
diff -r 8d8af7e92c5e -r 30182f9e1818 src/HOL/Analysis/Linear_Algebra.thy
--- a/src/HOL/Analysis/Linear_Algebra.thy	Sat Dec 31 11:09:19 2022 +0000
+++ b/src/HOL/Analysis/Linear_Algebra.thy	Sun Jan 01 00:45:55 2023 +0000
@@ -68,38 +68,17 @@
   fixes a :: "'a::ab_group_add"
   assumes "(\<lambda>x. a + x) ` A = (\<lambda>x. a + x) ` B"
   shows "A = B"
-proof -
-  have "(\<lambda>x. -a + x) ` ((\<lambda>x. a + x) ` A) = (\<lambda>x. - a + x) ` ((\<lambda>x. a + x) ` B)"
-    using assms by auto
-  then show ?thesis
-    using translation_assoc[of "-a" a A] translation_assoc[of "-a" a B] by auto
-qed
+  using assms translation_assoc by fastforce
 
 lemma translation_galois:
   fixes a :: "'a::ab_group_add"
   shows "T = ((\<lambda>x. a + x) ` S) \<longleftrightarrow> S = ((\<lambda>x. (- a) + x) ` T)"
-  using translation_assoc[of "-a" a S]
-  apply auto
-  using translation_assoc[of a "-a" T]
-  apply auto
-  done
+  by (metis add.right_inverse group_cancel.rule0 translation_invert translation_assoc)
 
 lemma translation_inverse_subset:
   assumes "((\<lambda>x. - a + x) ` V) \<le> (S :: 'n::ab_group_add set)"
   shows "V \<le> ((\<lambda>x. a + x) ` S)"
-proof -
-  {
-    fix x
-    assume "x \<in> V"
-    then have "x-a \<in> S" using assms by auto
-    then have "x \<in> {a + v |v. v \<in> S}"
-      apply auto
-      apply (rule exI[of _ "x-a"], simp)
-      done
-    then have "x \<in> ((\<lambda>x. a+x) ` S)" by auto
-  }
-  then show ?thesis by auto
-qed
+  by (metis assms subset_image_iff translation_galois)
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>More interesting properties of the norm\<close>
 
@@ -120,41 +99,25 @@
 qed
 
 lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x"
-  (is "?lhs \<longleftrightarrow> ?rhs")
-proof
-  assume ?lhs
-  then show ?rhs by simp
-next
-  assume ?rhs
-  then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0"
-    by simp
-  then have "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0"
-    by (simp add: inner_diff inner_commute)
-  then have "(x - y) \<bullet> (x - y) = 0"
-    by (simp add: field_simps inner_diff inner_commute)
-  then show "x = y" by simp
-qed
+  by (metis (no_types, opaque_lifting) inner_commute inner_diff_right inner_eq_zero_iff right_minus_eq)
 
 lemma norm_triangle_half_r:
-  "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
+  "norm (y - x1) < e/2 \<Longrightarrow> norm (y - x2) < e/2 \<Longrightarrow> norm (x1 - x2) < e"
   using dist_triangle_half_r unfolding dist_norm[symmetric] by auto
 
 lemma norm_triangle_half_l:
-  assumes "norm (x - y) < e / 2"
-    and "norm (x' - y) < e / 2"
+  assumes "norm (x - y) < e/2" and "norm (x' - y) < e/2"
   shows "norm (x - x') < e"
-  using dist_triangle_half_l[OF assms[unfolded dist_norm[symmetric]]]
-  unfolding dist_norm[symmetric] .
+  by (metis assms dist_norm dist_triangle_half_l)
 
 lemma abs_triangle_half_r:
   fixes y :: "'a::linordered_field"
-  shows "abs (y - x1) < e / 2 \<Longrightarrow> abs (y - x2) < e / 2 \<Longrightarrow> abs (x1 - x2) < e"
+  shows "abs (y - x1) < e/2 \<Longrightarrow> abs (y - x2) < e/2 \<Longrightarrow> abs (x1 - x2) < e"
   by linarith
 
 lemma abs_triangle_half_l:
   fixes y :: "'a::linordered_field"
-  assumes "abs (x - y) < e / 2"
-    and "abs (x' - y) < e / 2"
+  assumes "abs (x - y) < e/2" and "abs (x' - y) < e/2"
   shows "abs (x - x') < e"
   using assms by linarith
 
@@ -163,41 +126,15 @@
     and "finite S \<Longrightarrow> sum f (insert x S) = (if x \<in> S then sum f S else f x + sum f S)"
   by (auto simp add: insert_absorb)
 
-lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z"
-proof
-  assume "\<forall>x. x \<bullet> y = x \<bullet> z"
-  then have "\<forall>x. x \<bullet> (y - z) = 0"
-    by (simp add: inner_diff)
-  then have "(y - z) \<bullet> (y - z) = 0" ..
-  then show "y = z" by simp
-qed simp
-
-lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = y"
-proof
-  assume "\<forall>z. x \<bullet> z = y \<bullet> z"
-  then have "\<forall>z. (x - y) \<bullet> z = 0"
-    by (simp add: inner_diff)
-  then have "(x - y) \<bullet> (x - y) = 0" ..
-  then show "x = y" by simp
-qed simp
+lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z" and vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = y"
+  by (metis inner_commute vector_eq)+
 
 subsection \<open>Substandard Basis\<close>
 
 lemma ex_card:
   assumes "n \<le> card A"
   shows "\<exists>S\<subseteq>A. card S = n"
-proof (cases "finite A")
-  case True
-  from ex_bij_betw_nat_finite[OF this] obtain f where f: "bij_betw f {0..<card A} A" ..
-  moreover from f \<open>n \<le> card A\<close> have "{..< n} \<subseteq> {..< card A}" "inj_on f {..< n}"
-    by (auto simp: bij_betw_def intro: subset_inj_on)
-  ultimately have "f ` {..< n} \<subseteq> A" "card (f ` {..< n}) = n"
-    by (auto simp: bij_betw_def card_image)
-  then show ?thesis by blast
-next
-  case False
-  with \<open>n \<le> card A\<close> show ?thesis by force
-qed
+  by (meson assms obtain_subset_with_card_n)
 
