--- a/src/HOL/Multivariate_Analysis/Brouwer_Fixpoint.thy Sun Apr 25 23:22:29 2010 -0700
+++ b/src/HOL/Multivariate_Analysis/Brouwer_Fixpoint.thy Mon Apr 26 09:21:25 2010 -0700
@@ -22,8 +22,6 @@
imports Convex_Euclidean_Space
begin
-declare norm_scaleR[simp]
-
lemma brouwer_compactness_lemma:
assumes "compact s" "continuous_on s f" "\<not> (\<exists>x\<in>s. (f x = (0::real^'n)))"
obtains d where "0 < d" "\<forall>x\<in>s. d \<le> norm(f x)" proof(cases "s={}") case False
@@ -131,7 +129,7 @@
lemma image_lemma_2: assumes "finite s" "finite t" "card s = card t" "f ` s \<subseteq> t" "f ` s \<noteq> t" "b \<in> t"
shows "(card {s'. \<exists>a\<in>s. (s' = s - {a}) \<and> f ` s' = t - {b}} = 0) \<or>
(card {s'. \<exists>a\<in>s. (s' = s - {a}) \<and> f ` s' = t - {b}} = 2)" proof(cases "{a\<in>s. f ` (s - {a}) = t - {b}} = {}")
- case True thus ?thesis apply-apply(rule disjI1, rule image_lemma_0) using assms(1) by(auto simp add:card_0_eq)
+ case True thus ?thesis apply-apply(rule disjI1, rule image_lemma_0) using assms(1) by auto
next let ?M = "{a\<in>s. f ` (s - {a}) = t - {b}}"
case False then obtain a where "a\<in>?M" by auto hence a:"a\<in>s" "f ` (s - {a}) = t - {b}" by auto
have "f a \<in> t - {b}" using a and assms by auto
@@ -1691,11 +1689,11 @@
path_image(linepath(pathfinish g)(vector[(pathfinish g)$1,a$2 - 1])) \<union>
path_image(linepath(vector[(pathfinish g)$1,a$2 - 1])(vector[b$1 + 1,a$2 - 1])) \<union>
path_image(linepath(vector[b$1 + 1,a$2 - 1])(vector[b$1 + 1,b$2 + 3]))" using assms(1-2)
- by(auto simp add: pathstart_join pathfinish_join path_image_join path_image_linepath path_join path_linepath)
+ by(auto simp add: path_image_join path_linepath)
have abab: "{a..b} \<subseteq> {?a..?b}" by(auto simp add:vector_le_def forall_2 vector_2)
guess z apply(rule fashoda[of ?P1 ?P2 ?a ?b])
unfolding pathstart_join pathfinish_join pathstart_linepath pathfinish_linepath vector_2 proof-
- show "path ?P1" "path ?P2" using assms by(auto simp add: pathstart_join pathfinish_join path_join)
+ show "path ?P1" "path ?P2" using assms by auto
have "path_image ?P1 \<subseteq> {?a .. ?b}" unfolding P1P2 path_image_linepath apply(rule Un_least)+ defer 3
apply(rule_tac[1-4] convex_interval(1)[unfolded convex_contains_segment,rule_format])
unfolding mem_interval forall_2 vector_2 using ab startfin abab assms(3)
--- a/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy Sun Apr 25 23:22:29 2010 -0700
+++ b/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy Mon Apr 26 09:21:25 2010 -0700
@@ -24,7 +24,7 @@
lemma dest_vec1_simps[simp]: fixes a::"real^1"
shows "a$1 = 0 \<longleftrightarrow> a = 0" (*"a \<le> 1 \<longleftrightarrow> dest_vec1 a \<le> 1" "0 \<le> a \<longleftrightarrow> 0 \<le> dest_vec1 a"*)
"a \<le> b \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 b" "dest_vec1 (1::real^1) = 1"
- by(auto simp add:vector_component_simps forall_1 Cart_eq)
+ by(auto simp add: vector_le_def Cart_eq)
lemma norm_not_0:"(x::real^'n)\<noteq>0 \<Longrightarrow> norm x \<noteq> 0" by auto
@@ -50,7 +50,7 @@
lemma mem_interval_1: fixes x :: "real^1" shows
"(x \<in> {a .. b} \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b)"
"(x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"
-by(simp_all add: Cart_eq vector_less_def vector_le_def forall_1)
+by(simp_all add: Cart_eq vector_less_def vector_le_def)
lemma image_smult_interval:"(\<lambda>x. m *\<^sub>R (x::real^'n)) ` {a..b} =
(if {a..b} = {} then {} else if 0 \<le> m then {m *\<^sub>R a..m *\<^sub>R b} else {m *\<^sub>R b..m *\<^sub>R a})"
@@ -66,8 +66,7 @@
apply(rule_tac [!] allI)apply(rule_tac [!] impI)
apply(rule_tac[2] x="vec1 x" in exI)apply(rule_tac[4] x="vec1 x" in exI)
apply(rule_tac[6] x="vec1 x" in exI)apply(rule_tac[8] x="vec1 x" in exI)
- by (auto simp add: vector_less_def vector_le_def forall_1
- vec1_dest_vec1[unfolded One_nat_def])
+ by (auto simp add: vector_less_def vector_le_def)
lemma dest_vec1_setsum: assumes "finite S"
shows " dest_vec1 (setsum f S) = setsum (\<lambda>x. dest_vec1 (f x)) S"
@@ -90,7 +89,7 @@
using one_le_card_finite by auto
lemma real_dimindex_ge_1:"real (CARD('n::finite)) \<ge> 1"
- by(metis dimindex_ge_1 linorder_not_less real_eq_of_nat real_le_trans real_of_nat_1 real_of_nat_le_iff)
+ by(metis dimindex_ge_1 real_eq_of_nat real_of_nat_1 real_of_nat_le_iff)
lemma real_dimindex_gt_0:"real (CARD('n::finite)) > 0" apply(rule less_le_trans[OF _ real_dimindex_ge_1]) by auto
@@ -480,8 +479,8 @@
next
fix t u assume asm:"\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s" "finite (t::'a set)"
(*"finite t" "t \<subseteq> s" "\<forall>x\<in>t. (0::real) \<le> u x" "setsum u t = 1"*)
- from this(2) have "\<forall>u. t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s" apply(induct_tac t rule:finite_induct)
- prefer 3 apply (rule,rule) apply(erule conjE)+ proof-
+ from this(2) have "\<forall>u. t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s" apply(induct t rule:finite_induct)
+ prefer 2 apply (rule,rule) apply(erule conjE)+ proof-
fix x f u assume ind:"\<forall>u. f \<subseteq> s \<and> (\<forall>x\<in>f. 0 \<le> u x) \<and> setsum u f = 1 \<longrightarrow> (\<Sum>x\<in>f. u x *\<^sub>R x) \<in> s"
assume as:"finite f" "x \<notin> f" "insert x f \<subseteq> s" "\<forall>x\<in>insert x f. 0 \<le> u x" "setsum u (insert x f) = (1::real)"
show "(\<Sum>x\<in>insert x f. u x *\<^sub>R x) \<in> s" proof(cases "u x = 1")
@@ -776,7 +775,7 @@
lemma convex_cball:
fixes x :: "'a::real_normed_vector"
shows "convex(cball x e)"
-proof(auto simp add: convex_def Ball_def mem_cball)
+proof(auto simp add: convex_def Ball_def)
fix y z assume yz:"dist x y \<le> e" "dist x z \<le> e"
fix u v ::real assume uv:" 0 \<le> u" "0 \<le> v" "u + v = 1"
have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z" using uv yz
@@ -1144,7 +1143,7 @@
hence "x + y \<in> s" using `?lhs`[unfolded convex_def, THEN conjunct1]
apply(erule_tac x="2*\<^sub>R x" in ballE) apply(erule_tac x="2*\<^sub>R y" in ballE)
apply(erule_tac x="1/2" in allE) apply simp apply(erule_tac x="1/2" in allE) by auto }
- thus ?thesis unfolding convex_def cone_def by auto
+ thus ?thesis unfolding convex_def cone_def by blast
qed
lemma affine_dependent_biggerset: fixes s::"(real^'n) set"
@@ -1259,7 +1258,7 @@
fixes s :: "'a::real_normed_vector set"
assumes "open s"
shows "open(convex hull s)"
- unfolding open_contains_cball convex_hull_explicit unfolding mem_Collect_eq ball_simps(10)
+ unfolding open_contains_cball convex_hull_explicit unfolding mem_Collect_eq ball_simps(10)
proof(rule, rule) fix a
assume "\<exists>sa u. finite sa \<and> sa \<subseteq> s \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = a"
then obtain t u where obt:"finite t" "t\<subseteq>s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = a" by auto
@@ -1279,7 +1278,7 @@
hence "Min i \<le> b x" unfolding i_def apply(rule_tac Min_le) using obt(1) by auto
hence "x + (y - a) \<in> cball x (b x)" using y unfolding mem_cball dist_norm by auto
moreover from `x\<in>t` have "x\<in>s" using obt(2) by auto
- ultimately have "x + (y - a) \<in> s" using y and b[THEN bspec[where x=x]] unfolding subset_eq by auto }
+ ultimately have "x + (y - a) \<in> s" using y and b[THEN bspec[where x=x]] unfolding subset_eq by fast }
moreover
have *:"inj_on (\<lambda>v. v + (y - a)) t" unfolding inj_on_def by auto
have "(\<Sum>v\<in>(\<lambda>v. v + (y - a)) ` t. u (v - (y - a))) = 1"
@@ -1349,7 +1348,7 @@
show ?thesis unfolding caratheodory[of s]
proof(induct ("CARD('n) + 1"))
have *:"{x.\<exists>sa. finite sa \<and> sa \<subseteq> s \<and> card sa \<le> 0 \<and> x \<in> convex hull sa} = {}"
- using compact_empty by (auto simp add: convex_hull_empty)
+ using compact_empty by auto
case 0 thus ?case unfolding * by simp
next
case (Suc n)
@@ -1359,11 +1358,11 @@
fix x assume "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
then obtain t where t:"finite t" "t \<subseteq> s" "card t \<le> Suc n" "x \<in> convex hull t" by auto
show "x\<in>s" proof(cases "card t = 0")
- case True thus ?thesis using t(4) unfolding card_0_eq[OF t(1)] by(simp add: convex_hull_empty)
+ case True thus ?thesis using t(4) unfolding card_0_eq[OF t(1)] by simp
next
case False hence "card t = Suc 0" using t(3) `n=0` by auto
then obtain a where "t = {a}" unfolding card_Suc_eq by auto
- thus ?thesis using t(2,4) by (simp add: convex_hull_singleton)
+ thus ?thesis using t(2,4) by simp
qed
next
fix x assume "x\<in>s"
@@ -1398,7 +1397,7 @@
show ?P proof(cases "u={}")
case True hence "x=a" using t(4)[unfolded au] by auto
show ?P unfolding `x=a` apply(rule_tac x=a in exI, rule_tac x=a in exI, rule_tac x=1 in exI)
- using t and `n\<noteq>0` unfolding au by(auto intro!: exI[where x="{a}"] simp add: convex_hull_singleton)
+ using t and `n\<noteq>0` unfolding au by(auto intro!: exI[where x="{a}"])
next
