tuned proofs;
authorwenzelm
Wed Sep 25 12:42:56 2013 +0200 (2013-09-25)
changeset 5387308594daabcd9
parent 53872 6e69f9ca8f1c
child 53874 7cec5a4d5532
tuned proofs;
src/HOL/Library/Extended_Real.thy
     1.1 --- a/src/HOL/Library/Extended_Real.thy	Wed Sep 25 11:12:59 2013 +0200
     1.2 +++ b/src/HOL/Library/Extended_Real.thy	Wed Sep 25 12:42:56 2013 +0200
     1.3 @@ -24,23 +24,29 @@
     1.4  
     1.5  instantiation ereal :: uminus
     1.6  begin
     1.7 -  fun uminus_ereal where
     1.8 -    "- (ereal r) = ereal (- r)"
     1.9 -  | "- PInfty = MInfty"
    1.10 -  | "- MInfty = PInfty"
    1.11 -  instance ..
    1.12 +
    1.13 +fun uminus_ereal where
    1.14 +  "- (ereal r) = ereal (- r)"
    1.15 +| "- PInfty = MInfty"
    1.16 +| "- MInfty = PInfty"
    1.17 +
    1.18 +instance ..
    1.19 +
    1.20  end
    1.21  
    1.22  instantiation ereal :: infinity
    1.23  begin
    1.24 -  definition "(\<infinity>::ereal) = PInfty"
    1.25 -  instance ..
    1.26 +
    1.27 +definition "(\<infinity>::ereal) = PInfty"
    1.28 +instance ..
    1.29 +
    1.30  end
    1.31  
    1.32  declare [[coercion "ereal :: real \<Rightarrow> ereal"]]
    1.33  
    1.34  lemma ereal_uminus_uminus[simp]:
    1.35 -  fixes a :: ereal shows "- (- a) = a"
    1.36 +  fixes a :: ereal
    1.37 +  shows "- (- a) = a"
    1.38    by (cases a) simp_all
    1.39  
    1.40  lemma
    1.41 @@ -59,7 +65,7 @@
    1.42  
    1.43  lemma [code_unfold]:
    1.44    "\<infinity> = PInfty"
    1.45 -  "-PInfty = MInfty"
    1.46 +  "- PInfty = MInfty"
    1.47    by simp_all
    1.48  
    1.49  lemma inj_ereal[simp]: "inj_on ereal A"
    1.50 @@ -76,77 +82,97 @@
    1.51  lemmas ereal3_cases = ereal2_cases[case_product ereal_cases]
    1.52  
    1.53  lemma ereal_uminus_eq_iff[simp]:
    1.54 -  fixes a b :: ereal shows "-a = -b \<longleftrightarrow> a = b"
    1.55 +  fixes a b :: ereal
    1.56 +  shows "-a = -b \<longleftrightarrow> a = b"
    1.57    by (cases rule: ereal2_cases[of a b]) simp_all
    1.58  
    1.59  function of_ereal :: "ereal \<Rightarrow> real" where
    1.60 -"of_ereal (ereal r) = r" |
    1.61 -"of_ereal \<infinity> = 0" |
    1.62 -"of_ereal (-\<infinity>) = 0"
    1.63 +  "of_ereal (ereal r) = r"
    1.64 +| "of_ereal \<infinity> = 0"
    1.65 +| "of_ereal (-\<infinity>) = 0"
    1.66    by (auto intro: ereal_cases)
    1.67 -termination proof qed (rule wf_empty)
    1.68 +termination by default (rule wf_empty)
    1.69  
    1.70  defs (overloaded)
    1.71    real_of_ereal_def [code_unfold]: "real \<equiv> of_ereal"
    1.72  
    1.73  lemma real_of_ereal[simp]:
    1.74 -    "real (- x :: ereal) = - (real x)"
    1.75 -    "real (ereal r) = r"
    1.76 -    "real (\<infinity>::ereal) = 0"
    1.77 +  "real (- x :: ereal) = - (real x)"
    1.78 +  "real (ereal r) = r"
    1.79 +  "real (\<infinity>::ereal) = 0"
    1.80    by (cases x) (simp_all add: real_of_ereal_def)
    1.81  
    1.82  lemma range_ereal[simp]: "range ereal = UNIV - {\<infinity>, -\<infinity>}"
    1.83  proof safe
    1.84 -  fix x assume "x \<notin> range ereal" "x \<noteq> \<infinity>"
    1.85 -  then show "x = -\<infinity>" by (cases x) auto
    1.86 +  fix x
    1.87 +  assume "x \<notin> range ereal" "x \<noteq> \<infinity>"
    1.88 +  then show "x = -\<infinity>"
    1.89 +    by (cases x) auto
    1.90  qed auto
    1.91  
    1.92  lemma ereal_range_uminus[simp]: "range uminus = (UNIV::ereal set)"
    1.93  proof safe
    1.94 -  fix x :: ereal show "x \<in> range uminus" by (intro image_eqI[of _ _ "-x"]) auto
    1.95 +  fix x :: ereal
    1.96 +  show "x \<in> range uminus"
    1.97 +    by (intro image_eqI[of _ _ "-x"]) auto
    1.98  qed auto
    1.99  
   1.100  instantiation ereal :: abs
   1.101  begin
   1.102 -  function abs_ereal where
   1.103 -    "\<bar>ereal r\<bar> = ereal \<bar>r\<bar>"
   1.104 -  | "\<bar>-\<infinity>\<bar> = (\<infinity>::ereal)"
   1.105 -  | "\<bar>\<infinity>\<bar> = (\<infinity>::ereal)"
   1.106 -  by (auto intro: ereal_cases)
   1.107 -  termination proof qed (rule wf_empty)
   1.108 -  instance ..
   1.109 +
   1.110 +function abs_ereal where
   1.111 +  "\<bar>ereal r\<bar> = ereal \<bar>r\<bar>"
   1.112 +| "\<bar>-\<infinity>\<bar> = (\<infinity>::ereal)"
   1.113 +| "\<bar>\<infinity>\<bar> = (\<infinity>::ereal)"
   1.114 +by (auto intro: ereal_cases)
   1.115 +termination proof qed (rule wf_empty)
   1.116 +
   1.117 +instance ..
   1.118 +
   1.119  end
   1.120  
   1.121 -lemma abs_eq_infinity_cases[elim!]: "\<lbrakk> \<bar>x :: ereal\<bar> = \<infinity> ; x = \<infinity> \<Longrightarrow> P ; x = -\<infinity> \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P"
   1.122 -  by (cases x) auto
   1.123 +lemma abs_eq_infinity_cases[elim!]:
   1.124 +  fixes x :: ereal
   1.125 +  assumes "\<bar>x\<bar> = \<infinity>"
   1.126 +  obtains "x = \<infinity>" | "x = -\<infinity>"
   1.127 +  using assms by (cases x) auto
   1.128  
   1.129 -lemma abs_neq_infinity_cases[elim!]: "\<lbrakk> \<bar>x :: ereal\<bar> \<noteq> \<infinity> ; \<And>r. x = ereal r \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P"
   1.130 +lemma abs_neq_infinity_cases[elim!]:
   1.131 +  fixes x :: ereal
   1.132 +  assumes "\<bar>x\<bar> \<noteq> \<infinity>"
   1.133 +  obtains r where "x = ereal r"
   1.134 +  using assms by (cases x) auto
   1.135 +
   1.136 +lemma abs_ereal_uminus[simp]:
   1.137 +  fixes x :: ereal
   1.138 +  shows "\<bar>- x\<bar> = \<bar>x\<bar>"
   1.139    by (cases x) auto
   1.140  
   1.141 -lemma abs_ereal_uminus[simp]: "\<bar>- x\<bar> = \<bar>x::ereal\<bar>"
   1.142 -  by (cases x) auto
   1.143 +lemma ereal_infinity_cases:
   1.144 +  fixes a :: ereal
   1.145 +  shows "a \<noteq> \<infinity> \<Longrightarrow> a \<noteq> -\<infinity> \<Longrightarrow> \<bar>a\<bar> \<noteq> \<infinity>"
   1.146 +  by auto
   1.147  
   1.148 -lemma ereal_infinity_cases: "(a::ereal) \<noteq> \<infinity> \<Longrightarrow> a \<noteq> -\<infinity> \<Longrightarrow> \<bar>a\<bar> \<noteq> \<infinity>"
   1.149 -  by auto
   1.150  
   1.151  subsubsection "Addition"
   1.152  
   1.153 -instantiation ereal :: "{one, comm_monoid_add}"
   1.154 +instantiation ereal :: "{one,comm_monoid_add}"
   1.155  begin
   1.156  
   1.157  definition "0 = ereal 0"
   1.158  definition "1 = ereal 1"
   1.159  
   1.160  function plus_ereal where
   1.161 -"ereal r + ereal p = ereal (r + p)" |
   1.162 -"\<infinity> + a = (\<infinity>::ereal)" |
   1.163 -"a + \<infinity> = (\<infinity>::ereal)" |
   1.164 -"ereal r + -\<infinity> = - \<infinity>" |
   1.165 -"-\<infinity> + ereal p = -(\<infinity>::ereal)" |
   1.166 -"-\<infinity> + -\<infinity> = -(\<infinity>::ereal)"
   1.167 +  "ereal r + ereal p = ereal (r + p)"
   1.168 +| "\<infinity> + a = (\<infinity>::ereal)"
   1.169 +| "a + \<infinity> = (\<infinity>::ereal)"
   1.170 +| "ereal r + -\<infinity> = - \<infinity>"
   1.171 +| "-\<infinity> + ereal p = -(\<infinity>::ereal)"
   1.172 +| "-\<infinity> + -\<infinity> = -(\<infinity>::ereal)"
   1.173  proof -
   1.174    case (goal1 P x)
   1.175 -  then obtain a b where "x = (a, b)" by (cases x) auto
   1.176 +  then obtain a b where "x = (a, b)"
   1.177 +    by (cases x) auto
   1.178    with goal1 show P
   1.179     by (cases rule: ereal2_cases[of a b]) auto
   1.180  qed auto
   1.181 @@ -172,6 +198,7 @@
   1.182    show "a + b + c = a + (b + c)"
   1.183      by (cases rule: ereal3_cases[of a b c]) simp_all
   1.184  qed
   1.185 +
   1.186  end
   1.187  
   1.188  instance ereal :: numeral ..
   1.189 @@ -182,35 +209,37 @@
   1.190  lemma abs_ereal_zero[simp]: "\<bar>0\<bar> = (0::ereal)"
   1.191    unfolding zero_ereal_def abs_ereal.simps by simp
   1.192  
   1.193 -lemma ereal_uminus_zero[simp]:
   1.194 -  "- 0 = (0::ereal)"
   1.195 +lemma ereal_uminus_zero[simp]: "- 0 = (0::ereal)"
   1.196    by (simp add: zero_ereal_def)
   1.197  
   1.198  lemma ereal_uminus_zero_iff[simp]:
   1.199 -  fixes a :: ereal shows "-a = 0 \<longleftrightarrow> a = 0"
   1.200 +  fixes a :: ereal
   1.201 +  shows "-a = 0 \<longleftrightarrow> a = 0"
   1.202    by (cases a) simp_all
   1.203  
   1.204  lemma ereal_plus_eq_PInfty[simp]:
   1.205 -  fixes a b :: ereal shows "a + b = \<infinity> \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>"
   1.206 +  fixes a b :: ereal
   1.207 +  shows "a + b = \<infinity> \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>"
   1.208    by (cases rule: ereal2_cases[of a b]) auto
   1.209  
   1.210  lemma ereal_plus_eq_MInfty[simp]:
   1.211 -  fixes a b :: ereal shows "a + b = -\<infinity> \<longleftrightarrow>
   1.212 -    (a = -\<infinity> \<or> b = -\<infinity>) \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>"
   1.213 +  fixes a b :: ereal
   1.214 +  shows "a + b = -\<infinity> \<longleftrightarrow> (a = -\<infinity> \<or> b = -\<infinity>) \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>"
   1.215    by (cases rule: ereal2_cases[of a b]) auto
   1.216  
   1.217  lemma ereal_add_cancel_left:
   1.218 -  fixes a b :: ereal assumes "a \<noteq> -\<infinity>"
   1.219 -  shows "a + b = a + c \<longleftrightarrow> (a = \<infinity> \<or> b = c)"
   1.220 +  fixes a b :: ereal
   1.221 +  assumes "a \<noteq> -\<infinity>"
   1.222 +  shows "a + b = a + c \<longleftrightarrow> a = \<infinity> \<or> b = c"
   1.223    using assms by (cases rule: ereal3_cases[of a b c]) auto
   1.224  
   1.225  lemma ereal_add_cancel_right:
   1.226 -  fixes a b :: ereal assumes "a \<noteq> -\<infinity>"
   1.227 -  shows "b + a = c + a \<longleftrightarrow> (a = \<infinity> \<or> b = c)"
   1.228 +  fixes a b :: ereal
   1.229 +  assumes "a \<noteq> -\<infinity>"
   1.230 +  shows "b + a = c + a \<longleftrightarrow> a = \<infinity> \<or> b = c"
   1.231    using assms by (cases rule: ereal3_cases[of a b c]) auto
   1.232  
   1.233 -lemma ereal_real:
   1.234 -  "ereal (real x) = (if \<bar>x\<bar> = \<infinity> then 0 else x)"
   1.235 +lemma ereal_real: "ereal (real x) = (if \<bar>x\<bar> = \<infinity> then 0 else x)"
   1.236    by (cases x) simp_all
   1.237  
   1.238  lemma real_of_ereal_add:
   1.239 @@ -219,6 +248,7 @@
   1.240      (if (\<bar>a\<bar> = \<infinity>) \<and> (\<bar>b\<bar> = \<infinity>) \<or> (\<bar>a\<bar> \<noteq> \<infinity>) \<and> (\<bar>b\<bar> \<noteq> \<infinity>) then real a + real b else 0)"
   1.241    by (cases rule: ereal2_cases[of a b]) auto
   1.242  
   1.243 +
   1.244  subsubsection "Linear order on @{typ ereal}"
   1.245  
   1.246  instantiation ereal :: linorder
   1.247 @@ -250,7 +280,7 @@
   1.248  lemma ereal_infty_less_eq[simp]:
   1.249    fixes x :: ereal
   1.250    shows "\<infinity> \<le> x \<longleftrightarrow> x = \<infinity>"
   1.251 -  "x \<le> -\<infinity> \<longleftrightarrow> x = -\<infinity>"
   1.252 +    and "x \<le> -\<infinity> \<longleftrightarrow> x = -\<infinity>"
   1.253    by (auto simp add: less_eq_ereal_def)
   1.254  
   1.255  lemma ereal_less[simp]:
   1.256 @@ -282,10 +312,16 @@
   1.257      by (cases rule: ereal2_cases[of x y]) auto
   1.258    show "x \<le> y \<or> y \<le> x "
   1.259      by (cases rule: ereal2_cases[of x y]) auto
   1.260 -  { assume "x \<le> y" "y \<le> x" then show "x = y"
   1.261 -    by (cases rule: ereal2_cases[of x y]) auto }
   1.262 -  { assume "x \<le> y" "y \<le> z" then show "x \<le> z"
   1.263 -    by (cases rule: ereal3_cases[of x y z]) auto }
   1.264 +  {
   1.265 +    assume "x \<le> y" "y \<le> x"
   1.266 +    then show "x = y"
   1.267 +      by (cases rule: ereal2_cases[of x y]) auto
   1.268 +  }
   1.269 +  {
   1.270 +    assume "x \<le> y" "y \<le> z"
   1.271 +    then show "x \<le> z"
   1.272 +      by (cases rule: ereal3_cases[of x y z]) auto
   1.273 +  }
   1.274  qed
   1.275  
   1.276  end
   1.277 @@ -298,20 +334,25 @@
   1.278  
   1.279  instance ereal :: ordered_ab_semigroup_add
   1.280  proof
   1.281 -  fix a b c :: ereal assume "a \<le> b" then show "c + a \<le> c + b"
   1.282 +  fix a b c :: ereal
   1.283 +  assume "a \<le> b"
   1.284 +  then show "c + a \<le> c + b"
   1.285      by (cases rule: ereal3_cases[of a b c]) auto
   1.286  qed
   1.287  
   1.288  lemma real_of_ereal_positive_mono:
   1.289 -  fixes x y :: ereal shows "\<lbrakk>0 \<le> x; x \<le> y; y \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> real x \<le> real y"
   1.290 +  fixes x y :: ereal
   1.291 +  shows "0 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<noteq> \<infinity> \<Longrightarrow> real x \<le> real y"
   1.292    by (cases rule: ereal2_cases[of x y]) auto
   1.293  
   1.294  lemma ereal_MInfty_lessI[intro, simp]:
   1.295 -  fixes a :: ereal shows "a \<noteq> -\<infinity> \<Longrightarrow> -\<infinity> < a"
   1.296 +  fixes a :: ereal
   1.297 +  shows "a \<noteq> -\<infinity> \<Longrightarrow> -\<infinity> < a"
   1.298    by (cases a) auto
   1.299  
   1.300  lemma ereal_less_PInfty[intro, simp]:
   1.301 -  fixes a :: ereal shows "a \<noteq> \<infinity> \<Longrightarrow> a < \<infinity>"
   1.302 +  fixes a :: ereal
   1.303 +  shows "a \<noteq> \<infinity> \<Longrightarrow> a < \<infinity>"
   1.304    by (cases a) auto
   1.305  
   1.306  lemma ereal_less_ereal_Ex:
   1.307 @@ -321,12 +362,16 @@
   1.308  
   1.309  lemma less_PInf_Ex_of_nat: "x \<noteq> \<infinity> \<longleftrightarrow> (\<exists>n::nat. x < ereal (real n))"
   1.310  proof (cases x)
   1.311 -  case (real r) then show ?thesis
   1.312 +  case (real r)
   1.313 +  then show ?thesis
   1.314      using reals_Archimedean2[of r] by simp
   1.315  qed simp_all
   1.316  
   1.317  lemma ereal_add_mono:
   1.318 -  fixes a b c d :: ereal assumes "a \<le> b" "c \<le> d" shows "a + c \<le> b + d"
   1.319 +  fixes a b c d :: ereal
   1.320 +  assumes "a \<le> b"
   1.321 +    and "c \<le> d"
   1.322 +  shows "a + c \<le> b + d"
   1.323    using assms
   1.324    apply (cases a)
   1.325    apply (cases rule: ereal3_cases[of b c d], auto)
   1.326 @@ -334,31 +379,34 @@
   1.327    done
   1.328  
   1.329  lemma ereal_minus_le_minus[simp]:
   1.330 -  fixes a b :: ereal shows "- a \<le> - b \<longleftrightarrow> b \<le> a"
   1.331 +  fixes a b :: ereal
   1.332 +  shows "- a \<le> - b \<longleftrightarrow> b \<le> a"
   1.333    by (cases rule: ereal2_cases[of a b]) auto
   1.334  
   1.335  lemma ereal_minus_less_minus[simp]:
   1.336 -  fixes a b :: ereal shows "- a < - b \<longleftrightarrow> b < a"
   1.337 +  fixes a b :: ereal
   1.338 +  shows "- a < - b \<longleftrightarrow> b < a"
   1.339    by (cases rule: ereal2_cases[of a b]) auto
   1.340  
   1.341  lemma ereal_le_real_iff:
   1.342 -  "x \<le> real y \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> ereal x \<le> y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x \<le> 0))"
   1.343 +  "x \<le> real y \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> ereal x \<le> y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x \<le> 0)"
   1.344    by (cases y) auto
   1.345  
   1.346  lemma real_le_ereal_iff:
   1.347 -  "real y \<le> x \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y \<le> ereal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 \<le> x))"
   1.348 +  "real y \<le> x \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y \<le> ereal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 \<le> x)"
   1.349    by (cases y) auto
   1.350  
   1.351  lemma ereal_less_real_iff:
   1.352 -  "x < real y \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> ereal x < y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x < 0))"
   1.353 +  "x < real y \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> ereal x < y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x < 0)"
   1.354    by (cases y) auto
   1.355  
   1.356  lemma real_less_ereal_iff:
   1.357 -  "real y < x \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y < ereal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 < x))"
   1.358 +  "real y < x \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y < ereal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 < x)"
   1.359    by (cases y) auto
   1.360  
   1.361  lemma real_of_ereal_pos:
   1.362 -  fixes x :: ereal shows "0 \<le> x \<Longrightarrow> 0 \<le> real x" by (cases x) auto
   1.363 +  fixes x :: ereal
   1.364 +  shows "0 \<le> x \<Longrightarrow> 0 \<le> real x" by (cases x) auto
   1.365  
   1.366  lemmas real_of_ereal_ord_simps =
   1.367    ereal_le_real_iff real_le_ereal_iff ereal_less_real_iff real_less_ereal_iff
   1.368 @@ -372,35 +420,44 @@
   1.369  lemma abs_ereal_pos[simp]: "0 \<le> \<bar>x :: ereal\<bar>"
   1.370    by (cases x) auto
   1.371  
   1.372 -lemma real_of_ereal_le_0[simp]: "real (x :: ereal) \<le> 0 \<longleftrightarrow> (x \<le> 0 \<or> x = \<infinity>)"
   1.373 +lemma real_of_ereal_le_0[simp]: "real (x :: ereal) \<le> 0 \<longleftrightarrow> x \<le> 0 \<or> x = \<infinity>"
   1.374    by (cases x) auto
   1.375  
   1.376  lemma abs_real_of_ereal[simp]: "\<bar>real (x :: ereal)\<bar> = real \<bar>x\<bar>"
   1.377    by (cases x) auto
   1.378  
   1.379  lemma zero_less_real_of_ereal:
   1.380 -  fixes x :: ereal shows "0 < real x \<longleftrightarrow> (0 < x \<and> x \<noteq> \<infinity>)"
   1.381 +  fixes x :: ereal
   1.382 +  shows "0 < real x \<longleftrightarrow> 0 < x \<and> x \<noteq> \<infinity>"
   1.383    by (cases x) auto
   1.384  
   1.385  lemma ereal_0_le_uminus_iff[simp]:
   1.386 -  fixes a :: ereal shows "0 \<le> -a \<longleftrightarrow> a \<le> 0"
   1.387 +  fixes a :: ereal
   1.388 +  shows "0 \<le> - a \<longleftrightarrow> a \<le> 0"
   1.389    by (cases rule: ereal2_cases[of a]) auto
   1.390  
   1.391  lemma ereal_uminus_le_0_iff[simp]:
   1.392 -  fixes a :: ereal shows "-a \<le> 0 \<longleftrightarrow> 0 \<le> a"
   1.393 +  fixes a :: ereal
   1.394 +  shows "- a \<le> 0 \<longleftrightarrow> 0 \<le> a"
   1.395    by (cases rule: ereal2_cases[of a]) auto
   1.396  
   1.397  lemma ereal_add_strict_mono:
   1.398    fixes a b c d :: ereal
   1.399 -  assumes "a = b" "0 \<le> a" "a \<noteq> \<infinity>" "c < d"
   1.400 +  assumes "a = b"
   1.401 +    and "0 \<le> a"
   1.402 +    and "a \<noteq> \<infinity>"
   1.403 +    and "c < d"
   1.404    shows "a + c < b + d"
   1.405 -  using assms by (cases rule: ereal3_cases[case_product ereal_cases, of a b c d]) auto
   1.406 +  using assms
   1.407 +  by (cases rule: ereal3_cases[case_product ereal_cases, of a b c d]) auto
   1.408  
   1.409 -lemma ereal_less_add: 
   1.410 -  fixes a b c :: ereal shows "\<bar>a\<bar> \<noteq> \<infinity> \<Longrightarrow> c < b \<Longrightarrow> a + c < a + b"
   1.411 +lemma ereal_less_add:
   1.412 +  fixes a b c :: ereal
   1.413 +  shows "\<bar>a\<bar> \<noteq> \<infinity> \<Longrightarrow> c < b \<Longrightarrow> a + c < a + b"
   1.414    by (cases rule: ereal2_cases[of b c]) auto
   1.415  
   1.416 -lemma ereal_uminus_eq_reorder: "- a = b \<longleftrightarrow> a = (-b::ereal)" by auto
   1.417 +lemma ereal_uminus_eq_reorder: "- a = b \<longleftrightarrow> a = (-b::ereal)"
   1.418 +  by auto
   1.419  
   1.420  lemma ereal_uminus_less_reorder: "- a < b \<longleftrightarrow> -b < (a::ereal)"
   1.421    by (subst (3) ereal_uminus_uminus[symmetric]) (simp only: ereal_minus_less_minus)
   1.422 @@ -412,23 +469,39 @@
   1.423    ereal_uminus_eq_reorder ereal_uminus_less_reorder ereal_uminus_le_reorder
   1.424  
   1.425  lemma ereal_bot:
   1.426 -  fixes x :: ereal assumes "\<And>B. x \<le> ereal B" shows "x = - \<infinity>"
   1.427 +  fixes x :: ereal
   1.428 +  assumes "\<And>B. x \<le> ereal B"
   1.429 +  shows "x = - \<infinity>"
   1.430  proof (cases x)
   1.431 -  case (real r) with assms[of "r - 1"] show ?thesis by auto
   1.432 +  case (real r)
   1.433 +  with assms[of "r - 1"] show ?thesis
   1.434 +    by auto
   1.435  next
   1.436 -  case PInf with assms[of 0] show ?thesis by auto
   1.437 +  case PInf
   1.438 +  with assms[of 0] show ?thesis
   1.439 +    by auto
   1.440  next
   1.441 -  case MInf then show ?thesis by simp
   1.442 +  case MInf
   1.443 +  then show ?thesis
   1.444 +    by simp
   1.445  qed
   1.446  
   1.447  lemma ereal_top:
   1.448 -  fixes x :: ereal assumes "\<And>B. x \<ge> ereal B" shows "x = \<infinity>"
   1.449 +  fixes x :: ereal
   1.450 +  assumes "\<And>B. x \<ge> ereal B"
   1.451 +  shows "x = \<infinity>"
   1.452  proof (cases x)
   1.453 -  case (real r) with assms[of "r + 1"] show ?thesis by auto
   1.454 +  case (real r)
   1.455 +  with assms[of "r + 1"] show ?thesis
   1.456 +    by auto
   1.457  next
   1.458 -  case MInf with assms[of 0] show ?thesis by auto
   1.459 +  case MInf
   1.460 +  with assms[of 0] show ?thesis
   1.461 +    by auto
   1.462  next
   1.463 -  case PInf then show ?thesis by simp
   1.464 +  case PInf
   1.465 +  then show ?thesis
   1.466 +    by simp
   1.467  qed
   1.468  
   1.469  lemma
   1.470 @@ -449,32 +522,36 @@
   1.471    unfolding incseq_def by auto
   1.472  
   1.473  lemma ereal_add_nonneg_nonneg:
   1.474 -  fixes a b :: ereal shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a + b"
   1.475 +  fixes a b :: ereal
   1.476 +  shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a + b"
   1.477    using add_mono[of 0 a 0 b] by simp
   1.478  
   1.479 -lemma image_eqD: "f ` A = B \<Longrightarrow> (\<forall>x\<in>A. f x \<in> B)"
   1.480 +lemma image_eqD: "f ` A = B \<Longrightarrow> \<forall>x\<in>A. f x \<in> B"
   1.481    by auto
   1.482  
   1.483  lemma incseq_setsumI:
   1.484 -  fixes f :: "nat \<Rightarrow> 'a::{comm_monoid_add, ordered_ab_semigroup_add}"
   1.485 +  fixes f :: "nat \<Rightarrow> 'a::{comm_monoid_add,ordered_ab_semigroup_add}"
   1.486    assumes "\<And>i. 0 \<le> f i"
   1.487    shows "incseq (\<lambda>i. setsum f {..< i})"
   1.488  proof (intro incseq_SucI)
   1.489 -  fix n have "setsum f {..< n} + 0 \<le> setsum f {..<n} + f n"
   1.490 +  fix n
   1.491 +  have "setsum f {..< n} + 0 \<le> setsum f {..<n} + f n"
   1.492      using assms by (rule add_left_mono)
   1.493    then show "setsum f {..< n} \<le> setsum f {..< Suc n}"
   1.494      by auto
   1.495  qed
   1.496  
   1.497  lemma incseq_setsumI2:
   1.498 -  fixes f :: "'i \<Rightarrow> nat \<Rightarrow> 'a::{comm_monoid_add, ordered_ab_semigroup_add}"
   1.499 +  fixes f :: "'i \<Rightarrow> nat \<Rightarrow> 'a::{comm_monoid_add,ordered_ab_semigroup_add}"
   1.500    assumes "\<And>n. n \<in> A \<Longrightarrow> incseq (f n)"
   1.501    shows "incseq (\<lambda>i. \<Sum>n\<in>A. f n i)"
   1.502 -  using assms unfolding incseq_def by (auto intro: setsum_mono)
   1.503 +  using assms
   1.504 +  unfolding incseq_def by (auto intro: setsum_mono)
   1.505 +
   1.506  
   1.507  subsubsection "Multiplication"
   1.508  
   1.509 -instantiation ereal :: "{comm_monoid_mult, sgn}"
   1.510 +instantiation ereal :: "{comm_monoid_mult,sgn}"
   1.511  begin
   1.512  
   1.513  function sgn_ereal :: "ereal \<Rightarrow> ereal" where
   1.514 @@ -482,28 +559,31 @@
   1.515  | "sgn (\<infinity>::ereal) = 1"
   1.516  | "sgn (-\<infinity>::ereal) = -1"
   1.517  by (auto intro: ereal_cases)
   1.518 -termination proof qed (rule wf_empty)
   1.519 +termination by default (rule wf_empty)
   1.520  
   1.521  function times_ereal where
   1.522 -"ereal r * ereal p = ereal (r * p)" |
   1.523 -"ereal r * \<infinity> = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)" |
   1.524 -"\<infinity> * ereal r = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)" |
   1.525 -"ereal r * -\<infinity> = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)" |
   1.526 -"-\<infinity> * ereal r = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)" |
   1.527 -"(\<infinity>::ereal) * \<infinity> = \<infinity>" |
   1.528 -"-(\<infinity>::ereal) * \<infinity> = -\<infinity>" |
   1.529 -"(\<infinity>::ereal) * -\<infinity> = -\<infinity>" |
   1.530 -"-(\<infinity>::ereal) * -\<infinity> = \<infinity>"
   1.531 +  "ereal r * ereal p = ereal (r * p)"
   1.532 +| "ereal r * \<infinity> = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)"
   1.533 +| "\<infinity> * ereal r = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)"
   1.534 +| "ereal r * -\<infinity> = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)"
   1.535 +| "-\<infinity> * ereal r = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)"
   1.536 +| "(\<infinity>::ereal) * \<infinity> = \<infinity>"
   1.537 +| "-(\<infinity>::ereal) * \<infinity> = -\<infinity>"
   1.538 +| "(\<infinity>::ereal) * -\<infinity> = -\<infinity>"
   1.539 +| "-(\<infinity>::ereal) * -\<infinity> = \<infinity>"
   1.540  proof -
   1.541    case (goal1 P x)
   1.542 -  then obtain a b where "x = (a, b)" by (cases x) auto
   1.543 -  with goal1 show P by (cases rule: ereal2_cases[of a b]) auto
   1.544 +  then obtain a b where "x = (a, b)"
   1.545 +    by (cases x) auto
   1.546 +  with goal1 show P
   1.547 +    by (cases rule: ereal2_cases[of a b]) auto
   1.548  qed simp_all
   1.549  termination by (relation "{}") simp
   1.550  
   1.551  instance
   1.552  proof
   1.553 -  fix a b c :: ereal show "1 * a = a"
   1.554 +  fix a b c :: ereal
   1.555 +  show "1 * a = a"
   1.556      by (cases a) (simp_all add: one_ereal_def)
   1.557    show "a * b = b * a"
   1.558      by (cases rule: ereal2_cases[of a b]) simp_all
   1.559 @@ -511,36 +591,39 @@
   1.560      by (cases rule: ereal3_cases[of a b c])
   1.561         (simp_all add: zero_ereal_def zero_less_mult_iff)
   1.562  qed
   1.563 +
   1.564  end
   1.565  
   1.566  lemma real_ereal_1[simp]: "real (1::ereal) = 1"
   1.567    unfolding one_ereal_def by simp
   1.568  
   1.569  lemma real_of_ereal_le_1:
   1.570 -  fixes a :: ereal shows "a \<le> 1 \<Longrightarrow> real a \<le> 1"
   1.571 +  fixes a :: ereal
   1.572 +  shows "a \<le> 1 \<Longrightarrow> real a \<le> 1"
   1.573    by (cases a) (auto simp: one_ereal_def)
   1.574  
   1.575  lemma abs_ereal_one[simp]: "\<bar>1\<bar> = (1::ereal)"
   1.576    unfolding one_ereal_def by simp
   1.577  
   1.578  lemma ereal_mult_zero[simp]:
   1.579 -  fixes a :: ereal shows "a * 0 = 0"
   1.580 +  fixes a :: ereal
   1.581 +  shows "a * 0 = 0"
   1.582    by (cases a) (simp_all add: zero_ereal_def)
   1.583  
   1.584  lemma ereal_zero_mult[simp]:
   1.585 -  fixes a :: ereal shows "0 * a = 0"
   1.586 +  fixes a :: ereal
   1.587 +  shows "0 * a = 0"
   1.588    by (cases a) (simp_all add: zero_ereal_def)
   1.589  
   1.590 -lemma ereal_m1_less_0[simp]:
   1.591 -  "-(1::ereal) < 0"
   1.592 +lemma ereal_m1_less_0[simp]: "-(1::ereal) < 0"
   1.593    by (simp add: zero_ereal_def one_ereal_def)
   1.594  
   1.595 -lemma ereal_zero_m1[simp]:
   1.596 -  "1 \<noteq> (0::ereal)"
   1.597 +lemma ereal_zero_m1[simp]: "1 \<noteq> (0::ereal)"
   1.598    by (simp add: zero_ereal_def one_ereal_def)
   1.599  
   1.600  lemma ereal_times_0[simp]:
   1.601 -  fixes x :: ereal shows "0 * x = 0"
   1.602 +  fixes x :: ereal
   1.603 +  shows "0 * x = 0"
   1.604    by (cases x) (auto simp: zero_ereal_def)
   1.605  
   1.606  lemma ereal_times[simp]:
   1.607 @@ -549,21 +632,24 @@
   1.608    by (auto simp add: times_ereal_def one_ereal_def)
   1.609  
   1.610  lemma ereal_plus_1[simp]:
   1.611 -  "1 + ereal r = ereal (r + 1)" "ereal r + 1 = ereal (r + 1)"
   1.612 -  "1 + -(\<infinity>::ereal) = -\<infinity>" "-(\<infinity>::ereal) + 1 = -\<infinity>"
   1.613 +  "1 + ereal r = ereal (r + 1)"
   1.614 +  "ereal r + 1 = ereal (r + 1)"
   1.615 +  "1 + -(\<infinity>::ereal) = -\<infinity>"
   1.616 +  "-(\<infinity>::ereal) + 1 = -\<infinity>"
   1.617    unfolding one_ereal_def by auto
   1.618  
   1.619  lemma ereal_zero_times[simp]:
   1.620 -  fixes a b :: ereal shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
   1.621 +  fixes a b :: ereal
   1.622 +  shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
   1.623    by (cases rule: ereal2_cases[of a b]) auto
   1.624  
   1.625  lemma ereal_mult_eq_PInfty[simp]:
   1.626 -  shows "a * b = (\<infinity>::ereal) \<longleftrightarrow>
   1.627 +  "a * b = (\<infinity>::ereal) \<longleftrightarrow>
   1.628      (a = \<infinity> \<and> b > 0) \<or> (a > 0 \<and> b = \<infinity>) \<or> (a = -\<infinity> \<and> b < 0) \<or> (a < 0 \<and> b = -\<infinity>)"
   1.629    by (cases rule: ereal2_cases[of a b]) auto
   1.630  
   1.631  lemma ereal_mult_eq_MInfty[simp]:
   1.632 -  shows "a * b = -(\<infinity>::ereal) \<longleftrightarrow>
   1.633 +  "a * b = -(\<infinity>::ereal) \<longleftrightarrow>
   1.634      (a = \<infinity> \<and> b < 0) \<or> (a < 0 \<and> b = \<infinity>) \<or> (a = -\<infinity> \<and> b > 0) \<or> (a > 0 \<and> b = -\<infinity>)"
   1.635    by (cases rule: ereal2_cases[of a b]) auto
   1.636  
   1.637 @@ -574,11 +660,13 @@
   1.638    by (simp_all add: zero_ereal_def one_ereal_def)
   1.639  
   1.640  lemma ereal_mult_minus_left[simp]:
   1.641 -  fixes a b :: ereal shows "-a * b = - (a * b)"
   1.642 +  fixes a b :: ereal
   1.643 +  shows "-a * b = - (a * b)"
   1.644    by (cases rule: ereal2_cases[of a b]) auto
   1.645  
   1.646  lemma ereal_mult_minus_right[simp]:
   1.647 -  fixes a b :: ereal shows "a * -b = - (a * b)"
   1.648 +  fixes a b :: ereal
   1.649 +  shows "a * -b = - (a * b)"
   1.650    by (cases rule: ereal2_cases[of a b]) auto
   1.651  
   1.652  lemma ereal_mult_infty[simp]:
   1.653 @@ -590,26 +678,33 @@
   1.654    by (cases a) auto
   1.655  
   1.656  lemma ereal_mult_strict_right_mono:
   1.657 -  assumes "a < b" and "0 < c" "c < (\<infinity>::ereal)"
   1.658 +  assumes "a < b"
   1.659 +    and "0 < c"
   1.660 +    and "c < (\<infinity>::ereal)"
   1.661    shows "a * c < b * c"
   1.662    using assms
   1.663 -  by (cases rule: ereal3_cases[of a b c])
   1.664 -     (auto simp: zero_le_mult_iff)
   1.665 +  by (cases rule: ereal3_cases[of a b c]) (auto simp: zero_le_mult_iff)
   1.666  
   1.667  lemma ereal_mult_strict_left_mono:
   1.668 -  "\<lbrakk> a < b ; 0 < c ; c < (\<infinity>::ereal)\<rbrakk> \<Longrightarrow> c * a < c * b"
   1.669 -  using ereal_mult_strict_right_mono by (simp add: mult_commute[of c])
   1.670 +  "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c < (\<infinity>::ereal) \<Longrightarrow> c * a < c * b"
   1.671 +  using ereal_mult_strict_right_mono
   1.672 +  by (simp add: mult_commute[of c])
   1.673  
   1.674  lemma ereal_mult_right_mono:
   1.675 -  fixes a b c :: ereal shows "\<lbrakk>a \<le> b; 0 \<le> c\<rbrakk> \<Longrightarrow> a*c \<le> b*c"
   1.676 +  fixes a b c :: ereal
   1.677 +  shows "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
   1.678    using assms
   1.679 -  apply (cases "c = 0") apply simp
   1.680 -  by (cases rule: ereal3_cases[of a b c])
   1.681 -     (auto simp: zero_le_mult_iff)
   1.682 +  apply (cases "c = 0")
   1.683 +  apply simp
