# HG changeset patch
# User wenzelm
# Date 1354896040 -3600
# Node ID 79858bd9f5ef731148cf5577150fea70c8619796
# Parent  eb7b59cc8e081424a3d91e65a61320d45c1079b3# Parent  7c8ce63a3c0021dbf09a588abd9356ac68284dda
merged

diff -r 7c8ce63a3c00 -r 79858bd9f5ef src/HOL/IMP/HoareT.thy
--- a/src/HOL/IMP/HoareT.thy	Fri Dec 07 16:53:35 2012 +0100
+++ b/src/HOL/IMP/HoareT.thy	Fri Dec 07 17:00:40 2012 +0100
@@ -1,7 +1,9 @@
-header{* Hoare Logic for Total Correctness *}
+(* Author: Tobias Nipkow *)
 
 theory HoareT imports Hoare_Sound_Complete begin
 
+subsection "Hoare Logic for Total Correctness"
+
 text{* Note that this definition of total validity @{text"\<Turnstile>\<^sub>t"} only
 works if execution is deterministic (which it is in our case). *}
 
diff -r 7c8ce63a3c00 -r 79858bd9f5ef src/HOL/IMP/VC.thy
--- a/src/HOL/IMP/VC.thy	Fri Dec 07 16:53:35 2012 +0100
+++ b/src/HOL/IMP/VC.thy	Fri Dec 07 17:00:40 2012 +0100
@@ -1,8 +1,8 @@
-header "Verification Conditions"
+(* Author: Tobias Nipkow *)
 
 theory VC imports Hoare begin
 
-subsection "VCG via Weakest Preconditions"
+subsection "Verification Conditions"
 
 text{* Annotated commands: commands where loops are annotated with
 invariants. *}
@@ -46,7 +46,7 @@
 "strip (Aif b c\<^isub>1 c\<^isub>2) = (IF b THEN strip c\<^isub>1 ELSE strip c\<^isub>2)" |
 "strip (Awhile I b c) = (WHILE b DO strip c)"
 
-subsection "Soundness"
+text {* Soundness: *}
 
 lemma vc_sound: "\<forall>s. vc c Q s \<Longrightarrow> \<turnstile> {pre c Q} strip c {Q}"
 proof(induction c arbitrary: Q)
@@ -68,7 +68,7 @@
 by (metis strengthen_pre vc_sound)
 
 
-subsection "Completeness"
+text{* Completeness: *}
 
 lemma pre_mono:
   "\<forall>s. P s \<longrightarrow> P' s \<Longrightarrow> pre c P s \<Longrightarrow> pre c P' s"
@@ -122,7 +122,7 @@
 qed
 
 
-subsection "An Optimization"
+text{* An Optimization: *}
 
 fun vcpre :: "acom \<Rightarrow> assn \<Rightarrow> assn \<times> assn" where
 "vcpre ASKIP Q = (\<lambda>s. True, Q)" |
diff -r 7c8ce63a3c00 -r 79858bd9f5ef src/HOL/Limits.thy
--- a/src/HOL/Limits.thy	Fri Dec 07 16:53:35 2012 +0100
+++ b/src/HOL/Limits.thy	Fri Dec 07 17:00:40 2012 +0100
@@ -732,6 +732,9 @@
   "filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> filterlim f F2' F1'"
   unfolding filterlim_def by (metis filtermap_mono order_trans)
 
+lemma filterlim_ident: "LIM x F. x :> F"
+  by (simp add: filterlim_def filtermap_ident)
+
 lemma filterlim_cong:
   "F1 = F1' \<Longrightarrow> F2 = F2' \<Longrightarrow> eventually (\<lambda>x. f x = g x) F2 \<Longrightarrow> filterlim f F1 F2 = filterlim g F1' F2'"
   by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2)
@@ -1560,6 +1563,10 @@
     by (auto intro: order_trans simp: filterlim_def filtermap_filtermap)
 qed
 
+lemma tendsto_inverse_0_at_top:
+  "LIM x F. f x :> at_top \<Longrightarrow> ((\<lambda>x. inverse (f x) :: real) ---> 0) F"
+ by (metis at_top_le_at_infinity filterlim_at filterlim_inverse_at_iff filterlim_mono order_refl)
+
 text {*
 
 We only show rules for multiplication and addition when the functions are either against a real
@@ -1610,6 +1617,12 @@
   qed
 qed
 
+lemma filterlim_tendsto_pos_mult_at_bot:
+  assumes "(f ---> c) F" "0 < (c::real)" "filterlim g at_bot F"
+  shows "LIM x F. f x * g x :> at_bot"
+  using filterlim_tendsto_pos_mult_at_top[OF assms(1,2), of "\<lambda>x. - g x"] assms(3)
+  unfolding filterlim_uminus_at_bot by simp
+
 lemma filterlim_tendsto_add_at_top: 
   assumes f: "(f ---> c) F"
   assumes g: "LIM x F. g x :> at_top"
diff -r 7c8ce63a3c00 -r 79858bd9f5ef src/HOL/Multivariate_Analysis/Derivative.thy
--- a/src/HOL/Multivariate_Analysis/Derivative.thy	Fri Dec 07 16:53:35 2012 +0100
+++ b/src/HOL/Multivariate_Analysis/Derivative.thy	Fri Dec 07 17:00:40 2012 +0100
@@ -1698,6 +1698,13 @@
     by auto
 qed
 
+lemma has_vector_derivative_withinI_DERIV:
+  assumes f: "DERIV f x :> y" shows "(f has_vector_derivative y) (at x within s)"
+  unfolding has_vector_derivative_def real_scaleR_def
+  apply (rule has_derivative_at_within)
+  using DERIV_conv_has_derivative[THEN iffD1, OF f]
+  apply (subst mult_commute) .
+
 lemma vector_derivative_unique_at:
   assumes "(f has_vector_derivative f') (at x)"
   assumes "(f has_vector_derivative f'') (at x)"
diff -r 7c8ce63a3c00 -r 79858bd9f5ef src/HOL/Probability/Borel_Space.thy
--- a/src/HOL/Probability/Borel_Space.thy	Fri Dec 07 16:53:35 2012 +0100
+++ b/src/HOL/Probability/Borel_Space.thy	Fri Dec 07 17:00:40 2012 +0100
@@ -816,6 +816,10 @@
   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. log (g x) (f x)) \<in> borel_measurable M"
   unfolding log_def by auto
 
