src/HOL/Rings.thy
changeset 63325 1086d56cde86
parent 63040 eb4ddd18d635
child 63359 99b51ba8da1c
     1.1 --- a/src/HOL/Rings.thy	Mon Jun 20 17:51:47 2016 +0200
     1.2 +++ b/src/HOL/Rings.thy	Mon Jun 20 21:40:48 2016 +0200
     1.3 @@ -18,10 +18,9 @@
     1.4    assumes distrib_left[algebra_simps]: "a * (b + c) = a * b + a * c"
     1.5  begin
     1.6  
     1.7 -text\<open>For the \<open>combine_numerals\<close> simproc\<close>
     1.8 -lemma combine_common_factor:
     1.9 -  "a * e + (b * e + c) = (a + b) * e + c"
    1.10 -by (simp add: distrib_right ac_simps)
    1.11 +text \<open>For the \<open>combine_numerals\<close> simproc\<close>
    1.12 +lemma combine_common_factor: "a * e + (b * e + c) = (a + b) * e + c"
    1.13 +  by (simp add: distrib_right ac_simps)
    1.14  
    1.15  end
    1.16  
    1.17 @@ -30,8 +29,7 @@
    1.18    assumes mult_zero_right [simp]: "a * 0 = 0"
    1.19  begin
    1.20  
    1.21 -lemma mult_not_zero:
    1.22 -  "a * b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<and> b \<noteq> 0"
    1.23 +lemma mult_not_zero: "a * b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<and> b \<noteq> 0"
    1.24    by auto
    1.25  
    1.26  end
    1.27 @@ -45,11 +43,9 @@
    1.28  proof
    1.29    fix a :: 'a
    1.30    have "0 * a + 0 * a = 0 * a + 0" by (simp add: distrib_right [symmetric])
    1.31 -  thus "0 * a = 0" by (simp only: add_left_cancel)
    1.32 -next
    1.33 -  fix a :: 'a
    1.34 +  then show "0 * a = 0" by (simp only: add_left_cancel)
    1.35    have "a * 0 + a * 0 = a * 0 + 0" by (simp add: distrib_left [symmetric])
    1.36 -  thus "a * 0 = 0" by (simp only: add_left_cancel)
    1.37 +  then show "a * 0 = 0" by (simp only: add_left_cancel)
    1.38  qed
    1.39  
    1.40  end
    1.41 @@ -63,8 +59,8 @@
    1.42    fix a b c :: 'a
    1.43    show "(a + b) * c = a * c + b * c" by (simp add: distrib)
    1.44    have "a * (b + c) = (b + c) * a" by (simp add: ac_simps)
    1.45 -  also have "... = b * a + c * a" by (simp only: distrib)
    1.46 -  also have "... = a * b + a * c" by (simp add: ac_simps)
    1.47 +  also have "\<dots> = b * a + c * a" by (simp only: distrib)
    1.48 +  also have "\<dots> = a * b + a * c" by (simp add: ac_simps)
    1.49    finally show "a * (b + c) = a * b + a * c" by blast
    1.50  qed
    1.51  
    1.52 @@ -91,27 +87,23 @@
    1.53  begin
    1.54  
    1.55  lemma one_neq_zero [simp]: "1 \<noteq> 0"
    1.56 -by (rule not_sym) (rule zero_neq_one)
    1.57 +  by (rule not_sym) (rule zero_neq_one)
    1.58  
    1.59  definition of_bool :: "bool \<Rightarrow> 'a"
    1.60 -where
    1.61 -  "of_bool p = (if p then 1 else 0)"
    1.62 +  where "of_bool p = (if p then 1 else 0)"
    1.63  
    1.64  lemma of_bool_eq [simp, code]:
    1.65    "of_bool False = 0"
    1.66    "of_bool True = 1"
    1.67    by (simp_all add: of_bool_def)
    1.68  
    1.69 -lemma of_bool_eq_iff:
    1.70 -  "of_bool p = of_bool q \<longleftrightarrow> p = q"
    1.71 +lemma of_bool_eq_iff: "of_bool p = of_bool q \<longleftrightarrow> p = q"
    1.72    by (simp add: of_bool_def)
    1.73  
    1.74 -lemma split_of_bool [split]:
    1.75 -  "P (of_bool p) \<longleftrightarrow> (p \<longrightarrow> P 1) \<and> (\<not> p \<longrightarrow> P 0)"
    1.76 +lemma split_of_bool [split]: "P (of_bool p) \<longleftrightarrow> (p \<longrightarrow> P 1) \<and> (\<not> p \<longrightarrow> P 0)"
    1.77    by (cases p) simp_all
    1.78  
    1.79 -lemma split_of_bool_asm:
    1.80 -  "P (of_bool p) \<longleftrightarrow> \<not> (p \<and> \<not> P 1 \<or> \<not> p \<and> \<not> P 0)"
    1.81 +lemma split_of_bool_asm: "P (of_bool p) \<longleftrightarrow> \<not> (p \<and> \<not> P 1 \<or> \<not> p \<and> \<not> P 0)"
    1.82    by (cases p) simp_all
    1.83  
    1.84  end
    1.85 @@ -123,8 +115,8 @@
    1.86  class dvd = times
    1.87  begin
    1.88  
    1.89 -definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "dvd" 50) where
    1.90 -  "b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)"
    1.91 +definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "dvd" 50)
    1.92 +  where "b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)"
    1.93  
    1.94  lemma dvdI [intro?]: "a = b * k \<Longrightarrow> b dvd a"
    1.95    unfolding dvd_def ..
    1.96 @@ -139,8 +131,7 @@
    1.97  
    1.98  subclass dvd .
    1.99  
   1.100 -lemma dvd_refl [simp]:
   1.101 -  "a dvd a"
   1.102 +lemma dvd_refl [simp]: "a dvd a"
   1.103  proof
   1.104    show "a = a * 1" by simp
   1.105  qed
   1.106 @@ -155,32 +146,25 @@
   1.107    then show ?thesis ..
   1.108  qed
   1.109  
   1.110 -lemma subset_divisors_dvd:
   1.111 -  "{c. c dvd a} \<subseteq> {c. c dvd b} \<longleftrightarrow> a dvd b"
   1.112 +lemma subset_divisors_dvd: "{c. c dvd a} \<subseteq> {c. c dvd b} \<longleftrightarrow> a dvd b"
   1.113    by (auto simp add: subset_iff intro: dvd_trans)
   1.114  
   1.115 -lemma strict_subset_divisors_dvd:
   1.116 -  "{c. c dvd a} \<subset> {c. c dvd b} \<longleftrightarrow> a dvd b \<and> \<not> b dvd a"
   1.117 +lemma strict_subset_divisors_dvd: "{c. c dvd a} \<subset> {c. c dvd b} \<longleftrightarrow> a dvd b \<and> \<not> b dvd a"
   1.118    by (auto simp add: subset_iff intro: dvd_trans)
   1.119  
   1.120 -lemma one_dvd [simp]:
   1.121 -  "1 dvd a"
   1.122 +lemma one_dvd [simp]: "1 dvd a"
   1.123    by (auto intro!: dvdI)
   1.124  
   1.125 -lemma dvd_mult [simp]:
   1.126 -  "a dvd c \<Longrightarrow> a dvd (b * c)"
   1.127 +lemma dvd_mult [simp]: "a dvd c \<Longrightarrow> a dvd (b * c)"
   1.128    by (auto intro!: mult.left_commute dvdI elim!: dvdE)
   1.129  
   1.130 -lemma dvd_mult2 [simp]:
   1.131 -  "a dvd b \<Longrightarrow> a dvd (b * c)"
   1.132 +lemma dvd_mult2 [simp]: "a dvd b \<Longrightarrow> a dvd (b * c)"
   1.133    using dvd_mult [of a b c] by (simp add: ac_simps)
   1.134  
   1.135 -lemma dvd_triv_right [simp]:
   1.136 -  "a dvd b * a"
   1.137 +lemma dvd_triv_right [simp]: "a dvd b * a"
   1.138    by (rule dvd_mult) (rule dvd_refl)
   1.139  
   1.140 -lemma dvd_triv_left [simp]:
   1.141 -  "a dvd a * b"
   1.142 +lemma dvd_triv_left [simp]: "a dvd a * b"
   1.143    by (rule dvd_mult2) (rule dvd_refl)
   1.144  
   1.145  lemma mult_dvd_mono:
   1.146 @@ -194,12 +178,10 @@
   1.147    then show ?thesis ..
   1.148  qed
   1.149  
   1.150 -lemma dvd_mult_left:
   1.151 -  "a * b dvd c \<Longrightarrow> a dvd c"
   1.152 +lemma dvd_mult_left: "a * b dvd c \<Longrightarrow> a dvd c"
   1.153    by (simp add: dvd_def mult.assoc) blast
   1.154  
   1.155 -lemma dvd_mult_right:
   1.156 -  "a * b dvd c \<Longrightarrow> b dvd c"
   1.157 +lemma dvd_mult_right: "a * b dvd c \<Longrightarrow> b dvd c"
   1.158    using dvd_mult_left [of b a c] by (simp add: ac_simps)
   1.159  
   1.160  end
   1.161 @@ -209,18 +191,15 @@
   1.162  
   1.163  subclass semiring_1 ..
   1.164  
   1.165 -lemma dvd_0_left_iff [simp]:
   1.166 -  "0 dvd a \<longleftrightarrow> a = 0"
   1.167 +lemma dvd_0_left_iff [simp]: "0 dvd a \<longleftrightarrow> a = 0"
   1.168    by (auto intro: dvd_refl elim!: dvdE)
   1.169  
   1.170 -lemma dvd_0_right [iff]:
   1.171 -  "a dvd 0"
   1.172 +lemma dvd_0_right [iff]: "a dvd 0"
   1.173  proof
   1.174    show "0 = a * 0" by simp
   1.175  qed
   1.176  
   1.177 -lemma dvd_0_left:
   1.178 -  "0 dvd a \<Longrightarrow> a = 0"
   1.179 +lemma dvd_0_left: "0 dvd a \<Longrightarrow> a = 0"
   1.180    by simp
   1.181  
   1.182  lemma dvd_add [simp]:
   1.183 @@ -245,8 +224,8 @@
   1.184  
   1.185  end
   1.186  
   1.187 -class comm_semiring_1_cancel = comm_semiring + cancel_comm_monoid_add +
   1.188 -                               zero_neq_one + comm_monoid_mult +
   1.189 +class comm_semiring_1_cancel =
   1.190 +  comm_semiring + cancel_comm_monoid_add + zero_neq_one + comm_monoid_mult +
   1.191    assumes right_diff_distrib' [algebra_simps]: "a * (b - c) = a * b - a * c"
   1.192  begin
   1.193  
   1.194 @@ -254,16 +233,15 @@
   1.195  subclass comm_semiring_0_cancel ..
   1.196  subclass comm_semiring_1 ..
   1.197  
   1.198 -lemma left_diff_distrib' [algebra_simps]:
   1.199 -  "(b - c) * a = b * a - c * a"
   1.200 +lemma left_diff_distrib' [algebra_simps]: "(b - c) * a = b * a - c * a"
   1.201    by (simp add: algebra_simps)
   1.202  
   1.203 -lemma dvd_add_times_triv_left_iff [simp]:
   1.204 -  "a dvd c * a + b \<longleftrightarrow> a dvd b"
   1.205 +lemma dvd_add_times_triv_left_iff [simp]: "a dvd c * a + b \<longleftrightarrow> a dvd b"
   1.206  proof -
   1.207    have "a dvd a * c + b \<longleftrightarrow> a dvd b" (is "?P \<longleftrightarrow> ?Q")
   1.208    proof
   1.209 -    assume ?Q then show ?P by simp
   1.210 +    assume ?Q
   1.211 +    then show ?P by simp
   1.212    next
   1.213      assume ?P
   1.214      then obtain d where "a * c + b = a * d" ..
   1.215 @@ -275,23 +253,21 @@
   1.216    then show "a dvd c * a + b \<longleftrightarrow> a dvd b" by (simp add: ac_simps)
   1.217  qed
   1.218  
   1.219 -lemma dvd_add_times_triv_right_iff [simp]:
   1.220 -  "a dvd b + c * a \<longleftrightarrow> a dvd b"
   1.221 +lemma dvd_add_times_triv_right_iff [simp]: "a dvd b + c * a \<longleftrightarrow> a dvd b"
   1.222    using dvd_add_times_triv_left_iff [of a c b] by (simp add: ac_simps)
   1.223  
   1.224 -lemma dvd_add_triv_left_iff [simp]:
   1.225 -  "a dvd a + b \<longleftrightarrow> a dvd b"
   1.226 +lemma dvd_add_triv_left_iff [simp]: "a dvd a + b \<longleftrightarrow> a dvd b"
   1.227    using dvd_add_times_triv_left_iff [of a 1 b] by simp
   1.228  
   1.229 -lemma dvd_add_triv_right_iff [simp]:
   1.230 -  "a dvd b + a \<longleftrightarrow> a dvd b"
   1.231 +lemma dvd_add_triv_right_iff [simp]: "a dvd b + a \<longleftrightarrow> a dvd b"
   1.232    using dvd_add_times_triv_right_iff [of a b 1] by simp
   1.233  
   1.234  lemma dvd_add_right_iff:
   1.235    assumes "a dvd b"
   1.236    shows "a dvd b + c \<longleftrightarrow> a dvd c" (is "?P \<longleftrightarrow> ?Q")
   1.237  proof
   1.238 -  assume ?P then obtain d where "b + c = a * d" ..
   1.239 +  assume ?P
   1.240 +  then obtain d where "b + c = a * d" ..
   1.241    moreover from \<open>a dvd b\<close> obtain e where "b = a * e" ..
   1.242    ultimately have "a * e + c = a * d" by simp
   1.243    then have "a * e + c - a * e = a * d - a * e" by simp
   1.244 @@ -299,13 +275,12 @@
   1.245    then have "c = a * (d - e)" by (simp add: algebra_simps)
   1.246    then show ?Q ..