 lemma subspace_substandard: "subspace {x::'a::euclidean_space. (\<forall>i\<in>Basis. P i \<longrightarrow> x\<bullet>i = 0)}"
   by (auto simp: subspace_def inner_add_left)
@@ -271,7 +208,7 @@
 
 lemma norm_add_Pythagorean:
   assumes "orthogonal a b"
-    shows "norm(a + b) ^ 2 = norm a ^ 2 + norm b ^ 2"
+    shows "(norm (a + b))\<^sup>2 = (norm a)\<^sup>2 + (norm b)\<^sup>2"
 proof -
   from assms have "(a - (0 - b)) \<bullet> (a - (0 - b)) = a \<bullet> a - (0 - b \<bullet> b)"
     by (simp add: algebra_simps orthogonal_def inner_commute)
@@ -300,12 +237,7 @@
 
 lemma\<^marker>\<open>tag unimportant\<close>  orthogonal_transformation:
   "orthogonal_transformation f \<longleftrightarrow> linear f \<and> (\<forall>v. norm (f v) = norm v)"
-  unfolding orthogonal_transformation_def
-  apply auto
-  apply (erule_tac x=v in allE)+
-  apply (simp add: norm_eq_sqrt_inner)
-  apply (simp add: dot_norm linear_add[symmetric])
-  done
+  by (smt (verit, ccfv_threshold) dot_norm linear_add norm_eq_sqrt_inner orthogonal_transformation_def)
 
 lemma\<^marker>\<open>tag unimportant\<close>  orthogonal_transformation_id [simp]: "orthogonal_transformation (\<lambda>x. x)"
   by (simp add: linear_iff orthogonal_transformation_def)
@@ -431,15 +363,8 @@
 next
   fix h
   assume "\<forall>x y. inner (f x) y = inner x (h y)"
-  then have "\<forall>x y. inner x (g y) = inner x (h y)"
-    using assms by simp
-  then have "\<forall>x y. inner x (g y - h y) = 0"
-    by (simp add: inner_diff_right)
-  then have "\<forall>y. inner (g y - h y) (g y - h y) = 0"
-    by simp
-  then have "\<forall>y. h y = g y"
-    by simp
-  then show "h = g" by (simp add: ext)
+  then show "h = g"
+    by (metis assms ext vector_eq_ldot) 
 qed
 
 text \<open>TODO: The following lemmas about adjoints should hold for any
@@ -524,32 +449,18 @@
     proof -
       from Basis_le_norm[OF that, of x]
       show "norm (?g i) \<le> norm (f i) * norm x"
-        unfolding norm_scaleR  by (metis mult.commute mult_left_mono norm_ge_zero)
+        unfolding norm_scaleR by (metis mult.commute mult_left_mono norm_ge_zero)
     qed
     from sum_norm_le[of _ ?g, OF th]
     show "norm (f x) \<le> ?B * norm x"
-      unfolding th0 sum_distrib_right by metis
+      by (simp add: sum_distrib_right th0)
   qed
 qed
 
 lemma linear_conv_bounded_linear:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   shows "linear f \<longleftrightarrow> bounded_linear f"
-proof
-  assume "linear f"
-  then interpret f: linear f .
-  show "bounded_linear f"
-  proof
-    have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
-      using \<open>linear f\<close> by (rule linear_bounded)
-    then show "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
-      by (simp add: mult.commute)
-  qed
-next
-  assume "bounded_linear f"
-  then interpret f: bounded_linear f .
-  show "linear f" ..
-qed
+  by (metis mult.commute bounded_linear_axioms.intro bounded_linear_def linear_bounded)
 
 lemmas linear_linear = linear_conv_bounded_linear[symmetric]
 
@@ -562,17 +473,11 @@
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes lf: "linear f"
  obtains B where "B > 0" "\<And>x. norm (f x) \<le> B * norm x"
-proof -
-  have "\<exists>B > 0. \<forall>x. norm (f x) \<le> norm x * B"
-    using lf unfolding linear_conv_bounded_linear
-    by (rule bounded_linear.pos_bounded)
-  with that show ?thesis
-    by (auto simp: mult.commute)
-qed
+  by (metis bounded_linear.pos_bounded lf linear_linear mult.commute)
 
 lemma linear_invertible_bounded_below_pos:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
-  assumes "linear f" "linear g" "g \<circ> f = id"
+  assumes "linear f" "linear g" and gf: "g \<circ> f = id"
   obtains B where "B > 0" "\<And>x. B * norm x \<le> norm(f x)"
 proof -
   obtain B where "B > 0" and B: "\<And>x. norm (g x) \<le> B * norm x"
@@ -582,13 +487,8 @@
     show "0 < 1/B"
       by (simp add: \<open>B > 0\<close>)
     show "1/B * norm x \<le> norm (f x)" for x
-    proof -
-      have "1/B * norm x = 1/B * norm (g (f x))"
-        using assms by (simp add: pointfree_idE)
-      also have "\<dots> \<le> norm (f x)"
-        using B [of "f x"] by (simp add: \<open>B > 0\<close> mult.commute pos_divide_le_eq)
-      finally show ?thesis .
-    qed
+      by (smt (verit, ccfv_SIG) B \<open>0 < B\<close> gf comp_apply divide_inverse id_apply inverse_eq_divide 
+              less_divide_eq mult.commute)
   qed
 qed
 
@@ -663,15 +563,10 @@
   fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
   assumes bh: "bilinear h"
   shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
-proof -
-  have "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> norm x * norm y * B"
-    using bh [unfolded bilinear_conv_bounded_bilinear]
-    by (rule bounded_bilinear.pos_bounded)
-  then show ?thesis
-    by (simp only: ac_simps)
-qed
+  by (metis mult.assoc bh bilinear_conv_bounded_bilinear bounded_bilinear.pos_bounded mult.commute)
 
-lemma bounded_linear_imp_has_derivative: "bounded_linear f \<Longrightarrow> (f has_derivative f) net"
+lemma bounded_linear_imp_has_derivative: 
+  "bounded_linear f \<Longrightarrow> (f has_derivative f) net"
   by (auto simp add: has_derivative_def linear_diff linear_linear linear_def
       dest: bounded_linear.linear)
 