+lemma borel_measurable_exp[measurable]: "exp \<in> borel_measurable borel"
+  by (intro borel_measurable_continuous_on1 continuous_at_imp_continuous_on ballI
+            continuous_isCont[THEN iffD1] isCont_exp)
+
 lemma measurable_count_space_eq2_countable:
   fixes f :: "'a => 'c::countable"
   shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
diff -r 7c8ce63a3c00 -r 79858bd9f5ef src/HOL/Probability/Distributions.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Probability/Distributions.thy	Fri Dec 07 17:00:40 2012 +0100
@@ -0,0 +1,392 @@
+theory Distributions
+  imports Probability_Measure
+begin
+
+subsection {* Exponential distribution *}
+
+definition exponential_density :: "real \<Rightarrow> real \<Rightarrow> real" where
+  "exponential_density l x = (if x < 0 then 0 else l * exp (- x * l))"
+
+lemma borel_measurable_exponential_density[measurable]: "exponential_density l \<in> borel_measurable borel"
+  by (auto simp add: exponential_density_def[abs_def])
+
+lemma (in prob_space) exponential_distributed_params:
+  assumes D: "distributed M lborel X (exponential_density l)"
+  shows "0 < l"
+proof (cases l "0 :: real" rule: linorder_cases)
+  assume "l < 0"
+  have "emeasure lborel {0 <.. 1::real} \<le>
+    emeasure lborel {x :: real \<in> space lborel. 0 < x}"
+    by (rule emeasure_mono) (auto simp: greaterThan_def[symmetric])
+  also have "emeasure lborel {x :: real \<in> space lborel. 0 < x} = 0"
+  proof -
+    have "AE x in lborel. 0 \<le> exponential_density l x"
+      using assms by (auto simp: distributed_real_AE)
+    then have "AE x in lborel. x \<le> (0::real)"
+      apply eventually_elim 
+      using `l < 0`
+      apply (auto simp: exponential_density_def zero_le_mult_iff split: split_if_asm)
+      done
+    then show "emeasure lborel {x :: real \<in> space lborel. 0 < x} = 0"
+      by (subst (asm) AE_iff_measurable[OF _ refl]) (auto simp: not_le greaterThan_def[symmetric])
+  qed
+  finally show "0 < l" by simp
+next
+  assume "l = 0"
+  then have [simp]: "\<And>x. ereal (exponential_density l x) = 0"
+    by (simp add: exponential_density_def)
+  interpret X: prob_space "distr M lborel X"
+    using distributed_measurable[OF D] by (rule prob_space_distr)
+  from X.emeasure_space_1
+  show "0 < l"
+    by (simp add: emeasure_density distributed_distr_eq_density[OF D])
+qed assumption
+
+lemma
+  assumes [arith]: "0 < l"
+  shows emeasure_exponential_density_le0: "0 \<le> a \<Longrightarrow> emeasure (density lborel (exponential_density l)) {.. a} = 1 - exp (- a * l)"
+    and prob_space_exponential_density: "prob_space (density lborel (exponential_density l))"
+      (is "prob_space ?D")
+proof -
+  let ?f = "\<lambda>x. l * exp (- x * l)"
+  let ?F = "\<lambda>x. - exp (- x * l)"
+
+  have deriv: "\<And>x. DERIV ?F x :> ?f x" "\<And>x. 0 \<le> l * exp (- x * l)"
+    by (auto intro!: DERIV_intros simp: zero_le_mult_iff)
+
+  have "emeasure ?D (space ?D) = (\<integral>\<^isup>+ x. ereal (?f x) * indicator {0..} x \<partial>lborel)"
+    by (auto simp: emeasure_density exponential_density_def
+             intro!: positive_integral_cong split: split_indicator)
+  also have "\<dots> = ereal (0 - ?F 0)"
+  proof (rule positive_integral_FTC_atLeast)
+    have "((\<lambda>x. exp (l * x)) ---> 0) at_bot"
+      by (rule filterlim_compose[OF exp_at_bot filterlim_tendsto_pos_mult_at_bot[of _ l]])
+         (simp_all add: tendsto_const filterlim_ident)
+    then show "((\<lambda>x. - exp (- x * l)) ---> 0) at_top"
+      unfolding filterlim_at_top_mirror
+      by (simp add: tendsto_minus_cancel_left[symmetric] ac_simps)
+  qed (insert deriv, auto)
+  also have "\<dots> = 1" by (simp add: one_ereal_def)
+  finally have "emeasure ?D (space ?D) = 1" .
+  then show "prob_space ?D" by rule
+
+  assume "0 \<le> a"
+  have "emeasure ?D {..a} = (\<integral>\<^isup>+x. ereal (?f x) * indicator {0..a} x \<partial>lborel)"
+    by (auto simp add: emeasure_density intro!: positive_integral_cong split: split_indicator)
+       (auto simp: exponential_density_def)
+  also have "(\<integral>\<^isup>+x. ereal (?f x) * indicator {0..a} x \<partial>lborel) = ereal (?F a) - ereal (?F 0)"
+    using `0 \<le> a` deriv by (intro positive_integral_FTC_atLeastAtMost) auto
+  also have "\<dots> = 1 - exp (- a * l)"
+    by simp
+  finally show "emeasure ?D {.. a} = 1 - exp (- a * l)" .
+qed
+
+
+lemma (in prob_space) exponential_distributedD_le:
+  assumes D: "distributed M lborel X (exponential_density l)" and a: "0 \<le> a"
+  shows "\<P>(x in M. X x \<le> a) = 1 - exp (- a * l)"
+proof -
+  have "emeasure M {x \<in> space M. X x \<le> a } = emeasure (distr M lborel X) {.. a}"
+    using distributed_measurable[OF D] 
+    by (subst emeasure_distr) (auto intro!: arg_cong2[where f=emeasure])
+  also have "\<dots> = emeasure (density lborel (exponential_density l)) {.. a}"
+    unfolding distributed_distr_eq_density[OF D] ..
+  also have "\<dots> = 1 - exp (- a * l)"
+    using emeasure_exponential_density_le0[OF exponential_distributed_params[OF D] a] .
+  finally show ?thesis
+    by (auto simp: measure_def)
+qed
+
+lemma (in prob_space) exponential_distributedD_gt:
+  assumes D: "distributed M lborel X (exponential_density l)" and a: "0 \<le> a"
+  shows "\<P>(x in M. a < X x ) = exp (- a * l)"
+proof -
+  have "exp (- a * l) = 1 - \<P>(x in M. X x \<le> a)"
+    unfolding exponential_distributedD_le[OF D a] by simp
+  also have "\<dots> = prob (space M - {x \<in> space M. X x \<le> a })"
+    using distributed_measurable[OF D]
+    by (subst prob_compl) auto
+  also have "\<dots> = \<P>(x in M. a < X x )"
+    by (auto intro!: arg_cong[where f=prob] simp: not_le)
+  finally show ?thesis by simp
+qed
+
+lemma (in prob_space) exponential_distributed_memoryless:
+  assumes D: "distributed M lborel X (exponential_density l)" and a: "0 \<le> a"and t: "0 \<le> t"
+  shows "\<P>(x in M. a + t < X x \<bar> a < X x) = \<P>(x in M. t < X x)"
+proof -
+  have "\<P>(x in M. a + t < X x \<bar> a < X x) = \<P>(x in M. a + t < X x) / \<P>(x in M. a < X x)"
+    using `0 \<le> t` by (auto simp: cond_prob_def intro!: arg_cong[where f=prob] arg_cong2[where f="op /"])
+  also have "\<dots> = exp (- (a + t) * l) / exp (- a * l)"
+    using a t by (simp add: exponential_distributedD_gt[OF D])
+  also have "\<dots> = exp (- t * l)"
+    using exponential_distributed_params[OF D] by (auto simp: field_simps exp_add[symmetric])
+  finally show ?thesis
+    using t by (simp add: exponential_distributedD_gt[OF D])
+qed
+
+lemma exponential_distributedI:
+  assumes X[measurable]: "X \<in> borel_measurable M" and [arith]: "0 < l"
+    and X_distr: "\<And>a. 0 \<le> a \<Longrightarrow> emeasure M {x\<in>space M. X x \<le> a} = 1 - exp (- a * l)"
+  shows "distributed M lborel X (exponential_density l)"
+proof (rule distributedI_borel_atMost)
+  fix a :: real
+
+  { assume "a \<le> 0"  
+    with X have "emeasure M {x\<in>space M. X x \<le> a} \<le> emeasure M {x\<in>space M. X x \<le> 0}"
+      by (intro emeasure_mono) auto
+    then have "emeasure M {x\<in>space M. X x \<le> a} = 0"