   1.684 +  apply (cases rule: ereal3_cases[of a b c])
   1.685 +  apply (auto simp: zero_le_mult_iff)
   1.686 +  done
   1.687  
   1.688  lemma ereal_mult_left_mono:
   1.689 -  fixes a b c :: ereal shows "\<lbrakk>a \<le> b; 0 \<le> c\<rbrakk> \<Longrightarrow> c * a \<le> c * b"
   1.690 -  using ereal_mult_right_mono by (simp add: mult_commute[of c])
   1.691 +  fixes a b c :: ereal
   1.692 +  shows "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
   1.693 +  using ereal_mult_right_mono
   1.694 +  by (simp add: mult_commute[of c])
   1.695  
   1.696  lemma zero_less_one_ereal[simp]: "0 \<le> (1::ereal)"
   1.697    by (simp add: one_ereal_def zero_ereal_def)
   1.698 @@ -618,11 +713,13 @@
   1.699    by (cases rule: ereal2_cases[of a b]) (auto simp: mult_nonneg_nonneg)
   1.700  
   1.701  lemma ereal_right_distrib:
   1.702 -  fixes r a b :: ereal shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> r * (a + b) = r * a + r * b"
   1.703 +  fixes r a b :: ereal
   1.704 +  shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> r * (a + b) = r * a + r * b"
   1.705    by (cases rule: ereal3_cases[of r a b]) (simp_all add: field_simps)
   1.706  
   1.707  lemma ereal_left_distrib:
   1.708 -  fixes r a b :: ereal shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> (a + b) * r = a * r + b * r"
   1.709 +  fixes r a b :: ereal
   1.710 +  shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> (a + b) * r = a * r + b * r"
   1.711    by (cases rule: ereal3_cases[of r a b]) (simp_all add: field_simps)
   1.712  
   1.713  lemma ereal_mult_le_0_iff:
   1.714 @@ -657,7 +754,9 @@
   1.715  
   1.716  lemma ereal_distrib:
   1.717    fixes a b c :: ereal
   1.718 -  assumes "a \<noteq> \<infinity> \<or> b \<noteq> -\<infinity>" "a \<noteq> -\<infinity> \<or> b \<noteq> \<infinity>" "\<bar>c\<bar> \<noteq> \<infinity>"
   1.719 +  assumes "a \<noteq> \<infinity> \<or> b \<noteq> -\<infinity>"
   1.720 +    and "a \<noteq> -\<infinity> \<or> b \<noteq> \<infinity>"
   1.721 +    and "\<bar>c\<bar> \<noteq> \<infinity>"
   1.722    shows "(a + b) * c = a * c + b * c"
   1.723    using assms
   1.724    by (cases rule: ereal3_cases[of a b c]) (simp_all add: field_simps)
   1.725 @@ -670,74 +769,119 @@
   1.726  
   1.727  lemma ereal_le_epsilon:
   1.728    fixes x y :: ereal
   1.729 -  assumes "ALL e. 0 < e --> x <= y + e"
   1.730 -  shows "x <= y"
   1.731 -proof-
   1.732 -{ assume a: "EX r. y = ereal r"
   1.733 -  then obtain r where r_def: "y = ereal r" by auto
   1.734 -  { assume "x=(-\<infinity>)" hence ?thesis by auto }
   1.735 +  assumes "\<forall>e. 0 < e \<longrightarrow> x \<le> y + e"
   1.736 +  shows "x \<le> y"
   1.737 +proof -
   1.738 +  {
   1.739 +    assume a: "\<exists>r. y = ereal r"
   1.740 +    then obtain r where r_def: "y = ereal r"
   1.741 +      by auto
   1.742 +    {
   1.743 +      assume "x = -\<infinity>"
   1.744 +      then have ?thesis by auto
   1.745 +    }
   1.746 +    moreover
   1.747 +    {
   1.748 +      assume "x \<noteq> -\<infinity>"
   1.749 +      then obtain p where p_def: "x = ereal p"
   1.750 +      using a assms[rule_format, of 1]
   1.751 +        by (cases x) auto
   1.752 +      {
   1.753 +        fix e
   1.754 +        have "0 < e \<longrightarrow> p \<le> r + e"
   1.755 +          using assms[rule_format, of "ereal e"] p_def r_def by auto
   1.756 +      }
   1.757 +      then have "p \<le> r"
   1.758 +        apply (subst field_le_epsilon)
   1.759 +        apply auto
   1.760 +        done
   1.761 +      then have ?thesis
   1.762 +        using r_def p_def by auto
   1.763 +    }
   1.764 +    ultimately have ?thesis
   1.765 +      by blast
   1.766 +  }
   1.767    moreover
   1.768 -  { assume "~(x=(-\<infinity>))"
   1.769 -    then obtain p where p_def: "x = ereal p"
   1.770 -    using a assms[rule_format, of 1] by (cases x) auto
   1.771 -    { fix e have "0 < e --> p <= r + e"
   1.772 -      using assms[rule_format, of "ereal e"] p_def r_def by auto }
   1.773 -    hence "p <= r" apply (subst field_le_epsilon) by auto
   1.774 -    hence ?thesis using r_def p_def by auto
   1.775 -  } ultimately have ?thesis by blast
   1.776 -}
   1.777 -moreover
   1.778 -{ assume "y=(-\<infinity>) | y=\<infinity>" hence ?thesis
   1.779 -    using assms[rule_format, of 1] by (cases x) auto
   1.780 -} ultimately show ?thesis by (cases y) auto
   1.781 +  {
   1.782 +    assume "y = -\<infinity> | y = \<infinity>"
   1.783 +    then have ?thesis
   1.784 +      using assms[rule_format, of 1] by (cases x) auto
   1.785 +  }
   1.786 +  ultimately show ?thesis
   1.787 +    by (cases y) auto
   1.788  qed
   1.789  
   1.790 -
   1.791  lemma ereal_le_epsilon2:
   1.792    fixes x y :: ereal
   1.793 -  assumes "ALL e. 0 < e --> x <= y + ereal e"
   1.794 -  shows "x <= y"
   1.795 -proof-
   1.796 -{ fix e :: ereal assume "e>0"
   1.797 -  { assume "e=\<infinity>" hence "x<=y+e" by auto }
   1.798 -  moreover
   1.799 -  { assume "e~=\<infinity>"
   1.800 -    then obtain r where "e = ereal r" using `e>0` apply (cases e) by auto
   1.801 -    hence "x<=y+e" using assms[rule_format, of r] `e>0` by auto
   1.802 -  } ultimately have "x<=y+e" by blast
   1.803 -} then show ?thesis using ereal_le_epsilon by auto
   1.804 +  assumes "\<forall>e. 0 < e \<longrightarrow> x \<le> y + ereal e"
   1.805 +  shows "x \<le> y"
   1.806 +proof -
   1.807 +  {
   1.808 +    fix e :: ereal
   1.809 +    assume "e > 0"
   1.810 +    {
   1.811 +      assume "e = \<infinity>"
   1.812 +      then have "x \<le> y + e"
   1.813 +        by auto
   1.814 +    }
   1.815 +    moreover
   1.816 +    {
   1.817 +      assume "e \<noteq> \<infinity>"
   1.818 +      then obtain r where "e = ereal r"
   1.819 +        using `e > 0` by (cases e) auto
   1.820 +      then have "x \<le> y + e"
   1.821 +        using assms[rule_format, of r] `e>0` by auto
   1.822 +    }
   1.823 +    ultimately have "x \<le> y + e"
   1.824 +      by blast
   1.825 +  }
   1.826 +  then show ?thesis
   1.827 +    using ereal_le_epsilon by auto
   1.828  qed
   1.829  
   1.830  lemma ereal_le_real:
   1.831    fixes x y :: ereal
   1.832 -  assumes "ALL z. x <= ereal z --> y <= ereal z"
   1.833 -  shows "y <= x"
   1.834 -by (metis assms ereal_bot ereal_cases ereal_infty_less_eq(2) ereal_less_eq(1) linorder_le_cases)
   1.835 +  assumes "\<forall>z. x \<le> ereal z \<longrightarrow> y \<le> ereal z"
   1.836 +  shows "y \<le> x"
   1.837 +  by (metis assms ereal_bot ereal_cases ereal_infty_less_eq(2) ereal_less_eq(1) linorder_le_cases)
   1.838  
   1.839  lemma setprod_ereal_0:
   1.840    fixes f :: "'a \<Rightarrow> ereal"
   1.841 -  shows "(\<Prod>i\<in>A. f i) = 0 \<longleftrightarrow> (finite A \<and> (\<exists>i\<in>A. f i = 0))"
   1.842 -proof cases
   1.843 -  assume "finite A"
   1.844 +  shows "(\<Prod>i\<in>A. f i) = 0 \<longleftrightarrow> finite A \<and> (\<exists>i\<in>A. f i = 0)"
   1.845 +proof (cases "finite A")
   1.846 +  case True
   1.847    then show ?thesis by (induct A) auto
   1.848 -qed auto
   1.849 +next
   1.850 +  case False
   1.851 +  then show ?thesis by auto
   1.852 +qed
   1.853  
   1.854  lemma setprod_ereal_pos:
   1.855 -  fixes f :: "'a \<Rightarrow> ereal" assumes pos: "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i" shows "0 \<le> (\<Prod>i\<in>I. f i)"
   1.856 -proof cases
   1.857 -  assume "finite I" from this pos show ?thesis by induct auto
   1.858 -qed simp
   1.859 +  fixes f :: "'a \<Rightarrow> ereal"
   1.860 +  assumes pos: "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i"
   1.861 +  shows "0 \<le> (\<Prod>i\<in>I. f i)"
   1.862 +proof (cases "finite I")
   1.863 +  case True
   1.864 +  from this pos show ?thesis
   1.865 +    by induct auto
   1.866 +next
   1.867 +  case False
   1.868 +  then show ?thesis by simp
   1.869 +qed
   1.870  
   1.871  lemma setprod_PInf:
   1.872    fixes f :: "'a \<Rightarrow> ereal"
   1.873    assumes "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i"
   1.874    shows "(\<Prod>i\<in>I. f i) = \<infinity> \<longleftrightarrow> finite I \<and> (\<exists>i\<in>I. f i = \<infinity>) \<and> (\<forall>i\<in>I. f i \<noteq> 0)"
   1.875 -proof cases
   1.876 -  assume "finite I" from this assms show ?thesis
   1.877 +proof (cases "finite I")
   1.878 +  case True
   1.879 +  from this assms show ?thesis
   1.880    proof (induct I)
   1.881      case (insert i I)
   1.882 -    then have pos: "0 \<le> f i" "0 \<le> setprod f I" by (auto intro!: setprod_ereal_pos)
   1.883 -    from insert have "(\<Prod>j\<in>insert i I. f j) = \<infinity> \<longleftrightarrow> setprod f I * f i = \<infinity>" by auto
   1.884 +    then have pos: "0 \<le> f i" "0 \<le> setprod f I"
   1.885 +      by (auto intro!: setprod_ereal_pos)
   1.886 +    from insert have "(\<Prod>j\<in>insert i I. f j) = \<infinity> \<longleftrightarrow> setprod f I * f i = \<infinity>"
   1.887 +      by auto
   1.888      also have "\<dots> \<longleftrightarrow> (setprod f I = \<infinity> \<or> f i = \<infinity>) \<and> f i \<noteq> 0 \<and> setprod f I \<noteq> 0"
   1.889        using setprod_ereal_pos[of I f] pos
   1.890        by (cases rule: ereal2_cases[of "f i" "setprod f I"]) auto
   1.891 @@ -745,13 +889,22 @@
   1.892        using insert by (auto simp: setprod_ereal_0)
   1.893      finally show ?case .
   1.894    qed simp
   1.895 -qed simp
   1.896 +next
   1.897 +  case False
   1.898 +  then show ?thesis by simp
   1.899 +qed
   1.900  
   1.901  lemma setprod_ereal: "(\<Prod>i\<in>A. ereal (f i)) = ereal (setprod f A)"
   1.902 -proof cases
   1.903 -  assume "finite A" then show ?thesis
   1.904 +proof (cases "finite A")
   1.905 +  case True
   1.906 +  then show ?thesis
   1.907      by induct (auto simp: one_ereal_def)
   1.908 -qed (simp add: one_ereal_def)
   1.909 +next
   1.910 +  case False
   1.911 +  then show ?thesis
   1.912 +    by (simp add: one_ereal_def)
   1.913 +qed
   1.914 +
   1.915  
   1.916  subsubsection {* Power *}
   1.917  
   1.918 @@ -771,10 +924,12 @@
   1.919    by (induct n) (auto simp: one_ereal_def)
   1.920  
   1.921  lemma zero_le_power_ereal[simp]:
   1.922 -  fixes a :: ereal assumes "0 \<le> a"
   1.923 +  fixes a :: ereal
   1.924 +  assumes "0 \<le> a"
   1.925    shows "0 \<le> a ^ n"
   1.926    using assms by (induct n) (auto simp: ereal_zero_le_0_iff)
   1.927  
   1.928 +
   1.929  subsubsection {* Subtraction *}
   1.930  
   1.931  lemma ereal_minus_minus_image[simp]:
   1.932 @@ -783,25 +938,30 @@
   1.933    by (auto simp: image_iff)
   1.934  
   1.935  lemma ereal_uminus_lessThan[simp]:
   1.936 -  fixes a :: ereal shows "uminus ` {..<a} = {-a<..}"
   1.937 +  fixes a :: ereal
   1.938 +  shows "uminus ` {..<a} = {-a<..}"
   1.939  proof -
   1.940    {
   1.941 -    fix x assume "-a < x"
   1.942 -    then have "- x < - (- a)" by (simp del: ereal_uminus_uminus)
   1.943 -    then have "- x < a" by simp
   1.944 +    fix x
   1.945 +    assume "-a < x"
   1.946 +    then have "- x < - (- a)"
   1.947 +      by (simp del: ereal_uminus_uminus)
   1.948 +    then have "- x < a"
   1.949 +      by simp
   1.950    }
   1.951 -  then show ?thesis by (auto intro!: image_eqI)
   1.952 +  then show ?thesis
   1.953 +    by (auto intro!: image_eqI)
   1.954  qed
   1.955  
   1.956 -lemma ereal_uminus_greaterThan[simp]:
   1.957 -  "uminus ` {(a::ereal)<..} = {..<-a}"
   1.958 -  by (metis ereal_uminus_lessThan ereal_uminus_uminus
   1.959 -            ereal_minus_minus_image)
   1.960 +lemma ereal_uminus_greaterThan[simp]: "uminus ` {(a::ereal)<..} = {..<-a}"
   1.961 +  by (metis ereal_uminus_lessThan ereal_uminus_uminus ereal_minus_minus_image)
   1.962  
   1.963  instantiation ereal :: minus
   1.964  begin
   1.965 +
   1.966  definition "x - y = x + -(y::ereal)"
   1.967  instance ..
   1.968 +
   1.969  end
   1.970  
   1.971  lemma ereal_minus[simp]:
   1.972 @@ -815,8 +975,7 @@
   1.973    "0 - x = -x"
   1.974    by (simp_all add: minus_ereal_def)
   1.975  
   1.976 -lemma ereal_x_minus_x[simp]:
   1.977 -  "x - x = (if \<bar>x\<bar> = \<infinity> then \<infinity> else 0::ereal)"
   1.978 +lemma ereal_x_minus_x[simp]: "x - x = (if \<bar>x\<bar> = \<infinity> then \<infinity> else 0::ereal)"
   1.979    by (cases x) simp_all
   1.980  
   1.981  lemma ereal_eq_minus_iff:
   1.982 @@ -848,9 +1007,7 @@
   1.983  
   1.984  lemma ereal_le_minus_iff:
   1.985    fixes x y z :: ereal
   1.986 -  shows "x \<le> z - y \<longleftrightarrow>
   1.987 -    (y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = -\<infinity>) \<and>
   1.988 -    (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y \<le> z)"
   1.989 +  shows "x \<le> z - y \<longleftrightarrow> (y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = -\<infinity>) \<and> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y \<le> z)"
   1.990    by (cases rule: ereal3_cases[of x y z]) auto
   1.991  
   1.992  lemma ereal_le_minus:
   1.993 @@ -860,9 +1017,7 @@
   1.994  
   1.995  lemma ereal_minus_less_iff:
   1.996    fixes x y z :: ereal
   1.997 -  shows "x - y < z \<longleftrightarrow>
   1.998 -    y \<noteq> -\<infinity> \<and> (y = \<infinity> \<longrightarrow> x \<noteq> \<infinity> \<and> z \<noteq> -\<infinity>) \<and>
   1.999 -    (y \<noteq> \<infinity> \<longrightarrow> x < z + y)"
  1.1000 +  shows "x - y < z \<longleftrightarrow> y \<noteq> -\<infinity> \<and> (y = \<infinity> \<longrightarrow> x \<noteq> \<infinity> \<and> z \<noteq> -\<infinity>) \<and> (y \<noteq> \<infinity> \<longrightarrow> x < z + y)"
  1.1001    by (cases rule: ereal3_cases[of x y z]) auto
  1.1002  
  1.1003  lemma ereal_minus_less:
  1.1004 @@ -917,31 +1072,40 @@
  1.1005  
  1.1006  lemma ereal_between:
  1.1007    fixes x e :: ereal
  1.1008 -  assumes "\<bar>x\<bar> \<noteq> \<infinity>" "0 < e"
  1.1009 -  shows "x - e < x" "x < x + e"
  1.1010 -using assms apply (cases x, cases e) apply auto
  1.1011 -using assms apply (cases x, cases e) apply auto
  1.1012 -done
  1.1013 +  assumes "\<bar>x\<bar> \<noteq> \<infinity>"
  1.1014 +    and "0 < e"
  1.1015 +  shows "x - e < x"
  1.1016 +    and "x < x + e"
  1.1017 +  using assms
  1.1018 +  apply (cases x, cases e)
  1.1019 +  apply auto
  1.1020 +  using assms
  1.1021 +  apply (cases x, cases e)
  1.1022 +  apply auto
  1.1023 +  done
  1.1024  
  1.1025  lemma ereal_minus_eq_PInfty_iff:
  1.1026 -  fixes x y :: ereal shows "x - y = \<infinity> \<longleftrightarrow> y = -\<infinity> \<or> x = \<infinity>"
  1.1027 +  fixes x y :: ereal
  1.1028 +  shows "x - y = \<infinity> \<longleftrightarrow> y = -\<infinity> \<or> x = \<infinity>"
  1.1029    by (cases x y rule: ereal2_cases) simp_all
  1.1030  
  1.1031 +
  1.1032  subsubsection {* Division *}
  1.1033  
  1.1034  instantiation ereal :: inverse
  1.1035  begin
  1.1036  
  1.1037  function inverse_ereal where
  1.1038 -"inverse (ereal r) = (if r = 0 then \<infinity> else ereal (inverse r))" |
  1.1039 -"inverse (\<infinity>::ereal) = 0" |
  1.1040 -"inverse (-\<infinity>::ereal) = 0"
  1.1041 +  "inverse (ereal r) = (if r = 0 then \<infinity> else ereal (inverse r))"
  1.1042 +| "inverse (\<infinity>::ereal) = 0"
  1.1043 +| "inverse (-\<infinity>::ereal) = 0"
  1.1044    by (auto intro: ereal_cases)
  1.1045  termination by (relation "{}") simp
  1.1046  
  1.1047  definition "x / y = x * inverse (y :: ereal)"
  1.1048  
  1.1049  instance ..
