.1039  
  1.1040 -lemma less_neg_neg [simp]:
  1.1041 -  "a < - a \<longleftrightarrow> a < 0"
  1.1042 +lemma less_neg_neg [simp]: "a < - a \<longleftrightarrow> a < 0"
  1.1043    using neg_less_pos [of "- a"] by simp
  1.1044  
  1.1045 -lemma double_zero [simp]:
  1.1046 -  "a + a = 0 \<longleftrightarrow> a = 0"
  1.1047 +lemma double_zero [simp]: "a + a = 0 \<longleftrightarrow> a = 0"
  1.1048  proof
  1.1049 -  assume assm: "a + a = 0"
  1.1050 +  assume "a + a = 0"
  1.1051    then have a: "- a = a" by (rule minus_unique)
  1.1052    then show "a = 0" by (simp only: neg_equal_zero)
  1.1053 -qed simp
  1.1054 +next
  1.1055 +  assume "a = 0"
  1.1056 +  then show "a + a = 0" by simp
  1.1057 +qed
  1.1058  
  1.1059 -lemma double_zero_sym [simp]:
  1.1060 -  "0 = a + a \<longleftrightarrow> a = 0"
  1.1061 -  by (rule, drule sym) simp_all
  1.1062 +lemma double_zero_sym [simp]: "0 = a + a \<longleftrightarrow> a = 0"
  1.1063 +  apply (rule iffI)
  1.1064 +  apply (drule sym)
  1.1065 +  apply simp_all
  1.1066 +  done
  1.1067  
  1.1068 -lemma zero_less_double_add_iff_zero_less_single_add [simp]:
  1.1069 -  "0 < a + a \<longleftrightarrow> 0 < a"
  1.1070 +lemma zero_less_double_add_iff_zero_less_single_add [simp]: "0 < a + a \<longleftrightarrow> 0 < a"
  1.1071  proof
  1.1072    assume "0 < a + a"
  1.1073    then have "0 - a < a" by (simp only: diff_less_eq)
  1.1074 @@ -1121,32 +1067,27 @@
  1.1075    then show "0 < a + a" by simp
  1.1076  qed
  1.1077  
  1.1078 -lemma zero_le_double_add_iff_zero_le_single_add [simp]:
  1.1079 -  "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
  1.1080 +lemma zero_le_double_add_iff_zero_le_single_add [simp]: "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
  1.1081    by (auto simp add: le_less)
  1.1082  
  1.1083 -lemma double_add_less_zero_iff_single_add_less_zero [simp]:
  1.1084 -  "a + a < 0 \<longleftrightarrow> a < 0"
  1.1085 +lemma double_add_less_zero_iff_single_add_less_zero [simp]: "a + a < 0 \<longleftrightarrow> a < 0"
  1.1086  proof -
  1.1087    have "\<not> a + a < 0 \<longleftrightarrow> \<not> a < 0"
  1.1088      by (simp add: not_less)
  1.1089    then show ?thesis by simp
  1.1090  qed
  1.1091  
  1.1092 -lemma double_add_le_zero_iff_single_add_le_zero [simp]:
  1.1093 -  "a + a \<le> 0 \<longleftrightarrow> a \<le> 0"
  1.1094 +lemma double_add_le_zero_iff_single_add_le_zero [simp]: "a + a \<le> 0 \<longleftrightarrow> a \<le> 0"
  1.1095  proof -
  1.1096    have "\<not> a + a \<le> 0 \<longleftrightarrow> \<not> a \<le> 0"
  1.1097      by (simp add: not_le)
  1.1098    then show ?thesis by simp
  1.1099  qed
  1.1100  
  1.1101 -lemma minus_max_eq_min:
  1.1102 -  "- max x y = min (-x) (-y)"
  1.1103 +lemma minus_max_eq_min: "- max x y = min (- x) (- y)"
  1.1104    by (auto simp add: max_def min_def)
  1.1105  
  1.1106 -lemma minus_min_eq_max:
  1.1107 -  "- min x y = max (-x) (-y)"
  1.1108 +lemma minus_min_eq_max: "- min x y = max (- x) (- y)"
  1.1109    by (auto simp add: max_def min_def)
  1.1110  
  1.1111  end
  1.1112 @@ -1181,16 +1122,17 @@
  1.1113    unfolding neg_le_0_iff_le by simp
  1.1114  
  1.1115  lemma abs_of_nonneg [simp]:
  1.1116 -  assumes nonneg: "0 \<le> a" shows "\<bar>a\<bar> = a"
  1.1117 +  assumes nonneg: "0 \<le> a"
  1.1118 +  shows "\<bar>a\<bar> = a"
  1.1119  proof (rule antisym)
  1.1120 +  show "a \<le> \<bar>a\<bar>" by (rule abs_ge_self)
  1.1121    from nonneg le_imp_neg_le have "- a \<le> 0" by simp
  1.1122    from this nonneg have "- a \<le> a" by (rule order_trans)
  1.1123    then show "\<bar>a\<bar> \<le> a" by (auto intro: abs_leI)
  1.1124 -qed (rule abs_ge_self)
  1.1125 +qed
  1.1126  
  1.1127  lemma abs_idempotent [simp]: "\<bar>\<bar>a\<bar>\<bar> = \<bar>a\<bar>"
  1.1128 -by (rule antisym)
  1.1129 -   (auto intro!: abs_ge_self abs_leI order_trans [of "- \<bar>a\<bar>" 0 "\<bar>a\<bar>"])
  1.1130 +  by (rule antisym) (auto intro!: abs_ge_self abs_leI order_trans [of "- \<bar>a\<bar>" 0 "\<bar>a\<bar>"])
  1.1131  
  1.1132  lemma abs_eq_0 [simp]: "\<bar>a\<bar> = 0 \<longleftrightarrow> a = 0"
  1.1133  proof -
  1.1134 @@ -1206,27 +1148,27 @@
  1.1135  qed
  1.1136  
  1.1137  lemma abs_zero [simp]: "\<bar>0\<bar> = 0"
  1.1138 -by simp
  1.1139 +  by simp
  1.1140  
  1.1141  lemma abs_0_eq [simp]: "0 = \<bar>a\<bar> \<longleftrightarrow> a = 0"
  1.1142  proof -
  1.1143    have "0 = \<bar>a\<bar> \<longleftrightarrow> \<bar>a\<bar> = 0" by (simp only: eq_ac)
  1.1144 -  thus ?thesis by simp
  1.1145 +  then show ?thesis by simp
  1.1146  qed
  1.1147  
  1.1148  lemma abs_le_zero_iff [simp]: "\<bar>a\<bar> \<le> 0 \<longleftrightarrow> a = 0"
  1.1149  proof
  1.1150    assume "\<bar>a\<bar> \<le> 0"
  1.1151    then have "\<bar>a\<bar> = 0" by (rule antisym) simp
  1.1152 -  thus "a = 0" by simp
  1.1153 +  then show "a = 0" by simp
  1.1154  next
  1.1155    assume "a = 0"
  1.1156 -  thus "\<bar>a\<bar> \<le> 0" by simp
  1.1157 +  then show "\<bar>a\<bar> \<le> 0" by simp
  1.1158  qed
  1.1159  
  1.1160  lemma abs_le_self_iff [simp]: "\<bar>a\<bar> \<le> a \<longleftrightarrow> 0 \<le> a"
  1.1161  proof -
  1.1162 -  have "\<forall>a. (0::'a) \<le> \<bar>a\<bar>"
  1.1163 +  have "0 \<le> \<bar>a\<bar>"
  1.1164      using abs_ge_zero by blast
  1.1165    then have "\<bar>a\<bar> \<le> a \<Longrightarrow> 0 \<le> a"
  1.1166      using order.trans by blast
  1.1167 @@ -1235,12 +1177,12 @@
  1.1168  qed
  1.1169  
  1.1170  lemma zero_less_abs_iff [simp]: "0 < \<bar>a\<bar> \<longleftrightarrow> a \<noteq> 0"
  1.1171 -by (simp add: less_le)
  1.1172 +  by (simp add: less_le)
  1.1173  
  1.1174  lemma abs_not_less_zero [simp]: "\<not> \<bar>a\<bar> < 0"
  1.1175  proof -
  1.1176 -  have a: "\<And>x y. x \<le> y \<Longrightarrow> \<not> y < x" by auto
  1.1177 -  show ?thesis by (simp add: a)
  1.1178 +  have "x \<le> y \<Longrightarrow> \<not> y < x" for x y by auto
  1.1179 +  then show ?thesis by simp
  1.1180  qed
  1.1181  
  1.1182  lemma abs_ge_minus_self: "- a \<le> \<bar>a\<bar>"
  1.1183 @@ -1249,39 +1191,40 @@
  1.1184    then show ?thesis by simp
  1.1185  qed
  1.1186  
  1.1187 -lemma abs_minus_commute:
  1.1188 -  "\<bar>a - b\<bar> = \<bar>b - a\<bar>"
  1.1189 +lemma abs_minus_commute: "\<bar>a - b\<bar> = \<bar>b - a\<bar>"
  1.1190  proof -
  1.1191 -  have "\<bar>a - b\<bar> = \<bar>- (a - b)\<bar>" by (simp only: abs_minus_cancel)
  1.1192 -  also have "... = \<bar>b - a\<bar>" by simp
  1.1193 +  have "\<bar>a - b\<bar> = \<bar>- (a - b)\<bar>"
  1.1194 +    by (simp only: abs_minus_cancel)
  1.1195 +  also have "\<dots> = \<bar>b - a\<bar>" by simp
  1.1196    finally show ?thesis .
  1.1197  qed
  1.1198  
  1.1199  lemma abs_of_pos: "0 < a \<Longrightarrow> \<bar>a\<bar> = a"
  1.1200 -by (rule abs_of_nonneg, rule less_imp_le)
  1.1201 +  by (rule abs_of_nonneg) (rule less_imp_le)
  1.1202  
  1.1203  lemma abs_of_nonpos [simp]:
  1.1204 -  assumes "a \<le> 0" shows "\<bar>a\<bar> = - a"
  1.1205 +  assumes "a \<le> 0"
  1.1206 +  shows "\<bar>a\<bar> = - a"
  1.1207  proof -
  1.1208    let ?b = "- a"
  1.1209    have "- ?b \<le> 0 \<Longrightarrow> \<bar>- ?b\<bar> = - (- ?b)"
  1.1210 -  unfolding abs_minus_cancel [of "?b"]
  1.1211 -  unfolding neg_le_0_iff_le [of "?b"]
  1.1212 -  unfolding minus_minus by (erule abs_of_nonneg)
  1.1213 +    unfolding abs_minus_cancel [of ?b]
  1.1214 +    unfolding neg_le_0_iff_le [of ?b]
  1.1215 +    unfolding minus_minus by (erule abs_of_nonneg)
  1.1216    then show ?thesis using assms by auto
  1.1217  qed
  1.1218  
  1.1219  lemma abs_of_neg: "a < 0 \<Longrightarrow> \<bar>a\<bar> = - a"
  1.1220 -by (rule abs_of_nonpos, rule less_imp_le)
  1.1221 +  by (rule abs_of_nonpos) (rule less_imp_le)
  1.1222  
  1.1223  lemma abs_le_D1: "\<bar>a\<bar> \<le> b \<Longrightarrow> a \<le> b"
  1.1224 -by (insert abs_ge_self, blast intro: order_trans)
  1.1225 +  using abs_ge_self by (blast intro: order_trans)
  1.1226  
  1.1227  lemma abs_le_D2: "\<bar>a\<bar> \<le> b \<Longrightarrow> - a \<le> b"
  1.1228 -by (insert abs_le_D1 [of "- a"], simp)
  1.1229 +  using abs_le_D1 [of "- a"] by simp
  1.1230  
  1.1231  lemma abs_le_iff: "\<bar>a\<bar> \<le> b \<longleftrightarrow> a \<le> b \<and> - a \<le> b"
  1.1232 -by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2)
  1.1233 +  by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2)
  1.1234  
  1.1235  lemma abs_triangle_ineq2: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>a - b\<bar>"
  1.1236  proof -
  1.1237 @@ -1301,24 +1244,27 @@
  1.1238  
  1.1239  lemma abs_triangle_ineq4: "\<bar>a - b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
  1.1240  proof -
  1.1241 -  have "\<bar>a - b\<bar> = \<bar>a + - b\<bar>" by (simp add: algebra_simps)
  1.1242 -  also have "... \<le> \<bar>a\<bar> + \<bar>- b\<bar>" by (rule abs_triangle_ineq)
  1.1243 +  have "\<bar>a - b\<bar> = \<bar>a + - b\<bar>"
  1.1244 +    by (simp add: algebra_simps)
  1.1245 +  also have "\<dots> \<le> \<bar>a\<bar> + \<bar>- b\<bar>"
  1.1246 +    by (rule abs_triangle_ineq)
  1.1247    finally show ?thesis by simp
  1.1248  qed
  1.1249  
  1.1250  lemma abs_diff_triangle_ineq: "\<bar>a + b - (c + d)\<bar> \<le> \<bar>a - c\<bar> + \<bar>b - d\<bar>"
  1.1251  proof -
  1.1252 -  have "\<bar>a + b - (c+d)\<bar> = \<bar>(a-c) + (b-d)\<bar>" by (simp add: algebra_simps)
  1.1253 -  also have "... \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>" by (rule abs_triangle_ineq)
  1.1254 +  have "\<bar>a + b - (c + d)\<bar> = \<bar>(a - c) + (b - d)\<bar>"
  1.1255 +    by (simp add: algebra_simps)
  1.1256 +  also have "\<dots> \<le> \<bar>a - c\<bar> + \<bar>b - d\<bar>"
  1.1257 +    by (rule abs_triangle_ineq)
  1.1258    finally show ?thesis .
  1.1259  qed
  1.1260  
  1.1261 -lemma abs_add_abs [simp]:
  1.1262 -  "\<bar>\<bar>a\<bar> + \<bar>b\<bar>\<bar> = \<bar>a\<bar> + \<bar>b\<bar>" (is "?L = ?R")
  1.1263 +lemma abs_add_abs [simp]: "\<bar>\<bar>a\<bar> + \<bar>b\<bar>\<bar> = \<bar>a\<bar> + \<bar>b\<bar>"
  1.1264 +  (is "?L = ?R")
  1.1265  proof (rule antisym)
  1.1266 -  show "?L \<ge> ?R" by(rule abs_ge_self)
  1.1267 -next
  1.1268 -  have "?L \<le> \<bar>\<bar>a\<bar>\<bar> + \<bar>\<bar>b\<bar>\<bar>" by(rule abs_triangle_ineq)
  1.1269 +  show "?L \<ge> ?R" by (rule abs_ge_self)
  1.1270 +  have "?L \<le> \<bar>\<bar>a\<bar>\<bar> + \<bar>\<bar>b\<bar>\<bar>" by (rule abs_triangle_ineq)
  1.1271    also have "\<dots> = ?R" by simp
  1.1272    finally show "?L \<le> ?R" .
