--- a/Admin/components/components.sha1 Fri Aug 04 18:03:50 2017 +0200
+++ b/Admin/components/components.sha1 Mon Aug 07 15:10:37 2017 +0200
@@ -34,6 +34,7 @@
e1919e72416cbd7ac8de5455caba8901acc7b44d e-1.6-2.tar.gz
b98a98025d1f7e560ca6864a53296137dae736b4 e-1.6.tar.gz
c11b25c919e2ec44fe2b6ac2086337b456344e97 e-1.8.tar.gz
+6b962a6b4539b7ca4199977973c61a8c98a492e8 e-2.0.tar.gz
6d34b18ca0aa1e10bab6413045d079188c0e2dfb exec_process-1.0.1.tar.gz
8b9bffd10e396d965e815418295f2ee2849bea75 exec_process-1.0.2.tar.gz
e6aada354da11e533af2dee3dcdd96c06479b053 exec_process-1.0.3.tar.gz
--- a/Admin/components/main Fri Aug 04 18:03:50 2017 +0200
+++ b/Admin/components/main Mon Aug 07 15:10:37 2017 +0200
@@ -2,7 +2,7 @@
bash_process-1.2.1
csdp-6.x
cvc4-1.5
-e-1.8
+e-2.0
isabelle_fonts-20160830
jdk-8u131
jedit_build-20170319
--- a/NEWS Fri Aug 04 18:03:50 2017 +0200
+++ b/NEWS Mon Aug 07 15:10:37 2017 +0200
@@ -116,6 +116,11 @@
*** HOL ***
+* Command and antiquotation "value" with modified default strategy:
+terms without free variables are always evaluated using plain evaluation
+only, with no fallback on normalization by evaluation.
+Minor INCOMPATIBILITY.
+
* Notions of squarefreeness, n-th powers, and prime powers in
HOL-Computational_Algebra and HOL-Number_Theory.
--- a/src/Doc/Sledgehammer/document/root.tex Fri Aug 04 18:03:50 2017 +0200
+++ b/src/Doc/Sledgehammer/document/root.tex Mon Aug 07 15:10:37 2017 +0200
@@ -197,8 +197,8 @@
for Alt-Ergo, set the
environment variable \texttt{WHY3\_HOME} to the directory that contains the
\texttt{why3} executable.
-Sledgehammer has been tested with AgsyHOL 1.0, Alt-Ergo 0.95.2, E 1.6 to 1.8,
-LEO-II 1.3.4, Satallax 2.2 to 2.7, SPASS 3.8ds, and Vampire 0.6 to 3.0.%
+Sledgehammer has been tested with AgsyHOL 1.0, Alt-Ergo 0.95.2, E 1.6 to 2.0,
+LEO-II 1.3.4, Satallax 2.2 to 2.7, SPASS 3.8ds, and Vampire 0.6 to 4.0.%
\footnote{Following the rewrite of Vampire, the counter for version numbers was
reset to 0; hence the (new) Vampire versions 0.6, 1.0, 1.8, 2.6, and 3.0 are more
recent than 9.0 or 11.5.}%
@@ -206,7 +206,7 @@
versions might not work well with Sledgehammer. Ideally,
you should also set \texttt{E\_VERSION}, \texttt{LEO2\_VERSION},
\texttt{SATALLAX\_VERSION}, \texttt{SPASS\_VERSION}, or
-\texttt{VAMPIRE\_VERSION} to the prover's version number (e.g., ``3.0'').
+\texttt{VAMPIRE\_VERSION} to the prover's version number (e.g., ``4.0'').
Similarly, if you want to install CVC3, CVC4, veriT, or Z3, set the environment
variable \texttt{CVC3\_\allowbreak SOLVER}, \texttt{CVC4\_\allowbreak SOLVER},
--- a/src/HOL/ATP.thy Fri Aug 04 18:03:50 2017 +0200
+++ b/src/HOL/ATP.thy Mon Aug 07 15:10:37 2017 +0200
@@ -6,7 +6,7 @@
section \<open>Automatic Theorem Provers (ATPs)\<close>
theory ATP
-imports Meson
+ imports Meson
begin
subsection \<open>ATP problems and proofs\<close>
--- a/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy Fri Aug 04 18:03:50 2017 +0200
+++ b/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy Mon Aug 07 15:10:37 2017 +0200
@@ -1,30 +1,13 @@
(* Title: HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy
Author: Johannes Hölzl, TU München
Author: Robert Himmelmann, TU München
+ Huge cleanup by LCP
*)
theory Equivalence_Lebesgue_Henstock_Integration
imports Lebesgue_Measure Henstock_Kurzweil_Integration Complete_Measure Set_Integral
begin
-lemma finite_product_dependent: (*FIXME DELETE*)
- assumes "finite s"
- and "\<And>x. x \<in> s \<Longrightarrow> finite (t x)"
- shows "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}"
- using assms
-proof induct
- case (insert x s)
- have *: "{(i, j) |i j. i \<in> insert x s \<and> j \<in> t i} =
- (\<lambda>y. (x,y)) ` (t x) \<union> {(i, j) |i j. i \<in> s \<and> j \<in> t i}" by auto
- show ?case
- unfolding *
- apply (rule finite_UnI)
- using insert
- apply auto
- done
-qed auto
-
-
lemma le_left_mono: "x \<le> y \<Longrightarrow> y \<le> a \<longrightarrow> x \<le> (a::'a::preorder)"
by (auto intro: order_trans)
@@ -90,7 +73,7 @@
obtain d
where "gauge d"
and integral_f: "\<forall>p. p tagged_division_of cbox x y \<and> d fine p \<longrightarrow>
- norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - I) < e"
+ norm ((\<Sum>(x,k) \<in> p. content k *\<^sub>R f x) - I) < e"
using \<open>0<e\<close> f unfolding has_integral by auto
define C where "C X m = X \<inter> {x. ball x (1/Suc m) \<subseteq> d x}" for X m
@@ -194,14 +177,14 @@
then show "\<Union>{k. \<exists>x. (x, k) \<in> ?p} = cbox x y"
using p(1) by auto
qed
- ultimately have I: "norm ((\<Sum>(x, k)\<in>?p. content k *\<^sub>R f x) - I) < e"
+ ultimately have I: "norm ((\<Sum>(x,k) \<in> ?p. content k *\<^sub>R f x) - I) < e"
using integral_f by auto
- have "(\<Sum>(x, k)\<in>?p. content k *\<^sub>R f x) =
- (\<Sum>(x, k)\<in>?T ` (p \<inter> s). content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p - s. content k *\<^sub>R f x)"
+ have "(\<Sum>(x,k) \<in> ?p. content k *\<^sub>R f x) =
+ (\<Sum>(x,k) \<in> ?T ` (p \<inter> s). content k *\<^sub>R f x) + (\<Sum>(x,k) \<in> p - s. content k *\<^sub>R f x)"
using p(1)[THEN tagged_division_ofD(1)]
by (safe intro!: sum.union_inter_neutral) (auto simp: s_def T_def)
- also have "(\<Sum>(x, k)\<in>?T ` (p \<inter> s). content k *\<^sub>R f x) = (\<Sum>(x, k)\<in>p \<inter> s. content k *\<^sub>R f (T X k))"
+ also have "(\<Sum>(x,k) \<in> ?T ` (p \<inter> s). content k *\<^sub>R f x) = (\<Sum>(x,k) \<in> p \<inter> s. content k *\<^sub>R f (T X k))"
proof (subst sum.reindex_nontrivial, safe)
fix x1 x2 k assume 1: "(x1, k) \<in> p" "(x1, k) \<in> s" and 2: "(x2, k) \<in> p" "(x2, k) \<in> s"
and eq: "content k *\<^sub>R f (T X k) \<noteq> 0"
@@ -209,8 +192,8 @@
show "x1 = x2"
by (auto simp: content_eq_0_interior)
qed (use p in \<open>auto intro!: sum.cong\<close>)
- finally have eq: "(\<Sum>(x, k)\<in>?p. content k *\<^sub>R f x) =
- (\<Sum>(x, k)\<in>p \<inter> s. content k *\<^sub>R f (T X k)) + (\<Sum>(x, k)\<in>p - s. content k *\<^sub>R f x)" .
+ finally have eq: "(\<Sum>(x,k) \<in> ?p. content k *\<^sub>R f x) =
+ (\<Sum>(x,k) \<in> p \<inter> s. content k *\<^sub>R f (T X k)) + (\<Sum>(x,k) \<in> p - s. content k *\<^sub>R f x)" .
have in_T: "(x, k) \<in> s \<Longrightarrow> T X k \<in> X" for x k
using in_s[of x k] by (auto simp: C_def)
@@ -224,7 +207,7 @@
have [simp]: "finite p"
using tagged_division_ofD(1)[OF p(1)] .
- have "(M - 3*e) * (b - a) \<le> (\<Sum>(x, k)\<in>p \<inter> s. content k) * (b - a)"
+ have "(M - 3*e) * (b - a) \<le> (\<Sum>(x,k) \<in> p \<inter> s. content k) * (b - a)"
proof (intro mult_right_mono)
have fin: "?\<mu> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C X m = {}}) < \<infinity>" for X
using \<open>?\<mu> E < \<infinity>\<close> by (rule le_less_trans[rotated]) (auto intro!: emeasure_mono \<open>E \<in> sets ?L\<close>)
@@ -287,15 +270,15 @@
finally show "M - 3 * e \<le> (\<Sum>(x, y)\<in>p \<inter> s. content y)"
using \<open>0 < e\<close> by (simp add: split_beta)
qed (use \<open>a < b\<close> in auto)
- also have "\<dots> = (\<Sum>(x, k)\<in>p \<inter> s. content k * (b - a))"
+ also have "\<dots> = (\<Sum>(x,k) \<in> p \<inter> s. content k * (b - a))"
by (simp add: sum_distrib_right split_beta')
- also have "\<dots> \<le> (\<Sum>(x, k)\<in>p \<inter> s. content k * (f (T ?F k) - f (T ?E k)))"
+ also have "\<dots> \<le> (\<Sum>(x,k) \<in> p \<inter> s. content k * (f (T ?F k) - f (T ?E k)))"
using parts(3) by (auto intro!: sum_mono mult_left_mono diff_mono)
- also have "\<dots> = (\<Sum>(x, k)\<in>p \<inter> s. content k * f (T ?F k)) - (\<Sum>(x, k)\<in>p \<inter> s. content k * f (T ?E k))"
+ also have "\<dots> = (\<Sum>(x,k) \<in> p \<inter> s. content k * f (T ?F k)) - (\<Sum>(x,k) \<in> p \<inter> s. content k * f (T ?E k))"
by (auto intro!: sum.cong simp: field_simps sum_subtractf[symmetric])
- also have "\<dots> = (\<Sum>(x, k)\<in>?B. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>?A. content k *\<^sub>R f x)"
+ also have "\<dots> = (\<Sum>(x,k) \<in> ?B. content k *\<^sub>R f x) - (\<Sum>(x,k) \<in> ?A. content k *\<^sub>R f x)"
by (subst (1 2) parts) auto
- also have "\<dots> \<le> norm ((\<Sum>(x, k)\<in>?B. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>?A. content k *\<^sub>R f x))"
+ also have "\<dots> \<le> norm ((\<Sum>(x,k) \<in> ?B. content k *\<^sub>R f x) - (\<Sum>(x,k) \<in> ?A. content k *\<^sub>R f x))"
by auto
also have "\<dots> \<le> e + e"
using parts(1)[of ?E] parts(1)[of ?F] by (intro norm_diff_triangle_le[of _ I]) auto
@@ -574,11 +557,14 @@
shows "integral\<^sup>N lborel f = I"
proof -
from f_borel have "(\<lambda>x. ennreal (f x)) \<in> borel_measurable lborel" by auto
- from borel_measurable_implies_simple_function_sequence'[OF this] guess F . note F = this
+ from borel_measurable_implies_simple_function_sequence'[OF this]
+ obtain F where F: "\<And>i. simple_function lborel (F i)" "incseq F"
+ "\<And>i x. F i x < top" "\<And>x. (SUP i. F i x) = ennreal (f x)"
+ by blast
+ then have [measurable]: "\<And>i. F i \<in> borel_measurable lborel"
+ by (metis borel_measurable_simple_function)
let ?B = "\<lambda>i::nat. box (- (real i *\<^sub>R One)) (real i *\<^sub>R One) :: 'a set"
- note F(1)[THEN borel_measurable_simple_function, measurable]
-
have "0 \<le> I"
using I by (rule has_integral_nonneg) (simp add: nonneg)
@@ -1199,13 +1185,13 @@
assume "0 < e"
have "S \<in> lmeasurable"
using \<open>negligible S\<close> by (simp add: negligible_iff_null_sets fmeasurableI_null_sets)
- have e22: "0 < e / 2 / (2 * B * real DIM('M)) ^ DIM('N)"
+ have e22: "0 < e/2 / (2 * B * real DIM('M)) ^ DIM('N)"
using \<open>0 < e\<close> \<open>0 < B\<close> by (simp add: divide_simps)
obtain T
where "open T" "S \<subseteq> T" "T \<in> lmeasurable"
- and "measure lebesgue T \<le> measure lebesgue S + e / 2 / (2 * B * DIM('M)) ^ DIM('N)"
+ and "measure lebesgue T \<le> measure lebesgue S + e/2 / (2 * B * DIM('M)) ^ DIM('N)"
by (rule lmeasurable_outer_open [OF \<open>S \<in> lmeasurable\<close> e22])
- then have T: "measure lebesgue T \<le> e / 2 / (2 * B * DIM('M)) ^ DIM('N)"
+ then have T: "measure lebesgue T \<le> e/2 / (2 * B * DIM('M)) ^ DIM('N)"
using \<open>negligible S\<close> by (simp add: negligible_iff_null_sets measure_eq_0_null_sets)
have "\<exists>r. 0 < r \<and> r \<le> 1/2 \<and>
(x \<in> S \<longrightarrow> (\<forall>y. norm(y - x) < r
@@ -1286,7 +1272,7 @@
qed
have countbl: "countable (fbx ` \<D>)"
using \<open>countable \<D>\<close> by blast
- have "(\<Sum>k\<in>fbx`\<D>'. measure lebesgue k) \<le> e / 2" if "\<D>' \<subseteq> \<D>" "finite \<D>'" for \<D>'
+ have "(\<Sum>k\<in>fbx`\<D>'. measure lebesgue k) \<le> e/2" if "\<D>' \<subseteq> \<D>" "finite \<D>'" for \<D>'
proof -
have BM_ge0: "0 \<le> B * (DIM('M) * prj1 (vf X - uf X))" if "X \<in> \<D>'" for X
using \<open>0 < B\<close> \<open>\<D>' \<subseteq> \<D>\<close> that vu_pos by fastforce
@@ -1330,7 +1316,7 @@
qed
also have "\<dots> = (2 * B * DIM('M)) ^ DIM('N) * sum (measure lebesgue) \<D>'"
by (simp add: sum_distrib_left)
- also have "\<dots> \<le> e / 2"
+ also have "\<dots> \<le> e/2"
proof -
have div: "\<D>' division_of \<Union>\<D>'"
apply (auto simp: \<open>finite \<D>'\<close> \<open>{} \<notin> \<D>'\<close> division_of_def)
@@ -1363,13 +1349,13 @@
using \<open>0 < B\<close>
apply (simp add: algebra_simps)
done
- also have "\<dots> \<le> e / 2"
+ also have "\<dots> \<le> e/2"
using T \<open>0 < B\<close> by (simp add: field_simps)
finally show ?thesis .
qed
finally show ?thesis .
qed
- then have e2: "sum (measure lebesgue) \<G> \<le> e / 2" if "\<G> \<subseteq> fbx ` \<D>" "finite \<G>" for \<G>
+ then have e2: "sum (measure lebesgue) \<G> \<le> e/2" if "\<G> \<subseteq> fbx ` \<D>" "finite \<G>" for \<G>
by (metis finite_subset_image that)
show "\<exists>W\<in>lmeasurable. f ` S \<subseteq> W \<and> measure lebesgue W < e"
proof (intro bexI conjI)
@@ -1602,7 +1588,7 @@
finally show "(\<Sum>k\<in>d. norm (integral k f)) \<le> integral UNIV (\<lambda>x. norm (f x))" .
