1.1 --- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Wed Aug 17 18:05:31 2011 +0200
     1.2 +++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Wed Aug 17 09:59:10 2011 -0700
     1.3 @@ -4882,56 +4882,28 @@
     1.4  
     1.5  (* Moved interval_open_subset_closed a bit upwards *)
     1.6  
     1.7 -lemma open_interval_lemma: fixes x :: "real" shows
     1.8 - "a < x \<Longrightarrow> x < b ==> (\<exists>d>0. \<forall>x'. abs(x' - x) < d --> a < x' \<and> x' < b)"
     1.9 -  by(rule_tac x="min (x - a) (b - x)" in exI, auto)
    1.10 -
    1.11 -lemma open_interval[intro]: fixes a :: "'a::ordered_euclidean_space" shows "open {a<..<b}"
    1.12 +lemma open_interval[intro]:
    1.13 +  fixes a b :: "'a::ordered_euclidean_space" shows "open {a<..<b}"
    1.14  proof-
    1.15 -  { fix x assume x:"x\<in>{a<..<b}"
    1.16 -    { fix i assume "i<DIM('a)"
    1.17 -      hence "\<exists>d>0. \<forall>x'. abs (x' - (x$$i)) < d \<longrightarrow> a$$i < x' \<and> x' < b$$i"
    1.18 -        using x[unfolded mem_interval, THEN spec[where x=i]]
    1.19 -        using open_interval_lemma[of "a$$i" "x$$i" "b$$i"] by auto  }
    1.20 -    hence "\<forall>i\<in>{..<DIM('a)}. \<exists>d>0. \<forall>x'. abs (x' - (x$$i)) < d \<longrightarrow> a$$i < x' \<and> x' < b$$i" by auto
    1.21 -    from bchoice[OF this] guess d .. note d=this
    1.22 -    let ?d = "Min (d ` {..<DIM('a)})"
    1.23 -    have **:"finite (d ` {..<DIM('a)})" "d ` {..<DIM('a)} \<noteq> {}" by auto
    1.24 -    have "?d>0" using Min_gr_iff[OF **] using d by auto
    1.25 -    moreover
    1.26 -    { fix x' assume as:"dist x' x < ?d"
    1.27 -      { fix i assume i:"i<DIM('a)"
    1.28 -        hence "\<bar>x'$$i - x $$ i\<bar> < d i"
    1.29 -          using norm_bound_component_lt[OF as[unfolded dist_norm], of i]
    1.30 -          unfolding euclidean_simps Min_gr_iff[OF **] by auto
    1.31 -        hence "a $$ i < x' $$ i" "x' $$ i < b $$ i" using i and d[THEN bspec[where x=i]] by auto  }
    1.32 -      hence "a < x' \<and> x' < b" apply(subst(2) eucl_less,subst(1) eucl_less) by auto  }
    1.33 -    ultimately have "\<exists>e>0. \<forall>x'. dist x' x < e \<longrightarrow> x' \<in> {a<..<b}" by auto
    1.34 -  }
    1.35 -  thus ?thesis unfolding open_dist using open_interval_lemma by auto
    1.36 -qed
    1.37 -
    1.38 -lemma closed_interval[intro]: fixes a :: "'a::ordered_euclidean_space" shows "closed {a .. b}"
    1.39 +  have "open (\<Inter>i<DIM('a). (\<lambda>x. x$$i) -` {a$$i<..<b$$i})"
    1.40 +    by (intro open_INT finite_lessThan ballI continuous_open_vimage allI
    1.41 +      linear_continuous_at bounded_linear_euclidean_component
    1.42 +      open_real_greaterThanLessThan)
    1.43 +  also have "(\<Inter>i<DIM('a). (\<lambda>x. x$$i) -` {a$$i<..<b$$i}) = {a<..<b}"
    1.44 +    by (auto simp add: eucl_less [where 'a='a])
    1.45 +  finally show "open {a<..<b}" .
    1.46 +qed
    1.47 +
    1.48 +lemma closed_interval[intro]:
    1.49 +  fixes a b :: "'a::ordered_euclidean_space" shows "closed {a .. b}"
    1.50  proof-
    1.51 -  { fix x i assume i:"i<DIM('a)"
    1.52 -    assume as:"\<forall>e>0. \<exists>x'\<in>{a..b}. x' \<noteq> x \<and> dist x' x < e"(* and xab:"a$$i > x$$i \<or> b$$i < x$$i"*)
    1.53 -    { assume xa:"a$$i > x$$i"
    1.54 -      with as obtain y where y:"y\<in>{a..b}" "y \<noteq> x" "dist y x < a$$i - x$$i" by(erule_tac x="a$$i - x$$i" in allE)auto
    1.55 -      hence False unfolding mem_interval and dist_norm
    1.56 -        using component_le_norm[of "y-x" i, unfolded euclidean_simps] and xa using i
    1.57 -        by(auto elim!: allE[where x=i])
    1.58 -    } hence "a$$i \<le> x$$i" by(rule ccontr)auto
    1.59 -    moreover
    1.60 -    { assume xb:"b$$i < x$$i"
    1.61 -      with as obtain y where y:"y\<in>{a..b}" "y \<noteq> x" "dist y x < x$$i - b$$i"
    1.62 -        by(erule_tac x="x$$i - b$$i" in allE)auto
    1.63 -      hence False unfolding mem_interval and dist_norm
    1.64 -        using component_le_norm[of "y-x" i, unfolded euclidean_simps] and xb using i
    1.65 -        by(auto elim!: allE[where x=i])
    1.66 -    } hence "x$$i \<le> b$$i" by(rule ccontr)auto
    1.67 -    ultimately
    1.68 -    have "a $$ i \<le> x $$ i \<and> x $$ i \<le> b $$ i" by auto }
    1.69 -  thus ?thesis unfolding closed_limpt islimpt_approachable mem_interval by auto
    1.70 +  have "closed (\<Inter>i<DIM('a). (\<lambda>x. x$$i) -` {a$$i .. b$$i})"
    1.71 +    by (intro closed_INT ballI continuous_closed_vimage allI
    1.72 +      linear_continuous_at bounded_linear_euclidean_component
    1.73 +      closed_real_atLeastAtMost)
    1.74 +  also have "(\<Inter>i<DIM('a). (\<lambda>x. x$$i) -` {a$$i .. b$$i}) = {a .. b}"
    1.75 +    by (auto simp add: eucl_le [where 'a='a])
    1.76 +  finally show "closed {a .. b}" .
    1.77  qed
    1.78  
    1.79  lemma interior_closed_interval[intro]: fixes a :: "'a::ordered_euclidean_space" shows