  1.1050 +
  1.1051  end
  1.1052  
  1.1053  lemma real_of_ereal_inverse[simp]:
  1.1054 @@ -959,53 +1123,61 @@
  1.1055    unfolding divide_ereal_def by (auto simp: divide_real_def)
  1.1056  
  1.1057  lemma ereal_divide_same[simp]:
  1.1058 -  fixes x :: ereal shows "x / x = (if \<bar>x\<bar> = \<infinity> \<or> x = 0 then 0 else 1)"
  1.1059 -  by (cases x)
  1.1060 -     (simp_all add: divide_real_def divide_ereal_def one_ereal_def)
  1.1061 +  fixes x :: ereal
  1.1062 +  shows "x / x = (if \<bar>x\<bar> = \<infinity> \<or> x = 0 then 0 else 1)"
  1.1063 +  by (cases x) (simp_all add: divide_real_def divide_ereal_def one_ereal_def)
  1.1064  
  1.1065  lemma ereal_inv_inv[simp]:
  1.1066 -  fixes x :: ereal shows "inverse (inverse x) = (if x \<noteq> -\<infinity> then x else \<infinity>)"
  1.1067 +  fixes x :: ereal
  1.1068 +  shows "inverse (inverse x) = (if x \<noteq> -\<infinity> then x else \<infinity>)"
  1.1069    by (cases x) auto
  1.1070  
  1.1071  lemma ereal_inverse_minus[simp]:
  1.1072 -  fixes x :: ereal shows "inverse (- x) = (if x = 0 then \<infinity> else -inverse x)"
  1.1073 +  fixes x :: ereal
  1.1074 +  shows "inverse (- x) = (if x = 0 then \<infinity> else -inverse x)"
  1.1075    by (cases x) simp_all
  1.1076  
  1.1077  lemma ereal_uminus_divide[simp]:
  1.1078 -  fixes x y :: ereal shows "- x / y = - (x / y)"
  1.1079 +  fixes x y :: ereal
  1.1080 +  shows "- x / y = - (x / y)"
  1.1081    unfolding divide_ereal_def by simp
  1.1082  
  1.1083  lemma ereal_divide_Infty[simp]:
  1.1084 -  fixes x :: ereal shows "x / \<infinity> = 0" "x / -\<infinity> = 0"
  1.1085 +  fixes x :: ereal
  1.1086 +  shows "x / \<infinity> = 0" "x / -\<infinity> = 0"
  1.1087    unfolding divide_ereal_def by simp_all
  1.1088  
  1.1089 -lemma ereal_divide_one[simp]:
  1.1090 -  "x / 1 = (x::ereal)"
  1.1091 +lemma ereal_divide_one[simp]: "x / 1 = (x::ereal)"
  1.1092    unfolding divide_ereal_def by simp
  1.1093  
  1.1094 -lemma ereal_divide_ereal[simp]:
  1.1095 -  "\<infinity> / ereal r = (if 0 \<le> r then \<infinity> else -\<infinity>)"
  1.1096 +lemma ereal_divide_ereal[simp]: "\<infinity> / ereal r = (if 0 \<le> r then \<infinity> else -\<infinity>)"
  1.1097    unfolding divide_ereal_def by simp
  1.1098  
  1.1099  lemma zero_le_divide_ereal[simp]:
  1.1100 -  fixes a :: ereal assumes "0 \<le> a" "0 \<le> b"
  1.1101 +  fixes a :: ereal
  1.1102 +  assumes "0 \<le> a"
  1.1103 +    and "0 \<le> b"
  1.1104    shows "0 \<le> a / b"
  1.1105    using assms by (cases rule: ereal2_cases[of a b]) (auto simp: zero_le_divide_iff)
  1.1106  
  1.1107  lemma ereal_le_divide_pos:
  1.1108 -  fixes x y z :: ereal shows "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> x * y \<le> z"
  1.1109 +  fixes x y z :: ereal
  1.1110 +  shows "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> x * y \<le> z"
  1.1111    by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)
  1.1112  
  1.1113  lemma ereal_divide_le_pos:
  1.1114 -  fixes x y z :: ereal shows "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> z \<le> x * y"
  1.1115 +  fixes x y z :: ereal
  1.1116 +  shows "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> z \<le> x * y"
  1.1117    by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)
  1.1118  
  1.1119  lemma ereal_le_divide_neg:
  1.1120 -  fixes x y z :: ereal shows "x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> z \<le> x * y"
  1.1121 +  fixes x y z :: ereal
  1.1122 +  shows "x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> z \<le> x * y"
  1.1123    by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)
  1.1124  
  1.1125  lemma ereal_divide_le_neg:
  1.1126 -  fixes x y z :: ereal shows "x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> x * y \<le> z"
  1.1127 +  fixes x y z :: ereal
  1.1128 +  shows "x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> x * y \<le> z"
  1.1129    by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)
  1.1130  
  1.1131  lemma ereal_inverse_antimono_strict:
  1.1132 @@ -1015,31 +1187,37 @@
  1.1133  
  1.1134  lemma ereal_inverse_antimono:
  1.1135    fixes x y :: ereal
  1.1136 -  shows "0 \<le> x \<Longrightarrow> x <= y \<Longrightarrow> inverse y <= inverse x"
  1.1137 +  shows "0 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> inverse y \<le> inverse x"
  1.1138    by (cases rule: ereal2_cases[of x y]) auto
  1.1139  
  1.1140  lemma inverse_inverse_Pinfty_iff[simp]:
  1.1141 -  fixes x :: ereal shows "inverse x = \<infinity> \<longleftrightarrow> x = 0"
  1.1142 +  fixes x :: ereal
  1.1143 +  shows "inverse x = \<infinity> \<longleftrightarrow> x = 0"
  1.1144    by (cases x) auto
  1.1145  
  1.1146  lemma ereal_inverse_eq_0:
  1.1147 -  fixes x :: ereal shows "inverse x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity>"
  1.1148 +  fixes x :: ereal
  1.1149 +  shows "inverse x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity>"
  1.1150    by (cases x) auto
  1.1151  
  1.1152  lemma ereal_0_gt_inverse:
  1.1153 -  fixes x :: ereal shows "0 < inverse x \<longleftrightarrow> x \<noteq> \<infinity> \<and> 0 \<le> x"
  1.1154 +  fixes x :: ereal
  1.1155 +  shows "0 < inverse x \<longleftrightarrow> x \<noteq> \<infinity> \<and> 0 \<le> x"
  1.1156    by (cases x) auto
  1.1157  
  1.1158  lemma ereal_mult_less_right:
  1.1159    fixes a b c :: ereal
  1.1160 -  assumes "b * a < c * a" "0 < a" "a < \<infinity>"
  1.1161 +  assumes "b * a < c * a"
  1.1162 +    and "0 < a"
  1.1163 +    and "a < \<infinity>"
  1.1164    shows "b < c"
  1.1165    using assms
  1.1166    by (cases rule: ereal3_cases[of a b c])
  1.1167       (auto split: split_if_asm simp: zero_less_mult_iff zero_le_mult_iff)
  1.1168  
  1.1169  lemma ereal_power_divide:
  1.1170 -  fixes x y :: ereal shows "y \<noteq> 0 \<Longrightarrow> (x / y) ^ n = x^n / y^n"
  1.1171 +  fixes x y :: ereal
  1.1172 +  shows "y \<noteq> 0 \<Longrightarrow> (x / y) ^ n = x^n / y^n"
  1.1173    by (cases rule: ereal2_cases[of x y])
  1.1174       (auto simp: one_ereal_def zero_ereal_def power_divide not_le
  1.1175                   power_less_zero_eq zero_le_power_iff)
  1.1176 @@ -1047,36 +1225,47 @@
  1.1177  lemma ereal_le_mult_one_interval:
  1.1178    fixes x y :: ereal
  1.1179    assumes y: "y \<noteq> -\<infinity>"
  1.1180 -  assumes z: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y"
  1.1181 +  assumes z: "\<And>z. 0 < z \<Longrightarrow> z < 1 \<Longrightarrow> z * x \<le> y"
  1.1182    shows "x \<le> y"
  1.1183  proof (cases x)
  1.1184 -  case PInf with z[of "1 / 2"] show "x \<le> y" by (simp add: one_ereal_def)
  1.1185 +  case PInf
  1.1186 +  with z[of "1 / 2"] show "x \<le> y"
  1.1187 +    by (simp add: one_ereal_def)
  1.1188  next
  1.1189 -  case (real r) note r = this
  1.1190 +  case (real r)
  1.1191 +  note r = this
  1.1192    show "x \<le> y"
  1.1193    proof (cases y)
  1.1194 -    case (real p) note p = this
  1.1195 +    case (real p)
  1.1196 +    note p = this
  1.1197      have "r \<le> p"
  1.1198      proof (rule field_le_mult_one_interval)
  1.1199 -      fix z :: real assume "0 < z" and "z < 1"
  1.1200 -      with z[of "ereal z"]
  1.1201 -      show "z * r \<le> p" using p r by (auto simp: zero_le_mult_iff one_ereal_def)
  1.1202 +      fix z :: real
  1.1203 +      assume "0 < z" and "z < 1"
  1.1204 +      with z[of "ereal z"] show "z * r \<le> p"
  1.1205 +        using p r by (auto simp: zero_le_mult_iff one_ereal_def)
  1.1206      qed
  1.1207 -    then show "x \<le> y" using p r by simp
  1.1208 +    then show "x \<le> y"
  1.1209 +      using p r by simp
  1.1210    qed (insert y, simp_all)
  1.1211  qed simp
  1.1212  
  1.1213  lemma ereal_divide_right_mono[simp]:
  1.1214    fixes x y z :: ereal
  1.1215 -  assumes "x \<le> y" "0 < z" shows "x / z \<le> y / z"
  1.1216 -using assms by (cases x y z rule: ereal3_cases) (auto intro: divide_right_mono)
  1.1217 +  assumes "x \<le> y"
  1.1218 +    and "0 < z"
  1.1219 +  shows "x / z \<le> y / z"
  1.1220 +  using assms by (cases x y z rule: ereal3_cases) (auto intro: divide_right_mono)
  1.1221  
  1.1222  lemma ereal_divide_left_mono[simp]:
  1.1223    fixes x y z :: ereal
  1.1224 -  assumes "y \<le> x" "0 < z" "0 < x * y"
  1.1225 +  assumes "y \<le> x"
  1.1226 +    and "0 < z"
  1.1227 +    and "0 < x * y"
  1.1228    shows "z / x \<le> z / y"
  1.1229 -using assms by (cases x y z rule: ereal3_cases)
  1.1230 -  (auto intro: divide_left_mono simp: field_simps sign_simps split: split_if_asm)
  1.1231 +  using assms
  1.1232 +  by (cases x y z rule: ereal3_cases)
  1.1233 +    (auto intro: divide_left_mono simp: field_simps sign_simps split: split_if_asm)
  1.1234  
  1.1235  lemma ereal_divide_zero_left[simp]:
  1.1236    fixes a :: ereal
  1.1237 @@ -1088,13 +1277,16 @@
  1.1238    shows "b / c * a = b * a / c"
  1.1239    by (cases a b c rule: ereal3_cases) (auto simp: field_simps sign_simps)
  1.1240  
  1.1241 +
  1.1242  subsection "Complete lattice"
  1.1243  
  1.1244  instantiation ereal :: lattice
  1.1245  begin
  1.1246 +
  1.1247  definition [simp]: "sup x y = (max x y :: ereal)"
  1.1248  definition [simp]: "inf x y = (min x y :: ereal)"
  1.1249  instance by default simp_all
  1.1250 +
  1.1251  end
  1.1252  
  1.1253  instantiation ereal :: complete_lattice
  1.1254 @@ -1109,29 +1301,46 @@
  1.1255  lemma ereal_complete_Sup:
  1.1256    fixes S :: "ereal set"
  1.1257    shows "\<exists>x. (\<forall>y\<in>S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>S. y \<le> z) \<longrightarrow> x \<le> z)"
  1.1258 -proof cases
  1.1259 -  assume "\<exists>x. \<forall>a\<in>S. a \<le> ereal x"
  1.1260 -  then obtain y where y: "\<And>a. a\<in>S \<Longrightarrow> a \<le> ereal y" by auto
  1.1261 -  then have "\<infinity> \<notin> S" by force
  1.1262 +proof (cases "\<exists>x. \<forall>a\<in>S. a \<le> ereal x")
  1.1263 +  case True
  1.1264 +  then obtain y where y: "\<And>a. a\<in>S \<Longrightarrow> a \<le> ereal y"
  1.1265 +    by auto
  1.1266 +  then have "\<infinity> \<notin> S"
  1.1267 +    by force
  1.1268    show ?thesis
  1.1269 -  proof cases
  1.1270 -    assume "S \<noteq> {-\<infinity>} \<and> S \<noteq> {}"
  1.1271 -    with `\<infinity> \<notin> S` obtain x where x: "x \<in> S" "\<bar>x\<bar> \<noteq> \<infinity>" by auto
  1.1272 +  proof (cases "S \<noteq> {-\<infinity>} \<and> S \<noteq> {}")
  1.1273 +    case True
  1.1274 +    with `\<infinity> \<notin> S` obtain x where x: "x \<in> S" "\<bar>x\<bar> \<noteq> \<infinity>"
  1.1275 +      by auto
  1.1276      obtain s where s: "\<forall>x\<in>ereal -` S. x \<le> s" "\<And>z. (\<forall>x\<in>ereal -` S. x \<le> z) \<Longrightarrow> s \<le> z"
  1.1277      proof (atomize_elim, rule complete_real)
  1.1278 -      show "\<exists>x. x \<in> ereal -` S" using x by auto
  1.1279 -      show "\<exists>z. \<forall>x\<in>ereal -` S. x \<le> z" by (auto dest: y intro!: exI[of _ y])
  1.1280 +      show "\<exists>x. x \<in> ereal -` S"
  1.1281 +        using x by auto
  1.1282 +      show "\<exists>z. \<forall>x\<in>ereal -` S. x \<le> z"
  1.1283 +        by (auto dest: y intro!: exI[of _ y])
  1.1284      qed
  1.1285      show ?thesis
  1.1286      proof (safe intro!: exI[of _ "ereal s"])
  1.1287 -      fix y assume "y \<in> S" with s `\<infinity> \<notin> S` show "y \<le> ereal s"
  1.1288 +      fix y
  1.1289 +      assume "y \<in> S"
  1.1290 +      with s `\<infinity> \<notin> S` show "y \<le> ereal s"
  1.1291          by (cases y) auto
  1.1292      next
  1.1293 -      fix z assume "\<forall>y\<in>S. y \<le> z" with `S \<noteq> {-\<infinity>} \<and> S \<noteq> {}` show "ereal s \<le> z"
  1.1294 +      fix z
  1.1295 +      assume "\<forall>y\<in>S. y \<le> z"
  1.1296 +      with `S \<noteq> {-\<infinity>} \<and> S \<noteq> {}` show "ereal s \<le> z"
  1.1297          by (cases z) (auto intro!: s)
  1.1298      qed
  1.1299 -  qed (auto intro!: exI[of _ "-\<infinity>"])
  1.1300 -qed (fastforce intro!: exI[of _ \<infinity>] ereal_top intro: order_trans dest: less_imp_le simp: not_le)
  1.1301 +  next
  1.1302 +    case False
  1.1303 +    then show ?thesis
  1.1304 +      by (auto intro!: exI[of _ "-\<infinity>"])
  1.1305 +  qed
  1.1306 +next
  1.1307 +  case False
  1.1308 +  then show ?thesis
  1.1309 +    by (fastforce intro!: exI[of _ \<infinity>] ereal_top intro: order_trans dest: less_imp_le simp: not_le)
  1.1310 +qed
  1.1311  
  1.1312  lemma ereal_complete_uminus_eq:
  1.1313    fixes S :: "ereal set"
  1.1314 @@ -1141,23 +1350,24 @@
  1.1315  
  1.1316  lemma ereal_complete_Inf:
  1.1317    "\<exists>x. (\<forall>y\<in>S::ereal set. x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> x)"
  1.1318 -  using ereal_complete_Sup[of "uminus ` S"] unfolding ereal_complete_uminus_eq by auto
  1.1319 +  using ereal_complete_Sup[of "uminus ` S"]
  1.1320 +  unfolding ereal_complete_uminus_eq
  1.1321 +  by auto
  1.1322  
  1.1323  instance
  1.1324  proof
  1.1325    show "Sup {} = (bot::ereal)"
  1.1326 -  apply (auto simp: bot_ereal_def Sup_ereal_def)
  1.1327 -  apply (rule some1_equality)
  1.1328 -  apply (metis ereal_bot ereal_less_eq(2))
  1.1329 -  apply (metis ereal_less_eq(2))
  1.1330 -  done
  1.1331 -next
  1.1332 +    apply (auto simp: bot_ereal_def Sup_ereal_def)
  1.1333 +    apply (rule some1_equality)
  1.1334 +    apply (metis ereal_bot ereal_less_eq(2))
  1.1335 +    apply (metis ereal_less_eq(2))
  1.1336 +    done
  1.1337    show "Inf {} = (top::ereal)"
  1.1338 -  apply (auto simp: top_ereal_def Inf_ereal_def)
  1.1339 -  apply (rule some1_equality)
  1.1340 -  apply (metis ereal_top ereal_less_eq(1))
  1.1341 -  apply (metis ereal_less_eq(1))
  1.1342 -  done
  1.1343 +    apply (auto simp: top_ereal_def Inf_ereal_def)
  1.1344 +    apply (rule some1_equality)
  1.1345 +    apply (metis ereal_top ereal_less_eq(1))
  1.1346 +    apply (metis ereal_less_eq(1))
  1.1347 +    done
  1.1348  qed (auto intro: someI2_ex ereal_complete_Sup ereal_complete_Inf
  1.1349    simp: Sup_ereal_def Inf_ereal_def bot_ereal_def top_ereal_def)
  1.1350  
  1.1351 @@ -1183,74 +1393,89 @@
  1.1352    using ereal_Sup_uminus_image_eq[of "uminus ` S"] by simp
  1.1353  
  1.1354  lemma ereal_SUPR_uminus:
  1.1355 -  fixes f :: "'a => ereal"
  1.1356 +  fixes f :: "'a \<Rightarrow> ereal"
  1.1357    shows "(SUP i : R. -(f i)) = -(INF i : R. f i)"
  1.1358    using ereal_Sup_uminus_image_eq[of "f`R"]
  1.1359    by (simp add: SUP_def INF_def image_image)
  1.1360  
  1.1361  lemma ereal_INFI_uminus:
  1.1362 -  fixes f :: "'a => ereal"
  1.1363 -  shows "(INF i : R. -(f i)) = -(SUP i : R. f i)"
  1.1364 +  fixes f :: "'a \<Rightarrow> ereal"