+      using X_distr[of 0] by (simp add: one_ereal_def emeasure_le_0_iff) }
+  note eq_0 = this
+
+  have "\<And>x. \<not> 0 \<le> a \<Longrightarrow> ereal (exponential_density l x) * indicator {..a} x = 0"
+    by (simp add: exponential_density_def)
+  then show "(\<integral>\<^isup>+ x. exponential_density l x * indicator {..a} x \<partial>lborel) = ereal (if 0 \<le> a then 1 - exp (- a * l) else 0)"
+    using emeasure_exponential_density_le0[of l a]
+    by (auto simp: emeasure_density times_ereal.simps[symmetric] ereal_indicator
+             simp del: times_ereal.simps ereal_zero_times)
+  show "emeasure M {x\<in>space M. X x \<le> a} = ereal (if 0 \<le> a then 1 - exp (- a * l) else 0)"
+    using X_distr[of a] eq_0 by (auto simp: one_ereal_def)
+  show "AE x in lborel. 0 \<le> exponential_density l x "
+    by (auto simp: exponential_density_def intro!: AE_I2 mult_nonneg_nonneg)
+qed simp_all
+
+lemma (in prob_space) exponential_distributed_iff:
+  "distributed M lborel X (exponential_density l) \<longleftrightarrow>
+    (X \<in> borel_measurable M \<and> 0 < l \<and> (\<forall>a\<ge>0. \<P>(x in M. X x \<le> a) = 1 - exp (- a * l)))"
+  using
+    distributed_measurable[of M lborel X "exponential_density l"]
+    exponential_distributed_params[of X l]
+    emeasure_exponential_density_le0[of l]
+    exponential_distributedD_le[of X l]
+  by (auto intro!: exponential_distributedI simp: one_ereal_def emeasure_eq_measure)
+
+lemma borel_integral_x_exp:
+  "(\<integral>x. x * exp (- x) * indicator {0::real ..} x \<partial>lborel) = 1"
+proof (rule integral_monotone_convergence)
+  let ?f = "\<lambda>i x. x * exp (- x) * indicator {0::real .. i} x"
+  have "eventually (\<lambda>b::real. 0 \<le> b) at_top"
+    by (rule eventually_ge_at_top)
+  then have "eventually (\<lambda>b. 1 - (inverse (exp b) + b / exp b) = integral\<^isup>L lborel (?f b)) at_top"
+  proof eventually_elim
+   fix b :: real assume [simp]: "0 \<le> b"
+    have "(\<integral>x. (exp (-x)) * indicator {0 .. b} x \<partial>lborel) - (integral\<^isup>L lborel (?f b)) = 
+      (\<integral>x. (exp (-x) - x * exp (-x)) * indicator {0 .. b} x \<partial>lborel)"
+      by (subst integral_diff(2)[symmetric])
+         (auto intro!: borel_integrable_atLeastAtMost integral_cong split: split_indicator)
+    also have "\<dots> = b * exp (-b) - 0 * exp (- 0)"
+    proof (rule integral_FTC_atLeastAtMost)
+      fix x assume "0 \<le> x" "x \<le> b"
+      show "DERIV (\<lambda>x. x * exp (-x)) x :> exp (-x) - x * exp (-x)"
+        by (auto intro!: DERIV_intros)
+      show "isCont (\<lambda>x. exp (- x) - x * exp (- x)) x "
+        by (intro isCont_intros isCont_exp')
+    qed fact
+    also have "(\<integral>x. (exp (-x)) * indicator {0 .. b} x \<partial>lborel) = - exp (- b) - - exp (- 0)"
+      by (rule integral_FTC_atLeastAtMost) (auto intro!: DERIV_intros)
+    finally show "1 - (inverse (exp b) + b / exp b) = integral\<^isup>L lborel (?f b)"
+      by (auto simp: field_simps exp_minus inverse_eq_divide)
+  qed
+  then have "((\<lambda>i. integral\<^isup>L lborel (?f i)) ---> 1 - (0 + 0)) at_top"
+  proof (rule Lim_transform_eventually)
+    show "((\<lambda>i. 1 - (inverse (exp i) + i / exp i)) ---> 1 - (0 + 0 :: real)) at_top"
+      using tendsto_power_div_exp_0[of 1]
+      by (intro tendsto_intros tendsto_inverse_0_at_top exp_at_top) simp
+  qed
+  then show "(\<lambda>i. integral\<^isup>L lborel (?f i)) ----> 1"
+    by (intro filterlim_compose[OF _ filterlim_real_sequentially]) simp
+  show "AE x in lborel. mono (\<lambda>n::nat. x * exp (- x) * indicator {0..real n} x)"
+    by (auto simp: mono_def mult_le_0_iff zero_le_mult_iff split: split_indicator)
+  show "\<And>i::nat. integrable lborel (\<lambda>x. x * exp (- x) * indicator {0..real i} x)"
+    by (rule borel_integrable_atLeastAtMost) auto
+  show "AE x in lborel. (\<lambda>i. x * exp (- x) * indicator {0..real i} x) ----> x * exp (- x) * indicator {0..} x"
+    apply (intro AE_I2 Lim_eventually )
+    apply (rule_tac c="natfloor x + 1" in eventually_sequentiallyI)
+    apply (auto simp add: not_le dest!: ge_natfloor_plus_one_imp_gt[simplified] split: split_indicator)
+    done
+qed (auto intro!: borel_measurable_times borel_measurable_exp)
+
+lemma (in prob_space) exponential_distributed_expectation:
+  assumes D: "distributed M lborel X (exponential_density l)"
+  shows "expectation X = 1 / l"
+proof (subst distributed_integral[OF D, of "\<lambda>x. x", symmetric])
+  have "0 < l"
+   using exponential_distributed_params[OF D] .
+  have [simp]: "\<And>x. x * (l * (exp (- (x * l)) * indicator {0..} (x * l))) =
+    x * exponential_density l x"
+    using `0 < l`
+    by (auto split: split_indicator simp: zero_le_mult_iff exponential_density_def)
+  from borel_integral_x_exp `0 < l`
+  show "(\<integral> x. exponential_density l x * x \<partial>lborel) = 1 / l"
+    by (subst (asm) lebesgue_integral_real_affine[of "l" _ 0])
+       (simp_all add: borel_measurable_exp nonzero_eq_divide_eq ac_simps)
+qed simp
+
+subsection {* Uniform distribution *}
+
+lemma uniform_distrI:
+  assumes X: "X \<in> measurable M M'"
+    and A: "A \<in> sets M'" "emeasure M' A \<noteq> \<infinity>" "emeasure M' A \<noteq> 0"
+  assumes distr: "\<And>B. B \<in> sets M' \<Longrightarrow> emeasure M (X -` B \<inter> space M) = emeasure M' (A \<inter> B) / emeasure M' A"
+  shows "distr M M' X = uniform_measure M' A"
+  unfolding uniform_measure_def
+proof (intro measure_eqI)
+  let ?f = "\<lambda>x. indicator A x / emeasure M' A"
+  fix B assume B: "B \<in> sets (distr M M' X)"
+  with X have "emeasure M (X -` B \<inter> space M) = emeasure M' (A \<inter> B) / emeasure M' A"
+    by (simp add: distr[of B] measurable_sets)
+  also have "\<dots> = (1 / emeasure M' A) * emeasure M' (A \<inter> B)"
+     by simp
+  also have "\<dots> = (\<integral>\<^isup>+ x. (1 / emeasure M' A) * indicator (A \<inter> B) x \<partial>M')"
+    using A B
+    by (intro positive_integral_cmult_indicator[symmetric]) (auto intro!: zero_le_divide_ereal)
+  also have "\<dots> = (\<integral>\<^isup>+ x. ?f x * indicator B x \<partial>M')"
+    by (rule positive_integral_cong) (auto split: split_indicator)
+  finally show "emeasure (distr M M' X) B = emeasure (density M' ?f) B"
+    using A B X by (auto simp add: emeasure_distr emeasure_density)
+qed simp
+
+lemma uniform_distrI_borel:
+  fixes A :: "real set"
+  assumes X[measurable]: "X \<in> borel_measurable M" and A: "emeasure lborel A = ereal r" "0 < r"
+    and [measurable]: "A \<in> sets borel"
+  assumes distr: "\<And>a. emeasure M {x\<in>space M. X x \<le> a} = emeasure lborel (A \<inter> {.. a}) / r"
+  shows "distributed M lborel X (\<lambda>x. indicator A x / measure lborel A)"
+proof (rule distributedI_borel_atMost)
+  let ?f = "\<lambda>x. 1 / r * indicator A x"
+  fix a
+  have "emeasure lborel (A \<inter> {..a}) \<le> emeasure lborel A"
+    using A by (intro emeasure_mono) auto
+  also have "\<dots> < \<infinity>"
+    using A by simp