case False obtain ux vx b where obt:"ux\<ge>0" "vx\<ge>0" "ux + vx = 1" "b \<in> convex hull u" "x = ux *\<^sub>R a + vx *\<^sub>R b"
using t(4)[unfolded au convex_hull_insert[OF False]] by auto
@@ -1411,7 +1410,7 @@
qed
thus ?thesis using compact_convex_combinations[OF assms Suc] by simp
qed
- qed
+ qed
qed
lemma finite_imp_compact_convex_hull:
@@ -1851,7 +1850,7 @@
lemma convex_hull_scaling:
"convex hull ((\<lambda>x. c *\<^sub>R x) ` s) = (\<lambda>x. c *\<^sub>R x) ` (convex hull s)"
apply(cases "c=0") defer apply(rule convex_hull_bilemma[rule_format, of _ _ inverse]) apply(rule convex_hull_scaling_lemma)
- unfolding image_image scaleR_scaleR by(auto simp add:image_constant_conv convex_hull_eq_empty)
+ unfolding image_image scaleR_scaleR by(auto simp add:image_constant_conv)
lemma convex_hull_affinity:
"convex hull ((\<lambda>x. a + c *\<^sub>R x) ` s) = (\<lambda>x. a + c *\<^sub>R x) ` (convex hull s)"
@@ -2313,7 +2312,7 @@
hence "a < b" unfolding * using as(4) apply(rule_tac mult_left_less_imp_less) by(auto simp add: ring_simps)
hence "u * a + v * b \<le> b" unfolding ** using **(2) as(3) by(auto simp add: field_simps intro!:mult_right_mono) }
ultimately show "u *\<^sub>R x + v *\<^sub>R y \<in> s" apply- apply(rule assms[unfolded is_interval_def, rule_format, OF as(1,2)])
- using as(3-) dimindex_ge_1 apply- by(auto simp add: vector_component) qed
+ using as(3-) dimindex_ge_1 by auto qed
lemma is_interval_connected:
fixes s :: "(real ^ _) set"
@@ -2336,7 +2335,7 @@
hence *:"dest_vec1 a < dest_vec1 x" "dest_vec1 x < dest_vec1 b" apply(rule_tac [!] ccontr) unfolding not_less by auto
let ?halfl = "{z. inner (basis 1) z < dest_vec1 x} " and ?halfr = "{z. inner (basis 1) z > dest_vec1 x} "
{ fix y assume "y \<in> s" have "y \<in> ?halfr \<union> ?halfl" apply(rule ccontr)
- using as(6) `y\<in>s` by (auto simp add: inner_vector_def dest_vec1_eq) }
+ using as(6) `y\<in>s` by (auto simp add: inner_vector_def) }
moreover have "a\<in>?halfl" "b\<in>?halfr" using * by (auto simp add: inner_vector_def)
hence "?halfl \<inter> s \<noteq> {}" "?halfr \<inter> s \<noteq> {}" using as(2-3) by auto
ultimately show False apply(rule_tac notE[OF as(1)[unfolded connected_def]])
@@ -2374,7 +2373,7 @@
shows "\<exists>x\<in>{a..b}. (f x)$k = y"
apply(subst neg_equal_iff_equal[THEN sym]) unfolding vector_uminus_component[THEN sym]
apply(rule ivt_increasing_component_on_1) using assms using continuous_on_neg
- by(auto simp add:vector_uminus_component)
+ by auto
lemma ivt_decreasing_component_1: fixes f::"real^1 \<Rightarrow> real^'n"
shows "dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> \<forall>x\<in>{a .. b}. continuous (at x) f
@@ -2415,7 +2414,7 @@
unfolding i'(1) xi_def apply(rule_tac Min_le) unfolding image_iff
defer apply(rule_tac x=j in bexI) using i' by auto
have i01:"x$i \<le> 1" "x$i > 0" using Suc(2)[unfolded mem_interval,rule_format,of i] using i'(2) `x$i \<noteq> 0`
- by(auto simp add: Cart_lambda_beta)
+ by auto
show ?thesis proof(cases "x$i=1")
case True have "\<forall>j\<in>{i. x$i \<noteq> 0}. x$j = 1" apply(rule, rule ccontr) unfolding mem_Collect_eq proof-
fix j assume "x $ j \<noteq> 0" "x $ j \<noteq> 1"
@@ -2424,21 +2423,21 @@
hence "x$j \<ge> x$i" unfolding i'(1) xi_def apply(rule_tac Min_le) by auto
thus False using True Suc(2) j by(auto simp add: vector_le_def elim!:ballE[where x=j]) qed
thus "x\<in>convex hull ?points" apply(rule_tac hull_subset[unfolded subset_eq, rule_format])
- by(auto simp add: Cart_lambda_beta)
+ by auto
next let ?y = "\<lambda>j. if x$j = 0 then 0 else (x$j - x$i) / (1 - x$i)"
case False hence *:"x = x$i *\<^sub>R (\<chi> j. if x$j = 0 then 0 else 1) + (1 - x$i) *\<^sub>R (\<chi> j. ?y j)" unfolding Cart_eq
- by(auto simp add: Cart_lambda_beta vector_add_component vector_smult_component vector_minus_component field_simps)
+ by(auto simp add: field_simps)
{ fix j have "x$j \<noteq> 0 \<Longrightarrow> 0 \<le> (x $ j - x $ i) / (1 - x $ i)" "(x $ j - x $ i) / (1 - x $ i) \<le> 1"
apply(rule_tac divide_nonneg_pos) using i(1)[of j] using False i01
- using Suc(2)[unfolded mem_interval, rule_format, of j] by(auto simp add:field_simps Cart_lambda_beta)
+ using Suc(2)[unfolded mem_interval, rule_format, of j] by(auto simp add:field_simps)
hence "0 \<le> ?y j \<and> ?y j \<le> 1" by auto }
- moreover have "i\<in>{j. x$j \<noteq> 0} - {j. ((\<chi> j. ?y j)::real^'n) $ j \<noteq> 0}" using i01 by(auto simp add: Cart_lambda_beta)
+ moreover have "i\<in>{j. x$j \<noteq> 0} - {j. ((\<chi> j. ?y j)::real^'n) $ j \<noteq> 0}" using i01 by auto
hence "{j. x$j \<noteq> 0} \<noteq> {j. ((\<chi> j. ?y j)::real^'n) $ j \<noteq> 0}" by auto
- hence **:"{j. ((\<chi> j. ?y j)::real^'n) $ j \<noteq> 0} \<subset> {j. x$j \<noteq> 0}" apply - apply rule by(auto simp add: Cart_lambda_beta)
+ hence **:"{j. ((\<chi> j. ?y j)::real^'n) $ j \<noteq> 0} \<subset> {j. x$j \<noteq> 0}" apply - apply rule by auto
have "card {j. ((\<chi> j. ?y j)::real^'n) $ j \<noteq> 0} \<le> n" using less_le_trans[OF psubset_card_mono[OF _ **] Suc(4)] by auto
ultimately show ?thesis apply(subst *) apply(rule convex_convex_hull[unfolded convex_def, rule_format])
apply(rule_tac hull_subset[unfolded subset_eq, rule_format]) defer apply(rule Suc(1))
- unfolding mem_interval using i01 Suc(3) by (auto simp add: Cart_lambda_beta)
+ unfolding mem_interval using i01 Suc(3) by auto
qed qed qed } note * = this
show ?thesis apply rule defer apply(rule hull_minimal) unfolding subset_eq prefer 3 apply rule
apply(rule_tac n2="CARD('n)" in *) prefer 3 apply(rule card_mono) using 01 and convex_interval(1) prefer 5 apply - apply rule
@@ -2453,7 +2452,7 @@
prefer 3 apply(rule unit_interval_convex_hull) apply rule unfolding mem_Collect_eq proof-
fix x::"real^'n" assume as:"\<forall>i. x $ i = 0 \<or> x $ i = 1"
show "x \<in> (\<lambda>s. \<chi> i. if i \<in> s then 1 else 0) ` UNIV" apply(rule image_eqI[where x="{i. x$i = 1}"])
- unfolding Cart_eq using as by(auto simp add:Cart_lambda_beta) qed auto
+ unfolding Cart_eq using as by auto qed auto
subsection {* Hence any cube (could do any nonempty interval). *}
@@ -2464,23 +2463,23 @@
unfolding image_iff defer apply(erule bexE) proof-
fix y assume as:"y\<in>{x - ?d .. x + ?d}"
{ fix i::'n have "x $ i \<le> d + y $ i" "y $ i \<le> d + x $ i" using as[unfolded mem_interval, THEN spec[where x=i]]
- by(auto simp add: vector_component)
+ by auto
hence "1 \<ge> inverse d * (x $ i - y $ i)" "1 \<ge> inverse d * (y $ i - x $ i)"
apply(rule_tac[!] mult_left_le_imp_le[OF _ assms]) unfolding mult_assoc[THEN sym]
- using assms by(auto simp add: field_simps right_inverse)
+ using assms by(auto simp add: field_simps)
hence "inverse d * (x $ i * 2) \<le> 2 + inverse d * (y $ i * 2)"
"inverse d * (y $ i * 2) \<le> 2 + inverse d * (x $ i * 2)" by(auto simp add:field_simps) }
hence "inverse (2 * d) *\<^sub>R (y - (x - ?d)) \<in> {0..1}" unfolding mem_interval using assms
- by(auto simp add: Cart_eq vector_component_simps field_simps)
+ by(auto simp add: Cart_eq field_simps)
thus "\<exists>z\<in>{0..1}. y = x - ?d + (2 * d) *\<^sub>R z" apply- apply(rule_tac x="inverse (2 * d) *\<^sub>R (y - (x - ?d))" in bexI)
- using assms by(auto simp add: Cart_eq vector_le_def Cart_lambda_beta)
+ using assms by(auto simp add: Cart_eq vector_le_def)
next
fix y z assume as:"z\<in>{0..1}" "y = x - ?d + (2*d) *\<^sub>R z"
have "\<And>i. 0 \<le> d * z $ i \<and> d * z $ i \<le> d" using assms as(1)[unfolded mem_interval] apply(erule_tac x=i in allE)
apply rule apply(rule mult_nonneg_nonneg) prefer 3 apply(rule mult_right_le_one_le)
- using assms by(auto simp add: vector_component_simps Cart_eq)
+ using assms by(auto simp add: Cart_eq)
thus "y \<in> {x - ?d..x + ?d}" unfolding as(2) mem_interval apply- apply rule using as(1)[unfolded mem_interval]
- apply(erule_tac x=i in allE) using assms by(auto simp add: vector_component_simps Cart_eq) qed
+ apply(erule_tac x=i in allE) using assms by(auto simp add: Cart_eq) qed
obtain s where "finite s" "{0..1::real^'n} = convex hull s" using unit_cube_convex_hull by auto
thus ?thesis apply(rule_tac that[of "(\<lambda>y. x - ?d + (2 * d) *\<^sub>R y)` s"]) unfolding * and convex_hull_affinity by auto qed
@@ -2570,8 +2569,8 @@
have "0 < d" unfolding d_def using `e>0` dimge1 by(rule_tac divide_pos_pos, auto)
let ?d = "(\<chi> i. d)::real^'n"
obtain c where c:"finite c" "{x - ?d..x + ?d} = convex hull c" using cube_convex_hull[OF `d>0`, of x] by auto
- have "x\<in>{x - ?d..x + ?d}" using `d>0` unfolding mem_interval by(auto simp add:vector_component_simps)
- hence "c\<noteq>{}" using c by(auto simp add:convex_hull_empty)
+ have "x\<in>{x - ?d..x + ?d}" using `d>0` unfolding mem_interval by auto
+ hence "c\<noteq>{}" using c by auto
def k \<equiv> "Max (f ` c)"
have "convex_on {x - ?d..x + ?d} f" apply(rule convex_on_subset[OF assms(2)])