  1.1273  qed
  1.1274 @@ -1327,8 +1273,9 @@
  1.1275  
  1.1276  lemma dense_eq0_I:
  1.1277    fixes x::"'a::{dense_linorder,ordered_ab_group_add_abs}"
  1.1278 -  shows "(\<And>e. 0 < e \<Longrightarrow> \<bar>x\<bar> \<le> e) ==> x = 0"
  1.1279 -  apply (cases "\<bar>x\<bar> = 0", simp)
  1.1280 +  shows "(\<And>e. 0 < e \<Longrightarrow> \<bar>x\<bar> \<le> e) \<Longrightarrow> x = 0"
  1.1281 +  apply (cases "\<bar>x\<bar> = 0")
  1.1282 +  apply simp
  1.1283    apply (simp only: zero_less_abs_iff [symmetric])
  1.1284    apply (drule dense)
  1.1285    apply (auto simp add: not_less [symmetric])
  1.1286 @@ -1336,10 +1283,11 @@
  1.1287  
  1.1288  hide_fact (open) ab_diff_conv_add_uminus add_0 mult_1 ab_left_minus
  1.1289  
  1.1290 -lemmas add_0 = add_0_left \<comment> \<open>FIXME duplicate\<close>
  1.1291 -lemmas mult_1 = mult_1_left \<comment> \<open>FIXME duplicate\<close>
  1.1292 -lemmas ab_left_minus = left_minus \<comment> \<open>FIXME duplicate\<close>
  1.1293 -lemmas diff_diff_eq = diff_diff_add \<comment> \<open>FIXME duplicate\<close>
  1.1294 +lemmas add_0 = add_0_left (* FIXME duplicate *)
  1.1295 +lemmas mult_1 = mult_1_left (* FIXME duplicate *)
  1.1296 +lemmas ab_left_minus = left_minus (* FIXME duplicate *)
  1.1297 +lemmas diff_diff_eq = diff_diff_add (* FIXME duplicate *)
  1.1298 +
  1.1299  
  1.1300  subsection \<open>Canonically ordered monoids\<close>
  1.1301  
  1.1302 @@ -1358,14 +1306,14 @@
  1.1303  lemma not_less_zero[simp]: "\<not> n < 0"
  1.1304    by (auto simp: less_le)
  1.1305  
  1.1306 -lemma zero_less_iff_neq_zero: "(0 < n) \<longleftrightarrow> (n \<noteq> 0)"
  1.1307 +lemma zero_less_iff_neq_zero: "0 < n \<longleftrightarrow> n \<noteq> 0"
  1.1308    by (auto simp: less_le)
  1.1309  
  1.1310  text \<open>This theorem is useful with \<open>blast\<close>\<close>
  1.1311  lemma gr_zeroI: "(n = 0 \<Longrightarrow> False) \<Longrightarrow> 0 < n"
  1.1312    by (rule zero_less_iff_neq_zero[THEN iffD2]) iprover
  1.1313  
  1.1314 -lemma not_gr_zero[simp]: "(\<not> (0 < n)) \<longleftrightarrow> (n = 0)"
  1.1315 +lemma not_gr_zero[simp]: "\<not> 0 < n \<longleftrightarrow> n = 0"
  1.1316    by (simp add: zero_less_iff_neq_zero)
  1.1317  
  1.1318  subclass ordered_comm_monoid_add
  1.1319 @@ -1388,54 +1336,48 @@
  1.1320  
  1.1321  context
  1.1322    fixes a b
  1.1323 -  assumes "a \<le> b"
  1.1324 +  assumes le: "a \<le> b"
  1.1325  begin
  1.1326  
  1.1327 -lemma add_diff_inverse:
  1.1328 -  "a + (b - a) = b"
  1.1329 -  using \<open>a \<le> b\<close> by (auto simp add: le_iff_add)
  1.1330 +lemma add_diff_inverse: "a + (b - a) = b"
  1.1331 +  using le by (auto simp add: le_iff_add)
  1.1332  
  1.1333 -lemma add_diff_assoc:
  1.1334 -  "c + (b - a) = c + b - a"
  1.1335 -  using \<open>a \<le> b\<close> by (auto simp add: le_iff_add add.left_commute [of c])
  1.1336 +lemma add_diff_assoc: "c + (b - a) = c + b - a"
  1.1337 +  using le by (auto simp add: le_iff_add add.left_commute [of c])
  1.1338  
  1.1339 -lemma add_diff_assoc2:
  1.1340 -  "b - a + c = b + c - a"
  1.1341 -  using \<open>a \<le> b\<close> by (auto simp add: le_iff_add add.assoc)
  1.1342 +lemma add_diff_assoc2: "b - a + c = b + c - a"
  1.1343 +  using le by (auto simp add: le_iff_add add.assoc)
  1.1344  
  1.1345 -lemma diff_add_assoc:
  1.1346 -  "c + b - a = c + (b - a)"
  1.1347 -  using \<open>a \<le> b\<close> by (simp add: add.commute add_diff_assoc)
  1.1348 +lemma diff_add_assoc: "c + b - a = c + (b - a)"
  1.1349 +  using le by (simp add: add.commute add_diff_assoc)
  1.1350  
  1.1351 -lemma diff_add_assoc2:
  1.1352 -  "b + c - a = b - a + c"
  1.1353 -  using \<open>a \<le> b\<close>by (simp add: add.commute add_diff_assoc)
  1.1354 +lemma diff_add_assoc2: "b + c - a = b - a + c"
  1.1355 +  using le by (simp add: add.commute add_diff_assoc)
  1.1356  
  1.1357 -lemma diff_diff_right:
  1.1358 -  "c - (b - a) = c + a - b"
  1.1359 +lemma diff_diff_right: "c - (b - a) = c + a - b"
  1.1360    by (simp add: add_diff_inverse add_diff_cancel_left [of a c "b - a", symmetric] add.commute)
  1.1361  
  1.1362 -lemma diff_add:
  1.1363 -  "b - a + a = b"
  1.1364 +lemma diff_add: "b - a + a = b"
  1.1365    by (simp add: add.commute add_diff_inverse)
  1.1366  
  1.1367 -lemma le_add_diff:
  1.1368 -  "c \<le> b + c - a"
  1.1369 +lemma le_add_diff: "c \<le> b + c - a"
  1.1370    by (auto simp add: add.commute diff_add_assoc2 le_iff_add)
  1.1371  
  1.1372 -lemma le_imp_diff_is_add:
  1.1373 -  "a \<le> b \<Longrightarrow> b - a = c \<longleftrightarrow> b = c + a"
  1.1374 +lemma le_imp_diff_is_add: "a \<le> b \<Longrightarrow> b - a = c \<longleftrightarrow> b = c + a"
  1.1375    by (auto simp add: add.commute add_diff_inverse)
  1.1376  
  1.1377 -lemma le_diff_conv2:
  1.1378 -  "c \<le> b - a \<longleftrightarrow> c + a \<le> b" (is "?P \<longleftrightarrow> ?Q")
  1.1379 +lemma le_diff_conv2: "c \<le> b - a \<longleftrightarrow> c + a \<le> b"
  1.1380 +  (is "?P \<longleftrightarrow> ?Q")
  1.1381  proof
  1.1382    assume ?P
  1.1383 -  then have "c + a \<le> b - a + a" by (rule add_right_mono)
  1.1384 -  then show ?Q by (simp add: add_diff_inverse add.commute)
  1.1385 +  then have "c + a \<le> b - a + a"
  1.1386 +    by (rule add_right_mono)
  1.1387 +  then show ?Q
  1.1388 +    by (simp add: add_diff_inverse add.commute)
  1.1389  next
  1.1390    assume ?Q
  1.1391 -  then have "a + c \<le> a + (b - a)" by (simp add: add_diff_inverse add.commute)
  1.1392 +  then have "a + c \<le> a + (b - a)"
  1.1393 +    by (simp add: add_diff_inverse add.commute)
  1.1394    then show ?P by simp
  1.1395  qed
  1.1396  
  1.1397 @@ -1443,6 +1385,7 @@
  1.1398  
  1.1399  end
  1.1400  
  1.1401 +
  1.1402  subsection \<open>Tools setup\<close>
  1.1403  
  1.1404  lemma add_mono_thms_linordered_semiring:
  1.1405 @@ -1451,7 +1394,7 @@
  1.1406      and "i = j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
  1.1407      and "i \<le> j \<and> k = l \<Longrightarrow> i + k \<le> j + l"
  1.1408      and "i = j \<and> k = l \<Longrightarrow> i + k = j + l"
  1.1409 -by (rule add_mono, clarify+)+
  1.1410 +  by (rule add_mono, clarify+)+
  1.1411  
  1.1412  lemma add_mono_thms_linordered_field:
  1.1413    fixes i j k :: "'a::ordered_cancel_ab_semigroup_add"
  1.1414 @@ -1460,8 +1403,8 @@
  1.1415      and "i < j \<and> k \<le> l \<Longrightarrow> i + k < j + l"
  1.1416      and "i \<le> j \<and> k < l \<Longrightarrow> i + k < j + l"
  1.1417      and "i < j \<and> k < l \<Longrightarrow> i + k < j + l"
  1.1418 -by (auto intro: add_strict_right_mono add_strict_left_mono
  1.1419 -  add_less_le_mono add_le_less_mono add_strict_mono)
  1.1420 +  by (auto intro: add_strict_right_mono add_strict_left_mono
  1.1421 +      add_less_le_mono add_le_less_mono add_strict_mono)
  1.1422  
  1.1423  code_identifier
  1.1424    code_module Groups \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
     2.1 --- a/src/HOL/Rings.thy	Mon Jun 20 17:51:47 2016 +0200
     2.2 +++ b/src/HOL/Rings.thy	Mon Jun 20 21:40:48 2016 +0200
     2.3 @@ -18,10 +18,9 @@
     2.4    assumes distrib_left[algebra_simps]: "a * (b + c) = a * b + a * c"
     2.5  begin
     2.6  
     2.7 -text\<open>For the \<open>combine_numerals\<close> simproc\<close>
     2.8 -lemma combine_common_factor:
     2.9 -  "a * e + (b * e + c) = (a + b) * e + c"
    2.10 -by (simp add: distrib_right ac_simps)
    2.11 +text \<open>For the \<open>combine_numerals\<close> simproc\<close>
    2.12 +lemma combine_common_factor: "a * e + (b * e + c) = (a + b) * e + c"
    2.13 +  by (simp add: distrib_right ac_simps)
    2.14  
    2.15  end
    2.16  
    2.17 @@ -30,8 +29,7 @@
    2.18    assumes mult_zero_right [simp]: "a * 0 = 0"
    2.19  begin
    2.20  
    2.21 -lemma mult_not_zero:
    2.22 -  "a * b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<and> b \<noteq> 0"
    2.23 +lemma mult_not_zero: "a * b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<and> b \<noteq> 0"
    2.24    by auto
    2.25  
    2.26  end
    2.27 @@ -45,11 +43,9 @@
    2.28  proof
    2.29    fix a :: 'a
    2.30    have "0 * a + 0 * a = 0 * a + 0" by (simp add: distrib_right [symmetric])
    2.31 -  thus "0 * a = 0" by (simp only: add_left_cancel)
    2.32 -next
    2.33 -  fix a :: 'a
    2.34 +  then show "0 * a = 0" by (simp only: add_left_cancel)
    2.35    have "a * 0 + a * 0 = a * 0 + 0" by (simp add: distrib_left [symmetric])
    2.36 -  thus "a * 0 = 0" by (simp only: add_left_cancel)
    2.37 +  then show "a * 0 = 0" by (simp only: add_left_cancel)
    2.38  qed
    2.39  
    2.40  end
    2.41 @@ -63,8 +59,8 @@
    2.42    fix a b c :: 'a
    2.43    show "(a + b) * c = a * c + b * c" by (simp add: distrib)
    2.44    have "a * (b + c) = (b + c) * a" by (simp add: ac_simps)
    2.45 -  also have "... = b * a + c * a" by (simp only: distrib)
    2.46 -  also have "... = a * b + a * c" by (simp add: ac_simps)
    2.47 +  also have "\<dots> = b * a + c * a" by (simp only: distrib)
    2.48 +  also have "\<dots> = a * b + a * c" by (simp add: ac_simps)
    2.49    finally show "a * (b + c) = a * b + a * c" by blast
    2.50  qed
    2.51  
    2.52 @@ -91,27 +87,23 @@
    2.53  begin
    2.54  
    2.55  lemma one_neq_zero [simp]: "1 \<noteq> 0"
    2.56 -by (rule not_sym) (rule zero_neq_one)
    2.57 +  by (rule not_sym) (rule zero_neq_one)
    2.58  
    2.59  definition of_bool :: "bool \<Rightarrow> 'a"
    2.60 -where
    2.61 -  "of_bool p = (if p then 1 else 0)"
    2.62 +  where "of_bool p = (if p then 1 else 0)"
    2.63  
    2.64  lemma of_bool_eq [simp, code]:
    2.65    "of_bool False = 0"
    2.66    "of_bool True = 1"
    2.67    by (simp_all add: of_bool_def)
    2.68  
    2.69 -lemma of_bool_eq_iff:
    2.70 -  "of_bool p = of_bool q \<longleftrightarrow> p = q"
    2.71 +lemma of_bool_eq_iff: "of_bool p = of_bool q \<longleftrightarrow> p = q"
    2.72    by (simp add: of_bool_def)
    2.73  
    2.74 -lemma split_of_bool [split]:
    2.75 -  "P (of_bool p) \<longleftrightarrow> (p \<longrightarrow> P 1) \<and> (\<not> p \<longrightarrow> P 0)"
    2.76 +lemma split_of_bool [split]: "P (of_bool p) \<longleftrightarrow> (p \<longrightarrow> P 1) \<and> (\<not> p \<longrightarrow> P 0)"
    2.77    by (cases p) simp_all
    2.78  
    2.79 -lemma split_of_bool_asm:
    2.80 -  "P (of_bool p) \<longleftrightarrow> \<not> (p \<and> \<not> P 1 \<or> \<not> p \<and> \<not> P 0)"
    2.81 +lemma split_of_bool_asm: "P (of_bool p) \<longleftrightarrow> \<not> (p \<and> \<not> P 1 \<or> \<not> p \<and> \<not> P 0)"
    2.82    by (cases p) simp_all
    2.83  
    2.84  end
    2.85 @@ -123,8 +115,8 @@
    2.86  class dvd = times
    2.87  begin
    2.88  
    2.89 -definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "dvd" 50) where
    2.90 -  "b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)"
    2.91 +definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "dvd" 50)
    2.92 +  where "b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)"
    2.93  
    2.94  lemma dvdI [intro?]: "a = b * k \<Longrightarrow> b dvd a"
    2.95    unfolding dvd_def ..
    2.96 @@ -139,8 +131,7 @@
    2.97  
    2.98  subclass dvd .
    2.99  
   2.100 -lemma dvd_refl [simp]:
   2.101 -  "a dvd a"
   2.102 +lemma dvd_refl [simp]: "a dvd a"
   2.103  proof
   2.104    show "a = a * 1" by simp
   2.105  qed
   2.106 @@ -155,32 +146,25 @@
   2.107    then show ?thesis ..
   2.108  qed
   2.109  
   2.110 -lemma subset_divisors_dvd:
   2.111 -  "{c. c dvd a} \<subseteq> {c. c dvd b} \<longleftrightarrow> a dvd b"
   2.112 +lemma subset_divisors_dvd: "{c. c dvd a} \<subseteq> {c. c dvd b} \<longleftrightarrow> a dvd b"
   2.113    by (auto simp add: subset_iff intro: dvd_trans)
   2.114  
   2.115 -lemma strict_subset_divisors_dvd:
   2.116 -  "{c. c dvd a} \<subset> {c. c dvd b} \<longleftrightarrow> a dvd b \<and> \<not> b dvd a"
   2.117 +lemma strict_subset_divisors_dvd: "{c. c dvd a} \<subset> {c. c dvd b} \<longleftrightarrow> a dvd b \<and> \<not> b dvd a"
   2.118    by (auto simp add: subset_iff intro: dvd_trans)
   2.119  
   2.120 -lemma one_dvd [simp]:
   2.121 -  "1 dvd a"
   2.122 +lemma one_dvd [simp]: "1 dvd a"
   2.123    by (auto intro!: dvdI)
   2.124  
   2.125 -lemma dvd_mult [simp]:
   2.126 -  "a dvd c \<Longrightarrow> a dvd (b * c)"
   2.127 +lemma dvd_mult [simp]: "a dvd c \<Longrightarrow> a dvd (b * c)"
   2.128    by (auto intro!: mult.left_commute dvdI elim!: dvdE)
   2.129  
   2.130 -lemma dvd_mult2 [simp]:
   2.131 -  "a dvd b \<Longrightarrow> a dvd (b * c)"
   2.132 +lemma dvd_mult2 [simp]: "a dvd b \<Longrightarrow> a dvd (b * c)"
   2.133    using dvd_mult [of a b c] by (simp add: ac_simps)
   2.134  
   2.135 -lemma dvd_triv_right [simp]:
   2.136 -  "a dvd b * a"
   2.137 +lemma dvd_triv_right [simp]: "a dvd b * a"
   2.138    by (rule dvd_mult) (rule dvd_refl)
   2.139  
   2.140 -lemma dvd_triv_left [simp]:
   2.141 -  "a dvd a * b"
   2.142 +lemma dvd_triv_left [simp]: "a dvd a * b"
   2.143    by (rule dvd_mult2) (rule dvd_refl)
   2.144  
   2.145  lemma mult_dvd_mono:
   2.146 @@ -194,12 +178,10 @@
   2.147    then show ?thesis ..
   2.148  qed
   2.149  
   2.150 -lemma dvd_mult_left:
   2.151 -  "a * b dvd c \<Longrightarrow> a dvd c"
   2.152 +lemma dvd_mult_left: "a * b dvd c \<Longrightarrow> a dvd c"
   2.153    by (simp add: dvd_def mult.assoc) blast
   2.154  
   2.155 -lemma dvd_mult_right:
   2.156 -  "a * b dvd c \<Longrightarrow> b dvd c"
   2.157 +lemma dvd_mult_right: "a * b dvd c \<Longrightarrow> b dvd c"
   2.158    using dvd_mult_left [of b a c] by (simp add: ac_simps)
   2.159  
   2.160  end
   2.161 @@ -209,18 +191,15 @@
   2.162  
   2.163  subclass semiring_1 ..