+  finally have fin: "emeasure lborel (A \<inter> {..a}) \<noteq> \<infinity>"
+    by simp
+  from emeasure_eq_ereal_measure[OF this]
+  have fin_eq: "emeasure lborel (A \<inter> {..a}) / r = ereal (measure lborel (A \<inter> {..a}) / r)"
+    using A by simp
+  then show "emeasure M {x\<in>space M. X x \<le> a} = ereal (measure lborel (A \<inter> {..a}) / r)"
+    using distr by simp
+ 
+  have "(\<integral>\<^isup>+ x. ereal (indicator A x / measure lborel A * indicator {..a} x) \<partial>lborel) =
+    (\<integral>\<^isup>+ x. ereal (1 / measure lborel A) * indicator (A \<inter> {..a}) x \<partial>lborel)"
+    by (auto intro!: positive_integral_cong split: split_indicator)
+  also have "\<dots> = ereal (1 / measure lborel A) * emeasure lborel (A \<inter> {..a})"
+    using `A \<in> sets borel`
+    by (intro positive_integral_cmult_indicator) (auto simp: measure_nonneg)
+  also have "\<dots> = ereal (measure lborel (A \<inter> {..a}) / r)"
+    unfolding emeasure_eq_ereal_measure[OF fin] using A by (simp add: measure_def)
+  finally show "(\<integral>\<^isup>+ x. ereal (indicator A x / measure lborel A * indicator {..a} x) \<partial>lborel) =
+    ereal (measure lborel (A \<inter> {..a}) / r)" .
+qed (auto intro!: divide_nonneg_nonneg measure_nonneg)
+
+lemma (in prob_space) uniform_distrI_borel_atLeastAtMost:
+  fixes a b :: real
+  assumes X: "X \<in> borel_measurable M" and "a < b"
+  assumes distr: "\<And>t. a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow> \<P>(x in M. X x \<le> t) = (t - a) / (b - a)"
+  shows "distributed M lborel X (\<lambda>x. indicator {a..b} x / measure lborel {a..b})"
+proof (rule uniform_distrI_borel)
+  fix t
+  have "t < a \<or> (a \<le> t \<and> t \<le> b) \<or> b < t"
+    by auto
+  then show "emeasure M {x\<in>space M. X x \<le> t} = emeasure lborel ({a .. b} \<inter> {..t}) / (b - a)"
+  proof (elim disjE conjE)
+    assume "t < a" 
+    then have "emeasure M {x\<in>space M. X x \<le> t} \<le> emeasure M {x\<in>space M. X x \<le> a}"
+      using X by (auto intro!: emeasure_mono measurable_sets)
+    also have "\<dots> = 0"
+      using distr[of a] `a < b` by (simp add: emeasure_eq_measure)
+    finally have "emeasure M {x\<in>space M. X x \<le> t} = 0"
+      by (simp add: antisym measure_nonneg emeasure_le_0_iff)
+    with `t < a` show ?thesis by simp
+  next
+    assume bnds: "a \<le> t" "t \<le> b"
+    have "{a..b} \<inter> {..t} = {a..t}"
+      using bnds by auto
+    then show ?thesis using `a \<le> t` `a < b`
+      using distr[OF bnds] by (simp add: emeasure_eq_measure)
+  next
+    assume "b < t" 
+    have "1 = emeasure M {x\<in>space M. X x \<le> b}"
+      using distr[of b] `a < b` by (simp add: one_ereal_def emeasure_eq_measure)
+    also have "\<dots> \<le> emeasure M {x\<in>space M. X x \<le> t}"
+      using X `b < t` by (auto intro!: emeasure_mono measurable_sets)
+    finally have "emeasure M {x\<in>space M. X x \<le> t} = 1"
+       by (simp add: antisym emeasure_eq_measure one_ereal_def)
+    with `b < t` `a < b` show ?thesis by (simp add: measure_def one_ereal_def)
+  qed
+qed (insert X `a < b`, auto)
+
+lemma (in prob_space) uniform_distributed_measure:
+  fixes a b :: real
+  assumes D: "distributed M lborel X (\<lambda>x. indicator {a .. b} x / measure lborel {a .. b})"
+  assumes " a \<le> t" "t \<le> b"
+  shows "\<P>(x in M. X x \<le> t) = (t - a) / (b - a)"
+proof -
+  have "emeasure M {x \<in> space M. X x \<le> t} = emeasure (distr M lborel X) {.. t}"
+    using distributed_measurable[OF D]
+    by (subst emeasure_distr) (auto intro!: arg_cong2[where f=emeasure])
+  also have "\<dots> = (\<integral>\<^isup>+x. ereal (1 / (b - a)) * indicator {a .. t} x \<partial>lborel)"
+    using distributed_borel_measurable[OF D] `a \<le> t` `t \<le> b`
+    unfolding distributed_distr_eq_density[OF D]
+    by (subst emeasure_density)
+       (auto intro!: positive_integral_cong simp: measure_def split: split_indicator)
+  also have "\<dots> = ereal (1 / (b - a)) * (t - a)"
+    using `a \<le> t` `t \<le> b`
+    by (subst positive_integral_cmult_indicator) auto
+  finally show ?thesis
+    by (simp add: measure_def)
+qed
+
+lemma (in prob_space) uniform_distributed_bounds:
+  fixes a b :: real
+  assumes D: "distributed M lborel X (\<lambda>x. indicator {a .. b} x / measure lborel {a .. b})"
+  shows "a < b"
+proof (rule ccontr)
+  assume "\<not> a < b"
+  then have "{a .. b} = {} \<or> {a .. b} = {a .. a}" by simp
+  with uniform_distributed_params[OF D] show False 
+    by (auto simp: measure_def)
+qed
+
+lemma (in prob_space) uniform_distributed_iff:
+  fixes a b :: real
+  shows "distributed M lborel X (\<lambda>x. indicator {a..b} x / measure lborel {a..b}) \<longleftrightarrow> 
+    (X \<in> borel_measurable M \<and> a < b \<and> (\<forall>t\<in>{a .. b}. \<P>(x in M. X x \<le> t)= (t - a) / (b - a)))"
+  using
+    uniform_distributed_bounds[of X a b]
+    uniform_distributed_measure[of X a b]
+    distributed_measurable[of M lborel X]
+  by (auto intro!: uniform_distrI_borel_atLeastAtMost simp: one_ereal_def emeasure_eq_measure)
+
+lemma (in prob_space) uniform_distributed_expectation:
+  fixes a b :: real
+  assumes D: "distributed M lborel X (\<lambda>x. indicator {a .. b} x / measure lborel {a .. b})"
+  shows "expectation X = (a + b) / 2"
+proof (subst distributed_integral[OF D, of "\<lambda>x. x", symmetric])
+  have "a < b"
+    using uniform_distributed_bounds[OF D] .
+
+  have "(\<integral> x. indicator {a .. b} x / measure lborel {a .. b} * x \<partial>lborel) = 
+    (\<integral> x. (x / measure lborel {a .. b}) * indicator {a .. b} x \<partial>lborel)"
+    by (intro integral_cong) auto
+  also have "(\<integral> x. (x / measure lborel {a .. b}) * indicator {a .. b} x \<partial>lborel) = (a + b) / 2"
+  proof (subst integral_FTC_atLeastAtMost)
+    fix x
+    show "DERIV (\<lambda>x. x ^ 2 / (2 * measure lborel {a..b})) x :> x / measure lborel {a..b}"
+      using uniform_distributed_params[OF D]
+      by (auto intro!: DERIV_intros)
+    show "isCont (\<lambda>x. x / Sigma_Algebra.measure lborel {a..b}) x"
+      using uniform_distributed_params[OF D]
+      by (auto intro!: isCont_divide)
+    have *: "b\<twosuperior> / (2 * measure lborel {a..b}) - a\<twosuperior> / (2 * measure lborel {a..b}) =
+      (b*b - a * a) / (2 * (b - a))"
+      using `a < b`
+      by (auto simp: measure_def power2_eq_square diff_divide_distrib[symmetric])
+    show "b\<twosuperior> / (2 * measure lborel {a..b}) - a\<twosuperior> / (2 * measure lborel {a..b}) = (a + b) / 2"
+      using `a < b`
+      unfolding * square_diff_square_factored by (auto simp: field_simps)
+  qed (insert `a < b`, simp)
+  finally show "(\<integral> x. indicator {a .. b} x / measure lborel {a .. b} * x \<partial>lborel) = (a + b) / 2" .
+qed auto
+
+end
diff -r 7c8ce63a3c00 -r 79858bd9f5ef src/HOL/Probability/Information.thy
--- a/src/HOL/Probability/Information.thy	Fri Dec 07 16:53:35 2012 +0100
+++ b/src/HOL/Probability/Information.thy	Fri Dec 07 17:00:40 2012 +0100
@@ -932,24 +932,6 @@
   finally show ?thesis .
 qed
 