@@ -723,8 +618,7 @@
   assumes "pairwise orthogonal S"
     and "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y"
   shows "pairwise orthogonal (insert x S)"
-  using assms unfolding pairwise_def
-  by (auto simp add: orthogonal_commute)
+  using assms by (auto simp: pairwise_def orthogonal_commute)
 
 lemma basis_orthogonal:
   fixes B :: "'a::real_inner set"
@@ -735,9 +629,7 @@
 proof (induct rule: finite_induct)
   case empty
   then show ?case
-    apply (rule exI[where x="{}"])
-    apply (auto simp add: pairwise_def)
-    done
+    using pairwise_empty by blast
 next
   case (insert a B)
   note fB = \<open>finite B\<close> and aB = \<open>a \<notin> B\<close>
@@ -748,21 +640,15 @@
   let ?C = "insert ?a C"
   from C(1) have fC: "finite ?C"
     by simp
-  from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)"
-    by (simp add: card_insert_if)
+  have cC: "card ?C \<le> card (insert a B)"
+    using C aB card_insert_if local.insert(1) by fastforce
   {
     fix x k
     have th0: "\<And>(a::'a) b c. a - (b - c) = c + (a - b)"
       by (simp add: field_simps)
     have "x - k *\<^sub>R (a - (\<Sum>x\<in>C. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x)) \<in> span C \<longleftrightarrow> x - k *\<^sub>R a \<in> span C"
-      apply (simp only: scaleR_right_diff_distrib th0)
-      apply (rule span_add_eq)
-      apply (rule span_scale)
-      apply (rule span_sum)
-      apply (rule span_scale)
-      apply (rule span_base)
-      apply assumption
-      done
+      unfolding scaleR_right_diff_distrib th0
+      by (intro span_add_eq span_scale span_sum span_base)
   }
   then have SC: "span ?C = span (insert a B)"
     unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto
@@ -773,15 +659,14 @@
       by blast
     have fth: "finite (C - {y})"
       using C by simp
-    have "orthogonal ?a y"
+    have "y \<noteq> 0 \<Longrightarrow> \<forall>x\<in>C - {y}. x \<bullet> a * (x \<bullet> y) / (x \<bullet> x) = 0"
+      using \<open>pairwise orthogonal C\<close>
+      by (metis Cy DiffE div_0 insertCI mult_zero_right orthogonal_def pairwise_insert)
+    then have "orthogonal ?a y"
       unfolding orthogonal_def
       unfolding inner_diff inner_sum_left right_minus_eq
       unfolding sum.remove [OF \<open>finite C\<close> \<open>y \<in> C\<close>]
-      apply (clarsimp simp add: inner_commute[of y a])
-      apply (rule sum.neutral)
-      apply clarsimp
-      apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
-      using \<open>y \<in> C\<close> by auto
+      by (auto simp add: sum.neutral inner_commute[of y a])
   }
   with \<open>pairwise orthogonal C\<close> have CPO: "pairwise orthogonal ?C"
     by (rule pairwise_orthogonal_insert)
@@ -792,8 +677,7 @@
 
 lemma orthogonal_basis_exists:
   fixes V :: "('a::euclidean_space) set"
-  shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and>
-  (card B = dim V) \<and> pairwise orthogonal B"
+  shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and> (card B = dim V) \<and> pairwise orthogonal B"
 proof -
   from basis_exists[of V] obtain B where
     B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V"
@@ -807,18 +691,15 @@
     by (metis span_superset span_mono subset_trans)
   from span_mono[OF B(3)] C have SVC: "span V \<subseteq> span C"
     by (simp add: span_span)
-  from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
-  have iC: "independent C"
-    by (simp)
   from C fB have "card C \<le> dim V"
     by simp
   moreover have "dim V \<le> card C"
     using span_card_ge_dim[OF CSV SVC C(1)]
     by simp
-  ultimately have CdV: "card C = dim V"
+  ultimately have "card C = dim V"
     using C(1) by simp
-  from C B CSV CdV iC show ?thesis
-    by auto
+  with C B CSV show ?thesis
+    by (metis SVC card_eq_dim dim_span)
 qed
 
 text \<open>Low-dimensional subset is in a hyperplane (weak orthogonal complement).\<close>
@@ -831,22 +712,15 @@
   from sU obtain a where a: "a \<notin> span S"
     by blast
   from orthogonal_basis_exists obtain B where
-    B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B"
-    "card B = dim S" "pairwise orthogonal B"
+    B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B" "card B = dim S" "pairwise orthogonal B"
     by blast
   from B have fB: "finite B" "card B = dim S"
     using independent_bound by auto
-  from span_mono[OF B(2)] span_mono[OF B(3)]
   have sSB: "span S = span B"
-    by (simp add: span_span)
+    by (simp add: B span_eq)
   let ?a = "a - sum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B"
   have "sum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B \<in> span S"
-    unfolding sSB
-    apply (rule span_sum)
-    apply (rule span_scale)
-    apply (rule span_base)
-    apply assumption
-    done
+    by (simp add: sSB span_base span_mul span_sum)
   with a have a0:"?a  \<noteq> 0"
     by auto
   have "?a \<bullet> x = 0" if "x\<in>span B" for x
@@ -861,23 +735,19 @@
         by blast
       have fth: "finite (B - {x})"
         using fB by simp
-      have "?a \<bullet> x = 0"
+      have "(\<Sum>b\<in>B - {x}. a \<bullet> b * (b \<bullet> x) / (b \<bullet> b)) = 0" if "x \<noteq> 0"
+        by (smt (verit) B' B(5) DiffD2 divide_eq_0_iff inner_real_def inner_zero_right insertCI orthogonal_def pairwise_insert sum.neutral)
+      then have "?a \<bullet> x = 0"
         apply (subst B')
         using fB fth
         unfolding sum_clauses(2)[OF fth]
-        apply simp unfolding inner_simps
-        apply (clarsimp simp add: inner_add inner_sum_left)
-        apply (rule sum.neutral, rule ballI)
-        apply (simp only: inner_commute)
-        apply (auto simp add: x field_simps
-          intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])
-        done
+        by (auto simp add: inner_add_left inner_diff_left inner_sum_left)
     }
     then show "?a \<bullet> x = 0" if "x \<in> B" for x
       using that by blast
     qed
-  with a0 show ?thesis
-    unfolding sSB by (auto intro: exI[where x="?a"])
+  with a0 sSB show ?thesis
+    by blast
 qed
 
