# HG changeset patch
# User wenzelm
# Date 1378308997 -7200
# Node ID d4374a69ddff877c8140e832004563a9b195054a
# Parent  ed2b48af04d941cd6a526e5f24fc9d2dfa29bd8c
tuned proofs;

diff -r ed2b48af04d9 -r d4374a69ddff src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy
--- a/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Wed Sep 04 17:35:47 2013 +0200
+++ b/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Wed Sep 04 17:36:37 2013 +0200
@@ -34,8 +34,8 @@
   using assms convex_def[of S] by auto
 
 lemma mem_convex_alt:
-  assumes "convex S" "x : S" "y : S" "u>=0" "v>=0" "u+v>0"
-  shows "((u/(u+v)) *\<^sub>R x + (v/(u+v)) *\<^sub>R y) : S"
+  assumes "convex S" "x \<in> S" "y \<in> S" "u \<ge> 0" "v \<ge> 0" "u + v > 0"
+  shows "((u/(u+v)) *\<^sub>R x + (v/(u+v)) *\<^sub>R y) \<in> S"
   apply (subst mem_convex_2)
   using assms
   apply (auto simp add: algebra_simps zero_le_divide_iff)
@@ -74,20 +74,20 @@
   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
   assumes lf: "linear f"
     and fi: "inj_on f (span S)"
-  shows "dim (f ` S) = dim (S:: 'n::euclidean_space set)"
-proof -
-  obtain B where B_def: "B \<subseteq> S \<and> independent B \<and> S \<subseteq> span B \<and> card B = dim S"
+  shows "dim (f ` S) = dim (S::'n::euclidean_space set)"
+proof -
+  obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S"
     using basis_exists[of S] by auto
   then have "span S = span B"
     using span_mono[of B S] span_mono[of S "span B"] span_span[of B] by auto
   then have "independent (f ` B)"
-    using independent_injective_on_span_image[of B f] B_def assms by auto
+    using independent_injective_on_span_image[of B f] B assms by auto
   moreover have "card (f ` B) = card B"
-    using assms card_image[of f B] subset_inj_on[of f "span S" B] B_def span_inc by auto
+    using assms card_image[of f B] subset_inj_on[of f "span S" B] B span_inc by auto
   moreover have "(f ` B) \<subseteq> (f ` S)"
-    using B_def by auto
+    using B by auto
   ultimately have "dim (f ` S) \<ge> dim S"
-    using independent_card_le_dim[of "f ` B" "f ` S"] B_def by auto
+    using independent_card_le_dim[of "f ` B" "f ` S"] B by auto
   then show ?thesis
     using dim_image_le[of f S] assms by auto
 qed
@@ -220,8 +220,6 @@
     and C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "card C = dim T"
   shows "\<exists>f. linear f \<and> f ` B = C \<and> f ` S = T \<and> inj_on f S"
 proof -
-(* Proof is a modified copy of the proof of similar lemma subspace_isomorphism
-*)
   from B independent_bound have fB: "finite B"
     by blast
   from C independent_bound have fC: "finite C"
@@ -293,9 +291,6 @@
   then show ?thesis using closure_linear_image[of f S] assms by auto
 qed
 
-lemma closure_direct_sum: "closure (S \<times> T) = closure S \<times> closure T"
-  by (rule closure_Times)
-
 lemma closure_scaleR:
   fixes S :: "'a::real_normed_vector set"
   shows "(op *\<^sub>R c) ` (closure S) = closure ((op *\<^sub>R c) ` S)"
@@ -367,7 +362,7 @@
     by (auto simp add:norm_minus_commute)
 qed
 
-lemma norm_minus_eqI:"x = - y \<Longrightarrow> norm x = norm y" by auto
+lemma norm_minus_eqI: "x = - y \<Longrightarrow> norm x = norm y" by auto
 
 lemma Min_grI:
   assumes "finite A" "A \<noteq> {}" "\<forall>a\<in>A. x < a"
@@ -8668,7 +8663,7 @@
   have "(closure S) + (closure T) = (\<lambda>(x,y). x + y) ` (closure S \<times> closure T)"
     by (simp add: set_plus_image)
   also have "... = (\<lambda>(x,y). x + y) ` closure (S \<times> T)"
-    using closure_direct_sum by auto
+    using closure_Times by auto
   also have "... \<subseteq> closure (S + T)"
     using fst_snd_linear closure_linear_image[of "(\<lambda>(x,y). x + y)" "S \<times> T"]
     by (auto simp: set_plus_image)
diff -r ed2b48af04d9 -r d4374a69ddff src/HOL/Multivariate_Analysis/Linear_Algebra.thy
--- a/src/HOL/Multivariate_Analysis/Linear_Algebra.thy	Wed Sep 04 17:35:47 2013 +0200
+++ b/src/HOL/Multivariate_Analysis/Linear_Algebra.thy	Wed Sep 04 17:36:37 2013 +0200
@@ -17,11 +17,15 @@
 
 lemma square_bound_lemma: "(x::real) < (1 + x) * (1 + x)"
 proof -
-  have "(x + 1/2)\<^sup>2 + 3/4 > 0" using zero_le_power2[of "x+1/2"] by arith
-  then show ?thesis by (simp add: field_simps power2_eq_square)
+  have "(x + 1/2)\<^sup>2 + 3/4 > 0"
+    using zero_le_power2[of "x+1/2"] by arith
+  then show ?thesis
+    by (simp add: field_simps power2_eq_square)
 qed
 