qed
-lemma helplemma:
+lemma absdiff_norm_less:
assumes "sum (\<lambda>x. norm (f x - g x)) s < e"
and "finite s"
shows "\<bar>sum (\<lambda>x. norm(f x)) s - sum (\<lambda>x. norm(g x)) s\<bar> < e"
@@ -1613,28 +1599,27 @@
apply (rule norm_triangle_ineq3)
done
-text\<open>FIXME: needs refactoring and use of Sigma\<close>
-lemma bounded_variation_absolutely_integrable_interval:
+proposition bounded_variation_absolutely_integrable_interval:
fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
assumes f: "f integrable_on cbox a b"
- and *: "\<forall>d. d division_of (cbox a b) \<longrightarrow> sum (\<lambda>k. norm(integral k f)) d \<le> B"
+ and *: "\<And>d. d division_of (cbox a b) \<Longrightarrow> sum (\<lambda>K. norm(integral K f)) d \<le> B"
shows "f absolutely_integrable_on cbox a b"
proof -
- let ?f = "\<lambda>d. \<Sum>k\<in>d. norm (integral k f)" and ?D = "{d. d division_of (cbox a b)}"
+ let ?f = "\<lambda>d. \<Sum>K\<in>d. norm (integral K f)" and ?D = "{d. d division_of (cbox a b)}"
have D_1: "?D \<noteq> {}"
by (rule elementary_interval[of a b]) auto
have D_2: "bdd_above (?f`?D)"
by (metis * mem_Collect_eq bdd_aboveI2)
note D = D_1 D_2
let ?S = "SUP x:?D. ?f x"
- show ?thesis
- apply (rule absolutely_integrable_onI [OF f has_integral_integrable])
- apply (subst has_integral[of _ ?S])
- apply safe
- proof goal_cases
- case e: (1 e)
- then have "?S - e / 2 < ?S" by simp
- then obtain d where d: "d division_of (cbox a b)" "?S - e / 2 < (\<Sum>k\<in>d. norm (integral k f))"
+ have *: "\<exists>\<gamma>. gauge \<gamma> \<and>
+ (\<forall>p. p tagged_division_of cbox a b \<and>
+ \<gamma> fine p \<longrightarrow>
+ norm ((\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x)) - ?S) < e)"
+ if e: "e > 0" for e
+ proof -
+ have "?S - e/2 < ?S" using \<open>e > 0\<close> by simp
+ then obtain d where d: "d division_of (cbox a b)" "?S - e/2 < (\<Sum>k\<in>d. norm (integral k f))"
unfolding less_cSUP_iff[OF D] by auto
note d' = division_ofD[OF this(1)]
@@ -1642,126 +1627,118 @@
proof
fix x
have "\<exists>da>0. \<forall>xa\<in>\<Union>{i \<in> d. x \<notin> i}. da \<le> dist x xa"
- apply (rule separate_point_closed)
- apply (rule closed_Union)
- apply (rule finite_subset[OF _ d'(1)])
- using d'(4)
- apply auto
- done
+ proof (rule separate_point_closed)
+ show "closed (\<Union>{i \<in> d. x \<notin> i})"
+ apply (rule closed_Union)
+ apply (simp add: d'(1))
+ using d'(4) apply auto
+ done
+ show "x \<notin> \<Union>{i \<in> d. x \<notin> i}"
+ by auto
+ qed
then show "\<exists>e>0. \<forall>i\<in>d. x \<notin> i \<longrightarrow> ball x e \<inter> i = {}"
by force
qed
- then obtain k where k: "\<And>x. 0 < k x"
- "\<And>i x. \<lbrakk>i \<in> d; x \<notin> i\<rbrakk> \<Longrightarrow> ball x (k x) \<inter> i = {}"
+ then obtain k where k: "\<And>x. 0 < k x" "\<And>i x. \<lbrakk>i \<in> d; x \<notin> i\<rbrakk> \<Longrightarrow> ball x (k x) \<inter> i = {}"
by metis
have "e/2 > 0"
using e by auto
- from henstock_lemma[OF assms(1) this]
- obtain g where g: "gauge g"
- "\<And>p. \<lbrakk>p tagged_partial_division_of cbox a b; g fine p\<rbrakk>
- \<Longrightarrow> (\<Sum>(x, k)\<in>p. norm (content k *\<^sub>R f x - integral k f)) < e / 2"
+ with henstock_lemma[OF f]
+ obtain \<gamma> where g: "gauge \<gamma>"
+ "\<And>p. \<lbrakk>p tagged_partial_division_of cbox a b; \<gamma> fine p\<rbrakk>
+ \<Longrightarrow> (\<Sum>(x,k) \<in> p. norm (content k *\<^sub>R f x - integral k f)) < e/2"
by (metis (no_types, lifting))
- let ?g = "\<lambda>x. g x \<inter> ball x (k x)"
- show ?case
- apply (rule_tac x="?g" in exI)
- apply safe
- proof -
+ let ?g = "\<lambda>x. \<gamma> x \<inter> ball x (k x)"
+ show ?thesis
+ proof (intro exI conjI allI impI)
show "gauge ?g"
- using g(1) k(1)
- unfolding gauge_def
- by auto
+ using g(1) k(1) by (auto simp: gauge_def)
+ next
fix p
- assume "p tagged_division_of (cbox a b)" and "?g fine p"
- note p = this(1) conjunctD2[OF this(2)[unfolded fine_Int]]
+ assume "p tagged_division_of (cbox a b) \<and> ?g fine p"
+ then have p: "p tagged_division_of cbox a b" "\<gamma> fine p" "(\<lambda>x. ball x (k x)) fine p"
+ by (auto simp: fine_Int)
note p' = tagged_division_ofD[OF p(1)]
define p' where "p' = {(x,k) | x k. \<exists>i l. x \<in> i \<and> i \<in> d \<and> (x,l) \<in> p \<and> k = i \<inter> l}"
- have gp': "g fine p'"
- using p(2)
- unfolding p'_def fine_def
- by auto
+ have gp': "\<gamma> fine p'"
+ using p(2) by (auto simp: p'_def fine_def)
have p'': "p' tagged_division_of (cbox a b)"
- apply (rule tagged_division_ofI)
- proof -
+ proof (rule tagged_division_ofI)
show "finite p'"
- apply (rule finite_subset[of _ "(\<lambda>(k,(x,l)). (x,k \<inter> l)) `
- {(k,xl) | k xl. k \<in> d \<and> xl \<in> p}"])
- unfolding p'_def
- defer
- apply (rule finite_imageI,rule finite_product_dependent[OF d'(1) p'(1)])
- apply safe
- unfolding image_iff
- apply (rule_tac x="(i,x,l)" in bexI)
- apply auto
- done
- fix x k
- assume "(x, k) \<in> p'"
- then have "\<exists>i l. x \<in> i \<and> i \<in> d \<and> (x, l) \<in> p \<and> k = i \<inter> l"
+ proof (rule finite_subset)
+ show "p' \<subseteq> (\<lambda>(k, x, l). (x, k \<inter> l)) ` (d \<times> p)"
+ by (force simp: p'_def image_iff)
+ show "finite ((\<lambda>(k, x, l). (x, k \<inter> l)) ` (d \<times> p))"
+ by (simp add: d'(1) p'(1))
+ qed
+ next
+ fix x K
+ assume "(x, K) \<in> p'"
+ then have "\<exists>i l. x \<in> i \<and> i \<in> d \<and> (x, l) \<in> p \<and> K = i \<inter> l"
unfolding p'_def by auto
- then obtain i l where il: "x \<in> i" "i \<in> d" "(x, l) \<in> p" "k = i \<inter> l" by blast
- show "x \<in> k" and "k \<subseteq> cbox a b"
+ then obtain i l where il: "x \<in> i" "i \<in> d" "(x, l) \<in> p" "K = i \<inter> l" by blast
+ show "x \<in> K" and "K \<subseteq> cbox a b"
using p'(2-3)[OF il(3)] il by auto
- show "\<exists>a b. k = cbox a b"
+ show "\<exists>a b. K = cbox a b"
unfolding il using p'(4)[OF il(3)] d'(4)[OF il(2)]
- apply safe
- unfolding Int_interval
- apply auto
- done
+ by (meson Int_interval)
next
- fix x1 k1
- assume "(x1, k1) \<in> p'"
- then have "\<exists>i l. x1 \<in> i \<and> i \<in> d \<and> (x1, l) \<in> p \<and> k1 = i \<inter> l"
+ fix x1 K1
+ assume "(x1, K1) \<in> p'"
+ then have "\<exists>i l. x1 \<in> i \<and> i \<in> d \<and> (x1, l) \<in> p \<and> K1 = i \<inter> l"
unfolding p'_def by auto
- then obtain i1 l1 where il1: "x1 \<in> i1" "i1 \<in> d" "(x1, l1) \<in> p" "k1 = i1 \<inter> l1" by blast
- fix x2 k2
- assume "(x2,k2)\<in>p'"
- then have "\<exists>i l. x2 \<in> i \<and> i \<in> d \<and> (x2, l) \<in> p \<and> k2 = i \<inter> l"
+ then obtain i1 l1 where il1: "x1 \<in> i1" "i1 \<in> d" "(x1, l1) \<in> p" "K1 = i1 \<inter> l1" by blast
+ fix x2 K2
+ assume "(x2,K2) \<in> p'"
+ then have "\<exists>i l. x2 \<in> i \<and> i \<in> d \<and> (x2, l) \<in> p \<and> K2 = i \<inter> l"
unfolding p'_def by auto
- then obtain i2 l2 where il2: "x2 \<in> i2" "i2 \<in> d" "(x2, l2) \<in> p" "k2 = i2 \<inter> l2" by blast
- assume "(x1, k1) \<noteq> (x2, k2)"
+ then obtain i2 l2 where il2: "x2 \<in> i2" "i2 \<in> d" "(x2, l2) \<in> p" "K2 = i2 \<inter> l2" by blast
+ assume "(x1, K1) \<noteq> (x2, K2)"
then have "interior i1 \<inter> interior i2 = {} \<or> interior l1 \<inter> interior l2 = {}"
- using d'(5)[OF il1(2) il2(2)] p'(5)[OF il1(3) il2(3)]
- unfolding il1 il2
- by auto
- then show "interior k1 \<inter> interior k2 = {}"
+ using d'(5)[OF il1(2) il2(2)] p'(5)[OF il1(3) il2(3)] by (auto simp: il1 il2)
+ then show "interior K1 \<inter> interior K2 = {}"
unfolding il1 il2 by auto
next
have *: "\<forall>(x, X) \<in> p'. X \<subseteq> cbox a b"
- unfolding p'_def using d' by auto
- show "\<Union>{k. \<exists>x. (x, k) \<in> p'} = cbox a b"
- apply rule
- apply (rule Union_least)
- unfolding mem_Collect_eq
- apply (erule exE)
- apply (drule *[rule_format])
- apply safe
+ unfolding p'_def using d' by blast
+ have "y \<in> \<Union>{K. \<exists>x. (x, K) \<in> p'}" if y: "y \<in> cbox a b" for y
proof -
- fix y
- assume y: "y \<in> cbox a b"
- then have "\<exists>x l. (x, l) \<in> p \<and> y\<in>l"
- unfolding p'(6)[symmetric] by auto
- then obtain x l where xl: "(x, l) \<in> p" "y \<in> l" by metis
- then have "\<exists>k. k \<in> d \<and> y \<in> k"
+ obtain x l where xl: "(x, l) \<in> p" "y \<in> l"
+ using y unfolding p'(6)[symmetric] by auto
+ obtain i where i: "i \<in> d" "y \<in> i"
using y unfolding d'(6)[symmetric] by auto
- then obtain i where i: "i \<in> d" "y \<in> i" by metis
have "x \<in> i"
using fineD[OF p(3) xl(1)] using k(2) i xl by auto
- then show "y \<in> \<Union>{k. \<exists>x. (x, k) \<in> p'}"
- unfolding p'_def Union_iff
- apply (rule_tac x="i \<inter> l" in bexI)
- using i xl
- apply auto
- done
+ then show ?thesis
+ unfolding p'_def
+ by (rule_tac X="i \<inter> l" in UnionI) (use i xl in auto)
+ qed
+ show "\<Union>{K. \<exists>x. (x, K) \<in> p'} = cbox a b"
+ proof
+ show "\<Union>{k. \<exists>x. (x, k) \<in> p'} \<subseteq> cbox a b"
+ using * by auto
+ next
+ show "cbox a b \<subseteq> \<Union>{k. \<exists>x. (x, k) \<in> p'}"
+ proof
+ fix y
+ assume y: "y \<in> cbox a b"
+ obtain x L where xl: "(x, L) \<in> p" "y \<in> L"
+ using y unfolding p'(6)[symmetric] by auto
+ obtain I where i: "I \<in> d" "y \<in> I"
+ using y unfolding d'(6)[symmetric] by auto
+ have "x \<in> I"
+ using fineD[OF p(3) xl(1)] using k(2) i xl by auto
+ then show "y \<in> \<Union>{k. \<exists>x. (x, k) \<in> p'}"
+ apply (rule_tac X="I \<inter> L" in UnionI)
+ using i xl by (auto simp: p'_def)
+ qed
qed
qed
- then have "(\<Sum>(x, k)\<in>p'. norm (content k *\<^sub>R f x - integral k f)) < e / 2"
+ then have sum_less_e2: "(\<Sum>(x,K) \<in> p'. norm (content K *\<^sub>R f x - integral K f)) < e/2"
using g(2) gp' tagged_division_of_def by blast
- then have **: "\<bar>(\<Sum>(x,k)\<in>p'. norm (content k *\<^sub>R f x)) - (\<Sum>(x,k)\<in>p'. norm (integral k f))\<bar> < e / 2"
- unfolding split_def
- using p''
- by (force intro!: helplemma)
- have p'alt: "p' = {(x,(i \<inter> l)) | x i l. (x,l) \<in> p \<and> i \<in> d \<and> i \<inter> l \<noteq> {}}"
+ have p'alt: "p' = {(x, I \<inter> L) | x I L. (x,L) \<in> p \<and> I \<in> d \<and> I \<inter> L \<noteq> {}}"
proof (safe, goal_cases)
case prems: (2 _ _ x i l)
have "x \<in> i"
@@ -1773,297 +1750,247 @@
apply safe
apply (rule_tac x=x in exI)
apply (rule_tac x="i \<inter> l" in exI)
- apply safe
- using prems
apply auto
done
then show ?case
using prems(3) by auto
next
- fix x k
- assume "(x, k) \<in> p'"
- then have "\<exists>i l. x \<in> i \<and> i \<in> d \<and> (x, l) \<in> p \<and> k = i \<inter> l"
+ fix x K
+ assume "(x, K) \<in> p'"
+ then obtain i l where il: "x \<in> i" "i \<in> d" "(x, l) \<in> p" "K = i \<inter> l"
unfolding p'_def by auto
- then obtain i l where il: "x \<in> i" "i \<in> d" "(x, l) \<in> p" "k = i \<inter> l" by blast
- then show "\<exists>y i l. (x, k) = (y, i \<inter> l) \<and> (y, l) \<in> p \<and> i \<in> d \<and> i \<inter> l \<noteq> {}"
+ then show "\<exists>y i l. (x, K) = (y, i \<inter> l) \<and> (y, l) \<in> p \<and> i \<in> d \<and> i \<inter> l \<noteq> {}"
using p'(2) by fastforce
qed
- have sum_p': "(\<Sum>(x, k)\<in>p'. norm (integral k f)) = (\<Sum>k\<in>snd ` p'. norm (integral k f))"
+ have sum_p': "(\<Sum>(x,K) \<in> p'. norm (integral K f)) = (\<Sum>k\<in>snd ` p'. norm (integral k f))"
apply (subst sum.over_tagged_division_lemma[OF p'',of "\<lambda>k. norm (integral k f)"])
apply (auto intro: integral_null simp: content_eq_0_interior)
done
- note snd_p = division_ofD[OF division_of_tagged_division[OF p(1)]]
+ have snd_p_div: "snd ` p division_of cbox a b"
+ by (rule division_of_tagged_division[OF p(1)])
+ note snd_p = division_ofD[OF snd_p_div]
+ have fin_d_sndp: "finite (d \<times> snd ` p)"
+ by (simp add: d'(1) snd_p(1))
- have *: "\<And>sni sni' sf sf'. \<bar>sf' - sni'\<bar> < e / 2 \<longrightarrow> ?S - e / 2 < sni \<and> sni' \<le> ?S \<and>
- sni \<le> sni' \<and> sf' = sf \<longrightarrow> \<bar>sf - ?S\<bar> < e"
+ have *: "\<And>sni sni' sf sf'. \<lbrakk>\<bar>sf' - sni'\<bar> < e/2; ?S - e/2 < sni; sni' \<le> ?S;
+ sni \<le> sni'; sf' = sf\<rbrakk> \<Longrightarrow> \<bar>sf - ?S\<bar> < e"
by arith
- show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x)) - ?S) < e"
+ show "norm ((\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x)) - ?S) < e"
unfolding real_norm_def
- apply (rule *[rule_format,OF **])
- apply safe
- apply(rule d(2))
- proof goal_cases
- case 1
- show ?case
+ proof (rule *)
+ show "\<bar>(\<Sum>(x,K)\<in>p'. norm (content K *\<^sub>R f x)) - (\<Sum>(x,k)\<in>p'. norm (integral k f))\<bar> < e/2"
+ using p'' sum_less_e2 unfolding split_def by (force intro!: absdiff_norm_less)
+ show "(\<Sum>(x,k) \<in> p'. norm (integral k f)) \<le>?S"
by (auto simp: sum_p' division_of_tagged_division[OF p''] D intro!: cSUP_upper)
- next
- case 2
- have *: "{k \<inter> l | k l. k \<in> d \<and> l \<in> snd ` p} =
- (\<lambda>(k,l). k \<inter> l) ` {(k,l)|k l. k \<in> d \<and> l \<in> snd ` p}"
- by auto
- have "(\<Sum>k\<in>d. norm (integral k f)) \<le> (\<Sum>i\<in>d. \<Sum>l\<in>snd ` p. norm (integral (i \<inter> l) f))"
- proof (rule sum_mono, goal_cases)
- case k: (1 k)
- from d'(4)[OF this] guess u v by (elim exE) note uv=this
- define d' where "d' = {cbox u v \<inter> l |l. l \<in> snd ` p \<and> cbox u v \<inter> l \<noteq> {}}"
- note uvab = d'(2)[OF k[unfolded uv]]
- have "d' division_of cbox u v"
- apply (subst d'_def)
- apply (rule division_inter_1)
- apply (rule division_of_tagged_division[OF p(1)])
- apply (rule uvab)
- done
- then have "norm (integral k f) \<le> sum (\<lambda>k. norm (integral k f)) d'"
- unfolding uv
- apply (subst integral_combine_division_topdown[of _ _ d'])
- apply (rule integrable_on_subcbox[OF assms(1) uvab])
- apply assumption
- apply (rule sum_norm_le)
- apply auto
- done
- also have "\<dots> = (\<Sum>k\<in>{k \<inter> l |l. l \<in> snd ` p}. norm (integral k f))"
- apply (rule sum.mono_neutral_left)
- apply (subst Setcompr_eq_image)
- apply (rule finite_imageI)+
- apply fact
- unfolding d'_def uv
- apply blast
- proof (rule, goal_cases)
- case prems: (1 i)
- then have "i \<in> {cbox u v \<inter> l |l. l \<in> snd ` p}"
- by auto
- from this[unfolded mem_Collect_eq] guess l .. note l=this
- then have "cbox u v \<inter> l = {}"
- using prems by auto
- then show ?case
- using l by auto
- qed
- also have "\<dots> = (\<Sum>l\<in>snd ` p. norm (integral (k \<inter> l) f))"
- unfolding Setcompr_eq_image
- apply (rule sum.reindex_nontrivial [unfolded o_def])
- apply (rule finite_imageI)
- apply (rule p')
- proof goal_cases
- case prems: (1 l y)
- have "interior (k \<inter> l) \<subseteq> interior (l \<inter> y)"
- apply (subst(2) interior_Int)
- by (metis Int_lower2 Int_subset_iff interior_mono prems(4))
- then have *: "interior (k \<inter> l) = {}"
- using snd_p(5)[OF prems(1-3)] by auto
- from d'(4)[OF k] snd_p(4)[OF prems(1)] guess u1 v1 u2 v2 by (elim exE) note uv=this
- show ?case
- using *
- unfolding uv Int_interval content_eq_0_interior[symmetric]
- by auto
- qed
- finally show ?case .