   2.164  
   2.165 -lemma dvd_0_left_iff [simp]:
   2.166 -  "0 dvd a \<longleftrightarrow> a = 0"
   2.167 +lemma dvd_0_left_iff [simp]: "0 dvd a \<longleftrightarrow> a = 0"
   2.168    by (auto intro: dvd_refl elim!: dvdE)
   2.169  
   2.170 -lemma dvd_0_right [iff]:
   2.171 -  "a dvd 0"
   2.172 +lemma dvd_0_right [iff]: "a dvd 0"
   2.173  proof
   2.174    show "0 = a * 0" by simp
   2.175  qed
   2.176  
   2.177 -lemma dvd_0_left:
   2.178 -  "0 dvd a \<Longrightarrow> a = 0"
   2.179 +lemma dvd_0_left: "0 dvd a \<Longrightarrow> a = 0"
   2.180    by simp
   2.181  
   2.182  lemma dvd_add [simp]:
   2.183 @@ -245,8 +224,8 @@
   2.184  
   2.185  end
   2.186  
   2.187 -class comm_semiring_1_cancel = comm_semiring + cancel_comm_monoid_add +
   2.188 -                               zero_neq_one + comm_monoid_mult +
   2.189 +class comm_semiring_1_cancel =
   2.190 +  comm_semiring + cancel_comm_monoid_add + zero_neq_one + comm_monoid_mult +
   2.191    assumes right_diff_distrib' [algebra_simps]: "a * (b - c) = a * b - a * c"
   2.192  begin
   2.193  
   2.194 @@ -254,16 +233,15 @@
   2.195  subclass comm_semiring_0_cancel ..
   2.196  subclass comm_semiring_1 ..
   2.197  
   2.198 -lemma left_diff_distrib' [algebra_simps]:
   2.199 -  "(b - c) * a = b * a - c * a"
   2.200 +lemma left_diff_distrib' [algebra_simps]: "(b - c) * a = b * a - c * a"
   2.201    by (simp add: algebra_simps)
   2.202  
   2.203 -lemma dvd_add_times_triv_left_iff [simp]:
   2.204 -  "a dvd c * a + b \<longleftrightarrow> a dvd b"
   2.205 +lemma dvd_add_times_triv_left_iff [simp]: "a dvd c * a + b \<longleftrightarrow> a dvd b"
   2.206  proof -
   2.207    have "a dvd a * c + b \<longleftrightarrow> a dvd b" (is "?P \<longleftrightarrow> ?Q")
   2.208    proof
   2.209 -    assume ?Q then show ?P by simp
   2.210 +    assume ?Q
   2.211 +    then show ?P by simp
   2.212    next
   2.213      assume ?P
   2.214      then obtain d where "a * c + b = a * d" ..
   2.215 @@ -275,23 +253,21 @@
   2.216    then show "a dvd c * a + b \<longleftrightarrow> a dvd b" by (simp add: ac_simps)
   2.217  qed
   2.218  
   2.219 -lemma dvd_add_times_triv_right_iff [simp]:
   2.220 -  "a dvd b + c * a \<longleftrightarrow> a dvd b"
   2.221 +lemma dvd_add_times_triv_right_iff [simp]: "a dvd b + c * a \<longleftrightarrow> a dvd b"
   2.222    using dvd_add_times_triv_left_iff [of a c b] by (simp add: ac_simps)
   2.223  
   2.224 -lemma dvd_add_triv_left_iff [simp]:
   2.225 -  "a dvd a + b \<longleftrightarrow> a dvd b"
   2.226 +lemma dvd_add_triv_left_iff [simp]: "a dvd a + b \<longleftrightarrow> a dvd b"
   2.227    using dvd_add_times_triv_left_iff [of a 1 b] by simp
   2.228  
   2.229 -lemma dvd_add_triv_right_iff [simp]:
   2.230 -  "a dvd b + a \<longleftrightarrow> a dvd b"
   2.231 +lemma dvd_add_triv_right_iff [simp]: "a dvd b + a \<longleftrightarrow> a dvd b"
   2.232    using dvd_add_times_triv_right_iff [of a b 1] by simp
   2.233  
   2.234  lemma dvd_add_right_iff:
   2.235    assumes "a dvd b"
   2.236    shows "a dvd b + c \<longleftrightarrow> a dvd c" (is "?P \<longleftrightarrow> ?Q")
   2.237  proof
   2.238 -  assume ?P then obtain d where "b + c = a * d" ..
   2.239 +  assume ?P
   2.240 +  then obtain d where "b + c = a * d" ..
   2.241    moreover from \<open>a dvd b\<close> obtain e where "b = a * e" ..
   2.242    ultimately have "a * e + c = a * d" by simp
   2.243    then have "a * e + c - a * e = a * d - a * e" by simp
   2.244 @@ -299,13 +275,12 @@
   2.245    then have "c = a * (d - e)" by (simp add: algebra_simps)
   2.246    then show ?Q ..
   2.247  next
   2.248 -  assume ?Q with assms show ?P by simp
   2.249 +  assume ?Q
   2.250 +  with assms show ?P by simp
   2.251  qed
   2.252  
   2.253 -lemma dvd_add_left_iff:
   2.254 -  assumes "a dvd c"
   2.255 -  shows "a dvd b + c \<longleftrightarrow> a dvd b"
   2.256 -  using assms dvd_add_right_iff [of a c b] by (simp add: ac_simps)
   2.257 +lemma dvd_add_left_iff: "a dvd c \<Longrightarrow> a dvd b + c \<longleftrightarrow> a dvd b"
   2.258 +  using dvd_add_right_iff [of a c b] by (simp add: ac_simps)
   2.259  
   2.260  end
   2.261  
   2.262 @@ -317,44 +292,38 @@
   2.263  text \<open>Distribution rules\<close>
   2.264  
   2.265  lemma minus_mult_left: "- (a * b) = - a * b"
   2.266 -by (rule minus_unique) (simp add: distrib_right [symmetric])
   2.267 +  by (rule minus_unique) (simp add: distrib_right [symmetric])
   2.268  
   2.269  lemma minus_mult_right: "- (a * b) = a * - b"
   2.270 -by (rule minus_unique) (simp add: distrib_left [symmetric])
   2.271 +  by (rule minus_unique) (simp add: distrib_left [symmetric])
   2.272  
   2.273 -text\<open>Extract signs from products\<close>
   2.274 +text \<open>Extract signs from products\<close>
   2.275  lemmas mult_minus_left [simp] = minus_mult_left [symmetric]
   2.276  lemmas mult_minus_right [simp] = minus_mult_right [symmetric]
   2.277  
   2.278  lemma minus_mult_minus [simp]: "- a * - b = a * b"
   2.279 -by simp
   2.280 +  by simp
   2.281  
   2.282  lemma minus_mult_commute: "- a * b = a * - b"
   2.283 -by simp
   2.284 +  by simp
   2.285  
   2.286 -lemma right_diff_distrib [algebra_simps]:
   2.287 -  "a * (b - c) = a * b - a * c"
   2.288 +lemma right_diff_distrib [algebra_simps]: "a * (b - c) = a * b - a * c"
   2.289    using distrib_left [of a b "-c "] by simp
   2.290  
   2.291 -lemma left_diff_distrib [algebra_simps]:
   2.292 -  "(a - b) * c = a * c - b * c"
   2.293 +lemma left_diff_distrib [algebra_simps]: "(a - b) * c = a * c - b * c"
   2.294    using distrib_right [of a "- b" c] by simp
   2.295  
   2.296 -lemmas ring_distribs =
   2.297 -  distrib_left distrib_right left_diff_distrib right_diff_distrib
   2.298 +lemmas ring_distribs = distrib_left distrib_right left_diff_distrib right_diff_distrib
   2.299  
   2.300 -lemma eq_add_iff1:
   2.301 -  "a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d"
   2.302 -by (simp add: algebra_simps)
   2.303 +lemma eq_add_iff1: "a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d"
   2.304 +  by (simp add: algebra_simps)
   2.305  
   2.306 -lemma eq_add_iff2:
   2.307 -  "a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d"
   2.308 -by (simp add: algebra_simps)
   2.309 +lemma eq_add_iff2: "a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d"
   2.310 +  by (simp add: algebra_simps)
   2.311  
   2.312  end
   2.313  
   2.314 -lemmas ring_distribs =
   2.315 -  distrib_left distrib_right left_diff_distrib right_diff_distrib
   2.316 +lemmas ring_distribs = distrib_left distrib_right left_diff_distrib right_diff_distrib
   2.317  
   2.318  class comm_ring = comm_semiring + ab_group_add
   2.319  begin
   2.320 @@ -362,8 +331,7 @@
   2.321  subclass ring ..
   2.322  subclass comm_semiring_0_cancel ..
   2.323  
   2.324 -lemma square_diff_square_factored:
   2.325 -  "x * x - y * y = (x + y) * (x - y)"
   2.326 +lemma square_diff_square_factored: "x * x - y * y = (x + y) * (x - y)"
   2.327    by (simp add: algebra_simps)
   2.328  
   2.329  end
   2.330 @@ -373,8 +341,7 @@
   2.331  
   2.332  subclass semiring_1_cancel ..
   2.333  
   2.334 -lemma square_diff_one_factored:
   2.335 -  "x * x - 1 = (x + 1) * (x - 1)"
   2.336 +lemma square_diff_one_factored: "x * x - 1 = (x + 1) * (x - 1)"
   2.337    by (simp add: algebra_simps)
   2.338  
   2.339  end
   2.340 @@ -410,8 +377,7 @@
   2.341    then show "- x dvd y" ..
   2.342  qed
   2.343  
   2.344 -lemma dvd_diff [simp]:
   2.345 -  "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)"
   2.346 +lemma dvd_diff [simp]: "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)"
   2.347    using dvd_add [of x y "- z"] by simp
   2.348  
   2.349  end
   2.350 @@ -424,19 +390,20 @@
   2.351    assumes "a * b = 0"
   2.352    shows "a = 0 \<or> b = 0"
   2.353  proof (rule classical)
   2.354 -  assume "\<not> (a = 0 \<or> b = 0)"
   2.355 +  assume "\<not> ?thesis"
   2.356    then have "a \<noteq> 0" and "b \<noteq> 0" by auto
   2.357    with no_zero_divisors have "a * b \<noteq> 0" by blast
   2.358    with assms show ?thesis by simp
   2.359  qed
   2.360  
   2.361 -lemma mult_eq_0_iff [simp]:
   2.362 -  shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
   2.363 +lemma mult_eq_0_iff [simp]: "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
   2.364  proof (cases "a = 0 \<or> b = 0")
   2.365 -  case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
   2.366 +  case False
   2.367 +  then have "a \<noteq> 0" and "b \<noteq> 0" by auto
   2.368      then show ?thesis using no_zero_divisors by simp
   2.369  next
   2.370 -  case True then show ?thesis by auto
   2.371 +  case True
   2.372 +  then show ?thesis by auto
   2.373  qed
   2.374  
   2.375  end
   2.376 @@ -448,12 +415,10 @@
   2.377      and mult_cancel_left [simp]: "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
   2.378  begin
   2.379  
   2.380 -lemma mult_left_cancel:
   2.381 -  "c \<noteq> 0 \<Longrightarrow> c * a = c * b \<longleftrightarrow> a = b"
   2.382 +lemma mult_left_cancel: "c \<noteq> 0 \<Longrightarrow> c * a = c * b \<longleftrightarrow> a = b"
   2.383    by simp
   2.384  
   2.385 -lemma mult_right_cancel:
   2.386 -  "c \<noteq> 0 \<Longrightarrow> a * c = b * c \<longleftrightarrow> a = b"
   2.387 +lemma mult_right_cancel: "c \<noteq> 0 \<Longrightarrow> a * c = b * c \<longleftrightarrow> a = b"
   2.388    by simp
   2.389  
   2.390  end
   2.391 @@ -483,32 +448,27 @@
   2.392  
   2.393  subclass semiring_1_no_zero_divisors ..
   2.394  
   2.395 -lemma square_eq_1_iff:
   2.396 -  "x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1"
   2.397 +lemma square_eq_1_iff: "x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1"
   2.398  proof -
   2.399    have "(x - 1) * (x + 1) = x * x - 1"
   2.400      by (simp add: algebra_simps)
   2.401 -  hence "x * x = 1 \<longleftrightarrow> (x - 1) * (x + 1) = 0"
   2.402 +  then have "x * x = 1 \<longleftrightarrow> (x - 1) * (x + 1) = 0"
   2.403      by simp
   2.404 -  thus ?thesis
   2.405 +  then show ?thesis
   2.406      by (simp add: eq_neg_iff_add_eq_0)
   2.407  qed
   2.408  
   2.409 -lemma mult_cancel_right1 [simp]:
   2.410 -  "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
   2.411 -by (insert mult_cancel_right [of 1 c b], force)
   2.412 +lemma mult_cancel_right1 [simp]: "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
   2.413 +  using mult_cancel_right [of 1 c b] by auto
   2.414  
   2.415 -lemma mult_cancel_right2 [simp]:
   2.416 -  "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
   2.417 -by (insert mult_cancel_right [of a c 1], simp)
   2.418 +lemma mult_cancel_right2 [simp]: "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
   2.419 +  using mult_cancel_right [of a c 1] by simp
   2.420  
   2.421 -lemma mult_cancel_left1 [simp]:
   2.422 -  "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
   2.423 -by (insert mult_cancel_left [of c 1 b], force)
   2.424 +lemma mult_cancel_left1 [simp]: "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
   2.425 +  using mult_cancel_left [of c 1 b] by force
   2.426  
   2.427 -lemma mult_cancel_left2 [simp]:
   2.428 -  "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
   2.429 -by (insert mult_cancel_left [of c a 1], simp)
   2.430 +lemma mult_cancel_left2 [simp]: "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
   2.431 +  using mult_cancel_left [of c a 1] by simp
   2.432  
   2.433  end
   2.434  
   2.435 @@ -526,8 +486,7 @@
   2.436  
   2.437  subclass ring_1_no_zero_divisors ..
   2.438  
   2.439 -lemma dvd_mult_cancel_right [simp]:
   2.440 -  "a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b"
   2.441 +lemma dvd_mult_cancel_right [simp]: "a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b"
   2.442  proof -
   2.443    have "a * c dvd b * c \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
   2.444      unfolding dvd_def by (simp add: ac_simps)
   2.445 @@ -536,8 +495,7 @@
   2.446    finally show ?thesis .
   2.447  qed
   2.448  
   2.449 -lemma dvd_mult_cancel_left [simp]:
   2.450 -  "c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b"
   2.451 +lemma dvd_mult_cancel_left [simp]: "c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b"
   2.452  proof -
   2.453    have "c * a dvd c * b \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
   2.454      unfolding dvd_def by (simp add: ac_simps)
   2.455 @@ -562,15 +520,12 @@
   2.456  
   2.457  text \<open>
   2.458    The theory of partially ordered rings is taken from the books:
   2.459 -  \begin{itemize}
   2.460 -  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979
   2.461 -  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
   2.462 -  \end{itemize}
   2.463 +    \<^item> \<^emph>\<open>Lattice Theory\<close> by Garret Birkhoff, American Mathematical Society, 1979
   2.464 +    \<^item> \<^emph>\<open>Partially Ordered Algebraic Systems\<close>, Pergamon Press, 1963
   2.465 +
   2.466    Most of the used notions can also be looked up in
   2.467 -  \begin{itemize}
   2.468 -  \item @{url "http://www.mathworld.com"} by Eric Weisstein et. al.
   2.469 -  \item \emph{Algebra I} by van der Waerden, Springer.
   2.470 -  \end{itemize}
   2.471 +    \<^item> @{url "http://www.mathworld.com"} by Eric Weisstein et. al.