 lemma span_not_univ_subset_hyperplane:
@@ -890,19 +760,7 @@
   fixes S :: "'a::euclidean_space set"
   assumes d: "dim S < DIM('a)"
   shows "\<exists>a::'a. a \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
-proof -
-  {
-    assume "span S = UNIV"
-    then have "dim (span S) = dim (UNIV :: ('a) set)"
-      by simp
-    then have "dim S = DIM('a)"
-      by (metis Euclidean_Space.dim_UNIV dim_span)
-    with d have False by arith
-  }
-  then have th: "span S \<noteq> UNIV"
-    by blast
-  from span_not_univ_subset_hyperplane[OF th] show ?thesis .
-qed
+  using d dim_eq_full nless_le span_not_univ_subset_hyperplane by blast
 
 lemma linear_eq_stdbasis:
   fixes f :: "'a::euclidean_space \<Rightarrow> _"
@@ -910,8 +768,7 @@
     and lg: "linear g"
     and fg: "\<And>b. b \<in> Basis \<Longrightarrow> f b = g b"
   shows "f = g"
-  using linear_eq_on_span[OF lf lg, of Basis] fg
-  by auto
+  using linear_eq_on_span[OF lf lg, of Basis] fg by auto
 
 
 text \<open>Similar results for bilinear functions.\<close>
@@ -932,15 +789,9 @@
         span_add Ball_def
       intro: bilinear_ladd[OF bf])
   have sfg: "\<And>x. x \<in> B \<Longrightarrow> subspace {a. f x a = g x a}"
-    apply (auto simp add: subspace_def)
-    using bf bg unfolding bilinear_def linear_iff
-      apply (auto simp add: span_zero bilinear_rzero[OF bf] bilinear_rzero[OF bg]
-        span_add Ball_def
-      intro: bilinear_ladd[OF bf])
-    done
+    by (auto simp: subspace_def bf bg bilinear_rzero bilinear_radd bilinear_rmul)
   have "\<forall>y\<in> span C. f x y = g x y" if "x \<in> span B" for x
-    apply (rule span_induct [OF that sp])
-    using fg sfg span_induct by blast
+    using span_induct [OF that sp] fg sfg span_induct by blast
   then show ?thesis
     using SB TC assms by auto
 qed
@@ -972,8 +823,7 @@
   fixes x :: "'a::euclidean_space"
   shows "finite {\<bar>x \<bullet> i\<bar> |i. i \<in> Basis}"
     and "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis} \<noteq> {}"
-  unfolding infnorm_set_image
-  by auto
+  unfolding infnorm_set_image by auto
 
 lemma infnorm_pos_le:
   fixes x :: "'a::euclidean_space"
@@ -1010,16 +860,7 @@
   by (metis infnorm_neg minus_diff_eq)
 
 lemma absdiff_infnorm: "\<bar>infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
-proof -
-  have *: "\<And>(nx::real) n ny. nx \<le> n + ny \<Longrightarrow> ny \<le> n + nx \<Longrightarrow> \<bar>nx - ny\<bar> \<le> n"
-    by arith
-  show ?thesis
-  proof (rule *)
-    from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
-    show "infnorm x \<le> infnorm (x - y) + infnorm y" "infnorm y \<le> infnorm (x - y) + infnorm x"
-      by (simp_all add: field_simps infnorm_neg)
-  qed
-qed
+  by (smt (verit, del_insts) diff_add_cancel infnorm_sub infnorm_triangle)
 
 lemma real_abs_infnorm: "\<bar>infnorm x\<bar> = infnorm x"
   using infnorm_pos_le[of x] by arith
@@ -1065,9 +906,9 @@
     by (simp add: zero_le_mult_iff infnorm_pos_le)
   have "x \<bullet> x \<le> (\<Sum>b\<in>Basis. x \<bullet> b * (x \<bullet> b))"
     by (metis euclidean_inner order_refl)
-  also have "... \<le> DIM('a) * \<bar>infnorm x\<bar>\<^sup>2"
+  also have "\<dots> \<le> DIM('a) * \<bar>infnorm x\<bar>\<^sup>2"
     by (rule sum_bounded_above) (metis Basis_le_infnorm abs_le_square_iff power2_eq_square real_abs_infnorm)
-  also have "... \<le> (sqrt DIM('a) * infnorm x)\<^sup>2"
+  also have "\<dots> \<le> (sqrt DIM('a) * infnorm x)\<^sup>2"
     by (simp add: power_mult_distrib)
   finally show "x \<bullet> x \<le> (sqrt DIM('a) * infnorm x)\<^sup>2" .
 qed
@@ -1091,11 +932,11 @@
   then show ?thesis
     by auto
 next
-  case False
+  case False 
   from inner_eq_zero_iff[of "norm y *\<^sub>R x - norm x *\<^sub>R y"]
   have "?rhs \<longleftrightarrow>
       (norm y * (norm y * norm x * norm x - norm x * (x \<bullet> y)) -
-        norm x * (norm y * (y \<bullet> x) - norm x * norm y * norm y) =  0)"
+        norm x * (norm y * (y \<bullet> x) - norm x * norm y * norm y) = 0)"
     using False unfolding inner_simps
     by (auto simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
   also have "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)"
@@ -1108,20 +949,9 @@
 lemma norm_cauchy_schwarz_abs_eq:
   "\<bar>x \<bullet> y\<bar> = norm x * norm y \<longleftrightarrow>
     norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm x *\<^sub>R y = - norm y *\<^sub>R x"
-  (is "?lhs \<longleftrightarrow> ?rhs")
-proof -
-  have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> \<bar>x\<bar> = a \<longleftrightarrow> x = a \<or> x = - a"
-    by arith
-  have "?rhs \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm (- x) *\<^sub>R y = norm y *\<^sub>R (- x)"
-    by simp
-  also have "\<dots> \<longleftrightarrow> (x \<bullet> y = norm x * norm y \<or> (- x) \<bullet> y = norm x * norm y)"
-    unfolding norm_cauchy_schwarz_eq[symmetric]
-    unfolding norm_minus_cancel norm_scaleR ..
-  also have "\<dots> \<longleftrightarrow> ?lhs"
-    unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] inner_simps
-    by auto
-  finally show ?thesis ..
-qed
+  using norm_cauchy_schwarz_eq [symmetric, of x y]
+  using norm_cauchy_schwarz_eq [symmetric, of "-x" y] Cauchy_Schwarz_ineq2 [of x y]
+  by auto
 
