--- a/src/HOL/Multivariate_Analysis/Integration.thy Mon Apr 26 11:34:17 2010 +0200
+++ b/src/HOL/Multivariate_Analysis/Integration.thy Mon Apr 26 11:34:19 2010 +0200
@@ -1131,7 +1131,7 @@
guess d2 by(rule has_integralD[OF goal1(2) e]) note d2=this
guess p by(rule fine_division_exists[OF gauge_inter[OF d1(1) d2(1)],of a b]) note p=this
let ?c = "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" have "norm (k1 - k2) \<le> norm (?c - k2) + norm (?c - k1)"
- using norm_triangle_ineq4[of "k1 - ?c" "k2 - ?c"] by(auto simp add:group_simps norm_minus_commute)
+ using norm_triangle_ineq4[of "k1 - ?c" "k2 - ?c"] by(auto simp add:algebra_simps norm_minus_commute)
also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
apply(rule add_strict_mono) apply(rule_tac[!] d2(2) d1(2)) using p unfolding fine_def by auto
finally show False by auto
@@ -1244,7 +1244,7 @@
unfolding scaleR_right_distrib setsum_addf[of "\<lambda>(x,k). content k *\<^sub>R f x" "\<lambda>(x,k). content k *\<^sub>R g x" p,THEN sym]
by(rule setsum_cong2,auto)
have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) = norm (((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k) + ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - l))"
- unfolding * by(auto simp add:group_simps) also let ?res = "\<dots>"
+ unfolding * by(auto simp add:algebra_simps) also let ?res = "\<dots>"
from as have *:"d1 fine p" "d2 fine p" unfolding fine_inter by auto
have "?res < e/2 + e/2" apply(rule le_less_trans[OF norm_triangle_ineq])
apply(rule add_strict_mono) using d1(2)[OF as(1) *(1)] and d2(2)[OF as(1) *(2)] by auto
@@ -1268,7 +1268,7 @@
lemma has_integral_sub:
shows "(f has_integral k) s \<Longrightarrow> (g has_integral l) s \<Longrightarrow> ((\<lambda>x. f(x) - g(x)) has_integral (k - l)) s"
- using has_integral_add[OF _ has_integral_neg,of f k s g l] unfolding group_simps by auto
+ using has_integral_add[OF _ has_integral_neg,of f k s g l] unfolding algebra_simps by auto
lemma integral_0: "integral s (\<lambda>x::real^'n. 0::real^'m) = 0"
by(rule integral_unique has_integral_0)+
@@ -1703,7 +1703,7 @@
proof- guess p using fine_division_exists[OF d(1), of a' b] . note p=this
show ?thesis using norm_triangle_half_l[OF d(2)[of p1 p] d(2)[of p2 p]]
using as unfolding interval_split b'_def[symmetric] a'_def[symmetric]
- using p using assms by(auto simp add:group_simps)
+ using p using assms by(auto simp add:algebra_simps)
qed qed
show "?P {x. x $ k \<ge> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)
proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b} \<inter> {x. x $ k \<ge> c} \<and> d fine p1 \<and> p2 tagged_division_of {a..b} \<inter> {x. x $ k \<ge> c} \<and> d fine p2"
@@ -1711,7 +1711,7 @@
proof- guess p using fine_division_exists[OF d(1), of a b'] . note p=this
show ?thesis using norm_triangle_half_l[OF d(2)[of p p1] d(2)[of p p2]]
using as unfolding interval_split b'_def[symmetric] a'_def[symmetric]
- using p using assms by(auto simp add:group_simps) qed qed qed qed
+ using p using assms by(auto simp add:algebra_simps) qed qed qed qed
subsection {* Generalized notion of additivity. *}
@@ -1848,7 +1848,7 @@
lemma monoidal_monoid[intro]:
shows "monoidal ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a)"
- unfolding monoidal_def neutral_monoid by(auto simp add: group_simps)
+ unfolding monoidal_def neutral_monoid by(auto simp add: algebra_simps)
lemma operative_integral: fixes f::"real^'n \<Rightarrow> 'a::banach"
shows "operative (lifted(op +)) (\<lambda>i. if f integrable_on i then Some(integral i f) else None)"
@@ -2381,8 +2381,8 @@
have lem2:"\<And>s1 s2 i1 i2. norm(s2 - s1) \<le> e/2 \<Longrightarrow> norm(s1 - i1) < e / 4 \<Longrightarrow> norm(s2 - i2) < e / 4 \<Longrightarrow>norm(i1 - i2) < e"
proof- case goal1 have "norm (i1 - i2) \<le> norm (i1 - s1) + norm (s1 - s2) + norm (s2 - i2)"
using norm_triangle_ineq[of "i1 - s1" "s1 - i2"]
- using norm_triangle_ineq[of "s1 - s2" "s2 - i2"] by(auto simp add:group_simps)
- also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:group_simps)
+ using norm_triangle_ineq[of "s1 - s2" "s2 - i2"] by(auto simp add:algebra_simps)
+ also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:algebra_simps)
finally show ?case .