   2.472 +    \<^item> \<^emph>\<open>Algebra I\<close> by van der Waerden, Springer
   2.473  \<close>
   2.474  
   2.475  class divide =
   2.476 @@ -605,49 +560,45 @@
   2.477    assumes divide_zero [simp]: "a div 0 = 0"
   2.478  begin
   2.479  
   2.480 -lemma nonzero_mult_divide_cancel_left [simp]:
   2.481 -  "a \<noteq> 0 \<Longrightarrow> (a * b) div a = b"
   2.482 +lemma nonzero_mult_divide_cancel_left [simp]: "a \<noteq> 0 \<Longrightarrow> (a * b) div a = b"
   2.483    using nonzero_mult_divide_cancel_right [of a b] by (simp add: ac_simps)
   2.484  
   2.485  subclass semiring_no_zero_divisors_cancel
   2.486  proof
   2.487 -  fix a b c
   2.488 -  { fix a b c
   2.489 -    show "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
   2.490 -    proof (cases "c = 0")
   2.491 -      case True then show ?thesis by simp
   2.492 -    next
   2.493 -      case False
   2.494 -      { assume "a * c = b * c"
   2.495 -        then have "a * c div c = b * c div c"
   2.496 -          by simp
   2.497 -        with False have "a = b"
   2.498 -          by simp
   2.499 -      } then show ?thesis by auto
   2.500 -    qed
   2.501 -  }
   2.502 -  from this [of a c b]
   2.503 -  show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
   2.504 -    by (simp add: ac_simps)
   2.505 +  show *: "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b" for a b c
   2.506 +  proof (cases "c = 0")
   2.507 +    case True
   2.508 +    then show ?thesis by simp
   2.509 +  next
   2.510 +    case False
   2.511 +    {
   2.512 +      assume "a * c = b * c"
   2.513 +      then have "a * c div c = b * c div c"
   2.514 +        by simp
   2.515 +      with False have "a = b"
   2.516 +        by simp
   2.517 +    }
   2.518 +    then show ?thesis by auto
   2.519 +  qed
   2.520 +  show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b" for a b c
   2.521 +    using * [of a c b] by (simp add: ac_simps)
   2.522  qed
   2.523  
   2.524 -lemma div_self [simp]:
   2.525 -  assumes "a \<noteq> 0"
   2.526 -  shows "a div a = 1"
   2.527 -  using assms nonzero_mult_divide_cancel_left [of a 1] by simp
   2.528 +lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1"
   2.529 +  using nonzero_mult_divide_cancel_left [of a 1] by simp
   2.530  
   2.531 -lemma divide_zero_left [simp]:
   2.532 -  "0 div a = 0"
   2.533 +lemma divide_zero_left [simp]: "0 div a = 0"
   2.534  proof (cases "a = 0")
   2.535 -  case True then show ?thesis by simp
   2.536 +  case True
   2.537 +  then show ?thesis by simp
   2.538  next
   2.539 -  case False then have "a * 0 div a = 0"
   2.540 +  case False
   2.541 +  then have "a * 0 div a = 0"
   2.542      by (rule nonzero_mult_divide_cancel_left)
   2.543    then show ?thesis by simp
   2.544  qed
   2.545  
   2.546 -lemma divide_1 [simp]:
   2.547 -  "a div 1 = a"
   2.548 +lemma divide_1 [simp]: "a div 1 = a"
   2.549    using nonzero_mult_divide_cancel_left [of 1 a] by simp
   2.550  
   2.551  end
   2.552 @@ -668,11 +619,13 @@
   2.553    assumes "a \<noteq> 0"
   2.554    shows "a * b dvd a * c \<longleftrightarrow> b dvd c" (is "?P \<longleftrightarrow> ?Q")
   2.555  proof
   2.556 -  assume ?P then obtain d where "a * c = a * b * d" ..
   2.557 +  assume ?P
   2.558 +  then obtain d where "a * c = a * b * d" ..
   2.559    with assms have "c = b * d" by (simp add: ac_simps)
   2.560    then show ?Q ..
   2.561  next
   2.562 -  assume ?Q then obtain d where "c = b * d" ..
   2.563 +  assume ?Q
   2.564 +  then obtain d where "c = b * d" ..
   2.565    then have "a * c = a * b * d" by (simp add: ac_simps)
   2.566    then show ?P ..
   2.567  qed
   2.568 @@ -680,7 +633,7 @@
   2.569  lemma dvd_times_right_cancel_iff [simp]:
   2.570    assumes "a \<noteq> 0"
   2.571    shows "b * a dvd c * a \<longleftrightarrow> b dvd c" (is "?P \<longleftrightarrow> ?Q")
   2.572 -using dvd_times_left_cancel_iff [of a b c] assms by (simp add: ac_simps)
   2.573 +  using dvd_times_left_cancel_iff [of a b c] assms by (simp add: ac_simps)
   2.574  
   2.575  lemma div_dvd_iff_mult:
   2.576    assumes "b \<noteq> 0" and "b dvd a"
   2.577 @@ -702,7 +655,8 @@
   2.578    assumes "a dvd b" and "a dvd c"
   2.579    shows "b div a dvd c div a \<longleftrightarrow> b dvd c"
   2.580  proof (cases "a = 0")
   2.581 -  case True with assms show ?thesis by simp
   2.582 +  case True
   2.583 +  with assms show ?thesis by simp
   2.584  next
   2.585    case False
   2.586    moreover from assms obtain k l where "b = a * k" and "c = a * l"
   2.587 @@ -714,7 +668,8 @@
   2.588    assumes "c dvd a" and "c dvd b"
   2.589    shows "(a + b) div c = a div c + b div c"
   2.590  proof (cases "c = 0")
   2.591 -  case True then show ?thesis by simp
   2.592 +  case True
   2.593 +  then show ?thesis by simp
   2.594  next
   2.595    case False
   2.596    moreover from assms obtain k l where "a = c * k" and "b = c * l"
   2.597 @@ -729,7 +684,8 @@
   2.598    assumes "b dvd a" and "d dvd c"
   2.599    shows "(a div b) * (c div d) = (a * c) div (b * d)"
   2.600  proof (cases "b = 0 \<or> c = 0")
   2.601 -  case True with assms show ?thesis by auto
   2.602 +  case True
   2.603 +  with assms show ?thesis by auto
   2.604  next
   2.605    case False
   2.606    moreover from assms obtain k l where "a = b * k" and "c = d * l"
   2.607 @@ -748,42 +704,39 @@
   2.608  next
   2.609    assume "b div a = c"
   2.610    then have "b div a * a = c * a" by simp
   2.611 -  moreover from \<open>a \<noteq> 0\<close> \<open>a dvd b\<close> have "b div a * a = b"
   2.612 +  moreover from assms have "b div a * a = b"
   2.613      by (auto elim!: dvdE simp add: ac_simps)
   2.614    ultimately show "b = c * a" by simp
   2.615  qed
   2.616  
   2.617 -lemma dvd_div_mult_self [simp]:
   2.618 -  "a dvd b \<Longrightarrow> b div a * a = b"
   2.619 +lemma dvd_div_mult_self [simp]: "a dvd b \<Longrightarrow> b div a * a = b"
   2.620    by (cases "a = 0") (auto elim: dvdE simp add: ac_simps)
   2.621  
   2.622 -lemma dvd_mult_div_cancel [simp]:
   2.623 -  "a dvd b \<Longrightarrow> a * (b div a) = b"
   2.624 +lemma dvd_mult_div_cancel [simp]: "a dvd b \<Longrightarrow> a * (b div a) = b"
   2.625    using dvd_div_mult_self [of a b] by (simp add: ac_simps)
   2.626  
   2.627  lemma div_mult_swap:
   2.628    assumes "c dvd b"
   2.629    shows "a * (b div c) = (a * b) div c"
   2.630  proof (cases "c = 0")
   2.631 -  case True then show ?thesis by simp
   2.632 +  case True
   2.633 +  then show ?thesis by simp
   2.634  next
   2.635 -  case False from assms obtain d where "b = c * d" ..
   2.636 +  case False
   2.637 +  from assms obtain d where "b = c * d" ..
   2.638    moreover from False have "a * divide (d * c) c = ((a * d) * c) div c"
   2.639      by simp
   2.640    ultimately show ?thesis by (simp add: ac_simps)
   2.641  qed
   2.642  
   2.643 -lemma dvd_div_mult:
   2.644 -  assumes "c dvd b"
   2.645 -  shows "b div c * a = (b * a) div c"
   2.646 -  using assms div_mult_swap [of c b a] by (simp add: ac_simps)
   2.647 +lemma dvd_div_mult: "c dvd b \<Longrightarrow> b div c * a = (b * a) div c"
   2.648 +  using div_mult_swap [of c b a] by (simp add: ac_simps)
   2.649  
   2.650  lemma dvd_div_mult2_eq:
   2.651    assumes "b * c dvd a"
   2.652    shows "a div (b * c) = a div b div c"
   2.653 -using assms proof
   2.654 -  fix k
   2.655 -  assume "a = b * c * k"
   2.656 +proof -
   2.657 +  from assms obtain k where "a = b * c * k" ..
   2.658    then show ?thesis
   2.659      by (cases "b = 0 \<or> c = 0") (auto, simp add: ac_simps)
   2.660  qed
   2.661 @@ -808,15 +761,12 @@
   2.662  text \<open>Units: invertible elements in a ring\<close>
   2.663  
   2.664  abbreviation is_unit :: "'a \<Rightarrow> bool"
   2.665 -where
   2.666 -  "is_unit a \<equiv> a dvd 1"
   2.667 +  where "is_unit a \<equiv> a dvd 1"
   2.668  
   2.669 -lemma not_is_unit_0 [simp]:
   2.670 -  "\<not> is_unit 0"
   2.671 +lemma not_is_unit_0 [simp]: "\<not> is_unit 0"
   2.672    by simp
   2.673  
   2.674 -lemma unit_imp_dvd [dest]:
   2.675 -  "is_unit b \<Longrightarrow> b dvd a"
   2.676 +lemma unit_imp_dvd [dest]: "is_unit b \<Longrightarrow> b dvd a"
   2.677    by (rule dvd_trans [of _ 1]) simp_all
   2.678  
   2.679  lemma unit_dvdE:
   2.680 @@ -829,8 +779,7 @@
   2.681    ultimately show thesis using that by blast
   2.682  qed
   2.683  
   2.684 -lemma dvd_unit_imp_unit:
   2.685 -  "a dvd b \<Longrightarrow> is_unit b \<Longrightarrow> is_unit a"
   2.686 +lemma dvd_unit_imp_unit: "a dvd b \<Longrightarrow> is_unit b \<Longrightarrow> is_unit a"
   2.687    by (rule dvd_trans)
   2.688  
   2.689  lemma unit_div_1_unit [simp, intro]:
   2.690 @@ -849,27 +798,24 @@
   2.691  proof (rule that)
   2.692    define b where "b = 1 div a"
   2.693    then show "1 div a = b" by simp
   2.694 -  from b_def \<open>is_unit a\<close> show "is_unit b" by simp
   2.695 -  from \<open>is_unit a\<close> and \<open>is_unit b\<close> show "a \<noteq> 0" and "b \<noteq> 0" by auto
   2.696 -  from b_def \<open>is_unit a\<close> show "a * b = 1" by simp
   2.697 +  from assms b_def show "is_unit b" by simp
   2.698 +  with assms show "a \<noteq> 0" and "b \<noteq> 0" by auto
   2.699 +  from assms b_def show "a * b = 1" by simp
   2.700    then have "1 = a * b" ..
   2.701    with b_def \<open>b \<noteq> 0\<close> show "1 div b = a" by simp
   2.702 -  from \<open>is_unit a\<close> have "a dvd c" ..
   2.703 +  from assms have "a dvd c" ..
   2.704    then obtain d where "c = a * d" ..
   2.705    with \<open>a \<noteq> 0\<close> \<open>a * b = 1\<close> show "c div a = c * b"
   2.706      by (simp add: mult.assoc mult.left_commute [of a])
   2.707  qed
   2.708  
   2.709 -lemma unit_prod [intro]:
   2.710 -  "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a * b)"
   2.711 +lemma unit_prod [intro]: "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a * b)"
   2.712    by (subst mult_1_left [of 1, symmetric]) (rule mult_dvd_mono)
   2.713  
   2.714 -lemma is_unit_mult_iff:
   2.715 -  "is_unit (a * b) \<longleftrightarrow> is_unit a \<and> is_unit b" (is "?P \<longleftrightarrow> ?Q")
   2.716 +lemma is_unit_mult_iff: "is_unit (a * b) \<longleftrightarrow> is_unit a \<and> is_unit b"
   2.717    by (auto dest: dvd_mult_left dvd_mult_right)
   2.718  
   2.719 -lemma unit_div [intro]:
   2.720 -  "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a div b)"
   2.721 +lemma unit_div [intro]: "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a div b)"
   2.722    by (erule is_unitE [of b a]) (simp add: ac_simps unit_prod)
   2.723  
   2.724  lemma mult_unit_dvd_iff:
   2.725 @@ -894,7 +840,8 @@
   2.726    assume "a dvd c * b"
   2.727    with assms have "c * b dvd c * (b * (1 div b))"
   2.728      by (subst mult_assoc [symmetric]) simp
   2.729 -  also from \<open>is_unit b\<close> have "b * (1 div b) = 1" by (rule is_unitE) simp
   2.730 +  also from assms have "b * (1 div b) = 1"
   2.731 +    by (rule is_unitE) simp
   2.732    finally have "c * b dvd c" by simp
   2.733    with \<open>a dvd c * b\<close> show "a dvd c" by (rule dvd_trans)
   2.734  next
   2.735 @@ -902,52 +849,40 @@
   2.736    then show "a dvd c * b" by simp
   2.737  qed
   2.738  
   2.739 -lemma div_unit_dvd_iff:
   2.740 -  "is_unit b \<Longrightarrow> a div b dvd c \<longleftrightarrow> a dvd c"
   2.741 +lemma div_unit_dvd_iff: "is_unit b \<Longrightarrow> a div b dvd c \<longleftrightarrow> a dvd c"
   2.742    by (erule is_unitE [of _ a]) (auto simp add: mult_unit_dvd_iff)
   2.743  
   2.744 -lemma dvd_div_unit_iff:
   2.745 -  "is_unit b \<Longrightarrow> a dvd c div b \<longleftrightarrow> a dvd c"
   2.746 +lemma dvd_div_unit_iff: "is_unit b \<Longrightarrow> a dvd c div b \<longleftrightarrow> a dvd c"
   2.747    by (erule is_unitE [of _ c]) (simp add: dvd_mult_unit_iff)
   2.748  
   2.749  lemmas unit_dvd_iff = mult_unit_dvd_iff div_unit_dvd_iff
   2.750 -  dvd_mult_unit_iff dvd_div_unit_iff \<comment> \<open>FIXME consider fact collection\<close>
   2.751 +  dvd_mult_unit_iff dvd_div_unit_iff  (* FIXME consider named_theorems *)
   2.752  
   2.753 -lemma unit_mult_div_div [simp]:
   2.754 -  "is_unit a \<Longrightarrow> b * (1 div a) = b div a"
   2.755 +lemma unit_mult_div_div [simp]: "is_unit a \<Longrightarrow> b * (1 div a) = b div a"
   2.756    by (erule is_unitE [of _ b]) simp
   2.757  
   2.758 -lemma unit_div_mult_self [simp]:
   2.759 -  "is_unit a \<Longrightarrow> b div a * a = b"
   2.760 +lemma unit_div_mult_self [simp]: "is_unit a \<Longrightarrow> b div a * a = b"
   2.761    by (rule dvd_div_mult_self) auto
   2.762  
   2.763 -lemma unit_div_1_div_1 [simp]:
   2.764 -  "is_unit a \<Longrightarrow> 1 div (1 div a) = a"
   2.765 +lemma unit_div_1_div_1 [simp]: "is_unit a \<Longrightarrow> 1 div (1 div a) = a"
   2.766    by (erule is_unitE) simp
   2.767  
   2.768 -lemma unit_div_mult_swap:
   2.769 -  "is_unit c \<Longrightarrow> a * (b div c) = (a * b) div c"
   2.770 +lemma unit_div_mult_swap: "is_unit c \<Longrightarrow> a * (b div c) = (a * b) div c"
   2.771    by (erule unit_dvdE [of _ b]) (simp add: mult.left_commute [of _ c])
   2.772  
   2.773 -lemma unit_div_commute:
   2.774 -  "is_unit b \<Longrightarrow> (a div b) * c = (a * c) div b"
   2.775 +lemma unit_div_commute: "is_unit b \<Longrightarrow> (a div b) * c = (a * c) div b"
   2.776    using unit_div_mult_swap [of b c a] by (simp add: ac_simps)
   2.777  
   2.778 -lemma unit_eq_div1:
   2.779 -  "is_unit b \<Longrightarrow> a div b = c \<longleftrightarrow> a = c * b"
   2.780 +lemma unit_eq_div1: "is_unit b \<Longrightarrow> a div b = c \<longleftrightarrow> a = c * b"
   2.781    by (auto elim: is_unitE)
   2.782  
   2.783 -lemma unit_eq_div2:
   2.784 -  "is_unit b \<Longrightarrow> a = c div b \<longleftrightarrow> a * b = c"
   2.785 +lemma unit_eq_div2: "is_unit b \<Longrightarrow> a = c div b \<longleftrightarrow> a * b = c"
   2.786    using unit_eq_div1 [of b c a] by auto
   2.787  
   2.788 -lemma unit_mult_left_cancel:
   2.789 -  assumes "is_unit a"
   2.790 -  shows "a * b = a * c \<longleftrightarrow> b = c" (is "?P \<longleftrightarrow> ?Q")
   2.791 -  using assms mult_cancel_left [of a b c] by auto
   2.792 +lemma unit_mult_left_cancel: "is_unit a \<Longrightarrow> a * b = a * c \<longleftrightarrow> b = c"
   2.793 +  using mult_cancel_left [of a b c] by auto
   2.794  
   2.795 -lemma unit_mult_right_cancel:
   2.796 -  "is_unit a \<Longrightarrow> b * a = c * a \<longleftrightarrow> b = c"
   2.797 +lemma unit_mult_right_cancel: "is_unit a \<Longrightarrow> b * a = c * a \<longleftrightarrow> b = c"
   2.798    using unit_mult_left_cancel [of a b c] by (auto simp add: ac_simps)
   2.799  
   2.800  lemma unit_div_cancel:
   2.801 @@ -964,7 +899,8 @@
   2.802    assumes "is_unit b" and "is_unit c"
   2.803    shows "a div (b * c) = a div b div c"
   2.804  proof -
   2.805 -  from assms have "is_unit (b * c)" by (simp add: unit_prod)
   2.806 +  from assms have "is_unit (b * c)"
   2.807 +    by (simp add: unit_prod)
   2.808    then have "b * c dvd a"
   2.809      by (rule unit_imp_dvd)
   2.810    then show ?thesis
   2.811 @@ -1015,58 +951,57 @@
   2.812    values rather than associated elements.