 lemma norm_triangle_eq:
   fixes x y :: "'a::real_inner"
@@ -1137,9 +967,7 @@
   have "norm (x + y) = norm x + norm y \<longleftrightarrow> (norm (x + y))\<^sup>2 = (norm x + norm y)\<^sup>2"
     by simp
   also have "\<dots> \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
-    unfolding norm_cauchy_schwarz_eq[symmetric]
-    unfolding power2_norm_eq_inner inner_simps
-    by (simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
+    by (smt (verit, best) dot_norm inner_real_def inner_simps norm_cauchy_schwarz_eq power2_eq_square)
   finally show ?thesis .
 qed
 
@@ -1147,11 +975,7 @@
   fixes x y z :: "'a::real_inner"
   shows "dist x z = dist x y + dist y z \<longleftrightarrow>
     norm (x - y) *\<^sub>R (y - z) = norm (y - z) *\<^sub>R (x - y)"
-proof -
-  have *: "x - y + (y - z) = x - z" by auto
-  show ?thesis unfolding dist_norm norm_triangle_eq[of "x - y" "y - z", unfolded *]
-    by (auto simp:norm_minus_commute)
-qed
+  by (metis (no_types, lifting) add_diff_eq diff_add_cancel dist_norm norm_triangle_eq)
 
 subsection \<open>Collinearity\<close>
 
@@ -1163,7 +987,7 @@
 proof
   assume ?lhs
   then show ?rhs
-    unfolding collinear_def by (metis Groups.add_ac(2) diff_add_cancel)
+    unfolding collinear_def by (metis add.commute diff_add_cancel)
 next
   assume ?rhs
   then obtain u v where *: "\<And>x. x \<in> S \<Longrightarrow> \<exists>c. x = u + c *\<^sub>R v"
@@ -1192,8 +1016,7 @@
       using \<open>v \<noteq> 0\<close> by blast
   qed
   then show ?thesis
-    apply (clarsimp simp: collinear_def)
-    by (metis scaleR_zero_right vector_fraction_eq_iff)
+    by (metis collinear_def)
 qed
 
 lemma collinear_subset: "\<lbrakk>collinear T; S \<subseteq> T\<rbrakk> \<Longrightarrow> collinear S"
@@ -1206,9 +1029,7 @@
   by (simp add: collinear_def)
 
 lemma collinear_2 [iff]: "collinear {x, y}"
-  apply (simp add: collinear_def)
-  apply (rule exI[where x="x - y"])
-  by (metis minus_diff_eq scaleR_left.minus scaleR_one)
+  by (simp add: collinear_def) (metis minus_diff_eq scaleR_left.minus scaleR_one)
 
 lemma collinear_lemma: "collinear {0, x, y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *\<^sub>R x)"
   (is "?lhs \<longleftrightarrow> ?rhs")
@@ -1237,15 +1058,8 @@
     then obtain c where c: "y = c *\<^sub>R x"
       using False by blast
     show ?lhs
-      unfolding collinear_def c
-      apply (rule exI[where x=x])
-      apply auto
-          apply (rule exI[where x="- 1"], simp)
-         apply (rule exI[where x= "-c"], simp)
-        apply (rule exI[where x=1], simp)
-       apply (rule exI[where x="1 - c"], simp add: scaleR_left_diff_distrib)
-      apply (rule exI[where x="c - 1"], simp add: scaleR_left_diff_distrib)
-      done
+      apply (simp add: collinear_def c)
+      by (metis (mono_tags, lifting) scaleR_left.minus scaleR_left_diff_distrib scaleR_one)
   qed
 qed
 
@@ -1291,11 +1105,8 @@
   fixes x y :: "'a::real_inner"
   assumes "norm (x + y) = norm x + norm y"
   shows "collinear{0,x,y}"
-proof (cases "x = 0 \<or> y = 0")
-  case False
-  with assms show ?thesis
-    by (meson norm_cauchy_schwarz_abs_eq norm_cauchy_schwarz_equal norm_triangle_eq)
-qed (use collinear_lemma in blast)
+  using assms norm_cauchy_schwarz_abs_eq norm_cauchy_schwarz_equal norm_triangle_eq 
+  by blast
 
 
 subsection\<open>Properties of special hyperplanes\<close>
@@ -1315,7 +1126,7 @@
   proof -
     have "x = (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b)"
       by (simp add: euclidean_representation)
-    also have "... = (\<Sum>b \<in> Basis - {k}. (x \<bullet> b) *\<^sub>R b)"
+    also have "\<dots> = (\<Sum>b \<in> Basis - {k}. (x \<bullet> b) *\<^sub>R b)"
       by (auto simp: sum.remove [of _ k] inner_commute assms that)
     finally have "x = (\<Sum>b\<in>Basis - {k}. (x \<bullet> b) *\<^sub>R b)" .
     then show ?thesis
@@ -1331,11 +1142,7 @@
 lemma dim_special_hyperplane:
   fixes k :: "'n::euclidean_space"
   shows "k \<in> Basis \<Longrightarrow> dim {x. k \<bullet> x = 0} = DIM('n) - 1"
-apply (simp add: special_hyperplane_span)
-apply (rule dim_unique [OF subset_refl])
-apply (auto simp: independent_substdbasis)
-apply (metis member_remove remove_def span_base)
-done
+  by (metis Diff_subset card_Diff_singleton indep_card_eq_dim_span independent_substdbasis special_hyperplane_span)
 