   1.247  next
   1.248 -  assume ?Q with assms show ?P by simp
   1.249 +  assume ?Q
   1.250 +  with assms show ?P by simp
   1.251  qed
   1.252  
   1.253 -lemma dvd_add_left_iff:
   1.254 -  assumes "a dvd c"
   1.255 -  shows "a dvd b + c \<longleftrightarrow> a dvd b"
   1.256 -  using assms dvd_add_right_iff [of a c b] by (simp add: ac_simps)
   1.257 +lemma dvd_add_left_iff: "a dvd c \<Longrightarrow> a dvd b + c \<longleftrightarrow> a dvd b"
   1.258 +  using dvd_add_right_iff [of a c b] by (simp add: ac_simps)
   1.259  
   1.260  end
   1.261  
   1.262 @@ -317,44 +292,38 @@
   1.263  text \<open>Distribution rules\<close>
   1.264  
   1.265  lemma minus_mult_left: "- (a * b) = - a * b"
   1.266 -by (rule minus_unique) (simp add: distrib_right [symmetric])
   1.267 +  by (rule minus_unique) (simp add: distrib_right [symmetric])
   1.268  
   1.269  lemma minus_mult_right: "- (a * b) = a * - b"
   1.270 -by (rule minus_unique) (simp add: distrib_left [symmetric])
   1.271 +  by (rule minus_unique) (simp add: distrib_left [symmetric])
   1.272  
   1.273 -text\<open>Extract signs from products\<close>
   1.274 +text \<open>Extract signs from products\<close>
   1.275  lemmas mult_minus_left [simp] = minus_mult_left [symmetric]
   1.276  lemmas mult_minus_right [simp] = minus_mult_right [symmetric]
   1.277  
   1.278  lemma minus_mult_minus [simp]: "- a * - b = a * b"
   1.279 -by simp
   1.280 +  by simp
   1.281  
   1.282  lemma minus_mult_commute: "- a * b = a * - b"
   1.283 -by simp
   1.284 +  by simp
   1.285  
   1.286 -lemma right_diff_distrib [algebra_simps]:
   1.287 -  "a * (b - c) = a * b - a * c"
   1.288 +lemma right_diff_distrib [algebra_simps]: "a * (b - c) = a * b - a * c"
   1.289    using distrib_left [of a b "-c "] by simp
   1.290  
   1.291 -lemma left_diff_distrib [algebra_simps]:
   1.292 -  "(a - b) * c = a * c - b * c"
   1.293 +lemma left_diff_distrib [algebra_simps]: "(a - b) * c = a * c - b * c"
   1.294    using distrib_right [of a "- b" c] by simp
   1.295  
   1.296 -lemmas ring_distribs =
   1.297 -  distrib_left distrib_right left_diff_distrib right_diff_distrib
   1.298 +lemmas ring_distribs = distrib_left distrib_right left_diff_distrib right_diff_distrib
   1.299  
   1.300 -lemma eq_add_iff1:
   1.301 -  "a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d"
   1.302 -by (simp add: algebra_simps)
   1.303 +lemma eq_add_iff1: "a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d"
   1.304 +  by (simp add: algebra_simps)
   1.305  
   1.306 -lemma eq_add_iff2:
   1.307 -  "a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d"
   1.308 -by (simp add: algebra_simps)
   1.309 +lemma eq_add_iff2: "a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d"
   1.310 +  by (simp add: algebra_simps)
   1.311  
   1.312  end
   1.313  
   1.314 -lemmas ring_distribs =
   1.315 -  distrib_left distrib_right left_diff_distrib right_diff_distrib
   1.316 +lemmas ring_distribs = distrib_left distrib_right left_diff_distrib right_diff_distrib
   1.317  
   1.318  class comm_ring = comm_semiring + ab_group_add
   1.319  begin
   1.320 @@ -362,8 +331,7 @@
   1.321  subclass ring ..
   1.322  subclass comm_semiring_0_cancel ..
   1.323  
   1.324 -lemma square_diff_square_factored:
   1.325 -  "x * x - y * y = (x + y) * (x - y)"
   1.326 +lemma square_diff_square_factored: "x * x - y * y = (x + y) * (x - y)"
   1.327    by (simp add: algebra_simps)
   1.328  
   1.329  end
   1.330 @@ -373,8 +341,7 @@
   1.331  
   1.332  subclass semiring_1_cancel ..
   1.333  
   1.334 -lemma square_diff_one_factored:
   1.335 -  "x * x - 1 = (x + 1) * (x - 1)"
   1.336 +lemma square_diff_one_factored: "x * x - 1 = (x + 1) * (x - 1)"
   1.337    by (simp add: algebra_simps)
   1.338  
   1.339  end
   1.340 @@ -410,8 +377,7 @@
   1.341    then show "- x dvd y" ..
   1.342  qed
   1.343  
   1.344 -lemma dvd_diff [simp]:
   1.345 -  "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)"
   1.346 +lemma dvd_diff [simp]: "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)"
   1.347    using dvd_add [of x y "- z"] by simp
   1.348  
   1.349  end
   1.350 @@ -424,19 +390,20 @@
   1.351    assumes "a * b = 0"
   1.352    shows "a = 0 \<or> b = 0"
   1.353  proof (rule classical)
   1.354 -  assume "\<not> (a = 0 \<or> b = 0)"
   1.355 +  assume "\<not> ?thesis"
   1.356    then have "a \<noteq> 0" and "b \<noteq> 0" by auto
   1.357    with no_zero_divisors have "a * b \<noteq> 0" by blast
   1.358    with assms show ?thesis by simp
   1.359  qed
   1.360  
   1.361 -lemma mult_eq_0_iff [simp]:
   1.362 -  shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
   1.363 +lemma mult_eq_0_iff [simp]: "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
   1.364  proof (cases "a = 0 \<or> b = 0")
   1.365 -  case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
   1.366 +  case False
   1.367 +  then have "a \<noteq> 0" and "b \<noteq> 0" by auto
   1.368      then show ?thesis using no_zero_divisors by simp
   1.369  next
   1.370 -  case True then show ?thesis by auto
   1.371 +  case True
   1.372 +  then show ?thesis by auto
   1.373  qed
   1.374  
   1.375  end
   1.376 @@ -448,12 +415,10 @@
   1.377      and mult_cancel_left [simp]: "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
   1.378  begin
   1.379  
   1.380 -lemma mult_left_cancel:
   1.381 -  "c \<noteq> 0 \<Longrightarrow> c * a = c * b \<longleftrightarrow> a = b"
   1.382 +lemma mult_left_cancel: "c \<noteq> 0 \<Longrightarrow> c * a = c * b \<longleftrightarrow> a = b"
   1.383    by simp
   1.384  
   1.385 -lemma mult_right_cancel:
   1.386 -  "c \<noteq> 0 \<Longrightarrow> a * c = b * c \<longleftrightarrow> a = b"
   1.387 +lemma mult_right_cancel: "c \<noteq> 0 \<Longrightarrow> a * c = b * c \<longleftrightarrow> a = b"
   1.388    by simp
   1.389  
   1.390  end
   1.391 @@ -483,32 +448,27 @@
   1.392  
   1.393  subclass semiring_1_no_zero_divisors ..
   1.394  
   1.395 -lemma square_eq_1_iff:
   1.396 -  "x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1"
   1.397 +lemma square_eq_1_iff: "x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1"
   1.398  proof -
   1.399    have "(x - 1) * (x + 1) = x * x - 1"
   1.400      by (simp add: algebra_simps)
   1.401 -  hence "x * x = 1 \<longleftrightarrow> (x - 1) * (x + 1) = 0"
   1.402 +  then have "x * x = 1 \<longleftrightarrow> (x - 1) * (x + 1) = 0"
   1.403      by simp
   1.404 -  thus ?thesis
   1.405 +  then show ?thesis
   1.406      by (simp add: eq_neg_iff_add_eq_0)
   1.407  qed
   1.408  
   1.409 -lemma mult_cancel_right1 [simp]:
   1.410 -  "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
   1.411 -by (insert mult_cancel_right [of 1 c b], force)
   1.412 +lemma mult_cancel_right1 [simp]: "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
   1.413 +  using mult_cancel_right [of 1 c b] by auto
   1.414  
   1.415 -lemma mult_cancel_right2 [simp]:
   1.416 -  "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
   1.417 -by (insert mult_cancel_right [of a c 1], simp)
   1.418 +lemma mult_cancel_right2 [simp]: "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
   1.419 +  using mult_cancel_right [of a c 1] by simp
   1.420  
   1.421 -lemma mult_cancel_left1 [simp]:
   1.422 -  "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
   1.423 -by (insert mult_cancel_left [of c 1 b], force)
   1.424 +lemma mult_cancel_left1 [simp]: "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
   1.425 +  using mult_cancel_left [of c 1 b] by force
   1.426  
   1.427 -lemma mult_cancel_left2 [simp]:
   1.428 -  "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
   1.429 -by (insert mult_cancel_left [of c a 1], simp)
   1.430 +lemma mult_cancel_left2 [simp]: "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
   1.431 +  using mult_cancel_left [of c a 1] by simp
   1.432  
   1.433  end
   1.434  
   1.435 @@ -526,8 +486,7 @@
   1.436  
   1.437  subclass ring_1_no_zero_divisors ..
   1.438  
   1.439 -lemma dvd_mult_cancel_right [simp]:
   1.440 -  "a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b"
   1.441 +lemma dvd_mult_cancel_right [simp]: "a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b"
   1.442  proof -
   1.443    have "a * c dvd b * c \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
   1.444      unfolding dvd_def by (simp add: ac_simps)
   1.445 @@ -536,8 +495,7 @@
   1.446    finally show ?thesis .
   1.447  qed
   1.448  
   1.449 -lemma dvd_mult_cancel_left [simp]:
   1.450 -  "c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b"
   1.451 +lemma dvd_mult_cancel_left [simp]: "c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b"
   1.452  proof -
   1.453    have "c * a dvd c * b \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
   1.454      unfolding dvd_def by (simp add: ac_simps)
   1.455 @@ -562,15 +520,12 @@
   1.456  
   1.457  text \<open>
   1.458    The theory of partially ordered rings is taken from the books:
   1.459 -  \begin{itemize}
   1.460 -  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979
   1.461 -  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
   1.462 -  \end{itemize}
   1.463 +    \<^item> \<^emph>\<open>Lattice Theory\<close> by Garret Birkhoff, American Mathematical Society, 1979
   1.464 +    \<^item> \<^emph>\<open>Partially Ordered Algebraic Systems\<close>, Pergamon Press, 1963
   1.465 +
   1.466    Most of the used notions can also be looked up in
   1.467 -  \begin{itemize}
   1.468 -  \item @{url "http://www.mathworld.com"} by Eric Weisstein et. al.
   1.469 -  \item \emph{Algebra I} by van der Waerden, Springer.
   1.470 -  \end{itemize}
   1.471 +    \<^item> @{url "http://www.mathworld.com"} by Eric Weisstein et. al.
   1.472 +    \<^item> \<^emph>\<open>Algebra I\<close> by van der Waerden, Springer
   1.473  \<close>
   1.474  
   1.475  class divide =
   1.476 @@ -605,49 +560,45 @@
   1.477    assumes divide_zero [simp]: "a div 0 = 0"
   1.478  begin
   1.479  
   1.480 -lemma nonzero_mult_divide_cancel_left [simp]:
   1.481 -  "a \<noteq> 0 \<Longrightarrow> (a * b) div a = b"
   1.482 +lemma nonzero_mult_divide_cancel_left [simp]: "a \<noteq> 0 \<Longrightarrow> (a * b) div a = b"
   1.483    using nonzero_mult_divide_cancel_right [of a b] by (simp add: ac_simps)
   1.484  
   1.485  subclass semiring_no_zero_divisors_cancel
   1.486  proof
   1.487 -  fix a b c
   1.488 -  { fix a b c
   1.489 -    show "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
   1.490 -    proof (cases "c = 0")
   1.491 -      case True then show ?thesis by simp
   1.492 -    next
   1.493 -      case False
   1.494 -      { assume "a * c = b * c"
   1.495 -        then have "a * c div c = b * c div c"
   1.496 -          by simp
   1.497 -        with False have "a = b"
   1.498 -          by simp
   1.499 -      } then show ?thesis by auto
   1.500 -    qed
   1.501 -  }
   1.502 -  from this [of a c b]
   1.503 -  show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
   1.504 -    by (simp add: ac_simps)
   1.505 +  show *: "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b" for a b c
   1.506 +  proof (cases "c = 0")
   1.507 +    case True
   1.508 +    then show ?thesis by simp
   1.509 +  next
   1.510 +    case False
   1.511 +    {
   1.512 +      assume "a * c = b * c"
   1.513 +      then have "a * c div c = b * c div c"
   1.514 +        by simp
   1.515 +      with False have "a = b"
   1.516 +        by simp
   1.517 +    }
   1.518 +    then show ?thesis by auto
   1.519 +  qed
   1.520 +  show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b" for a b c
   1.521 +    using * [of a c b] by (simp add: ac_simps)
   1.522  qed
   1.523  
   1.524 -lemma div_self [simp]:
   1.525 -  assumes "a \<noteq> 0"
   1.526 -  shows "a div a = 1"
   1.527 -  using assms nonzero_mult_divide_cancel_left [of a 1] by simp
   1.528 +lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1"
   1.529 +  using nonzero_mult_divide_cancel_left [of a 1] by simp
   1.530  
   1.531 -lemma divide_zero_left [simp]:
   1.532 -  "0 div a = 0"
   1.533 +lemma divide_zero_left [simp]: "0 div a = 0"
   1.534  proof (cases "a = 0")
   1.535 -  case True then show ?thesis by simp
   1.536 +  case True
   1.537 +  then show ?thesis by simp
   1.538  next
   1.539 -  case False then have "a * 0 div a = 0"
   1.540 +  case False
   1.541 +  then have "a * 0 div a = 0"
   1.542      by (rule nonzero_mult_divide_cancel_left)
   1.543    then show ?thesis by simp
   1.544  qed
   1.545  
   1.546 -lemma divide_1 [simp]:
   1.547 -  "a div 1 = a"
   1.548 +lemma divide_1 [simp]: "a div 1 = a"
   1.549    using nonzero_mult_divide_cancel_left [of 1 a] by simp
   1.550  
   1.551  end
   1.552 @@ -668,11 +619,13 @@
   1.553    assumes "a \<noteq> 0"
   1.554    shows "a * b dvd a * c \<longleftrightarrow> b dvd c" (is "?P \<longleftrightarrow> ?Q")
   1.555  proof
   1.556 -  assume ?P then obtain d where "a * c = a * b * d" ..