- qed
- also have "\<dots> = (\<Sum>(i,l) \<in> d \<times> (snd ` p). norm (integral (i\<inter>l) f))"
- apply (subst sum_Sigma_product[symmetric])
- apply fact
- using p'(1)
- apply auto
- done
- also have "\<dots> = (\<Sum>x\<in>{(i, l) |i l. i \<in> d \<and> l \<in> snd ` p}. norm (integral (case_prod op \<inter> x) f))"
- by (force simp: split_def Sigma_def intro!: sum.cong)
- also have "\<dots> = (\<Sum>k\<in>{i \<inter> l |i l. i \<in> d \<and> l \<in> snd ` p}. norm (integral k f))"
- unfolding *
- apply (rule sum.reindex_nontrivial [symmetric, unfolded o_def])
- apply (rule finite_product_dependent)
- apply fact
- apply (rule finite_imageI)
- apply (rule p')
- unfolding split_paired_all mem_Collect_eq split_conv o_def
+ show "(\<Sum>k\<in>d. norm (integral k f)) \<le> (\<Sum>(x,k) \<in> p'. norm (integral k f))"
proof -
- note * = division_ofD(4,5)[OF division_of_tagged_division,OF p(1)]
- fix l1 l2 k1 k2
- assume as:
- "(l1, k1) \<noteq> (l2, k2)"
- "l1 \<inter> k1 = l2 \<inter> k2"
- "\<exists>i l. (l1, k1) = (i, l) \<and> i \<in> d \<and> l \<in> snd ` p"
- "\<exists>i l. (l2, k2) = (i, l) \<and> i \<in> d \<and> l \<in> snd ` p"
- then have "l1 \<in> d" and "k1 \<in> snd ` p"
- by auto from d'(4)[OF this(1)] *(1)[OF this(2)]
- guess u1 v1 u2 v2 by (elim exE) note uv=this
- have "l1 \<noteq> l2 \<or> k1 \<noteq> k2"
- using as by auto
- then have "interior k1 \<inter> interior k2 = {} \<or> interior l1 \<inter> interior l2 = {}"
- by (metis Pair_inject \<open>k1 \<in> snd ` p\<close> \<open>l1 \<in> d\<close> as(4) d'(5) snd_p(5))
- moreover have "interior (l1 \<inter> k1) = interior (l2 \<inter> k2)"
- using as(2) by auto
- ultimately have "interior(l1 \<inter> k1) = {}"
- by auto
- then show "norm (integral (l1 \<inter> k1) f) = 0"
- unfolding uv Int_interval
- unfolding content_eq_0_interior[symmetric]
- by auto
- qed
- also have "\<dots> = (\<Sum>(x, k)\<in>p'. norm (integral k f))"
- unfolding sum_p'
- apply (rule sum.mono_neutral_right)
- apply (subst *)
- apply (rule finite_imageI[OF finite_product_dependent])
- apply fact
- apply (rule finite_imageI[OF p'(1)])
- apply safe
- proof goal_cases
- case (2 i ia l a b)
- then have "ia \<inter> b = {}"
- unfolding p'alt image_iff Bex_def not_ex
- apply (erule_tac x="(a, ia \<inter> b)" in allE)
- apply auto
- done
- then show ?case
+ have *: "{k \<inter> l | k l. k \<in> d \<and> l \<in> snd ` p} = (\<lambda>(k,l). k \<inter> l) ` (d \<times> snd ` p)"
by auto
- next
- case (1 x a b)
- then show ?case
- unfolding p'_def
+ have "(\<Sum>K\<in>d. norm (integral K f)) \<le> (\<Sum>i\<in>d. \<Sum>l\<in>snd ` p. norm (integral (i \<inter> l) f))"
+ proof (rule sum_mono)
+ fix K assume k: "K \<in> d"
+ from d'(4)[OF this] obtain u v where uv: "K = cbox u v" by metis
+ define d' where "d' = {cbox u v \<inter> l |l. l \<in> snd ` p \<and> cbox u v \<inter> l \<noteq> {}}"
+ have uvab: "cbox u v \<subseteq> cbox a b"
+ using d(1) k uv by blast
+ have "d' division_of cbox u v"
+ unfolding d'_def by (rule division_inter_1 [OF snd_p_div uvab])
+ moreover then have "norm (\<Sum>i\<in>d'. integral i f) \<le> (\<Sum>k\<in>d'. norm (integral k f))"
+ by (simp add: sum_norm_le)
+ ultimately have "norm (integral K f) \<le> sum (\<lambda>k. norm (integral k f)) d'"
+ apply (subst integral_combine_division_topdown[of _ _ d'])
+ apply (auto simp: uv intro: integrable_on_subcbox[OF assms(1) uvab])
+ done
+ also have "\<dots> = (\<Sum>I\<in>{K \<inter> L |L. L \<in> snd ` p}. norm (integral I f))"
+ proof -
+ have *: "norm (integral I f) = 0"
+ if "I \<in> {cbox u v \<inter> l |l. l \<in> snd ` p}"
+ "I \<notin> {cbox u v \<inter> l |l. l \<in> snd ` p \<and> cbox u v \<inter> l \<noteq> {}}" for I
+ using that by auto
+ show ?thesis
+ apply (rule sum.mono_neutral_left)
+ apply (simp add: snd_p(1))
+ unfolding d'_def uv using * by auto
+ qed
+ also have "\<dots> = (\<Sum>l\<in>snd ` p. norm (integral (K \<inter> l) f))"
+ proof -
+ have *: "norm (integral (K \<inter> l) f) = 0"
+ if "l \<in> snd ` p" "y \<in> snd ` p" "l \<noteq> y" "K \<inter> l = K \<inter> y" for l y
+ proof -
+ have "interior (K \<inter> l) \<subseteq> interior (l \<inter> y)"
+ by (metis Int_lower2 interior_mono le_inf_iff that(4))
+ then have "interior (K \<inter> l) = {}"
+ by (simp add: snd_p(5) that)
+ moreover from d'(4)[OF k] snd_p(4)[OF that(1)]
+ obtain u1 v1 u2 v2
+ where uv: "K = cbox u1 u2" "l = cbox v1 v2" by metis
+ ultimately show ?thesis
+ using that integral_null
+ unfolding uv Int_interval content_eq_0_interior
+ by (metis (mono_tags, lifting) norm_eq_zero)
+ qed
+ show ?thesis
+ unfolding Setcompr_eq_image
+ apply (rule sum.reindex_nontrivial [unfolded o_def])
+ apply (rule finite_imageI)
+ apply (rule p')
+ using * by auto
+ qed
+ finally show "norm (integral K f) \<le> (\<Sum>l\<in>snd ` p. norm (integral (K \<inter> l) f))" .
+ qed
+ also have "\<dots> = (\<Sum>(i,l) \<in> d \<times> snd ` p. norm (integral (i\<inter>l) f))"
+ by (simp add: sum.cartesian_product)
+ also have "\<dots> = (\<Sum>x \<in> d \<times> snd ` p. norm (integral (case_prod op \<inter> x) f))"
+ by (force simp: split_def intro!: sum.cong)
+ also have "\<dots> = (\<Sum>k\<in>{i \<inter> l |i l. i \<in> d \<and> l \<in> snd ` p}. norm (integral k f))"
proof -
- assume "(a, b) \<in> {(x, k) |x k. \<exists>i l. x \<in> i \<and> i \<in> d \<and> (x, l) \<in> p \<and> k = i \<inter> l}"
- then have "\<exists>n N. (a, b) = (n, N) \<and> (\<exists>Na Nb. n \<in> Na \<and> Na \<in> d \<and> (n, Nb) \<in> p \<and> N = Na \<inter> Nb)"
- by force
- then show ?thesis
- by (metis (no_types) image_iff snd_conv)
+ have eq0: " (integral (l1 \<inter> k1) f) = 0"
+ if "l1 \<inter> k1 = l2 \<inter> k2" "(l1, k1) \<noteq> (l2, k2)"
+ "l1 \<in> d" "(j1,k1) \<in> p" "l2 \<in> d" "(j2,k2) \<in> p"
+ for l1 l2 k1 k2 j1 j2
+ proof -
+ obtain u1 v1 u2 v2 where uv: "l1 = cbox u1 u2" "k1 = cbox v1 v2"
+ using \<open>(j1, k1) \<in> p\<close> \<open>l1 \<in> d\<close> d'(4) p'(4) by blast
+ have "l1 \<noteq> l2 \<or> k1 \<noteq> k2"
+ using that by auto
+ then have "interior k1 \<inter> interior k2 = {} \<or> interior l1 \<inter> interior l2 = {}"
+ by (meson d'(5) old.prod.inject p'(5) that(3) that(4) that(5) that(6))
+ moreover have "interior (l1 \<inter> k1) = interior (l2 \<inter> k2)"
+ by (simp add: that(1))
+ ultimately have "interior(l1 \<inter> k1) = {}"
+ by auto
+ then show ?thesis
+ unfolding uv Int_interval content_eq_0_interior[symmetric] by auto
+ qed
+ show ?thesis
+ unfolding *
+ apply (rule sum.reindex_nontrivial [OF fin_d_sndp, symmetric, unfolded o_def])
+ apply clarsimp
+ by (metis eq0 fst_conv snd_conv)
qed
+ also have "\<dots> = (\<Sum>(x,k) \<in> p'. norm (integral k f))"
+ proof -
+ have 0: "integral (ia \<inter> snd (a, b)) f = 0"
+ if "ia \<inter> snd (a, b) \<notin> snd ` p'" "ia \<in> d" "(a, b) \<in> p" for ia a b
+ proof -
+ have "ia \<inter> b = {}"
+ using that unfolding p'alt image_iff Bex_def not_ex
+ apply (erule_tac x="(a, ia \<inter> b)" in allE)
+ apply auto
+ done
+ then show ?thesis by auto
+ qed
+ have 1: "\<exists>i l. snd (a, b) = i \<inter> l \<and> i \<in> d \<and> l \<in> snd ` p" if "(a, b) \<in> p'" for a b
+ using that
+ apply (clarsimp simp: p'_def image_iff)
+ by (metis (no_types, hide_lams) snd_conv)
+ show ?thesis
+ unfolding sum_p'
+ apply (rule sum.mono_neutral_right)
+ apply (metis * finite_imageI[OF fin_d_sndp])
+ using 0 1 by auto
+ qed
+ finally show ?thesis .
qed
- finally show ?case .
- next
- case 3
- let ?S = "{(x, i \<inter> l) |x i l. (x, l) \<in> p \<and> i \<in> d}"
- have *: "?S = (\<lambda>(xl,i). (fst xl, snd xl \<inter> i)) ` (p \<times> d)"
- apply safe
- unfolding image_iff
- apply (rule_tac x="((x,l),i)" in bexI)
- apply auto
- done
- note pdfin = finite_cartesian_product[OF p'(1) d'(1)]
- have "(\<Sum>(x, k)\<in>p'. norm (content k *\<^sub>R f x)) = (\<Sum>(x, k)\<in>?S. \<bar>content k\<bar> * norm (f x))"
- unfolding norm_scaleR
- apply (rule sum.mono_neutral_left)
- apply (subst *)
- apply (rule finite_imageI)
- apply fact
- unfolding p'alt
- apply blast
- apply safe
- apply (rule_tac x=x in exI)
- apply (rule_tac x=i in exI)
- apply (rule_tac x=l in exI)
- apply auto
- done
- also have "\<dots> = (\<Sum>((x,l),i)\<in>p \<times> d. \<bar>content (l \<inter> i)\<bar> * norm (f x))"
- unfolding *
- apply (subst sum.reindex_nontrivial)
- apply fact
- unfolding split_paired_all
- unfolding o_def split_def snd_conv fst_conv mem_Sigma_iff prod.inject
- apply (elim conjE)
+ show "(\<Sum>(x,k) \<in> p'. norm (content k *\<^sub>R f x)) = (\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x))"
proof -
- fix x1 l1 k1 x2 l2 k2
- assume as: "(x1, l1) \<in> p" "(x2, l2) \<in> p" "k1 \<in> d" "k2 \<in> d"
- "x1 = x2" "l1 \<inter> k1 = l2 \<inter> k2" "\<not> ((x1 = x2 \<and> l1 = l2) \<and> k1 = k2)"
- from d'(4)[OF as(3)] p'(4)[OF as(1)] guess u1 v1 u2 v2 by (elim exE) note uv=this
- from as have "l1 \<noteq> l2 \<or> k1 \<noteq> k2"
- by auto
- then have "interior k1 \<inter> interior k2 = {} \<or> interior l1 \<inter> interior l2 = {}"
- apply -
- apply (erule disjE)
- apply (rule disjI2)
- defer
- apply (rule disjI1)
- apply (rule d'(5)[OF as(3-4)])
- apply assumption
- apply (rule p'(5)[OF as(1-2)])
- apply auto
- done
- moreover have "interior (l1 \<inter> k1) = interior (l2 \<inter> k2)"
- unfolding as ..