 proposition dim_hyperplane:
   fixes a :: "'a::euclidean_space"
@@ -1358,20 +1165,17 @@
     using \<open>independent B\<close> independent_bound by blast
   have "UNIV \<subseteq> span (insert a B)"
   proof fix y::'a
-    obtain r z where z: "y = r *\<^sub>R a + z" "a \<bullet> z = 0"
-      apply (rule_tac r="(a \<bullet> y) / (a \<bullet> a)" and z = "y - ((a \<bullet> y) / (a \<bullet> a)) *\<^sub>R a" in that)
-      using assms
-      by (auto simp: algebra_simps)
-    show "y \<in> span (insert a B)"
-      by (metis (mono_tags, lifting) z Bsub span_eq_iff
-         add_diff_cancel_left' mem_Collect_eq span0 span_breakdown_eq span_subspace subspB)
+    obtain r z where "y = r *\<^sub>R a + z" "a \<bullet> z = 0"
+      by (metis add.commute diff_add_cancel vector_sub_project_orthogonal)
+    then show "y \<in> span (insert a B)"
+      by (metis (mono_tags, lifting) Bsub add_diff_cancel_left'
+          mem_Collect_eq span0 span_breakdown_eq span_eq subspB)
   qed
-  then have dima: "DIM('a) = dim(insert a B)"
+  then have "DIM('a) = dim(insert a B)"
     by (metis independent_Basis span_Basis dim_eq_card top.extremum_uniqueI)
   then show ?thesis
-    by (metis (mono_tags, lifting) Bsub Diff_insert_absorb \<open>a \<notin> span B\<close> ind card0
-        card_Diff_singleton dim_span indep_card_eq_dim_span insertI1 subsetCE
-        subspB)
+    by (metis One_nat_def \<open>a \<notin> span B\<close> \<open>finite B\<close> card0 card_insert_disjoint 
+        diff_Suc_Suc diff_zero dim_eq_card_independent ind span_base)
 qed
 
 lemma lowdim_eq_hyperplane:
@@ -1379,14 +1183,10 @@
   assumes "dim S = DIM('a) - 1"
   obtains a where "a \<noteq> 0" and "span S = {x. a \<bullet> x = 0}"
 proof -
-  have dimS: "dim S < DIM('a)"
-    by (simp add: assms)
-  then obtain b where b: "b \<noteq> 0" "span S \<subseteq> {a. b \<bullet> a = 0}"
-    using lowdim_subset_hyperplane [of S] by fastforce
-  show ?thesis
-    apply (rule that[OF b(1)])
-    apply (rule subspace_dim_equal)
-    by (auto simp: assms b dim_hyperplane subspace_hyperplane)
+  obtain b where b: "b \<noteq> 0" "span S \<subseteq> {a. b \<bullet> a = 0}"
+    by (metis DIM_positive assms diff_less zero_less_one lowdim_subset_hyperplane)
+  then show ?thesis
+    by (metis assms dim_hyperplane dim_span dim_subset subspace_dim_equal subspace_hyperplane subspace_span that)
 qed
 
 lemma dim_eq_hyperplane:
@@ -1409,10 +1209,9 @@
       using a by (force simp: span_explicit)
     then have "a \<bullet> a = a \<bullet> (\<Sum>v\<in>T. U v *\<^sub>R v)"
       by simp
-    also have "... = 0"
+    also have "\<dots> = 0"
       apply (simp add: inner_sum_right)
-      apply (rule comm_monoid_add_class.sum.neutral)
-      by (metis "0" DiffE \<open>T \<subseteq> S - {a}\<close> mult_not_zero singletonI subsetCE \<open>a \<in> S\<close>)
+      by (smt (verit) "0" DiffE \<open>T \<subseteq> S - {a}\<close> in_mono insertCI mult_not_zero sum.neutral that(1))
     finally show ?thesis
       using \<open>0 \<notin> S\<close> \<open>a \<in> S\<close> by auto
   qed
@@ -1424,14 +1223,8 @@
   fixes S :: "'a::euclidean_space set"
   assumes "pairwise orthogonal S"
     shows "finite S"
-proof -
-  have "independent (S - {0})"
-    apply (rule pairwise_orthogonal_independent)
-     apply (metis Diff_iff assms pairwise_def)
-    by blast
-  then show ?thesis
-    by (meson independent_imp_finite infinite_remove)
-qed
+  by (metis Set.set_insert assms finite_insert independent_bound pairwise_insert 
+            pairwise_orthogonal_independent)
 
 lemma subspace_orthogonal_to_vector: "subspace {y. orthogonal x y}"
   by (simp add: subspace_def orthogonal_clauses)
@@ -1457,7 +1250,7 @@
   proof -
     have "a \<bullet> x = (\<Sum>y\<in>S. if y = x then y \<bullet> a else 0)"
       by (simp add: \<open>finite S\<close> inner_commute that)
-    also have "... =  (\<Sum>b\<in>S. b \<bullet> a * (b \<bullet> x) / (b \<bullet> b))"
+    also have "\<dots> =  (\<Sum>b\<in>S. b \<bullet> a * (b \<bullet> x) / (b \<bullet> b))"
       apply (rule sum.cong [OF refl], simp)
       by (meson S orthogonal_def pairwise_def that)
    finally show ?thesis
@@ -1487,20 +1280,15 @@
       (auto simp: Gram_Schmidt_step a'_def insert.prems orthogonal_commute
         pairwise_orthogonal_insert span_clauses)
   have orthS: "\<And>x. x \<in> S \<Longrightarrow> a' \<bullet> x = 0"
-    apply (simp add: a'_def)
-    using Gram_Schmidt_step [OF \<open>pairwise orthogonal S\<close>]
-    apply (force simp: orthogonal_def inner_commute span_superset [THEN subsetD])
-    done
+    using Gram_Schmidt_step a'_def insert.prems orthogonal_commute orthogonal_def span_base by blast
   have "span (S \<union> insert a' U) = span (insert a' (S \<union> T))"
     using spanU by simp
-  also have "... = span (insert a (S \<union> T))"
-    apply (rule eq_span_insert_eq)
-    apply (simp add: a'_def span_neg span_sum span_base span_mul)
-    done
-  also have "... = span (S \<union> insert a T)"
+  also have "\<dots> = span (insert a (S \<union> T))"
+    by (simp add: a'_def span_neg span_sum span_base span_mul eq_span_insert_eq)
+  also have "\<dots> = span (S \<union> insert a T)"
     by simp
   finally show ?case
-    by (rule_tac x="insert a' U" in exI) (use orthU in auto)
+    using orthU by blast
 qed
 