apply(rule subset_trans[OF _ e(1)]) unfolding subset_eq mem_cball proof
@@ -2579,7 +2578,7 @@
have e:"e = setsum (\<lambda>i. d) (UNIV::'n set)" unfolding setsum_constant d_def using dimge1
by (metis eq_divide_imp mult_frac_num real_dimindex_gt_0 real_eq_of_nat real_less_def real_mult_commute)
show "dist x z \<le> e" unfolding dist_norm e apply(rule_tac order_trans[OF norm_le_l1], rule setsum_mono)
- using z[unfolded mem_interval] apply(erule_tac x=i in allE) by(auto simp add:field_simps vector_component_simps) qed
+ using z[unfolded mem_interval] apply(erule_tac x=i in allE) by auto qed
hence k:"\<forall>y\<in>{x - ?d..x + ?d}. f y \<le> k" unfolding c(2) apply(rule_tac convex_on_convex_hull_bound) apply assumption
unfolding k_def apply(rule, rule Max_ge) using c(1) by auto
have "d \<le> e" unfolding d_def apply(rule mult_imp_div_pos_le) using `e>0` dimge1 unfolding mult_le_cancel_left1 using real_dimindex_ge_1 by auto
@@ -2588,9 +2587,9 @@
hence "\<forall>y\<in>cball x d. abs (f y) \<le> k + 2 * abs (f x)" apply(rule_tac convex_bounds_lemma) apply assumption proof
fix y assume y:"y\<in>cball x d"
{ fix i::'n have "x $ i - d \<le> y $ i" "y $ i \<le> x $ i + d"
- using order_trans[OF component_le_norm y[unfolded mem_cball dist_norm], of i] by(auto simp add: vector_component) }
+ using order_trans[OF component_le_norm y[unfolded mem_cball dist_norm], of i] by auto }
thus "f y \<le> k" apply(rule_tac k[rule_format]) unfolding mem_cball mem_interval dist_norm
- by(auto simp add: vector_component_simps) qed
+ by auto qed
hence "continuous_on (ball x d) f" apply(rule_tac convex_on_bounded_continuous)
apply(rule open_ball, rule convex_on_subset[OF conv], rule ball_subset_cball)
apply force
@@ -2716,17 +2715,17 @@
unfolding as(1) by(auto simp add:algebra_simps)
show "norm (a - x) *\<^sub>R (x - b) = norm (x - b) *\<^sub>R (a - x)"
unfolding norm_minus_commute[of x a] * Cart_eq using as(2,3)
- by(auto simp add: vector_component_simps field_simps)
+ by(auto simp add: field_simps)
next assume as:"dist a b = dist a x + dist x b"
have "norm (a - x) / norm (a - b) \<le> 1" unfolding divide_le_eq_1_pos[OF Fal2] unfolding as[unfolded dist_norm] norm_ge_zero by auto
thus "\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1" apply(rule_tac x="dist a x / dist a b" in exI)
unfolding dist_norm Cart_eq apply- apply rule defer apply(rule, rule divide_nonneg_pos) prefer 4 proof rule
fix i::'n have "((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) $ i =
((norm (a - b) - norm (a - x)) * (a $ i) + norm (a - x) * (b $ i)) / norm (a - b)"
- using Fal by(auto simp add:vector_component_simps field_simps)
+ using Fal by(auto simp add: field_simps)
also have "\<dots> = x$i" apply(rule divide_eq_imp[OF Fal])
unfolding as[unfolded dist_norm] using as[unfolded dist_triangle_eq Cart_eq,rule_format, of i]
- by(auto simp add:field_simps vector_component_simps)
+ by(auto simp add:field_simps)
finally show "x $ i = ((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) $ i" by auto
qed(insert Fal2, auto) qed qed
@@ -2735,7 +2734,7 @@
"between (b,a) (midpoint a b)" (is ?t2)
proof- have *:"\<And>x y z. x = (1/2::real) *\<^sub>R z \<Longrightarrow> y = (1/2) *\<^sub>R z \<Longrightarrow> norm z = norm x + norm y" by auto
show ?t1 ?t2 unfolding between midpoint_def dist_norm apply(rule_tac[!] *)
- by(auto simp add:field_simps Cart_eq vector_component_simps) qed
+ by(auto simp add:field_simps Cart_eq) qed
lemma between_mem_convex_hull:
"between (a,b) x \<longleftrightarrow> x \<in> convex hull {a,b}"
@@ -2754,7 +2753,7 @@
have *:"y = (1 - (1 - e)) *\<^sub>R ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) + (1 - e) *\<^sub>R x" using `e>0` by (auto simp add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
have "dist c ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) = abs(1/e) * norm (e *\<^sub>R c - y + (1 - e) *\<^sub>R x)"
unfolding dist_norm unfolding norm_scaleR[THEN sym] apply(rule norm_eqI) using `e>0`
- by(auto simp add:vector_component_simps Cart_eq field_simps)
+ by(auto simp add: Cart_eq field_simps)
also have "\<dots> = abs(1/e) * norm (x - e *\<^sub>R (x - c) - y)" by(auto intro!:norm_eqI simp add: algebra_simps)
also have "\<dots> < d" using as[unfolded dist_norm] and `e>0`
by(auto simp add:pos_divide_less_eq[OF `e>0`] real_mult_commute)
@@ -2826,11 +2825,11 @@
fix x::"real^'n" and e assume "0<e" and as:"\<forall>xa. dist x xa < e \<longrightarrow> (\<forall>x. 0 \<le> xa $ x) \<and> setsum (op $ xa) UNIV \<le> 1"
show "(\<forall>xa. 0 < x $ xa) \<and> setsum (op $ x) UNIV < 1" apply(rule,rule) proof-
fix i::'n show "0 < x $ i" using as[THEN spec[where x="x - (e / 2) *\<^sub>R basis i"]] and `e>0`
- unfolding dist_norm by(auto simp add: norm_basis vector_component_simps basis_component elim:allE[where x=i])
+ unfolding dist_norm by(auto simp add: norm_basis elim:allE[where x=i])
next guess a using UNIV_witness[where 'a='n] ..
have **:"dist x (x + (e / 2) *\<^sub>R basis a) < e" using `e>0` and norm_basis[of a]
- unfolding dist_norm by(auto simp add: vector_component_simps basis_component intro!: mult_strict_left_mono_comm)
- have "\<And>i. (x + (e / 2) *\<^sub>R basis a) $ i = x$i + (if i = a then e/2 else 0)" by(auto simp add:vector_component_simps)
+ unfolding dist_norm by(auto intro!: mult_strict_left_mono_comm)
+ have "\<And>i. (x + (e / 2) *\<^sub>R basis a) $ i = x$i + (if i = a then e/2 else 0)" by auto
hence *:"setsum (op $ (x + (e / 2) *\<^sub>R basis a)) UNIV = setsum (\<lambda>i. x$i + (if a = i then e/2 else 0)) UNIV" by(rule_tac setsum_cong, auto)
have "setsum (op $ x) UNIV < setsum (op $ (x + (e / 2) *\<^sub>R basis a)) UNIV" unfolding * setsum_addf
using `0<e` dimindex_ge_1 by(auto simp add: setsum_delta')
@@ -2847,13 +2846,13 @@
fix y assume y:"dist x y < min (Min (op $ x ` UNIV)) ?d"
have "setsum (op $ y) UNIV \<le> setsum (\<lambda>i. x$i + ?d) UNIV" proof(rule setsum_mono)
fix i::'n have "abs (y$i - x$i) < ?d" apply(rule le_less_trans) using component_le_norm[of "y - x" i]
- using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2] by(auto simp add:vector_component_simps norm_minus_commute)
+ using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2] by(auto simp add: norm_minus_commute)
thus "y $ i \<le> x $ i + ?d" by auto qed
also have "\<dots> \<le> 1" unfolding setsum_addf setsum_constant card_enum real_eq_of_nat using dimindex_ge_1 by(auto simp add: Suc_le_eq)
finally show "(\<forall>i. 0 \<le> y $ i) \<and> setsum (op $ y) UNIV \<le> 1" apply- proof(rule,rule)
fix i::'n have "norm (x - y) < x$i" using y[unfolded min_less_iff_conj dist_norm, THEN conjunct1]
- using Min_gr_iff[of "op $ x ` dimset x"] dimindex_ge_1 by auto
- thus "0 \<le> y$i" using component_le_norm[of "x - y" i] and as(1)[rule_format, of i] by(auto simp add: vector_component_simps)
+ by auto
+ thus "0 \<le> y$i" using component_le_norm[of "x - y" i] and as(1)[rule_format, of i] by auto
qed auto qed auto qed
lemma interior_std_simplex_nonempty: obtains a::"real^'n" where
@@ -3040,9 +3039,6 @@
apply(erule_tac x="1-x" in ballE, erule_tac x="1-y" in ballE)
unfolding mem_interval_1 by auto
-(** move this **)
-declare vector_scaleR_component[simp]
-
lemma simple_path_join_loop:
assumes "injective_path g1" "injective_path g2" "pathfinish g2 = pathstart g1"
"(path_image g1 \<inter> path_image g2) \<subseteq> {pathstart g1,pathstart g2}"
@@ -3390,7 +3386,7 @@
hence "\<psi> k \<noteq> \<psi> (Suc k)" using \<psi>[unfolded bij_betw_def inj_on_def, THEN conjunct1, THEN bspec[where x=k]] by auto
hence **:"?basis k + ?basis (Suc k) \<in> {x. 0 < inner (?basis (Suc k)) x} \<inter> (?A k)"
"?basis k - ?basis (Suc k) \<in> {x. 0 > inner (?basis (Suc k)) x} \<inter> ({x. 0 < inner (?basis (Suc k)) x} \<union> (?A k))" using d
- by(auto simp add: inner_basis vector_component_simps intro!:bexI[where x=k])
+ by(auto simp add: inner_basis intro!:bexI[where x=k])
show ?thesis unfolding * Un_assoc apply(rule path_connected_Un) defer apply(rule path_connected_Un)
prefer 5 apply(rule_tac[1-2] convex_imp_path_connected, rule convex_halfspace_lt, rule convex_halfspace_gt)
apply(rule Suc(1)) using d ** False by auto
@@ -3404,7 +3400,7 @@
apply(rule_tac[5] x=" ?basis 1 + ?basis 2" in nequals0I)
apply(rule_tac[6] x="-?basis 1 + ?basis 2" in nequals0I)
apply(rule_tac[7] x="-?basis 1 - ?basis 2" in nequals0I)
- using d unfolding *** by(auto intro!: convex_halfspace_gt convex_halfspace_lt, auto simp add:vector_component_simps inner_basis)
+ using d unfolding *** by(auto intro!: convex_halfspace_gt convex_halfspace_lt, auto simp add: inner_basis)
qed qed auto qed note lem = this
have ***:"\<And>x::real^'n. (\<exists>i\<in>{1..CARD('n)}. inner (basis (\<psi> i)) x \<noteq> 0) \<longleftrightarrow> (\<exists>i. inner (basis i) x \<noteq> 0)"
@@ -3432,7 +3428,7 @@