- ultimately have "interior (l1 \<inter> k1) = {}"
- by auto
- then show "\<bar>content (l1 \<inter> k1)\<bar> * norm (f x1) = 0"
- unfolding uv Int_interval
- unfolding content_eq_0_interior[symmetric]
- by auto
- qed safe
- also have "\<dots> = (\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x))"
- apply (subst sum_Sigma_product[symmetric])
- apply (rule p')
- apply (rule d')
- apply (rule sum.cong)
- apply (rule refl)
- unfolding split_paired_all split_conv
- proof -
- fix x l
- assume as: "(x, l) \<in> p"
- note xl = p'(2-4)[OF this]
- from this(3) guess u v by (elim exE) note uv=this
- have "(\<Sum>i\<in>d. \<bar>content (l \<inter> i)\<bar>) = (\<Sum>k\<in>d. content (k \<inter> cbox u v))"
- by (simp add: Int_commute uv)
- also have "\<dots> = sum content {k \<inter> cbox u v| k. k \<in> d}"
- unfolding Setcompr_eq_image
- apply (rule sum.reindex_nontrivial [unfolded o_def, symmetric])
- apply (rule d')
- proof goal_cases
- case prems: (1 k y)
- from d'(4)[OF this(1)] d'(4)[OF this(2)]
- guess u1 v1 u2 v2 by (elim exE) note uv=this
- have "{} = interior ((k \<inter> y) \<inter> cbox u v)"
- apply (subst interior_Int)
- using d'(5)[OF prems(1-3)]
- apply auto
+ let ?S = "{(x, i \<inter> l) |x i l. (x, l) \<in> p \<and> i \<in> d}"
+ have *: "?S = (\<lambda>(xl,i). (fst xl, snd xl \<inter> i)) ` (p \<times> d)"
+ by force
+ have fin_pd: "finite (p \<times> d)"
+ using finite_cartesian_product[OF p'(1) d'(1)] by metis
+ have "(\<Sum>(x,k) \<in> p'. norm (content k *\<^sub>R f x)) = (\<Sum>(x,k) \<in> ?S. \<bar>content k\<bar> * norm (f x))"
+ unfolding norm_scaleR
+ apply (rule sum.mono_neutral_left)
+ apply (subst *)
+ apply (rule finite_imageI [OF fin_pd])
+ unfolding p'alt apply auto
+ by fastforce
+ also have "\<dots> = (\<Sum>((x,l),i)\<in>p \<times> d. \<bar>content (l \<inter> i)\<bar> * norm (f x))"
+ proof -
+ have "\<bar>content (l1 \<inter> k1)\<bar> * norm (f x1) = 0"
+ if "(x1, l1) \<in> p" "(x2, l2) \<in> p" "k1 \<in> d" "k2 \<in> d"
+ "x1 = x2" "l1 \<inter> k1 = l2 \<inter> k2" "x1 \<noteq> x2 \<or> l1 \<noteq> l2 \<or> k1 \<noteq> k2"
+ for x1 l1 k1 x2 l2 k2
+ proof -
+ obtain u1 v1 u2 v2 where uv: "k1 = cbox u1 u2" "l1 = cbox v1 v2"
+ by (meson \<open>(x1, l1) \<in> p\<close> \<open>k1 \<in> d\<close> d(1) division_ofD(4) p'(4))
+ have "l1 \<noteq> l2 \<or> k1 \<noteq> k2"
+ using that by auto
+ then have "interior k1 \<inter> interior k2 = {} \<or> interior l1 \<inter> interior l2 = {}"
+ apply (rule disjE)
+ using that p'(5) d'(5) by auto
+ moreover have "interior (l1 \<inter> k1) = interior (l2 \<inter> k2)"
+ unfolding that ..
+ ultimately have "interior (l1 \<inter> k1) = {}"
+ by auto
+ then show "\<bar>content (l1 \<inter> k1)\<bar> * norm (f x1) = 0"
+ unfolding uv Int_interval content_eq_0_interior[symmetric] by auto
+ qed
+ then show ?thesis
+ unfolding *
+ apply (subst sum.reindex_nontrivial [OF fin_pd])
+ unfolding split_paired_all o_def split_def prod.inject
+ apply force+
done
- also have "\<dots> = interior (y \<inter> (k \<inter> cbox u v))"
- by auto
- also have "\<dots> = interior (k \<inter> cbox u v)"
- unfolding prems(4) by auto
- finally show ?case
- unfolding uv Int_interval content_eq_0_interior ..
qed
- also have "\<dots> = sum content {cbox u v \<inter> k |k. k \<in> d \<and> cbox u v \<inter> k \<noteq> {}}"
- apply (rule sum.mono_neutral_right)
- unfolding Setcompr_eq_image
- apply (rule finite_imageI)
- apply (rule d')
- apply blast
- apply safe
- apply (rule_tac x=k in exI)
- proof goal_cases
- case prems: (1 i k)
- from d'(4)[OF this(1)] guess a b by (elim exE) note ab=this
- have "interior (k \<inter> cbox u v) \<noteq> {}"
- using prems(2)
- unfolding ab Int_interval content_eq_0_interior
- by auto
- then show ?case
- using prems(1)
- using interior_subset[of "k \<inter> cbox u v"]
- by auto
+ also have "\<dots> = (\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x))"
+ proof -
+ have sumeq: "(\<Sum>i\<in>d. content (l \<inter> i) * norm (f x)) = content l * norm (f x)"
+ if "(x, l) \<in> p" for x l
+ proof -
+ note xl = p'(2-4)[OF that]
+ then obtain u v where uv: "l = cbox u v" by blast
+ have "(\<Sum>i\<in>d. \<bar>content (l \<inter> i)\<bar>) = (\<Sum>k\<in>d. content (k \<inter> cbox u v))"
+ by (simp add: Int_commute uv)
+ also have "\<dots> = sum content {k \<inter> cbox u v| k. k \<in> d}"
+ proof -
+ have eq0: "content (k \<inter> cbox u v) = 0"
+ if "k \<in> d" "y \<in> d" "k \<noteq> y" and eq: "k \<inter> cbox u v = y \<inter> cbox u v" for k y
+ proof -
+ from d'(4)[OF that(1)] d'(4)[OF that(2)]
+ obtain \<alpha> \<beta> where \<alpha>: "k \<inter> cbox u v = cbox \<alpha> \<beta>"
+ by (meson Int_interval)
+ have "{} = interior ((k \<inter> y) \<inter> cbox u v)"
+ by (simp add: d'(5) that)
+ also have "\<dots> = interior (y \<inter> (k \<inter> cbox u v))"
+ by auto
+ also have "\<dots> = interior (k \<inter> cbox u v)"
+ unfolding eq by auto
+ finally show ?thesis
+ unfolding \<alpha> content_eq_0_interior ..
+ qed
+ then show ?thesis
+ unfolding Setcompr_eq_image
+ apply (rule sum.reindex_nontrivial [OF \<open>finite d\<close>, unfolded o_def, symmetric])
+ by auto
+ qed
+ also have "\<dots> = sum content {cbox u v \<inter> k |k. k \<in> d \<and> cbox u v \<inter> k \<noteq> {}}"
+ apply (rule sum.mono_neutral_right)
+ unfolding Setcompr_eq_image
+ apply (rule finite_imageI [OF \<open>finite d\<close>])
+ apply (fastforce simp: inf.commute)+
+ done
+ finally show "(\<Sum>i\<in>d. content (l \<inter> i) * norm (f x)) = content l * norm (f x)"
+ unfolding sum_distrib_right[symmetric] real_scaleR_def
+ apply (subst(asm) additive_content_division[OF division_inter_1[OF d(1)]])
+ using xl(2)[unfolded uv] unfolding uv apply auto
+ done
+ qed
+ show ?thesis
+ by (subst sum_Sigma_product[symmetric]) (auto intro!: sumeq sum.cong p' d')
qed
- finally show "(\<Sum>i\<in>d. \<bar>content (l \<inter> i)\<bar> * norm (f x)) = content l *\<^sub>R norm (f x)"
- unfolding sum_distrib_right[symmetric] real_scaleR_def
- apply (subst(asm) additive_content_division[OF division_inter_1[OF d(1)]])
- using xl(2)[unfolded uv]
- unfolding uv
- apply auto
- done
+ finally show ?thesis .
qed
- finally show ?case .
- qed
- qed
+ qed (rule d)
+ qed
qed
+ then show ?thesis
+ using absolutely_integrable_onI [OF f has_integral_integrable] has_integral[of _ ?S]
+ by blast
qed
+
lemma bounded_variation_absolutely_integrable:
fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
assumes "f integrable_on UNIV"
@@ -2169,43 +2096,40 @@
using \<open>e > 0\<close> by auto
from * [OF this] obtain d1 where
d1: "gauge d1" "\<forall>p. p tagged_division_of (cbox a b) \<and> d1 fine p \<longrightarrow>
- norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x)) - integral (cbox a b) (\<lambda>x. norm (f x))) < e / 2"
+ norm ((\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x)) - integral (cbox a b) (\<lambda>x. norm (f x))) < e/2"
by auto
from henstock_lemma [OF f(1) \<open>e/2>0\<close>] obtain d2 where
d2: "gauge d2" "\<forall>p. p tagged_partial_division_of (cbox a b) \<and> d2 fine p \<longrightarrow>
- (\<Sum>(x, k)\<in>p. norm (content k *\<^sub>R f x - integral k f)) < e / 2" .
+ (\<Sum>(x,k) \<in> p. norm (content k *\<^sub>R f x - integral k f)) < e/2" .
obtain p where
p: "p tagged_division_of (cbox a b)" "d1 fine p" "d2 fine p"
by (rule fine_division_exists [OF gauge_Int [OF d1(1) d2(1)], of a b])
(auto simp add: fine_Int)
- have *: "\<And>sf sf' si di. sf' = sf \<longrightarrow> si \<le> ?S \<longrightarrow> \<bar>sf - si\<bar> < e / 2 \<longrightarrow>
- \<bar>sf' - di\<bar> < e / 2 \<longrightarrow> di < ?S + e"
+ have *: "\<And>sf sf' si di. sf' = sf \<longrightarrow> si \<le> ?S \<longrightarrow> \<bar>sf - si\<bar> < e/2 \<longrightarrow>
+ \<bar>sf' - di\<bar> < e/2 \<longrightarrow> di < ?S + e"
by arith
show "integral (cbox a b) (\<lambda>x. if x \<in> UNIV then norm (f x) else 0) < ?S + e"
apply (subst if_P)
apply rule
proof (rule *[rule_format])
- show "\<bar>(\<Sum>(x,k)\<in>p. norm (content k *\<^sub>R f x)) - (\<Sum>(x,k)\<in>p. norm (integral k f))\<bar> < e / 2"
+ show "\<bar>(\<Sum>(x,k)\<in>p. norm (content k *\<^sub>R f x)) - (\<Sum>(x,k)\<in>p. norm (integral k f))\<bar> < e/2"
unfolding split_def
- apply (rule helplemma)
+ apply (rule absdiff_norm_less)
using d2(2)[rule_format,of p]
using p(1,3)
unfolding tagged_division_of_def split_def
apply auto
done
- show "\<bar>(\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x)) - integral (cbox a b) (\<lambda>x. norm(f x))\<bar> < e / 2"
+ show "\<bar>(\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x)) - integral (cbox a b) (\<lambda>x. norm(f x))\<bar> < e/2"
using d1(2)[rule_format,OF conjI[OF p(1,2)]]
by (simp only: real_norm_def)
- show "(\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x)) = (\<Sum>(x, k)\<in>p. norm (content k *\<^sub>R f x))"
+ show "(\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x)) = (\<Sum>(x,k) \<in> p. norm (content k *\<^sub>R f x))"
apply (rule sum.cong)
apply (rule refl)
unfolding split_paired_all split_conv
apply (drule tagged_division_ofD(4)[OF p(1)])
- unfolding norm_scaleR
- apply (subst abs_of_nonneg)
- apply auto
- done
- show "(\<Sum>(x, k)\<in>p. norm (integral k f)) \<le> ?S"
+ by simp
+ show "(\<Sum>(x,k) \<in> p. norm (integral k f)) \<le> ?S"
using partial_division_of_tagged_division[of p "cbox a b"] p(1)
apply (subst sum.over_tagged_division_lemma[OF p(1)])
apply (simp add: content_eq_0_interior)
@@ -2678,11 +2602,12 @@
using f nonneg by (intro DERIV_nonneg_imp_nondecreasing[of x y F]) (auto intro: order_trans)
then have F_le_T: "a \<le> x \<Longrightarrow> F x \<le> T" for x
by (intro tendsto_lowerbound[OF lim])
- (auto simp: trivial_limit_at_top_linorder eventually_at_top_linorder)
+ (auto simp: eventually_at_top_linorder)
have "(SUP i::nat. ?f i x) = ?fR x" for x
proof (rule LIMSEQ_unique[OF LIMSEQ_SUP])
- from reals_Archimedean2[of "x - a"] guess n ..
+ obtain n where "x - a < real n"
+ using reals_Archimedean2[of "x - a"] ..