   2.813  \<close>
   2.814  
   2.815 -lemma unit_factor_dvd [simp]:
   2.816 -  "a \<noteq> 0 \<Longrightarrow> unit_factor a dvd b"
   2.817 +lemma unit_factor_dvd [simp]: "a \<noteq> 0 \<Longrightarrow> unit_factor a dvd b"
   2.818    by (rule unit_imp_dvd) simp
   2.819  
   2.820 -lemma unit_factor_self [simp]:
   2.821 -  "unit_factor a dvd a"
   2.822 +lemma unit_factor_self [simp]: "unit_factor a dvd a"
   2.823    by (cases "a = 0") simp_all
   2.824  
   2.825 -lemma normalize_mult_unit_factor [simp]:
   2.826 -  "normalize a * unit_factor a = a"
   2.827 +lemma normalize_mult_unit_factor [simp]: "normalize a * unit_factor a = a"
   2.828    using unit_factor_mult_normalize [of a] by (simp add: ac_simps)
   2.829  
   2.830 -lemma normalize_eq_0_iff [simp]:
   2.831 -  "normalize a = 0 \<longleftrightarrow> a = 0" (is "?P \<longleftrightarrow> ?Q")
   2.832 +lemma normalize_eq_0_iff [simp]: "normalize a = 0 \<longleftrightarrow> a = 0"
   2.833 +  (is "?P \<longleftrightarrow> ?Q")
   2.834  proof
   2.835    assume ?P
   2.836    moreover have "unit_factor a * normalize a = a" by simp
   2.837    ultimately show ?Q by simp
   2.838  next
   2.839 -  assume ?Q then show ?P by simp
   2.840 +  assume ?Q
   2.841 +  then show ?P by simp
   2.842  qed
   2.843  
   2.844 -lemma unit_factor_eq_0_iff [simp]:
   2.845 -  "unit_factor a = 0 \<longleftrightarrow> a = 0" (is "?P \<longleftrightarrow> ?Q")
   2.846 +lemma unit_factor_eq_0_iff [simp]: "unit_factor a = 0 \<longleftrightarrow> a = 0"
   2.847 +  (is "?P \<longleftrightarrow> ?Q")
   2.848  proof
   2.849    assume ?P
   2.850    moreover have "unit_factor a * normalize a = a" by simp
   2.851    ultimately show ?Q by simp
   2.852  next
   2.853 -  assume ?Q then show ?P by simp
   2.854 +  assume ?Q
   2.855 +  then show ?P by simp
   2.856  qed
   2.857  
   2.858  lemma is_unit_unit_factor:
   2.859 -  assumes "is_unit a" shows "unit_factor a = a"
   2.860 +  assumes "is_unit a"
   2.861 +  shows "unit_factor a = a"
   2.862  proof -
   2.863    from assms have "normalize a = 1" by (rule is_unit_normalize)
   2.864    moreover from unit_factor_mult_normalize have "unit_factor a * normalize a = a" .
   2.865    ultimately show ?thesis by simp
   2.866  qed
   2.867  
   2.868 -lemma unit_factor_1 [simp]:
   2.869 -  "unit_factor 1 = 1"
   2.870 +lemma unit_factor_1 [simp]: "unit_factor 1 = 1"
   2.871    by (rule is_unit_unit_factor) simp
   2.872  
   2.873 -lemma normalize_1 [simp]:
   2.874 -  "normalize 1 = 1"
   2.875 +lemma normalize_1 [simp]: "normalize 1 = 1"
   2.876    by (rule is_unit_normalize) simp
   2.877  
   2.878 -lemma normalize_1_iff:
   2.879 -  "normalize a = 1 \<longleftrightarrow> is_unit a" (is "?P \<longleftrightarrow> ?Q")
   2.880 +lemma normalize_1_iff: "normalize a = 1 \<longleftrightarrow> is_unit a"
   2.881 +  (is "?P \<longleftrightarrow> ?Q")
   2.882  proof
   2.883 -  assume ?Q then show ?P by (rule is_unit_normalize)
   2.884 +  assume ?Q
   2.885 +  then show ?P by (rule is_unit_normalize)
   2.886  next
   2.887    assume ?P
   2.888    then have "a \<noteq> 0" by auto
   2.889 @@ -1079,32 +1014,34 @@
   2.890    ultimately show ?Q by simp
   2.891  qed
   2.892  
   2.893 -lemma div_normalize [simp]:
   2.894 -  "a div normalize a = unit_factor a"
   2.895 +lemma div_normalize [simp]: "a div normalize a = unit_factor a"
   2.896  proof (cases "a = 0")
   2.897 -  case True then show ?thesis by simp
   2.898 +  case True
   2.899 +  then show ?thesis by simp
   2.900  next
   2.901 -  case False then have "normalize a \<noteq> 0" by simp
   2.902 +  case False
   2.903 +  then have "normalize a \<noteq> 0" by simp
   2.904    with nonzero_mult_divide_cancel_right
   2.905    have "unit_factor a * normalize a div normalize a = unit_factor a" by blast
   2.906    then show ?thesis by simp
   2.907  qed
   2.908  
   2.909 -lemma div_unit_factor [simp]:
   2.910 -  "a div unit_factor a = normalize a"
   2.911 +lemma div_unit_factor [simp]: "a div unit_factor a = normalize a"
   2.912  proof (cases "a = 0")
   2.913 -  case True then show ?thesis by simp
   2.914 +  case True
   2.915 +  then show ?thesis by simp
   2.916  next
   2.917 -  case False then have "unit_factor a \<noteq> 0" by simp
   2.918 +  case False
   2.919 +  then have "unit_factor a \<noteq> 0" by simp
   2.920    with nonzero_mult_divide_cancel_left
   2.921    have "unit_factor a * normalize a div unit_factor a = normalize a" by blast
   2.922    then show ?thesis by simp
   2.923  qed
   2.924  
   2.925 -lemma normalize_div [simp]:
   2.926 -  "normalize a div a = 1 div unit_factor a"
   2.927 +lemma normalize_div [simp]: "normalize a div a = 1 div unit_factor a"
   2.928  proof (cases "a = 0")
   2.929 -  case True then show ?thesis by simp
   2.930 +  case True
   2.931 +  then show ?thesis by simp
   2.932  next
   2.933    case False
   2.934    have "normalize a div a = normalize a div (unit_factor a * normalize a)"
   2.935 @@ -1114,62 +1051,64 @@
   2.936    finally show ?thesis .
   2.937  qed
   2.938  
   2.939 -lemma mult_one_div_unit_factor [simp]:
   2.940 -  "a * (1 div unit_factor b) = a div unit_factor b"
   2.941 +lemma mult_one_div_unit_factor [simp]: "a * (1 div unit_factor b) = a div unit_factor b"
   2.942    by (cases "b = 0") simp_all
   2.943  
   2.944 -lemma normalize_mult:
   2.945 -  "normalize (a * b) = normalize a * normalize b"
   2.946 +lemma normalize_mult: "normalize (a * b) = normalize a * normalize b"
   2.947  proof (cases "a = 0 \<or> b = 0")
   2.948 -  case True then show ?thesis by auto
   2.949 +  case True
   2.950 +  then show ?thesis by auto
   2.951  next
   2.952    case False
   2.953    from unit_factor_mult_normalize have "unit_factor (a * b) * normalize (a * b) = a * b" .
   2.954 -  then have "normalize (a * b) = a * b div unit_factor (a * b)" by simp
   2.955 -  also have "\<dots> = a * b div unit_factor (b * a)" by (simp add: ac_simps)
   2.956 +  then have "normalize (a * b) = a * b div unit_factor (a * b)"
   2.957 +    by simp
   2.958 +  also have "\<dots> = a * b div unit_factor (b * a)"
   2.959 +    by (simp add: ac_simps)
   2.960    also have "\<dots> = a * b div unit_factor b div unit_factor a"
   2.961      using False by (simp add: unit_factor_mult is_unit_div_mult2_eq [symmetric])
   2.962    also have "\<dots> = a * (b div unit_factor b) div unit_factor a"
   2.963      using False by (subst unit_div_mult_swap) simp_all
   2.964    also have "\<dots> = normalize a * normalize b"
   2.965 -    using False by (simp add: mult.commute [of a] mult.commute [of "normalize a"] unit_div_mult_swap [symmetric])
   2.966 +    using False
   2.967 +    by (simp add: mult.commute [of a] mult.commute [of "normalize a"] unit_div_mult_swap [symmetric])
   2.968    finally show ?thesis .
   2.969  qed
   2.970  
   2.971 -lemma unit_factor_idem [simp]:
   2.972 -  "unit_factor (unit_factor a) = unit_factor a"
   2.973 +lemma unit_factor_idem [simp]: "unit_factor (unit_factor a) = unit_factor a"
   2.974    by (cases "a = 0") (auto intro: is_unit_unit_factor)
   2.975  
   2.976 -lemma normalize_unit_factor [simp]:
   2.977 -  "a \<noteq> 0 \<Longrightarrow> normalize (unit_factor a) = 1"
   2.978 +lemma normalize_unit_factor [simp]: "a \<noteq> 0 \<Longrightarrow> normalize (unit_factor a) = 1"
   2.979    by (rule is_unit_normalize) simp
   2.980  
   2.981 -lemma normalize_idem [simp]:
   2.982 -  "normalize (normalize a) = normalize a"
   2.983 +lemma normalize_idem [simp]: "normalize (normalize a) = normalize a"
   2.984  proof (cases "a = 0")
   2.985 -  case True then show ?thesis by simp
   2.986 +  case True
   2.987 +  then show ?thesis by simp
   2.988  next
   2.989    case False
   2.990 -  have "normalize a = normalize (unit_factor a * normalize a)" by simp
   2.991 +  have "normalize a = normalize (unit_factor a * normalize a)"
   2.992 +    by simp
   2.993    also have "\<dots> = normalize (unit_factor a) * normalize (normalize a)"
   2.994      by (simp only: normalize_mult)
   2.995 -  finally show ?thesis using False by simp_all
   2.996 +  finally show ?thesis
   2.997 +    using False by simp_all
   2.998  qed
   2.999  
  2.1000  lemma unit_factor_normalize [simp]:
  2.1001    assumes "a \<noteq> 0"
  2.1002    shows "unit_factor (normalize a) = 1"
  2.1003  proof -
  2.1004 -  from assms have "normalize a \<noteq> 0" by simp
  2.1005 +  from assms have *: "normalize a \<noteq> 0"
  2.1006 +    by simp
  2.1007    have "unit_factor (normalize a) * normalize (normalize a) = normalize a"
  2.1008      by (simp only: unit_factor_mult_normalize)
  2.1009    then have "unit_factor (normalize a) * normalize a = normalize a"
  2.1010      by simp
  2.1011 -  with \<open>normalize a \<noteq> 0\<close>
  2.1012 -  have "unit_factor (normalize a) * normalize a div normalize a = normalize a div normalize a"
  2.1013 +  with * have "unit_factor (normalize a) * normalize a div normalize a = normalize a div normalize a"
  2.1014      by simp
  2.1015 -  with \<open>normalize a \<noteq> 0\<close>
  2.1016 -  show ?thesis by simp
  2.1017 +  with * show ?thesis
  2.1018 +    by simp
  2.1019  qed
  2.1020  
  2.1021  lemma dvd_unit_factor_div:
  2.1022 @@ -1196,8 +1135,7 @@
  2.1023      by (cases "b = 0") (simp_all add: normalize_mult)
  2.1024  qed
  2.1025  
  2.1026 -lemma normalize_dvd_iff [simp]:
  2.1027 -  "normalize a dvd b \<longleftrightarrow> a dvd b"
  2.1028 +lemma normalize_dvd_iff [simp]: "normalize a dvd b \<longleftrightarrow> a dvd b"
  2.1029  proof -
  2.1030    have "normalize a dvd b \<longleftrightarrow> unit_factor a * normalize a dvd b"
  2.1031      using mult_unit_dvd_iff [of "unit_factor a" "normalize a" b]
  2.1032 @@ -1205,8 +1143,7 @@
  2.1033    then show ?thesis by simp
  2.1034  qed
  2.1035  
  2.1036 -lemma dvd_normalize_iff [simp]:
  2.1037 -  "a dvd normalize b \<longleftrightarrow> a dvd b"
  2.1038 +lemma dvd_normalize_iff [simp]: "a dvd normalize b \<longleftrightarrow> a dvd b"
  2.1039  proof -
  2.1040    have "a dvd normalize  b \<longleftrightarrow> a dvd normalize b * unit_factor b"
  2.1041      using dvd_mult_unit_iff [of "unit_factor b" a "normalize b"]
  2.1042 @@ -1226,36 +1163,38 @@
  2.1043    assumes "a dvd b" and "b dvd a"
  2.1044    shows "normalize a = normalize b"
  2.1045  proof (cases "a = 0 \<or> b = 0")
  2.1046 -  case True with assms show ?thesis by auto
  2.1047 +  case True
  2.1048 +  with assms show ?thesis by auto
  2.1049  next
  2.1050    case False
  2.1051    from \<open>a dvd b\<close> obtain c where b: "b = a * c" ..
  2.1052    moreover from \<open>b dvd a\<close> obtain d where a: "a = b * d" ..
  2.1053 -  ultimately have "b * 1 = b * (c * d)" by (simp add: ac_simps)
  2.1054 +  ultimately have "b * 1 = b * (c * d)"
  2.1055 +    by (simp add: ac_simps)
  2.1056    with False have "1 = c * d"
  2.1057      unfolding mult_cancel_left by simp
  2.1058 -  then have "is_unit c" and "is_unit d" by auto
  2.1059 -  with a b show ?thesis by (simp add: normalize_mult is_unit_normalize)
  2.1060 +  then have "is_unit c" and "is_unit d"
  2.1061 +    by auto
  2.1062 +  with a b show ?thesis
  2.1063 +    by (simp add: normalize_mult is_unit_normalize)
  2.1064  qed
  2.1065  
  2.1066 -lemma associatedD1:
  2.1067 -  "normalize a = normalize b \<Longrightarrow> a dvd b"
  2.1068 +lemma associatedD1: "normalize a = normalize b \<Longrightarrow> a dvd b"
  2.1069    using dvd_normalize_iff [of _ b, symmetric] normalize_dvd_iff [of a _, symmetric]
  2.1070    by simp
  2.1071  
  2.1072 -lemma associatedD2:
  2.1073 -  "normalize a = normalize b \<Longrightarrow> b dvd a"
  2.1074 +lemma associatedD2: "normalize a = normalize b \<Longrightarrow> b dvd a"
  2.1075    using dvd_normalize_iff [of _ a, symmetric] normalize_dvd_iff [of b _, symmetric]
  2.1076    by simp
  2.1077  
  2.1078 -lemma associated_unit:
  2.1079 -  "normalize a = normalize b \<Longrightarrow> is_unit a \<Longrightarrow> is_unit b"
  2.1080 +lemma associated_unit: "normalize a = normalize b \<Longrightarrow> is_unit a \<Longrightarrow> is_unit b"
  2.1081    using dvd_unit_imp_unit by (auto dest!: associatedD1 associatedD2)
  2.1082  
  2.1083 -lemma associated_iff_dvd:
  2.1084 -  "normalize a = normalize b \<longleftrightarrow> a dvd b \<and> b dvd a" (is "?P \<longleftrightarrow> ?Q")
  2.1085 +lemma associated_iff_dvd: "normalize a = normalize b \<longleftrightarrow> a dvd b \<and> b dvd a"
  2.1086 +  (is "?P \<longleftrightarrow> ?Q")
  2.1087  proof
  2.1088 -  assume ?Q then show ?P by (auto intro!: associatedI)
  2.1089 +  assume ?Q
  2.1090 +  then show ?P by (auto intro!: associatedI)
  2.1091  next
  2.1092    assume ?P
  2.1093    then have "unit_factor a * normalize a = unit_factor a * normalize b"
  2.1094 @@ -1264,7 +1203,8 @@
  2.1095      by (simp add: ac_simps)
  2.1096    show ?Q
  2.1097    proof (cases "a = 0 \<or> b = 0")
  2.1098 -    case True with \<open>?P\<close> show ?thesis by auto
  2.1099 +    case True
  2.1100 +    with \<open>?P\<close> show ?thesis by auto
  2.1101    next
  2.1102      case False
  2.1103      then have "b dvd normalize b * unit_factor a" and "normalize b * unit_factor a dvd b"
  2.1104 @@ -1291,38 +1231,38 @@
  2.1105    assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
  2.1106  begin
  2.1107  
  2.1108 -lemma mult_mono:
  2.1109 -  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
  2.1110 -apply (erule mult_right_mono [THEN order_trans], assumption)
  2.1111 -apply (erule mult_left_mono, assumption)
  2.1112 -done
  2.1113 +lemma mult_mono: "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
  2.1114 +  apply (erule (1) mult_right_mono [THEN order_trans])
  2.1115 +  apply (erule (1) mult_left_mono)
  2.1116 +  done
  2.1117  
  2.1118 -lemma mult_mono':
  2.1119 -  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
  2.1120 -apply (rule mult_mono)
  2.1121 -apply (fast intro: order_trans)+
  2.1122 -done
  2.1123 +lemma mult_mono': "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
  2.1124 +  apply (rule mult_mono)
  2.1125 +  apply (fast intro: order_trans)+
  2.1126 +  done
  2.1127  
  2.1128  end
  2.1129  
  2.1130  class ordered_semiring_0 = semiring_0 + ordered_semiring
  2.1131  begin
  2.1132  
  2.1133 -lemma mult_nonneg_nonneg[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
  2.1134 -using mult_left_mono [of 0 b a] by simp
  2.1135 +lemma mult_nonneg_nonneg [simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
  2.1136 +  using mult_left_mono [of 0 b a] by simp
  2.1137  
  2.1138  lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
  2.1139 -using mult_left_mono [of b 0 a] by simp
  2.1140 +  using mult_left_mono [of b 0 a] by simp
  2.1141  
  2.1142  lemma mult_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> b \<Longrightarrow> a * b \<le> 0"
  2.1143 -using mult_right_mono [of a 0 b] by simp
  2.1144 +  using mult_right_mono [of a 0 b] by simp
  2.1145  
  2.1146  text \<open>Legacy - use \<open>mult_nonpos_nonneg\<close>\<close>
  2.1147  lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0"
  2.1148 -by (drule mult_right_mono [of b 0], auto)
  2.1149 +  apply (drule mult_right_mono [of b 0])
  2.1150 +  apply auto
  2.1151 +  done
  2.1152  
  2.1153  lemma split_mult_neg_le: "(0 \<le> a \<and> b \<le> 0) \<or> (a \<le> 0 \<and> 0 \<le> b) \<Longrightarrow> a * b \<le> 0"
  2.1154 -by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
  2.1155 +  by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
  2.1156  
  2.1157  end
  2.1158  
  2.1159 @@ -1341,44 +1281,34 @@
  2.1160  
  2.1161  subclass ordered_cancel_comm_monoid_add ..