apply(rule, rule continuous_at_within_inv[unfolded o_def inverse_eq_divide]) apply(rule continuous_at_within)
apply(rule continuous_at_norm[unfolded o_def]) by auto
thus ?thesis unfolding * ** using path_connected_punctured_universe[OF assms]
- by(auto intro!: path_connected_continuous_image continuous_on_intros continuous_on_mul) qed
+ by(auto intro!: path_connected_continuous_image continuous_on_intros) qed
lemma connected_sphere: "2 \<le> CARD('n) \<Longrightarrow> connected {x::real^'n. norm(x - a) = r}"
using path_connected_sphere path_connected_imp_connected by auto
--- a/src/HOL/Multivariate_Analysis/Derivative.thy Sun Apr 25 23:22:29 2010 -0700
+++ b/src/HOL/Multivariate_Analysis/Derivative.thy Mon Apr 26 09:21:25 2010 -0700
@@ -93,7 +93,7 @@
subsection {* Derivatives on real = Derivatives on @{typ "real^1"} *}
-lemma dist_vec1_0[simp]: "dist(vec1 (x::real)) 0 = norm x" unfolding vector_dist_norm by(auto simp add:vec1_dest_vec1_simps)
+lemma dist_vec1_0[simp]: "dist(vec1 (x::real)) 0 = norm x" unfolding vector_dist_norm by auto
lemma bounded_linear_vec1_dest_vec1: fixes f::"real \<Rightarrow> real"
shows "linear (vec1 \<circ> f \<circ> dest_vec1) = bounded_linear f" (is "?l = ?r") proof-
@@ -281,6 +281,8 @@
subsection {* differentiability. *}
+no_notation Deriv.differentiable (infixl "differentiable" 60)
+
definition differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" (infixr "differentiable" 30) where
"f differentiable net \<equiv> (\<exists>f'. (f has_derivative f') net)"
@@ -754,11 +756,11 @@
lemma onorm_vec1: fixes f::"real \<Rightarrow> real"
shows "onorm (\<lambda>x. vec1 (f (dest_vec1 x))) = onorm f" proof-
have "\<forall>x::real^1. norm x = 1 \<longleftrightarrow> x\<in>{vec1 -1, vec1 (1::real)}" unfolding forall_vec1 by(auto simp add:Cart_eq)
- hence 1:"{x. norm x = 1} = {vec1 -1, vec1 (1::real)}" by(auto simp add:norm_vec1)
+ hence 1:"{x. norm x = 1} = {vec1 -1, vec1 (1::real)}" by auto
have 2:"{norm (vec1 (f (dest_vec1 x))) |x. norm x = 1} = (\<lambda>x. norm (vec1 (f (dest_vec1 x)))) ` {x. norm x=1}" by auto
have "\<forall>x::real. norm x = 1 \<longleftrightarrow> x\<in>{-1, 1}" by auto hence 3:"{x. norm x = 1} = {-1, (1::real)}" by auto
have 4:"{norm (f x) |x. norm x = 1} = (\<lambda>x. norm (f x)) ` {x. norm x=1}" by auto
- show ?thesis unfolding onorm_def 1 2 3 4 by(simp add:Sup_finite_Max norm_vec1) qed
+ show ?thesis unfolding onorm_def 1 2 3 4 by(simp add:Sup_finite_Max) qed
lemma differentiable_bound_real: fixes f::"real \<Rightarrow> real"
assumes "convex s" "\<forall>x\<in>s. (f has_derivative f' x) (at x within s)" "\<forall>x\<in>s. onorm(f' x) \<le> B" and x:"x\<in>s" and y:"y\<in>s"
@@ -1013,7 +1015,7 @@
unfolding ph' * apply(rule diff_chain_within) defer apply(rule bounded_linear.has_derivative[OF assms(3)])
apply(rule has_derivative_intros) defer apply(rule has_derivative_sub[where g'="\<lambda>x.0",unfolded diff_0_right])
apply(rule has_derivative_at_within) using assms(5) and `u\<in>s` `a\<in>s`
- by(auto intro!: has_derivative_intros derivative_linear)
+ by(auto intro!: has_derivative_intros derivative_linear)
have **:"bounded_linear (\<lambda>x. f' u x - f' a x)" "bounded_linear (\<lambda>x. f' a x - f' u x)" apply(rule_tac[!] bounded_linear_sub)
apply(rule_tac[!] derivative_linear) using assms(5) `u\<in>s` `a\<in>s` by auto
have "onorm (\<lambda>v. v - g' (f' u v)) \<le> onorm g' * onorm (\<lambda>w. f' a w - f' u w)" unfolding * apply(rule onorm_compose)
--- a/src/HOL/Multivariate_Analysis/Determinants.thy Sun Apr 25 23:22:29 2010 -0700
+++ b/src/HOL/Multivariate_Analysis/Determinants.thy Mon Apr 26 09:21:25 2010 -0700
@@ -813,7 +813,7 @@
lemma cramer:
fixes A ::"real^'n^'n"
assumes d0: "det A \<noteq> 0"
- shows "A *v x = b \<longleftrightarrow> x = (\<chi> k. det(\<chi> i j. if j=k then b$i else A$i$j :: real^'n^'n) / det A)"
+ shows "A *v x = b \<longleftrightarrow> x = (\<chi> k. det(\<chi> i j. if j=k then b$i else A$i$j) / det A)"
proof-
from d0 obtain B where B: "A ** B = mat 1" "B ** A = mat 1"
unfolding invertible_det_nz[symmetric] invertible_def by blast
@@ -821,7 +821,7 @@
hence "A *v (B *v b) = b" by (simp add: matrix_vector_mul_assoc)
then have xe: "\<exists>x. A*v x = b" by blast
{fix x assume x: "A *v x = b"
- have "x = (\<chi> k. det(\<chi> i j. if j=k then b$i else A$i$j :: real^'n^'n) / det A)"
+ have "x = (\<chi> k. det(\<chi> i j. if j=k then b$i else A$i$j) / det A)"
unfolding x[symmetric]
using d0 by (simp add: Cart_eq cramer_lemma field_simps)}
with xe show ?thesis by auto
@@ -993,15 +993,15 @@
moreover
{assume "x = 0" "y \<noteq> 0"
then have "dist (?g x) (?g y) = dist x y"
- apply (simp add: dist_norm norm_mul)
+ apply (simp add: dist_norm)
apply (rule f1[rule_format])
- by(simp add: norm_mul field_simps)}
+ by(simp add: field_simps)}
moreover
{assume "x \<noteq> 0" "y = 0"
then have "dist (?g x) (?g y) = dist x y"
- apply (simp add: dist_norm norm_mul)
+ apply (simp add: dist_norm)
apply (rule f1[rule_format])
- by(simp add: norm_mul field_simps)}
+ by(simp add: field_simps)}
moreover
{assume z: "x \<noteq> 0" "y \<noteq> 0"
have th00: "x = norm x *s (inverse (norm x) *s x)" "y = norm y *s (inverse (norm y) *s y)" "norm x *s f ((inverse (norm x) *s x)) = norm x *s f (inverse (norm x) *s x)"
@@ -1013,7 +1013,7 @@
"norm (f (inverse (norm x) *s x) - f (inverse (norm y) *s y)) =
norm (inverse (norm x) *s x - inverse (norm y) *s y)"
using z
- by (auto simp add: vector_smult_assoc field_simps norm_mul intro: f1[rule_format] fd1[rule_format, unfolded dist_norm])
+ by (auto simp add: vector_smult_assoc field_simps intro: f1[rule_format] fd1[rule_format, unfolded dist_norm])
from z th0[OF th00] have "dist (?g x) (?g y) = dist x y"
by (simp add: dist_norm)}
ultimately have "dist (?g x) (?g y) = dist x y" by blast}
@@ -1047,7 +1047,7 @@
by (simp add: nat_number setprod_numseg mult_commute)
lemma det_1: "det (A::'a::comm_ring_1^1^1) = A$1$1"
- by (simp add: det_def permutes_sing sign_id UNIV_1)
+ by (simp add: det_def sign_id UNIV_1)
lemma det_2: "det (A::'a::comm_ring_1^2^2) = A$1$1 * A$2$2 - A$1$2 * A$2$1"
proof-
@@ -1057,7 +1057,7 @@
unfolding setsum_over_permutations_insert[OF f12]
unfolding permutes_sing
apply (simp add: sign_swap_id sign_id swap_id_eq)
- by (simp add: arith_simps(31)[symmetric] of_int_minus of_int_1 del: arith_simps(31))
+ by (simp add: arith_simps(31)[symmetric] del: arith_simps(31))
qed
lemma det_3: "det (A::'a::comm_ring_1^3^3) =
@@ -1078,7 +1078,7 @@
unfolding permutes_sing
apply (simp add: sign_swap_id permutation_swap_id sign_compose sign_id swap_id_eq)
- apply (simp add: arith_simps(31)[symmetric] of_int_minus of_int_1 del: arith_simps(31))
+ apply (simp add: arith_simps(31)[symmetric] del: arith_simps(31))
by (simp add: ring_simps)
qed
--- a/src/HOL/Multivariate_Analysis/Euclidean_Space.thy Sun Apr 25 23:22:29 2010 -0700
+++ b/src/HOL/Multivariate_Analysis/Euclidean_Space.thy Mon Apr 26 09:21:25 2010 -0700
@@ -670,7 +670,7 @@
subsection{* The collapse of the general concepts to dimension one. *}
lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
- by (simp add: Cart_eq forall_1)
+ by (simp add: Cart_eq)
lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
apply auto
@@ -775,8 +775,7 @@
lemma sqrt_even_pow2: assumes n: "even n"
shows "sqrt(2 ^ n) = 2 ^ (n div 2)"
proof-
- from n obtain m where m: "n = 2*m" unfolding even_nat_equiv_def2
- by (auto simp add: nat_number)
+ from n obtain m where m: "n = 2*m" unfolding even_mult_two_ex ..
from m have "sqrt(2 ^ n) = sqrt ((2 ^ m) ^ 2)"
by (simp only: power_mult[symmetric] mult_commute)
then show ?thesis using m by simp
@@ -785,7 +784,7 @@
lemma real_div_sqrt: "0 <= x ==> x / sqrt(x) = sqrt(x)"
apply (cases "x = 0", simp_all)
using sqrt_divide_self_eq[of x]
- apply (simp add: inverse_eq_divide real_sqrt_ge_0_iff field_simps)
+ apply (simp add: inverse_eq_divide field_simps)
done
text{* Hence derive more interesting properties of the norm. *}
@@ -798,8 +797,8 @@
by (rule norm_zero)
lemma norm_mul[simp]: "norm(a *s x) = abs(a) * norm x"
- by (simp add: norm_vector_def vector_component setL2_right_distrib
- abs_mult cong: strong_setL2_cong)
+ by (simp add: norm_vector_def setL2_right_distrib abs_mult)
+
lemma norm_eq_0_dot: "(norm x = 0) \<longleftrightarrow> (inner x x = (0::real))"
by (simp add: norm_vector_def setL2_def power2_eq_square)
lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero)
@@ -866,10 +865,14 @@
by (auto simp add: norm_eq_sqrt_inner)
lemma real_abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> (x::real)^2 \<le> y^2"
-proof-
- have "x^2 \<le> y^2 \<longleftrightarrow> (x -y) * (y + x) \<le> 0" by (simp add: ring_simps power2_eq_square)
- also have "\<dots> \<longleftrightarrow> \<bar>x\<bar> \<le> \<bar>y\<bar>" apply (simp add: zero_compare_simps real_abs_def not_less) by arith
-finally show ?thesis ..