 
@@ -1524,14 +1312,12 @@
   obtains U where "U \<inter> (insert 0 S) = {}" "pairwise orthogonal (S \<union> U)"
                   "span (S \<union> U) = span (S \<union> T)"
 proof -
-  obtain U where "pairwise orthogonal (S \<union> U)" "span (S \<union> U) = span (S \<union> T)"
+  obtain U where U: "pairwise orthogonal (S \<union> U)" "span (S \<union> U) = span (S \<union> T)"
     using orthogonal_extension assms by blast
-  then show ?thesis
-    apply (rule_tac U = "U - (insert 0 S)" in that)
-      apply blast
-     apply (force simp: pairwise_def)
-    apply (metis Un_Diff_cancel Un_insert_left span_redundant span_zero)
-    done
+  moreover have "pairwise orthogonal (S \<union> (U - insert 0 S))"
+    by (smt (verit, best) Un_Diff_Int Un_iff U pairwise_def)
+  ultimately show ?thesis
+    by (metis Diff_disjoint Un_Diff_cancel Un_insert_left inf_commute span_insert_0 that)
 qed
 
 subsection\<open>Decomposing a vector into parts in orthogonal subspaces\<close>
@@ -1542,27 +1328,14 @@
   fixes S :: "'a :: euclidean_space set"
   assumes "subspace S"
   obtains B where "B \<subseteq> S" "pairwise orthogonal B" "span B = S"
-proof -
-  obtain B where "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S"
-    using basis_exists by blast
-  with orthogonal_extension [of "{}" B]
-  show ?thesis
-    by (metis Un_empty_left assms pairwise_empty span_superset span_subspace that)
-qed
+  by (metis assms basis_orthogonal basis_subspace_exists span_eq)
 
 lemma orthogonal_basis_subspace:
   fixes S :: "'a :: euclidean_space set"
   assumes "subspace S"
   obtains B where "0 \<notin> B" "B \<subseteq> S" "pairwise orthogonal B" "independent B"
                   "card B = dim S" "span B = S"
-proof -
-  obtain B where "B \<subseteq> S" "pairwise orthogonal B" "span B = S"
-    using assms orthogonal_spanningset_subspace by blast
-  then show ?thesis
-    apply (rule_tac B = "B - {0}" in that)
-    apply (auto simp: indep_card_eq_dim_span pairwise_subset pairwise_orthogonal_independent elim: pairwise_subset)
-    done
-qed
+  by (metis assms dependent_zero orthogonal_basis_exists span_eq span_eq_iff)
 
 proposition orthonormal_basis_subspace:
   fixes S :: "'a :: euclidean_space set"
@@ -1609,7 +1382,7 @@
   obtain B where "B \<subseteq> span S" and orthB: "pairwise orthogonal B"
              and "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"
              and "independent B" "card B = dim S" "span B = span S"
-    by (rule orthonormal_basis_subspace [of "span S", OF subspace_span]) (auto)
+    by (metis dim_span orthonormal_basis_subspace subspace_span)
   with assms obtain u where spanBT: "span B \<subseteq> span T" and "u \<notin> span B" "u \<in> span T"
     by auto
   obtain C where orthBC: "pairwise orthogonal (B \<union> C)" and spanBC: "span (B \<union> C) = span (B \<union> {u})"
@@ -1630,9 +1403,8 @@
     then obtain x where "x \<in> C" "x \<noteq> 0" "x \<notin> B"
       by blast
     then have "x \<in> span T"
-      by (metis (no_types, lifting) Un_insert_right Un_upper2 \<open>u \<in> span T\<close> spanBT spanBC
-          \<open>u \<in> span T\<close> insert_subset span_superset span_mono
-          span_span subsetCE subset_trans sup_bot.comm_neutral)
+      by (smt (verit, ccfv_SIG) Set.set_insert  \<open>u \<in> span T\<close> empty_subsetI insert_subset 
+          le_sup_iff spanBC spanBT span_mono span_span span_superset subset_trans)
     moreover have "orthogonal x y" if "y \<in> span B" for y
       using that
     proof (rule span_induct)
@@ -1652,8 +1424,7 @@
   obtains x where "x \<noteq> 0" "\<And>y. y \<in> span S \<Longrightarrow> orthogonal x y"
 proof -
   have "span S \<subset> UNIV"
-  by (metis (mono_tags) UNIV_I assms inner_eq_zero_iff less_le lowdim_subset_hyperplane
-      mem_Collect_eq top.extremum_strict top.not_eq_extremum)
+    by (metis assms dim_eq_full order_less_imp_not_less top.not_eq_extremum)
   with orthogonal_to_subspace_exists_gen [of S UNIV] that show ?thesis
     by (auto)
 qed
@@ -1683,11 +1454,8 @@
   have orth: "orthogonal (x - ?a) w" if "w \<in> span S" for w
     by (simp add: Gram_Schmidt_step \<open>pairwise orthogonal T\<close> \<open>span T = span S\<close>
         orthogonal_commute that)
-  show ?thesis
-    apply (rule_tac y = "?a" and z = "x - ?a" in that)
-      apply (meson \<open>T \<subseteq> span S\<close> span_scale span_sum subsetCE)
-     apply (fact orth, simp)
-    done
+  with that[of ?a "x-?a"] \<open>T \<subseteq> span S\<close> show ?thesis
+    by (simp add: span_mul span_sum subsetD)
 qed
 
 lemma orthogonal_subspace_decomp_unique:
@@ -1702,15 +1470,15 @@
   moreover have "\<And>a b. \<lbrakk>a \<in> span S; b \<in> span T\<rbrakk> \<Longrightarrow> orthogonal a b"
     by (meson orth orthogonal_commute orthogonal_to_span)
   ultimately have "0 = x' - x"
-    by (metis (full_types) add_diff_cancel_left' ST diff_right_commute orthogonal_clauses(10) orthogonal_clauses(5) orthogonal_self)
+    using assms
+    by (metis add.commute add_diff_cancel_right' diff_right_commute orthogonal_self span_diff)
   with assms show ?thesis by auto
 qed
 