-lemma square_continuous: "0 < (e::real) ==> \<exists>d. 0 < d \<and> (\<forall>y. abs(y - x) < d \<longrightarrow> abs(y * y - x * x) < e)"
+lemma square_continuous:
+  fixes e :: real
+  shows "e > 0 \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>y. abs (y - x) < d \<longrightarrow> abs (y * y - x * x) < e)"
   using isCont_power[OF isCont_ident, of x, unfolded isCont_def LIM_eq, rule_format, of e 2]
   apply (auto simp add: power2_eq_square)
   apply (rule_tac x="s" in exI)
@@ -30,7 +34,7 @@
   apply auto
   done
 
-lemma real_le_lsqrt: "0 <= x \<Longrightarrow> 0 <= y \<Longrightarrow> x <= y\<^sup>2 ==> sqrt x <= y"
+lemma real_le_lsqrt: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y\<^sup>2 \<Longrightarrow> sqrt x \<le> y"
   using real_sqrt_le_iff[of x "y\<^sup>2"] by simp
 
 lemma real_le_rsqrt: "x\<^sup>2 \<le> y \<Longrightarrow> x \<le> sqrt y"
@@ -41,46 +45,49 @@
 
 lemma sqrt_even_pow2:
   assumes n: "even n"
-  shows "sqrt(2 ^ n) = 2 ^ (n div 2)"
+  shows "sqrt (2 ^ n) = 2 ^ (n div 2)"
 proof -
-  from n obtain m where m: "n = 2*m" unfolding even_mult_two_ex ..
-  from m  have "sqrt(2 ^ n) = sqrt ((2 ^ m)\<^sup>2)"
+  from n obtain m where m: "n = 2 * m"
+    unfolding even_mult_two_ex ..
+  from m have "sqrt (2 ^ n) = sqrt ((2 ^ m)\<^sup>2)"
     by (simp only: power_mult[symmetric] mult_commute)
-  then show ?thesis  using m by simp
+  then show ?thesis
+    using m by simp
 qed
 
-lemma real_div_sqrt: "0 <= x ==> x / sqrt(x) = sqrt(x)"
-  apply (cases "x = 0", simp_all)
+lemma real_div_sqrt: "0 \<le> x \<Longrightarrow> x / sqrt x = sqrt x"
+  apply (cases "x = 0")
+  apply simp_all
   using sqrt_divide_self_eq[of x]
   apply (simp add: inverse_eq_divide field_simps)
   done
 
 text{* Hence derive more interesting properties of the norm. *}
 
-lemma norm_eq_0_dot: "(norm x = 0) \<longleftrightarrow> (inner x x = (0::real))"
+lemma norm_eq_0_dot: "norm x = 0 \<longleftrightarrow> x \<bullet> x = (0::real)"
   by simp (* TODO: delete *)
 
-lemma norm_cauchy_schwarz: "inner x y <= norm x * norm y"
+lemma norm_cauchy_schwarz: "x \<bullet> y \<le> norm x * norm y"
   (* TODO: move to Inner_Product.thy *)
   using Cauchy_Schwarz_ineq2[of x y] by auto
 
 lemma norm_triangle_sub:
   fixes x y :: "'a::real_normed_vector"
-  shows "norm x \<le> norm y  + norm (x - y)"
+  shows "norm x \<le> norm y + norm (x - y)"
   using norm_triangle_ineq[of "y" "x - y"] by (simp add: field_simps)
 
-lemma norm_le: "norm(x) <= norm(y) \<longleftrightarrow> x \<bullet> x <= y \<bullet> y"
-  by (simp add: norm_eq_sqrt_inner) 
-
-lemma norm_lt: "norm(x) < norm(y) \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
+lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> x \<bullet> x \<le> y \<bullet> y"
   by (simp add: norm_eq_sqrt_inner)
 
-lemma norm_eq: "norm(x) = norm (y) \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
+lemma norm_lt: "norm x < norm y \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
+  by (simp add: norm_eq_sqrt_inner)
+
+lemma norm_eq: "norm x = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
   apply (subst order_eq_iff)
   apply (auto simp: norm_le)
   done
 