  1.1365 +  shows "(INF i : R. - f i) = - (SUP i : R. f i)"
  1.1366    using ereal_SUPR_uminus[of _ "\<lambda>x. - f x"] by simp
  1.1367  
  1.1368  lemma ereal_image_uminus_shift:
  1.1369 -  fixes X Y :: "ereal set" shows "uminus ` X = Y \<longleftrightarrow> X = uminus ` Y"
  1.1370 +  fixes X Y :: "ereal set"
  1.1371 +  shows "uminus ` X = Y \<longleftrightarrow> X = uminus ` Y"
  1.1372  proof
  1.1373    assume "uminus ` X = Y"
  1.1374    then have "uminus ` uminus ` X = uminus ` Y"
  1.1375      by (simp add: inj_image_eq_iff)
  1.1376 -  then show "X = uminus ` Y" by (simp add: image_image)
  1.1377 +  then show "X = uminus ` Y"
  1.1378 +    by (simp add: image_image)
  1.1379  qed (simp add: image_image)
  1.1380  
  1.1381  lemma Inf_ereal_iff:
  1.1382    fixes z :: ereal
  1.1383 -  shows "(!!x. x:X ==> z <= x) ==> (EX x:X. x<y) <-> Inf X < y"
  1.1384 -  by (metis complete_lattice_class.Inf_greatest complete_lattice_class.Inf_lower less_le_not_le linear
  1.1385 -            order_less_le_trans)
  1.1386 +  shows "(\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> (\<exists>x\<in>X. x < y) \<longleftrightarrow> Inf X < y"
  1.1387 +  by (metis complete_lattice_class.Inf_greatest complete_lattice_class.Inf_lower
  1.1388 +      less_le_not_le linear order_less_le_trans)
  1.1389  
  1.1390  lemma Sup_eq_MInfty:
  1.1391 -  fixes S :: "ereal set" shows "Sup S = -\<infinity> \<longleftrightarrow> S = {} \<or> S = {-\<infinity>}"
  1.1392 +  fixes S :: "ereal set"
  1.1393 +  shows "Sup S = -\<infinity> \<longleftrightarrow> S = {} \<or> S = {-\<infinity>}"
  1.1394    unfolding bot_ereal_def[symmetric] by auto
  1.1395  
  1.1396  lemma Inf_eq_PInfty:
  1.1397 -  fixes S :: "ereal set" shows "Inf S = \<infinity> \<longleftrightarrow> S = {} \<or> S = {\<infinity>}"
  1.1398 +  fixes S :: "ereal set"
  1.1399 +  shows "Inf S = \<infinity> \<longleftrightarrow> S = {} \<or> S = {\<infinity>}"
  1.1400    using Sup_eq_MInfty[of "uminus`S"]
  1.1401    unfolding ereal_Sup_uminus_image_eq ereal_image_uminus_shift by simp
  1.1402  
  1.1403 -lemma Inf_eq_MInfty: 
  1.1404 -  fixes S :: "ereal set" shows "-\<infinity> \<in> S \<Longrightarrow> Inf S = -\<infinity>"
  1.1405 +lemma Inf_eq_MInfty:
  1.1406 +  fixes S :: "ereal set"
  1.1407 +  shows "-\<infinity> \<in> S \<Longrightarrow> Inf S = -\<infinity>"
  1.1408    unfolding bot_ereal_def[symmetric] by auto
  1.1409  
  1.1410  lemma Sup_eq_PInfty:
  1.1411 -  fixes S :: "ereal set" shows "\<infinity> \<in> S \<Longrightarrow> Sup S = \<infinity>"
  1.1412 +  fixes S :: "ereal set"
  1.1413 +  shows "\<infinity> \<in> S \<Longrightarrow> Sup S = \<infinity>"
  1.1414    unfolding top_ereal_def[symmetric] by auto
  1.1415  
  1.1416  lemma Sup_ereal_close:
  1.1417    fixes e :: ereal
  1.1418 -  assumes "0 < e" and S: "\<bar>Sup S\<bar> \<noteq> \<infinity>" "S \<noteq> {}"
  1.1419 +  assumes "0 < e"
  1.1420 +    and S: "\<bar>Sup S\<bar> \<noteq> \<infinity>" "S \<noteq> {}"
  1.1421    shows "\<exists>x\<in>S. Sup S - e < x"
  1.1422    using assms by (cases e) (auto intro!: less_Sup_iff[THEN iffD1])
  1.1423  
  1.1424  lemma Inf_ereal_close:
  1.1425 -  fixes e :: ereal assumes "\<bar>Inf X\<bar> \<noteq> \<infinity>" "0 < e"
  1.1426 +  fixes e :: ereal
  1.1427 +  assumes "\<bar>Inf X\<bar> \<noteq> \<infinity>"
  1.1428 +    and "0 < e"
  1.1429    shows "\<exists>x\<in>X. x < Inf X + e"
  1.1430  proof (rule Inf_less_iff[THEN iffD1])
  1.1431 -  show "Inf X < Inf X + e" using assms
  1.1432 -    by (cases e) auto
  1.1433 +  show "Inf X < Inf X + e"
  1.1434 +    using assms by (cases e) auto
  1.1435  qed
  1.1436  
  1.1437  lemma SUP_nat_Infty: "(SUP i::nat. ereal (real i)) = \<infinity>"
  1.1438  proof -
  1.1439 -  { fix x ::ereal assume "x \<noteq> \<infinity>"
  1.1440 +  {
  1.1441 +    fix x :: ereal
  1.1442 +    assume "x \<noteq> \<infinity>"
  1.1443      then have "\<exists>k::nat. x < ereal (real k)"
  1.1444      proof (cases x)
  1.1445 -      case MInf then show ?thesis by (intro exI[of _ 0]) auto
  1.1446 +      case MInf
  1.1447 +      then show ?thesis
  1.1448 +        by (intro exI[of _ 0]) auto
  1.1449      next
  1.1450        case (real r)
  1.1451        moreover obtain k :: nat where "r < real k"
  1.1452          using ex_less_of_nat by (auto simp: real_eq_of_nat)
  1.1453 -      ultimately show ?thesis by auto
  1.1454 -    qed simp }
  1.1455 +      ultimately show ?thesis
  1.1456 +        by auto
  1.1457 +    qed simp
  1.1458 +  }
  1.1459    then show ?thesis
  1.1460      using SUP_eq_top_iff[of UNIV "\<lambda>n::nat. ereal (real n)"]
  1.1461      by (auto simp: top_ereal_def)
  1.1462 @@ -1259,96 +1484,136 @@
  1.1463  lemma Inf_less:
  1.1464    fixes x :: ereal
  1.1465    assumes "(INF i:A. f i) < x"
  1.1466 -  shows "EX i. i : A & f i <= x"
  1.1467 -proof(rule ccontr)
  1.1468 -  assume "~ (EX i. i : A & f i <= x)"
  1.1469 -  hence "ALL i:A. f i > x" by auto
  1.1470 -  hence "(INF i:A. f i) >= x" apply (subst INF_greatest) by auto
  1.1471 -  thus False using assms by auto
  1.1472 +  shows "\<exists>i. i \<in> A \<and> f i \<le> x"
  1.1473 +proof (rule ccontr)
  1.1474 +  assume "\<not> ?thesis"
  1.1475 +  then have "\<forall>i\<in>A. f i > x"
  1.1476 +    by auto
  1.1477 +  then have "(INF i:A. f i) \<ge> x"
  1.1478 +    by (subst INF_greatest) auto
  1.1479 +  then show False
  1.1480 +    using assms by auto
  1.1481  qed
  1.1482  
  1.1483  lemma SUP_ereal_le_addI:
  1.1484    fixes f :: "'i \<Rightarrow> ereal"
  1.1485 -  assumes "\<And>i. f i + y \<le> z" and "y \<noteq> -\<infinity>"
  1.1486 +  assumes "\<And>i. f i + y \<le> z"
  1.1487 +    and "y \<noteq> -\<infinity>"
  1.1488    shows "SUPR UNIV f + y \<le> z"
  1.1489  proof (cases y)
  1.1490    case (real r)
  1.1491 -  then have "\<And>i. f i \<le> z - y" using assms by (simp add: ereal_le_minus_iff)
  1.1492 -  then have "SUPR UNIV f \<le> z - y" by (rule SUP_least)
  1.1493 -  then show ?thesis using real by (simp add: ereal_le_minus_iff)
  1.1494 +  then have "\<And>i. f i \<le> z - y"
  1.1495 +    using assms by (simp add: ereal_le_minus_iff)
  1.1496 +  then have "SUPR UNIV f \<le> z - y"
  1.1497 +    by (rule SUP_least)
  1.1498 +  then show ?thesis
  1.1499 +    using real by (simp add: ereal_le_minus_iff)
  1.1500  qed (insert assms, auto)
  1.1501  
  1.1502  lemma SUPR_ereal_add:
  1.1503    fixes f g :: "nat \<Rightarrow> ereal"
  1.1504 -  assumes "incseq f" "incseq g" and pos: "\<And>i. f i \<noteq> -\<infinity>" "\<And>i. g i \<noteq> -\<infinity>"
  1.1505 +  assumes "incseq f"
  1.1506 +    and "incseq g"
  1.1507 +    and pos: "\<And>i. f i \<noteq> -\<infinity>" "\<And>i. g i \<noteq> -\<infinity>"
  1.1508    shows "(SUP i. f i + g i) = SUPR UNIV f + SUPR UNIV g"
  1.1509  proof (rule SUP_eqI)
  1.1510 -  fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> f i + g i \<le> y"
  1.1511 -  have f: "SUPR UNIV f \<noteq> -\<infinity>" using pos
  1.1512 -    unfolding SUP_def Sup_eq_MInfty by (auto dest: image_eqD)
  1.1513 -  { fix j
  1.1514 -    { fix i
  1.1515 +  fix y
  1.1516 +  assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> f i + g i \<le> y"
  1.1517 +  have f: "SUPR UNIV f \<noteq> -\<infinity>"
  1.1518 +    using pos
  1.1519 +    unfolding SUP_def Sup_eq_MInfty
  1.1520 +    by (auto dest: image_eqD)
  1.1521 +  {
  1.1522 +    fix j
  1.1523 +    {
  1.1524 +      fix i
  1.1525        have "f i + g j \<le> f i + g (max i j)"
  1.1526 -        using `incseq g`[THEN incseqD] by (rule add_left_mono) auto
  1.1527 +        using `incseq g`[THEN incseqD]
  1.1528 +        by (rule add_left_mono) auto
  1.1529        also have "\<dots> \<le> f (max i j) + g (max i j)"
  1.1530 -        using `incseq f`[THEN incseqD] by (rule add_right_mono) auto
  1.1531 +        using `incseq f`[THEN incseqD]
  1.1532 +        by (rule add_right_mono) auto
  1.1533        also have "\<dots> \<le> y" using * by auto
  1.1534 -      finally have "f i + g j \<le> y" . }
  1.1535 +      finally have "f i + g j \<le> y" .
  1.1536 +    }
  1.1537      then have "SUPR UNIV f + g j \<le> y"
  1.1538        using assms(4)[of j] by (intro SUP_ereal_le_addI) auto
  1.1539 -    then have "g j + SUPR UNIV f \<le> y" by (simp add: ac_simps) }
  1.1540 +    then have "g j + SUPR UNIV f \<le> y" by (simp add: ac_simps)
  1.1541 +  }
  1.1542    then have "SUPR UNIV g + SUPR UNIV f \<le> y"
  1.1543      using f by (rule SUP_ereal_le_addI)
  1.1544 -  then show "SUPR UNIV f + SUPR UNIV g \<le> y" by (simp add: ac_simps)
  1.1545 +  then show "SUPR UNIV f + SUPR UNIV g \<le> y"
  1.1546 +    by (simp add: ac_simps)
  1.1547  qed (auto intro!: add_mono SUP_upper)
  1.1548  
  1.1549  lemma SUPR_ereal_add_pos:
  1.1550    fixes f g :: "nat \<Rightarrow> ereal"
  1.1551 -  assumes inc: "incseq f" "incseq g" and pos: "\<And>i. 0 \<le> f i" "\<And>i. 0 \<le> g i"
  1.1552 +  assumes inc: "incseq f" "incseq g"
  1.1553 +    and pos: "\<And>i. 0 \<le> f i" "\<And>i. 0 \<le> g i"
  1.1554    shows "(SUP i. f i + g i) = SUPR UNIV f + SUPR UNIV g"
  1.1555  proof (intro SUPR_ereal_add inc)
  1.1556 -  fix i show "f i \<noteq> -\<infinity>" "g i \<noteq> -\<infinity>" using pos[of i] by auto
  1.1557 +  fix i
  1.1558 +  show "f i \<noteq> -\<infinity>" "g i \<noteq> -\<infinity>"
  1.1559 +    using pos[of i] by auto
  1.1560  qed
  1.1561  
  1.1562  lemma SUPR_ereal_setsum:
  1.1563    fixes f g :: "'a \<Rightarrow> nat \<Rightarrow> ereal"
  1.1564 -  assumes "\<And>n. n \<in> A \<Longrightarrow> incseq (f n)" and pos: "\<And>n i. n \<in> A \<Longrightarrow> 0 \<le> f n i"
  1.1565 +  assumes "\<And>n. n \<in> A \<Longrightarrow> incseq (f n)"
  1.1566 +    and pos: "\<And>n i. n \<in> A \<Longrightarrow> 0 \<le> f n i"
  1.1567    shows "(SUP i. \<Sum>n\<in>A. f n i) = (\<Sum>n\<in>A. SUPR UNIV (f n))"
  1.1568 -proof cases
  1.1569 -  assume "finite A" then show ?thesis using assms
  1.1570 +proof (cases "finite A")
  1.1571 +  case True
  1.1572 +  then show ?thesis using assms
  1.1573      by induct (auto simp: incseq_setsumI2 setsum_nonneg SUPR_ereal_add_pos)
  1.1574 -qed simp
  1.1575 +next
  1.1576 +  case False
  1.1577 +  then show ?thesis by simp
  1.1578 +qed
  1.1579  
  1.1580  lemma SUPR_ereal_cmult:
  1.1581 -  fixes f :: "nat \<Rightarrow> ereal" assumes "\<And>i. 0 \<le> f i" "0 \<le> c"
  1.1582 +  fixes f :: "nat \<Rightarrow> ereal"
  1.1583 +  assumes "\<And>i. 0 \<le> f i"
  1.1584 +    and "0 \<le> c"
  1.1585    shows "(SUP i. c * f i) = c * SUPR UNIV f"
  1.1586  proof (rule SUP_eqI)
  1.1587 -  fix i have "f i \<le> SUPR UNIV f" by (rule SUP_upper) auto
  1.1588 +  fix i
  1.1589 +  have "f i \<le> SUPR UNIV f"
  1.1590 +    by (rule SUP_upper) auto
  1.1591    then show "c * f i \<le> c * SUPR UNIV f"
  1.1592      using `0 \<le> c` by (rule ereal_mult_left_mono)
  1.1593  next
  1.1594 -  fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> c * f i \<le> y"
  1.1595 +  fix y
  1.1596 +  assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> c * f i \<le> y"
  1.1597    show "c * SUPR UNIV f \<le> y"
  1.1598 -  proof cases
  1.1599 -    assume c: "0 < c \<and> c \<noteq> \<infinity>"
  1.1600 +  proof (cases "0 < c \<and> c \<noteq> \<infinity>")
  1.1601 +    case True
  1.1602      with * have "SUPR UNIV f \<le> y / c"
  1.1603        by (intro SUP_least) (auto simp: ereal_le_divide_pos)
  1.1604 -    with c show ?thesis
  1.1605 +    with True show ?thesis
  1.1606        by (auto simp: ereal_le_divide_pos)
  1.1607    next
  1.1608 -    { assume "c = \<infinity>" have ?thesis
  1.1609 -      proof cases
  1.1610 -        assume **: "\<forall>i. f i = 0"
  1.1611 -        then have "range f = {0}" by auto
  1.1612 -        with ** show "c * SUPR UNIV f \<le> y" using *
  1.1613 -          by (auto simp: SUP_def min_max.sup_absorb1)
  1.1614 +    case False
  1.1615 +    {
  1.1616 +      assume "c = \<infinity>"
  1.1617 +      have ?thesis
  1.1618 +      proof (cases "\<forall>i. f i = 0")
  1.1619 +        case True
  1.1620 +        then have "range f = {0}"
  1.1621 +          by auto
  1.1622 +        with True show "c * SUPR UNIV f \<le> y"
  1.1623 +          using * by (auto simp: SUP_def min_max.sup_absorb1)
  1.1624        next
  1.1625 -        assume "\<not> (\<forall>i. f i = 0)"
  1.1626 -        then obtain i where "f i \<noteq> 0" by auto
  1.1627 -        with *[of i] `c = \<infinity>` `0 \<le> f i` show ?thesis by (auto split: split_if_asm)
  1.1628 -      qed }
  1.1629 -    moreover assume "\<not> (0 < c \<and> c \<noteq> \<infinity>)"
  1.1630 -    ultimately show ?thesis using * `0 \<le> c` by auto
  1.1631 +        case False
  1.1632 +        then obtain i where "f i \<noteq> 0"
  1.1633 +          by auto
  1.1634 +        with *[of i] `c = \<infinity>` `0 \<le> f i` show ?thesis
  1.1635 +          by (auto split: split_if_asm)
  1.1636 +      qed
  1.1637 +    }
  1.1638 +    moreover note False
  1.1639 +    ultimately show ?thesis
  1.1640 +      using * `0 \<le> c` by auto
  1.1641    qed
  1.1642  qed
  1.1643  
  1.1644 @@ -1359,15 +1624,21 @@
  1.1645    unfolding SUP_def Sup_eq_top_iff[where 'a=ereal, unfolded top_ereal_def]
  1.1646    apply simp
  1.1647  proof safe
  1.1648 -  fix x :: ereal assume "x \<noteq> \<infinity>"
  1.1649 +  fix x :: ereal
  1.1650 +  assume "x \<noteq> \<infinity>"
  1.1651    show "\<exists>i\<in>A. x < f i"
  1.1652    proof (cases x)
  1.1653 -    case PInf with `x \<noteq> \<infinity>` show ?thesis by simp
  1.1654 +    case PInf
  1.1655 +    with `x \<noteq> \<infinity>` show ?thesis
  1.1656 +      by simp
  1.1657    next
  1.1658 -    case MInf with assms[of "0"] show ?thesis by force
  1.1659 +    case MInf
  1.1660 +    with assms[of "0"] show ?thesis
  1.1661 +      by force
  1.1662    next
  1.1663      case (real r)
  1.1664 -    with less_PInf_Ex_of_nat[of x] obtain n :: nat where "x < ereal (real n)" by auto
  1.1665 +    with less_PInf_Ex_of_nat[of x] obtain n :: nat where "x < ereal (real n)"
  1.1666 +      by auto
  1.1667      moreover obtain i where "i \<in> A" "ereal (real n) \<le> f i"
  1.1668        using assms ..
  1.1669      ultimately show ?thesis
  1.1670 @@ -1382,7 +1653,8 @@
  1.1671    case (real r)
  1.1672    have "\<forall>n::nat. \<exists>x. x \<in> A \<and> Sup A < x + 1 / ereal (real n)"
  1.1673    proof
  1.1674 -    fix n ::nat have "\<exists>x\<in>A. Sup A - 1 / ereal (real n) < x"
  1.1675 +    fix n :: nat
  1.1676 +    have "\<exists>x\<in>A. Sup A - 1 / ereal (real n) < x"
  1.1677        using assms real by (intro Sup_ereal_close) (auto simp: one_ereal_def)
  1.1678      then obtain x where "x \<in> A" "Sup A - 1 / ereal (real n) < x" ..