-lemma (in prob_space) uniform_distributed_params:
-  assumes X: "distributed M MX X (\<lambda>x. indicator A x / measure MX A)"
-  shows "A \<in> sets MX" "measure MX A \<noteq> 0"
-proof -
-  interpret X: prob_space "distr M MX X"
-    using distributed_measurable[OF X] by (rule prob_space_distr)
-
-  show "measure MX A \<noteq> 0"
-  proof
-    assume "measure MX A = 0"
-    with X.emeasure_space_1 X.prob_space distributed_distr_eq_density[OF X]
-    show False
-      by (simp add: emeasure_density zero_ereal_def[symmetric])
-  qed
-  with measure_notin_sets[of A MX] show "A \<in> sets MX"
-    by blast
-qed
-
 lemma (in information_space) entropy_uniform:
   assumes X: "distributed M MX X (\<lambda>x. indicator A x / measure MX A)" (is "distributed _ _ _ ?f")
   shows "entropy b MX X = log b (measure MX A)"
diff -r 7c8ce63a3c00 -r 79858bd9f5ef src/HOL/Probability/Lebesgue_Measure.thy
--- a/src/HOL/Probability/Lebesgue_Measure.thy	Fri Dec 07 16:53:35 2012 +0100
+++ b/src/HOL/Probability/Lebesgue_Measure.thy	Fri Dec 07 17:00:40 2012 +0100
@@ -1020,4 +1020,159 @@
   shows "integral\<^isup>L lborel f = \<integral>x. f (p2e x) \<partial>((\<Pi>\<^isub>M i\<in>{..<DIM('a)}. lborel))"
   using f by (simp add: borel_fubini_positiv_integral lebesgue_integral_def)
 