-lemma norm_eq_1: "norm(x) = 1 \<longleftrightarrow> x \<bullet> x = 1"
+lemma norm_eq_1: "norm x = 1 \<longleftrightarrow> x \<bullet> x = 1"
   by (simp add: norm_eq_sqrt_inner)
 
 text{* Squaring equations and inequalities involving norms.  *}
@@ -88,7 +95,7 @@
 lemma dot_square_norm: "x \<bullet> x = (norm x)\<^sup>2"
   by (simp only: power2_norm_eq_inner) (* TODO: move? *)
 
-lemma norm_eq_square: "norm(x) = a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x = a\<^sup>2"
+lemma norm_eq_square: "norm x = a \<longleftrightarrow> 0 \<le> a \<and> x \<bullet> x = a\<^sup>2"
   by (auto simp add: norm_eq_sqrt_inner)
 
 lemma real_abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> (x::real)\<^sup>2 \<le> y\<^sup>2"
@@ -102,13 +109,13 @@
   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\<^sup>2"
+lemma norm_le_square: "norm x \<le> a \<longleftrightarrow> 0 \<le> a \<and> x \<bullet> x \<le> a\<^sup>2"
   apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
   using norm_ge_zero[of x]
   apply arith
   done
 
-lemma norm_ge_square: "norm(x) >= a \<longleftrightarrow> a <= 0 \<or> x \<bullet> x >= a\<^sup>2"
+lemma norm_ge_square: "norm x \<ge> a \<longleftrightarrow> a \<le> 0 \<or> x \<bullet> x \<ge> a\<^sup>2"
   apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
   using norm_ge_zero[of x]
   apply arith
@@ -116,16 +123,17 @@
 
 lemma norm_lt_square: "norm(x) < a \<longleftrightarrow> 0 < a \<and> x \<bullet> x < a\<^sup>2"
   by (metis not_le norm_ge_square)
+
 lemma norm_gt_square: "norm(x) > a \<longleftrightarrow> a < 0 \<or> x \<bullet> x > a\<^sup>2"
   by (metis norm_le_square not_less)
 
 text{* Dot product in terms of the norm rather than conversely. *}
 
-lemmas inner_simps = inner_add_left inner_add_right inner_diff_right inner_diff_left 
+lemmas inner_simps = inner_add_left inner_add_right inner_diff_right inner_diff_left
   inner_scaleR_left inner_scaleR_right
 
 lemma dot_norm: "x \<bullet> y = ((norm (x + y))\<^sup>2 - (norm x)\<^sup>2 - (norm y)\<^sup>2) / 2"
-  unfolding power2_norm_eq_inner inner_simps inner_commute by auto 
+  unfolding power2_norm_eq_inner inner_simps inner_commute by auto
 
 lemma dot_norm_neg: "x \<bullet> y = (((norm x)\<^sup>2 + (norm y)\<^sup>2) - (norm (x - y))\<^sup>2) / 2"
   unfolding power2_norm_eq_inner inner_simps inner_commute
@@ -133,32 +141,37 @@
 
 text{* Equality of vectors in terms of @{term "op \<bullet>"} products.    *}
 
-lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x" (is "?lhs \<longleftrightarrow> ?rhs")
+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)
+  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
 
 lemma norm_triangle_half_r:
-  shows "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
-  using dist_triangle_half_r unfolding dist_norm[THEN sym] by auto
+  "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" "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[THEN sym]]]
-  unfolding dist_norm[THEN sym] .
-
-lemma norm_triangle_le: "norm(x) + norm y <= e ==> norm(x + y) <= e"
+  using dist_triangle_half_l[OF assms[unfolded dist_norm[symmetric]]]
+  unfolding dist_norm[symmetric] .
+
+lemma norm_triangle_le: "norm x + norm y \<le> e \<Longrightarrow> norm (x + y) \<le> e"
   by (rule norm_triangle_ineq [THEN order_trans])
 
-lemma norm_triangle_lt: "norm(x) + norm(y) < e ==> norm(x + y) < e"
+lemma norm_triangle_lt: "norm x + norm y < e \<Longrightarrow> norm (x + y) < e"
   by (rule norm_triangle_ineq [THEN le_less_trans])
 
 lemma setsum_clauses:
@@ -191,7 +204,8 @@
 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 "\<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
@@ -199,7 +213,8 @@
 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 "\<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
@@ -237,31 +252,35 @@
   where "linear f \<longleftrightarrow> (\<forall>x y. f(x + y) = f x + f y) \<and> (\<forall>c x. f(c *\<^sub>R x) = c *\<^sub>R f x)"
 
 lemma linearI:
-  assumes "\<And>x y. f (x + y) = f x + f y" "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
+  assumes "\<And>x y. f (x + y) = f x + f y"
+    and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
   shows "linear f"
   using assms unfolding linear_def by auto
 