  1.1679      then show "\<exists>x. x \<in> A \<and> Sup A < x + 1 / ereal (real n)"
  1.1680 @@ -1392,48 +1664,63 @@
  1.1681      where f: "\<forall>x. f x \<in> A \<and> Sup A < f x + 1 / ereal (real x)" ..
  1.1682    have "SUPR UNIV f = Sup A"
  1.1683    proof (rule SUP_eqI)
  1.1684 -    fix i show "f i \<le> Sup A" using f
  1.1685 -      by (auto intro!: complete_lattice_class.Sup_upper)
  1.1686 +    fix i
  1.1687 +    show "f i \<le> Sup A"
  1.1688 +      using f by (auto intro!: complete_lattice_class.Sup_upper)
  1.1689    next
  1.1690 -    fix y assume bound: "\<And>i. i \<in> UNIV \<Longrightarrow> f i \<le> y"
  1.1691 +    fix y
  1.1692 +    assume bound: "\<And>i. i \<in> UNIV \<Longrightarrow> f i \<le> y"
  1.1693      show "Sup A \<le> y"
  1.1694      proof (rule ereal_le_epsilon, intro allI impI)
  1.1695 -      fix e :: ereal assume "0 < e"
  1.1696 +      fix e :: ereal
  1.1697 +      assume "0 < e"
  1.1698        show "Sup A \<le> y + e"
  1.1699        proof (cases e)
  1.1700          case (real r)
  1.1701 -        hence "0 < r" using `0 < e` by auto
  1.1702 -        then obtain n ::nat where *: "1 / real n < r" "0 < n"
  1.1703 -          using ex_inverse_of_nat_less by (auto simp: real_eq_of_nat inverse_eq_divide)
  1.1704 -        have "Sup A \<le> f n + 1 / ereal (real n)" using f[THEN spec, of n]
  1.1705 +        then have "0 < r"
  1.1706 +          using `0 < e` by auto
  1.1707 +        then obtain n :: nat where *: "1 / real n < r" "0 < n"
  1.1708 +          using ex_inverse_of_nat_less
  1.1709 +          by (auto simp: real_eq_of_nat inverse_eq_divide)
  1.1710 +        have "Sup A \<le> f n + 1 / ereal (real n)"
  1.1711 +          using f[THEN spec, of n]
  1.1712            by auto
  1.1713 -        also have "1 / ereal (real n) \<le> e" using real * by (auto simp: one_ereal_def )
  1.1714 -        with bound have "f n + 1 / ereal (real n) \<le> y + e" by (rule add_mono) simp
  1.1715 +        also have "1 / ereal (real n) \<le> e"
  1.1716 +          using real *
  1.1717 +          by (auto simp: one_ereal_def )
  1.1718 +        with bound have "f n + 1 / ereal (real n) \<le> y + e"
  1.1719 +          by (rule add_mono) simp
  1.1720          finally show "Sup A \<le> y + e" .
  1.1721        qed (insert `0 < e`, auto)
  1.1722      qed
  1.1723    qed
  1.1724 -  with f show ?thesis by (auto intro!: exI[of _ f])
  1.1725 +  with f show ?thesis
  1.1726 +    by (auto intro!: exI[of _ f])
  1.1727  next
  1.1728    case PInf
  1.1729 -  from `A \<noteq> {}` obtain x where "x \<in> A" by auto
  1.1730 +  from `A \<noteq> {}` obtain x where "x \<in> A"
  1.1731 +    by auto
  1.1732    show ?thesis
  1.1733 -  proof cases
  1.1734 -    assume *: "\<infinity> \<in> A"
  1.1735 -    then have "\<infinity> \<le> Sup A" by (intro complete_lattice_class.Sup_upper)
  1.1736 -    with * show ?thesis by (auto intro!: exI[of _ "\<lambda>x. \<infinity>"])
  1.1737 +  proof (cases "\<infinity> \<in> A")
  1.1738 +    case True
  1.1739 +    then have "\<infinity> \<le> Sup A"
  1.1740 +      by (intro complete_lattice_class.Sup_upper)
  1.1741 +    with True show ?thesis
  1.1742 +      by (auto intro!: exI[of _ "\<lambda>x. \<infinity>"])
  1.1743    next
  1.1744 -    assume "\<infinity> \<notin> A"
  1.1745 +    case False
  1.1746      have "\<exists>x\<in>A. 0 \<le> x"
  1.1747 -      by (metis Infty_neq_0 PInf complete_lattice_class.Sup_least ereal_infty_less_eq2 linorder_linear)
  1.1748 -    then obtain x where "x \<in> A" "0 \<le> x" by auto
  1.1749 +      by (metis Infty_neq_0 PInf complete_lattice_class.Sup_least
  1.1750 +          ereal_infty_less_eq2 linorder_linear)
  1.1751 +    then obtain x where "x \<in> A" and "0 \<le> x"
  1.1752 +      by auto
  1.1753      have "\<forall>n::nat. \<exists>f. f \<in> A \<and> x + ereal (real n) \<le> f"
  1.1754      proof (rule ccontr)
  1.1755        assume "\<not> ?thesis"
  1.1756        then have "\<exists>n::nat. Sup A \<le> x + ereal (real n)"
  1.1757          by (simp add: Sup_le_iff not_le less_imp_le Ball_def) (metis less_imp_le)
  1.1758        then show False using `x \<in> A` `\<infinity> \<notin> A` PInf
  1.1759 -        by(cases x) auto
  1.1760 +        by (cases x) auto
  1.1761      qed
  1.1762      from choice[OF this] obtain f :: "nat \<Rightarrow> ereal"
  1.1763        where f: "\<forall>z. f z \<in> A \<and> x + ereal (real z) \<le> f z" ..
  1.1764 @@ -1444,20 +1731,26 @@
  1.1765          using f[THEN spec, of n] `0 \<le> x`
  1.1766          by (cases rule: ereal2_cases[of "f n" x]) (auto intro!: exI[of _ n])
  1.1767      qed
  1.1768 -    then show ?thesis using f PInf by (auto intro!: exI[of _ f])
  1.1769 +    then show ?thesis
  1.1770 +      using f PInf by (auto intro!: exI[of _ f])
  1.1771    qed
  1.1772  next
  1.1773    case MInf
  1.1774 -  with `A \<noteq> {}` have "A = {-\<infinity>}" by (auto simp: Sup_eq_MInfty)
  1.1775 -  then show ?thesis using MInf by (auto intro!: exI[of _ "\<lambda>x. -\<infinity>"])
  1.1776 +  with `A \<noteq> {}` have "A = {-\<infinity>}"
  1.1777 +    by (auto simp: Sup_eq_MInfty)
  1.1778 +  then show ?thesis
  1.1779 +    using MInf by (auto intro!: exI[of _ "\<lambda>x. -\<infinity>"])
  1.1780  qed
  1.1781  
  1.1782  lemma SUPR_countable_SUPR:
  1.1783    "A \<noteq> {} \<Longrightarrow> \<exists>f::nat \<Rightarrow> ereal. range f \<subseteq> g`A \<and> SUPR A g = SUPR UNIV f"
  1.1784 -  using Sup_countable_SUPR[of "g`A"] by (auto simp: SUP_def)
  1.1785 +  using Sup_countable_SUPR[of "g`A"]
  1.1786 +  by (auto simp: SUP_def)
  1.1787  
  1.1788  lemma Sup_ereal_cadd:
  1.1789 -  fixes A :: "ereal set" assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>"
  1.1790 +  fixes A :: "ereal set"
  1.1791 +  assumes "A \<noteq> {}"
  1.1792 +    and "a \<noteq> -\<infinity>"
  1.1793    shows "Sup ((\<lambda>x. a + x) ` A) = a + Sup A"
  1.1794  proof (rule antisym)
  1.1795    have *: "\<And>a::ereal. \<And>A. Sup ((\<lambda>x. a + x) ` A) \<le> a + Sup A"
  1.1796 @@ -1465,37 +1758,46 @@
  1.1797    then show "Sup ((\<lambda>x. a + x) ` A) \<le> a + Sup A" .
  1.1798    show "a + Sup A \<le> Sup ((\<lambda>x. a + x) ` A)"
  1.1799    proof (cases a)
  1.1800 -    case PInf with `A \<noteq> {}` show ?thesis by (auto simp: image_constant min_max.sup_absorb1)
  1.1801 +    case PInf with `A \<noteq> {}`
  1.1802 +    show ?thesis
  1.1803 +      by (auto simp: image_constant min_max.sup_absorb1)
  1.1804    next
  1.1805      case (real r)
  1.1806      then have **: "op + (- a) ` op + a ` A = A"
  1.1807        by (auto simp: image_iff ac_simps zero_ereal_def[symmetric])
  1.1808 -    from *[of "-a" "(\<lambda>x. a + x) ` A"] real show ?thesis unfolding **
  1.1809 +    from *[of "-a" "(\<lambda>x. a + x) ` A"] real show ?thesis
  1.1810 +      unfolding **
  1.1811        by (cases rule: ereal2_cases[of "Sup A" "Sup (op + a ` A)"]) auto
  1.1812    qed (insert `a \<noteq> -\<infinity>`, auto)
  1.1813  qed
  1.1814  
  1.1815  lemma Sup_ereal_cminus:
  1.1816 -  fixes A :: "ereal set" assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>"
  1.1817 +  fixes A :: "ereal set"
  1.1818 +  assumes "A \<noteq> {}"
  1.1819 +    and "a \<noteq> -\<infinity>"
  1.1820    shows "Sup ((\<lambda>x. a - x) ` A) = a - Inf A"
  1.1821    using Sup_ereal_cadd[of "uminus ` A" a] assms
  1.1822 -  by (simp add: comp_def image_image minus_ereal_def
  1.1823 -                 ereal_Sup_uminus_image_eq)
  1.1824 +  by (simp add: comp_def image_image minus_ereal_def ereal_Sup_uminus_image_eq)
  1.1825  
  1.1826  lemma SUPR_ereal_cminus:
  1.1827    fixes f :: "'i \<Rightarrow> ereal"
  1.1828 -  fixes A assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>"
  1.1829 +  fixes A
  1.1830 +  assumes "A \<noteq> {}"
  1.1831 +    and "a \<noteq> -\<infinity>"
  1.1832    shows "(SUP x:A. a - f x) = a - (INF x:A. f x)"
  1.1833    using Sup_ereal_cminus[of "f`A" a] assms
  1.1834    unfolding SUP_def INF_def image_image by auto
  1.1835  
  1.1836  lemma Inf_ereal_cminus:
  1.1837 -  fixes A :: "ereal set" assumes "A \<noteq> {}" and "\<bar>a\<bar> \<noteq> \<infinity>"
  1.1838 +  fixes A :: "ereal set"
  1.1839 +  assumes "A \<noteq> {}"
  1.1840 +    and "\<bar>a\<bar> \<noteq> \<infinity>"
  1.1841    shows "Inf ((\<lambda>x. a - x) ` A) = a - Sup A"
  1.1842  proof -
  1.1843    {
  1.1844      fix x
  1.1845 -    have "-a - -x = -(a - x)" using assms by (cases x) auto
  1.1846 +    have "-a - -x = -(a - x)"
  1.1847 +      using assms by (cases x) auto
  1.1848    } note * = this
  1.1849    then have "(\<lambda>x. -a - x)`uminus`A = uminus ` (\<lambda>x. a - x) ` A"
  1.1850      by (auto simp: image_image)
  1.1851 @@ -1505,25 +1807,32 @@
  1.1852  qed
  1.1853  
  1.1854  lemma INFI_ereal_cminus:
  1.1855 -  fixes a :: ereal assumes "A \<noteq> {}" and "\<bar>a\<bar> \<noteq> \<infinity>"
  1.1856 +  fixes a :: ereal
  1.1857 +  assumes "A \<noteq> {}"
  1.1858 +    and "\<bar>a\<bar> \<noteq> \<infinity>"
  1.1859    shows "(INF x:A. a - f x) = a - (SUP x:A. f x)"
  1.1860    using Inf_ereal_cminus[of "f`A" a] assms
  1.1861    unfolding SUP_def INF_def image_image
  1.1862    by auto
  1.1863  
  1.1864  lemma uminus_ereal_add_uminus_uminus:
  1.1865 -  fixes a b :: ereal shows "a \<noteq> \<infinity> \<Longrightarrow> b \<noteq> \<infinity> \<Longrightarrow> - (- a + - b) = a + b"
  1.1866 +  fixes a b :: ereal
  1.1867 +  shows "a \<noteq> \<infinity> \<Longrightarrow> b \<noteq> \<infinity> \<Longrightarrow> - (- a + - b) = a + b"
  1.1868    by (cases rule: ereal2_cases[of a b]) auto
  1.1869  
  1.1870  lemma INFI_ereal_add:
  1.1871    fixes f :: "nat \<Rightarrow> ereal"
  1.1872 -  assumes "decseq f" "decseq g" and fin: "\<And>i. f i \<noteq> \<infinity>" "\<And>i. g i \<noteq> \<infinity>"
  1.1873 +  assumes "decseq f" "decseq g"
  1.1874 +    and fin: "\<And>i. f i \<noteq> \<infinity>" "\<And>i. g i \<noteq> \<infinity>"
  1.1875    shows "(INF i. f i + g i) = INFI UNIV f + INFI UNIV g"
  1.1876  proof -
  1.1877    have INF_less: "(INF i. f i) < \<infinity>" "(INF i. g i) < \<infinity>"
  1.1878      using assms unfolding INF_less_iff by auto
  1.1879 -  { fix i from fin[of i] have "- ((- f i) + (- g i)) = f i + g i"
  1.1880 -      by (rule uminus_ereal_add_uminus_uminus) }
  1.1881 +  {
  1.1882 +    fix i
  1.1883 +    from fin[of i] have "- ((- f i) + (- g i)) = f i + g i"
  1.1884 +      by (rule uminus_ereal_add_uminus_uminus)
  1.1885 +  }
  1.1886    then have "(INF i. f i + g i) = (INF i. - ((- f i) + (- g i)))"
  1.1887      by simp
  1.1888    also have "\<dots> = INFI UNIV f + INFI UNIV g"
  1.1889 @@ -1534,6 +1843,7 @@
  1.1890    finally show ?thesis .