qed
show ?case unfolding vector_dist_norm apply(rule lem2) defer
@@ -2399,7 +2399,7 @@
also have "\<dots> = 2 / real M" unfolding real_divide_def by auto
finally show "norm (g n x - g m x) \<le> 2 / real M"
using norm_triangle_le[of "g n x - f x" "f x - g m x" "2 / real M"]
- by(auto simp add:group_simps simp add:norm_minus_commute)
+ by(auto simp add:algebra_simps simp add:norm_minus_commute)
qed qed qed
from this[unfolded convergent_eq_cauchy[THEN sym]] guess s .. note s=this
@@ -2413,8 +2413,8 @@
have lem:"\<And>sf sg i. norm(sf - sg) \<le> e / 3 \<Longrightarrow> norm(i - s) < e / 3 \<Longrightarrow> norm(sg - i) < e / 3 \<Longrightarrow> norm(sf - s) < e"
proof- case goal1 have "norm (sf - s) \<le> norm (sf - sg) + norm (sg - i) + norm (i - s)"
using norm_triangle_ineq[of "sf - sg" "sg - s"]
- using norm_triangle_ineq[of "sg - i" " i - s"] by(auto simp add:group_simps)
- also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:group_simps)
+ using norm_triangle_ineq[of "sg - i" " i - s"] by(auto simp add:algebra_simps)
+ also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:algebra_simps)
finally show ?case .
qed
show ?case apply(rule_tac x=g' in exI) apply(rule,rule g')
@@ -2956,7 +2956,7 @@
have ball:"\<forall>xa\<in>k. xa \<in> ball x (d (dest_vec1 x))" using as(2)[unfolded fine_def,rule_format,OF `(x,k)\<in>p`,unfolded split_conv subset_eq] .