+lemma integrable_on_borel_integrable:
+  fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real"
+  assumes f_borel: "f \<in> borel_measurable borel" and nonneg: "\<And>x. 0 \<le> f x"
+  assumes f: "f integrable_on UNIV" 
+  shows "integrable lborel f"
+proof -
+  have "(\<integral>\<^isup>+ x. ereal (f x) \<partial>lborel) \<noteq> \<infinity>" 
+    using has_integral_iff_positive_integral_lborel[OF f_borel nonneg] f
+    by (auto simp: integrable_on_def)
+  moreover have "(\<integral>\<^isup>+ x. ereal (- f x) \<partial>lborel) = 0"
+    using f_borel nonneg by (subst positive_integral_0_iff_AE) auto
+  ultimately show ?thesis
+    using f_borel by (auto simp: integrable_def)
+qed
+
+subsection {* Fundamental Theorem of Calculus for the Lebesgue integral *}
+
+lemma borel_integrable_atLeastAtMost:
+  fixes a b :: real
+  assumes f: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x"
+  shows "integrable lborel (\<lambda>x. f x * indicator {a .. b} x)" (is "integrable _ ?f")
+proof cases
+  assume "a \<le> b"
+
+  from isCont_Lb_Ub[OF `a \<le> b`, of f] f
+  obtain M L where
+    bounds: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> f x \<le> M" "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> L \<le> f x"
+    by metis
+
+  show ?thesis
+  proof (rule integrable_bound)
+    show "integrable lborel (\<lambda>x. max \<bar>M\<bar> \<bar>L\<bar> * indicator {a..b} x)"
+      by (rule integral_cmul_indicator) simp_all
+    show "AE x in lborel. \<bar>?f x\<bar> \<le> max \<bar>M\<bar> \<bar>L\<bar> * indicator {a..b} x"
+    proof (rule AE_I2)
+      fix x show "\<bar>?f x\<bar> \<le> max \<bar>M\<bar> \<bar>L\<bar> * indicator {a..b} x"
+        using bounds[of x] by (auto split: split_indicator)
+    qed
+
+    let ?g = "\<lambda>x. if x = a then f a else if x = b then f b else if x \<in> {a <..< b} then f x else 0"
+    from f have "continuous_on {a <..< b} f"
+      by (subst continuous_on_eq_continuous_at) (auto simp add: continuous_isCont)
+    then have "?g \<in> borel_measurable borel"
+      using borel_measurable_continuous_on_open[of "{a <..< b }" f "\<lambda>x. x" borel 0]
+      by (auto intro!: measurable_If[where P="\<lambda>x. x = a"] measurable_If[where P="\<lambda>x. x = b"])
+    also have "?g = ?f"
+      using `a \<le> b` by (auto intro!: ext split: split_indicator)
+    finally show "?f \<in> borel_measurable lborel"
+      by simp
+  qed
+qed simp
+
+lemma integral_FTC_atLeastAtMost:
+  fixes a b :: real
+  assumes "a \<le> b"
+    and F: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> DERIV F x :> f x"
+    and f: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x"
+  shows "integral\<^isup>L lborel (\<lambda>x. f x * indicator {a .. b} x) = F b - F a"
+proof -
+  let ?f = "\<lambda>x. f x * indicator {a .. b} x"
+  have "(?f has_integral (\<integral>x. ?f x \<partial>lborel)) UNIV"
+    using borel_integrable_atLeastAtMost[OF f]
+    by (rule borel_integral_has_integral)
+  moreover
+  have "(f has_integral F b - F a) {a .. b}"
+    by (intro fundamental_theorem_of_calculus has_vector_derivative_withinI_DERIV ballI assms) auto
+  then have "(?f has_integral F b - F a) {a .. b}"
+    by (subst has_integral_eq_eq[where g=f]) auto
+  then have "(?f has_integral F b - F a) UNIV"
+    by (intro has_integral_on_superset[where t=UNIV and s="{a..b}"]) auto
+  ultimately show "integral\<^isup>L lborel ?f = F b - F a"
+    by (rule has_integral_unique)
+qed
+
+text {*
+
+For the positive integral we replace continuity with Borel-measurability. 
+
+*}
+
+lemma positive_integral_FTC_atLeastAtMost:
+  assumes f_borel: "f \<in> borel_measurable borel"
+  assumes f: "\<And>x. x \<in> {a..b} \<Longrightarrow> DERIV F x :> f x" "\<And>x. x \<in> {a..b} \<Longrightarrow> 0 \<le> f x" and "a \<le> b"
+  shows "(\<integral>\<^isup>+x. f x * indicator {a .. b} x \<partial>lborel) = F b - F a"
+proof -
+  have i: "(f has_integral F b - F a) {a..b}"
+    by (intro fundamental_theorem_of_calculus ballI has_vector_derivative_withinI_DERIV assms)
+  have i: "((\<lambda>x. f x * indicator {a..b} x) has_integral F b - F a) {a..b}"
+    by (rule has_integral_eq[OF _ i]) auto
+  have i: "((\<lambda>x. f x * indicator {a..b} x) has_integral F b - F a) UNIV"
+    by (rule has_integral_on_superset[OF _ _ i]) auto
+  then have "(\<integral>\<^isup>+x. ereal (f x * indicator {a .. b} x) \<partial>lborel) = F b - F a"
+    using f f_borel
+    by (subst has_integral_iff_positive_integral_lborel[symmetric]) (auto split: split_indicator)
+  also have "(\<integral>\<^isup>+x. ereal (f x * indicator {a .. b} x) \<partial>lborel) = (\<integral>\<^isup>+x. ereal (f x) * indicator {a .. b} x \<partial>lborel)"
+    by (auto intro!: positive_integral_cong simp: indicator_def)
+  finally show ?thesis by simp
+qed
+
+lemma positive_integral_FTC_atLeast:
+  fixes f :: "real \<Rightarrow> real"
+  assumes f_borel: "f \<in> borel_measurable borel"
+  assumes f: "\<And>x. a \<le> x \<Longrightarrow> DERIV F x :> f x" 
+  assumes nonneg: "\<And>x. a \<le> x \<Longrightarrow> 0 \<le> f x"
+  assumes lim: "(F ---> T) at_top"
+  shows "(\<integral>\<^isup>+x. ereal (f x) * indicator {a ..} x \<partial>lborel) = T - F a"
+proof -
+  let ?f = "\<lambda>(i::nat) (x::real). ereal (f x) * indicator {a..a + real i} x"
+  let ?fR = "\<lambda>x. ereal (f x) * indicator {a ..} x"
+  have "\<And>x. (SUP i::nat. ?f i x) = ?fR x"
+  proof (rule SUP_Lim_ereal)
+    show "\<And>x. incseq (\<lambda>i. ?f i x)"
+      using nonneg by (auto simp: incseq_def le_fun_def split: split_indicator)
+
+    fix x
+    from reals_Archimedean2[of "x - a"] guess n ..
+    then have "eventually (\<lambda>n. ?f n x = ?fR x) sequentially"
+      by (auto intro!: eventually_sequentiallyI[where c=n] split: split_indicator)
+    then show "(\<lambda>n. ?f n x) ----> ?fR x"
+      by (rule Lim_eventually)
+  qed
+  then have "integral\<^isup>P lborel ?fR = (\<integral>\<^isup>+ x. (SUP i::nat. ?f i x) \<partial>lborel)"
+    by simp
+  also have "\<dots> = (SUP i::nat. (\<integral>\<^isup>+ x. ?f i x \<partial>lborel))"
+  proof (rule positive_integral_monotone_convergence_SUP)
+    show "incseq ?f"
+      using nonneg by (auto simp: incseq_def le_fun_def split: split_indicator)
+    show "\<And>i. (?f i) \<in> borel_measurable lborel"
+      using f_borel by auto
+    show "\<And>i x. 0 \<le> ?f i x"
+      using nonneg by (auto split: split_indicator)
+  qed
+  also have "\<dots> = (SUP i::nat. F (a + real i) - F a)"
+    by (subst positive_integral_FTC_atLeastAtMost[OF f_borel f nonneg]) auto
+  also have "\<dots> = T - F a"
+  proof (rule SUP_Lim_ereal)
+    show "incseq (\<lambda>n. ereal (F (a + real n) - F a))"
+    proof (simp add: incseq_def, safe)
+      fix m n :: nat assume "m \<le> n"
+      with f nonneg show "F (a + real m) \<le> F (a + real n)"
+        by (intro DERIV_nonneg_imp_nondecreasing[where f=F])
+           (simp, metis add_increasing2 order_refl order_trans real_of_nat_ge_zero)
+    qed 
+    have "(\<lambda>x. F (a + real x)) ----> T"
+      apply (rule filterlim_compose[OF lim filterlim_tendsto_add_at_top])
+      apply (rule LIMSEQ_const_iff[THEN iffD2, OF refl])
+      apply (rule filterlim_real_sequentially)
+      done
+    then show "(\<lambda>n. ereal (F (a + real n) - F a)) ----> ereal (T - F a)"
+      unfolding lim_ereal
+      by (intro tendsto_diff) auto
+  qed
+  finally show ?thesis .
+qed
+
 end
diff -r 7c8ce63a3c00 -r 79858bd9f5ef src/HOL/Probability/Measure_Space.thy
--- a/src/HOL/Probability/Measure_Space.thy	Fri Dec 07 16:53:35 2012 +0100
+++ b/src/HOL/Probability/Measure_Space.thy	Fri Dec 07 17:00:40 2012 +0100
@@ -443,6 +443,12 @@
 