  1.1891  qed
  1.1892  
  1.1893 +
  1.1894  subsection "Relation to @{typ enat}"
  1.1895  
  1.1896  definition "ereal_of_enat n = (case n of enat n \<Rightarrow> ereal (real n) | \<infinity> \<Rightarrow> \<infinity>)"
  1.1897 @@ -1546,50 +1856,41 @@
  1.1898    "ereal_of_enat \<infinity> = \<infinity>"
  1.1899    by (simp_all add: ereal_of_enat_def)
  1.1900  
  1.1901 -lemma ereal_of_enat_le_iff[simp]:
  1.1902 -  "ereal_of_enat m \<le> ereal_of_enat n \<longleftrightarrow> m \<le> n"
  1.1903 -by (cases m n rule: enat2_cases) auto
  1.1904 +lemma ereal_of_enat_le_iff[simp]: "ereal_of_enat m \<le> ereal_of_enat n \<longleftrightarrow> m \<le> n"
  1.1905 +  by (cases m n rule: enat2_cases) auto
  1.1906  
  1.1907 -lemma ereal_of_enat_less_iff[simp]:
  1.1908 -  "ereal_of_enat m < ereal_of_enat n \<longleftrightarrow> m < n"
  1.1909 -by (cases m n rule: enat2_cases) auto
  1.1910 +lemma ereal_of_enat_less_iff[simp]: "ereal_of_enat m < ereal_of_enat n \<longleftrightarrow> m < n"
  1.1911 +  by (cases m n rule: enat2_cases) auto
  1.1912  
  1.1913 -lemma numeral_le_ereal_of_enat_iff[simp]:
  1.1914 -  shows "numeral m \<le> ereal_of_enat n \<longleftrightarrow> numeral m \<le> n"
  1.1915 -by (cases n) (auto dest: natceiling_le intro: natceiling_le_eq[THEN iffD1])
  1.1916 +lemma numeral_le_ereal_of_enat_iff[simp]: "numeral m \<le> ereal_of_enat n \<longleftrightarrow> numeral m \<le> n"
  1.1917 +  by (cases n) (auto dest: natceiling_le intro: natceiling_le_eq[THEN iffD1])
  1.1918  
  1.1919 -lemma numeral_less_ereal_of_enat_iff[simp]:
  1.1920 -  shows "numeral m < ereal_of_enat n \<longleftrightarrow> numeral m < n"
  1.1921 -by (cases n) (auto simp: real_of_nat_less_iff[symmetric])
  1.1922 +lemma numeral_less_ereal_of_enat_iff[simp]: "numeral m < ereal_of_enat n \<longleftrightarrow> numeral m < n"
  1.1923 +  by (cases n) (auto simp: real_of_nat_less_iff[symmetric])
  1.1924  
  1.1925 -lemma ereal_of_enat_ge_zero_cancel_iff[simp]:
  1.1926 -  "0 \<le> ereal_of_enat n \<longleftrightarrow> 0 \<le> n"
  1.1927 -by (cases n) (auto simp: enat_0[symmetric])
  1.1928 +lemma ereal_of_enat_ge_zero_cancel_iff[simp]: "0 \<le> ereal_of_enat n \<longleftrightarrow> 0 \<le> n"
  1.1929 +  by (cases n) (auto simp: enat_0[symmetric])
  1.1930  
  1.1931 -lemma ereal_of_enat_gt_zero_cancel_iff[simp]:
  1.1932 -  "0 < ereal_of_enat n \<longleftrightarrow> 0 < n"
  1.1933 -by (cases n) (auto simp: enat_0[symmetric])
  1.1934 +lemma ereal_of_enat_gt_zero_cancel_iff[simp]: "0 < ereal_of_enat n \<longleftrightarrow> 0 < n"
  1.1935 +  by (cases n) (auto simp: enat_0[symmetric])
  1.1936  
  1.1937 -lemma ereal_of_enat_zero[simp]:
  1.1938 -  "ereal_of_enat 0 = 0"
  1.1939 -by (auto simp: enat_0[symmetric])
  1.1940 +lemma ereal_of_enat_zero[simp]: "ereal_of_enat 0 = 0"
  1.1941 +  by (auto simp: enat_0[symmetric])
  1.1942  
  1.1943 -lemma ereal_of_enat_inf[simp]:
  1.1944 -  "ereal_of_enat n = \<infinity> \<longleftrightarrow> n = \<infinity>"
  1.1945 +lemma ereal_of_enat_inf[simp]: "ereal_of_enat n = \<infinity> \<longleftrightarrow> n = \<infinity>"
  1.1946    by (cases n) auto
  1.1947  
  1.1948 -
  1.1949 -lemma ereal_of_enat_add:
  1.1950 -  "ereal_of_enat (m + n) = ereal_of_enat m + ereal_of_enat n"
  1.1951 -by (cases m n rule: enat2_cases) auto
  1.1952 +lemma ereal_of_enat_add: "ereal_of_enat (m + n) = ereal_of_enat m + ereal_of_enat n"
  1.1953 +  by (cases m n rule: enat2_cases) auto
  1.1954  
  1.1955  lemma ereal_of_enat_sub:
  1.1956 -  assumes "n \<le> m" shows "ereal_of_enat (m - n) = ereal_of_enat m - ereal_of_enat n "
  1.1957 -using assms by (cases m n rule: enat2_cases) auto
  1.1958 +  assumes "n \<le> m"
  1.1959 +  shows "ereal_of_enat (m - n) = ereal_of_enat m - ereal_of_enat n "
  1.1960 +  using assms by (cases m n rule: enat2_cases) auto
  1.1961  
  1.1962  lemma ereal_of_enat_mult:
  1.1963    "ereal_of_enat (m * n) = ereal_of_enat m * ereal_of_enat n"
  1.1964 -by (cases m n rule: enat2_cases) auto
  1.1965 +  by (cases m n rule: enat2_cases) auto
  1.1966  
  1.1967  lemmas ereal_of_enat_pushin = ereal_of_enat_add ereal_of_enat_sub ereal_of_enat_mult
  1.1968  lemmas ereal_of_enat_pushout = ereal_of_enat_pushin[symmetric]
  1.1969 @@ -1607,6 +1908,7 @@
  1.1970  
  1.1971  instance
  1.1972    by default (simp add: open_ereal_generated)
  1.1973 +
  1.1974  end
  1.1975  
  1.1976  lemma open_PInfty: "open A \<Longrightarrow> \<infinity> \<in> A \<Longrightarrow> (\<exists>x. {ereal x<..} \<subseteq> A)"
  1.1977 @@ -1618,8 +1920,13 @@
  1.1978    with Int show ?case
  1.1979      by (intro exI[of _ "max x z"]) fastforce
  1.1980  next
  1.1981 -  { fix x have "x \<noteq> \<infinity> \<Longrightarrow> \<exists>t. x \<le> ereal t" by (cases x) auto }
  1.1982 -  moreover case (Basis S)
  1.1983 +  case (Basis S)
  1.1984 +  {
  1.1985 +    fix x
  1.1986 +    have "x \<noteq> \<infinity> \<Longrightarrow> \<exists>t. x \<le> ereal t"
  1.1987 +      by (cases x) auto
  1.1988 +  }
  1.1989 +  moreover note Basis
  1.1990    ultimately show ?case
  1.1991      by (auto split: ereal.split)
  1.1992  qed (fastforce simp add: vimage_Union)+
  1.1993 @@ -1633,8 +1940,13 @@
  1.1994    with Int show ?case
  1.1995      by (intro exI[of _ "min x z"]) fastforce
  1.1996  next
  1.1997 -  { fix x have "x \<noteq> - \<infinity> \<Longrightarrow> \<exists>t. ereal t \<le> x" by (cases x) auto }
  1.1998 -  moreover case (Basis S)
  1.1999 +  case (Basis S)
  1.2000 +  {
  1.2001 +    fix x
  1.2002 +    have "x \<noteq> - \<infinity> \<Longrightarrow> \<exists>t. ereal t \<le> x"
  1.2003 +      by (cases x) auto
  1.2004 +  }
  1.2005 +  moreover note Basis
  1.2006    ultimately show ?case
  1.2007      by (auto split: ereal.split)
  1.2008  qed (fastforce simp add: vimage_Union)+
  1.2009 @@ -1642,13 +1954,18 @@
  1.2010  lemma open_ereal_vimage: "open S \<Longrightarrow> open (ereal -` S)"
  1.2011    unfolding open_ereal_generated
  1.2012  proof (induct rule: generate_topology.induct)
  1.2013 -  case (Int A B) then show ?case by auto
  1.2014 +  case (Int A B)
  1.2015 +  then show ?case
  1.2016 +    by auto
  1.2017  next
  1.2018 -  { fix x have
  1.2019 +  case (Basis S)
  1.2020 +  {
  1.2021 +    fix x have
  1.2022        "ereal -` {..<x} = (case x of PInfty \<Rightarrow> UNIV | MInfty \<Rightarrow> {} | ereal r \<Rightarrow> {..<r})"
  1.2023        "ereal -` {x<..} = (case x of PInfty \<Rightarrow> {} | MInfty \<Rightarrow> UNIV | ereal r \<Rightarrow> {r<..})"
  1.2024 -      by (induct x) auto }
  1.2025 -  moreover case (Basis S)
  1.2026 +      by (induct x) auto
  1.2027 +  }
  1.2028 +  moreover note Basis
  1.2029    ultimately show ?case
  1.2030      by (auto split: ereal.split)
  1.2031  qed (fastforce simp add: vimage_Union)+
  1.2032 @@ -1657,16 +1974,32 @@
  1.2033    unfolding open_generated_order[where 'a=real]
  1.2034  proof (induct rule: generate_topology.induct)
  1.2035    case (Basis S)
  1.2036 -  moreover { fix x have "ereal ` {..< x} = { -\<infinity> <..< ereal x }" by auto (case_tac xa, auto) }
  1.2037 -  moreover { fix x have "ereal ` {x <..} = { ereal x <..< \<infinity> }" by auto (case_tac xa, auto) }
  1.2038 +  moreover {
  1.2039 +    fix x
  1.2040 +    have "ereal ` {..< x} = { -\<infinity> <..< ereal x }"
  1.2041 +      apply auto
  1.2042 +      apply (case_tac xa)
  1.2043 +      apply auto
  1.2044 +      done
  1.2045 +  }
  1.2046 +  moreover {
  1.2047 +    fix x
  1.2048 +    have "ereal ` {x <..} = { ereal x <..< \<infinity> }"
  1.2049 +      apply auto
  1.2050 +      apply (case_tac xa)
  1.2051 +      apply auto
  1.2052 +      done
  1.2053 +  }
  1.2054    ultimately show ?case
  1.2055       by auto
  1.2056  qed (auto simp add: image_Union image_Int)
  1.2057  
  1.2058 -lemma open_ereal_def: "open A \<longleftrightarrow> open (ereal -` A) \<and> (\<infinity> \<in> A \<longrightarrow> (\<exists>x. {ereal x <..} \<subseteq> A)) \<and> (-\<infinity> \<in> A \<longrightarrow> (\<exists>x. {..<ereal x} \<subseteq> A))"
  1.2059 +lemma open_ereal_def:
  1.2060 +  "open A \<longleftrightarrow> open (ereal -` A) \<and> (\<infinity> \<in> A \<longrightarrow> (\<exists>x. {ereal x <..} \<subseteq> A)) \<and> (-\<infinity> \<in> A \<longrightarrow> (\<exists>x. {..<ereal x} \<subseteq> A))"
  1.2061    (is "open A \<longleftrightarrow> ?rhs")
  1.2062  proof
  1.2063 -  assume "open A" then show ?rhs
  1.2064 +  assume "open A"
  1.2065 +  then show ?rhs
  1.2066      using open_PInfty open_MInfty open_ereal_vimage by auto
  1.2067  next
  1.2068    assume "?rhs"
  1.2069 @@ -1678,14 +2011,23 @@
  1.2070      by (subst *) (auto simp: open_Un)
  1.2071  qed
  1.2072  
  1.2073 -lemma open_PInfty2: assumes "open A" "\<infinity> \<in> A" obtains x where "{ereal x<..} \<subseteq> A"
  1.2074 +lemma open_PInfty2:
  1.2075 +  assumes "open A"
  1.2076 +    and "\<infinity> \<in> A"
  1.2077 +  obtains x where "{ereal x<..} \<subseteq> A"
  1.2078    using open_PInfty[OF assms] by auto
  1.2079  
  1.2080 -lemma open_MInfty2: assumes "open A" "-\<infinity> \<in> A" obtains x where "{..<ereal x} \<subseteq> A"
  1.2081 +lemma open_MInfty2:
  1.2082 +  assumes "open A"
  1.2083 +    and "-\<infinity> \<in> A"
  1.2084 +  obtains x where "{..<ereal x} \<subseteq> A"
  1.2085    using open_MInfty[OF assms] by auto
  1.2086  
  1.2087 -lemma ereal_openE: assumes "open A" obtains x y where
  1.2088 -  "open (ereal -` A)" "\<infinity> \<in> A \<Longrightarrow> {ereal x<..} \<subseteq> A" "-\<infinity> \<in> A \<Longrightarrow> {..<ereal y} \<subseteq> A"
  1.2089 +lemma ereal_openE:
  1.2090 +  assumes "open A"
  1.2091 +  obtains x y where "open (ereal -` A)"
  1.2092 +    and "\<infinity> \<in> A \<Longrightarrow> {ereal x<..} \<subseteq> A"
  1.2093 +    and "-\<infinity> \<in> A \<Longrightarrow> {..<ereal y} \<subseteq> A"
  1.2094    using assms open_ereal_def by auto
  1.2095  
  1.2096  lemmas open_ereal_lessThan = open_lessThan[where 'a=ereal]
  1.2097 @@ -1695,60 +2037,76 @@
  1.2098  lemmas closed_ereal_atMost = closed_atMost[where 'a=ereal]
  1.2099  lemmas closed_ereal_atLeastAtMost = closed_atLeastAtMost[where 'a=ereal]
  1.2100  lemmas closed_ereal_singleton = closed_singleton[where 'a=ereal]
  1.2101 -  
  1.2102 +
  1.2103  lemma ereal_open_cont_interval:
  1.2104    fixes S :: "ereal set"
  1.2105 -  assumes "open S" "x \<in> S" "\<bar>x\<bar> \<noteq> \<infinity>"
  1.2106 -  obtains e where "e>0" "{x-e <..< x+e} \<subseteq> S"
  1.2107 -proof-
  1.2108 -  from `open S` have "open (ereal -` S)" by (rule ereal_openE)
  1.2109 -  then obtain e where "0 < e" and e: "\<And>y. dist y (real x) < e \<Longrightarrow> ereal y \<in> S"
  1.2110 +  assumes "open S"
  1.2111 +    and "x \<in> S"
  1.2112 +    and "\<bar>x\<bar> \<noteq> \<infinity>"
  1.2113 +  obtains e where "e > 0" and "{x-e <..< x+e} \<subseteq> S"
  1.2114 +proof -
  1.2115 +  from `open S`
  1.2116 +  have "open (ereal -` S)"
  1.2117 +    by (rule ereal_openE)
  1.2118 +  then obtain e where "e > 0" and e: "\<And>y. dist y (real x) < e \<Longrightarrow> ereal y \<in> S"
  1.2119      using assms unfolding open_dist by force
  1.2120    show thesis
  1.2121    proof (intro that subsetI)
  1.2122 -    show "0 < ereal e" using `0 < e` by auto
  1.2123 -    fix y assume "y \<in> {x - ereal e<..<x + ereal e}"
  1.2124 +    show "0 < ereal e"
  1.2125 +      using `0 < e` by auto
  1.2126 +    fix y
  1.2127 +    assume "y \<in> {x - ereal e<..<x + ereal e}"
  1.2128      with assms obtain t where "y = ereal t" "dist t (real x) < e"
  1.2129 -      apply (cases y) by (auto simp: dist_real_def)
  1.2130 -    then show "y \<in> S" using e[of t] by auto
  1.2131 +      by (cases y) (auto simp: dist_real_def)
  1.2132 +    then show "y \<in> S"
  1.2133 +      using e[of t] by auto
  1.2134    qed
  1.2135  qed
  1.2136  
  1.2137  lemma ereal_open_cont_interval2:
  1.2138    fixes S :: "ereal set"
  1.2139 -  assumes "open S" "x \<in> S" and x: "\<bar>x\<bar> \<noteq> \<infinity>"
  1.2140 -  obtains a b where "a < x" "x < b" "{a <..< b} \<subseteq> S"
  1.2141 +  assumes "open S"
  1.2142 +    and "x \<in> S"
  1.2143 +    and x: "\<bar>x\<bar> \<noteq> \<infinity>"
  1.2144 +  obtains a b where "a < x" and "x < b" and "{a <..< b} \<subseteq> S"
  1.2145  proof -
  1.2146    obtain e where "0 < e" "{x - e<..<x + e} \<subseteq> S"
  1.2147      using assms by (rule ereal_open_cont_interval)
  1.2148 -  with that[of "x-e" "x+e"] ereal_between[OF x, of e]
  1.2149 -  show thesis by auto
  1.2150 +  with that[of "x - e" "x + e"] ereal_between[OF x, of e]
  1.2151 +  show thesis
  1.2152 +    by auto
  1.2153  qed
  1.2154  
  1.2155 +
  1.2156  subsubsection {* Convergent sequences *}
  1.2157  
  1.2158 -lemma lim_ereal[simp]:
  1.2159 -  "((\<lambda>n. ereal (f n)) ---> ereal x) net \<longleftrightarrow> (f ---> x) net" (is "?l = ?r")
  1.2160 +lemma lim_ereal[simp]: "((\<lambda>n. ereal (f n)) ---> ereal x) net \<longleftrightarrow> (f ---> x) net"
  1.2161 +  (is "?l = ?r")
  1.2162  proof (intro iffI topological_tendstoI)
  1.2163 -  fix S assume "?l" "open S" "x \<in> S"
  1.2164 +  fix S
  1.2165 +  assume "?l" and "open S" and "x \<in> S"
  1.2166    then show "eventually (\<lambda>x. f x \<in> S) net"
  1.2167      using `?l`[THEN topological_tendstoD, OF open_ereal, OF `open S`]
  1.2168      by (simp add: inj_image_mem_iff)
  1.2169  next
  1.2170 -  fix S assume "?r" "open S" "ereal x \<in> S"
  1.2171 +  fix S
  1.2172 +  assume "?r" and "open S" and "ereal x \<in> S"
  1.2173    show "eventually (\<lambda>x. ereal (f x) \<in> S) net"
  1.2174      using `?r`[THEN topological_tendstoD, OF open_ereal_vimage, OF `open S`]
  1.2175 -    using `ereal x \<in> S` by auto
  1.2176 +    using `ereal x \<in> S`
  1.2177 +    by auto
  1.2178  qed
  1.2179  
  1.2180  lemma lim_real_of_ereal[simp]:
  1.2181    assumes lim: "(f ---> ereal x) net"
  1.2182    shows "((\<lambda>x. real (f x)) ---> x) net"
  1.2183  proof (intro topological_tendstoI)
  1.2184 -  fix S assume "open S" "x \<in> S"
  1.2185 +  fix S
  1.2186 +  assume "open S" and "x \<in> S"
  1.2187    then have S: "open S" "ereal x \<in> ereal ` S"
  1.2188      by (simp_all add: inj_image_mem_iff)
  1.2189 -  have "\<forall>x. f x \<in> ereal ` S \<longrightarrow> real (f x) \<in> S" by auto
  1.2190 +  have "\<forall>x. f x \<in> ereal ` S \<longrightarrow> real (f x) \<in> S"
  1.2191 +    by auto
  1.2192    from this lim[THEN topological_tendstoD, OF open_ereal, OF S]
  1.2193    show "eventually (\<lambda>x. real (f x) \<in> S) net"
  1.2194      by (rule eventually_mono)
  1.2195 @@ -1756,10 +2114,12 @@
  1.2196  
  1.2197  lemma tendsto_PInfty: "(f ---> \<infinity>) F \<longleftrightarrow> (\<forall>r. eventually (\<lambda>x. ereal r < f x) F)"
  1.2198  proof -
  1.2199 -  { fix l :: ereal assume "\<forall>r. eventually (\<lambda>x. ereal r < f x) F"
  1.2200 -    from this[THEN spec, of "real l"]
  1.2201 -    have "l \<noteq> \<infinity> \<Longrightarrow> eventually (\<lambda>x. l < f x) F"
  1.2202 -      by (cases l) (auto elim: eventually_elim1) }
  1.2203 +  {
  1.2204 +    fix l :: ereal
  1.2205 +    assume "\<forall>r. eventually (\<lambda>x. ereal r < f x) F"
  1.2206 +    from this[THEN spec, of "real l"] have "l \<noteq> \<infinity> \<Longrightarrow> eventually (\<lambda>x. l < f x) F"
  1.2207 +      by (cases l) (auto elim: eventually_elim1)
  1.2208 +  }
  1.2209    then show ?thesis
  1.2210      by (auto simp: order_tendsto_iff)
  1.2211  qed
  1.2212 @@ -1772,20 +2132,26 @@
  1.2213    from open_MInfty[OF this] obtain B where "{..<ereal B} \<subseteq> S" ..