have "norm ((v$1 - u$1) *\<^sub>R f' x - (f v - f u)) \<le> norm (f u - f x - (u$1 - x$1) *\<^sub>R f' x) + norm (f v - f x - (v$1 - x$1) *\<^sub>R f' x)"
apply(rule order_trans[OF _ norm_triangle_ineq4]) apply(rule eq_refl) apply(rule arg_cong[where f=norm])
- unfolding scaleR.diff_left by(auto simp add:group_simps)
+ unfolding scaleR.diff_left by(auto simp add:algebra_simps)
also have "... \<le> e * norm (dest_vec1 u - dest_vec1 x) + e * norm (dest_vec1 v - dest_vec1 x)"
apply(rule add_mono) apply(rule d(2)[of "x$1" "u$1",unfolded o_def vec1_dest_vec1]) prefer 4
apply(rule d(2)[of "x$1" "v$1",unfolded o_def vec1_dest_vec1])
@@ -3098,7 +3098,7 @@
proof(rule,rule,rule d,safe) case goal1 show ?case proof(cases "y < x")
case False have "f \<circ> dest_vec1 integrable_on {vec1 a..vec1 y}" apply(rule integrable_subinterval,rule integrable_continuous)
apply(rule continuous_on_o_dest_vec1 assms)+ unfolding not_less using assms(2) goal1 by auto
- hence *:"?I a y - ?I a x = ?I x y" unfolding group_simps apply(subst eq_commute) apply(rule integral_combine)
+ hence *:"?I a y - ?I a x = ?I x y" unfolding algebra_simps apply(subst eq_commute) apply(rule integral_combine)
using False unfolding not_less using assms(2) goal1 by auto
have **:"norm (y - x) = content {vec1 x..vec1 y}" apply(subst content_1) using False unfolding not_less by auto
show ?thesis unfolding ** apply(rule has_integral_bound[where f="(\<lambda>u. f u - f x) o dest_vec1"]) unfolding * unfolding o_def
@@ -3112,7 +3112,7 @@
qed(insert e,auto)
next case True have "f \<circ> dest_vec1 integrable_on {vec1 a..vec1 x}" apply(rule integrable_subinterval,rule integrable_continuous)
apply(rule continuous_on_o_dest_vec1 assms)+ unfolding not_less using assms(2) goal1 by auto
- hence *:"?I a x - ?I a y = ?I y x" unfolding group_simps apply(subst eq_commute) apply(rule integral_combine)
+ hence *:"?I a x - ?I a y = ?I y x" unfolding algebra_simps apply(subst eq_commute) apply(rule integral_combine)
using True using assms(2) goal1 by auto
have **:"norm (y - x) = content {vec1 y..vec1 x}" apply(subst content_1) using True unfolding not_less by auto
have ***:"\<And>fy fx c::'a. fx - fy - (y - x) *\<^sub>R c = -(fy - fx - (x - y) *\<^sub>R c)" unfolding scaleR_left.diff by auto
@@ -3194,7 +3194,7 @@
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] by auto
qed note ** = d(2)[OF this] have *:"inj_on (\<lambda>(x, k). (g x, g ` k)) p" using inj(1) unfolding inj_on_def by fastsimp
- have "(\<Sum>(x, k)\<in>(\<lambda>(x, k). (g x, g ` k)) ` p. content k *\<^sub>R f x) - i = r *\<^sub>R (\<Sum>(x, k)\<in>p. content k *\<^sub>R f (g x)) - i" (is "?l = _") unfolding group_simps add_left_cancel
+ have "(\<Sum>(x, k)\<in>(\<lambda>(x, k). (g x, g ` k)) ` p. content k *\<^sub>R f x) - i = r *\<^sub>R (\<Sum>(x, k)\<in>p. content k *\<^sub>R f (g x)) - i" (is "?l = _") unfolding algebra_simps add_left_cancel
unfolding setsum_reindex[OF *] apply(subst scaleR_right.setsum) defer apply(rule setsum_cong2) unfolding o_def split_paired_all split_conv
apply(drule p(4)) apply safe unfolding assms(7)[rule_format] using p by auto
also have "... = r *\<^sub>R ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f (g x)) - (1 / r) *\<^sub>R i)" (is "_ = ?r") unfolding scaleR.diff_right scaleR.scaleR_left[THEN sym]
@@ -3332,7 +3332,7 @@
lemma norm_triangle_le_sub: "norm x + norm y \<le> e \<Longrightarrow> norm (x - y) \<le> e"