-lemma linear_compose_cmul: "linear f ==> linear (\<lambda>x. c *\<^sub>R f x)"
+lemma linear_compose_cmul: "linear f \<Longrightarrow> linear (\<lambda>x. c *\<^sub>R f x)"
   by (simp add: linear_def algebra_simps)
 
-lemma linear_compose_neg: "linear f ==> linear (\<lambda>x. -(f(x)))"
+lemma linear_compose_neg: "linear f \<Longrightarrow> linear (\<lambda>x. - f x)"
   by (simp add: linear_def)
 
-lemma linear_compose_add: "linear f \<Longrightarrow> linear g ==> linear (\<lambda>x. f(x) + g(x))"
+lemma linear_compose_add: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (\<lambda>x. f x + g x)"
   by (simp add: linear_def algebra_simps)
 
-lemma linear_compose_sub: "linear f \<Longrightarrow> linear g ==> linear (\<lambda>x. f x - g x)"
+lemma linear_compose_sub: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (\<lambda>x. f x - g x)"
   by (simp add: linear_def algebra_simps)
 
-lemma linear_compose: "linear f \<Longrightarrow> linear g ==> linear (g o f)"
+lemma linear_compose: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (g \<circ> f)"
   by (simp add: linear_def)
 
-lemma linear_id: "linear id" by (simp add: linear_def id_def)
-
-lemma linear_zero: "linear (\<lambda>x. 0)" by (simp add: linear_def)
+lemma linear_id: "linear id"
+  by (simp add: linear_def id_def)
+
+lemma linear_zero: "linear (\<lambda>x. 0)"
+  by (simp add: linear_def)
 
 lemma linear_compose_setsum:
-  assumes fS: "finite S" and lS: "\<forall>a \<in> S. linear (f a)"
+  assumes fS: "finite S"
+    and lS: "\<forall>a \<in> S. linear (f a)"
   shows "linear(\<lambda>x. setsum (\<lambda>a. f a x) S)"
   using lS
   apply (induct rule: finite_induct[OF fS])
@@ -275,88 +294,100 @@
   apply simp
   done
 
-lemma linear_cmul: "linear f ==> f(c *\<^sub>R x) = c *\<^sub>R f x"
+lemma linear_cmul: "linear f \<Longrightarrow> f (c *\<^sub>R x) = c *\<^sub>R f x"
   by (simp add: linear_def)
 
-lemma linear_neg: "linear f ==> f (-x) = - f x"
+lemma linear_neg: "linear f \<Longrightarrow> f (- x) = - f x"
   using linear_cmul [where c="-1"] by simp
 
-lemma linear_add: "linear f ==> f(x + y) = f x + f y"
+lemma linear_add: "linear f \<Longrightarrow> f(x + y) = f x + f y"
   by (metis linear_def)
 
-lemma linear_sub: "linear f ==> f(x - y) = f x - f y"
+lemma linear_sub: "linear f \<Longrightarrow> f(x - y) = f x - f y"
   by (simp add: diff_minus linear_add linear_neg)
 
 lemma linear_setsum:
-  assumes lf: "linear f" and fS: "finite S"
-  shows "f (setsum g S) = setsum (f o g) S"
-  using fS
-proof (induct rule: finite_induct)
+  assumes lin: "linear f"
+    and fin: "finite S"
+  shows "f (setsum g S) = setsum (f \<circ> g) S"
+  using fin
+proof induct
   case empty
-  then show ?case by (simp add: linear_0[OF lf])
+  then show ?case
+    by (simp add: linear_0[OF lin])
 next
   case (insert x F)
-  have "f (setsum g (insert x F)) = f (g x + setsum g F)" using insert.hyps
-    by simp
-  also have "\<dots> = f (g x) + f (setsum g F)" using linear_add[OF lf] by simp
-  also have "\<dots> = setsum (f o g) (insert x F)" using insert.hyps by simp
+  have "f (setsum g (insert x F)) = f (g x + setsum g F)"
+    using insert.hyps by simp
+  also have "\<dots> = f (g x) + f (setsum g F)"
+    using linear_add[OF lin] by simp
+  also have "\<dots> = setsum (f \<circ> g) (insert x F)"
+    using insert.hyps by simp
   finally show ?case .
 qed
 
 lemma linear_setsum_mul:
-  assumes lf: "linear f" and fS: "finite S"
+  assumes lin: "linear f"
+    and fin: "finite S"
   shows "f (setsum (\<lambda>i. c i *\<^sub>R v i) S) = setsum (\<lambda>i. c i *\<^sub>R f (v i)) S"
-  using linear_setsum[OF lf fS, of "\<lambda>i. c i *\<^sub>R v i" , unfolded o_def] linear_cmul[OF lf]
+  using linear_setsum[OF lin fin, of "\<lambda>i. c i *\<^sub>R v i" , unfolded o_def] linear_cmul[OF lin]
   by simp
 