  2.1162  
  2.1163 -lemma mult_left_less_imp_less:
  2.1164 -  "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
  2.1165 -by (force simp add: mult_left_mono not_le [symmetric])
  2.1166 +lemma mult_left_less_imp_less: "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
  2.1167 +  by (force simp add: mult_left_mono not_le [symmetric])
  2.1168  
  2.1169 -lemma mult_right_less_imp_less:
  2.1170 -  "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
  2.1171 -by (force simp add: mult_right_mono not_le [symmetric])
  2.1172 +lemma mult_right_less_imp_less: "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
  2.1173 +  by (force simp add: mult_right_mono not_le [symmetric])
  2.1174  
  2.1175 -lemma less_eq_add_cancel_left_greater_eq_zero [simp]:
  2.1176 -  "a \<le> a + b \<longleftrightarrow> 0 \<le> b"
  2.1177 +lemma less_eq_add_cancel_left_greater_eq_zero [simp]: "a \<le> a + b \<longleftrightarrow> 0 \<le> b"
  2.1178    using add_le_cancel_left [of a 0 b] by simp
  2.1179  
  2.1180 -lemma less_eq_add_cancel_left_less_eq_zero [simp]:
  2.1181 -  "a + b \<le> a \<longleftrightarrow> b \<le> 0"
  2.1182 +lemma less_eq_add_cancel_left_less_eq_zero [simp]: "a + b \<le> a \<longleftrightarrow> b \<le> 0"
  2.1183    using add_le_cancel_left [of a b 0] by simp
  2.1184  
  2.1185 -lemma less_eq_add_cancel_right_greater_eq_zero [simp]:
  2.1186 -  "a \<le> b + a \<longleftrightarrow> 0 \<le> b"
  2.1187 +lemma less_eq_add_cancel_right_greater_eq_zero [simp]: "a \<le> b + a \<longleftrightarrow> 0 \<le> b"
  2.1188    using add_le_cancel_right [of 0 a b] by simp
  2.1189  
  2.1190 -lemma less_eq_add_cancel_right_less_eq_zero [simp]:
  2.1191 -  "b + a \<le> a \<longleftrightarrow> b \<le> 0"
  2.1192 +lemma less_eq_add_cancel_right_less_eq_zero [simp]: "b + a \<le> a \<longleftrightarrow> b \<le> 0"
  2.1193    using add_le_cancel_right [of b a 0] by simp
  2.1194  
  2.1195 -lemma less_add_cancel_left_greater_zero [simp]:
  2.1196 -  "a < a + b \<longleftrightarrow> 0 < b"
  2.1197 +lemma less_add_cancel_left_greater_zero [simp]: "a < a + b \<longleftrightarrow> 0 < b"
  2.1198    using add_less_cancel_left [of a 0 b] by simp
  2.1199  
  2.1200 -lemma less_add_cancel_left_less_zero [simp]:
  2.1201 -  "a + b < a \<longleftrightarrow> b < 0"
  2.1202 +lemma less_add_cancel_left_less_zero [simp]: "a + b < a \<longleftrightarrow> b < 0"
  2.1203    using add_less_cancel_left [of a b 0] by simp
  2.1204  
  2.1205 -lemma less_add_cancel_right_greater_zero [simp]:
  2.1206 -  "a < b + a \<longleftrightarrow> 0 < b"
  2.1207 +lemma less_add_cancel_right_greater_zero [simp]: "a < b + a \<longleftrightarrow> 0 < b"
  2.1208    using add_less_cancel_right [of 0 a b] by simp
  2.1209  
  2.1210 -lemma less_add_cancel_right_less_zero [simp]:
  2.1211 -  "b + a < a \<longleftrightarrow> b < 0"
  2.1212 +lemma less_add_cancel_right_less_zero [simp]: "b + a < a \<longleftrightarrow> b < 0"
  2.1213    using add_less_cancel_right [of b a 0] by simp
  2.1214  
  2.1215  end
  2.1216 @@ -1392,7 +1322,8 @@
  2.1217  proof-
  2.1218    from assms have "u * x + v * y \<le> u * a + v * a"
  2.1219      by (simp add: add_mono mult_left_mono)
  2.1220 -  thus ?thesis using assms unfolding distrib_right[symmetric] by simp
  2.1221 +  with assms show ?thesis
  2.1222 +    unfolding distrib_right[symmetric] by simp
  2.1223  qed
  2.1224  
  2.1225  end
  2.1226 @@ -1416,80 +1347,79 @@
  2.1227      using mult_strict_right_mono by (cases "c = 0") auto
  2.1228  qed
  2.1229  
  2.1230 -lemma mult_left_le_imp_le:
  2.1231 -  "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
  2.1232 -by (force simp add: mult_strict_left_mono _not_less [symmetric])
  2.1233 +lemma mult_left_le_imp_le: "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
  2.1234 +  by (auto simp add: mult_strict_left_mono _not_less [symmetric])
  2.1235  
  2.1236 -lemma mult_right_le_imp_le:
  2.1237 -  "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
  2.1238 -by (force simp add: mult_strict_right_mono not_less [symmetric])
  2.1239 +lemma mult_right_le_imp_le: "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
  2.1240 +  by (auto simp add: mult_strict_right_mono not_less [symmetric])
  2.1241  
  2.1242  lemma mult_pos_pos[simp]: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
  2.1243 -using mult_strict_left_mono [of 0 b a] by simp
  2.1244 +  using mult_strict_left_mono [of 0 b a] by simp
  2.1245  
  2.1246  lemma mult_pos_neg: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
  2.1247 -using mult_strict_left_mono [of b 0 a] by simp
  2.1248 +  using mult_strict_left_mono [of b 0 a] by simp
  2.1249  
  2.1250  lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0"
  2.1251 -using mult_strict_right_mono [of a 0 b] by simp
  2.1252 +  using mult_strict_right_mono [of a 0 b] by simp
  2.1253  
  2.1254  text \<open>Legacy - use \<open>mult_neg_pos\<close>\<close>
  2.1255  lemma mult_pos_neg2: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0"
  2.1256 -by (drule mult_strict_right_mono [of b 0], auto)
  2.1257 -
  2.1258 -lemma zero_less_mult_pos:
  2.1259 -  "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
  2.1260 -apply (cases "b\<le>0")
  2.1261 - apply (auto simp add: le_less not_less)
  2.1262 -apply (drule_tac mult_pos_neg [of a b])
  2.1263 - apply (auto dest: less_not_sym)
  2.1264 -done
  2.1265 +  apply (drule mult_strict_right_mono [of b 0])
  2.1266 +  apply auto
  2.1267 +  done
  2.1268  
  2.1269 -lemma zero_less_mult_pos2:
  2.1270 -  "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
  2.1271 -apply (cases "b\<le>0")
  2.1272 - apply (auto simp add: le_less not_less)
  2.1273 -apply (drule_tac mult_pos_neg2 [of a b])
  2.1274 - apply (auto dest: less_not_sym)
  2.1275 -done
  2.1276 +lemma zero_less_mult_pos: "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
  2.1277 +  apply (cases "b \<le> 0")
  2.1278 +   apply (auto simp add: le_less not_less)
  2.1279 +  apply (drule_tac mult_pos_neg [of a b])
  2.1280 +   apply (auto dest: less_not_sym)
  2.1281 +  done
  2.1282  
  2.1283 -text\<open>Strict monotonicity in both arguments\<close>
  2.1284 +lemma zero_less_mult_pos2: "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
  2.1285 +  apply (cases "b \<le> 0")
  2.1286 +   apply (auto simp add: le_less not_less)
  2.1287 +  apply (drule_tac mult_pos_neg2 [of a b])
  2.1288 +   apply (auto dest: less_not_sym)
  2.1289 +  done
  2.1290 +
  2.1291 +text \<open>Strict monotonicity in both arguments\<close>
  2.1292  lemma mult_strict_mono:
  2.1293    assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"
  2.1294    shows "a * c < b * d"
  2.1295 -  using assms apply (cases "c=0")
  2.1296 -  apply (simp)
  2.1297 +  using assms
  2.1298 +  apply (cases "c = 0")
  2.1299 +  apply simp
  2.1300    apply (erule mult_strict_right_mono [THEN less_trans])
  2.1301 -  apply (force simp add: le_less)
  2.1302 -  apply (erule mult_strict_left_mono, assumption)
  2.1303 +  apply (auto simp add: le_less)
  2.1304 +  apply (erule (1) mult_strict_left_mono)
  2.1305    done
  2.1306  
  2.1307 -text\<open>This weaker variant has more natural premises\<close>
  2.1308 +text \<open>This weaker variant has more natural premises\<close>
  2.1309  lemma mult_strict_mono':
  2.1310    assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
  2.1311    shows "a * c < b * d"
  2.1312 -by (rule mult_strict_mono) (insert assms, auto)
  2.1313 +  by (rule mult_strict_mono) (insert assms, auto)
  2.1314  
  2.1315  lemma mult_less_le_imp_less:
  2.1316    assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"
  2.1317    shows "a * c < b * d"
  2.1318 -  using assms apply (subgoal_tac "a * c < b * c")
  2.1319 +  using assms
  2.1320 +  apply (subgoal_tac "a * c < b * c")
  2.1321    apply (erule less_le_trans)
  2.1322    apply (erule mult_left_mono)
  2.1323    apply simp
  2.1324 -  apply (erule mult_strict_right_mono)
  2.1325 -  apply assumption
  2.1326 +  apply (erule (1) mult_strict_right_mono)
  2.1327    done
  2.1328  
  2.1329  lemma mult_le_less_imp_less:
  2.1330    assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"
  2.1331    shows "a * c < b * d"
  2.1332 -  using assms apply (subgoal_tac "a * c \<le> b * c")
  2.1333 +  using assms
  2.1334 +  apply (subgoal_tac "a * c \<le> b * c")
  2.1335    apply (erule le_less_trans)
  2.1336    apply (erule mult_strict_left_mono)
  2.1337    apply simp
  2.1338 -  apply (erule mult_right_mono)
  2.1339 -  apply simp
  2.1340 +  apply (erule (1) mult_right_mono)
  2.1341    done
  2.1342  
  2.1343  end
  2.1344 @@ -1504,9 +1434,9 @@
  2.1345    shows "u * x + v * y < a"
  2.1346  proof -
  2.1347    from assms have "u * x + v * y < u * a + v * a"
  2.1348 -    by (cases "u = 0")
  2.1349 -       (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)
  2.1350 -  thus ?thesis using assms unfolding distrib_right[symmetric] by simp
  2.1351 +    by (cases "u = 0") (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)
  2.1352 +  with assms show ?thesis
  2.1353 +    unfolding distrib_right[symmetric] by simp
  2.1354  qed
  2.1355  
  2.1356  end
  2.1357 @@ -1519,8 +1449,8 @@
  2.1358  proof
  2.1359    fix a b c :: 'a
  2.1360    assume "a \<le> b" "0 \<le> c"
  2.1361 -  thus "c * a \<le> c * b" by (rule comm_mult_left_mono)
  2.1362 -  thus "a * c \<le> b * c" by (simp only: mult.commute)
  2.1363 +  then show "c * a \<le> c * b" by (rule comm_mult_left_mono)
  2.1364 +  then show "a * c \<le> b * c" by (simp only: mult.commute)
  2.1365  qed
  2.1366  
  2.1367  end
  2.1368 @@ -1542,15 +1472,15 @@
  2.1369  proof
  2.1370    fix a b c :: 'a
  2.1371    assume "a < b" "0 < c"
  2.1372 -  thus "c * a < c * b" by (rule comm_mult_strict_left_mono)
  2.1373 -  thus "a * c < b * c" by (simp only: mult.commute)
  2.1374 +  then show "c * a < c * b" by (rule comm_mult_strict_left_mono)
  2.1375 +  then show "a * c < b * c" by (simp only: mult.commute)
  2.1376  qed
  2.1377  
  2.1378  subclass ordered_cancel_comm_semiring
  2.1379  proof
  2.1380    fix a b c :: 'a
  2.1381    assume "a \<le> b" "0 \<le> c"
  2.1382 -  thus "c * a \<le> c * b"
  2.1383 +  then show "c * a \<le> c * b"
  2.1384      unfolding le_less
  2.1385      using mult_strict_left_mono by (cases "c = 0") auto
  2.1386  qed
  2.1387 @@ -1562,40 +1492,33 @@
  2.1388  
  2.1389  subclass ordered_ab_group_add ..
  2.1390  
  2.1391 -lemma less_add_iff1:
  2.1392 -  "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
  2.1393 -by (simp add: algebra_simps)
  2.1394 +lemma less_add_iff1: "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
  2.1395 +  by (simp add: algebra_simps)
  2.1396  
  2.1397 -lemma less_add_iff2:
  2.1398 -  "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
  2.1399 -by (simp add: algebra_simps)
  2.1400 +lemma less_add_iff2: "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
  2.1401 +  by (simp add: algebra_simps)
  2.1402  
  2.1403 -lemma le_add_iff1:
  2.1404 -  "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
  2.1405 -by (simp add: algebra_simps)
  2.1406 +lemma le_add_iff1: "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
  2.1407 +  by (simp add: algebra_simps)
  2.1408  
  2.1409 -lemma le_add_iff2:
  2.1410 -  "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
  2.1411 -by (simp add: algebra_simps)
  2.1412 +lemma le_add_iff2: "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
  2.1413 +  by (simp add: algebra_simps)
  2.1414  
  2.1415 -lemma mult_left_mono_neg:
  2.1416 -  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
  2.1417 +lemma mult_left_mono_neg: "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
  2.1418    apply (drule mult_left_mono [of _ _ "- c"])
  2.1419    apply simp_all
  2.1420    done
  2.1421  
  2.1422 -lemma mult_right_mono_neg:
  2.1423 -  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
  2.1424 +lemma mult_right_mono_neg: "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
  2.1425    apply (drule mult_right_mono [of _ _ "- c"])
  2.1426    apply simp_all
  2.1427    done
  2.1428  
  2.1429  lemma mult_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
  2.1430 -using mult_right_mono_neg [of a 0 b] by simp
  2.1431 +  using mult_right_mono_neg [of a 0 b] by simp
  2.1432  
  2.1433 -lemma split_mult_pos_le:
  2.1434 -  "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
  2.1435 -by (auto simp add: mult_nonpos_nonpos)
  2.1436 +lemma split_mult_pos_le: "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
  2.1437 +  by (auto simp add: mult_nonpos_nonpos)
  2.1438  
  2.1439  end
  2.1440  
  2.1441 @@ -1608,12 +1531,12 @@
  2.1442  proof
  2.1443    fix a b
  2.1444    show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
  2.1445 -    by (auto simp add: abs_if not_le not_less algebra_simps simp del: add.commute dest: add_neg_neg add_nonneg_nonneg)
  2.1446 +    by (auto simp add: abs_if not_le not_less algebra_simps
  2.1447 +        simp del: add.commute dest: add_neg_neg add_nonneg_nonneg)
  2.1448  qed (auto simp add: abs_if)
  2.1449  
  2.1450  lemma zero_le_square [simp]: "0 \<le> a * a"
  2.1451 -  using linear [of 0 a]
  2.1452 -  by (auto simp add: mult_nonpos_nonpos)
  2.1453 +  using linear [of 0 a] by (auto simp add: mult_nonpos_nonpos)
  2.1454  
  2.1455  lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"
  2.1456    by (simp add: not_less)
  2.1457 @@ -1621,12 +1544,10 @@
  2.1458  proposition abs_eq_iff: "\<bar>x\<bar> = \<bar>y\<bar> \<longleftrightarrow> x = y \<or> x = -y"
  2.1459    by (auto simp add: abs_if split: if_split_asm)
  2.1460  
  2.1461 -lemma sum_squares_ge_zero:
  2.1462 -  "0 \<le> x * x + y * y"
  2.1463 +lemma sum_squares_ge_zero: "0 \<le> x * x + y * y"
  2.1464    by (intro add_nonneg_nonneg zero_le_square)
  2.1465  
  2.1466 -lemma not_sum_squares_lt_zero:
  2.1467 -  "\<not> x * x + y * y < 0"
  2.1468 +lemma not_sum_squares_lt_zero: "\<not> x * x + y * y < 0"
  2.1469    by (simp add: not_less sum_squares_ge_zero)
  2.1470  
  2.1471  end
  2.1472 @@ -1638,40 +1559,49 @@
  2.1473  subclass linordered_ring ..