 lemma emeasure_not_MInf[simp]: "emeasure M A \<noteq> - \<infinity>"
   using emeasure_nonneg[of M A] by auto
+
+lemma emeasure_le_0_iff: "emeasure M A \<le> 0 \<longleftrightarrow> emeasure M A = 0"
+  using emeasure_nonneg[of M A] by auto
+
+lemma emeasure_less_0_iff: "emeasure M A < 0 \<longleftrightarrow> False"
+  using emeasure_nonneg[of M A] by auto
   
 lemma emeasure_countably_additive: "countably_additive (sets M) (emeasure M)"
   by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)
diff -r 7c8ce63a3c00 -r 79858bd9f5ef src/HOL/Probability/Probability.thy
--- a/src/HOL/Probability/Probability.thy	Fri Dec 07 16:53:35 2012 +0100
+++ b/src/HOL/Probability/Probability.thy	Fri Dec 07 17:00:40 2012 +0100
@@ -7,6 +7,7 @@
   Projective_Limit
   Independent_Family
   Information
+  Distributions
 begin
 
 end
diff -r 7c8ce63a3c00 -r 79858bd9f5ef src/HOL/Probability/Probability_Measure.thy
--- a/src/HOL/Probability/Probability_Measure.thy	Fri Dec 07 16:53:35 2012 +0100
+++ b/src/HOL/Probability/Probability_Measure.thy	Fri Dec 07 17:00:40 2012 +0100
@@ -843,6 +843,81 @@
     done
 qed
 
+lemma distributedI_real:
+  fixes f :: "'a \<Rightarrow> real"
+  assumes gen: "sets M1 = sigma_sets (space M1) E" and "Int_stable E"
+    and A: "range A \<subseteq> E" "(\<Union>i::nat. A i) = space M1" "\<And>i. emeasure (distr M M1 X) (A i) \<noteq> \<infinity>"
+    and X: "X \<in> measurable M M1"
+    and f: "f \<in> borel_measurable M1" "AE x in M1. 0 \<le> f x"
+    and eq: "\<And>A. A \<in> E \<Longrightarrow> emeasure M (X -` A \<inter> space M) = (\<integral>\<^isup>+ x. f x * indicator A x \<partial>M1)"
+  shows "distributed M M1 X f"
+  unfolding distributed_def
+proof (intro conjI)
+  show "distr M M1 X = density M1 f"
+  proof (rule measure_eqI_generator_eq[where A=A])
+    { fix A assume A: "A \<in> E"
+      then have "A \<in> sigma_sets (space M1) E" by auto
+      then have "A \<in> sets M1"
+        using gen by simp
+      with f A eq[of A] X show "emeasure (distr M M1 X) A = emeasure (density M1 f) A"
+        by (simp add: emeasure_distr emeasure_density borel_measurable_ereal
+                      times_ereal.simps[symmetric] ereal_indicator
+                 del: times_ereal.simps) }
+    note eq_E = this
+    show "Int_stable E" by fact
+    { fix e assume "e \<in> E"
+      then have "e \<in> sigma_sets (space M1) E" by auto
+      then have "e \<in> sets M1" unfolding gen .
+      then have "e \<subseteq> space M1" by (rule sets.sets_into_space) }
+    then show "E \<subseteq> Pow (space M1)" by auto
+    show "sets (distr M M1 X) = sigma_sets (space M1) E"
+      "sets (density M1 (\<lambda>x. ereal (f x))) = sigma_sets (space M1) E"
+      unfolding gen[symmetric] by auto
+  qed fact+
+qed (insert X f, auto)
+
+lemma distributedI_borel_atMost:
+  fixes f :: "real \<Rightarrow> real"
+  assumes [measurable]: "X \<in> borel_measurable M"
+    and [measurable]: "f \<in> borel_measurable borel" and f[simp]: "AE x in lborel. 0 \<le> f x"
+    and g_eq: "\<And>a. (\<integral>\<^isup>+x. f x * indicator {..a} x \<partial>lborel)  = ereal (g a)"
+    and M_eq: "\<And>a. emeasure M {x\<in>space M. X x \<le> a} = ereal (g a)"
+  shows "distributed M lborel X f"
+proof (rule distributedI_real)
+  show "sets lborel = sigma_sets (space lborel) (range atMost)"
+    by (simp add: borel_eq_atMost)
+  show "Int_stable (range atMost :: real set set)"
+    by (auto simp: Int_stable_def)
+  have vimage_eq: "\<And>a. (X -` {..a} \<inter> space M) = {x\<in>space M. X x \<le> a}" by auto
+  def A \<equiv> "\<lambda>i::nat. {.. real i}"
+  then show "range A \<subseteq> range atMost" "(\<Union>i. A i) = space lborel"
+    "\<And>i. emeasure (distr M lborel X) (A i) \<noteq> \<infinity>"
+    by (auto simp: real_arch_simple emeasure_distr vimage_eq M_eq)
+
+  fix A :: "real set" assume "A \<in> range atMost"
+  then obtain a where A: "A = {..a}" by auto
+  show "emeasure M (X -` A \<inter> space M) = (\<integral>\<^isup>+x. f x * indicator A x \<partial>lborel)"
+    unfolding vimage_eq A M_eq g_eq ..
+qed auto
+
+lemma (in prob_space) uniform_distributed_params:
+  assumes X: "distributed M MX X (\<lambda>x. indicator A x / measure MX A)"
+  shows "A \<in> sets MX" "measure MX A \<noteq> 0"
+proof -
+  interpret X: prob_space "distr M MX X"
+    using distributed_measurable[OF X] by (rule prob_space_distr)
+
+  show "measure MX A \<noteq> 0"
+  proof
+    assume "measure MX A = 0"
+    with X.emeasure_space_1 X.prob_space distributed_distr_eq_density[OF X]
+    show False
+      by (simp add: emeasure_density zero_ereal_def[symmetric])
+  qed
+  with measure_notin_sets[of A MX] show "A \<in> sets MX"
+    by blast
+qed
+
 lemma prob_space_uniform_measure:
   assumes A: "emeasure M A \<noteq> 0" "emeasure M A \<noteq> \<infinity>"
   shows "prob_space (uniform_measure M A)"
diff -r 7c8ce63a3c00 -r 79858bd9f5ef src/HOL/Relation.thy
--- a/src/HOL/Relation.thy	Fri Dec 07 16:53:35 2012 +0100
+++ b/src/HOL/Relation.thy	Fri Dec 07 17:00:40 2012 +0100
@@ -919,7 +919,7 @@
 
 subsubsection {* Image of a set under a relation *}
 
-definition Image :: "('a \<times> 'b) set \<Rightarrow> 'a set \<Rightarrow> 'b set" (infixl "``" 90)
+definition Image :: "('a \<times> 'b) set \<Rightarrow> 'a set \<Rightarrow> 'b set" (infixr "``" 90)
 where
   "r `` s = {y. \<exists>x\<in>s. (x, y) \<in> r}"
 
diff -r 7c8ce63a3c00 -r 79858bd9f5ef src/HOL/Rings.thy
--- a/src/HOL/Rings.thy	Fri Dec 07 16:53:35 2012 +0100
+++ b/src/HOL/Rings.thy	Fri Dec 07 17:00:40 2012 +0100
@@ -95,7 +95,7 @@
 class dvd = times
 begin
 
-definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "dvd" 50) where
+definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "dvd" 50) where
   "b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)"
 
 lemma dvdI [intro?]: "a = b * k \<Longrightarrow> b dvd a"
diff -r 7c8ce63a3c00 -r 79858bd9f5ef src/HOL/Set_Interval.thy
--- a/src/HOL/Set_Interval.thy	Fri Dec 07 16:53:35 2012 +0100
+++ b/src/HOL/Set_Interval.thy	Fri Dec 07 17:00:40 2012 +0100
@@ -351,6 +351,9 @@
 lemma Int_greaterThanLessThan[simp]: "{a<..<b} Int {c<..<d} = {max a c <..< min b d}"
 by auto
 
+lemma Int_atMost[simp]: "{..a} \<inter> {..b} = {.. min a b}"
+  by (auto simp: min_def)
+
 end