 lemma linear_injective_0:
-  assumes lf: "linear f"
+  assumes lin: "linear f"
   shows "inj f \<longleftrightarrow> (\<forall>x. f x = 0 \<longrightarrow> x = 0)"
 proof -
-  have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)" by (simp add: inj_on_def)
-  also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)" by simp
+  have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)"
+    by (simp add: inj_on_def)
+  also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)"
+    by simp
   also have "\<dots> \<longleftrightarrow> (\<forall> x y. f (x - y) = 0 \<longrightarrow> x - y = 0)"
-    by (simp add: linear_sub[OF lf])
-  also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)" by auto
+    by (simp add: linear_sub[OF lin])
+  also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)"
+    by auto
   finally show ?thesis .
 qed
 
 
 subsection {* Bilinear functions. *}
 
-definition "bilinear f \<longleftrightarrow> (\<forall>x. linear(\<lambda>y. f x y)) \<and> (\<forall>y. linear(\<lambda>x. f x y))"
-
-lemma bilinear_ladd: "bilinear h ==> h (x + y) z = (h x z) + (h y z)"
+definition "bilinear f \<longleftrightarrow> (\<forall>x. linear (\<lambda>y. f x y)) \<and> (\<forall>y. linear (\<lambda>x. f x y))"
+
+lemma bilinear_ladd: "bilinear h \<Longrightarrow> h (x + y) z = h x z + h y z"
   by (simp add: bilinear_def linear_def)
 
-lemma bilinear_radd: "bilinear h ==> h x (y + z) = (h x y) + (h x z)"
+lemma bilinear_radd: "bilinear h \<Longrightarrow> h x (y + z) = h x y + h x z"
   by (simp add: bilinear_def linear_def)
 
-lemma bilinear_lmul: "bilinear h ==> h (c *\<^sub>R x) y = c *\<^sub>R (h x y)"
+lemma bilinear_lmul: "bilinear h \<Longrightarrow> h (c *\<^sub>R x) y = c *\<^sub>R h x y"
   by (simp add: bilinear_def linear_def)
 
-lemma bilinear_rmul: "bilinear h ==> h x (c *\<^sub>R y) = c *\<^sub>R (h x y)"
+lemma bilinear_rmul: "bilinear h \<Longrightarrow> h x (c *\<^sub>R y) = c *\<^sub>R h x y"
   by (simp add: bilinear_def linear_def)
 
-lemma bilinear_lneg: "bilinear h ==> h (- x) y = -(h x y)"
+lemma bilinear_lneg: "bilinear h \<Longrightarrow> h (- x) y = - h x y"
   by (simp only: scaleR_minus1_left [symmetric] bilinear_lmul)
 
-lemma bilinear_rneg: "bilinear h ==> h x (- y) = - h x y"
+lemma bilinear_rneg: "bilinear h \<Longrightarrow> h x (- y) = - h x y"
   by (simp only: scaleR_minus1_left [symmetric] bilinear_rmul)
 
-lemma  (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
+lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
   using add_imp_eq[of x y 0] by auto
 
-lemma bilinear_lzero: assumes "bilinear h" shows "h 0 x = 0"
+lemma bilinear_lzero:
+  assumes "bilinear h"
+  shows "h 0 x = 0"
   using bilinear_ladd [OF assms, of 0 0 x] by (simp add: eq_add_iff field_simps)
 
-lemma bilinear_rzero: assumes "bilinear h" shows "h x 0 = 0"
+lemma bilinear_rzero:
+  assumes "bilinear h"
+  shows "h x 0 = 0"
   using bilinear_radd [OF assms, of x 0 0 ] by (simp add: eq_add_iff field_simps)
 
-lemma bilinear_lsub: "bilinear h ==> h (x - y) z = h x z - h y z"
+lemma bilinear_lsub: "bilinear h \<Longrightarrow> h (x - y) z = h x z - h y z"
   by (simp  add: diff_minus bilinear_ladd bilinear_lneg)
 
-lemma bilinear_rsub: "bilinear h ==> h z (x - y) = h z x - h z y"
+lemma bilinear_rsub: "bilinear h \<Longrightarrow> h z (x - y) = h z x - h z y"
   by (simp  add: diff_minus bilinear_radd bilinear_rneg)
 
 lemma bilinear_setsum:
@@ -367,7 +398,8 @@
 proof -
   have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. h (f x) (setsum g T)) S"
     apply (rule linear_setsum[unfolded o_def])
-    using bh fS apply (auto simp add: bilinear_def)
+    using bh fS
+    apply (auto simp add: bilinear_def)
     done
   also have "\<dots> = setsum (\<lambda>x. setsum (\<lambda>y. h (f x) (g y)) T) S"
     apply (rule setsum_cong, simp)
@@ -375,7 +407,8 @@
     using bh fT
     apply (auto simp add: bilinear_def)
     done
-  finally show ?thesis unfolding setsum_cartesian_product .
+  finally show ?thesis
+    unfolding setsum_cartesian_product .
 qed
 