  2.1474  
  2.1475  lemma mult_strict_left_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
  2.1476 -using mult_strict_left_mono [of b a "- c"] by simp
  2.1477 +  using mult_strict_left_mono [of b a "- c"] by simp
  2.1478  
  2.1479  lemma mult_strict_right_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
  2.1480 -using mult_strict_right_mono [of b a "- c"] by simp
  2.1481 +  using mult_strict_right_mono [of b a "- c"] by simp
  2.1482  
  2.1483  lemma mult_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
  2.1484 -using mult_strict_right_mono_neg [of a 0 b] by simp
  2.1485 +  using mult_strict_right_mono_neg [of a 0 b] by simp
  2.1486  
  2.1487  subclass ring_no_zero_divisors
  2.1488  proof
  2.1489    fix a b
  2.1490 -  assume "a \<noteq> 0" then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
  2.1491 -  assume "b \<noteq> 0" then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
  2.1492 +  assume "a \<noteq> 0"
  2.1493 +  then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
  2.1494 +  assume "b \<noteq> 0"
  2.1495 +  then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
  2.1496    have "a * b < 0 \<or> 0 < a * b"
  2.1497    proof (cases "a < 0")
  2.1498 -    case True note A' = this
  2.1499 -    show ?thesis proof (cases "b < 0")
  2.1500 -      case True with A'
  2.1501 -      show ?thesis by (auto dest: mult_neg_neg)
  2.1502 +    case A': True
  2.1503 +    show ?thesis
  2.1504 +    proof (cases "b < 0")
  2.1505 +      case True
  2.1506 +      with A' show ?thesis by (auto dest: mult_neg_neg)
  2.1507      next
  2.1508 -      case False with B have "0 < b" by auto
  2.1509 +      case False
  2.1510 +      with B have "0 < b" by auto
  2.1511        with A' show ?thesis by (auto dest: mult_strict_right_mono)
  2.1512      qed
  2.1513    next
  2.1514 -    case False with A have A': "0 < a" by auto
  2.1515 -    show ?thesis proof (cases "b < 0")
  2.1516 -      case True with A'
  2.1517 -      show ?thesis by (auto dest: mult_strict_right_mono_neg)
  2.1518 +    case False
  2.1519 +    with A have A': "0 < a" by auto
  2.1520 +    show ?thesis
  2.1521 +    proof (cases "b < 0")
  2.1522 +      case True
  2.1523 +      with A' show ?thesis
  2.1524 +        by (auto dest: mult_strict_right_mono_neg)
  2.1525      next
  2.1526 -      case False with B have "0 < b" by auto
  2.1527 +      case False
  2.1528 +      with B have "0 < b" by auto
  2.1529        with A' show ?thesis by auto
  2.1530      qed
  2.1531    qed
  2.1532 -  then show "a * b \<noteq> 0" by (simp add: neq_iff)
  2.1533 +  then show "a * b \<noteq> 0"
  2.1534 +    by (simp add: neq_iff)
  2.1535  qed
  2.1536  
  2.1537  lemma zero_less_mult_iff: "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
  2.1538 @@ -1681,84 +1611,66 @@
  2.1539  lemma zero_le_mult_iff: "0 \<le> a * b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0"
  2.1540    by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)
  2.1541  
  2.1542 -lemma mult_less_0_iff:
  2.1543 -  "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
  2.1544 -  apply (insert zero_less_mult_iff [of "-a" b])
  2.1545 -  apply force
  2.1546 -  done
  2.1547 +lemma mult_less_0_iff: "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
  2.1548 +  using zero_less_mult_iff [of "- a" b] by auto
  2.1549  
  2.1550 -lemma mult_le_0_iff:
  2.1551 -  "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
  2.1552 -  apply (insert zero_le_mult_iff [of "-a" b])
  2.1553 -  apply force
  2.1554 -  done
  2.1555 +lemma mult_le_0_iff: "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
  2.1556 +  using zero_le_mult_iff [of "- a" b] by auto
  2.1557  
  2.1558 -text\<open>Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
  2.1559 -   also with the relations \<open>\<le>\<close> and equality.\<close>
  2.1560 +text \<open>
  2.1561 +  Cancellation laws for @{term "c * a < c * b"} and @{term "a * c < b * c"},
  2.1562 +  also with the relations \<open>\<le>\<close> and equality.
  2.1563 +\<close>
  2.1564  
  2.1565 -text\<open>These ``disjunction'' versions produce two cases when the comparison is
  2.1566 - an assumption, but effectively four when the comparison is a goal.\<close>
  2.1567 +text \<open>
  2.1568 +  These ``disjunction'' versions produce two cases when the comparison is
  2.1569 +  an assumption, but effectively four when the comparison is a goal.
  2.1570 +\<close>
  2.1571  
  2.1572 -lemma mult_less_cancel_right_disj:
  2.1573 -  "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
  2.1574 +lemma mult_less_cancel_right_disj: "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
  2.1575    apply (cases "c = 0")
  2.1576 -  apply (auto simp add: neq_iff mult_strict_right_mono
  2.1577 -                      mult_strict_right_mono_neg)
  2.1578 -  apply (auto simp add: not_less
  2.1579 -                      not_le [symmetric, of "a*c"]
  2.1580 -                      not_le [symmetric, of a])
  2.1581 +  apply (auto simp add: neq_iff mult_strict_right_mono mult_strict_right_mono_neg)
  2.1582 +  apply (auto simp add: not_less not_le [symmetric, of "a*c"] not_le [symmetric, of a])
  2.1583    apply (erule_tac [!] notE)
  2.1584 -  apply (auto simp add: less_imp_le mult_right_mono
  2.1585 -                      mult_right_mono_neg)
  2.1586 +  apply (auto simp add: less_imp_le mult_right_mono mult_right_mono_neg)
  2.1587    done
  2.1588  
  2.1589 -lemma mult_less_cancel_left_disj:
  2.1590 -  "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
  2.1591 +lemma mult_less_cancel_left_disj: "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
  2.1592    apply (cases "c = 0")
  2.1593 -  apply (auto simp add: neq_iff mult_strict_left_mono
  2.1594 -                      mult_strict_left_mono_neg)
  2.1595 -  apply (auto simp add: not_less
  2.1596 -                      not_le [symmetric, of "c*a"]
  2.1597 -                      not_le [symmetric, of a])
  2.1598 +  apply (auto simp add: neq_iff mult_strict_left_mono mult_strict_left_mono_neg)
  2.1599 +  apply (auto simp add: not_less not_le [symmetric, of "c * a"] not_le [symmetric, of a])
  2.1600    apply (erule_tac [!] notE)
  2.1601 -  apply (auto simp add: less_imp_le mult_left_mono
  2.1602 -                      mult_left_mono_neg)
  2.1603 +  apply (auto simp add: less_imp_le mult_left_mono mult_left_mono_neg)
  2.1604    done
  2.1605  
  2.1606 -text\<open>The ``conjunction of implication'' lemmas produce two cases when the
  2.1607 -comparison is a goal, but give four when the comparison is an assumption.\<close>
  2.1608 +text \<open>
  2.1609 +  The ``conjunction of implication'' lemmas produce two cases when the
  2.1610 +  comparison is a goal, but give four when the comparison is an assumption.
  2.1611 +\<close>
  2.1612  
  2.1613 -lemma mult_less_cancel_right:
  2.1614 -  "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
  2.1615 +lemma mult_less_cancel_right: "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
  2.1616    using mult_less_cancel_right_disj [of a c b] by auto
  2.1617  
  2.1618 -lemma mult_less_cancel_left:
  2.1619 -  "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
  2.1620 +lemma mult_less_cancel_left: "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
  2.1621    using mult_less_cancel_left_disj [of c a b] by auto
  2.1622  
  2.1623 -lemma mult_le_cancel_right:
  2.1624 -   "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
  2.1625 -by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
  2.1626 +lemma mult_le_cancel_right: "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
  2.1627 +  by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
  2.1628  
  2.1629 -lemma mult_le_cancel_left:
  2.1630 -  "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
  2.1631 -by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
  2.1632 +lemma mult_le_cancel_left: "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
  2.1633 +  by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
  2.1634  
  2.1635 -lemma mult_le_cancel_left_pos:
  2.1636 -  "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"
  2.1637 -by (auto simp: mult_le_cancel_left)
  2.1638 +lemma mult_le_cancel_left_pos: "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"
  2.1639 +  by (auto simp: mult_le_cancel_left)
  2.1640  
  2.1641 -lemma mult_le_cancel_left_neg:
  2.1642 -  "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a"
  2.1643 -by (auto simp: mult_le_cancel_left)
  2.1644 +lemma mult_le_cancel_left_neg: "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a"
  2.1645 +  by (auto simp: mult_le_cancel_left)
  2.1646  
  2.1647 -lemma mult_less_cancel_left_pos:
  2.1648 -  "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"
  2.1649 -by (auto simp: mult_less_cancel_left)
  2.1650 +lemma mult_less_cancel_left_pos: "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"
  2.1651 +  by (auto simp: mult_less_cancel_left)
  2.1652  
  2.1653 -lemma mult_less_cancel_left_neg:
  2.1654 -  "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a"
  2.1655 -by (auto simp: mult_less_cancel_left)
  2.1656 +lemma mult_less_cancel_left_neg: "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a"
  2.1657 +  by (auto simp: mult_less_cancel_left)
  2.1658  
  2.1659  end
  2.1660  
  2.1661 @@ -1783,19 +1695,19 @@
  2.1662  begin
  2.1663  
  2.1664  subclass zero_neq_one
  2.1665 -  proof qed (insert zero_less_one, blast)
  2.1666 +  by standard (insert zero_less_one, blast)
  2.1667  
  2.1668  subclass comm_semiring_1
  2.1669 -  proof qed (rule mult_1_left)
  2.1670 +  by standard (rule mult_1_left)
  2.1671  
  2.1672  lemma zero_le_one [simp]: "0 \<le> 1"
  2.1673 -by (rule zero_less_one [THEN less_imp_le])
  2.1674 +  by (rule zero_less_one [THEN less_imp_le])
  2.1675  
  2.1676  lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"
  2.1677 -by (simp add: not_le)
  2.1678 +  by (simp add: not_le)
  2.1679  
  2.1680  lemma not_one_less_zero [simp]: "\<not> 1 < 0"
  2.1681 -by (simp add: not_less)
  2.1682 +  by (simp add: not_less)
  2.1683  
  2.1684  lemma mult_left_le: "c \<le> 1 \<Longrightarrow> 0 \<le> a \<Longrightarrow> a * c \<le> a"
  2.1685    using mult_left_mono[of c 1 a] by simp
  2.1686 @@ -1812,8 +1724,7 @@
  2.1687    assumes le_add_diff_inverse2 [simp]: "b \<le> a \<Longrightarrow> a - b + b = a"
  2.1688  begin
  2.1689  
  2.1690 -subclass linordered_nonzero_semiring
  2.1691 -  proof qed
  2.1692 +subclass linordered_nonzero_semiring ..
  2.1693  
  2.1694  text \<open>Addition is the inverse of subtraction.\<close>
  2.1695  
  2.1696 @@ -1823,31 +1734,31 @@
  2.1697  lemma add_diff_inverse: "\<not> a < b \<Longrightarrow> b + (a - b) = a"
  2.1698    by simp
  2.1699  
  2.1700 -lemma add_le_imp_le_diff:
  2.1701 -  shows "i + k \<le> n \<Longrightarrow> i \<le> n - k"
  2.1702 +lemma add_le_imp_le_diff: "i + k \<le> n \<Longrightarrow> i \<le> n - k"
  2.1703    apply (subst add_le_cancel_right [where c=k, symmetric])
  2.1704    apply (frule le_add_diff_inverse2)
  2.1705    apply (simp only: add.assoc [symmetric])
  2.1706 -  using add_implies_diff by fastforce
  2.1707 +  using add_implies_diff apply fastforce
  2.1708 +  done
  2.1709  
  2.1710  lemma add_le_add_imp_diff_le:
  2.1711 -  assumes a1: "i + k \<le> n"
  2.1712 -      and a2: "n \<le> j + k"
  2.1713 -  shows "\<lbrakk>i + k \<le> n; n \<le> j + k\<rbrakk> \<Longrightarrow> n - k \<le> j"
  2.1714 +  assumes 1: "i + k \<le> n"
  2.1715 +    and 2: "n \<le> j + k"
  2.1716 +  shows "i + k \<le> n \<Longrightarrow> n \<le> j + k \<Longrightarrow> n - k \<le> j"
  2.1717  proof -
  2.1718    have "n - (i + k) + (i + k) = n"
  2.1719 -    using a1 by simp
  2.1720 +    using 1 by simp
  2.1721    moreover have "n - k = n - k - i + i"
  2.1722 -    using a1 by (simp add: add_le_imp_le_diff)
  2.1723 +    using 1 by (simp add: add_le_imp_le_diff)
  2.1724    ultimately show ?thesis
  2.1725 -    using a2
  2.1726 +    using 2
  2.1727      apply (simp add: add.assoc [symmetric])
  2.1728 -    apply (rule add_le_imp_le_diff [of _ k "j+k", simplified add_diff_cancel_right'])
  2.1729 -    by (simp add: add.commute diff_diff_add)
  2.1730 +    apply (rule add_le_imp_le_diff [of _ k "j + k", simplified add_diff_cancel_right'])
  2.1731 +    apply (simp add: add.commute diff_diff_add)
  2.1732 +    done
  2.1733  qed
  2.1734  
  2.1735 -lemma less_1_mult:
  2.1736 -  "1 < m \<Longrightarrow> 1 < n \<Longrightarrow> 1 < m * n"
  2.1737 +lemma less_1_mult: "1 < m \<Longrightarrow> 1 < n \<Longrightarrow> 1 < m * n"
  2.1738    using mult_strict_mono [of 1 m 1 n] by (simp add: less_trans [OF zero_less_one])
  2.1739  
  2.1740  end
  2.1741 @@ -1864,90 +1775,73 @@
  2.1742  subclass linordered_semidom
  2.1743  proof
  2.1744    have "0 \<le> 1 * 1" by (rule zero_le_square)
  2.1745 -  thus "0 < 1" by (simp add: le_less)
  2.1746 -  show "\<And>b a. b \<le> a \<Longrightarrow> a - b + b = a"
  2.1747 +  then show "0 < 1" by (simp add: le_less)
  2.1748 +  show "b \<le> a \<Longrightarrow> a - b + b = a" for a b
  2.1749      by simp
  2.1750  qed
  2.1751  
  2.1752  lemma linorder_neqE_linordered_idom:
  2.1753 -  assumes "x \<noteq> y" obtains "x < y" | "y < x"
  2.1754 +  assumes "x \<noteq> y"
  2.1755 +  obtains "x < y" | "y < x"
  2.1756    using assms by (rule neqE)
  2.1757  
  2.1758  text \<open>These cancellation simprules also produce two cases when the comparison is a goal.\<close>
  2.1759  
  2.1760 -lemma mult_le_cancel_right1:
  2.1761 -  "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
  2.1762 -by (insert mult_le_cancel_right [of 1 c b], simp)
  2.1763 +lemma mult_le_cancel_right1: "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
  2.1764 +  using mult_le_cancel_right [of 1 c b] by simp
  2.1765  
  2.1766 -lemma mult_le_cancel_right2:
  2.1767 -  "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
  2.1768 -by (insert mult_le_cancel_right [of a c 1], simp)
  2.1769 +lemma mult_le_cancel_right2: "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
  2.1770 +  using mult_le_cancel_right [of a c 1] by simp
  2.1771  
  2.1772 -lemma mult_le_cancel_left1:
  2.1773 -  "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
  2.1774 -by (insert mult_le_cancel_left [of c 1 b], simp)
  2.1775 +lemma mult_le_cancel_left1: "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
  2.1776 +  using mult_le_cancel_left [of c 1 b] by simp
  2.1777  
  2.1778 -lemma mult_le_cancel_left2:
  2.1779 -  "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
  2.1780 -by (insert mult_le_cancel_left [of c a 1], simp)
  2.1781 +lemma mult_le_cancel_left2: "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
  2.1782 +  using mult_le_cancel_left [of c a 1] by simp
  2.1783  
  2.1784 -lemma mult_less_cancel_right1:
  2.1785 -  "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
  2.1786 -by (insert mult_less_cancel_right [of 1 c b], simp)
  2.1787 +lemma mult_less_cancel_right1: "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
  2.1788 +  using mult_less_cancel_right [of 1 c b] by simp
  2.1789  
  2.1790 -lemma mult_less_cancel_right2:
  2.1791 -  "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
  2.1792 -by (insert mult_less_cancel_right [of a c 1], simp)
  2.1793 +lemma mult_less_cancel_right2: "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
  2.1794 +  using mult_less_cancel_right [of a c 1] by simp
  2.1795  
  2.1796 -lemma mult_less_cancel_left1:
  2.1797 -  "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
  2.1798 -by (insert mult_less_cancel_left [of c 1 b], simp)
  2.1799 +lemma mult_less_cancel_left1: "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
  2.1800 +  using mult_less_cancel_left [of c 1 b] by simp
  2.1801  
  2.1802 -lemma mult_less_cancel_left2:
  2.1803 -  "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
  2.1804 -by (insert mult_less_cancel_left [of c a 1], simp)
  2.1805 +lemma mult_less_cancel_left2: "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
  2.1806 +  using mult_less_cancel_left [of c a 1] by simp
  2.1807  
  2.1808 -lemma sgn_sgn [simp]:
  2.1809 -  "sgn (sgn a) = sgn a"
  2.1810 -unfolding sgn_if by simp
  2.1811 +lemma sgn_sgn [simp]: "sgn (sgn a) = sgn a"
  2.1812 +  unfolding sgn_if by simp
  2.1813  
  2.1814 -lemma sgn_0_0:
  2.1815 -  "sgn a = 0 \<longleftrightarrow> a = 0"
  2.1816 -unfolding sgn_if by simp
  2.1817 +lemma sgn_0_0: "sgn a = 0 \<longleftrightarrow> a = 0"
  2.1818 +  unfolding sgn_if by simp
  2.1819  
  2.1820 -lemma sgn_1_pos:
  2.1821 -  "sgn a = 1 \<longleftrightarrow> a > 0"
  2.1822 -unfolding sgn_if by simp
  2.1823 +lemma sgn_1_pos: "sgn a = 1 \<longleftrightarrow> a > 0"
  2.1824 +  unfolding sgn_if by simp
  2.1825  
  2.1826 -lemma sgn_1_neg:
  2.1827 -  "sgn a = - 1 \<longleftrightarrow> a < 0"
  2.1828 -unfolding sgn_if by auto
  2.1829 +lemma sgn_1_neg: "sgn a = - 1 \<longleftrightarrow> a < 0"
  2.1830 +  unfolding sgn_if by auto
  2.1831  
  2.1832 -lemma sgn_pos [simp]:
  2.1833 -  "0 < a \<Longrightarrow> sgn a = 1"
  2.1834 -unfolding sgn_1_pos .
  2.1835 +lemma sgn_pos [simp]: "0 < a \<Longrightarrow> sgn a = 1"
  2.1836 +  by (simp only: sgn_1_pos)
  2.1837  
  2.1838 -lemma sgn_neg [simp]:
  2.1839 -  "a < 0 \<Longrightarrow> sgn a = - 1"
  2.1840 -unfolding sgn_1_neg .
  2.1841 +lemma sgn_neg [simp]: "a < 0 \<Longrightarrow> sgn a = - 1"
  2.1842 +  by (simp only: sgn_1_neg)
  2.1843  
  2.1844 -lemma sgn_times:
  2.1845 -  "sgn (a * b) = sgn a * sgn b"
  2.1846 -by (auto simp add: sgn_if zero_less_mult_iff)
  2.1847 +lemma sgn_times: "sgn (a * b) = sgn a * sgn b"
  2.1848 +  by (auto simp add: sgn_if zero_less_mult_iff)
  2.1849  
  2.1850  lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"
  2.1851 -unfolding sgn_if abs_if by auto
  2.1852 +  unfolding sgn_if abs_if by auto
  2.1853  
  2.1854 -lemma sgn_greater [simp]:
  2.1855 -  "0 < sgn a \<longleftrightarrow> 0 < a"
  2.1856 +lemma sgn_greater [simp]: "0 < sgn a \<longleftrightarrow> 0 < a"
  2.1857    unfolding sgn_if by auto
  2.1858  
  2.1859 -lemma sgn_less [simp]:
  2.1860 -  "sgn a < 0 \<longleftrightarrow> a < 0"
  2.1861 +lemma sgn_less [simp]: "sgn a < 0 \<longleftrightarrow> a < 0"
  2.1862    unfolding sgn_if by auto
  2.1863  
  2.1864 -lemma abs_sgn_eq:
  2.1865 -  "\<bar>sgn a\<bar> = (if a = 0 then 0 else 1)"
  2.1866 +lemma abs_sgn_eq: "\<bar>sgn a\<bar> = (if a = 0 then 0 else 1)"
  2.1867    by (simp add: sgn_if)
  2.1868  
  2.1869  lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k"
  2.1870 @@ -1956,36 +1850,31 @@
  2.1871  lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k"
  2.1872    by (simp add: abs_if)
  2.1873  
  2.1874 -lemma dvd_if_abs_eq:
  2.1875 -  "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"
  2.1876 -by(subst abs_dvd_iff[symmetric]) simp
  2.1877 +lemma dvd_if_abs_eq: "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"
  2.1878 +  by (subst abs_dvd_iff [symmetric]) simp
  2.1879  
  2.1880 -text \<open>The following lemmas can be proven in more general structures, but
  2.1881 -are dangerous as simp rules in absence of @{thm neg_equal_zero},
  2.1882 -@{thm neg_less_pos}, @{thm neg_less_eq_nonneg}.\<close>
  2.1883 +text \<open>
  2.1884 +  The following lemmas can be proven in more general structures, but
  2.1885 +  are dangerous as simp rules in absence of @{thm neg_equal_zero},
  2.1886 +  @{thm neg_less_pos}, @{thm neg_less_eq_nonneg}.
  2.1887 +\<close>
  2.1888  
  2.1889 -lemma equation_minus_iff_1 [simp, no_atp]:
  2.1890 -  "1 = - a \<longleftrightarrow> a = - 1"
  2.1891 +lemma equation_minus_iff_1 [simp, no_atp]: "1 = - a \<longleftrightarrow> a = - 1"
  2.1892    by (fact equation_minus_iff)
  2.1893  
  2.1894 -lemma minus_equation_iff_1 [simp, no_atp]:
  2.1895 -  "- a = 1 \<longleftrightarrow> a = - 1"
  2.1896 +lemma minus_equation_iff_1 [simp, no_atp]: "- a = 1 \<longleftrightarrow> a = - 1"
  2.1897    by (subst minus_equation_iff, auto)
  2.1898  
  2.1899 -lemma le_minus_iff_1 [simp, no_atp]:
  2.1900 -  "1 \<le> - b \<longleftrightarrow> b \<le> - 1"
  2.1901 +lemma le_minus_iff_1 [simp, no_atp]: "1 \<le> - b \<longleftrightarrow> b \<le> - 1"
  2.1902    by (fact le_minus_iff)
  2.1903  
  2.1904 -lemma minus_le_iff_1 [simp, no_atp]:
  2.1905 -  "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"
  2.1906 +lemma minus_le_iff_1 [simp, no_atp]: "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"
  2.1907    by (fact minus_le_iff)
  2.1908  
  2.1909 -lemma less_minus_iff_1 [simp, no_atp]:
  2.1910 -  "1 < - b \<longleftrightarrow> b < - 1"
  2.1911 +lemma less_minus_iff_1 [simp, no_atp]: "1 < - b \<longleftrightarrow> b < - 1"
  2.1912    by (fact less_minus_iff)
  2.1913  
  2.1914 -lemma minus_less_iff_1 [simp, no_atp]:
  2.1915 -  "- a < 1 \<longleftrightarrow> - 1 < a"
  2.1916 +lemma minus_less_iff_1 [simp, no_atp]: "- a < 1 \<longleftrightarrow> - 1 < a"
  2.1917    by (fact minus_less_iff)
  2.1918  
  2.1919  end
  2.1920 @@ -1993,15 +1882,16 @@
  2.1921  text \<open>Simprules for comparisons where common factors can be cancelled.\<close>
  2.1922  
  2.1923  lemmas mult_compare_simps =
  2.1924 -    mult_le_cancel_right mult_le_cancel_left
  2.1925 -    mult_le_cancel_right1 mult_le_cancel_right2
  2.1926 -    mult_le_cancel_left1 mult_le_cancel_left2
  2.1927 -    mult_less_cancel_right mult_less_cancel_left
  2.1928 -    mult_less_cancel_right1 mult_less_cancel_right2
  2.1929 -    mult_less_cancel_left1 mult_less_cancel_left2
  2.1930 -    mult_cancel_right mult_cancel_left
  2.1931 -    mult_cancel_right1 mult_cancel_right2
  2.1932 -    mult_cancel_left1 mult_cancel_left2
  2.1933 +  mult_le_cancel_right mult_le_cancel_left
  2.1934 +  mult_le_cancel_right1 mult_le_cancel_right2
  2.1935 +  mult_le_cancel_left1 mult_le_cancel_left2
  2.1936 +  mult_less_cancel_right mult_less_cancel_left
  2.1937 +  mult_less_cancel_right1 mult_less_cancel_right2
  2.1938 +  mult_less_cancel_left1 mult_less_cancel_left2
  2.1939 +  mult_cancel_right mult_cancel_left
  2.1940 +  mult_cancel_right1 mult_cancel_right2
  2.1941 +  mult_cancel_left1 mult_cancel_left2
  2.1942 +
  2.1943  
  2.1944  text \<open>Reasoning about inequalities with division\<close>
  2.1945  
  2.1946 @@ -2012,7 +1902,7 @@
  2.1947  proof -
  2.1948    have "a + 0 < a + 1"
  2.1949      by (blast intro: zero_less_one add_strict_left_mono)
  2.1950 -  thus ?thesis by simp
  2.1951 +  then show ?thesis by simp
  2.1952  qed
  2.1953  
  2.1954  end
  2.1955 @@ -2020,12 +1910,10 @@
  2.1956  context linordered_idom
  2.1957  begin
  2.1958  
  2.1959 -lemma mult_right_le_one_le:
  2.1960 -  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"
  2.1961 +lemma mult_right_le_one_le: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"
  2.1962    by (rule mult_left_le)
  2.1963  
  2.1964 -lemma mult_left_le_one_le:
  2.1965 -  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"
  2.1966 +lemma mult_left_le_one_le: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"
  2.1967    by (auto simp add: mult_le_cancel_right2)
  2.1968  
  2.1969  end
  2.1970 @@ -2035,12 +1923,10 @@
  2.1971  context linordered_idom
  2.1972  begin
  2.1973  
  2.1974 -lemma mult_sgn_abs:
  2.1975 -  "sgn x * \<bar>x\<bar> = x"
  2.1976 +lemma mult_sgn_abs: "sgn x * \<bar>x\<bar> = x"
  2.1977    unfolding abs_if sgn_if by auto
  2.1978  
  2.1979 -lemma abs_one [simp]:
  2.1980 -  "\<bar>1\<bar> = 1"
  2.1981 +lemma abs_one [simp]: "\<bar>1\<bar> = 1"
  2.1982    by (simp add: abs_if)
  2.1983  
  2.1984  end
  2.1985 @@ -2052,57 +1938,54 @@
  2.1986  context linordered_idom
  2.1987  begin
  2.1988  
  2.1989 -subclass ordered_ring_abs proof
  2.1990 -qed (auto simp add: abs_if not_less mult_less_0_iff)
  2.1991 +subclass ordered_ring_abs
  2.1992 +  by standard (auto simp add: abs_if not_less mult_less_0_iff)
  2.1993  
  2.1994 -lemma abs_mult:
  2.1995 -  "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
  2.1996 +lemma abs_mult: "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
  2.1997    by (rule abs_eq_mult) auto
  2.1998  
  2.1999 -lemma abs_mult_self [simp]:
  2.2000 -  "\<bar>a\<bar> * \<bar>a\<bar> = a * a"
  2.2001 +lemma abs_mult_self [simp]: "\<bar>a\<bar> * \<bar>a\<bar> = a * a"
  2.2002    by (simp add: abs_if)
  2.2003  
  2.2004  lemma abs_mult_less:
  2.2005 -  "\<bar>a\<bar> < c \<Longrightarrow> \<bar>b\<bar> < d \<Longrightarrow> \<bar>a\<bar> * \<bar>b\<bar> < c * d"
  2.2006 +  assumes ac: "\<bar>a\<bar> < c"
  2.2007 +    and bd: "\<bar>b\<bar> < d"
  2.2008 +  shows "\<bar>a\<bar> * \<bar>b\<bar> < c * d"
  2.2009  proof -
  2.2010 -  assume ac: "\<bar>a\<bar> < c"
  2.2011 -  hence cpos: "0<c" by (blast intro: le_less_trans abs_ge_zero)
  2.2012 -  assume "\<bar>b\<bar> < d"
  2.2013 -  thus ?thesis by (simp add: ac cpos mult_strict_mono)
  2.2014 +  from ac have "0 < c"
  2.2015 +    by (blast intro: le_less_trans abs_ge_zero)
  2.2016 +  with bd show ?thesis by (simp add: ac mult_strict_mono)
  2.2017  qed
  2.2018  
  2.2019 -lemma abs_less_iff:
  2.2020 -  "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b"
  2.2021 +lemma abs_less_iff: "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b"
  2.2022    by (simp add: less_le abs_le_iff) (auto simp add: abs_if)
  2.2023  
  2.2024 -lemma abs_mult_pos:
  2.2025 -  "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"
  2.2026 +lemma abs_mult_pos: "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"
  2.2027    by (simp add: abs_mult)
  2.2028  
  2.2029 -lemma abs_diff_less_iff:
  2.2030 -  "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r"
  2.2031 +lemma abs_diff_less_iff: "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r"
  2.2032    by (auto simp add: diff_less_eq ac_simps abs_less_iff)
  2.2033  
  2.2034 -lemma abs_diff_le_iff:
  2.2035 -   "\<bar>x - a\<bar> \<le> r \<longleftrightarrow> a - r \<le> x \<and> x \<le> a + r"
  2.2036 +lemma abs_diff_le_iff: "\<bar>x - a\<bar> \<le> r \<longleftrightarrow> a - r \<le> x \<and> x \<le> a + r"
  2.2037    by (auto simp add: diff_le_eq ac_simps abs_le_iff)
  2.2038  
  2.2039  lemma abs_add_one_gt_zero: "0 < 1 + \<bar>x\<bar>"
  2.2040 -  by (force simp: abs_if not_less intro: zero_less_one add_strict_increasing less_trans)
  2.2041 +  by (auto simp: abs_if not_less intro: zero_less_one add_strict_increasing less_trans)
  2.2042  
  2.2043  end
  2.2044  
  2.2045  subsection \<open>Dioids\<close>
  2.2046  
  2.2047 -text \<open>Dioids are the alternative extensions of semirings, a semiring can either be a ring or a dioid
  2.2048 -but never both.\<close>
  2.2049 +text \<open>
  2.2050 +  Dioids are the alternative extensions of semirings, a semiring can
  2.2051 +  either be a ring or a dioid but never both.
  2.2052 +\<close>
  2.2053  
  2.2054  class dioid = semiring_1 + canonically_ordered_monoid_add
  2.2055  begin
  2.2056  
  2.2057  subclass ordered_semiring
  2.2058 -  proof qed (auto simp: le_iff_add distrib_left distrib_right)
  2.2059 +  by standard (auto simp: le_iff_add distrib_left distrib_right)
  2.2060  
  2.2061  end
  2.2062