 lemma vector_in_orthogonal_spanningset:
   fixes a :: "'a::euclidean_space"
   obtains S where "a \<in> S" "pairwise orthogonal S" "span S = UNIV"
-  by (metis UNIV_I Un_iff empty_iff insert_subset orthogonal_extension pairwise_def
-      pairwise_orthogonal_insert span_UNIV subsetI subset_antisym)
+  by (metis UnI1 Un_UNIV_right insertI1 orthogonal_extension pairwise_singleton span_UNIV)
 
 lemma vector_in_orthogonal_basis:
   fixes a :: "'a::euclidean_space"
@@ -1755,17 +1523,15 @@
       using \<open>pairwise orthogonal S\<close> by (auto simp: pairwise_def orthogonal_def)
     show "\<And>x. x \<in> (\<lambda>x. x /\<^sub>R norm x) ` S \<Longrightarrow> norm x = 1"
       using \<open>0 \<notin> S\<close> by (auto simp: field_split_simps)
-    then show "independent ?S"
+    then show ind: "independent ?S"
       by (metis \<open>pairwise orthogonal ((\<lambda>x. x /\<^sub>R norm x) ` S)\<close> norm_zero pairwise_orthogonal_independent zero_neq_one)
     have "inj_on (\<lambda>x. x /\<^sub>R norm x) S"
       unfolding inj_on_def
       by (metis (full_types) S(1) \<open>0 \<notin> S\<close> inverse_nonzero_iff_nonzero norm_eq_zero orthogonal_scaleR orthogonal_self pairwise_def)
     then show "card ?S = DIM('a)"
       by (simp add: card_image S)
-    show "span ?S = UNIV"
-      by (metis (no_types) \<open>0 \<notin> S\<close> \<open>finite S\<close> \<open>span S = UNIV\<close>
-          field_class.field_inverse_zero inverse_inverse_eq less_irrefl span_image_scale
-          zero_less_norm_iff)
+    then show "span ?S = UNIV"
+      by (metis ind dim_eq_card dim_eq_full)
   qed
 qed
 
@@ -1786,12 +1552,10 @@
     by (auto simp add: span_Un image_def)
   also have "dim \<dots> = dim {x + y |x y. x \<in> span A \<and> y \<in> span B}"
     by (auto intro!: arg_cong [where f=dim])
-  also have "... = dim {x + y |x y. x \<in> span A \<and> y \<in> span B} + dim(span A \<inter> span B)"
-    by (auto simp: dest: 0)
-  also have "... = dim (span A) + dim (span B)"
-    by (rule dim_sums_Int) (auto)
-  also have "... = dim A + dim B"
-    by (simp)
+  also have "\<dots> = dim {x + y |x y. x \<in> span A \<and> y \<in> span B} + dim(span A \<inter> span B)"
+    by (auto dest: 0)
+  also have "\<dots> = dim A + dim B"
+    using dim_sums_Int by fastforce
   finally show ?thesis .
 qed
 
@@ -1810,17 +1574,16 @@
     proof -
       obtain y z where "x = y + z" "y \<in> span A" and orth: "\<And>w. w \<in> span A \<Longrightarrow> orthogonal z w"
         using orthogonal_subspace_decomp_exists [of A x] that by auto
+      moreover
       have "y \<in> span B"
         using \<open>y \<in> span A\<close> assms(3) span_mono by blast
-      then have "z \<in> {a \<in> B. \<forall>x. x \<in> A \<longrightarrow> orthogonal x a}"
-        apply simp
-        using \<open>x = y + z\<close> assms(1) assms(2) orth orthogonal_commute span_add_eq
-          span_eq_iff that by blast
+      ultimately have "z \<in> B \<and> (\<forall>x. x \<in> A \<longrightarrow> orthogonal x z)"
+        using assms by (metis orthogonal_commute span_add_eq span_eq_iff that)
       then have z: "z \<in> span {y \<in> B. \<forall>x\<in>A. orthogonal x y}"
-        by (meson span_superset subset_iff)
+        by (simp add: span_base)
       then show ?thesis
-        apply (auto simp: span_Un image_def  \<open>x = y + z\<close> \<open>y \<in> span A\<close>)
-        using \<open>y \<in> span A\<close> add.commute by blast
+        by (smt (verit, best) \<open>x = y + z\<close> \<open>y \<in> span A\<close> le_sup_iff span_add_eq span_subspace_induct 
+            span_superset subset_iff subspace_span)
     qed
     show "span B \<subseteq> span ({y \<in> B. \<forall>x\<in>A. orthogonal x y} \<union> A)"
       by (rule span_minimal) (auto intro: * span_minimal)
@@ -1845,12 +1608,8 @@
 qed
 
 lemma linear_continuous_at:
-  assumes "bounded_linear f"
-  shows "continuous (at a) f"
-  unfolding continuous_at using assms
-  apply (rule bounded_linear.tendsto)
-  apply (rule tendsto_ident_at)
-  done
+  "bounded_linear f \<Longrightarrow>continuous (at a) f"
+  by (simp add: bounded_linear.isUCont isUCont_isCont)
 
 lemma linear_continuous_within:
   "bounded_linear f \<Longrightarrow> continuous (at x within s) f"
@@ -1869,21 +1628,7 @@
     using linear_bounded_pos [OF \<open>linear h\<close>] by blast
   show ?thesis
     unfolding tendsto_iff
-  proof (intro allI impI)
-    show "\<forall>\<^sub>F x in F. dist (h (f x)) (h l) < e" if "e > 0" for e
-    proof -
-      have "\<forall>\<^sub>F x in F. dist (f x) l < e/B"
-        by (simp add: \<open>0 < B\<close> assms(1) tendstoD that)
-      then show ?thesis
-        unfolding dist_norm
-      proof (rule eventually_mono)
-        show "norm (h (f x) - h l) < e" if "norm (f x - l) < e / B" for x
-          using that B
-          apply (simp add: field_split_simps)
-          by (metis \<open>linear h\<close> le_less_trans linear_diff)
-      qed
-    qed
-  qed
+      by (simp add: assms bounded_linear.tendsto linear_linear tendstoD)
 qed
 
 lemma linear_continuous_compose:




