--- a/src/HOL/Multivariate_Analysis/Integration.thy Tue Dec 04 18:00:37 2012 +0100
+++ b/src/HOL/Multivariate_Analysis/Integration.thy Tue Dec 04 18:00:40 2012 +0100
@@ -8,10 +8,6 @@
"~~/src/HOL/Library/Indicator_Function"
begin
-declare [[smt_certificates = "Integration.certs"]]
-declare [[smt_read_only_certificates = true]]
-declare [[smt_oracle = false]]
-
(*declare not_less[simp] not_le[simp]*)
lemmas scaleR_simps = scaleR_zero_left scaleR_minus_left scaleR_left_diff_distrib
@@ -2770,16 +2766,21 @@
lemma has_integral_component_le: fixes f g::"'a::ordered_euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "(f has_integral i) s" "(g has_integral j) s" "\<forall>x\<in>s. (f x)$$k \<le> (g x)$$k"
shows "i$$k \<le> j$$k"
-proof- have lem:"\<And>a b i (j::'b). \<And>g f::'a \<Rightarrow> 'b. (f has_integral i) ({a..b}) \<Longrightarrow>
+proof -
+ have lem:"\<And>a b i (j::'b). \<And>g f::'a \<Rightarrow> 'b. (f has_integral i) ({a..b}) \<Longrightarrow>
(g has_integral j) ({a..b}) \<Longrightarrow> \<forall>x\<in>{a..b}. (f x)$$k \<le> (g x)$$k \<Longrightarrow> i$$k \<le> j$$k"
- proof(rule ccontr) case goal1 hence *:"0 < (i$$k - j$$k) / 3" by auto
+ proof (rule ccontr)
+ case goal1
+ then have *: "0 < (i$$k - j$$k) / 3" by auto
guess d1 using goal1(1)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d1=this[rule_format]
guess d2 using goal1(2)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d2=this[rule_format]
guess p using fine_division_exists[OF gauge_inter[OF d1(1) d2(1)], of a b] unfolding fine_inter .
note p = this(1) conjunctD2[OF this(2)] note le_less_trans[OF component_le_norm, of _ _ k] term g
note this[OF d1(2)[OF conjI[OF p(1,2)]]] this[OF d2(2)[OF conjI[OF p(1,3)]]]
- thus False unfolding euclidean_simps using rsum_component_le[OF p(1) goal1(3)] apply simp
- using [[z3_with_extensions]] by smt
+ thus False
+ unfolding euclidean_simps
+ using rsum_component_le[OF p(1) goal1(3)]
+ by (simp add: abs_real_def split: split_if_asm)
qed let ?P = "\<exists>a b. s = {a..b}"
{ presume "\<not> ?P \<Longrightarrow> ?thesis" thus ?thesis proof(cases ?P)
case True then guess a b apply-by(erule exE)+ note s=this
@@ -2793,7 +2794,8 @@
note ab = conjunctD2[OF this[unfolded Un_subset_iff]]
guess w1 using B(2)[OF ab(1)] .. note w1=conjunctD2[OF this]
guess w2 using B(4)[OF ab(2)] .. note w2=conjunctD2[OF this]
- have *:"\<And>w1 w2 j i::real .\<bar>w1 - i\<bar> < (i - j) / 3 \<Longrightarrow> \<bar>w2 - j\<bar> < (i - j) / 3 \<Longrightarrow> w1 \<le> w2 \<Longrightarrow> False" using [[z3_with_extensions]] by smt
+ have *:"\<And>w1 w2 j i::real .\<bar>w1 - i\<bar> < (i - j) / 3 \<Longrightarrow> \<bar>w2 - j\<bar> < (i - j) / 3 \<Longrightarrow> w1 \<le> w2 \<Longrightarrow> False"
+ by (simp add: abs_real_def split: split_if_asm)
note le_less_trans[OF component_le_norm[of _ k]] note this[OF w1(2)] this[OF w2(2)] moreover
have "w1$$k \<le> w2$$k" apply(rule lem[OF w1(1) w2(1)]) using assms by auto ultimately
show False unfolding euclidean_simps by(rule *) qed
@@ -3022,7 +3024,8 @@
also have "... < e" apply(subst setsum_over_tagged_division_lemma[OF p[THEN conjunct1]])
proof- case goal1 have "content ({u..v} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}) \<le> content {u..v}"
unfolding interval_doublesplit[OF k] apply(rule content_subset) unfolding interval_doublesplit[THEN sym,OF k] by auto
- thus ?case unfolding goal1 unfolding interval_doublesplit[OF k] using content_pos_le by smt
+ thus ?case unfolding goal1 unfolding interval_doublesplit[OF k]
+ by (blast intro: antisym)
next have *:"setsum content {l \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d} |l. l \<in> snd ` p \<and> l \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d} \<noteq> {}} \<ge> 0"
apply(rule setsum_nonneg,rule) unfolding mem_Collect_eq image_iff apply(erule exE bexE conjE)+ unfolding split_paired_all
proof- fix x l a b assume as:"x = l \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}" "(a, b) \<in> p" "l = snd (a, b)"
@@ -3516,7 +3519,7 @@
proof safe fix x and e::real assume as:"x\<in>k" "e>0"
from k(2)[unfolded k content_eq_0] guess i ..
hence i:"c$$i = d$$i" "i<DIM('a)" using s(3)[OF k(1),unfolded k] unfolding interval_ne_empty by auto
- hence xi:"x$$i = d$$i" using as unfolding k mem_interval by smt
+ hence xi:"x$$i = d$$i" using as unfolding k mem_interval by (metis antisym)
def y \<equiv> "(\<chi>\<chi> j. if j = i then if c$$i \<le> (a$$i + b$$i) / 2 then c$$i +
min e (b$$i - c$$i) / 2 else c$$i - min e (c$$i - a$$i) / 2 else x$$j)::'a"
show "\<exists>x'\<in>\<Union>(s - {k}). x' \<noteq> x \<and> dist x' x < e" apply(rule_tac x=y in bexI)
@@ -3524,7 +3527,8 @@
hence "d \<in> {a..b}" using s(2)[OF k(1)] unfolding k by auto note di = this[unfolded mem_interval,THEN spec[where x=i]]
hence xyi:"y$$i \<noteq> x$$i" unfolding y_def unfolding i xi euclidean_lambda_beta'[OF i(2)] if_P[OF refl]
apply(cases) apply(subst if_P,assumption) unfolding if_not_P not_le using as(2)
- using assms(2)[unfolded content_eq_0] using i(2) using [[z3_with_extensions]] by smt+
+ using assms(2)[unfolded content_eq_0] using i(2)
+ by (auto simp add: not_le min_def)
thus "y \<noteq> x" unfolding euclidean_eq[where 'a='a] using i by auto
have *:"{..<DIM('a)} = insert i ({..<DIM('a)} - {i})" using i by auto
have "norm (y - x) < e + setsum (\<lambda>i. 0) {..<DIM('a)}" apply(rule le_less_trans[OF norm_le_l1])
@@ -4710,7 +4714,8 @@
proof- fix a b c d::"'n::ordered_euclidean_space" assume as:"ball 0 (max B1 B2) \<subseteq> {a..b}" "ball 0 (max B1 B2) \<subseteq> {c..d}"
have **:"ball 0 B1 \<subseteq> ball (0::'n::ordered_euclidean_space) (max B1 B2)" "ball 0 B2 \<subseteq> ball (0::'n::ordered_euclidean_space) (max B1 B2)" by auto
have *:"\<And>ga gc ha hc fa fc::real. abs(ga - i) < e / 3 \<and> abs(gc - i) < e / 3 \<and> abs(ha - j) < e / 3 \<and>
- abs(hc - j) < e / 3 \<and> abs(i - j) < e / 3 \<and> ga \<le> fa \<and> fa \<le> ha \<and> gc \<le> fc \<and> fc \<le> hc\<Longrightarrow> abs(fa - fc) < e" using [[z3_with_extensions]] by smt
+ abs(hc - j) < e / 3 \<and> abs(i - j) < e / 3 \<and> ga \<le> fa \<and> fa \<le> ha \<and> gc \<le> fc \<and> fc \<le> hc\<Longrightarrow> abs(fa - fc) < e"
+ by (simp add: abs_real_def split: split_if_asm)
show "norm (integral {a..b} (\<lambda>x. if x \<in> s then f x else 0) - integral {c..d} (\<lambda>x. if x \<in> s then f x else 0)) < e"
unfolding real_norm_def apply(rule *, safe) unfolding real_norm_def[THEN sym]
apply(rule B1(2),rule order_trans,rule **,rule as(1))
@@ -6057,7 +6062,4 @@
using assms(3)[rule_format,OF x] unfolding real_norm_def abs_le_iff by auto
qed qed(insert n,auto) qed qed qed
-declare [[smt_certificates = ""]]
-declare [[smt_read_only_certificates = false]]
-
end