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) \<longlonglongrightarrow> ?fR x"
--- a/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy Fri Aug 04 18:03:50 2017 +0200
+++ b/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy Mon Aug 07 15:10:37 2017 +0200
@@ -9,6 +9,10 @@
Lebesgue_Measure Tagged_Division
begin
+lemma eps_leI:
+ assumes "(\<And>e::'a::linordered_idom. 0 < e \<Longrightarrow> x < y + e)" shows "x \<le> y"
+ by (metis add_diff_eq assms diff_diff_add diff_gt_0_iff_gt linorder_not_less order_less_irrefl)
+
(*FIXME DELETE*)
lemma conjunctD2: assumes "a \<and> b" shows a b using assms by auto
@@ -1260,7 +1264,7 @@
lemma has_integral_separate_sides:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
- assumes "(f has_integral i) (cbox a b)"
+ assumes f: "(f has_integral i) (cbox a b)"
and "e > 0"
and k: "k \<in> Basis"
obtains d where "gauge d"
@@ -1268,18 +1272,23 @@
p2 tagged_division_of (cbox a b \<inter> {x. x\<bullet>k \<ge> c}) \<and> d fine p2 \<longrightarrow>
norm ((sum (\<lambda>(x,k). content k *\<^sub>R f x) p1 + sum (\<lambda>(x,k). content k *\<^sub>R f x) p2) - i) < e"
proof -
- guess d using has_integralD[OF assms(1-2)] . note d=this
+ obtain \<gamma> where d: "gauge \<gamma>"
+ "\<And>p. \<lbrakk>p tagged_division_of cbox a b; \<gamma> fine p\<rbrakk>
+ \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - i) < e"
+ using has_integralD[OF f \<open>e > 0\<close>] by metis
{ fix p1 p2
- assume "p1 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<le> c}" "d fine p1"
- note p1=tagged_division_ofD[OF this(1)] this
- assume "p2 tagged_division_of (cbox a b) \<inter> {x. c \<le> x \<bullet> k}" "d fine p2"
- note p2=tagged_division_ofD[OF this(1)] this
- note tagged_division_union_interval[OF p1(7) p2(7)] note p12 = tagged_division_ofD[OF this] this
+ assume tdiv1: "p1 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<le> c}" and "\<gamma> fine p1"
+ note p1=tagged_division_ofD[OF this(1)]
+ assume tdiv2: "p2 tagged_division_of (cbox a b) \<inter> {x. c \<le> x \<bullet> k}" and "\<gamma> fine p2"
+ note p2=tagged_division_ofD[OF this(1)]
+ note tagged_division_union_interval[OF tdiv1 tdiv2]
+ note p12 = tagged_division_ofD[OF this] this
{ fix a b
assume ab: "(a, b) \<in> p1 \<inter> p2"
have "(a, b) \<in> p1"
using ab by auto
- with p1 obtain u v where uv: "b = cbox u v" by auto
+ obtain u v where uv: "b = cbox u v"
+ using \<open>(a, b) \<in> p1\<close> p1(4) by moura
have "b \<subseteq> {x. x\<bullet>k = c}"
using ab p1(3)[of a b] p2(3)[of a b] by fastforce
moreover
@@ -1288,25 +1297,23 @@
assume "\<not> ?thesis"
then obtain x where x: "x \<in> interior {x::'a. x\<bullet>k = c}"
by auto
- then guess e unfolding mem_interior .. note e=this
+ then obtain \<epsilon> where "0 < \<epsilon>" and \<epsilon>: "ball x \<epsilon> \<subseteq> {x. x \<bullet> k = c}"
+ using mem_interior by metis
have x: "x\<bullet>k = c"
using x interior_subset by fastforce
- have *: "\<And>i. i \<in> Basis \<Longrightarrow> \<bar>(x - (x + (e / 2) *\<^sub>R k)) \<bullet> i\<bar> = (if i = k then e/2 else 0)"
- using e k by (auto simp: inner_simps inner_not_same_Basis)
- have "(\<Sum>i\<in>Basis. \<bar>(x - (x + (e / 2 ) *\<^sub>R k)) \<bullet> i\<bar>) =
- (\<Sum>i\<in>Basis. (if i = k then e / 2 else 0))"
+ have *: "\<And>i. i \<in> Basis \<Longrightarrow> \<bar>(x - (x + (\<epsilon> / 2) *\<^sub>R k)) \<bullet> i\<bar> = (if i = k then \<epsilon>/2 else 0)"
+ using \<open>0 < \<epsilon>\<close> k by (auto simp: inner_simps inner_not_same_Basis)
+ have "(\<Sum>i\<in>Basis. \<bar>(x - (x + (\<epsilon> / 2 ) *\<^sub>R k)) \<bullet> i\<bar>) =
+ (\<Sum>i\<in>Basis. (if i = k then \<epsilon> / 2 else 0))"
using "*" by (blast intro: sum.cong)
- also have "\<dots> < e"
- apply (subst sum.delta)
- using e
- apply auto
- done
- finally have "x + (e/2) *\<^sub>R k \<in> ball x e"
+ also have "\<dots> < \<epsilon>"
+ by (subst sum.delta) (use \<open>0 < \<epsilon>\<close> in auto)
+ finally have "x + (\<epsilon>/2) *\<^sub>R k \<in> ball x \<epsilon>"
unfolding mem_ball dist_norm by(rule le_less_trans[OF norm_le_l1])
- then have "x + (e/2) *\<^sub>R k \<in> {x. x\<bullet>k = c}"
- using e by auto
+ then have "x + (\<epsilon>/2) *\<^sub>R k \<in> {x. x\<bullet>k = c}"
+ using \<epsilon> by auto
then show False
- unfolding mem_Collect_eq using e x k by (auto simp: inner_simps)
+ using \<open>0 < \<epsilon>\<close> x k by (auto simp: inner_simps)
qed
ultimately have "content b = 0"
unfolding uv content_eq_0_interior
@@ -1318,11 +1325,11 @@
norm ((\<Sum>(x, k)\<in>p1 \<union> p2. content k *\<^sub>R f x) - i)"
by (subst sum.union_inter_neutral) (auto simp: p1 p2)
also have "\<dots> < e"
- by (rule k d(2) p12 fine_Un p1 p2)+
+ using d(2) p12 by (simp add: fine_Un k \<open>\<gamma> fine p1\<close> \<open>\<gamma> fine p2\<close>)
finally have "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - i) < e" .
}
then show ?thesis
- by (auto intro: that[of d] d elim: )
+ using d(1) that by auto
qed
lemma integrable_split [intro]:
@@ -2463,53 +2470,39 @@
proof safe
fix a b :: 'b
show "\<exists>g. (\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b"
- if "box a b = {}"
+ if "box a b = {}" for a b
apply (rule_tac x=f in exI)
- using assms that
- apply (auto simp: content_eq_0_interior)
- done
+ using assms that by (auto simp: content_eq_0_interior)
{
- fix c g
- fix k :: 'b
- assume as: "\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e" "g integrable_on cbox a b"
+ fix c g and k :: 'b
+ assume fg: "\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e" and g: "g integrable_on cbox a b"
assume k: "k \<in> Basis"
show "\<exists>g. (\<forall>x\<in>cbox a b \<inter> {x. x \<bullet> k \<le> c}. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c}"
- "\<exists>g. (\<forall>x\<in>cbox a b \<inter> {x. c \<le> x \<bullet> k}. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b \<inter> {x. c \<le> x \<bullet> k}"
- apply (rule_tac[!] x=g in exI)
- using as(1) integrable_split[OF as(2) k]
- apply auto
- done
+ "\<exists>g. (\<forall>x\<in>cbox a b \<inter> {x. c \<le> x \<bullet> k}. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b \<inter> {x. c \<le> x \<bullet> k}"
+ apply (rule_tac[!] x=g in exI)
+ using fg integrable_split[OF g k] by auto
}
- fix c k g1 g2
- assume as: "\<forall>x\<in>cbox a b \<inter> {x. x \<bullet> k \<le> c}. norm (f x - g1 x) \<le> e" "g1 integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c}"
- "\<forall>x\<in>cbox a b \<inter> {x. c \<le> x \<bullet> k}. norm (f x - g2 x) \<le> e" "g2 integrable_on cbox a b \<inter> {x. c \<le> x \<bullet> k}"
- assume k: "k \<in> Basis"
- let ?g = "\<lambda>x. if x\<bullet>k = c then f x else if x\<bullet>k \<le> c then g1 x else g2 x"
show "\<exists>g. (\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b"
- apply (rule_tac x="?g" in exI)
- apply safe
- proof goal_cases
- case (1 x)
- then show ?case
- apply -
- apply (cases "x\<bullet>k=c")
- apply (case_tac "x\<bullet>k < c")
- using as assms
- apply auto
- done
- next
- case 2
- presume "?g integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c}"
- and "?g integrable_on cbox a b \<inter> {x. x \<bullet> k \<ge> c}"
- then guess h1 h2 unfolding integrable_on_def by auto
- from has_integral_split[OF this k] show ?case
- unfolding integrable_on_def by auto
- next
- show "?g integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c}" "?g integrable_on cbox a b \<inter> {x. x \<bullet> k \<ge> c}"
- apply(rule_tac[!] integrable_spike[OF negligible_standard_hyperplane[of k c]])
- using k as(2,4)
- apply auto
- done
+ if fg1: "\<forall>x\<in>cbox a b \<inter> {x. x \<bullet> k \<le> c}. norm (f x - g1 x) \<le> e"
+ and g1: "g1 integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c}"
+ and fg2: "\<forall>x\<in>cbox a b \<inter> {x. c \<le> x \<bullet> k}. norm (f x - g2 x) \<le> e"
+ and g2: "g2 integrable_on cbox a b \<inter> {x. c \<le> x \<bullet> k}"
+ and k: "k \<in> Basis"
+ for c k g1 g2
+ proof -
+ let ?g = "\<lambda>x. if x\<bullet>k = c then f x else if x\<bullet>k \<le> c then g1 x else g2 x"
+ show "\<exists>g. (\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b"
+ proof (intro exI conjI ballI)
+ show "norm (f x - ?g x) \<le> e" if "x \<in> cbox a b" for x
+ by (auto simp: that assms fg1 fg2)
+ show "?g integrable_on cbox a b"
+ proof -
+ have "?g integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c}" "?g integrable_on cbox a b \<inter> {x. x \<bullet> k \<ge> c}"
+ by(rule integrable_spike[OF negligible_standard_hyperplane[of k c]], use k g1 g2 in auto)+
+ with has_integral_split[OF _ _ k] show ?thesis
+ unfolding integrable_on_def by blast
+ qed
+ qed
qed
qed
@@ -2524,18 +2517,16 @@
lemma approximable_on_division:
fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
assumes "0 \<le> e"
- and "d division_of (cbox a b)"
- and "\<forall>i\<in>d. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i"
+ and d: "d division_of (cbox a b)"
+ and f: "\<forall>i\<in>d. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i"
obtains g where "\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e" "g integrable_on cbox a b"
proof -
- note * = comm_monoid_set.operative_division[OF comm_monoid_set_and operative_approximable[OF assms(1)] assms(2)]
- from assms(3) this[unfolded comm_monoid_set_F_and, of f] division_of_finite[OF assms(2)]
- guess g by auto
- then show thesis
- apply -
- apply (rule that[of g])
- apply auto
- done
+ note * = comm_monoid_set.operative_division
+ [OF comm_monoid_set_and operative_approximable[OF \<open>0 \<le> e\<close>] d]
+ have "finite d"
+ by (rule division_of_finite[OF d])
+ with f *[unfolded comm_monoid_set_F_and, of f] that show thesis
+ by auto
qed
lemma integrable_continuous:
@@ -2669,18 +2660,22 @@
lemma fundamental_theorem_of_calculus:
fixes f :: "real \<Rightarrow> 'a::banach"
assumes "a \<le> b"
- and "\<forall>x\<in>{a .. b}. (f has_vector_derivative f' x) (at x within {a .. b})"
+ and vecd: "\<forall>x\<in>{a .. b}. (f has_vector_derivative f' x) (at x within {a .. b})"
shows "(f' has_integral (f b - f a)) {a .. b}"
unfolding has_integral_factor_content box_real[symmetric]
proof safe
fix e :: real
- assume e: "e > 0"
- note assm = assms(2)[unfolded has_vector_derivative_def has_derivative_within_alt]
- have *: "\<And>P Q. \<forall>x\<in>{a .. b}. P x \<and> (\<forall>e>0. \<exists>d>0. Q x e d) \<Longrightarrow> \<forall>x. \<exists>(d::real)>0. x\<in>{a .. b} \<longrightarrow> Q x e d"
- using e by blast
- note this[OF assm,unfolded gauge_existence_lemma]
- from choice[OF this,unfolded Ball_def[symmetric]] guess d ..
- note d=conjunctD2[OF this[rule_format],rule_format]
+ assume "e > 0"
+ then have "\<forall>x. \<exists>d>0.
+ x \<in> {a..b} \<longrightarrow>
+ (\<forall>y\<in>{a..b}.
+ norm (y - x) < d \<longrightarrow> norm (f y - f x - (y - x) *\<^sub>R f' x) \<le> e * norm (y - x))"
+ using vecd[unfolded has_vector_derivative_def has_derivative_within_alt] by blast
+ then obtain d where d: "\<And>x. 0 < d x"
+ "\<And>x y. \<lbrakk>x \<in> {a..b}; y \<in> {a..b}; norm (y - x) < d x\<rbrakk>
+ \<Longrightarrow> norm (f y - f x - (y - x) *\<^sub>R f' x) \<le> e * norm (y - x)"
+ by metis
+
show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f' x) - (f b - f a)) \<le> e * content (cbox a b))"
apply (rule_tac x="\<lambda>x. ball x (d x)" in exI)
@@ -2701,7 +2696,7 @@
fix x k
assume "(x, k) \<in> p"
note xk = tagged_division_ofD(2-4)[OF as(1) this]
- from this(3) guess u v by (elim exE) note k=this
+ then obtain u v where k: "k = cbox u v" by blast
have *: "u \<le> v"
using xk unfolding k by auto
have ball: "\<forall>xa\<in>k. xa \<in> ball x (d x)"
@@ -2907,10 +2902,9 @@
done
}
assume noteq: "{k \<in> s. content k \<noteq> 0} \<noteq> s"
- then obtain k where k: "k \<in> s" "content k = 0"
- by auto
- from s(4)[OF k(1)] guess c d by (elim exE) note k=k this
- from k have "card s > 0"
+ then obtain k c d where k: "k \<in> s" "content k = 0" "k = cbox c d"
+ using s(4) by blast
+ then have "card s > 0"
unfolding card_gt_0_iff using assm(1) by auto
then have card: "card (s - {k}) < card s"
using assm(1) k(1)
@@ -2933,9 +2927,9 @@
fix x
fix e :: real
assume as: "x \<in> k" "e > 0"
- from k(2)[unfolded k content_eq_0] guess i ..
- then have i:"c\<bullet>i = d\<bullet>i" "i\<in>Basis"
- using s(3)[OF k(1),unfolded k] unfolding box_ne_empty by auto
+ obtain i where i: "c\<bullet>i = d\<bullet>i" "i\<in>Basis"
+ using k(2) s(3)[OF k(1)] unfolding box_ne_empty k
+ by (metis dual_order.antisym content_eq_0)
then have xi: "x\<bullet>i = d\<bullet>i"
using as unfolding k mem_box by (metis antisym)
define y where "y = (\<Sum>j\<in>Basis. (if j = i then if c\<bullet>i \<le> (a\<bullet>i + b\<bullet>i) / 2 then c\<bullet>i +
@@ -3114,42 +3108,30 @@
f integrable_on cbox u v"
shows "f integrable_on cbox a b"
proof -
- have "\<forall>x. \<exists>d. x\<in>cbox a b \<longrightarrow> d>0 \<and> (\<forall>u v. x \<in> cbox u v \<and> cbox u v \<subseteq> ball x d \<and> cbox u v \<subseteq> cbox a b \<longrightarrow>
+ have "\<forall>x. \<exists>d>0. x\<in>cbox a b \<longrightarrow> (\<forall>u v. x \<in> cbox u v \<and> cbox u v \<subseteq> ball x d \<and> cbox u v \<subseteq> cbox a b \<longrightarrow>
f integrable_on cbox u v)"
- using assms by auto
- note this[unfolded gauge_existence_lemma]
- from choice[OF this] guess d .. note d=this[rule_format]
- guess p
- apply (rule fine_division_exists[OF gauge_ball_dependent,of d a b])
- using d
+ using assms by (metis zero_less_one)
+ then obtain d where d: "\<And>x. 0 < d x"
+ "\<And>x u v. \<lbrakk>x \<in> cbox a b; x \<in> cbox u v; cbox u v \<subseteq> ball x (d x); cbox u v \<subseteq> cbox a b\<rbrakk>
+ \<Longrightarrow> f integrable_on cbox u v"
+ by metis
+ obtain p where p: "p tagged_division_of cbox a b" "(\<lambda>x. ball x (d x)) fine p"
+ using fine_division_exists[OF gauge_ball_dependent,of d a b] d(1) by blast
+ then have sndp: "snd ` p division_of cbox a b"
+ by (metis division_of_tagged_division)
+ have "f integrable_on k" if "(x, k) \<in> p" for x k
+ using tagged_division_ofD(2-4)[OF p(1) that] fineD[OF p(2) that] d[of x] by auto
+ then show ?thesis
+ unfolding comm_monoid_set.operative_division[OF comm_monoid_set_and operative_integrable sndp, symmetric]
+ comm_monoid_set_F_and
by auto
- note p=this(1-2)
- note division_of_tagged_division[OF this(1)]
- note * = comm_monoid_set.operative_division[OF comm_monoid_set_and operative_integrable, OF this, symmetric, of f]
- show ?thesis
- unfolding * comm_monoid_set_F_and
- apply safe
- unfolding snd_conv
- proof -
- fix x k
- assume "(x, k) \<in> p"
- note tagged_division_ofD(2-4)[OF p(1) this] fineD[OF p(2) this]
- then show "f integrable_on k"
- apply safe
- apply (rule d[THEN conjunct2,rule_format,of x])
- apply (auto intro: order.trans)
- done
- qed
qed
subsection \<open>Second FTC or existence of antiderivative.\<close>
lemma integrable_const[intro]: "(\<lambda>x. c) integrable_on cbox a b"
- unfolding integrable_on_def
- apply rule
- apply (rule has_integral_const)
- done
+ unfolding integrable_on_def by blast
lemma integral_has_vector_derivative_continuous_at:
fixes f :: "real \<Rightarrow> 'a::banach"
@@ -3239,22 +3221,18 @@
assumes "continuous_on {a .. b} f"
obtains g where "\<forall>u\<in>{a .. b}. \<forall>v \<in> {a .. b}. u \<le> v \<longrightarrow> (f has_integral (g v - g u)) {u .. v}"
proof -
- from antiderivative_continuous[OF assms] guess g . note g=this
- show ?thesis
- apply (rule that[of g])
- apply safe
- proof goal_cases
- case prems: (1 u v)
+ obtain g
+ where g: "\<And>x. x \<in> {a..b} \<Longrightarrow> (g has_vector_derivative f x) (at x within {a..b})"
+ using antiderivative_continuous[OF assms] by metis
+ have "(f has_integral g v - g u) {u..v}" if "u \<in> {a..b}" "v \<in> {a..b}" "u \<le> v" for u v
+ proof -
have "\<forall>x\<in>cbox u v. (g has_vector_derivative f x) (at x within cbox u v)"
- apply rule
- apply (rule has_vector_derivative_within_subset)
- apply (rule g[rule_format])
- using prems(1,2)
- apply auto
- done
- then show ?case
- using fundamental_theorem_of_calculus[OF prems(3), of g f] by auto
+ by (meson g has_vector_derivative_within_subset interval_subset_is_interval is_interval_closed_interval subsetCE that(1) that(2))
+ then show ?thesis
+ by (simp add: fundamental_theorem_of_calculus that(3))
qed
+ then show ?thesis
+ using that by blast
qed
@@ -3268,7 +3246,7 @@
and "\<forall>u v. \<exists>w z. g ` cbox u v = cbox w z"
and h: "\<forall>u v. \<exists>w z. h ` cbox u v = cbox w z"
and "\<forall>u v. content(g ` cbox u v) = r * content (cbox u v)"
- and "(f has_integral i) (cbox a b)"
+ and intfi: "(f has_integral i) (cbox a b)"
shows "((\<lambda>x. f(g x)) has_integral (1 / r) *\<^sub>R i) (h ` cbox a b)"
proof -
show ?thesis when *: "cbox a b \<noteq> {} \<Longrightarrow> ?thesis"
@@ -3282,22 +3260,11 @@
unfolding prems assms(8)[unfolded prems has_integral_empty_eq] by auto
qed
assume "cbox a b \<noteq> {}"
- from assms(6)[rule_format,of a b] guess w z by (elim exE) note wz=this
+ obtain w z where wz: "h ` cbox a b = cbox w z"
+ using h by blast
have inj: "inj g" "inj h"
- unfolding inj_on_def
- apply safe
- apply(rule_tac[!] ccontr)
- using assms(2)
- apply(erule_tac x=x in allE)
- using assms(2)
- apply(erule_tac x=y in allE)
- defer
- using assms(3)
- apply (erule_tac x=x in allE)
- using assms(3)
- apply(erule_tac x=y in allE)
- apply auto
- done
+ apply (metis assms(2) injI)
+ by (metis assms(3) injI)
from h obtain ha hb where h_eq: "h ` cbox a b = cbox ha hb" by blast
show ?thesis
unfolding h_eq has_integral
@@ -3305,12 +3272,17 @@
proof safe
fix e :: real
assume e: "e > 0"
- with assms(1) have "e * r > 0" by simp
- from assms(8)[unfolded has_integral,rule_format,OF this] guess d by (elim exE conjE) note d=this[rule_format]
+ with \<open>0 < r\<close> have "e * r > 0" by simp
+ with intfi[unfolded has_integral]
+ obtain d where d: "gauge d"
+ "\<And>p. p tagged_division_of cbox a b \<and> d fine p
+ \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - i) < e * r"
+ by metis
define d' where "d' x = {y. g y \<in> d (g x)}" for x
have d': "\<And>x. d' x = {y. g y \<in> (d (g x))}"
unfolding d'_def ..
- show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of h ` cbox a b \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f (g x)) - (1 / r) *\<^sub>R i) < e)"
+ show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of h ` cbox a b \<and> d fine p
+ \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f (g x)) - (1 / r) *\<^sub>R i) < e)"
proof (rule_tac x=d' in exI, safe)
show "gauge d'"
using d(1)
@@ -3377,16 +3349,10 @@
assume "x \<in> cbox a b"
then have "h x \<in> \<Union>{k. \<exists>x. (x, k) \<in> p}"
using p(6) by auto
- then guess X unfolding Union_iff .. note X=this
- from this(1) guess y unfolding mem_Collect_eq ..
+ then obtain X y where "h x \<in> X" "(y, X) \<in> p" by blast
then show "x \<in> \<Union>{k. \<exists>x. (x, k) \<in> (\<lambda>(x, k). (g x, g ` k)) ` p}"
- apply -
- apply (rule_tac X="g ` X" in UnionI)
- defer
- apply (rule_tac x="h x" in image_eqI)
- using X(2) assms(3)[rule_format,of x]
- apply auto
- done
+ apply (clarsimp simp: )
+ by (metis (no_types, lifting) assms(3) image_eqI pair_imageI)
qed
note ** = d(2)[OF this]
have *: "inj_on (\<lambda>(x, k). (g x, g ` k)) p"
@@ -3626,8 +3592,8 @@
lemma fundamental_theorem_of_calculus_interior:
fixes f :: "real \<Rightarrow> 'a::real_normed_vector"
assumes "a \<le> b"
- and "continuous_on {a .. b} f"
- and "\<forall>x\<in>{a <..< b}. (f has_vector_derivative f'(x)) (at x)"
+ and contf: "continuous_on {a .. b} f"
+ and derf: "\<And>x. x \<in> {a <..< b} \<Longrightarrow> (f has_vector_derivative f'(x)) (at x)"
shows "(f' has_integral (f b - f a)) {a .. b}"
proof -
{
@@ -3655,29 +3621,30 @@
{ presume "\<And>e. e > 0 \<Longrightarrow> ?P e" then show ?thesis unfolding has_integral_factor_content_real by auto }
fix e :: real
assume e: "e > 0"
- note assms(3)[unfolded has_vector_derivative_def has_derivative_at_alt ball_conj_distrib]
- note conjunctD2[OF this]
- note bounded=this(1) and this(2)
- from this(2) have "\<forall>x\<in>box a b. \<exists>d>0. \<forall>y. norm (y - x) < d \<longrightarrow>
- norm (f y - f x - (y - x) *\<^sub>R f' x) \<le> e/2 * norm (y - x)"
- apply -
- apply safe
- apply (erule_tac x=x in ballE)
- apply (erule_tac x="e/2" in allE)
- using e
- apply auto
- done
- note this[unfolded bgauge_existence_lemma]
- from choice[OF this] guess d ..
- note conjunctD2[OF this[rule_format]]
- note d = this[rule_format]
+ note derf_exp = derf[unfolded has_vector_derivative_def has_derivative_at_alt]
+ have bounded: "\<And>x. x \<in> {a<..<b} \<Longrightarrow> bounded_linear (\<lambda>u. u *\<^sub>R f' x)"
+ using derf_exp by simp
+ have "\<forall>x \<in> box a b. \<exists>d>0. \<forall>y. norm (y - x) < d \<longrightarrow> norm (f y - f x - (y - x) *\<^sub>R f' x) \<le> e/2 * norm (y - x)"
+ (is "\<forall>x \<in> box a b. ?Q x")
+ proof
+ fix x assume x: "x \<in> box a b"
+ show "?Q x"
+ apply (rule allE [where x="e/2", OF derf_exp [THEN conjunct2, of x]])
+ using x e by auto
+ qed
+ from this [unfolded bgauge_existence_lemma]
+ obtain d where d: "\<And>x. 0 < d x"
+ "\<And>x y. \<lbrakk>x \<in> box a b; norm (y - x) < d x\<rbrakk>
+ \<Longrightarrow> norm (f y - f x - (y - x) *\<^sub>R f' x) \<le> e / 2 * norm (y - x)"
+ by metis
have "bounded (f ` cbox a b)"
apply (rule compact_imp_bounded compact_continuous_image)+
using compact_cbox assms
apply auto
done
- from this[unfolded bounded_pos] guess B .. note B = this[rule_format]
-
+ from this[unfolded bounded_pos] obtain B
+ where "0 < B" and B: "\<And>x. x \<in> f ` cbox a b \<Longrightarrow> norm x \<le> B"
+ by metis
have "\<exists>da. 0 < da \<and> (\<forall>c. a \<le> c \<and> {a .. c} \<subseteq> {a .. b} \<and> {a .. c} \<subseteq> ball a da \<longrightarrow>
norm (content {a .. c} *\<^sub>R f' a - (f c - f a)) \<le> (e * (b - a)) / 4)"
proof -
@@ -3700,7 +3667,7 @@
apply (auto simp add: field_simps)
done
qed
- then guess l .. note l = conjunctD2[OF this]
+ then obtain l where l: "0 < l" "norm (l *\<^sub>R f' a) \<le> e * (b - a) / 8" by metis
show ?thesis
apply (rule_tac x="min k l" in exI)
apply safe
@@ -3715,24 +3682,28 @@
by (rule norm_triangle_ineq4)
also have "\<dots> \<le> e * (b - a) / 8 + e * (b - a) / 8"
proof (rule add_mono)
- have "\<bar>c - a\<bar> \<le> \<bar>l\<bar>"
- using as' by auto
- then show "norm ((c - a) *\<^sub>R f' a) \<le> e * (b - a) / 8"
- apply -
- apply (rule order_trans[OF _ l(2)])
+ have "norm ((c - a) *\<^sub>R f' a) \<le> norm (l *\<^sub>R f' a)"
unfolding norm_scaleR
apply (rule mult_right_mono)
- apply auto
- done
+ using as' by auto
+ also have "... \<le> e * (b - a) / 8"
+ by (rule l)
+ finally show "norm ((c - a) *\<^sub>R f' a) \<le> e * (b - a) / 8" .
next
- show "norm (f c - f a) \<le> e * (b - a) / 8"
- apply (rule less_imp_le)
- apply (cases "a = c")
- defer
- apply (rule k(2)[unfolded dist_norm])
- using as' e ab
- apply (auto simp add: field_simps)
- done
+ have "norm (f c - f a) < e * (b - a) / 8"
+ proof (cases "a = c")
+ case True
+ then show ?thesis
+ using \<open>0 < e * (b - a) / 8\<close> by auto
+ next
+ case False
+ show ?thesis
+ apply (rule k(2)[unfolded dist_norm])
+ using as' False
+ apply (auto simp add: field_simps)
+ done
+ qed
+ then show "norm (f c - f a) \<le> e * (b - a) / 8" by simp
qed
finally show "norm (content {a .. c} *\<^sub>R f' a - (f c - f a)) \<le> e * (b - a) / 4"
unfolding content_real[OF as(1)] by auto
@@ -3748,20 +3719,23 @@
note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this]
note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0"
using e ab by (auto simp add: field_simps)
- from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
- have "\<exists>l. 0 < l \<and> norm (l *\<^sub>R f' b) \<le> (e * (b - a)) / 8"
+ from *[OF this] obtain k
+ where k: "0 < k"
+ "\<And>x. \<lbrakk>x \<in> {a..b}; 0 < dist x b \<and> dist x b < k\<rbrakk>
+ \<Longrightarrow> dist (f x) (f b) < e * (b - a) / 8"
+ by blast
+ obtain l where l: "0 < l" "norm (l *\<^sub>R f' b) \<le> (e * (b - a)) / 8"
proof (cases "f' b = 0")
case True
- thus ?thesis using ab e by auto
+ thus ?thesis using ab e that by auto
next
case False
then show ?thesis
- apply (rule_tac x="(e * (b - a)) / 8 / norm (f' b)" in exI)
+ apply (rule_tac l="(e * (b - a)) / 8 / norm (f' b)" in that)
using ab e
apply (auto simp add: field_simps)
done
qed
- then guess l .. note l = conjunctD2[OF this]
show ?thesis
apply (rule_tac x="min k l" in exI)
apply safe
@@ -3818,7 +3792,7 @@
note p = tagged_division_ofD[OF as(1)]
have pA: "p = (p \<inter> ?A) \<union> (p - ?A)" "finite (p \<inter> ?A)" "finite (p - ?A)" "(p \<inter> ?A) \<inter> (p - ?A) = {}"
using as by auto
- note * = additive_tagged_division_1'[OF assms(1) as(1), symmetric]
+ note * = additive_tagged_division_1[OF assms(1) as(1), symmetric]
have **: "\<And>n1 s1 n2 s2::real. n2 \<le> s2 / 2 \<Longrightarrow> n1 - s1 \<le> s2 / 2 \<Longrightarrow> n1 + n2 \<le> s1 + s2"
by arith
show ?case
@@ -3842,11 +3816,12 @@
assume xk: "(x, k) \<in> p"
"e * (Sup k - Inf k) / 2 <
norm (content k *\<^sub>R f' x - (f (Sup k) - f (Inf k)))"
- from p(4)[OF this(1)] guess u v by (elim exE) note k=this
+ obtain u v where k: "k = cbox u v"
+ using p(4) xk(1) by blast
then have "u \<le> v" and uv: "{u, v} \<subseteq> cbox u v"
using p(2)[OF xk(1)] by auto
- note result = xk(2)[unfolded k box_real interval_bounds_real[OF this(1)] content_real[OF this(1)]]
-
+ then have result: "e * (v - u) / 2 < norm ((v - u) *\<^sub>R f' x - (f v - f u))"
+ using xk(2)[unfolded k box_real interval_bounds_real content_real] by auto
assume as': "x \<noteq> a" "x \<noteq> b"
then have "x \<in> box a b"
using p(2-3)[OF xk(1)] by (auto simp: mem_box)
@@ -3910,7 +3885,8 @@
assume "(x, k) \<in> p \<inter> {t. fst t \<in> {a, b}} - p \<inter> {t. fst t \<in> {a, b} \<and> content (snd t) \<noteq> 0}"
then have xk: "(x, k) \<in> p" "content k = 0"
by auto
- from p(4)[OF xk(1)] guess u v by (elim exE) note uv=this
+ then obtain u v where uv: "k = cbox u v"
+ using p(4) by blast
have "k \<noteq> {}"
using p(2)[OF xk(1)] by auto
then have *: "u = v"
@@ -3952,9 +3928,10 @@
let ?B = "\<lambda>x. {t \<in> p. fst t = x \<and> content (snd t) \<noteq> 0}"
have pa: "\<exists>v. k = cbox a v \<and> a \<le> v" if "(a, k) \<in> p" for k
proof -
- guess u v using p(4)[OF that] by (elim exE) note uv=this
- have *: "u \<le> v"
- using p(2)[OF that] unfolding uv by auto
+ obtain u v where uv: "k = cbox u v"
+ using \<open>(a, k) \<in> p\<close> p(4) by blast
+ then have *: "u \<le> v"
+ using p(2)[OF that] by auto
have u: "u = a"
proof (rule ccontr)
have "u \<in> cbox u v"
@@ -3975,7 +3952,8 @@
qed
have pb: "\<exists>v. k = cbox v b \<and> b \<ge> v" if "(b, k) \<in> p" for k
proof -
- guess u v using p(4)[OF that] by (elim exE) note uv=this
+ obtain u v where uv: "k = cbox u v"
+ using \<open>(b, k) \<in> p\<close> p(4) by blast
have *: "u \<le> v"
using p(2)[OF that] unfolding uv by auto
have u: "v = b"
@@ -6814,68 +6792,72 @@
and "\<forall>x\<in>s. norm (f x) \<le> g x"
shows "norm (integral s f) \<le> integral s g"
proof -
- have *: "\<And>x y. (\<forall>e::real. 0 < e \<longrightarrow> x < y + e) \<Longrightarrow> x \<le> y"
- apply (rule ccontr)
- apply (erule_tac x="x - y" in allE)
- apply auto
- done
have norm: "norm ig < dia + e"
- if "norm sg \<le> dsa"
- and "\<bar>dsa - dia\<bar> < e / 2"
- and "norm (sg - ig) < e / 2"
+ if "norm sg \<le> dsa" and "\<bar>dsa - dia\<bar> < e / 2" and "norm (sg - ig) < e / 2"
for e dsa dia and sg ig :: 'a
apply (rule le_less_trans[OF norm_triangle_sub[of ig sg]])
apply (subst real_sum_of_halves[of e,symmetric])
unfolding add.assoc[symmetric]
apply (rule add_le_less_mono)
- defer
- apply (subst norm_minus_commute)
- apply (rule that(3))
+ defer
+ apply (subst norm_minus_commute)
+ apply (rule that(3))
apply (rule order_trans[OF that(1)])
using that(2)
apply arith
done
have lem: "norm (integral(cbox a b) f) \<le> integral (cbox a b) g"
- if "f integrable_on cbox a b"
- and "g integrable_on cbox a b"
- and "\<forall>x\<in>cbox a b. norm (f x) \<le> g x"
+ if f: "f integrable_on cbox a b"
+ and g: "g integrable_on cbox a b"
+ and nle: "\<And>x. x \<in> cbox a b \<Longrightarrow> norm (f x) \<le> g x"
for f :: "'n \<Rightarrow> 'a" and g a b
- proof (rule *[rule_format])
+ proof (rule eps_leI)
fix e :: real
assume "e > 0"
- then have *: "e/2 > 0"
+ then have e: "e/2 > 0"
by auto
- from integrable_integral[OF that(1),unfolded has_integral[of f],rule_format,OF *]
- guess d1 .. note d1 = conjunctD2[OF this,rule_format]
- from integrable_integral[OF that(2),unfolded has_integral[of g],rule_format,OF *]
- guess d2 .. note d2 = conjunctD2[OF this,rule_format]
- note gauge_Int[OF d1(1) d2(1)]
- from fine_division_exists[OF this, of a b] guess p . note p=this
+ with integrable_integral[OF f,unfolded has_integral[of f]]
+ obtain \<gamma> where \<gamma>: "gauge \<gamma>"
+ "\<And>p. p tagged_division_of cbox a b \<and> \<gamma> fine p
+ \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - integral (cbox a b) f) < e / 2"
+ by meson
+ moreover
+ from integrable_integral[OF g,unfolded has_integral[of g]] e
+ obtain \<delta> where \<delta>: "gauge \<delta>"
+ "\<And>p. p tagged_division_of cbox a b \<and> \<delta> fine p
+ \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - integral (cbox a b) g) < e / 2"
+ by meson
+ ultimately have "gauge (\<lambda>x. \<gamma> x \<inter> \<delta> x)"
+ using gauge_Int by blast
+ with fine_division_exists obtain p
+ where p: "p tagged_division_of cbox a b" "(\<lambda>x. \<gamma> x \<inter> \<delta> x) fine p"
+ by metis
+ have "\<gamma> fine p" "\<delta> fine p"
+ using fine_Int p(2) by blast+
show "norm (integral (cbox a b) f) < integral (cbox a b) g + e"
- apply (rule norm)
- defer
- apply (rule d2(2)[OF conjI[OF p(1)],unfolded real_norm_def])
- defer
- apply (rule d1(2)[OF conjI[OF p(1)]])
- defer
- apply (rule sum_norm_le)
- proof safe
- fix x k
- assume "(x, k) \<in> p"
- note as = tagged_division_ofD(2-4)[OF p(1) this]
- from this(3) guess u v by (elim exE) note uv=this
- show "norm (content k *\<^sub>R f x) \<le> content k *\<^sub>R g x"
- unfolding uv norm_scaleR
- unfolding abs_of_nonneg[OF content_pos_le] real_scaleR_def
- apply (rule mult_left_mono)
- using that(3) as
- apply auto
- done
- qed (insert p[unfolded fine_Int], auto)
+ proof (rule norm)
+ have "norm (content K *\<^sub>R f x) \<le> content K *\<^sub>R g x" if "(x, K) \<in> p" for x K
+ proof-
+ have K: "x \<in> K" "K \<subseteq> cbox a b"
+ using \<open>(x, K) \<in> p\<close> p(1) by blast+
+ obtain u v where "K = cbox u v"
+ using \<open>(x, K) \<in> p\<close> p(1) by blast
+ moreover have "content K * norm (f x) \<le> content K * g x"
+ by (metis K subsetD dual_order.antisym measure_nonneg mult_zero_left nle not_le real_mult_le_cancel_iff2)
+ then show ?thesis
+ by simp
+ qed
+ then show "norm (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) \<le> (\<Sum>(x, k)\<in>p. content k *\<^sub>R g x)"
+ by (simp add: sum_norm_le split_def)
+ show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - integral (cbox a b) f) < e / 2"
+ using \<open>\<gamma> fine p\<close> \<gamma> p(1) by simp
+ show "\<bar>(\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - integral (cbox a b) g\<bar> < e / 2"
+ using \<open>\<delta> fine p\<close> \<delta> p(1) by simp
+ qed
qed
{ presume "\<And>e. 0 < e \<Longrightarrow> norm (integral s f) < integral s g + e"
- then show ?thesis by (rule *[rule_format]) auto }
+ then show ?thesis by (rule eps_leI) auto }
fix e :: real
assume "e > 0"
then have e: "e/2 > 0"
@@ -6903,7 +6885,6 @@
defer
apply (rule w(2)[unfolded real_norm_def])
apply (rule z(2))
- apply safe
apply (case_tac "x \<in> s")
unfolding if_P
apply (rule assms(3)[rule_format])
@@ -6911,6 +6892,7 @@
done
qed
+
lemma integral_norm_bound_integral_component:
fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
fixes g :: "'n \<Rightarrow> 'b::euclidean_space"
--- a/src/HOL/Analysis/Tagged_Division.thy Fri Aug 04 18:03:50 2017 +0200
+++ b/src/HOL/Analysis/Tagged_Division.thy Mon Aug 07 15:10:37 2017 +0200
@@ -1836,15 +1836,6 @@
lemma bgauge_existence_lemma: "(\<forall>x\<in>s. \<exists>d::real. 0 < d \<and> q d x) \<longleftrightarrow> (\<forall>x. \<exists>d>0. x\<in>s \<longrightarrow> q d x)"
by (meson zero_less_one)
-lemma additive_tagged_division_1':
- fixes f :: "real \<Rightarrow> 'a::real_normed_vector"
- assumes "a \<le> b"
- and "p tagged_division_of {a..b}"
- shows "sum (\<lambda>(x,k). f (Sup k) - f(Inf k)) p = f b - f a"
- using additive_tagged_division_1[OF _ assms(2), of f]
- using assms(1)
- by auto
-
subsection \<open>Fine-ness of a partition w.r.t. a gauge.\<close>
definition fine (infixr "fine" 46)
--- a/src/HOL/Equiv_Relations.thy Fri Aug 04 18:03:50 2017 +0200
+++ b/src/HOL/Equiv_Relations.thy Mon Aug 07 15:10:37 2017 +0200
@@ -5,7 +5,7 @@
section \<open>Equivalence Relations in Higher-Order Set Theory\<close>
theory Equiv_Relations
- imports Groups_Big Relation
+ imports Groups_Big
begin
subsection \<open>Equivalence relations -- set version\<close>
--- a/src/HOL/Groups_Big.thy Fri Aug 04 18:03:50 2017 +0200
+++ b/src/HOL/Groups_Big.thy Mon Aug 07 15:10:37 2017 +0200
@@ -8,7 +8,7 @@
section \<open>Big sum and product over finite (non-empty) sets\<close>
theory Groups_Big
- imports Finite_Set Power
+ imports Power
begin
subsection \<open>Generic monoid operation over a set\<close>
--- a/src/HOL/Lattices_Big.thy Fri Aug 04 18:03:50 2017 +0200
+++ b/src/HOL/Lattices_Big.thy Mon Aug 07 15:10:37 2017 +0200
@@ -6,7 +6,7 @@
section \<open>Big infimum (minimum) and supremum (maximum) over finite (non-empty) sets\<close>
theory Lattices_Big
-imports Finite_Set Option
+ imports Option
begin
subsection \<open>Generic lattice operations over a set\<close>
--- a/src/HOL/List.thy Fri Aug 04 18:03:50 2017 +0200
+++ b/src/HOL/List.thy Mon Aug 07 15:10:37 2017 +0200
@@ -4796,7 +4796,7 @@
qed
lemma card_lists_distinct_length_eq:
- assumes "k < card A"
+ assumes "finite A" "k \<le> card A"
shows "card {xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> A} = \<Prod>{card A - k + 1 .. card A}"
using assms
proof (induct k)
@@ -4808,25 +4808,32 @@
let "?k_list" = "\<lambda>k xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> A"
have inj_Cons: "\<And>A. inj_on (\<lambda>(xs, n). n # xs) A" by (rule inj_onI) auto
- from Suc have "k < card A" by simp
- moreover have "finite A" using assms by (simp add: card_ge_0_finite)
+ from Suc have "k \<le> card A" by simp
+ moreover note \<open>finite A\<close>
moreover have "finite {xs. ?k_list k xs}"
by (rule finite_subset) (use finite_lists_length_eq[OF \<open>finite A\<close>, of k] in auto)
moreover have "\<And>i j. i \<noteq> j \<longrightarrow> {i} \<times> (A - set i) \<inter> {j} \<times> (A - set j) = {}"
by auto
- moreover have "\<And>i. i \<in>Collect (?k_list k) \<Longrightarrow> card (A - set i) = card A - k"
+ moreover have "\<And>i. i \<in> {xs. ?k_list k xs} \<Longrightarrow> card (A - set i) = card A - k"
by (simp add: card_Diff_subset distinct_card)
moreover have "{xs. ?k_list (Suc k) xs} =
(\<lambda>(xs, n). n#xs) ` \<Union>((\<lambda>xs. {xs} \<times> (A - set xs)) ` {xs. ?k_list k xs})"
by (auto simp: length_Suc_conv)
- moreover
- have "Suc (card A - Suc k) = card A - k" using Suc.prems by simp
+ moreover have "Suc (card A - Suc k) = card A - k" using Suc.prems by simp
then have "(card A - k) * \<Prod>{Suc (card A - k)..card A} = \<Prod>{Suc (card A - Suc k)..card A}"
by (subst prod.insert[symmetric]) (simp add: atLeastAtMost_insertL)+
ultimately show ?case
by (simp add: card_image inj_Cons card_UN_disjoint Suc.hyps algebra_simps)
qed
+lemma card_lists_distinct_length_eq':
+ assumes "k < card A"
+ shows "card {xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> A} = \<Prod>{card A - k + 1 .. card A}"
+proof -
+ from \<open>k < card A\<close> have "finite A" and "k \<le> card A" using card_infinite by force+
+ from this show ?thesis by (rule card_lists_distinct_length_eq)
+qed
+
lemma infinite_UNIV_listI: "~ finite(UNIV::'a list set)"
apply (rule notI)
apply (drule finite_maxlen)
--- a/src/HOL/Option.thy Fri Aug 04 18:03:50 2017 +0200
+++ b/src/HOL/Option.thy Mon Aug 07 15:10:37 2017 +0200
@@ -5,7 +5,7 @@
section \<open>Datatype option\<close>
theory Option
-imports Lifting Finite_Set
+ imports Lifting
begin
datatype 'a option =
--- a/src/HOL/Partial_Function.thy Fri Aug 04 18:03:50 2017 +0200
+++ b/src/HOL/Partial_Function.thy Mon Aug 07 15:10:37 2017 +0200
@@ -5,8 +5,8 @@
section \<open>Partial Function Definitions\<close>
theory Partial_Function
-imports Complete_Partial_Order Fun_Def_Base Option
-keywords "partial_function" :: thy_decl
+ imports Complete_Partial_Order Option
+ keywords "partial_function" :: thy_decl
begin
named_theorems partial_function_mono "monotonicity rules for partial function definitions"
--- a/src/HOL/Tools/ATP/atp_systems.ML Fri Aug 04 18:03:50 2017 +0200
+++ b/src/HOL/Tools/ATP/atp_systems.ML Mon Aug 07 15:10:37 2017 +0200
@@ -207,8 +207,6 @@
(* E *)
-fun is_e_at_least_1_8 () = string_ord (getenv "E_VERSION", "1.8") <> LESS
-
val e_smartN = "smart"
val e_autoN = "auto"
val e_fun_weightN = "fun_weight"
@@ -277,23 +275,19 @@
(if gen_prec then "--precedence='" ^ e_ord_precedence ord_info ^ "' " else "")
end
+val e_tff0 = TFF Monomorphic
+
val e_config : atp_config =
- {exec = fn full_proofs => (["E_HOME"],
- if full_proofs orelse not (is_e_at_least_1_8 ()) then ["eproof_ram", "eproof"]
- else ["eprover"]),
- arguments = fn ctxt => fn full_proofs => fn heuristic => fn timeout => fn file_name =>
+ {exec = fn _ => (["E_HOME"], ["eprover"]),
+ arguments = fn ctxt => fn _ => fn heuristic => fn timeout => fn file_name =>
fn ({is_lpo, gen_weights, gen_prec, ...}, ord_info, sel_weights) =>
- (if is_e_at_least_1_8 () then "--auto-schedule " else "") ^
- "--tstp-in --tstp-out --silent " ^
+ "--auto-schedule --tstp-in --tstp-out --silent " ^
e_selection_weight_arguments ctxt heuristic sel_weights ^
e_term_order_info_arguments gen_weights gen_prec ord_info ^
"--term-ordering=" ^ (if is_lpo then "LPO4" else "KBO6") ^ " " ^
"--cpu-limit=" ^ string_of_int (to_secs 2 timeout) ^
- (if full_proofs orelse not (is_e_at_least_1_8 ()) then
- " --output-level=5 --pcl-shell-level=" ^ (if full_proofs then "0" else "2")
- else
- " --proof-object=1") ^
- " " ^ file_name,
+ " --proof-object=1 " ^
+ file_name,
proof_delims =
[("# SZS output start CNFRefutation", "# SZS output end CNFRefutation")] @
tstp_proof_delims,
@@ -306,14 +300,14 @@
let val heuristic = Config.get ctxt e_selection_heuristic in
(* FUDGE *)
if heuristic = e_smartN then
- [(0.15, (((128, meshN), FOF, "mono_tags??", combsN, false), e_fun_weightN)),
- (0.15, (((128, mashN), FOF, "mono_guards??", combsN, false), e_sym_offset_weightN)),
- (0.15, (((91, mepoN), FOF, "mono_tags??", combsN, false), e_autoN)),
- (0.15, (((1000, meshN), FOF, "poly_guards??", combsN, false), e_sym_offset_weightN)),
- (0.15, (((256, mepoN), FOF, "mono_tags??", liftingN, false), e_fun_weightN)),
- (0.25, (((64, mashN), FOF, "mono_guards??", combsN, false), e_fun_weightN))]
+ [(0.15, (((128, meshN), e_tff0, "mono_native", combsN, false), e_fun_weightN)),
+ (0.15, (((128, mashN), e_tff0, "mono_native", combsN, false), e_sym_offset_weightN)),
+ (0.15, (((91, mepoN), e_tff0, "mono_native", combsN, false), e_autoN)),
+ (0.15, (((1000, meshN), e_tff0, "poly_guards??", combsN, false), e_sym_offset_weightN)),
+ (0.15, (((256, mepoN), e_tff0, "mono_native", liftingN, false), e_fun_weightN)),
+ (0.25, (((64, mashN), e_tff0, "mono_native", combsN, false), e_fun_weightN))]
else
- [(1.0, (((500, ""), FOF, "mono_tags??", combsN, false), heuristic))]
+ [(1.0, (((500, ""), e_tff0, "mono_native", combsN, false), heuristic))]
end,
best_max_mono_iters = default_max_mono_iters,
best_max_new_mono_instances = default_max_new_mono_instances}
@@ -456,9 +450,8 @@
val spass_extra_options =
Attrib.setup_config_string @{binding atp_spass_extra_options} (K "")
-(* FIXME: Make "SPASS_NEW_HOME" legacy. *)
val spass_config : atp_config =
- {exec = K (["SPASS_NEW_HOME", "SPASS_HOME"], ["SPASS"]),
+ {exec = K (["SPASS_HOME"], ["SPASS"]),
arguments = fn _ => fn full_proofs => fn extra_options => fn timeout =>
fn file_name => fn _ =>
"-Isabelle=1 " ^ (if full_proofs then "-CNFRenaming=0 -Splits=0 " else "") ^
@@ -710,7 +703,7 @@
(K (((60, ""), agsyhol_thf0, "mono_native_higher", keep_lamsN, false), "") (* FUDGE *))
val remote_e =
remotify_atp e "E" ["2.0", "1.9.1", "1.8"]
- (K (((750, ""), FOF, "mono_tags??", combsN, false), "") (* FUDGE *))
+ (K (((750, ""), e_tff0, "mono_native", combsN, false), "") (* FUDGE *))
val remote_iprover =
remotify_atp iprover "iProver" ["0.99"]
(K (((150, ""), FOF, "mono_guards??", liftingN, false), "") (* FUDGE *))
@@ -724,7 +717,7 @@
remotify_atp satallax "Satallax" ["2.7", "2.3", "2"]
(K (((60, ""), satallax_thf0, "mono_native_higher", keep_lamsN, false), "") (* FUDGE *))
val remote_vampire =
- remotify_atp vampire "Vampire" ["4.0", "3.0", "2.6"]
+ remotify_atp vampire "Vampire" ["4.2", "4.0"]
(K (((400, ""), vampire_tff0, "mono_native", combs_or_liftingN, false), remote_vampire_full_proof_command) (* FUDGE *))
val remote_e_sine =
remote_atp e_sineN "SInE" ["0.4"] [] (#known_failures e_config) Conjecture
--- a/src/HOL/Tools/value_command.ML Fri Aug 04 18:03:50 2017 +0200
+++ b/src/HOL/Tools/value_command.ML Mon Aug 07 15:10:37 2017 +0200
@@ -17,9 +17,7 @@
fun default_value ctxt t =
if null (Term.add_frees t [])
- then case try (Code_Evaluation.dynamic_value_strict ctxt) t of
- SOME t' => t'
- | NONE => Nbe.dynamic_value ctxt t
+ then Code_Evaluation.dynamic_value_strict ctxt t
else Nbe.dynamic_value ctxt t;
structure Evaluator = Theory_Data
--- a/src/HOL/ex/Adhoc_Overloading_Examples.thy Fri Aug 04 18:03:50 2017 +0200
+++ b/src/HOL/ex/Adhoc_Overloading_Examples.thy Mon Aug 07 15:10:37 2017 +0200
@@ -38,7 +38,7 @@
adhoc_overloading
vars term_vars
-value "vars (Fun ''f'' [Var 0, Var 1])"
+value [nbe] "vars (Fun ''f'' [Var 0, Var 1])"
fun rule_vars :: "('a, 'b) term \<times> ('a, 'b) term \<Rightarrow> 'b set" where
"rule_vars (l, r) = vars l \<union> vars r"
@@ -46,7 +46,7 @@
adhoc_overloading
vars rule_vars
-value "vars (Var 1, Var 0)"
+value [nbe] "vars (Var 1, Var 0)"
definition trs_vars :: "(('a, 'b) term \<times> ('a, 'b) term) set \<Rightarrow> 'b set" where
"trs_vars R = \<Union>(rule_vars ` R)"
@@ -54,7 +54,7 @@
adhoc_overloading
vars trs_vars
-value "vars {(Var 1, Var 0)}"
+value [nbe] "vars {(Var 1, Var 0)}"
text \<open>Sometimes it is necessary to add explicit type constraints
before a variant can be determined.\<close>
--- a/src/HOL/ex/Normalization_by_Evaluation.thy Fri Aug 04 18:03:50 2017 +0200
+++ b/src/HOL/ex/Normalization_by_Evaluation.thy Mon Aug 07 15:10:37 2017 +0200
@@ -68,20 +68,20 @@
lemma "map (%f. f True) ([id, g, Not] @ fs) = [True, g True, False] @ map (%f. f True) fs"
by normalization rule
lemma "rev [a, b, c] = [c, b, a]" by normalization
-value "rev (a#b#cs) = rev cs @ [b, a]"
-value "map (%F. F [a,b,c::'x]) (map map [f,g,h])"
-value "map (%F. F ([a,b,c] @ ds)) (map map ([f,g,h]@fs))"
-value "map (%F. F [Z,S Z,S(S Z)]) (map map [S,add (S Z),mul (S(S Z)),id])"
+value [nbe] "rev (a#b#cs) = rev cs @ [b, a]"
+value [nbe] "map (%F. F [a,b,c::'x]) (map map [f,g,h])"
+value [nbe] "map (%F. F ([a,b,c] @ ds)) (map map ([f,g,h]@fs))"
+value [nbe] "map (%F. F [Z,S Z,S(S Z)]) (map map [S,add (S Z),mul (S(S Z)),id])"
lemma "map (%x. case x of None \<Rightarrow> False | Some y \<Rightarrow> True) [None, Some ()] = [False, True]"
by normalization
-value "case xs of [] \<Rightarrow> True | x#xs \<Rightarrow> False"
-value "map (%x. case x of None \<Rightarrow> False | Some y \<Rightarrow> True) xs = P"
+value [nbe] "case xs of [] \<Rightarrow> True | x#xs \<Rightarrow> False"
+value [nbe] "map (%x. case x of None \<Rightarrow> False | Some y \<Rightarrow> True) xs = P"
lemma "let x = y in [x, x] = [y, y]" by normalization
lemma "Let y (%x. [x,x]) = [y, y]" by normalization
-value "case n of Z \<Rightarrow> True | S x \<Rightarrow> False"
+value [nbe] "case n of Z \<Rightarrow> True | S x \<Rightarrow> False"
lemma "(%(x,y). add x y) (S z,S z) = S (add z (S z))" by normalization
-value "filter (%x. x) ([True,False,x]@xs)"
-value "filter Not ([True,False,x]@xs)"
+value [nbe] "filter (%x. x) ([True,False,x]@xs)"
+value [nbe] "filter Not ([True,False,x]@xs)"
lemma "[x,y,z] @ [a,b,c] = [x, y, z, a, b, c]" by normalization
lemma "(%(xs, ys). xs @ ys) ([a, b, c], [d, e, f]) = [a, b, c, d, e, f]" by normalization
@@ -106,7 +106,7 @@
lemma "[Suc 0, 0] = [Suc 0, 0]" by normalization
lemma "max (Suc 0) 0 = Suc 0" by normalization
lemma "(42::rat) / 1704 = 1 / 284 + 3 / 142" by normalization
-value "Suc 0 \<in> set ms"
+value [nbe] "Suc 0 \<in> set ms"
(* non-left-linear patterns, equality by extensionality *)
@@ -115,13 +115,13 @@
lemma "(f o g) x = f (g x)" by normalization
lemma "(f o id) x = f x" by normalization
lemma "(id :: bool \<Rightarrow> bool) = id" by normalization
-value "(\<lambda>x. x)"
+value [nbe] "(\<lambda>x. x)"
(* Church numerals: *)
-value "(%m n f x. m f (n f x)) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))"
-value "(%m n f x. m (n f) x) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))"
-value "(%m n. n m) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))"
+value [nbe] "(%m n f x. m f (n f x)) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))"
+value [nbe] "(%m n f x. m (n f) x) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))"
+value [nbe] "(%m n. n m) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))"
(* handling of type classes in connection with equality *)
--- a/src/HOL/ex/Transitive_Closure_Table_Ex.thy Fri Aug 04 18:03:50 2017 +0200
+++ b/src/HOL/ex/Transitive_Closure_Table_Ex.thy Mon Aug 07 15:10:37 2017 +0200
@@ -24,7 +24,7 @@
values "{x. test\<^sup>*\<^sup>* A x}"
values "{x. test\<^sup>*\<^sup>* x C}"
-value "test\<^sup>*\<^sup>* A C"
-value "test\<^sup>*\<^sup>* C A"
+value [nbe] "test\<^sup>*\<^sup>* A C"
+value [nbe] "test\<^sup>*\<^sup>* C A"
end
--- a/src/Pure/System/isabelle_tool.scala Fri Aug 04 18:03:50 2017 +0200
+++ b/src/Pure/System/isabelle_tool.scala Mon Aug 07 15:10:37 2017 +0200
@@ -115,6 +115,7 @@
Options.isabelle_tool,
Profiling_Report.isabelle_tool,
Remote_DMG.isabelle_tool,
+ Server.isabelle_tool,
Update_Cartouches.isabelle_tool,
Update_Header.isabelle_tool,
Update_Then.isabelle_tool,
--- a/src/Pure/System/system_channel.scala Fri Aug 04 18:03:50 2017 +0200
+++ b/src/Pure/System/system_channel.scala Mon Aug 07 15:10:37 2017 +0200
@@ -18,7 +18,7 @@
class System_Channel private
{
- private val server = new ServerSocket(0, 2, InetAddress.getByName("127.0.0.1"))
+ private val server = new ServerSocket(0, 50, InetAddress.getByName("127.0.0.1"))
val server_name: String = "127.0.0.1:" + server.getLocalPort
override def toString: String = server_name
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Pure/Tools/server.scala Mon Aug 07 15:10:37 2017 +0200
@@ -0,0 +1,180 @@
+/* Title: Pure/Tools/server.scala
+ Author: Makarius
+
+Resident Isabelle servers.
+*/
+
+package isabelle
+
+
+import java.io.{BufferedReader, BufferedWriter, InputStreamReader, OutputStreamWriter,
+ IOException}
+import java.net.{Socket, ServerSocket, InetAddress}
+
+
+object Server
+{
+ /* per-user servers */
+
+ object Data
+ {
+ val database = Path.explode("$ISABELLE_HOME_USER/servers.db")
+
+ val name = SQL.Column.string("name", primary_key = true)
+ val port = SQL.Column.int("port")
+ val password = SQL.Column.string("password")
+ val table = SQL.Table("isabelle_servers", List(name, port, password))
+
+ sealed case class Entry(name: String, port: Int, password: String)
+ {
+ def print: String =
+ "server " + quote(name) + " = 127.0.0.1:" + port + " (password " + quote(password) + ")"
+
+ def active: Boolean =
+ try { (new Socket(InetAddress.getByName("127.0.0.1"), port)).close; true }
+ catch { case _: IOException => false }
+ }
+ }
+
+ def list(db: SQLite.Database): List[Data.Entry] =
+ if (db.tables.contains(Data.table.name)) {
+ db.using_statement(Data.table.select())(stmt =>
+ stmt.execute_query().iterator(res =>
+ Data.Entry(
+ res.string(Data.name),
+ res.int(Data.port),
+ res.string(Data.password))).toList.sortBy(_.name))
+ }
+ else Nil
+
+ def find(db: SQLite.Database, name: String): Option[Data.Entry] =
+ list(db).find(entry => entry.name == name && entry.active)
+
+ def start(name: String = "", port: Int = 0): (Data.Entry, Option[Thread]) =
+ {
+ using(SQLite.open_database(Data.database))(db =>
+ db.transaction {
+ find(db, name) match {
+ case Some(entry) => (entry, None)
+ case None =>
+ val server = new Server(port)
+ val entry = Data.Entry(name, server.port, server.password)
+
+ Isabelle_System.bash("chmod 600 " + File.bash_path(Data.database)).check
+ db.create_table(Data.table)
+ db.using_statement(Data.table.delete(Data.name.where_equal(name)))(_.execute)
+ db.using_statement(Data.table.insert())(stmt =>
+ {
+ stmt.string(1) = entry.name
+ stmt.int(2) = entry.port
+ stmt.string(3) = entry.password
+ stmt.execute()
+ })
+
+ (entry, Some(server.thread))
+ }
+ })
+ }
+
+ def stop(name: String = ""): Boolean =
+ {
+ using(SQLite.open_database(Data.database))(db =>
+ db.transaction {
+ find(db, name) match {
+ case Some(entry) =>
+ // FIXME shutdown server
+ db.using_statement(Data.table.delete(Data.name.where_equal(name)))(_.execute)
+ true
+ case None =>
+ false
+ }
+ })
+ }
+
+
+ /* Isabelle tool wrapper */
+
+ val isabelle_tool =
+ Isabelle_Tool("server", "manage resident Isabelle servers", args =>
+ {
+ var operation_list = false
+ var name = ""
+ var port = 0
+
+ val getopts =
+ Getopts("""
+Usage: isabelle server [OPTIONS]
+
+ Options are:
+ -L list servers
+ -n NAME explicit server name
+ -p PORT explicit server port
+
+ Manage resident Isabelle servers.
+""",
+ "L" -> (_ => operation_list = true),
+ "n:" -> (arg => name = arg),
+ "p:" -> (arg => port = Value.Int.parse(arg)))
+
+ val more_args = getopts(args)
+ if (more_args.nonEmpty) getopts.usage()
+
+ if (operation_list) {
+ for (entry <- using(SQLite.open_database(Data.database))(list(_)) if entry.active)
+ Console.println(entry.print)
+ }
+ else {
+ val (entry, thread) = start(name, port)
+ Console.println(entry.print)
+ thread.foreach(_.join)
+ }
+ })
+}
+
+class Server private(_port: Int)
+{
+ private val server_socket = new ServerSocket(_port, 50, InetAddress.getByName("127.0.0.1"))
+ def port: Int = server_socket.getLocalPort
+ def close { server_socket.close }
+
+ val password: String = Library.UUID()
+
+ private def handle_connection(socket: Socket)
+ {
+ val reader = new BufferedReader(new InputStreamReader(socket.getInputStream, UTF8.charset))
+ val writer = new BufferedWriter(new OutputStreamWriter(socket.getOutputStream, UTF8.charset))
+
+ def println(s: String)
+ {
+ writer.write(s)
+ writer.newLine()
+ writer.flush()
+ }
+
+ reader.readLine() match {
+ case null =>
+ case bad if bad != password => println("BAD PASSWORD")
+ case _ =>
+ var finished = false
+ while (!finished) {
+ reader.readLine() match {
+ case null => println("FINISHED"); finished = true
+ case line => println("ECHO " + line)
+ }
+ }
+ }
+ }
+
+ lazy val thread: Thread =
+ Standard_Thread.fork("server") {
+ var finished = false
+ while (!finished) {
+ Exn.capture(server_socket.accept) match {
+ case Exn.Res(socket) =>
+ Standard_Thread.fork("server_connection")
+ { try { handle_connection(socket) } finally { socket.close } }
+ case Exn.Exn(_) => finished = true
+ }
+ }
+ }
+}
--- a/src/Pure/build-jars Fri Aug 04 18:03:50 2017 +0200
+++ b/src/Pure/build-jars Mon Aug 07 15:10:37 2017 +0200
@@ -140,6 +140,7 @@
Tools/main.scala
Tools/print_operation.scala
Tools/profiling_report.scala
+ Tools/server.scala
Tools/simplifier_trace.scala
Tools/spell_checker.scala
Tools/task_statistics.scala
--- a/src/Pure/library.scala Fri Aug 04 18:03:50 2017 +0200
+++ b/src/Pure/library.scala Mon Aug 07 15:10:37 2017 +0200
@@ -259,4 +259,9 @@
def proper_list[A](list: List[A]): Option[List[A]] =
if (list == null || list.isEmpty) None else Some(list)
+
+
+ /* UUID */
+
+ def UUID(): String = java.util.UUID.randomUUID().toString
}
--- a/src/Tools/VSCode/src/grammar.scala Fri Aug 04 18:03:50 2017 +0200
+++ b/src/Tools/VSCode/src/grammar.scala Mon Aug 07 15:10:37 2017 +0200
@@ -9,8 +9,6 @@
import isabelle._
-import java.util.UUID
-
object Grammar
{
@@ -47,7 +45,7 @@
"name": "Isabelle",
"scopeName": "source.isabelle",
"fileTypes": ["thy"],
- "uuid": """ + JSON.Format(UUID.randomUUID().toString) + """,
+ "uuid": """ + JSON.Format(Library.UUID()) + """,
"repository": {
"comment": {
"patterns": [
--- a/src/Tools/jEdit/src/plugin.scala Fri Aug 04 18:03:50 2017 +0200
+++ b/src/Tools/jEdit/src/plugin.scala Mon Aug 07 15:10:37 2017 +0200
@@ -12,7 +12,6 @@
import javax.swing.JOptionPane
import java.io.{File => JFile}
-import java.util.UUID
import org.gjt.sp.jedit.{jEdit, EBMessage, EBPlugin, Buffer, View, Debug, PerspectiveManager}
import org.gjt.sp.jedit.textarea.JEditTextArea
@@ -398,7 +397,7 @@
/* HTTP server */
- val http_root: String = "/" + UUID.randomUUID().toString
+ val http_root: String = "/" + Library.UUID()
val http_server: HTTP.Server = HTTP.server(Document_Model.http_handlers(http_root))