+proof
+ assume "\<bar>x\<bar> \<le> \<bar>y\<bar>"
+ then have "\<bar>x\<bar>\<twosuperior> \<le> \<bar>y\<bar>\<twosuperior>" by (rule power_mono, simp)
+ then show "x\<twosuperior> \<le> y\<twosuperior>" by simp
+next
+ assume "x\<twosuperior> \<le> y\<twosuperior>"
+ then have "sqrt (x\<twosuperior>) \<le> sqrt (y\<twosuperior>)" by (rule real_sqrt_le_mono)
+ then show "\<bar>x\<bar> \<le> \<bar>y\<bar>" by simp
qed
lemma norm_le_square: "norm(x) <= a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x <= a^2"
@@ -1179,7 +1182,7 @@
assumes fS: "finite S"
shows "setsum f S *s v = setsum (\<lambda>x. f x *s v) S"
proof(induct rule: finite_induct[OF fS])
- case 1 then show ?case by (simp add: vector_smult_lzero)
+ case 1 then show ?case by simp
next
case (2 x F)
from "2.hyps" have "setsum f (insert x F) *s v = (f x + setsum f F) *s v"
@@ -1398,7 +1401,7 @@
apply (subgoal_tac "vector [v$1] = v")
apply simp
apply (vector vector_def)
- apply (simp add: forall_1)
+ apply simp
done
lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
@@ -1536,7 +1539,7 @@
unfolding norm_mul
apply (simp only: mult_commute)
apply (rule mult_mono)
- by (auto simp add: ring_simps norm_ge_zero) }
+ by (auto simp add: ring_simps) }
then have th: "\<forall>i\<in> ?S. norm ((x$i) *s f (basis i :: real ^'m)) \<le> norm (f (basis i)) * norm x" by metis
from real_setsum_norm_le[OF fS, of "\<lambda>i. (x$i) *s (f (basis i))", OF th]
have "norm (f x) \<le> ?B * norm x" unfolding th0 setsum_left_distrib by metis}
@@ -1553,9 +1556,9 @@
let ?K = "\<bar>B\<bar> + 1"
have Kp: "?K > 0" by arith
{assume C: "B < 0"
- have "norm (1::real ^ 'n) > 0" by (simp add: zero_less_norm_iff)
+ have "norm (1::real ^ 'n) > 0" by simp
with C have "B * norm (1:: real ^ 'n) < 0"
- by (simp add: zero_compare_simps)
+ by (simp add: mult_less_0_iff)
with B[rule_format, of 1] norm_ge_zero[of "f 1"] have False by simp
}
then have Bp: "B \<ge> 0" by ferrack
@@ -1677,11 +1680,11 @@
apply (rule real_setsum_norm_le)
using fN fM
apply simp
- apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh] norm_mul ring_simps)
+ apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh] ring_simps)
apply (rule mult_mono)
- apply (auto simp add: norm_ge_zero zero_le_mult_iff component_le_norm)
+ apply (auto simp add: zero_le_mult_iff component_le_norm)
apply (rule mult_mono)
- apply (auto simp add: norm_ge_zero zero_le_mult_iff component_le_norm)
+ apply (auto simp add: zero_le_mult_iff component_le_norm)
done}
then show ?thesis by metis
qed
@@ -1701,7 +1704,7 @@
have "B * norm x * norm y \<le> ?K * norm x * norm y"
apply -
apply (rule mult_right_mono, rule mult_right_mono)
- by (auto simp add: norm_ge_zero)
+ by auto
then have "norm (h x y) \<le> ?K * norm x * norm y"
using B[rule_format, of x y] by simp}
with Kp show ?thesis by blast
@@ -2006,8 +2009,8 @@
have uv': "u = 0 \<longrightarrow> v \<noteq> 0" using u v uv by arith
have "a = a * (u + v)" unfolding uv by simp
hence th: "u * a + v * a = a" by (simp add: ring_simps)
- from xa u have "u \<noteq> 0 \<Longrightarrow> u*x < u*a" by (simp add: mult_compare_simps)
- from ya v have "v \<noteq> 0 \<Longrightarrow> v * y < v * a" by (simp add: mult_compare_simps)
+ from xa u have "u \<noteq> 0 \<Longrightarrow> u*x < u*a" by (simp add: mult_strict_left_mono)
+ from ya v have "v \<noteq> 0 \<Longrightarrow> v * y < v * a" by (simp add: mult_strict_left_mono)
from xa ya u v have "u * x + v * y < u * a + v * a"
apply (cases "u = 0", simp_all add: uv')
apply(rule mult_strict_left_mono)
@@ -2048,7 +2051,7 @@
assumes x: "0 <= (x::real)" and y:"0 <= y" and z: "0 <= z" and xy: "x^2 <= y^2 + z^2"
shows "x <= y + z"
proof-
- have "y^2 + z^2 \<le> y^2 + 2*y*z + z^2" using z y by (simp add: zero_compare_simps)
+ have "y^2 + z^2 \<le> y^2 + 2*y*z + z^2" using z y by (simp add: mult_nonneg_nonneg)
with xy have th: "x ^2 \<le> (y+z)^2" by (simp add: power2_eq_square ring_simps)
from y z have yz: "y + z \<ge> 0" by arith
from power2_le_imp_le[OF th yz] show ?thesis .
@@ -2100,10 +2103,10 @@
{assume x0: "x \<noteq> 0"
hence n0: "norm x \<noteq> 0" by (metis norm_eq_zero)
let ?c = "1/ norm x"
- have "norm (?c*s x) = 1" using x0 by (simp add: n0 norm_mul)
+ have "norm (?c*s x) = 1" using x0 by (simp add: n0)
with H have "norm (f(?c*s x)) \<le> b" by blast
hence "?c * norm (f x) \<le> b"
- by (simp add: linear_cmul[OF lf] norm_mul)
+ by (simp add: linear_cmul[OF lf])
hence "norm (f x) \<le> b * norm x"
using n0 norm_ge_zero[of x] by (auto simp add: field_simps)}
ultimately have "norm (f x) \<le> b * norm x" by blast}
@@ -2221,18 +2224,24 @@
where "dest_vec1 x \<equiv> (x$1)"
lemma vec1_component[simp]: "(vec1 x)$1 = x"
- by (simp add: )
-
-lemma vec1_dest_vec1[simp]: "vec1(dest_vec1 x) = x" "dest_vec1(vec1 y) = y"
- by (simp_all add: Cart_eq forall_1)
-
-lemma forall_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (vec1 x))" by (metis vec1_dest_vec1)
-
-lemma exists_vec1: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P(vec1 x))" by (metis vec1_dest_vec1)
-
-lemma vec1_eq[simp]: "vec1 x = vec1 y \<longleftrightarrow> x = y" by (metis vec1_dest_vec1)
-
-lemma dest_vec1_eq[simp]: "dest_vec1 x = dest_vec1 y \<longleftrightarrow> x = y" by (metis vec1_dest_vec1)
+ by simp
+
+lemma vec1_dest_vec1: "vec1(dest_vec1 x) = x" "dest_vec1(vec1 y) = y"
+ by (simp_all add: Cart_eq)
+
+declare vec1_dest_vec1(1) [simp]
+
+lemma forall_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (vec1 x))"
+ by (metis vec1_dest_vec1(1))
+
+lemma exists_vec1: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P(vec1 x))"
+ by (metis vec1_dest_vec1(1))
+
+lemma vec1_eq[simp]: "vec1 x = vec1 y \<longleftrightarrow> x = y"
+ by (metis vec1_dest_vec1(2))
+
+lemma dest_vec1_eq[simp]: "dest_vec1 x = dest_vec1 y \<longleftrightarrow> x = y"
+ by (metis vec1_dest_vec1(1))
lemma vec_in_image_vec: "vec x \<in> (vec ` S) \<longleftrightarrow> x \<in> S" by auto
@@ -2260,8 +2269,8 @@
lemma dest_vec1_sum: assumes fS: "finite S"
shows "dest_vec1(setsum f S) = setsum (dest_vec1 o f) S"
apply (induct rule: finite_induct[OF fS])
- apply (simp add: dest_vec1_vec)
- apply (auto simp add:vector_minus_component)
+ apply simp
+ apply auto
done
lemma norm_vec1: "norm(vec1 x) = abs(x)"
@@ -2270,7 +2279,7 @@
lemma dist_vec1: "dist(vec1 x) (vec1 y) = abs(x - y)"
by (simp only: dist_real vec1_component)
lemma abs_dest_vec1: "norm x = \<bar>dest_vec1 x\<bar>"
- by (metis vec1_dest_vec1 norm_vec1)
+ by (metis vec1_dest_vec1(1) norm_vec1)
lemmas vec1_dest_vec1_simps = forall_vec1 vec_add[THEN sym] dist_vec1 vec_sub[THEN sym] vec1_dest_vec1 norm_vec1 vector_smult_component
vec1_eq vec_cmul[THEN sym] smult_conv_scaleR[THEN sym] o_def dist_real_def norm_vec1 real_norm_def
@@ -2298,7 +2307,7 @@
shows "f = (\<lambda>x. vec1(row 1 (matrix f) \<bullet> x))"
apply (rule ext)
apply (subst matrix_works[OF lf, symmetric])
- apply (simp add: Cart_eq matrix_vector_mult_def row_def inner_vector_def mult_commute forall_1)
+ apply (simp add: Cart_eq matrix_vector_mult_def row_def inner_vector_def mult_commute)
done
lemma dest_vec1_eq_0: "dest_vec1 x = 0 \<longleftrightarrow> x = 0"
@@ -2366,13 +2375,13 @@
by (induct rule: finite_induct[OF fS], simp_all add: vec_0[symmetric] del: vec_0)
lemma pastecart_vec[simp]: "pastecart (vec x) (vec x) = vec x"
- by (simp add: pastecart_eq fstcart_pastecart sndcart_pastecart)
+ by (simp add: pastecart_eq)
lemma pastecart_add[simp]:"pastecart (x1::'a::{plus,times}^_) y1 + pastecart x2 y2 = pastecart (x1 + x2) (y1 + y2)"
- by (simp add: pastecart_eq fstcart_pastecart sndcart_pastecart)
+ by (simp add: pastecart_eq)
lemma pastecart_cmul[simp]: "pastecart (c *s (x1::'a::{plus,times}^_)) (c *s y1) = c *s pastecart x1 y1"
- by (simp add: pastecart_eq fstcart_pastecart sndcart_pastecart)
+ by (simp add: pastecart_eq)
lemma pastecart_neg[simp]: "pastecart (- (x::'a::ring_1^_)) (- y) = - pastecart x y"
unfolding vector_sneg_minus1 pastecart_cmul ..