   1.557 +  assume ?P
   1.558 +  then obtain d where "a * c = a * b * d" ..
   1.559    with assms have "c = b * d" by (simp add: ac_simps)
   1.560    then show ?Q ..
   1.561  next
   1.562 -  assume ?Q then obtain d where "c = b * d" ..
   1.563 +  assume ?Q
   1.564 +  then obtain d where "c = b * d" ..
   1.565    then have "a * c = a * b * d" by (simp add: ac_simps)
   1.566    then show ?P ..
   1.567  qed
   1.568 @@ -680,7 +633,7 @@
   1.569  lemma dvd_times_right_cancel_iff [simp]:
   1.570    assumes "a \<noteq> 0"
   1.571    shows "b * a dvd c * a \<longleftrightarrow> b dvd c" (is "?P \<longleftrightarrow> ?Q")
   1.572 -using dvd_times_left_cancel_iff [of a b c] assms by (simp add: ac_simps)
   1.573 +  using dvd_times_left_cancel_iff [of a b c] assms by (simp add: ac_simps)
   1.574  
   1.575  lemma div_dvd_iff_mult:
   1.576    assumes "b \<noteq> 0" and "b dvd a"
   1.577 @@ -702,7 +655,8 @@
   1.578    assumes "a dvd b" and "a dvd c"
   1.579    shows "b div a dvd c div a \<longleftrightarrow> b dvd c"
   1.580  proof (cases "a = 0")
   1.581 -  case True with assms show ?thesis by simp
   1.582 +  case True
   1.583 +  with assms show ?thesis by simp
   1.584  next
   1.585    case False
   1.586    moreover from assms obtain k l where "b = a * k" and "c = a * l"
   1.587 @@ -714,7 +668,8 @@
   1.588    assumes "c dvd a" and "c dvd b"
   1.589    shows "(a + b) div c = a div c + b div c"
   1.590  proof (cases "c = 0")
   1.591 -  case True then show ?thesis by simp
   1.592 +  case True
   1.593 +  then show ?thesis by simp
   1.594  next
   1.595    case False
   1.596    moreover from assms obtain k l where "a = c * k" and "b = c * l"
   1.597 @@ -729,7 +684,8 @@
   1.598    assumes "b dvd a" and "d dvd c"
   1.599    shows "(a div b) * (c div d) = (a * c) div (b * d)"
   1.600  proof (cases "b = 0 \<or> c = 0")
   1.601 -  case True with assms show ?thesis by auto
   1.602 +  case True
   1.603 +  with assms show ?thesis by auto
   1.604  next
   1.605    case False
   1.606    moreover from assms obtain k l where "a = b * k" and "c = d * l"
   1.607 @@ -748,42 +704,39 @@
   1.608  next
   1.609    assume "b div a = c"
   1.610    then have "b div a * a = c * a" by simp
   1.611 -  moreover from \<open>a \<noteq> 0\<close> \<open>a dvd b\<close> have "b div a * a = b"
   1.612 +  moreover from assms have "b div a * a = b"
   1.613      by (auto elim!: dvdE simp add: ac_simps)
   1.614    ultimately show "b = c * a" by simp
   1.615  qed
   1.616  
   1.617 -lemma dvd_div_mult_self [simp]:
   1.618 -  "a dvd b \<Longrightarrow> b div a * a = b"
   1.619 +lemma dvd_div_mult_self [simp]: "a dvd b \<Longrightarrow> b div a * a = b"
   1.620    by (cases "a = 0") (auto elim: dvdE simp add: ac_simps)
   1.621  
   1.622 -lemma dvd_mult_div_cancel [simp]:
   1.623 -  "a dvd b \<Longrightarrow> a * (b div a) = b"
   1.624 +lemma dvd_mult_div_cancel [simp]: "a dvd b \<Longrightarrow> a * (b div a) = b"
   1.625    using dvd_div_mult_self [of a b] by (simp add: ac_simps)
   1.626  
   1.627  lemma div_mult_swap:
   1.628    assumes "c dvd b"
   1.629    shows "a * (b div c) = (a * b) div c"
   1.630  proof (cases "c = 0")
   1.631 -  case True then show ?thesis by simp
   1.632 +  case True
   1.633 +  then show ?thesis by simp
   1.634  next
   1.635 -  case False from assms obtain d where "b = c * d" ..
   1.636 +  case False
   1.637 +  from assms obtain d where "b = c * d" ..
   1.638    moreover from False have "a * divide (d * c) c = ((a * d) * c) div c"
   1.639      by simp
   1.640    ultimately show ?thesis by (simp add: ac_simps)
   1.641  qed
   1.642  
   1.643 -lemma dvd_div_mult:
   1.644 -  assumes "c dvd b"
   1.645 -  shows "b div c * a = (b * a) div c"
   1.646 -  using assms div_mult_swap [of c b a] by (simp add: ac_simps)
   1.647 +lemma dvd_div_mult: "c dvd b \<Longrightarrow> b div c * a = (b * a) div c"
   1.648 +  using div_mult_swap [of c b a] by (simp add: ac_simps)
   1.649  
   1.650  lemma dvd_div_mult2_eq:
   1.651    assumes "b * c dvd a"
   1.652    shows "a div (b * c) = a div b div c"
   1.653 -using assms proof
   1.654 -  fix k
   1.655 -  assume "a = b * c * k"
   1.656 +proof -
   1.657 +  from assms obtain k where "a = b * c * k" ..
   1.658    then show ?thesis
   1.659      by (cases "b = 0 \<or> c = 0") (auto, simp add: ac_simps)
   1.660  qed
   1.661 @@ -808,15 +761,12 @@
   1.662  text \<open>Units: invertible elements in a ring\<close>
   1.663  
   1.664  abbreviation is_unit :: "'a \<Rightarrow> bool"
   1.665 -where
   1.666 -  "is_unit a \<equiv> a dvd 1"
   1.667 +  where "is_unit a \<equiv> a dvd 1"
   1.668  
   1.669 -lemma not_is_unit_0 [simp]:
   1.670 -  "\<not> is_unit 0"
   1.671 +lemma not_is_unit_0 [simp]: "\<not> is_unit 0"
   1.672    by simp
   1.673  
   1.674 -lemma unit_imp_dvd [dest]:
   1.675 -  "is_unit b \<Longrightarrow> b dvd a"
   1.676 +lemma unit_imp_dvd [dest]: "is_unit b \<Longrightarrow> b dvd a"
   1.677    by (rule dvd_trans [of _ 1]) simp_all
   1.678  
   1.679  lemma unit_dvdE:
   1.680 @@ -829,8 +779,7 @@
   1.681    ultimately show thesis using that by blast
   1.682  qed
   1.683  
   1.684 -lemma dvd_unit_imp_unit:
   1.685 -  "a dvd b \<Longrightarrow> is_unit b \<Longrightarrow> is_unit a"
   1.686 +lemma dvd_unit_imp_unit: "a dvd b \<Longrightarrow> is_unit b \<Longrightarrow> is_unit a"
   1.687    by (rule dvd_trans)
   1.688  
   1.689  lemma unit_div_1_unit [simp, intro]:
   1.690 @@ -849,27 +798,24 @@
   1.691  proof (rule that)
   1.692    define b where "b = 1 div a"
   1.693    then show "1 div a = b" by simp
   1.694 -  from b_def \<open>is_unit a\<close> show "is_unit b" by simp
   1.695 -  from \<open>is_unit a\<close> and \<open>is_unit b\<close> show "a \<noteq> 0" and "b \<noteq> 0" by auto
   1.696 -  from b_def \<open>is_unit a\<close> show "a * b = 1" by simp
   1.697 +  from assms b_def show "is_unit b" by simp
   1.698 +  with assms show "a \<noteq> 0" and "b \<noteq> 0" by auto
   1.699 +  from assms b_def show "a * b = 1" by simp
   1.700    then have "1 = a * b" ..
   1.701    with b_def \<open>b \<noteq> 0\<close> show "1 div b = a" by simp
   1.702 -  from \<open>is_unit a\<close> have "a dvd c" ..
   1.703 +  from assms have "a dvd c" ..
   1.704    then obtain d where "c = a * d" ..
   1.705    with \<open>a \<noteq> 0\<close> \<open>a * b = 1\<close> show "c div a = c * b"
   1.706      by (simp add: mult.assoc mult.left_commute [of a])
   1.707  qed
   1.708  
   1.709 -lemma unit_prod [intro]:
   1.710 -  "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a * b)"
   1.711 +lemma unit_prod [intro]: "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a * b)"
   1.712    by (subst mult_1_left [of 1, symmetric]) (rule mult_dvd_mono)
   1.713  
   1.714 -lemma is_unit_mult_iff:
   1.715 -  "is_unit (a * b) \<longleftrightarrow> is_unit a \<and> is_unit b" (is "?P \<longleftrightarrow> ?Q")
   1.716 +lemma is_unit_mult_iff: "is_unit (a * b) \<longleftrightarrow> is_unit a \<and> is_unit b"
   1.717    by (auto dest: dvd_mult_left dvd_mult_right)
   1.718  
   1.719 -lemma unit_div [intro]:
   1.720 -  "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a div b)"
   1.721 +lemma unit_div [intro]: "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a div b)"
   1.722    by (erule is_unitE [of b a]) (simp add: ac_simps unit_prod)
   1.723  
   1.724  lemma mult_unit_dvd_iff:
   1.725 @@ -894,7 +840,8 @@
   1.726    assume "a dvd c * b"
   1.727    with assms have "c * b dvd c * (b * (1 div b))"
   1.728      by (subst mult_assoc [symmetric]) simp
   1.729 -  also from \<open>is_unit b\<close> have "b * (1 div b) = 1" by (rule is_unitE) simp
   1.730 +  also from assms have "b * (1 div b) = 1"
   1.731 +    by (rule is_unitE) simp
   1.732    finally have "c * b dvd c" by simp
   1.733    with \<open>a dvd c * b\<close> show "a dvd c" by (rule dvd_trans)
   1.734  next
   1.735 @@ -902,52 +849,40 @@
   1.736    then show "a dvd c * b" by simp
   1.737  qed
   1.738  
   1.739 -lemma div_unit_dvd_iff:
   1.740 -  "is_unit b \<Longrightarrow> a div b dvd c \<longleftrightarrow> a dvd c"
   1.741 +lemma div_unit_dvd_iff: "is_unit b \<Longrightarrow> a div b dvd c \<longleftrightarrow> a dvd c"
   1.742    by (erule is_unitE [of _ a]) (auto simp add: mult_unit_dvd_iff)
   1.743  
   1.744 -lemma dvd_div_unit_iff:
   1.745 -  "is_unit b \<Longrightarrow> a dvd c div b \<longleftrightarrow> a dvd c"
   1.746 +lemma dvd_div_unit_iff: "is_unit b \<Longrightarrow> a dvd c div b \<longleftrightarrow> a dvd c"
   1.747    by (erule is_unitE [of _ c]) (simp add: dvd_mult_unit_iff)
   1.748  
   1.749  lemmas unit_dvd_iff = mult_unit_dvd_iff div_unit_dvd_iff
   1.750 -  dvd_mult_unit_iff dvd_div_unit_iff \<comment> \<open>FIXME consider fact collection\<close>
   1.751 +  dvd_mult_unit_iff dvd_div_unit_iff  (* FIXME consider named_theorems *)
   1.752  
   1.753 -lemma unit_mult_div_div [simp]:
   1.754 -  "is_unit a \<Longrightarrow> b * (1 div a) = b div a"
   1.755 +lemma unit_mult_div_div [simp]: "is_unit a \<Longrightarrow> b * (1 div a) = b div a"
   1.756    by (erule is_unitE [of _ b]) simp
   1.757  
   1.758 -lemma unit_div_mult_self [simp]:
   1.759 -  "is_unit a \<Longrightarrow> b div a * a = b"
   1.760 +lemma unit_div_mult_self [simp]: "is_unit a \<Longrightarrow> b div a * a = b"
   1.761    by (rule dvd_div_mult_self) auto
   1.762  
   1.763 -lemma unit_div_1_div_1 [simp]:
   1.764 -  "is_unit a \<Longrightarrow> 1 div (1 div a) = a"
   1.765 +lemma unit_div_1_div_1 [simp]: "is_unit a \<Longrightarrow> 1 div (1 div a) = a"
   1.766    by (erule is_unitE) simp
   1.767  
   1.768 -lemma unit_div_mult_swap:
   1.769 -  "is_unit c \<Longrightarrow> a * (b div c) = (a * b) div c"
   1.770 +lemma unit_div_mult_swap: "is_unit c \<Longrightarrow> a * (b div c) = (a * b) div c"
   1.771    by (erule unit_dvdE [of _ b]) (simp add: mult.left_commute [of _ c])
   1.772  
   1.773 -lemma unit_div_commute:
   1.774 -  "is_unit b \<Longrightarrow> (a div b) * c = (a * c) div b"
   1.775 +lemma unit_div_commute: "is_unit b \<Longrightarrow> (a div b) * c = (a * c) div b"
   1.776    using unit_div_mult_swap [of b c a] by (simp add: ac_simps)
   1.777  
   1.778 -lemma unit_eq_div1:
   1.779 -  "is_unit b \<Longrightarrow> a div b = c \<longleftrightarrow> a = c * b"
   1.780 +lemma unit_eq_div1: "is_unit b \<Longrightarrow> a div b = c \<longleftrightarrow> a = c * b"
   1.781    by (auto elim: is_unitE)
   1.782  
   1.783 -lemma unit_eq_div2:
   1.784 -  "is_unit b \<Longrightarrow> a = c div b \<longleftrightarrow> a * b = c"
   1.785 +lemma unit_eq_div2: "is_unit b \<Longrightarrow> a = c div b \<longleftrightarrow> a * b = c"
   1.786    using unit_eq_div1 [of b c a] by auto
   1.787  
   1.788 -lemma unit_mult_left_cancel:
   1.789 -  assumes "is_unit a"
   1.790 -  shows "a * b = a * c \<longleftrightarrow> b = c" (is "?P \<longleftrightarrow> ?Q")
   1.791 -  using assms mult_cancel_left [of a b c] by auto
   1.792 +lemma unit_mult_left_cancel: "is_unit a \<Longrightarrow> a * b = a * c \<longleftrightarrow> b = c"
   1.793 +  using mult_cancel_left [of a b c] by auto
   1.794  
   1.795 -lemma unit_mult_right_cancel:
   1.796 -  "is_unit a \<Longrightarrow> b * a = c * a \<longleftrightarrow> b = c"
   1.797 +lemma unit_mult_right_cancel: "is_unit a \<Longrightarrow> b * a = c * a \<longleftrightarrow> b = c"
   1.798    using unit_mult_left_cancel [of a b c] by (auto simp add: ac_simps)
   1.799  
   1.800  lemma unit_div_cancel:
   1.801 @@ -964,7 +899,8 @@
   1.802    assumes "is_unit b" and "is_unit c"
   1.803    shows "a div (b * c) = a div b div c"
   1.804  proof -
   1.805 -  from assms have "is_unit (b * c)" by (simp add: unit_prod)
   1.806 +  from assms have "is_unit (b * c)"
   1.807 +    by (simp add: unit_prod)
   1.808    then have "b * c dvd a"
   1.809      by (rule unit_imp_dvd)
   1.810    then show ?thesis
   1.811 @@ -1015,58 +951,57 @@
   1.812    values rather than associated elements.
   1.813  \<close>
   1.814  
   1.815 -lemma unit_factor_dvd [simp]:
   1.816 -  "a \<noteq> 0 \<Longrightarrow> unit_factor a dvd b"
   1.817 +lemma unit_factor_dvd [simp]: "a \<noteq> 0 \<Longrightarrow> unit_factor a dvd b"
   1.818    by (rule unit_imp_dvd) simp
   1.819  
   1.820 -lemma unit_factor_self [simp]:
   1.821 -  "unit_factor a dvd a"
   1.822 +lemma unit_factor_self [simp]: "unit_factor a dvd a"
   1.823    by (cases "a = 0") simp_all
   1.824  
   1.825 -lemma normalize_mult_unit_factor [simp]:
   1.826 -  "normalize a * unit_factor a = a"
   1.827 +lemma normalize_mult_unit_factor [simp]: "normalize a * unit_factor a = a"
   1.828    using unit_factor_mult_normalize [of a] by (simp add: ac_simps)
   1.829  
   1.830 -lemma normalize_eq_0_iff [simp]:
   1.831 -  "normalize a = 0 \<longleftrightarrow> a = 0" (is "?P \<longleftrightarrow> ?Q")
   1.832 +lemma normalize_eq_0_iff [simp]: "normalize a = 0 \<longleftrightarrow> a = 0"
   1.833 +  (is "?P \<longleftrightarrow> ?Q")
   1.834  proof
   1.835    assume ?P
   1.836    moreover have "unit_factor a * normalize a = a" by simp
   1.837    ultimately show ?Q by simp
   1.838  next
   1.839 -  assume ?Q then show ?P by simp
   1.840 +  assume ?Q
   1.841 +  then show ?P by simp
   1.842  qed
   1.843  
   1.844 -lemma unit_factor_eq_0_iff [simp]:
   1.845 -  "unit_factor a = 0 \<longleftrightarrow> a = 0" (is "?P \<longleftrightarrow> ?Q")
   1.846 +lemma unit_factor_eq_0_iff [simp]: "unit_factor a = 0 \<longleftrightarrow> a = 0"
   1.847 +  (is "?P \<longleftrightarrow> ?Q")
   1.848  proof
   1.849    assume ?P
   1.850    moreover have "unit_factor a * normalize a = a" by simp
   1.851    ultimately show ?Q by simp
   1.852  next
   1.853 -  assume ?Q then show ?P by simp
   1.854 +  assume ?Q
   1.855 +  then show ?P by simp
   1.856  qed
   1.857  
   1.858  lemma is_unit_unit_factor:
   1.859 -  assumes "is_unit a" shows "unit_factor a = a"
   1.860 +  assumes "is_unit a"
   1.861 +  shows "unit_factor a = a"
   1.862  proof -
   1.863    from assms have "normalize a = 1" by (rule is_unit_normalize)
   1.864    moreover from unit_factor_mult_normalize have "unit_factor a * normalize a = a" .
   1.865    ultimately show ?thesis by simp
   1.866  qed
   1.867  
   1.868 -lemma unit_factor_1 [simp]:
   1.869 -  "unit_factor 1 = 1"
   1.870 +lemma unit_factor_1 [simp]: "unit_factor 1 = 1"
   1.871    by (rule is_unit_unit_factor) simp
   1.872  
   1.873 -lemma normalize_1 [simp]:
   1.874 -  "normalize 1 = 1"
   1.875 +lemma normalize_1 [simp]: "normalize 1 = 1"
   1.876    by (rule is_unit_normalize) simp
   1.877  
   1.878 -lemma normalize_1_iff:
   1.879 -  "normalize a = 1 \<longleftrightarrow> is_unit a" (is "?P \<longleftrightarrow> ?Q")
   1.880 +lemma normalize_1_iff: "normalize a = 1 \<longleftrightarrow> is_unit a"
   1.881 +  (is "?P \<longleftrightarrow> ?Q")
   1.882  proof
   1.883 -  assume ?Q then show ?P by (rule is_unit_normalize)
   1.884 +  assume ?Q
   1.885 +  then show ?P by (rule is_unit_normalize)
   1.886  next
   1.887    assume ?P
   1.888    then have "a \<noteq> 0" by auto
   1.889 @@ -1079,32 +1014,34 @@
   1.890    ultimately show ?Q by simp
   1.891  qed
   1.892  
   1.893 -lemma div_normalize [simp]:
   1.894 -  "a div normalize a = unit_factor a"
   1.895 +lemma div_normalize [simp]: "a div normalize a = unit_factor a"
   1.896  proof (cases "a = 0")
   1.897 -  case True then show ?thesis by simp
   1.898 +  case True
   1.899 +  then show ?thesis by simp
   1.900  next
   1.901 -  case False then have "normalize a \<noteq> 0" by simp
   1.902 +  case False
   1.903 +  then have "normalize a \<noteq> 0" by simp
   1.904    with nonzero_mult_divide_cancel_right
   1.905    have "unit_factor a * normalize a div normalize a = unit_factor a" by blast
   1.906    then show ?thesis by simp
   1.907  qed
   1.908  
   1.909 -lemma div_unit_factor [simp]:
   1.910 -  "a div unit_factor a = normalize a"
   1.911 +lemma div_unit_factor [simp]: "a div unit_factor a = normalize a"
   1.912  proof (cases "a = 0")
   1.913 -  case True then show ?thesis by simp
   1.914 +  case True
   1.915 +  then show ?thesis by simp
   1.916  next
   1.917 -  case False then have "unit_factor a \<noteq> 0" by simp
   1.918 +  case False
   1.919 +  then have "unit_factor a \<noteq> 0" by simp
   1.920    with nonzero_mult_divide_cancel_left
   1.921    have "unit_factor a * normalize a div unit_factor a = normalize a" by blast
   1.922    then show ?thesis by simp
   1.923  qed
   1.924  
   1.925 -lemma normalize_div [simp]:
   1.926 -  "normalize a div a = 1 div unit_factor a"
   1.927 +lemma normalize_div [simp]: "normalize a div a = 1 div unit_factor a"
   1.928  proof (cases "a = 0")
   1.929 -  case True then show ?thesis by simp
   1.930 +  case True
   1.931 +  then show ?thesis by simp
   1.932  next
   1.933    case False
   1.934    have "normalize a div a = normalize a div (unit_factor a * normalize a)"
   1.935 @@ -1114,62 +1051,64 @@
   1.936    finally show ?thesis .
   1.937  qed
   1.938  
   1.939 -lemma mult_one_div_unit_factor [simp]:
   1.940 -  "a * (1 div unit_factor b) = a div unit_factor b"
   1.941 +lemma mult_one_div_unit_factor [simp]: "a * (1 div unit_factor b) = a div unit_factor b"
   1.942    by (cases "b = 0") simp_all
   1.943  
   1.944 -lemma normalize_mult:
   1.945 -  "normalize (a * b) = normalize a * normalize b"
   1.946 +lemma normalize_mult: "normalize (a * b) = normalize a * normalize b"
   1.947  proof (cases "a = 0 \<or> b = 0")
   1.948 -  case True then show ?thesis by auto
   1.949 +  case True
   1.950 +  then show ?thesis by auto
   1.951  next
   1.952    case False
   1.953    from unit_factor_mult_normalize have "unit_factor (a * b) * normalize (a * b) = a * b" .
   1.954 -  then have "normalize (a * b) = a * b div unit_factor (a * b)" by simp
   1.955 -  also have "\<dots> = a * b div unit_factor (b * a)" by (simp add: ac_simps)
   1.956 +  then have "normalize (a * b) = a * b div unit_factor (a * b)"
   1.957 +    by simp
   1.958 +  also have "\<dots> = a * b div unit_factor (b * a)"
   1.959 +    by (simp add: ac_simps)
   1.960    also have "\<dots> = a * b div unit_factor b div unit_factor a"
   1.961      using False by (simp add: unit_factor_mult is_unit_div_mult2_eq [symmetric])
   1.962    also have "\<dots> = a * (b div unit_factor b) div unit_factor a"
   1.963      using False by (subst unit_div_mult_swap) simp_all
   1.964    also have "\<dots> = normalize a * normalize b"
   1.965 -    using False by (simp add: mult.commute [of a] mult.commute [of "normalize a"] unit_div_mult_swap [symmetric])
   1.966 +    using False
   1.967 +    by (simp add: mult.commute [of a] mult.commute [of "normalize a"] unit_div_mult_swap [symmetric])
   1.968    finally show ?thesis .
   1.969  qed
   1.970  
   1.971 -lemma unit_factor_idem [simp]:
   1.972 -  "unit_factor (unit_factor a) = unit_factor a"
   1.973 +lemma unit_factor_idem [simp]: "unit_factor (unit_factor a) = unit_factor a"
   1.974    by (cases "a = 0") (auto intro: is_unit_unit_factor)
   1.975  
   1.976 -lemma normalize_unit_factor [simp]:
   1.977 -  "a \<noteq> 0 \<Longrightarrow> normalize (unit_factor a) = 1"
   1.978 +lemma normalize_unit_factor [simp]: "a \<noteq> 0 \<Longrightarrow> normalize (unit_factor a) = 1"
   1.979    by (rule is_unit_normalize) simp
   1.980  
   1.981 -lemma normalize_idem [simp]:
   1.982 -  "normalize (normalize a) = normalize a"
   1.983 +lemma normalize_idem [simp]: "normalize (normalize a) = normalize a"
   1.984  proof (cases "a = 0")
   1.985 -  case True then show ?thesis by simp
   1.986 +  case True
   1.987 +  then show ?thesis by simp
   1.988  next
   1.989    case False
   1.990 -  have "normalize a = normalize (unit_factor a * normalize a)" by simp
   1.991 +  have "normalize a = normalize (unit_factor a * normalize a)"
   1.992 +    by simp
   1.993    also have "\<dots> = normalize (unit_factor a) * normalize (normalize a)"
   1.994      by (simp only: normalize_mult)
   1.995 -  finally show ?thesis using False by simp_all
   1.996 +  finally show ?thesis
   1.997 +    using False by simp_all
   1.998  qed
   1.999  
  1.1000  lemma unit_factor_normalize [simp]:
  1.1001    assumes "a \<noteq> 0"
  1.1002    shows "unit_factor (normalize a) = 1"
  1.1003  proof -
  1.1004 -  from assms have "normalize a \<noteq> 0" by simp
  1.1005 +  from assms have *: "normalize a \<noteq> 0"
  1.1006 +    by simp
  1.1007    have "unit_factor (normalize a) * normalize (normalize a) = normalize a"
  1.1008      by (simp only: unit_factor_mult_normalize)
  1.1009    then have "unit_factor (normalize a) * normalize a = normalize a"
  1.1010      by simp
  1.1011 -  with \<open>normalize a \<noteq> 0\<close>
  1.1012 -  have "unit_factor (normalize a) * normalize a div normalize a = normalize a div normalize a"
  1.1013 +  with * have "unit_factor (normalize a) * normalize a div normalize a = normalize a div normalize a"
  1.1014      by simp
  1.1015 -  with \<open>normalize a \<noteq> 0\<close>
  1.1016 -  show ?thesis by simp
  1.1017 +  with * show ?thesis
  1.1018 +    by simp
  1.1019  qed
  1.1020  
  1.1021  lemma dvd_unit_factor_div:
  1.1022 @@ -1196,8 +1135,7 @@
  1.1023      by (cases "b = 0") (simp_all add: normalize_mult)
  1.1024  qed
  1.1025  
  1.1026 -lemma normalize_dvd_iff [simp]:
  1.1027 -  "normalize a dvd b \<longleftrightarrow> a dvd b"
  1.1028 +lemma normalize_dvd_iff [simp]: "normalize a dvd b \<longleftrightarrow> a dvd b"
  1.1029  proof -
  1.1030    have "normalize a dvd b \<longleftrightarrow> unit_factor a * normalize a dvd b"
  1.1031      using mult_unit_dvd_iff [of "unit_factor a" "normalize a" b]
  1.1032 @@ -1205,8 +1143,7 @@
  1.1033    then show ?thesis by simp
  1.1034  qed
  1.1035  
  1.1036 -lemma dvd_normalize_iff [simp]:
  1.1037 -  "a dvd normalize b \<longleftrightarrow> a dvd b"
  1.1038 +lemma dvd_normalize_iff [simp]: "a dvd normalize b \<longleftrightarrow> a dvd b"
  1.1039  proof -
  1.1040    have "a dvd normalize  b \<longleftrightarrow> a dvd normalize b * unit_factor b"
  1.1041      using dvd_mult_unit_iff [of "unit_factor b" a "normalize b"]
  1.1042 @@ -1226,36 +1163,38 @@
  1.1043    assumes "a dvd b" and "b dvd a"
  1.1044    shows "normalize a = normalize b"
  1.1045  proof (cases "a = 0 \<or> b = 0")
  1.1046 -  case True with assms show ?thesis by auto
  1.1047 +  case True
  1.1048 +  with assms show ?thesis by auto
  1.1049  next
  1.1050    case False
  1.1051    from \<open>a dvd b\<close> obtain c where b: "b = a * c" ..
  1.1052    moreover from \<open>b dvd a\<close> obtain d where a: "a = b * d" ..
  1.1053 -  ultimately have "b * 1 = b * (c * d)" by (simp add: ac_simps)
  1.1054 +  ultimately have "b * 1 = b * (c * d)"
  1.1055 +    by (simp add: ac_simps)
  1.1056    with False have "1 = c * d"
  1.1057      unfolding mult_cancel_left by simp
  1.1058 -  then have "is_unit c" and "is_unit d" by auto
  1.1059 -  with a b show ?thesis by (simp add: normalize_mult is_unit_normalize)
  1.1060 +  then have "is_unit c" and "is_unit d"
  1.1061 +    by auto
  1.1062 +  with a b show ?thesis
  1.1063 +    by (simp add: normalize_mult is_unit_normalize)
  1.1064  qed
  1.1065  
  1.1066 -lemma associatedD1:
  1.1067 -  "normalize a = normalize b \<Longrightarrow> a dvd b"
  1.1068 +lemma associatedD1: "normalize a = normalize b \<Longrightarrow> a dvd b"
  1.1069    using dvd_normalize_iff [of _ b, symmetric] normalize_dvd_iff [of a _, symmetric]
  1.1070    by simp
  1.1071  
  1.1072 -lemma associatedD2:
  1.1073 -  "normalize a = normalize b \<Longrightarrow> b dvd a"
  1.1074 +lemma associatedD2: "normalize a = normalize b \<Longrightarrow> b dvd a"
  1.1075    using dvd_normalize_iff [of _ a, symmetric] normalize_dvd_iff [of b _, symmetric]
  1.1076    by simp
  1.1077  
  1.1078 -lemma associated_unit:
  1.1079 -  "normalize a = normalize b \<Longrightarrow> is_unit a \<Longrightarrow> is_unit b"
  1.1080 +lemma associated_unit: "normalize a = normalize b \<Longrightarrow> is_unit a \<Longrightarrow> is_unit b"
  1.1081    using dvd_unit_imp_unit by (auto dest!: associatedD1 associatedD2)
  1.1082  
  1.1083 -lemma associated_iff_dvd:
  1.1084 -  "normalize a = normalize b \<longleftrightarrow> a dvd b \<and> b dvd a" (is "?P \<longleftrightarrow> ?Q")
  1.1085 +lemma associated_iff_dvd: "normalize a = normalize b \<longleftrightarrow> a dvd b \<and> b dvd a"
  1.1086 +  (is "?P \<longleftrightarrow> ?Q")
  1.1087  proof
  1.1088 -  assume ?Q then show ?P by (auto intro!: associatedI)
  1.1089 +  assume ?Q
  1.1090 +  then show ?P by (auto intro!: associatedI)
  1.1091  next
  1.1092    assume ?P
  1.1093    then have "unit_factor a * normalize a = unit_factor a * normalize b"
  1.1094 @@ -1264,7 +1203,8 @@
  1.1095      by (simp add: ac_simps)
  1.1096    show ?Q
  1.1097    proof (cases "a = 0 \<or> b = 0")
  1.1098 -    case True with \<open>?P\<close> show ?thesis by auto
  1.1099 +    case True
  1.1100 +    with \<open>?P\<close> show ?thesis by auto
  1.1101    next
  1.1102      case False
  1.1103      then have "b dvd normalize b * unit_factor a" and "normalize b * unit_factor a dvd b"
  1.1104 @@ -1291,38 +1231,38 @@
  1.1105    assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
  1.1106  begin
  1.1107  
  1.1108 -lemma mult_mono:
  1.1109 -  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
  1.1110 -apply (erule mult_right_mono [THEN order_trans], assumption)
  1.1111 -apply (erule mult_left_mono, assumption)
  1.1112 -done
  1.1113 +lemma mult_mono: "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
  1.1114 +  apply (erule (1) mult_right_mono [THEN order_trans])
  1.1115 +  apply (erule (1) mult_left_mono)
  1.1116 +  done
  1.1117  
  1.1118 -lemma mult_mono':
  1.1119 -  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
  1.1120 -apply (rule mult_mono)
  1.1121 -apply (fast intro: order_trans)+
  1.1122 -done
  1.1123 +lemma mult_mono': "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
  1.1124 +  apply (rule mult_mono)
  1.1125 +  apply (fast intro: order_trans)+
  1.1126 +  done
  1.1127  
  1.1128  end
  1.1129  
  1.1130  class ordered_semiring_0 = semiring_0 + ordered_semiring
  1.1131  begin
  1.1132  
  1.1133 -lemma mult_nonneg_nonneg[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
  1.1134 -using mult_left_mono [of 0 b a] by simp
  1.1135 +lemma mult_nonneg_nonneg [simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
  1.1136 +  using mult_left_mono [of 0 b a] by simp
  1.1137  
  1.1138  lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
  1.1139 -using mult_left_mono [of b 0 a] by simp
  1.1140 +  using mult_left_mono [of b 0 a] by simp
  1.1141  
  1.1142  lemma mult_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> b \<Longrightarrow> a * b \<le> 0"
  1.1143 -using mult_right_mono [of a 0 b] by simp
  1.1144 +  using mult_right_mono [of a 0 b] by simp
  1.1145  
  1.1146  text \<open>Legacy - use \<open>mult_nonpos_nonneg\<close>\<close>
  1.1147  lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0"
  1.1148 -by (drule mult_right_mono [of b 0], auto)
  1.1149 +  apply (drule mult_right_mono [of b 0])
  1.1150 +  apply auto
  1.1151 +  done
  1.1152  
  1.1153  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"
  1.1154 -by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
  1.1155 +  by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
  1.1156  
  1.1157  end
  1.1158  
  1.1159 @@ -1341,44 +1281,34 @@
  1.1160  
  1.1161  subclass ordered_cancel_comm_monoid_add ..
  1.1162  
  1.1163 -lemma mult_left_less_imp_less:
  1.1164 -  "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
  1.1165 -by (force simp add: mult_left_mono not_le [symmetric])
  1.1166 +lemma mult_left_less_imp_less: "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
  1.1167 +  by (force simp add: mult_left_mono not_le [symmetric])
  1.1168  
  1.1169 -lemma mult_right_less_imp_less:
  1.1170 -  "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
  1.1171 -by (force simp add: mult_right_mono not_le [symmetric])
  1.1172 +lemma mult_right_less_imp_less: "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
  1.1173 +  by (force simp add: mult_right_mono not_le [symmetric])
  1.1174  
  1.1175 -lemma less_eq_add_cancel_left_greater_eq_zero [simp]:
  1.1176 -  "a \<le> a + b \<longleftrightarrow> 0 \<le> b"
  1.1177 +lemma less_eq_add_cancel_left_greater_eq_zero [simp]: "a \<le> a + b \<longleftrightarrow> 0 \<le> b"
  1.1178    using add_le_cancel_left [of a 0 b] by simp
  1.1179  
  1.1180 -lemma less_eq_add_cancel_left_less_eq_zero [simp]:
  1.1181 -  "a + b \<le> a \<longleftrightarrow> b \<le> 0"
  1.1182 +lemma less_eq_add_cancel_left_less_eq_zero [simp]: "a + b \<le> a \<longleftrightarrow> b \<le> 0"
  1.1183    using add_le_cancel_left [of a b 0] by simp
  1.1184  
  1.1185 -lemma less_eq_add_cancel_right_greater_eq_zero [simp]:
  1.1186 -  "a \<le> b + a \<longleftrightarrow> 0 \<le> b"
  1.1187 +lemma less_eq_add_cancel_right_greater_eq_zero [simp]: "a \<le> b + a \<longleftrightarrow> 0 \<le> b"
  1.1188    using add_le_cancel_right [of 0 a b] by simp
  1.1189  
  1.1190 -lemma less_eq_add_cancel_right_less_eq_zero [simp]:
  1.1191 -  "b + a \<le> a \<longleftrightarrow> b \<le> 0"
  1.1192 +lemma less_eq_add_cancel_right_less_eq_zero [simp]: "b + a \<le> a \<longleftrightarrow> b \<le> 0"
  1.1193    using add_le_cancel_right [of b a 0] by simp
  1.1194  
  1.1195 -lemma less_add_cancel_left_greater_zero [simp]:
  1.1196 -  "a < a + b \<longleftrightarrow> 0 < b"
  1.1197 +lemma less_add_cancel_left_greater_zero [simp]: "a < a + b \<longleftrightarrow> 0 < b"
  1.1198    using add_less_cancel_left [of a 0 b] by simp
  1.1199  
  1.1200 -lemma less_add_cancel_left_less_zero [simp]:
  1.1201 -  "a + b < a \<longleftrightarrow> b < 0"
  1.1202 +lemma less_add_cancel_left_less_zero [simp]: "a + b < a \<longleftrightarrow> b < 0"
  1.1203    using add_less_cancel_left [of a b 0] by simp
  1.1204  
  1.1205 -lemma less_add_cancel_right_greater_zero [simp]:
  1.1206 -  "a < b + a \<longleftrightarrow> 0 < b"
  1.1207 +lemma less_add_cancel_right_greater_zero [simp]: "a < b + a \<longleftrightarrow> 0 < b"
  1.1208    using add_less_cancel_right [of 0 a b] by simp
  1.1209  
  1.1210 -lemma less_add_cancel_right_less_zero [simp]:
  1.1211 -  "b + a < a \<longleftrightarrow> b < 0"
  1.1212 +lemma less_add_cancel_right_less_zero [simp]: "b + a < a \<longleftrightarrow> b < 0"
  1.1213    using add_less_cancel_right [of b a 0] by simp
  1.1214  
  1.1215  end
  1.1216 @@ -1392,7 +1322,8 @@
  1.1217  proof-
  1.1218    from assms have "u * x + v * y \<le> u * a + v * a"
  1.1219      by (simp add: add_mono mult_left_mono)
  1.1220 -  thus ?thesis using assms unfolding distrib_right[symmetric] by simp
  1.1221 +  with assms show ?thesis
  1.1222 +    unfolding distrib_right[symmetric] by simp
  1.1223  qed
  1.1224  
  1.1225  end
  1.1226 @@ -1416,80 +1347,79 @@
  1.1227      using mult_strict_right_mono by (cases "c = 0") auto
  1.1228  qed
  1.1229  
  1.1230 -lemma mult_left_le_imp_le:
  1.1231 -  "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
  1.1232 -by (force simp add: mult_strict_left_mono _not_less [symmetric])
  1.1233 +lemma mult_left_le_imp_le: "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
  1.1234 +  by (auto simp add: mult_strict_left_mono _not_less [symmetric])
  1.1235  
  1.1236 -lemma mult_right_le_imp_le:
  1.1237 -  "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
  1.1238 -by (force simp add: mult_strict_right_mono not_less [symmetric])
  1.1239 +lemma mult_right_le_imp_le: "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
  1.1240 +  by (auto simp add: mult_strict_right_mono not_less [symmetric])
  1.1241  
  1.1242  lemma mult_pos_pos[simp]: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
  1.1243 -using mult_strict_left_mono [of 0 b a] by simp
  1.1244 +  using mult_strict_left_mono [of 0 b a] by simp
  1.1245  
  1.1246  lemma mult_pos_neg: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
  1.1247 -using mult_strict_left_mono [of b 0 a] by simp
  1.1248 +  using mult_strict_left_mono [of b 0 a] by simp
  1.1249  
  1.1250  lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0"
  1.1251 -using mult_strict_right_mono [of a 0 b] by simp
  1.1252 +  using mult_strict_right_mono [of a 0 b] by simp
  1.1253  
  1.1254  text \<open>Legacy - use \<open>mult_neg_pos\<close>\<close>
  1.1255  lemma mult_pos_neg2: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0"
  1.1256 -by (drule mult_strict_right_mono [of b 0], auto)
  1.1257 -
  1.1258 -lemma zero_less_mult_pos:
  1.1259 -  "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
  1.1260 -apply (cases "b\<le>0")
  1.1261 - apply (auto simp add: le_less not_less)
  1.1262 -apply (drule_tac mult_pos_neg [of a b])
  1.1263 - apply (auto dest: less_not_sym)
  1.1264 -done
  1.1265 +  apply (drule mult_strict_right_mono [of b 0])
  1.1266 +  apply auto
  1.1267 +  done
  1.1268  
  1.1269 -lemma zero_less_mult_pos2:
  1.1270 -  "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
  1.1271 -apply (cases "b\<le>0")
  1.1272 - apply (auto simp add: le_less not_less)
  1.1273 -apply (drule_tac mult_pos_neg2 [of a b])
  1.1274 - apply (auto dest: less_not_sym)
  1.1275 -done
  1.1276 +lemma zero_less_mult_pos: "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
  1.1277 +  apply (cases "b \<le> 0")
  1.1278 +   apply (auto simp add: le_less not_less)
  1.1279 +  apply (drule_tac mult_pos_neg [of a b])
  1.1280 +   apply (auto dest: less_not_sym)
  1.1281 +  done
  1.1282  
  1.1283 -text\<open>Strict monotonicity in both arguments\<close>
  1.1284 +lemma zero_less_mult_pos2: "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
  1.1285 +  apply (cases "b \<le> 0")
  1.1286 +   apply (auto simp add: le_less not_less)
  1.1287 +  apply (drule_tac mult_pos_neg2 [of a b])
  1.1288 +   apply (auto dest: less_not_sym)
  1.1289 +  done
  1.1290 +
  1.1291 +text \<open>Strict monotonicity in both arguments\<close>
  1.1292  lemma mult_strict_mono:
  1.1293    assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"
  1.1294    shows "a * c < b * d"
  1.1295 -  using assms apply (cases "c=0")
  1.1296 -  apply (simp)
  1.1297 +  using assms
  1.1298 +  apply (cases "c = 0")
  1.1299 +  apply simp
  1.1300    apply (erule mult_strict_right_mono [THEN less_trans])
  1.1301 -  apply (force simp add: le_less)
  1.1302 -  apply (erule mult_strict_left_mono, assumption)
  1.1303 +  apply (auto simp add: le_less)
  1.1304 +  apply (erule (1) mult_strict_left_mono)
  1.1305    done
  1.1306  
  1.1307 -text\<open>This weaker variant has more natural premises\<close>
  1.1308 +text \<open>This weaker variant has more natural premises\<close>
  1.1309  lemma mult_strict_mono':
  1.1310    assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
  1.1311    shows "a * c < b * d"
  1.1312 -by (rule mult_strict_mono) (insert assms, auto)
  1.1313 +  by (rule mult_strict_mono) (insert assms, auto)
  1.1314  
  1.1315  lemma mult_less_le_imp_less:
  1.1316    assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"
  1.1317    shows "a * c < b * d"
  1.1318 -  using assms apply (subgoal_tac "a * c < b * c")
  1.1319 +  using assms
  1.1320 +  apply (subgoal_tac "a * c < b * c")
  1.1321    apply (erule less_le_trans)
  1.1322    apply (erule mult_left_mono)
  1.1323    apply simp
  1.1324 -  apply (erule mult_strict_right_mono)
  1.1325 -  apply assumption
  1.1326 +  apply (erule (1) mult_strict_right_mono)
  1.1327    done
  1.1328  
  1.1329  lemma mult_le_less_imp_less:
  1.1330    assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"
  1.1331    shows "a * c < b * d"
  1.1332 -  using assms apply (subgoal_tac "a * c \<le> b * c")
  1.1333 +  using assms
  1.1334 +  apply (subgoal_tac "a * c \<le> b * c")
  1.1335    apply (erule le_less_trans)
  1.1336    apply (erule mult_strict_left_mono)
  1.1337    apply simp
  1.1338 -  apply (erule mult_right_mono)
  1.1339 -  apply simp
  1.1340 +  apply (erule (1) mult_right_mono)
  1.1341    done
  1.1342  
  1.1343  end
  1.1344 @@ -1504,9 +1434,9 @@
  1.1345    shows "u * x + v * y < a"
  1.1346  proof -
  1.1347    from assms have "u * x + v * y < u * a + v * a"
  1.1348 -    by (cases "u = 0")
  1.1349 -       (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)
  1.1350 -  thus ?thesis using assms unfolding distrib_right[symmetric] by simp
  1.1351 +    by (cases "u = 0") (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)
  1.1352 +  with assms show ?thesis
  1.1353 +    unfolding distrib_right[symmetric] by simp
  1.1354  qed
  1.1355  
  1.1356  end
  1.1357 @@ -1519,8 +1449,8 @@
  1.1358  proof
  1.1359    fix a b c :: 'a
  1.1360    assume "a \<le> b" "0 \<le> c"
  1.1361 -  thus "c * a \<le> c * b" by (rule comm_mult_left_mono)
  1.1362 -  thus "a * c \<le> b * c" by (simp only: mult.commute)
  1.1363 +  then show "c * a \<le> c * b" by (rule comm_mult_left_mono)
  1.1364 +  then show "a * c \<le> b * c" by (simp only: mult.commute)
  1.1365  qed
  1.1366  
  1.1367  end
  1.1368 @@ -1542,15 +1472,15 @@
  1.1369  proof
  1.1370    fix a b c :: 'a
  1.1371    assume "a < b" "0 < c"
  1.1372 -  thus "c * a < c * b" by (rule comm_mult_strict_left_mono)
  1.1373 -  thus "a * c < b * c" by (simp only: mult.commute)
  1.1374 +  then show "c * a < c * b" by (rule comm_mult_strict_left_mono)
  1.1375 +  then show "a * c < b * c" by (simp only: mult.commute)
  1.1376  qed
  1.1377  
  1.1378  subclass ordered_cancel_comm_semiring
  1.1379  proof
  1.1380    fix a b c :: 'a
  1.1381    assume "a \<le> b" "0 \<le> c"
  1.1382 -  thus "c * a \<le> c * b"
  1.1383 +  then show "c * a \<le> c * b"
  1.1384      unfolding le_less
  1.1385      using mult_strict_left_mono by (cases "c = 0") auto
  1.1386  qed
  1.1387 @@ -1562,40 +1492,33 @@
  1.1388  
  1.1389  subclass ordered_ab_group_add ..
  1.1390  
  1.1391 -lemma less_add_iff1:
  1.1392 -  "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
  1.1393 -by (simp add: algebra_simps)
  1.1394 +lemma less_add_iff1: "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
  1.1395 +  by (simp add: algebra_simps)
  1.1396  
  1.1397 -lemma less_add_iff2:
  1.1398 -  "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
  1.1399 -by (simp add: algebra_simps)
  1.1400 +lemma less_add_iff2: "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
  1.1401 +  by (simp add: algebra_simps)
  1.1402  
  1.1403 -lemma le_add_iff1:
  1.1404 -  "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
  1.1405 -by (simp add: algebra_simps)
  1.1406 +lemma le_add_iff1: "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
  1.1407 +  by (simp add: algebra_simps)
  1.1408  
  1.1409 -lemma le_add_iff2:
  1.1410 -  "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
  1.1411 -by (simp add: algebra_simps)
  1.1412 +lemma le_add_iff2: "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
  1.1413 +  by (simp add: algebra_simps)
  1.1414  
  1.1415 -lemma mult_left_mono_neg:
  1.1416 -  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
  1.1417 +lemma mult_left_mono_neg: "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
  1.1418    apply (drule mult_left_mono [of _ _ "- c"])
  1.1419    apply simp_all
  1.1420    done
  1.1421  
  1.1422 -lemma mult_right_mono_neg:
  1.1423 -  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
  1.1424 +lemma mult_right_mono_neg: "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
  1.1425    apply (drule mult_right_mono [of _ _ "- c"])
  1.1426    apply simp_all
  1.1427    done
  1.1428  
  1.1429  lemma mult_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
  1.1430 -using mult_right_mono_neg [of a 0 b] by simp
  1.1431 +  using mult_right_mono_neg [of a 0 b] by simp
  1.1432  
  1.1433 -lemma split_mult_pos_le:
  1.1434 -  "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
  1.1435 -by (auto simp add: mult_nonpos_nonpos)
  1.1436 +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"
  1.1437 +  by (auto simp add: mult_nonpos_nonpos)
  1.1438  
  1.1439  end
  1.1440  
  1.1441 @@ -1608,12 +1531,12 @@
  1.1442  proof
  1.1443    fix a b
  1.1444    show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
  1.1445 -    by (auto simp add: abs_if not_le not_less algebra_simps simp del: add.commute dest: add_neg_neg add_nonneg_nonneg)
  1.1446 +    by (auto simp add: abs_if not_le not_less algebra_simps
  1.1447 +        simp del: add.commute dest: add_neg_neg add_nonneg_nonneg)
  1.1448  qed (auto simp add: abs_if)
  1.1449  
  1.1450  lemma zero_le_square [simp]: "0 \<le> a * a"
  1.1451 -  using linear [of 0 a]
  1.1452 -  by (auto simp add: mult_nonpos_nonpos)
  1.1453 +  using linear [of 0 a] by (auto simp add: mult_nonpos_nonpos)
  1.1454  
  1.1455  lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"
  1.1456    by (simp add: not_less)
  1.1457 @@ -1621,12 +1544,10 @@
  1.1458  proposition abs_eq_iff: "\<bar>x\<bar> = \<bar>y\<bar> \<longleftrightarrow> x = y \<or> x = -y"
  1.1459    by (auto simp add: abs_if split: if_split_asm)
  1.1460  
  1.1461 -lemma sum_squares_ge_zero:
  1.1462 -  "0 \<le> x * x + y * y"
  1.1463 +lemma sum_squares_ge_zero: "0 \<le> x * x + y * y"
  1.1464    by (intro add_nonneg_nonneg zero_le_square)
  1.1465  
  1.1466 -lemma not_sum_squares_lt_zero:
  1.1467 -  "\<not> x * x + y * y < 0"
  1.1468 +lemma not_sum_squares_lt_zero: "\<not> x * x + y * y < 0"
  1.1469    by (simp add: not_less sum_squares_ge_zero)
  1.1470  
  1.1471  end
  1.1472 @@ -1638,40 +1559,49 @@
  1.1473  subclass linordered_ring ..
  1.1474  
  1.1475  lemma mult_strict_left_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
  1.1476 -using mult_strict_left_mono [of b a "- c"] by simp
  1.1477 +  using mult_strict_left_mono [of b a "- c"] by simp
  1.1478  
  1.1479  lemma mult_strict_right_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
  1.1480 -using mult_strict_right_mono [of b a "- c"] by simp
  1.1481 +  using mult_strict_right_mono [of b a "- c"] by simp
  1.1482  
  1.1483  lemma mult_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
  1.1484 -using mult_strict_right_mono_neg [of a 0 b] by simp
  1.1485 +  using mult_strict_right_mono_neg [of a 0 b] by simp
  1.1486  
  1.1487  subclass ring_no_zero_divisors
  1.1488  proof
  1.1489    fix a b
  1.1490 -  assume "a \<noteq> 0" then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
  1.1491 -  assume "b \<noteq> 0" then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
  1.1492 +  assume "a \<noteq> 0"
  1.1493 +  then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
  1.1494 +  assume "b \<noteq> 0"
  1.1495 +  then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
  1.1496    have "a * b < 0 \<or> 0 < a * b"
  1.1497    proof (cases "a < 0")
  1.1498 -    case True note A' = this
  1.1499 -    show ?thesis proof (cases "b < 0")
  1.1500 -      case True with A'
  1.1501 -      show ?thesis by (auto dest: mult_neg_neg)
  1.1502 +    case A': True
  1.1503 +    show ?thesis
  1.1504 +    proof (cases "b < 0")
  1.1505 +      case True
  1.1506 +      with A' show ?thesis by (auto dest: mult_neg_neg)
  1.1507      next
  1.1508 -      case False with B have "0 < b" by auto
  1.1509 +      case False
  1.1510 +      with B have "0 < b" by auto
  1.1511        with A' show ?thesis by (auto dest: mult_strict_right_mono)
  1.1512      qed
  1.1513    next
  1.1514 -    case False with A have A': "0 < a" by auto
  1.1515 -    show ?thesis proof (cases "b < 0")
  1.1516 -      case True with A'
  1.1517 -      show ?thesis by (auto dest: mult_strict_right_mono_neg)
  1.1518 +    case False
  1.1519 +    with A have A': "0 < a" by auto
  1.1520 +    show ?thesis
  1.1521 +    proof (cases "b < 0")
  1.1522 +      case True
  1.1523 +      with A' show ?thesis
  1.1524 +        by (auto dest: mult_strict_right_mono_neg)
  1.1525      next
  1.1526 -      case False with B have "0 < b" by auto
  1.1527 +      case False
  1.1528 +      with B have "0 < b" by auto
  1.1529        with A' show ?thesis by auto
  1.1530      qed
  1.1531    qed
  1.1532 -  then show "a * b \<noteq> 0" by (simp add: neq_iff)
  1.1533 +  then show "a * b \<noteq> 0"
  1.1534 +    by (simp add: neq_iff)
  1.1535  qed
  1.1536  
  1.1537  lemma zero_less_mult_iff: "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
  1.1538 @@ -1681,84 +1611,66 @@
  1.1539  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"
  1.1540    by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)
  1.1541  
  1.1542 -lemma mult_less_0_iff:
  1.1543 -  "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
  1.1544 -  apply (insert zero_less_mult_iff [of "-a" b])
  1.1545 -  apply force
  1.1546 -  done
  1.1547 +lemma mult_less_0_iff: "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
  1.1548 +  using zero_less_mult_iff [of "- a" b] by auto
  1.1549  
  1.1550 -lemma mult_le_0_iff:
  1.1551 -  "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
  1.1552 -  apply (insert zero_le_mult_iff [of "-a" b])
  1.1553 -  apply force
  1.1554 -  done
  1.1555 +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"
  1.1556 +  using zero_le_mult_iff [of "- a" b] by auto
  1.1557  
  1.1558 -text\<open>Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
  1.1559 -   also with the relations \<open>\<le>\<close> and equality.\<close>
  1.1560 +text \<open>
  1.1561 +  Cancellation laws for @{term "c * a < c * b"} and @{term "a * c < b * c"},
  1.1562 +  also with the relations \<open>\<le>\<close> and equality.
  1.1563 +\<close>
  1.1564  
  1.1565 -text\<open>These ``disjunction'' versions produce two cases when the comparison is
  1.1566 - an assumption, but effectively four when the comparison is a goal.\<close>
  1.1567 +text \<open>
  1.1568 +  These ``disjunction'' versions produce two cases when the comparison is
  1.1569 +  an assumption, but effectively four when the comparison is a goal.
  1.1570 +\<close>
  1.1571  
  1.1572 -lemma mult_less_cancel_right_disj:
  1.1573 -  "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
  1.1574 +lemma mult_less_cancel_right_disj: "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
  1.1575    apply (cases "c = 0")
  1.1576 -  apply (auto simp add: neq_iff mult_strict_right_mono
  1.1577 -                      mult_strict_right_mono_neg)
  1.1578 -  apply (auto simp add: not_less
  1.1579 -                      not_le [symmetric, of "a*c"]
  1.1580 -                      not_le [symmetric, of a])
  1.1581 +  apply (auto simp add: neq_iff mult_strict_right_mono mult_strict_right_mono_neg)
  1.1582 +  apply (auto simp add: not_less not_le [symmetric, of "a*c"] not_le [symmetric, of a])
  1.1583    apply (erule_tac [!] notE)
  1.1584 -  apply (auto simp add: less_imp_le mult_right_mono
  1.1585 -                      mult_right_mono_neg)
  1.1586 +  apply (auto simp add: less_imp_le mult_right_mono mult_right_mono_neg)
  1.1587    done
  1.1588  
  1.1589 -lemma mult_less_cancel_left_disj:
  1.1590 -  "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
  1.1591 +lemma mult_less_cancel_left_disj: "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
  1.1592    apply (cases "c = 0")
  1.1593 -  apply (auto simp add: neq_iff mult_strict_left_mono
  1.1594 -                      mult_strict_left_mono_neg)
  1.1595 -  apply (auto simp add: not_less
  1.1596 -                      not_le [symmetric, of "c*a"]
  1.1597 -                      not_le [symmetric, of a])
  1.1598 +  apply (auto simp add: neq_iff mult_strict_left_mono mult_strict_left_mono_neg)
  1.1599 +  apply (auto simp add: not_less not_le [symmetric, of "c * a"] not_le [symmetric, of a])
  1.1600    apply (erule_tac [!] notE)
  1.1601 -  apply (auto simp add: less_imp_le mult_left_mono
  1.1602 -                      mult_left_mono_neg)
  1.1603 +  apply (auto simp add: less_imp_le mult_left_mono mult_left_mono_neg)
  1.1604    done
  1.1605  
  1.1606 -text\<open>The ``conjunction of implication'' lemmas produce two cases when the
  1.1607 -comparison is a goal, but give four when the comparison is an assumption.\<close>
  1.1608 +text \<open>
  1.1609 +  The ``conjunction of implication'' lemmas produce two cases when the
  1.1610 +  comparison is a goal, but give four when the comparison is an assumption.
  1.1611 +\<close>
  1.1612  
  1.1613 -lemma mult_less_cancel_right:
  1.1614 -  "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
  1.1615 +lemma mult_less_cancel_right: "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
  1.1616    using mult_less_cancel_right_disj [of a c b] by auto
  1.1617  
  1.1618 -lemma mult_less_cancel_left:
  1.1619 -  "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
  1.1620 +lemma mult_less_cancel_left: "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
  1.1621    using mult_less_cancel_left_disj [of c a b] by auto
  1.1622  
  1.1623 -lemma mult_le_cancel_right:
  1.1624 -   "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
  1.1625 -by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
  1.1626 +lemma mult_le_cancel_right: "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
  1.1627 +  by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
  1.1628  
  1.1629 -lemma mult_le_cancel_left:
  1.1630 -  "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
  1.1631 -by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
  1.1632 +lemma mult_le_cancel_left: "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
  1.1633 +  by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
  1.1634  
  1.1635 -lemma mult_le_cancel_left_pos:
  1.1636 -  "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"
  1.1637 -by (auto simp: mult_le_cancel_left)
  1.1638 +lemma mult_le_cancel_left_pos: "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"
  1.1639 +  by (auto simp: mult_le_cancel_left)
  1.1640  
  1.1641 -lemma mult_le_cancel_left_neg:
  1.1642 -  "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a"
  1.1643 -by (auto simp: mult_le_cancel_left)
  1.1644 +lemma mult_le_cancel_left_neg: "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a"
  1.1645 +  by (auto simp: mult_le_cancel_left)
  1.1646  
  1.1647 -lemma mult_less_cancel_left_pos:
  1.1648 -  "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"
  1.1649 -by (auto simp: mult_less_cancel_left)
  1.1650 +lemma mult_less_cancel_left_pos: "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"
  1.1651 +  by (auto simp: mult_less_cancel_left)
  1.1652  
  1.1653 -lemma mult_less_cancel_left_neg:
  1.1654 -  "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a"
  1.1655 -by (auto simp: mult_less_cancel_left)
  1.1656 +lemma mult_less_cancel_left_neg: "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a"
  1.1657 +  by (auto simp: mult_less_cancel_left)
  1.1658  
  1.1659  end
  1.1660  
  1.1661 @@ -1783,19 +1695,19 @@
  1.1662  begin
  1.1663  
  1.1664  subclass zero_neq_one
  1.1665 -  proof qed (insert zero_less_one, blast)
  1.1666 +  by standard (insert zero_less_one, blast)
  1.1667  
  1.1668  subclass comm_semiring_1
  1.1669 -  proof qed (rule mult_1_left)
  1.1670 +  by standard (rule mult_1_left)
  1.1671  
  1.1672  lemma zero_le_one [simp]: "0 \<le> 1"
  1.1673 -by (rule zero_less_one [THEN less_imp_le])
  1.1674 +  by (rule zero_less_one [THEN less_imp_le])
  1.1675  
  1.1676  lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"
  1.1677 -by (simp add: not_le)
  1.1678 +  by (simp add: not_le)
  1.1679  
  1.1680  lemma not_one_less_zero [simp]: "\<not> 1 < 0"
  1.1681 -by (simp add: not_less)
  1.1682 +  by (simp add: not_less)
  1.1683  
  1.1684  lemma mult_left_le: "c \<le> 1 \<Longrightarrow> 0 \<le> a \<Longrightarrow> a * c \<le> a"
  1.1685    using mult_left_mono[of c 1 a] by simp
  1.1686 @@ -1812,8 +1724,7 @@
  1.1687    assumes le_add_diff_inverse2 [simp]: "b \<le> a \<Longrightarrow> a - b + b = a"
  1.1688  begin
  1.1689  
  1.1690 -subclass linordered_nonzero_semiring
  1.1691 -  proof qed
  1.1692 +subclass linordered_nonzero_semiring ..
  1.1693  
  1.1694  text \<open>Addition is the inverse of subtraction.\<close>
  1.1695  
  1.1696 @@ -1823,31 +1734,31 @@
  1.1697  lemma add_diff_inverse: "\<not> a < b \<Longrightarrow> b + (a - b) = a"
  1.1698    by simp
  1.1699  
  1.1700 -lemma add_le_imp_le_diff:
  1.1701 -  shows "i + k \<le> n \<Longrightarrow> i \<le> n - k"
  1.1702 +lemma add_le_imp_le_diff: "i + k \<le> n \<Longrightarrow> i \<le> n - k"
  1.1703    apply (subst add_le_cancel_right [where c=k, symmetric])
  1.1704    apply (frule le_add_diff_inverse2)
  1.1705    apply (simp only: add.assoc [symmetric])
  1.1706 -  using add_implies_diff by fastforce
  1.1707 +  using add_implies_diff apply fastforce
  1.1708 +  done
  1.1709  
  1.1710  lemma add_le_add_imp_diff_le:
  1.1711 -  assumes a1: "i + k \<le> n"
  1.1712 -      and a2: "n \<le> j + k"
  1.1713 -  shows "\<lbrakk>i + k \<le> n; n \<le> j + k\<rbrakk> \<Longrightarrow> n - k \<le> j"
  1.1714 +  assumes 1: "i + k \<le> n"
  1.1715 +    and 2: "n \<le> j + k"
  1.1716 +  shows "i + k \<le> n \<Longrightarrow> n \<le> j + k \<Longrightarrow> n - k \<le> j"
  1.1717  proof -
  1.1718    have "n - (i + k) + (i + k) = n"
  1.1719 -    using a1 by simp
  1.1720 +    using 1 by simp
  1.1721    moreover have "n - k = n - k - i + i"
  1.1722 -    using a1 by (simp add: add_le_imp_le_diff)
  1.1723 +    using 1 by (simp add: add_le_imp_le_diff)
  1.1724    ultimately show ?thesis
  1.1725 -    using a2
  1.1726 +    using 2
  1.1727      apply (simp add: add.assoc [symmetric])
  1.1728 -    apply (rule add_le_imp_le_diff [of _ k "j+k", simplified add_diff_cancel_right'])
  1.1729 -    by (simp add: add.commute diff_diff_add)
  1.1730 +    apply (rule add_le_imp_le_diff [of _ k "j + k", simplified add_diff_cancel_right'])
  1.1731 +    apply (simp add: add.commute diff_diff_add)
  1.1732 +    done
  1.1733  qed
  1.1734  
  1.1735 -lemma less_1_mult:
  1.1736 -  "1 < m \<Longrightarrow> 1 < n \<Longrightarrow> 1 < m * n"
  1.1737 +lemma less_1_mult: "1 < m \<Longrightarrow> 1 < n \<Longrightarrow> 1 < m * n"
  1.1738    using mult_strict_mono [of 1 m 1 n] by (simp add: less_trans [OF zero_less_one])
  1.1739  
  1.1740  end
  1.1741 @@ -1864,90 +1775,73 @@
  1.1742  subclass linordered_semidom
  1.1743  proof
  1.1744    have "0 \<le> 1 * 1" by (rule zero_le_square)
  1.1745 -  thus "0 < 1" by (simp add: le_less)
  1.1746 -  show "\<And>b a. b \<le> a \<Longrightarrow> a - b + b = a"
  1.1747 +  then show "0 < 1" by (simp add: le_less)
  1.1748 +  show "b \<le> a \<Longrightarrow> a - b + b = a" for a b
  1.1749      by simp
  1.1750  qed
  1.1751  
  1.1752  lemma linorder_neqE_linordered_idom:
  1.1753 -  assumes "x \<noteq> y" obtains "x < y" | "y < x"
  1.1754 +  assumes "x \<noteq> y"
  1.1755 +  obtains "x < y" | "y < x"
  1.1756    using assms by (rule neqE)
  1.1757  
  1.1758  text \<open>These cancellation simprules also produce two cases when the comparison is a goal.\<close>
  1.1759  
  1.1760 -lemma mult_le_cancel_right1:
  1.1761 -  "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
  1.1762 -by (insert mult_le_cancel_right [of 1 c b], simp)
  1.1763 +lemma mult_le_cancel_right1: "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
  1.1764 +  using mult_le_cancel_right [of 1 c b] by simp
  1.1765  
  1.1766 -lemma mult_le_cancel_right2:
  1.1767 -  "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
  1.1768 -by (insert mult_le_cancel_right [of a c 1], simp)
  1.1769 +lemma mult_le_cancel_right2: "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
  1.1770 +  using mult_le_cancel_right [of a c 1] by simp
  1.1771  
  1.1772 -lemma mult_le_cancel_left1:
  1.1773 -  "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
  1.1774 -by (insert mult_le_cancel_left [of c 1 b], simp)
  1.1775 +lemma mult_le_cancel_left1: "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
  1.1776 +  using mult_le_cancel_left [of c 1 b] by simp
  1.1777  
  1.1778 -lemma mult_le_cancel_left2:
  1.1779 -  "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
  1.1780 -by (insert mult_le_cancel_left [of c a 1], simp)
  1.1781 +lemma mult_le_cancel_left2: "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
  1.1782 +  using mult_le_cancel_left [of c a 1] by simp
  1.1783  
  1.1784 -lemma mult_less_cancel_right1:
  1.1785 -  "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
  1.1786 -by (insert mult_less_cancel_right [of 1 c b], simp)
  1.1787 +lemma mult_less_cancel_right1: "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
  1.1788 +  using mult_less_cancel_right [of 1 c b] by simp
  1.1789  
  1.1790 -lemma mult_less_cancel_right2:
  1.1791 -  "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
  1.1792 -by (insert mult_less_cancel_right [of a c 1], simp)
  1.1793 +lemma mult_less_cancel_right2: "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
  1.1794 +  using mult_less_cancel_right [of a c 1] by simp
  1.1795  
  1.1796 -lemma mult_less_cancel_left1:
  1.1797 -  "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
  1.1798 -by (insert mult_less_cancel_left [of c 1 b], simp)
  1.1799 +lemma mult_less_cancel_left1: "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
  1.1800 +  using mult_less_cancel_left [of c 1 b] by simp
  1.1801  
  1.1802 -lemma mult_less_cancel_left2:
  1.1803 -  "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
  1.1804 -by (insert mult_less_cancel_left [of c a 1], simp)
  1.1805 +lemma mult_less_cancel_left2: "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
  1.1806 +  using mult_less_cancel_left [of c a 1] by simp
  1.1807  
  1.1808 -lemma sgn_sgn [simp]:
  1.1809 -  "sgn (sgn a) = sgn a"
  1.1810 -unfolding sgn_if by simp
  1.1811 +lemma sgn_sgn [simp]: "sgn (sgn a) = sgn a"
  1.1812 +  unfolding sgn_if by simp
  1.1813  
  1.1814 -lemma sgn_0_0:
  1.1815 -  "sgn a = 0 \<longleftrightarrow> a = 0"
  1.1816 -unfolding sgn_if by simp
  1.1817 +lemma sgn_0_0: "sgn a = 0 \<longleftrightarrow> a = 0"
  1.1818 +  unfolding sgn_if by simp
  1.1819  
  1.1820 -lemma sgn_1_pos:
  1.1821 -  "sgn a = 1 \<longleftrightarrow> a > 0"
  1.1822 -unfolding sgn_if by simp
  1.1823 +lemma sgn_1_pos: "sgn a = 1 \<longleftrightarrow> a > 0"
  1.1824 +  unfolding sgn_if by simp
  1.1825  
  1.1826 -lemma sgn_1_neg:
  1.1827 -  "sgn a = - 1 \<longleftrightarrow> a < 0"
  1.1828 -unfolding sgn_if by auto
  1.1829 +lemma sgn_1_neg: "sgn a = - 1 \<longleftrightarrow> a < 0"
  1.1830 +  unfolding sgn_if by auto
  1.1831  
  1.1832 -lemma sgn_pos [simp]:
  1.1833 -  "0 < a \<Longrightarrow> sgn a = 1"
  1.1834 -unfolding sgn_1_pos .
  1.1835 +lemma sgn_pos [simp]: "0 < a \<Longrightarrow> sgn a = 1"
  1.1836 +  by (simp only: sgn_1_pos)
  1.1837  
  1.1838 -lemma sgn_neg [simp]:
  1.1839 -  "a < 0 \<Longrightarrow> sgn a = - 1"
  1.1840 -unfolding sgn_1_neg .
  1.1841 +lemma sgn_neg [simp]: "a < 0 \<Longrightarrow> sgn a = - 1"
  1.1842 +  by (simp only: sgn_1_neg)
  1.1843  
  1.1844 -lemma sgn_times:
  1.1845 -  "sgn (a * b) = sgn a * sgn b"
  1.1846 -by (auto simp add: sgn_if zero_less_mult_iff)
  1.1847 +lemma sgn_times: "sgn (a * b) = sgn a * sgn b"
  1.1848 +  by (auto simp add: sgn_if zero_less_mult_iff)
  1.1849  
  1.1850  lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"
  1.1851 -unfolding sgn_if abs_if by auto
  1.1852 +  unfolding sgn_if abs_if by auto
  1.1853  
  1.1854 -lemma sgn_greater [simp]:
  1.1855 -  "0 < sgn a \<longleftrightarrow> 0 < a"
  1.1856 +lemma sgn_greater [simp]: "0 < sgn a \<longleftrightarrow> 0 < a"
  1.1857    unfolding sgn_if by auto
  1.1858  
  1.1859 -lemma sgn_less [simp]:
  1.1860 -  "sgn a < 0 \<longleftrightarrow> a < 0"
  1.1861 +lemma sgn_less [simp]: "sgn a < 0 \<longleftrightarrow> a < 0"
  1.1862    unfolding sgn_if by auto
  1.1863  
  1.1864 -lemma abs_sgn_eq:
  1.1865 -  "\<bar>sgn a\<bar> = (if a = 0 then 0 else 1)"
  1.1866 +lemma abs_sgn_eq: "\<bar>sgn a\<bar> = (if a = 0 then 0 else 1)"
  1.1867    by (simp add: sgn_if)
  1.1868  
  1.1869  lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k"
  1.1870 @@ -1956,36 +1850,31 @@
  1.1871  lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k"
  1.1872    by (simp add: abs_if)
  1.1873  
  1.1874 -lemma dvd_if_abs_eq:
  1.1875 -  "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"
  1.1876 -by(subst abs_dvd_iff[symmetric]) simp
  1.1877 +lemma dvd_if_abs_eq: "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"
  1.1878 +  by (subst abs_dvd_iff [symmetric]) simp
  1.1879  
  1.1880 -text \<open>The following lemmas can be proven in more general structures, but
  1.1881 -are dangerous as simp rules in absence of @{thm neg_equal_zero},
  1.1882 -@{thm neg_less_pos}, @{thm neg_less_eq_nonneg}.\<close>
  1.1883 +text \<open>
  1.1884 +  The following lemmas can be proven in more general structures, but
  1.1885 +  are dangerous as simp rules in absence of @{thm neg_equal_zero},
  1.1886 +  @{thm neg_less_pos}, @{thm neg_less_eq_nonneg}.
  1.1887 +\<close>
  1.1888  
  1.1889 -lemma equation_minus_iff_1 [simp, no_atp]:
  1.1890 -  "1 = - a \<longleftrightarrow> a = - 1"
  1.1891 +lemma equation_minus_iff_1 [simp, no_atp]: "1 = - a \<longleftrightarrow> a = - 1"
  1.1892    by (fact equation_minus_iff)
  1.1893  
  1.1894 -lemma minus_equation_iff_1 [simp, no_atp]:
  1.1895 -  "- a = 1 \<longleftrightarrow> a = - 1"
  1.1896 +lemma minus_equation_iff_1 [simp, no_atp]: "- a = 1 \<longleftrightarrow> a = - 1"
  1.1897    by (subst minus_equation_iff, auto)
  1.1898  
  1.1899 -lemma le_minus_iff_1 [simp, no_atp]:
  1.1900 -  "1 \<le> - b \<longleftrightarrow> b \<le> - 1"
  1.1901 +lemma le_minus_iff_1 [simp, no_atp]: "1 \<le> - b \<longleftrightarrow> b \<le> - 1"
  1.1902    by (fact le_minus_iff)
  1.1903  
  1.1904 -lemma minus_le_iff_1 [simp, no_atp]:
  1.1905 -  "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"
  1.1906 +lemma minus_le_iff_1 [simp, no_atp]: "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"
  1.1907    by (fact minus_le_iff)
  1.1908  
  1.1909 -lemma less_minus_iff_1 [simp, no_atp]:
  1.1910 -  "1 < - b \<longleftrightarrow> b < - 1"
  1.1911 +lemma less_minus_iff_1 [simp, no_atp]: "1 < - b \<longleftrightarrow> b < - 1"
  1.1912    by (fact less_minus_iff)
  1.1913  
  1.1914 -lemma minus_less_iff_1 [simp, no_atp]:
  1.1915 -  "- a < 1 \<longleftrightarrow> - 1 < a"
  1.1916 +lemma minus_less_iff_1 [simp, no_atp]: "- a < 1 \<longleftrightarrow> - 1 < a"
  1.1917    by (fact minus_less_iff)
  1.1918  
  1.1919  end
  1.1920 @@ -1993,15 +1882,16 @@
  1.1921  text \<open>Simprules for comparisons where common factors can be cancelled.\<close>
  1.1922  
  1.1923  lemmas mult_compare_simps =
  1.1924 -    mult_le_cancel_right mult_le_cancel_left
  1.1925 -    mult_le_cancel_right1 mult_le_cancel_right2
  1.1926 -    mult_le_cancel_left1 mult_le_cancel_left2
  1.1927 -    mult_less_cancel_right mult_less_cancel_left
  1.1928 -    mult_less_cancel_right1 mult_less_cancel_right2
  1.1929 -    mult_less_cancel_left1 mult_less_cancel_left2
  1.1930 -    mult_cancel_right mult_cancel_left
  1.1931 -    mult_cancel_right1 mult_cancel_right2
  1.1932 -    mult_cancel_left1 mult_cancel_left2
  1.1933 +  mult_le_cancel_right mult_le_cancel_left
  1.1934 +  mult_le_cancel_right1 mult_le_cancel_right2
  1.1935 +  mult_le_cancel_left1 mult_le_cancel_left2
  1.1936 +  mult_less_cancel_right mult_less_cancel_left
  1.1937 +  mult_less_cancel_right1 mult_less_cancel_right2
  1.1938 +  mult_less_cancel_left1 mult_less_cancel_left2
  1.1939 +  mult_cancel_right mult_cancel_left
  1.1940 +  mult_cancel_right1 mult_cancel_right2
  1.1941 +  mult_cancel_left1 mult_cancel_left2
  1.1942 +
  1.1943  
  1.1944  text \<open>Reasoning about inequalities with division\<close>
  1.1945  
  1.1946 @@ -2012,7 +1902,7 @@
  1.1947  proof -
  1.1948    have "a + 0 < a + 1"
  1.1949      by (blast intro: zero_less_one add_strict_left_mono)
  1.1950 -  thus ?thesis by simp
  1.1951 +  then show ?thesis by simp
  1.1952  qed
  1.1953  
  1.1954  end
  1.1955 @@ -2020,12 +1910,10 @@
  1.1956  context linordered_idom
  1.1957  begin
  1.1958  
  1.1959 -lemma mult_right_le_one_le:
  1.1960 -  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"
  1.1961 +lemma mult_right_le_one_le: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"
  1.1962    by (rule mult_left_le)
  1.1963  
  1.1964 -lemma mult_left_le_one_le:
  1.1965 -  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"
  1.1966 +lemma mult_left_le_one_le: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"
  1.1967    by (auto simp add: mult_le_cancel_right2)
  1.1968  
  1.1969  end
  1.1970 @@ -2035,12 +1923,10 @@
  1.1971  context linordered_idom
  1.1972  begin
  1.1973  
  1.1974 -lemma mult_sgn_abs:
  1.1975 -  "sgn x * \<bar>x\<bar> = x"
  1.1976 +lemma mult_sgn_abs: "sgn x * \<bar>x\<bar> = x"
  1.1977    unfolding abs_if sgn_if by auto
  1.1978  
  1.1979 -lemma abs_one [simp]:
  1.1980 -  "\<bar>1\<bar> = 1"
  1.1981 +lemma abs_one [simp]: "\<bar>1\<bar> = 1"
  1.1982    by (simp add: abs_if)
  1.1983  
  1.1984  end
  1.1985 @@ -2052,57 +1938,54 @@
  1.1986  context linordered_idom
  1.1987  begin
  1.1988  
  1.1989 -subclass ordered_ring_abs proof
  1.1990 -qed (auto simp add: abs_if not_less mult_less_0_iff)
  1.1991 +subclass ordered_ring_abs
  1.1992 +  by standard (auto simp add: abs_if not_less mult_less_0_iff)
  1.1993  
  1.1994 -lemma abs_mult:
  1.1995 -  "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
  1.1996 +lemma abs_mult: "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
  1.1997    by (rule abs_eq_mult) auto
  1.1998  
  1.1999 -lemma abs_mult_self [simp]:
  1.2000 -  "\<bar>a\<bar> * \<bar>a\<bar> = a * a"
  1.2001 +lemma abs_mult_self [simp]: "\<bar>a\<bar> * \<bar>a\<bar> = a * a"
  1.2002    by (simp add: abs_if)
  1.2003  
  1.2004  lemma abs_mult_less:
  1.2005 -  "\<bar>a\<bar> < c \<Longrightarrow> \<bar>b\<bar> < d \<Longrightarrow> \<bar>a\<bar> * \<bar>b\<bar> < c * d"
  1.2006 +  assumes ac: "\<bar>a\<bar> < c"
  1.2007 +    and bd: "\<bar>b\<bar> < d"
  1.2008 +  shows "\<bar>a\<bar> * \<bar>b\<bar> < c * d"
  1.2009  proof -
  1.2010 -  assume ac: "\<bar>a\<bar> < c"
  1.2011 -  hence cpos: "0<c" by (blast intro: le_less_trans abs_ge_zero)
  1.2012 -  assume "\<bar>b\<bar> < d"
  1.2013 -  thus ?thesis by (simp add: ac cpos mult_strict_mono)
  1.2014 +  from ac have "0 < c"
  1.2015 +    by (blast intro: le_less_trans abs_ge_zero)
  1.2016 +  with bd show ?thesis by (simp add: ac mult_strict_mono)
  1.2017  qed
  1.2018  
  1.2019 -lemma abs_less_iff:
  1.2020 -  "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b"
  1.2021 +lemma abs_less_iff: "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b"
  1.2022    by (simp add: less_le abs_le_iff) (auto simp add: abs_if)
  1.2023  
  1.2024 -lemma abs_mult_pos:
  1.2025 -  "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"
  1.2026 +lemma abs_mult_pos: "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"
  1.2027    by (simp add: abs_mult)
  1.2028  
  1.2029 -lemma abs_diff_less_iff:
  1.2030 -  "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r"
  1.2031 +lemma abs_diff_less_iff: "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r"
  1.2032    by (auto simp add: diff_less_eq ac_simps abs_less_iff)
  1.2033  
  1.2034 -lemma abs_diff_le_iff:
  1.2035 -   "\<bar>x - a\<bar> \<le> r \<longleftrightarrow> a - r \<le> x \<and> x \<le> a + r"
  1.2036 +lemma abs_diff_le_iff: "\<bar>x - a\<bar> \<le> r \<longleftrightarrow> a - r \<le> x \<and> x \<le> a + r"
  1.2037    by (auto simp add: diff_le_eq ac_simps abs_le_iff)
  1.2038  
  1.2039  lemma abs_add_one_gt_zero: "0 < 1 + \<bar>x\<bar>"
  1.2040 -  by (force simp: abs_if not_less intro: zero_less_one add_strict_increasing less_trans)
  1.2041 +  by (auto simp: abs_if not_less intro: zero_less_one add_strict_increasing less_trans)
  1.2042  
  1.2043  end
  1.2044  
  1.2045  subsection \<open>Dioids\<close>
  1.2046  
  1.2047 -text \<open>Dioids are the alternative extensions of semirings, a semiring can either be a ring or a dioid
  1.2048 -but never both.\<close>
  1.2049 +text \<open>
  1.2050 +  Dioids are the alternative extensions of semirings, a semiring can
  1.2051 +  either be a ring or a dioid but never both.
  1.2052 +\<close>
  1.2053  
  1.2054  class dioid = semiring_1 + canonically_ordered_monoid_add
  1.2055  begin
  1.2056  
  1.2057  subclass ordered_semiring
  1.2058 -  proof qed (auto simp: le_iff_add distrib_left distrib_right)
  1.2059 +  by standard (auto simp: le_iff_add distrib_left distrib_right)
  1.2060  
  1.2061  end
  1.2062