  1.2214    moreover
  1.2215    assume "\<forall>r::real. eventually (\<lambda>z. f z < r) F"
  1.2216 -  then have "eventually (\<lambda>z. f z \<in> {..< B}) F" by auto
  1.2217 -  ultimately show "eventually (\<lambda>z. f z \<in> S) F" by (auto elim!: eventually_elim1)
  1.2218 +  then have "eventually (\<lambda>z. f z \<in> {..< B}) F"
  1.2219 +    by auto
  1.2220 +  ultimately show "eventually (\<lambda>z. f z \<in> S) F"
  1.2221 +    by (auto elim!: eventually_elim1)
  1.2222  next
  1.2223 -  fix x assume "\<forall>S. open S \<longrightarrow> -\<infinity> \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F"
  1.2224 -  from this[rule_format, of "{..< ereal x}"]
  1.2225 -  show "eventually (\<lambda>y. f y < ereal x) F" by auto
  1.2226 +  fix x
  1.2227 +  assume "\<forall>S. open S \<longrightarrow> -\<infinity> \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F"
  1.2228 +  from this[rule_format, of "{..< ereal x}"] show "eventually (\<lambda>y. f y < ereal x) F"
  1.2229 +    by auto
  1.2230  qed
  1.2231  
  1.2232  lemma Lim_PInfty: "f ----> \<infinity> \<longleftrightarrow> (\<forall>B. \<exists>N. \<forall>n\<ge>N. f n \<ge> ereal B)"
  1.2233    unfolding tendsto_PInfty eventually_sequentially
  1.2234  proof safe
  1.2235 -  fix r assume "\<forall>r. \<exists>N. \<forall>n\<ge>N. ereal r \<le> f n"
  1.2236 -  then obtain N where "\<forall>n\<ge>N. ereal (r + 1) \<le> f n" by blast
  1.2237 -  moreover have "ereal r < ereal (r + 1)" by auto
  1.2238 +  fix r
  1.2239 +  assume "\<forall>r. \<exists>N. \<forall>n\<ge>N. ereal r \<le> f n"
  1.2240 +  then obtain N where "\<forall>n\<ge>N. ereal (r + 1) \<le> f n"
  1.2241 +    by blast
  1.2242 +  moreover have "ereal r < ereal (r + 1)"
  1.2243 +    by auto
  1.2244    ultimately show "\<exists>N. \<forall>n\<ge>N. ereal r < f n"
  1.2245      by (blast intro: less_le_trans)
  1.2246  qed (blast intro: less_imp_le)
  1.2247 @@ -1793,9 +2159,12 @@
  1.2248  lemma Lim_MInfty: "f ----> -\<infinity> \<longleftrightarrow> (\<forall>B. \<exists>N. \<forall>n\<ge>N. ereal B \<ge> f n)"
  1.2249    unfolding tendsto_MInfty eventually_sequentially
  1.2250  proof safe
  1.2251 -  fix r assume "\<forall>r. \<exists>N. \<forall>n\<ge>N. f n \<le> ereal r"
  1.2252 -  then obtain N where "\<forall>n\<ge>N. f n \<le> ereal (r - 1)" by blast
  1.2253 -  moreover have "ereal (r - 1) < ereal r" by auto
  1.2254 +  fix r
  1.2255 +  assume "\<forall>r. \<exists>N. \<forall>n\<ge>N. f n \<le> ereal r"
  1.2256 +  then obtain N where "\<forall>n\<ge>N. f n \<le> ereal (r - 1)"
  1.2257 +    by blast
  1.2258 +  moreover have "ereal (r - 1) < ereal r"
  1.2259 +    by auto
  1.2260    ultimately show "\<exists>N. \<forall>n\<ge>N. f n < ereal r"
  1.2261      by (blast intro: le_less_trans)
  1.2262  qed (blast intro: less_imp_le)
  1.2263 @@ -1807,38 +2176,43 @@
  1.2264    using LIMSEQ_le_const[of f l "ereal B"] by auto
  1.2265  
  1.2266  lemma tendsto_explicit:
  1.2267 -  "f ----> f0 <-> (ALL S. open S --> f0 : S --> (EX N. ALL n>=N. f n : S))"
  1.2268 +  "f ----> f0 \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> f0 \<in> S \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. f n \<in> S))"
  1.2269    unfolding tendsto_def eventually_sequentially by auto
  1.2270  
  1.2271 -lemma Lim_bounded_PInfty2:
  1.2272 -  "f ----> l \<Longrightarrow> ALL n>=N. f n <= ereal B \<Longrightarrow> l ~= \<infinity>"
  1.2273 +lemma Lim_bounded_PInfty2: "f ----> l \<Longrightarrow> \<forall>n\<ge>N. f n \<le> ereal B \<Longrightarrow> l \<noteq> \<infinity>"
  1.2274    using LIMSEQ_le_const2[of f l "ereal B"] by fastforce
  1.2275  
  1.2276 -lemma Lim_bounded_ereal: "f ----> (l :: 'a::linorder_topology) \<Longrightarrow> ALL n>=M. f n <= C \<Longrightarrow> l<=C"
  1.2277 +lemma Lim_bounded_ereal: "f ----> (l :: 'a::linorder_topology) \<Longrightarrow> \<forall>n\<ge>M. f n \<le> C \<Longrightarrow> l \<le> C"
  1.2278    by (intro LIMSEQ_le_const2) auto
  1.2279  
  1.2280  lemma Lim_bounded2_ereal:
  1.2281 -  assumes lim:"f ----> (l :: 'a::linorder_topology)" and ge: "ALL n>=N. f n >= C"
  1.2282 -  shows "l>=C"
  1.2283 +  assumes lim:"f ----> (l :: 'a::linorder_topology)"
  1.2284 +    and ge: "\<forall>n\<ge>N. f n \<ge> C"
  1.2285 +  shows "l \<ge> C"
  1.2286    using ge
  1.2287    by (intro tendsto_le[OF trivial_limit_sequentially lim tendsto_const])
  1.2288       (auto simp: eventually_sequentially)
  1.2289  
  1.2290  lemma real_of_ereal_mult[simp]:
  1.2291 -  fixes a b :: ereal shows "real (a * b) = real a * real b"
  1.2292 +  fixes a b :: ereal
  1.2293 +  shows "real (a * b) = real a * real b"
  1.2294    by (cases rule: ereal2_cases[of a b]) auto
  1.2295  
  1.2296  lemma real_of_ereal_eq_0:
  1.2297 -  fixes x :: ereal shows "real x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity> \<or> x = 0"
  1.2298 +  fixes x :: ereal
  1.2299 +  shows "real x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity> \<or> x = 0"
  1.2300    by (cases x) auto
  1.2301  
  1.2302  lemma tendsto_ereal_realD:
  1.2303    fixes f :: "'a \<Rightarrow> ereal"
  1.2304 -  assumes "x \<noteq> 0" and tendsto: "((\<lambda>x. ereal (real (f x))) ---> x) net"
  1.2305 +  assumes "x \<noteq> 0"
  1.2306 +    and tendsto: "((\<lambda>x. ereal (real (f x))) ---> x) net"
  1.2307    shows "(f ---> x) net"
  1.2308  proof (intro topological_tendstoI)
  1.2309 -  fix S assume S: "open S" "x \<in> S"
  1.2310 -  with `x \<noteq> 0` have "open (S - {0})" "x \<in> S - {0}" by auto
  1.2311 +  fix S
  1.2312 +  assume S: "open S" "x \<in> S"
  1.2313 +  with `x \<noteq> 0` have "open (S - {0})" "x \<in> S - {0}"
  1.2314 +    by auto
  1.2315    from tendsto[THEN topological_tendstoD, OF this]
  1.2316    show "eventually (\<lambda>x. f x \<in> S) net"
  1.2317      by (rule eventually_rev_mp) (auto simp: ereal_real)
  1.2318 @@ -1849,22 +2223,25 @@
  1.2319    assumes x: "\<bar>x\<bar> \<noteq> \<infinity>" and tendsto: "(f ---> x) net"
  1.2320    shows "((\<lambda>x. ereal (real (f x))) ---> x) net"
  1.2321  proof (intro topological_tendstoI)
  1.2322 -  fix S assume "open S" "x \<in> S"
  1.2323 -  with x have "open (S - {\<infinity>, -\<infinity>})" "x \<in> S - {\<infinity>, -\<infinity>}" by auto
  1.2324 +  fix S
  1.2325 +  assume "open S" and "x \<in> S"
  1.2326 +  with x have "open (S - {\<infinity>, -\<infinity>})" "x \<in> S - {\<infinity>, -\<infinity>}"
  1.2327 +    by auto
  1.2328    from tendsto[THEN topological_tendstoD, OF this]
  1.2329    show "eventually (\<lambda>x. ereal (real (f x)) \<in> S) net"
  1.2330      by (elim eventually_elim1) (auto simp: ereal_real)
  1.2331  qed
  1.2332  
  1.2333  lemma ereal_mult_cancel_left:
  1.2334 -  fixes a b c :: ereal shows "a * b = a * c \<longleftrightarrow>
  1.2335 -    ((\<bar>a\<bar> = \<infinity> \<and> 0 < b * c) \<or> a = 0 \<or> b = c)"
  1.2336 -  by (cases rule: ereal3_cases[of a b c])
  1.2337 -     (simp_all add: zero_less_mult_iff)
  1.2338 +  fixes a b c :: ereal
  1.2339 +  shows "a * b = a * c \<longleftrightarrow> (\<bar>a\<bar> = \<infinity> \<and> 0 < b * c) \<or> a = 0 \<or> b = c"
  1.2340 +  by (cases rule: ereal3_cases[of a b c]) (simp_all add: zero_less_mult_iff)
  1.2341  
  1.2342  lemma ereal_inj_affinity:
  1.2343    fixes m t :: ereal
  1.2344 -  assumes "\<bar>m\<bar> \<noteq> \<infinity>" "m \<noteq> 0" "\<bar>t\<bar> \<noteq> \<infinity>"
  1.2345 +  assumes "\<bar>m\<bar> \<noteq> \<infinity>"
  1.2346 +    and "m \<noteq> 0"
  1.2347 +    and "\<bar>t\<bar> \<noteq> \<infinity>"
  1.2348    shows "inj_on (\<lambda>x. m * x + t) A"
  1.2349    using assms
  1.2350    by (cases rule: ereal2_cases[of m t])
  1.2351 @@ -1902,108 +2279,136 @@
  1.2352  lemma ereal_mult_m1[simp]: "x * ereal (-1) = -x"
  1.2353    by (cases x) auto
  1.2354  
  1.2355 -lemma ereal_real': assumes "\<bar>x\<bar> \<noteq> \<infinity>" shows "ereal (real x) = x"
  1.2356 +lemma ereal_real':
  1.2357 +  assumes "\<bar>x\<bar> \<noteq> \<infinity>"
  1.2358 +  shows "ereal (real x) = x"
  1.2359    using assms by auto
  1.2360  
  1.2361 -lemma real_ereal_id: "real o ereal = id"
  1.2362 -proof-
  1.2363 -  { fix x have "(real o ereal) x = id x" by auto }
  1.2364 -  then show ?thesis using ext by blast
  1.2365 +lemma real_ereal_id: "real \<circ> ereal = id"
  1.2366 +proof -
  1.2367 +  {
  1.2368 +    fix x
  1.2369 +    have "(real o ereal) x = id x"
  1.2370 +      by auto
  1.2371 +  }
  1.2372 +  then show ?thesis
  1.2373 +    using ext by blast
  1.2374  qed
  1.2375  
  1.2376  lemma open_image_ereal: "open(UNIV-{ \<infinity> , (-\<infinity> :: ereal)})"
  1.2377 -by (metis range_ereal open_ereal open_UNIV)
  1.2378 +  by (metis range_ereal open_ereal open_UNIV)
  1.2379  
  1.2380  lemma ereal_le_distrib:
  1.2381 -  fixes a b c :: ereal shows "c * (a + b) \<le> c * a + c * b"
  1.2382 +  fixes a b c :: ereal
  1.2383 +  shows "c * (a + b) \<le> c * a + c * b"
  1.2384    by (cases rule: ereal3_cases[of a b c])
  1.2385       (auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff)
  1.2386  
  1.2387  lemma ereal_pos_distrib:
  1.2388 -  fixes a b c :: ereal assumes "0 \<le> c" "c \<noteq> \<infinity>" shows "c * (a + b) = c * a + c * b"
  1.2389 -  using assms by (cases rule: ereal3_cases[of a b c])
  1.2390 -                 (auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff)
  1.2391 +  fixes a b c :: ereal
  1.2392 +  assumes "0 \<le> c"
  1.2393 +    and "c \<noteq> \<infinity>"
  1.2394 +  shows "c * (a + b) = c * a + c * b"
  1.2395 +  using assms
  1.2396 +  by (cases rule: ereal3_cases[of a b c])
  1.2397 +    (auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff)
  1.2398  
  1.2399  lemma ereal_pos_le_distrib:
  1.2400 -fixes a b c :: ereal
  1.2401 -assumes "c>=0"
  1.2402 -shows "c * (a + b) <= c * a + c * b"
  1.2403 -  using assms by (cases rule: ereal3_cases[of a b c])
  1.2404 -                 (auto simp add: field_simps)
  1.2405 +  fixes a b c :: ereal
  1.2406 +  assumes "c \<ge> 0"
  1.2407 +  shows "c * (a + b) \<le> c * a + c * b"
  1.2408 +  using assms
  1.2409 +  by (cases rule: ereal3_cases[of a b c]) (auto simp add: field_simps)
  1.2410  
  1.2411 -lemma ereal_max_mono:
  1.2412 -  "[| (a::ereal) <= b; c <= d |] ==> max a c <= max b d"
  1.2413 +lemma ereal_max_mono: "(a::ereal) \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> max a c \<le> max b d"
  1.2414    by (metis sup_ereal_def sup_mono)
  1.2415  
  1.2416 -
  1.2417 -lemma ereal_max_least:
  1.2418 -  "[| (a::ereal) <= x; c <= x |] ==> max a c <= x"
  1.2419 +lemma ereal_max_least: "(a::ereal) \<le> x \<Longrightarrow> c \<le> x \<Longrightarrow> max a c \<le> x"
  1.2420    by (metis sup_ereal_def sup_least)
  1.2421  
  1.2422  lemma ereal_LimI_finite:
  1.2423    fixes x :: ereal
  1.2424    assumes "\<bar>x\<bar> \<noteq> \<infinity>"
  1.2425 -  assumes "!!r. 0 < r ==> EX N. ALL n>=N. u n < x + r & x < u n + r"
  1.2426 +    and "\<And>r. 0 < r \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. u n < x + r \<and> x < u n + r"
  1.2427    shows "u ----> x"
  1.2428  proof (rule topological_tendstoI, unfold eventually_sequentially)
  1.2429 -  obtain rx where rx_def: "x=ereal rx" using assms by (cases x) auto
  1.2430 -  fix S assume "open S" "x : S"
  1.2431 -  then have "open (ereal -` S)" unfolding open_ereal_def by auto
  1.2432 -  with `x \<in> S` obtain r where "0 < r" and dist: "!!y. dist y rx < r ==> ereal y \<in> S"
  1.2433 -    unfolding open_real_def rx_def by auto
  1.2434 +  obtain rx where rx: "x = ereal rx"
  1.2435 +    using assms by (cases x) auto
  1.2436 +  fix S
  1.2437 +  assume "open S" and "x \<in> S"
  1.2438 +  then have "open (ereal -` S)"
  1.2439 +    unfolding open_ereal_def by auto
  1.2440 +  with `x \<in> S` obtain r where "0 < r" and dist: "\<And>y. dist y rx < r \<Longrightarrow> ereal y \<in> S"
  1.2441 +    unfolding open_real_def rx by auto
  1.2442    then obtain n where
  1.2443 -    upper: "!!N. n <= N ==> u N < x + ereal r" and
  1.2444 -    lower: "!!N. n <= N ==> x < u N + ereal r" using assms(2)[of "ereal r"] by auto
  1.2445 -  show "EX N. ALL n>=N. u n : S"
  1.2446 +    upper: "\<And>N. n \<le> N \<Longrightarrow> u N < x + ereal r" and
  1.2447 +    lower: "\<And>N. n \<le> N \<Longrightarrow> x < u N + ereal r"
  1.2448 +    using assms(2)[of "ereal r"] by auto
  1.2449 +  show "\<exists>N. \<forall>n\<ge>N. u n \<in> S"
  1.2450    proof (safe intro!: exI[of _ n])
  1.2451 -    fix N assume "n <= N"
  1.2452 +    fix N
  1.2453 +    assume "n \<le> N"
  1.2454      from upper[OF this] lower[OF this] assms `0 < r`
  1.2455 -    have "u N ~: {\<infinity>,(-\<infinity>)}" by auto
  1.2456 -    then obtain ra where ra_def: "(u N) = ereal ra" by (cases "u N") auto
  1.2457 -    hence "rx < ra + r" and "ra < rx + r"
  1.2458 -       using rx_def assms `0 < r` lower[OF `n <= N`] upper[OF `n <= N`] by auto
  1.2459 -    hence "dist (real (u N)) rx < r"
  1.2460 -      using rx_def ra_def
  1.2461 +    have "u N \<notin> {\<infinity>,(-\<infinity>)}"
  1.2462 +      by auto
  1.2463 +    then obtain ra where ra_def: "(u N) = ereal ra"
  1.2464 +      by (cases "u N") auto
  1.2465 +    then have "rx < ra + r" and "ra < rx + r"
  1.2466 +      using rx assms `0 < r` lower[OF `n \<le> N`] upper[OF `n \<le> N`]
  1.2467 +      by auto
  1.2468 +    then have "dist (real (u N)) rx < r"
  1.2469 +      using rx ra_def
  1.2470        by (auto simp: dist_real_def abs_diff_less_iff field_simps)
  1.2471 -    from dist[OF this] show "u N : S" using `u N  ~: {\<infinity>, -\<infinity>}`
  1.2472 +    from dist[OF this] show "u N \<in> S"
  1.2473 +      using `u N  \<notin> {\<infinity>, -\<infinity>}`
  1.2474        by (auto simp: ereal_real split: split_if_asm)
  1.2475    qed
  1.2476  qed
  1.2477  
  1.2478  lemma tendsto_obtains_N:
  1.2479    assumes "f ----> f0"
  1.2480 -  assumes "open S" "f0 : S"
  1.2481 -  obtains N where "ALL n>=N. f n : S"
  1.2482 +  assumes "open S"
  1.2483 +    and "f0 \<in> S"
  1.2484 +  obtains N where "\<forall>n\<ge>N. f n \<in> S"
  1.2485    using assms using tendsto_def
  1.2486    using tendsto_explicit[of f f0] assms by auto
  1.2487  
  1.2488  lemma ereal_LimI_finite_iff:
  1.2489    fixes x :: ereal
  1.2490    assumes "\<bar>x\<bar> \<noteq> \<infinity>"
  1.2491 -  shows "u ----> x <-> (ALL r. 0 < r --> (EX N. ALL n>=N. u n < x + r & x < u n + r))"
  1.2492 -  (is "?lhs <-> ?rhs")
  1.2493 +  shows "u ----> x \<longleftrightarrow> (\<forall>r. 0 < r \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. u n < x + r \<and> x < u n + r))"
  1.2494 +  (is "?lhs \<longleftrightarrow> ?rhs")
  1.2495  proof
  1.2496    assume lim: "u ----> x"
  1.2497 -  { fix r assume "(r::ereal)>0"
  1.2498 -    then obtain N where N_def: "ALL n>=N. u n : {x - r <..< x + r}"
  1.2499 +  {
  1.2500 +    fix r :: ereal
  1.2501 +    assume "r > 0"
  1.2502 +    then obtain N where "\<forall>n\<ge>N. u n \<in> {x - r <..< x + r}"
  1.2503         apply (subst tendsto_obtains_N[of u x "{x - r <..< x + r}"])
  1.2504 -       using lim ereal_between[of x r] assms `r>0` by auto
  1.2505 -    hence "EX N. ALL n>=N. u n < x + r & x < u n + r"
  1.2506 -      using ereal_minus_less[of r x] by (cases r) auto
  1.2507 -  } then show "?rhs" by auto
  1.2508 +       using lim ereal_between[of x r] assms `r > 0`
  1.2509 +       apply auto
  1.2510 +       done
  1.2511 +    then have "\<exists>N. \<forall>n\<ge>N. u n < x + r \<and> x < u n + r"
  1.2512 +      using ereal_minus_less[of r x]
  1.2513 +      by (cases r) auto
  1.2514 +  }
  1.2515 +  then show ?rhs
  1.2516 +    by auto
  1.2517  next
  1.2518 -  assume ?rhs then show "u ----> x"
  1.2519 +  assume ?rhs
  1.2520 +  then show "u ----> x"
  1.2521      using ereal_LimI_finite[of x] assms by auto
  1.2522  qed
  1.2523  
  1.2524  lemma ereal_Limsup_uminus:
  1.2525 -  fixes f :: "'a => ereal"
  1.2526 -  shows "Limsup net (\<lambda>x. - (f x)) = -(Liminf net f)"
  1.2527 +  fixes f :: "'a \<Rightarrow> ereal"
  1.2528 +  shows "Limsup net (\<lambda>x. - (f x)) = - Liminf net f"
  1.2529    unfolding Limsup_def Liminf_def ereal_SUPR_uminus ereal_INFI_uminus ..
  1.2530  
  1.2531  lemma liminf_bounded_iff:
  1.2532    fixes x :: "nat \<Rightarrow> ereal"
  1.2533 -  shows "C \<le> liminf x \<longleftrightarrow> (\<forall>B<C. \<exists>N. \<forall>n\<ge>N. B < x n)" (is "?lhs <-> ?rhs")
  1.2534 +  shows "C \<le> liminf x \<longleftrightarrow> (\<forall>B<C. \<exists>N. \<forall>n\<ge>N. B < x n)"
  1.2535 +  (is "?lhs \<longleftrightarrow> ?rhs")
  1.2536    unfolding le_Liminf_iff eventually_sequentially ..
  1.2537  
  1.2538  lemma
  1.2539 @@ -2012,6 +2417,7 @@
  1.2540      and ereal_decseq_uminus[simp]: "decseq (\<lambda>i. - X i) = incseq X"
  1.2541    unfolding incseq_def decseq_def by auto
  1.2542  
  1.2543 +
  1.2544  subsubsection {* Tests for code generator *}
  1.2545  
  1.2546  (* A small list of simple arithmetic expressions *)