@@ -2384,7 +2393,7 @@
fixes f:: "'d \<Rightarrow> 'a::semiring_1^_"
assumes fS: "finite S"
shows "pastecart (setsum f S) (setsum g S) = setsum (\<lambda>i. pastecart (f i) (g i)) S"
- by (simp add: pastecart_eq fstcart_setsum[OF fS] sndcart_setsum[OF fS] fstcart_pastecart sndcart_pastecart)
+ by (simp add: pastecart_eq fstcart_setsum[OF fS] sndcart_setsum[OF fS])
lemma setsum_Plus:
"\<lbrakk>finite A; finite B\<rbrakk> \<Longrightarrow>
@@ -2402,7 +2411,7 @@
lemma norm_fstcart: "norm(fstcart x) <= norm (x::real ^('n::finite + 'm::finite))"
proof-
have th0: "norm x = norm (pastecart (fstcart x) (sndcart x))"
- by (simp add: pastecart_fst_snd)
+ by simp
have th1: "fstcart x \<bullet> fstcart x \<le> pastecart (fstcart x) (sndcart x) \<bullet> pastecart (fstcart x) (sndcart x)"
by (simp add: inner_vector_def setsum_UNIV_sum pastecart_def setsum_nonneg)
then show ?thesis
@@ -2417,7 +2426,7 @@
lemma norm_sndcart: "norm(sndcart x) <= norm (x::real ^('n::finite + 'm::finite))"
proof-
have th0: "norm x = norm (pastecart (fstcart x) (sndcart x))"
- by (simp add: pastecart_fst_snd)
+ by simp
have th1: "sndcart x \<bullet> sndcart x \<le> pastecart (fstcart x) (sndcart x) \<bullet> pastecart (fstcart x) (sndcart x)"
by (simp add: inner_vector_def setsum_UNIV_sum pastecart_def setsum_nonneg)
then show ?thesis
@@ -2519,7 +2528,7 @@
lemma real_arch_inv: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
using reals_Archimedean
- apply (auto simp add: field_simps inverse_positive_iff_positive)
+ apply (auto simp add: field_simps)
apply (subgoal_tac "inverse (real n) > 0")
apply arith
apply simp
@@ -2732,7 +2741,8 @@
"0 \<in> span S"
"x\<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
"x \<in> span S \<Longrightarrow> c *s x \<in> span S"
- by (metis span_def hull_subset subset_eq subspace_span subspace_def)+
+ by (metis span_def hull_subset subset_eq)
+ (metis subspace_span subspace_def)+
lemma span_induct: assumes SP: "\<And>x. x \<in> S ==> P x"
and P: "subspace P" and x: "x \<in> span S" shows "P x"
@@ -2830,7 +2840,7 @@
(* Individual closure properties. *)
-lemma span_superset: "x \<in> S ==> x \<in> span S" by (metis span_clauses)
+lemma span_superset: "x \<in> S ==> x \<in> span S" by (metis span_clauses(1))
lemma span_0: "0 \<in> span S" by (metis subspace_span subspace_0)
@@ -2847,8 +2857,7 @@
by (metis subspace_span subspace_sub)
lemma span_setsum: "finite A \<Longrightarrow> \<forall>x \<in> A. f x \<in> span S ==> setsum f A \<in> span S"
- apply (rule subspace_setsum)
- by (metis subspace_span subspace_setsum)+
+ by (rule subspace_setsum, rule subspace_span)
lemma span_add_eq: "(x::'a::ring_1^_) \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
apply (auto simp only: span_add span_sub)
@@ -3110,7 +3119,7 @@
from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0" by auto
have s0: "setsum (\<lambda>v. ?u v *s v) ?S = 0"
using fS aS
- apply (simp add: vector_smult_lneg vector_smult_lid setsum_clauses ring_simps )
+ apply (simp add: vector_smult_lneg setsum_clauses ring_simps)
apply (subst (2) ua[symmetric])
apply (rule setsum_cong2)
by auto
@@ -3479,7 +3488,7 @@
lemma basis_card_eq_dim:
"B \<subseteq> (V:: (real ^'n) set) \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V"
- by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_mono independent_bound)
+ by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_bound)
lemma dim_unique: "(B::(real ^'n) set) \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> card B = n \<Longrightarrow> dim V = n"
by (metis basis_card_eq_dim)
@@ -3669,7 +3678,7 @@
apply (subst Cy)
using C(1) fth
apply (simp only: setsum_clauses) unfolding smult_conv_scaleR
- apply (auto simp add: inner_simps inner_eq_zero_iff inner_commute[of y a] dot_lsum[OF fth])
+ apply (auto simp add: inner_simps inner_commute[of y a] dot_lsum[OF fth])
apply (rule setsum_0')
apply clarsimp
apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
@@ -3686,7 +3695,7 @@
using C(1) fth
apply (simp only: setsum_clauses) unfolding smult_conv_scaleR
apply (subst inner_commute[of x])
- apply (auto simp add: inner_simps inner_eq_zero_iff inner_commute[of x a] dot_rsum[OF fth])
+ apply (auto simp add: inner_simps inner_commute[of x a] dot_rsum[OF fth])
apply (rule setsum_0')
apply clarsimp
apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
@@ -3759,10 +3768,10 @@
apply (subst B') using fB fth
unfolding setsum_clauses(2)[OF fth]
apply simp unfolding inner_simps smult_conv_scaleR
- apply (clarsimp simp add: inner_simps inner_eq_zero_iff smult_conv_scaleR dot_lsum)
+ apply (clarsimp simp add: inner_simps smult_conv_scaleR dot_lsum)
apply (rule setsum_0', rule ballI)
unfolding inner_commute
- by (auto simp add: x field_simps inner_eq_zero_iff intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])}
+ by (auto simp add: x field_simps intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])}
then show "\<forall>x \<in> B. ?a \<bullet> x = 0" by blast
qed
with a0 show ?thesis unfolding sSB by (auto intro: exI[where x="?a"])
@@ -3915,7 +3924,7 @@
{assume xb: "x \<in> b"
have h0: "0 = ?h x"
apply (rule conjunct2[OF h, rule_format])
- apply (metis span_superset insertI1 xb x)
+ apply (metis span_superset x)
apply simp
apply (metis span_superset xb)
done
@@ -4262,8 +4271,7 @@
{fix y have "?P y"
proof(rule span_induct_alt[of ?P "columns A"])
show "\<exists>x\<Colon>real ^ 'm. setsum (\<lambda>i. (x$i) *s column i A) ?U = 0"
- apply (rule exI[where x=0])
- by (simp add: zero_index vector_smult_lzero)
+ by (rule exI[where x=0], simp)
next
fix c y1 y2 assume y1: "y1 \<in> columns A" and y2: "?P y2"
from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
@@ -4687,7 +4695,7 @@
hence d2: "(sqrt (real ?d))^2 = real ?d"
by (auto intro: real_sqrt_pow2)
have th: "sqrt (real ?d) * infnorm x \<ge> 0"
- by (simp add: zero_le_mult_iff real_sqrt_ge_0_iff infnorm_pos_le)
+ by (simp add: zero_le_mult_iff infnorm_pos_le)
have th1: "x \<bullet> x \<le> (sqrt (real ?d) * infnorm x)^2"
unfolding power_mult_distrib d2
unfolding real_of_nat_def inner_vector_def
@@ -4861,4 +4869,3 @@
done
end
-
\ No newline at end of file
--- a/src/HOL/Multivariate_Analysis/Integration.thy Sun Apr 25 23:22:29 2010 -0700
+++ b/src/HOL/Multivariate_Analysis/Integration.thy Mon Apr 26 09:21:25 2010 -0700
@@ -933,7 +933,7 @@
lemma has_integral_vec1: assumes "(f has_integral k) {a..b}"
shows "((\<lambda>x. vec1 (f x)) has_integral (vec1 k)) {a..b}"
proof- have *:"\<And>p. (\<Sum>(x, k)\<in>p. content k *\<^sub>R vec1 (f x)) - vec1 k = vec1 ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k)"
- unfolding vec_sub Cart_eq by(auto simp add:vec1_dest_vec1_simps split_beta)
+ unfolding vec_sub Cart_eq by(auto simp add: split_beta)
show ?thesis using assms unfolding has_integral apply safe
apply(erule_tac x=e in allE,safe) apply(rule_tac x=d in exI,safe)
apply(erule_tac x=p in allE,safe) unfolding * norm_vector_1 by auto qed
@@ -1356,7 +1356,7 @@
lemma has_integral_eq_eq:
shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> ((f has_integral k) s \<longleftrightarrow> (g has_integral k) s)"
- using has_integral_eq[of s f g] has_integral_eq[of s g f] by auto
+ using has_integral_eq[of s f g] has_integral_eq[of s g f] by rule auto
lemma has_integral_null[dest]:
assumes "content({a..b}) = 0" shows "(f has_integral 0) ({a..b})"
@@ -1653,7 +1653,7 @@
proof- have *:"{a..b} = ({a..b} \<inter> {x. x$k \<le> c}) \<union> ({a..b} \<inter> {x. x$k \<ge> c})" by auto
show ?thesis apply(subst *) apply(rule tagged_division_union[OF assms])
unfolding interval_split interior_closed_interval
- by(auto simp add: vector_less_def Cart_lambda_beta elim!:allE[where x=k]) qed
+ by(auto simp add: vector_less_def elim!:allE[where x=k]) qed
lemma has_integral_separate_sides: fixes f::"real^'m \<Rightarrow> 'a::real_normed_vector"
assumes "(f has_integral i) ({a..b})" "e>0"
@@ -1743,11 +1743,11 @@
subsection {* Using additivity of lifted function to encode definedness. *}
lemma forall_option: "(\<forall>x. P x) \<longleftrightarrow> P None \<and> (\<forall>x. P(Some x))"
- by (metis map_of.simps option.nchotomy)
+ by (metis option.nchotomy)
lemma exists_option:
"(\<exists>x. P x) \<longleftrightarrow> P None \<or> (\<exists>x. P(Some x))"
- by (metis map_of.simps option.nchotomy)
+ by (metis option.nchotomy)
fun lifted where
"lifted (opp::'a\<Rightarrow>'a\<Rightarrow>'b) (Some x) (Some y) = Some(opp x y)" |
@@ -1842,9 +1842,8 @@
lemma operative_content[intro]: "operative (op +) content"
unfolding operative_def content_split[THEN sym] neutral_add by auto
-lemma neutral_monoid[simp]: "neutral ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a) = 0"
- unfolding neutral_def apply(rule some_equality) defer
- apply(erule_tac x=0 in allE) by auto
+lemma neutral_monoid: "neutral ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a) = 0"
+ by (rule neutral_add) (* FIXME: duplicate *)
lemma monoidal_monoid[intro]:
shows "monoidal ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a)"
@@ -1941,7 +1940,7 @@
apply(rule_tac x="(k,(interval_lowerbound l)$k)" in exI) defer
apply(rule_tac x="(k,(interval_upperbound l)$k)" in exI)
unfolding division_points_def unfolding interval_bounds[OF ab]
- apply (auto simp add:interval_bounds) unfolding * by auto
+ apply auto unfolding * by auto
thus "?D1 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) by auto
have *:"interval_lowerbound ({a..b} \<inter> {x. x $ k \<ge> interval_lowerbound l $ k}) $ k = interval_lowerbound l $ k"
@@ -1952,7 +1951,7 @@
apply(rule_tac x="(k,(interval_lowerbound l)$k)" in exI) defer
apply(rule_tac x="(k,(interval_upperbound l)$k)" in exI)
unfolding division_points_def unfolding interval_bounds[OF ab]
- apply (auto simp add:interval_bounds) unfolding * by auto
+ apply auto unfolding * by auto
thus "?D2 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) by auto qed
subsection {* Preservation by divisions and tagged divisions. *}
@@ -2254,7 +2253,7 @@
assumes "p tagged_division_of {a..b}" "\<forall>x\<in>{a..b}. (f x)$i \<le> (g x)$i"
shows "(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p)$i \<le> (setsum (\<lambda>(x,k). content k *\<^sub>R g x) p)$i"
unfolding setsum_component apply(rule setsum_mono)
- apply(rule mp) defer apply assumption apply(induct_tac x,rule) unfolding split_conv
+ apply(rule mp) defer apply assumption unfolding split_paired_all apply rule unfolding split_conv
proof- fix a b assume ab:"(a,b) \<in> p" note assm = tagged_division_ofD(2-4)[OF assms(1) ab]
from this(3) guess u v apply-by(erule exE)+ note b=this
show "(content b *\<^sub>R f a) $ i \<le> (content b *\<^sub>R g a) $ i" unfolding b
@@ -2903,7 +2902,7 @@
shows "setsum (\<lambda>(x,k). f(interval_upperbound k) - f(interval_lowerbound k)) p = f b - f a"
proof- let ?f = "(\<lambda>k::(real^1) set. if k = {} then 0 else f(interval_upperbound k) - f(interval_lowerbound k))"
have *:"operative op + ?f" unfolding operative_1_lt[OF monoidal_monoid] interval_eq_empty_1
- by(auto simp add:not_less interval_bound_1 vector_less_def)
+ by(auto simp add:not_less vector_less_def)
have **:"{a..b} \<noteq> {}" using assms(1) by auto note operative_tagged_division[OF monoidal_monoid * assms(2)]
note * = this[unfolded if_not_P[OF **] interval_bound_1[OF assms(1)],THEN sym ]
show ?thesis unfolding * apply(subst setsum_iterate[THEN sym]) defer
@@ -3340,7 +3339,7 @@
proof- { presume *:"a < b \<Longrightarrow> ?thesis"
show ?thesis proof(cases,rule *,assumption)
assume "\<not> a < b" hence "a = b" using assms(1) by auto
- hence *:"{vec a .. vec b} = {vec b}" "f b - f a = 0" apply(auto simp add: Cart_simps) by smt
+ hence *:"{vec a .. vec b} = {vec b}" "f b - f a = 0" by(auto simp add: Cart_eq vector_le_def order_antisym)
show ?thesis unfolding *(2) apply(rule has_integral_null) unfolding content_eq_0_1 using * `a=b` by auto
qed } assume ab:"a < b"
let ?P = "\<lambda>e. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {vec1 a..vec1 b} \<and> d fine p \<longrightarrow>
@@ -3422,7 +3421,7 @@
hence "\<forall>i. u$i \<le> v$i" and uv:"{u,v}\<subseteq>{u..v}" using p(2)[OF as(1)] by auto note this(1) this(1)[unfolded forall_1]
note result = as(2)[unfolded k interval_bounds[OF this(1)] content_1[OF this(2)]]
- assume as':"x \<noteq> vec1 a" "x \<noteq> vec1 b" hence "x$1 \<in> {a<..<b}" using p(2-3)[OF as(1)] by(auto simp add:Cart_simps) note * = d(2)[OF this]
+ assume as':"x \<noteq> vec1 a" "x \<noteq> vec1 b" hence "x$1 \<in> {a<..<b}" using p(2-3)[OF as(1)] by(auto simp add: Cart_eq) note * = d(2)[OF this]
have "norm ((v$1 - u$1) *\<^sub>R f' (x$1) - (f (v$1) - f (u$1))) =
norm ((f (u$1) - f (x$1) - (u$1 - x$1) *\<^sub>R f' (x$1)) - (f (v$1) - f (x$1) - (v$1 - x$1) *\<^sub>R f' (x$1)))"
apply(rule arg_cong[of _ _ norm]) unfolding scaleR_left.diff by auto
@@ -3669,7 +3668,7 @@
from indefinite_integral_continuous_left[OF assms(1) `a<b` this `e>0`] guess d . note d=this
show ?thesis apply(rule,rule,rule d,safe) apply(subst dist_commute)
unfolding `x=b` vector_dist_norm apply(rule d(2)[rule_format]) unfolding norm_real by auto
- next assume "a<x \<and> x<b" hence xl:"a<x" "x\<le>b" and xr:"a\<le>x" "x<b" by(auto simp add:Cart_simps)
+ next assume "a<x \<and> x<b" hence xl:"a<x" "x\<le>b" and xr:"a\<le>x" "x<b" by(auto simp add: vector_less_def)
from indefinite_integral_continuous_left [OF assms(1) xl `e>0`] guess d1 . note d1=this
from indefinite_integral_continuous_right[OF assms(1) xr `e>0`] guess d2 . note d2=this
show ?thesis apply(rule_tac x="min d1 d2" in exI)
@@ -3726,7 +3725,7 @@
unfolding o_def using assms(5) defer apply-apply(rule)
proof- fix t assume as:"t\<in>{0..1} - {t. (1 - t) *\<^sub>R c + t *\<^sub>R x \<in> k}"
have *:"c - t *\<^sub>R c + t *\<^sub>R x \<in> s - k" apply safe apply(rule conv[unfolded scaleR_simps])
- using `x\<in>s` `c\<in>s` as by(auto simp add:scaleR_simps)
+ using `x\<in>s` `c\<in>s` as by(auto simp add: algebra_simps)
have "(f \<circ> (\<lambda>t. (1 - t) *\<^sub>R c + t *\<^sub>R x) has_derivative (\<lambda>x. 0) \<circ> (\<lambda>z. (0 - z *\<^sub>R c) + z *\<^sub>R x)) (at t within {0..1})"
apply(rule diff_chain_within) apply(rule has_derivative_add)
unfolding scaleR_simps apply(rule has_derivative_sub) apply(rule has_derivative_const)
@@ -3949,7 +3948,7 @@
lemma has_integral_spike_set_eq: fixes f::"real^'n \<Rightarrow> 'a::banach"
assumes "negligible((s - t) \<union> (t - s))" shows "((f has_integral y) s \<longleftrightarrow> (f has_integral y) t)"
- unfolding has_integral_restrict_univ[THEN sym,of f] apply(rule has_integral_spike_eq[OF assms]) by auto
+ unfolding has_integral_restrict_univ[THEN sym,of f] apply(rule has_integral_spike_eq[OF assms]) by (safe, auto split: split_if_asm)
lemma has_integral_spike_set[dest]: fixes f::"real^'n \<Rightarrow> 'a::banach"
assumes "negligible((s - t) \<union> (t - s))" "(f has_integral y) s"
--- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Sun Apr 25 23:22:29 2010 -0700
+++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Mon Apr 26 09:21:25 2010 -0700
@@ -48,16 +48,17 @@
"\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"
"\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"
using openin[of U] unfolding istopology_def Collect_def mem_def
- by (metis mem_def subset_eq)+
+ unfolding subset_eq Ball_def mem_def by auto
lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"
unfolding topspace_def by blast
lemma openin_empty[simp]: "openin U {}" by (simp add: openin_clauses)
lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)"
- by (simp add: openin_clauses)
-
-lemma openin_Union[intro]: "(\<forall>S \<in>K. openin U S) \<Longrightarrow> openin U (\<Union> K)" by (simp add: openin_clauses)
+ using openin_clauses by simp
+
+lemma openin_Union[intro]: "(\<forall>S \<in>K. openin U S) \<Longrightarrow> openin U (\<Union> K)"
+ using openin_clauses by simp
lemma openin_Un[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<union> T)"
using openin_Union[of "{S,T}" U] by auto
@@ -946,7 +947,7 @@
by (metis frontier_def closure_closed Diff_subset)
lemma frontier_empty[simp]: "frontier {} = {}"
- by (simp add: frontier_def closure_empty)
+ by (simp add: frontier_def)
lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
proof-
@@ -954,7 +955,7 @@
hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto
hence "closed S" using closure_subset_eq by auto
}
- thus ?thesis using frontier_subset_closed[of S] by auto
+ thus ?thesis using frontier_subset_closed[of S] ..
qed
lemma frontier_complement: "frontier(- S) = frontier S"
@@ -1588,7 +1589,7 @@
(* FIXME: Only one congruence rule for tendsto can be used at a time! *)
-lemma Lim_cong_within[cong add]:
+lemma Lim_cong_within(*[cong add]*):
fixes a :: "'a::metric_space"
fixes l :: "'b::metric_space" (* TODO: generalize *)
shows "(\<And>x. x \<noteq> a \<Longrightarrow> f x = g x) ==> ((\<lambda>x. f x) ---> l) (at a within S) \<longleftrightarrow> ((g ---> l) (at a within S))"
@@ -1667,7 +1668,7 @@
{ fix e::real assume "e>0"
hence "\<exists>N::nat. \<forall>n::nat\<ge>N. inverse (real n) < e"
using real_arch_inv[of e] apply auto apply(rule_tac x=n in exI)
- by (metis not_le le_imp_inverse_le not_less real_of_nat_gt_zero_cancel_iff real_of_nat_less_iff xt1(7))
+ by (metis le_imp_inverse_le not_less real_of_nat_gt_zero_cancel_iff real_of_nat_less_iff xt1(7))
}
thus ?thesis unfolding Lim_sequentially dist_norm by simp
qed
@@ -1702,7 +1703,7 @@
apply (simp add: interior_def, safe)
apply (force simp add: open_contains_cball)
apply (rule_tac x="ball x e" in exI)
- apply (simp add: open_ball centre_in_ball subset_trans [OF ball_subset_cball])
+ apply (simp add: subset_trans [OF ball_subset_cball])
done
lemma islimpt_ball:
@@ -1877,14 +1878,14 @@
lemma frontier_ball:
fixes a :: "'a::real_normed_vector"
shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"
- apply (simp add: frontier_def closure_ball interior_open open_ball order_less_imp_le)
+ apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)
apply (simp add: expand_set_eq)
by arith
lemma frontier_cball:
fixes a :: "'a::{real_normed_vector, perfect_space}"
shows "frontier(cball a e) = {x. dist a x = e}"
- apply (simp add: frontier_def interior_cball closed_cball closure_closed order_less_imp_le)
+ apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)
apply (simp add: expand_set_eq)
by arith
@@ -2004,9 +2005,10 @@
lemma bounded_ball[simp,intro]: "bounded(ball x e)"
by (metis ball_subset_cball bounded_cball bounded_subset)
-lemma finite_imp_bounded[intro]: assumes "finite S" shows "bounded S"
+lemma finite_imp_bounded[intro]:
+ fixes S :: "'a::metric_space set" assumes "finite S" shows "bounded S"
proof-
- { fix a F assume as:"bounded F"
+ { fix a and F :: "'a set" assume as:"bounded F"
then obtain x e where "\<forall>y\<in>F. dist x y \<le> e" unfolding bounded_def by auto
hence "\<forall>y\<in>(insert a F). dist x y \<le> max e (dist x a)" by auto
hence "bounded (insert a F)" unfolding bounded_def by (intro exI)
@@ -2216,7 +2218,7 @@
apply (rule allI, rule impI, rule ext)
apply (erule conjE)
apply (induct_tac x)
-apply (simp add: nat_rec_0)
+apply simp
apply (erule_tac x="n" in allE)
apply (simp)
done
@@ -2648,7 +2650,8 @@
{ fix x y assume "x\<in>t" "y\<in>t" "f x = f y"
hence "x \<in> f x" "y \<in> f x \<longrightarrow> y = x" using f[THEN bspec[where x=x]] and `t\<subseteq>s` by auto
hence "x = y" using `f x = f y` and f[THEN bspec[where x=y]] and `y\<in>t` and `t\<subseteq>s` by auto }
- hence "infinite (f ` t)" using assms(2) using finite_imageD[unfolded inj_on_def, of f t] by auto
+ hence "inj_on f t" unfolding inj_on_def by simp
+ hence "infinite (f ` t)" using assms(2) using finite_imageD by auto
moreover
{ fix x assume "x\<in>t" "f x \<notin> g"
from g(3) assms(3) `x\<in>t` obtain h where "h\<in>g" and "x\<in>h" by auto
@@ -3143,7 +3146,7 @@
using `?lhs`[unfolded continuous_within Lim_within] by auto
{ fix y assume "y\<in>f ` (ball x d \<inter> s)"
hence "y \<in> ball (f x) e" using d(2) unfolding dist_nz[THEN sym]
- apply (auto simp add: dist_commute mem_ball) apply(erule_tac x=xa in ballE) apply auto using `e>0` by auto
+ apply (auto simp add: dist_commute) apply(erule_tac x=xa in ballE) apply auto using `e>0` by auto
}
hence "\<exists>d>0. f ` (ball x d \<inter> s) \<subseteq> ball (f x) e" using `d>0` unfolding subset_eq ball_def by (auto simp add: dist_commute) }
thus ?rhs by auto
@@ -3875,7 +3878,7 @@
fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
assumes "connected s" "continuous_on s f" "open {x \<in> s. f x = a}" "\<exists>x \<in> s. f x = a"
shows "\<forall>x \<in> s. f x = a"
-using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by auto
+using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by fast
text{* Some arithmetical combinations (more to prove). *}
@@ -4387,7 +4390,7 @@
using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda>n. sndcart (x n)"], THEN spec[where x="sndcart l"]]
unfolding Lim_sequentially by auto
hence "l \<in> {pastecart x y |x y. x \<in> s \<and> y \<in> t}" apply- unfolding mem_Collect_eq apply(rule_tac x="fstcart l" in exI,rule_tac x="sndcart l" in exI) by auto }
- thus ?thesis unfolding closed_sequential_limits by auto
+ thus ?thesis unfolding closed_sequential_limits by blast
qed
lemma compact_pastecart:
@@ -4478,7 +4481,7 @@
have "{x - y | x y . x\<in>s \<and> y\<in>s} \<noteq> {}" using `s \<noteq> {}` by auto
then obtain x where x:"x\<in>{x - y |x y. x \<in> s \<and> y \<in> s}" "\<forall>y\<in>{x - y |x y. x \<in> s \<and> y \<in> s}. norm y \<le> norm x"
using compact_differences[OF assms(1) assms(1)]
- using distance_attains_sup[where 'a="'a", unfolded dist_norm, of "{x - y | x y . x\<in>s \<and> y\<in>s}" 0] by(auto simp add: norm_minus_cancel)
+ using distance_attains_sup[where 'a="'a", unfolded dist_norm, of "{x - y | x y . x\<in>s \<and> y\<in>s}" 0] by auto
from x(1) obtain a b where "a\<in>s" "b\<in>s" "x = a - b" by auto
thus ?thesis using x(2)[unfolded `x = a - b`] by blast
qed
@@ -4499,7 +4502,7 @@
hence "norm (x - y) \<le> 2 * a" using norm_triangle_ineq[of x "-y", unfolded norm_minus_cancel] a[THEN bspec[where x=x]] a[THEN bspec[where x=y]] by (auto simp add: ring_simps) }
note * = this
{ fix x y assume "x\<in>s" "y\<in>s" hence "s \<noteq> {}" by auto
- have "norm(x - y) \<le> diameter s" unfolding diameter_def using `s\<noteq>{}` *[OF `x\<in>s` `y\<in>s`] `x\<in>s` `y\<in>s`
+ have "norm(x - y) \<le> diameter s" unfolding diameter_def using `s\<noteq>{}` *[OF `x\<in>s` `y\<in>s`] `x\<in>s` `y\<in>s`
by simp (blast intro!: Sup_upper *) }
moreover
{ fix d::real assume "d>0" "d < diameter s"
@@ -4530,10 +4533,10 @@
proof-
have b:"bounded s" using assms(1) by (rule compact_imp_bounded)
then obtain x y where xys:"x\<in>s" "y\<in>s" and xy:"\<forall>u\<in>s. \<forall>v\<in>s. norm (u - v) \<le> norm (x - y)" using compact_sup_maxdistance[OF assms] by auto
- hence "diameter s \<le> norm (x - y)"
- by (force simp add: diameter_def intro!: Sup_least)
+ hence "diameter s \<le> norm (x - y)"
+ unfolding diameter_def by clarsimp (rule Sup_least, fast+)
thus ?thesis
- by (metis b diameter_bounded_bound order_antisym xys)
+ by (metis b diameter_bounded_bound order_antisym xys)
qed
text{* Related results with closure as the conclusion. *}
@@ -4722,7 +4725,7 @@
lemma mem_interval_1: fixes x :: "real^1" shows
"(x \<in> {a .. b} \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b)"
"(x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"
-by(simp_all add: Cart_eq vector_less_def vector_le_def forall_1)
+by(simp_all add: Cart_eq vector_less_def vector_le_def)
lemma vec1_interval:fixes a::"real" shows
"vec1 ` {a .. b} = {vec1 a .. vec1 b}"
@@ -4748,7 +4751,7 @@
have "a$i < b$i" using as[THEN spec[where x=i]] by auto
hence "a$i < ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i < b$i"
unfolding vector_smult_component and vector_add_component
- by (auto simp add: less_divide_eq_number_of1) }
+ by auto }
hence "{a <..< b} \<noteq> {}" using mem_interval(1)[of "?x" a b] by auto }
ultimately show ?th1 by blast
@@ -4763,7 +4766,7 @@
have "a$i \<le> b$i" using as[THEN spec[where x=i]] by auto
hence "a$i \<le> ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i \<le> b$i"
unfolding vector_smult_component and vector_add_component
- by (auto simp add: less_divide_eq_number_of1) }
+ by auto }
hence "{a .. b} \<noteq> {}" using mem_interval(2)[of "?x" a b] by auto }
ultimately show ?th2 by blast
qed
@@ -4826,13 +4829,13 @@
{ fix j
have "c $ j < ?x $ j \<and> ?x $ j < d $ j" unfolding Cart_lambda_beta
apply(cases "j=i") using as(2)[THEN spec[where x=j]]
- by (auto simp add: less_divide_eq_number_of1 as2) }
+ by (auto simp add: as2) }
hence "?x\<in>{c<..<d}" unfolding mem_interval by auto
moreover
have "?x\<notin>{a .. b}"
unfolding mem_interval apply auto apply(rule_tac x=i in exI)
using as(2)[THEN spec[where x=i]] and as2
- by (auto simp add: less_divide_eq_number_of1)
+ by auto
ultimately have False using as by auto }
hence "a$i \<le> c$i" by(rule ccontr)auto
moreover
@@ -4841,13 +4844,13 @@
{ fix j
have "d $ j > ?x $ j \<and> ?x $ j > c $ j" unfolding Cart_lambda_beta
apply(cases "j=i") using as(2)[THEN spec[where x=j]]
- by (auto simp add: less_divide_eq_number_of1 as2) }
+ by (auto simp add: as2) }
hence "?x\<in>{c<..<d}" unfolding mem_interval by auto
moreover
have "?x\<notin>{a .. b}"
unfolding mem_interval apply auto apply(rule_tac x=i in exI)
using as(2)[THEN spec[where x=i]] and as2
- by (auto simp add: less_divide_eq_number_of1)
+ by auto
ultimately have False using as by auto }
hence "b$i \<ge> d$i" by(rule ccontr)auto
ultimately
@@ -4878,7 +4881,7 @@
lemma inter_interval: fixes a :: "'a::linorder^'n" shows
"{a .. b} \<inter> {c .. d} = {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}"
unfolding expand_set_eq and Int_iff and mem_interval
- by (auto simp add: less_divide_eq_number_of1 intro!: bexI)
+ by auto
(* Moved interval_open_subset_closed a bit upwards *)
@@ -4918,7 +4921,7 @@
using open_interval[of "vec1 a" "vec1 b"] unfolding open_contains_ball
apply-apply(rule,erule_tac x="vec1 x" in ballE) apply(erule exE,rule_tac x=e in exI)
unfolding subset_eq mem_ball apply(rule) defer apply(rule,erule conjE,erule_tac x="vec1 xa" in ballE)
- by(auto simp add: vec1_dest_vec1_simps vector_less_def forall_1)
+ by(auto simp add: dist_vec1 dist_real_def vector_less_def)
lemma closed_interval[intro]: fixes a :: "real^'n" shows "closed {a .. b}"
proof-
@@ -4999,7 +5002,7 @@
have "a $ i < ((1 / 2) *\<^sub>R (a + b)) $ i \<and> ((1 / 2) *\<^sub>R (a + b)) $ i < b $ i"
using assms[unfolded interval_ne_empty, THEN spec[where x=i]]
unfolding vector_smult_component and vector_add_component
- by(auto simp add: less_divide_eq_number_of1) }
+ by auto }
thus ?thesis unfolding mem_interval by auto
qed
@@ -5041,7 +5044,7 @@
x + (inverse (real n + 1)) *\<^sub>R (((1 / 2) *\<^sub>R (a + b)) - x)"
by (auto simp add: algebra_simps)
hence "f n < b" and "a < f n" using open_closed_interval_convex[OF open_interval_midpoint[OF assms] as *] unfolding f_def by auto
- hence False using fn unfolding f_def using xc by(auto simp add: vector_mul_lcancel vector_ssub_ldistrib) }
+ hence False using fn unfolding f_def using xc by(auto simp add: vector_ssub_ldistrib) }
moreover
{ assume "\<not> (f ---> x) sequentially"
{ fix e::real assume "e>0"
@@ -5457,7 +5460,7 @@
then obtain y where y:"y\<in>t" "g y = x" by auto
then obtain x' where x':"x'\<in>s" "f x' = y" using assms(3) by auto
hence "x \<in> s" unfolding g_def using someI2[of "\<lambda>b. b\<in>s \<and> f b = y" x' "\<lambda>x. x\<in>s"] unfolding y(2)[THEN sym] and g_def by auto }
- ultimately have "x\<in>s \<longleftrightarrow> x \<in> g ` t" by auto }
+ ultimately have "x\<in>s \<longleftrightarrow> x \<in> g ` t" .. }
hence "g ` t = s" by auto
ultimately
show ?thesis unfolding homeomorphism_def homeomorphic_def
@@ -5599,7 +5602,7 @@
let ?S' = "{x::real^'m. x\<in>s \<and> norm x = norm a}"
let ?S'' = "{x::real^'m. norm x = norm a}"
- have "?S'' = frontier(cball 0 (norm a))" unfolding frontier_cball and dist_norm by (auto simp add: norm_minus_cancel)
+ have "?S'' = frontier(cball 0 (norm a))" unfolding frontier_cball and dist_norm by auto
hence "compact ?S''" using compact_frontier[OF compact_cball, of 0 "norm a"] by auto
moreover have "?S' = s \<inter> ?S''" by auto
ultimately have "compact ?S'" using closed_inter_compact[of s ?S''] using s(1) by auto
@@ -5646,7 +5649,7 @@
lemma subspace_substandard:
"subspace {x::real^_. (\<forall>i. P i \<longrightarrow> x$i = 0)}"
- unfolding subspace_def by(auto simp add: vector_add_component vector_smult_component elim!: ballE)
+ unfolding subspace_def by auto
lemma closed_substandard:
"closed {x::real^'n. \<forall>i. P i --> x$i = 0}" (is "closed ?A")
@@ -5663,7 +5666,7 @@
then obtain B where BB:"B \<in> ?Bs" and B:"B = {x::real^'n. inner (basis i) x = 0}" by auto
hence "x $ i = 0" unfolding B using x unfolding inner_basis by auto }
hence "x\<in>?A" by auto }
- ultimately have "x\<in>?A \<longleftrightarrow> x\<in> \<Inter>?Bs" by auto }
+ ultimately have "x\<in>?A \<longleftrightarrow> x\<in> \<Inter>?Bs" .. }
hence "?A = \<Inter> ?Bs" by auto
thus ?thesis by(auto simp add: closed_Inter closed_hyperplane)
qed
@@ -5839,23 +5842,23 @@
case False
{ fix y assume "a \<le> y" "y \<le> b" "m > 0"
hence "m *\<^sub>R a + c \<le> m *\<^sub>R y + c" "m *\<^sub>R y + c \<le> m *\<^sub>R b + c"
- unfolding vector_le_def by(auto simp add: vector_smult_component vector_add_component)
+ unfolding vector_le_def by auto
} moreover
{ fix y assume "a \<le> y" "y \<le> b" "m < 0"
hence "m *\<^sub>R b + c \<le> m *\<^sub>R y + c" "m *\<^sub>R y + c \<le> m *\<^sub>R a + c"
- unfolding vector_le_def by(auto simp add: vector_smult_component vector_add_component mult_left_mono_neg elim!:ballE)
+ unfolding vector_le_def by(auto simp add: mult_left_mono_neg)
} moreover
{ fix y assume "m > 0" "m *\<^sub>R a + c \<le> y" "y \<le> m *\<^sub>R b + c"
hence "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}"
unfolding image_iff Bex_def mem_interval vector_le_def
- apply(auto simp add: vector_smult_component vector_add_component vector_minus_component vector_smult_assoc pth_3[symmetric]
+ apply(auto simp add: vector_smult_assoc pth_3[symmetric]
intro!: exI[where x="(1 / m) *\<^sub>R (y - c)"])
by(auto simp add: pos_le_divide_eq pos_divide_le_eq real_mult_commute diff_le_iff)
} moreover
{ fix y assume "m *\<^sub>R b + c \<le> y" "y \<le> m *\<^sub>R a + c" "m < 0"
hence "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}"
unfolding image_iff Bex_def mem_interval vector_le_def
- apply(auto simp add: vector_smult_component vector_add_component vector_minus_component vector_smult_assoc pth_3[symmetric]
+ apply(auto simp add: vector_smult_assoc pth_3[symmetric]
intro!: exI[where x="(1 / m) *\<^sub>R (y - c)"])
by(auto simp add: neg_le_divide_eq neg_divide_le_eq real_mult_commute diff_le_iff)
}