apply(subst(asm)(2) norm_minus_cancel[THEN sym])
- apply(drule norm_triangle_le) by(auto simp add:group_simps)
+ apply(drule norm_triangle_le) by(auto simp add:algebra_simps)
lemma fundamental_theorem_of_calculus_interior:
assumes"a \<le> b" "continuous_on {a..b} f" "\<forall>x\<in>{a<..<b}. (f has_vector_derivative f'(x)) (at x)"
@@ -3641,11 +3641,11 @@
proof safe show "0 < ?d" using d(1) assms(3) unfolding Cart_simps by auto
fix t::"_^1" assume as:"c \<le> t" "t$1 < c$1 + ?d"
have *:"integral{a..c} f = integral{a..b} f - integral{c..b} f"
- "integral{a..t} f = integral{a..b} f - integral{t..b} f" unfolding group_simps
+ "integral{a..t} f = integral{a..b} f - integral{t..b} f" unfolding algebra_simps
apply(rule_tac[!] integral_combine) using assms as unfolding Cart_simps by auto
have "(- c)$1 - d < (- t)$1 \<and> - t \<le> - c" using as by auto note d(2)[rule_format,OF this]
thus "norm (integral {a..c} f - integral {a..t} f) < e" unfolding *
- unfolding integral_reflect apply-apply(subst norm_minus_commute) by(auto simp add:group_simps) qed qed
+ unfolding integral_reflect apply-apply(subst norm_minus_commute) by(auto simp add:algebra_simps) qed qed
declare dest_vec1_eq[simp del] not_less[simp] not_le[simp]
@@ -3715,7 +3715,7 @@
apply safe apply(rule conv) using assms(4,7) by auto
have *:"\<And>t xa. (1 - t) *\<^sub>R c + t *\<^sub>R x = (1 - xa) *\<^sub>R c + xa *\<^sub>R x \<Longrightarrow> t = xa"
proof- case goal1 hence "(t - xa) *\<^sub>R x = (t - xa) *\<^sub>R c"
- unfolding scaleR_simps by(auto simp add:group_simps)
+ unfolding scaleR_simps by(auto simp add:algebra_simps)
thus ?case using `x\<noteq>c` by auto qed
have as2:"finite {t. ((1 - t) *\<^sub>R c + t *\<^sub>R x) \<in> k}" using assms(2)
apply(rule finite_surj[where f="\<lambda>z. SOME t. (1-t) *\<^sub>R c + t *\<^sub>R x = z"])
@@ -4390,7 +4390,7 @@
have *:"\<And>ir ip i cr cp. norm((cp + cr) - i) < e \<Longrightarrow> norm(cr - ir) < k \<Longrightarrow>
ip + ir = i \<Longrightarrow> norm(cp - ip) \<le> e + k"
proof- case goal1 thus ?case using norm_triangle_le[of "cp + cr - i" "- (cr - ir)"]
- unfolding goal1(3)[THEN sym] norm_minus_cancel by(auto simp add:group_simps) qed
+ unfolding goal1(3)[THEN sym] norm_minus_cancel by(auto simp add:algebra_simps) qed
have "?x = norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p. integral k f))"
unfolding split_def setsum_subtractf ..
@@ -4501,7 +4501,7 @@
norm(c - d) < e / 4 \<longrightarrow> norm(a - d) < e"
proof safe case goal1 thus ?case using norm_triangle_lt[of "a - b" "b - c" "3* e/4"]
norm_triangle_lt[of "a - b + (b - c)" "c - d" e] unfolding norm_minus_cancel
- by(auto simp add:group_simps) qed
+ by(auto simp add:algebra_simps) qed
show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - i) < e" apply(rule *[rule_format,where
b="\<Sum>(x, k)\<in>p. content k *\<^sub>R f (m x) x" and c="\<Sum>(x, k)\<in>p. integral k (f (m x))"])
proof safe case goal1
@@ -5152,7 +5152,7 @@
assumes "f absolutely_integrable_on s" "g absolutely_integrable_on s"
shows "(\<lambda>x. f(x) - g(x)) absolutely_integrable_on s"
using absolutely_integrable_add[OF assms(1) absolutely_integrable_neg[OF assms(2)]]
- unfolding group_simps .
+ unfolding algebra_simps .
lemma absolutely_integrable_linear: fixes f::"real^'m \<Rightarrow> real^'n" and h::"real^'n \<Rightarrow> real^'p"
assumes "f absolutely_integrable_on s" "bounded_linear h"