 
@@ -388,13 +421,19 @@
   shows "adjoint f = g"
   unfolding adjoint_def
 proof (rule some_equality)
-  show "\<forall>x y. inner (f x) y = inner x (g y)" using assms .
+  show "\<forall>x y. inner (f x) y = inner x (g y)"
+    by (rule assms)
 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
+  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)
 qed
 
@@ -418,7 +457,7 @@
       unfolding linear_setsum[OF lf finite_Basis]
       by (simp add: linear_cmul[OF lf])
     finally show "f x \<bullet> y = x \<bullet> ?w"
-        by (simp add: inner_setsum_left inner_setsum_right mult_commute)
+      by (simp add: inner_setsum_left inner_setsum_right mult_commute)
   qed
   then show ?thesis
     unfolding adjoint_def choice_iff
@@ -445,18 +484,22 @@
   shows "adjoint (adjoint f) = f"
   by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])
 
+
 subsection {* Interlude: Some properties of real sets *}
 
-lemma seq_mono_lemma: assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n" and "\<forall>n \<ge> m. e n <= e m"
+lemma seq_mono_lemma:
+  assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n"
+    and "\<forall>n \<ge> m. e n \<le> e m"
   shows "\<forall>n \<ge> m. d n < e m"
-  using assms apply auto
+  using assms
+  apply auto
   apply (erule_tac x="n" in allE)
   apply (erule_tac x="n" in allE)
   apply auto
   done
 
-
-lemma infinite_enumerate: assumes fS: "infinite S"
+lemma infinite_enumerate:
+  assumes fS: "infinite S"
   shows "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)"
   unfolding subseq_def
   using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
@@ -467,53 +510,57 @@
   apply auto
   done
 
-
 lemma triangle_lemma:
-  assumes x: "0 <= (x::real)" and y:"0 <= y" and z: "0 <= z" and xy: "x\<^sup>2 <= y\<^sup>2 + z\<^sup>2"
-  shows "x <= y + z"
+  fixes x y z :: real
+  assumes x: "0 \<le> x"
+    and y: "0 \<le> y"
+    and z: "0 \<le> z"
+    and xy: "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2"
+  shows "x \<le> y + z"
 proof -
-  have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2*y*z + z\<^sup>2" using z y by (simp add: mult_nonneg_nonneg)
-  with xy have th: "x\<^sup>2 \<le> (y+z)\<^sup>2" by (simp add: power2_eq_square field_simps)
-  from y z have yz: "y + z \<ge> 0" by arith
+  have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2 *y * z + z\<^sup>2"
+    using z y by (simp add: mult_nonneg_nonneg)
+  with xy have th: "x\<^sup>2 \<le> (y + z)\<^sup>2"
+    by (simp add: power2_eq_square field_simps)
+  from y z have yz: "y + z \<ge> 0"
+    by arith
   from power2_le_imp_le[OF th yz] show ?thesis .
 qed
 
 
 subsection {* A generic notion of "hull" (convex, affine, conic hull and closure). *}
 
-definition hull :: "('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "hull" 75)
-  where "S hull s = Inter {t. S t \<and> s \<subseteq> t}"
+definition hull :: "('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "hull" 75)
+  where "S hull s = \<Inter>{t. S t \<and> s \<subseteq> t}"
 
 lemma hull_same: "S s \<Longrightarrow> S hull s = s"
   unfolding hull_def by auto
 
-lemma hull_in: "(\<And>T. Ball T S ==> S (Inter T)) ==> S (S hull s)"
+lemma hull_in: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> S (S hull s)"
   unfolding hull_def Ball_def by auto
 
-lemma hull_eq: "(\<And>T. Ball T S ==> S (Inter T)) ==> (S hull s) = s \<longleftrightarrow> S s"
+lemma hull_eq: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> (S hull s) = s \<longleftrightarrow> S s"
   using hull_same[of S s] hull_in[of S s] by metis
 
-
 lemma hull_hull: "S hull (S hull s) = S hull s"
   unfolding hull_def by blast
 
 lemma hull_subset[intro]: "s \<subseteq> (S hull s)"
   unfolding hull_def by blast
 
-lemma hull_mono: " s \<subseteq> t ==> (S hull s) \<subseteq> (S hull t)"
+lemma hull_mono: "s \<subseteq> t \<Longrightarrow> (S hull s) \<subseteq> (S hull t)"
   unfolding hull_def by blast
 
-lemma hull_antimono: "\<forall>x. S x \<longrightarrow> T x ==> (T hull s) \<subseteq> (S hull s)"
+lemma hull_antimono: "\<forall>x. S x \<longrightarrow> T x \<Longrightarrow> (T hull s) \<subseteq> (S hull s)"
   unfolding hull_def by blast
 
-lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> S t ==> (S hull s) \<subseteq> t"
+lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (S hull s) \<subseteq> t"
   unfolding hull_def by blast
 
-lemma subset_hull: "S t ==> S hull s \<subseteq> t \<longleftrightarrow>  s \<subseteq> t"
+lemma subset_hull: "S t \<Longrightarrow> S hull s \<subseteq> t \<longleftrightarrow> s \<subseteq> t"
   unfolding hull_def by blast
 
-lemma hull_unique: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow>
-    (\<And>t'. s \<subseteq> t' \<Longrightarrow> S t' \<Longrightarrow> t \<subseteq> t') \<Longrightarrow> (S hull s = t)"
+lemma hull_unique: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (\<And>t'. s \<subseteq> t' \<Longrightarrow> S t' \<Longrightarrow> t \<subseteq> t') \<Longrightarrow> (S hull s = t)"
   unfolding hull_def by auto
 
 lemma hull_induct: "(\<And>x. x\<in> S \<Longrightarrow> P x) \<Longrightarrow> Q {x. P x} \<Longrightarrow> \<forall>x\<in> Q hull S. P x"
@@ -527,7 +574,7 @@
   unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2)
 
 lemma hull_union:
-  assumes T: "\<And>T. Ball T S ==> S (Inter T)"
+  assumes T: "\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)"
   shows "S hull (s \<union> t) = S hull (S hull s \<union> S hull t)"
   apply rule
   apply (rule hull_mono)
@@ -541,13 +588,13 @@
 lemma hull_redundant_eq: "a \<in> (S hull s) \<longleftrightarrow> (S hull (insert a s) = S hull s)"
   unfolding hull_def by blast
 
-lemma hull_redundant: "a \<in> (S hull s) ==> (S hull (insert a s) = S hull s)"
+lemma hull_redundant: "a \<in> (S hull s) \<Longrightarrow> (S hull (insert a s) = S hull s)"
   by (metis hull_redundant_eq)
 
 
 subsection {* Archimedean properties and useful consequences *}
 
-lemma real_arch_simple: "\<exists>n. x <= real (n::nat)"
+lemma real_arch_simple: "\<exists>n. x \<le> real (n::nat)"
   unfolding real_of_nat_def by (rule ex_le_of_nat)
 
 lemma real_arch_inv: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
@@ -558,60 +605,77 @@
   apply simp
   done
 
-lemma real_pow_lbound: "0 <= x ==> 1 + real n * x <= (1 + x) ^ n"
+lemma real_pow_lbound: "0 \<le> x \<Longrightarrow> 1 + real n * x \<le> (1 + x) ^ n"
 proof (induct n)
   case 0
   then show ?case by simp
 next
   case (Suc n)
-  then have h: "1 + real n * x \<le> (1 + x) ^ n" by simp
-  from h have p: "1 \<le> (1 + x) ^ n" using Suc.prems by simp
-  from h have "1 + real n * x + x \<le> (1 + x) ^ n + x" by simp
-  also have "\<dots> \<le> (1 + x) ^ Suc n" apply (subst diff_le_0_iff_le[symmetric])
+  then have h: "1 + real n * x \<le> (1 + x) ^ n"
+    by simp
+  from h have p: "1 \<le> (1 + x) ^ n"
+    using Suc.prems by simp
+  from h have "1 + real n * x + x \<le> (1 + x) ^ n + x"
+    by simp
+  also have "\<dots> \<le> (1 + x) ^ Suc n"
+    apply (subst diff_le_0_iff_le[symmetric])
     apply (simp add: field_simps)
-    using mult_left_mono[OF p Suc.prems] apply simp
+    using mult_left_mono[OF p Suc.prems]
+    apply simp
     done
-  finally show ?case  by (simp add: real_of_nat_Suc field_simps)
+  finally show ?case
+    by (simp add: real_of_nat_Suc field_simps)
 qed
 
-lemma real_arch_pow: assumes x: "1 < (x::real)" shows "\<exists>n. y < x^n"
+lemma real_arch_pow:
+  fixes x :: real
+  assumes x: "1 < x"
+  shows "\<exists>n. y < x^n"
 proof -
-  from x have x0: "x - 1 > 0" by arith
+  from x have x0: "x - 1 > 0"
+    by arith
   from reals_Archimedean3[OF x0, rule_format, of y]
-  obtain n::nat where n:"y < real n * (x - 1)" by metis
+  obtain n :: nat where n: "y < real n * (x - 1)" by metis
   from x0 have x00: "x- 1 \<ge> 0" by arith
   from real_pow_lbound[OF x00, of n] n
   have "y < x^n" by auto
   then show ?thesis by metis
 qed
 
-lemma real_arch_pow2: "\<exists>n. (x::real) < 2^ n"
+lemma real_arch_pow2:
+  fixes x :: real
+  shows "\<exists>n. x < 2^ n"
   using real_arch_pow[of 2 x] by simp
 
 lemma real_arch_pow_inv:
-  assumes y: "(y::real) > 0" and x1: "x < 1"
+  fixes x y :: real
+  assumes y: "y > 0"
+    and x1: "x < 1"
   shows "\<exists>n. x^n < y"
-proof -
-  { assume x0: "x > 0"
-    from x0 x1 have ix: "1 < 1/x" by (simp add: field_simps)
-    from real_arch_pow[OF ix, of "1/y"]
-    obtain n where n: "1/y < (1/x)^n" by blast
-    then have ?thesis using y x0
-      by (auto simp add: field_simps power_divide) }
-  moreover
-  { assume "\<not> x > 0"
-    with y x1 have ?thesis apply auto by (rule exI[where x=1], auto) }
-  ultimately show ?thesis by metis
+proof (cases "x > 0")
+  case True
+  with x1 have ix: "1 < 1/x" by (simp add: field_simps)
+  from real_arch_pow[OF ix, of "1/y"]
+  obtain n where n: "1/y < (1/x)^n" by blast
+  then show ?thesis using y `x > 0`
+    by (auto simp add: field_simps power_divide)
+next
+  case False
+  with y x1 show ?thesis
+    apply auto
+    apply (rule exI[where x=1])
+    apply auto
+    done
 qed
 
 lemma forall_pos_mono:
-  "(\<And>d e::real. d < e \<Longrightarrow> P d ==> P e) \<Longrightarrow>
-    (\<And>n::nat. n \<noteq> 0 ==> P(inverse(real n))) \<Longrightarrow> (\<And>e. 0 < e ==> P e)"
+  "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
+    (\<And>n::nat. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))) \<Longrightarrow> (\<And>e. 0 < e \<Longrightarrow> P e)"
   by (metis real_arch_inv)
 
 lemma forall_pos_mono_1:
-  "(\<And>d e::real. d < e \<Longrightarrow> P d ==> P e) \<Longrightarrow>
-    (\<And>n. P(inverse(real (Suc n)))) ==> 0 < e ==> P e"
+  "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
+    (\<And>n. P(inverse(real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e"
   apply (rule forall_pos_mono)
   apply auto
   apply (atomize)
@@ -620,15 +684,20 @@
   done
 
 lemma real_archimedian_rdiv_eq_0:
-  assumes x0: "x \<ge> 0" and c: "c \<ge> 0" and xc: "\<forall>(m::nat)>0. real m * x \<le> c"
+  assumes x0: "x \<ge> 0"
+    and c: "c \<ge> 0"
+    and xc: "\<forall>(m::nat)>0. real m * x \<le> c"
   shows "x = 0"
-proof -
-  { assume "x \<noteq> 0" with x0 have xp: "x > 0" by arith
-    from reals_Archimedean3[OF xp, rule_format, of c]
-    obtain n::nat where n: "c < real n * x" by blast
-    with xc[rule_format, of n] have "n = 0" by arith
-    with n c have False by simp }
-  then show ?thesis by blast
+proof (rule ccontr)
+  assume "x \<noteq> 0"
+  with x0 have xp: "x > 0" by arith
+  from reals_Archimedean3[OF xp, rule_format, of c]
+  obtain n :: nat where n: "c < real n * x"
+    by blast
+  with xc[rule_format, of n] have "n = 0"
+    by arith
+  with n c show False
+    by simp
 qed
 
 
@@ -639,15 +708,17 @@
 
 definition (in real_vector) "span S = (subspace hull S)"
 definition (in real_vector) "dependent S \<longleftrightarrow> (\<exists>a \<in> S. a \<in> span(S - {a}))"
-abbreviation (in real_vector) "independent s == ~(dependent s)"
+abbreviation (in real_vector) "independent s \<equiv> \<not> dependent s"
 
 text {* Closure properties of subspaces. *}
 
-lemma subspace_UNIV[simp]: "subspace(UNIV)" by (simp add: subspace_def)
-
-lemma (in real_vector) subspace_0: "subspace S ==> 0 \<in> S" by (metis subspace_def)
-
-lemma (in real_vector) subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S ==> x + y \<in> S"
+lemma subspace_UNIV[simp]: "subspace UNIV"
+  by (simp add: subspace_def)
+
+lemma (in real_vector) subspace_0: "subspace S \<Longrightarrow> 0 \<in> S"
+  by (metis subspace_def)
+
+lemma (in real_vector) subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x + y \<in> S"
   by (metis subspace_def)
 
 lemma (in real_vector) subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *\<^sub>R x \<in> S"
@@ -660,7 +731,8 @@
   by (metis diff_minus subspace_add subspace_neg)
 
 lemma (in real_vector) subspace_setsum:
-  assumes sA: "subspace A" and fB: "finite B"
+  assumes sA: "subspace A"
+    and fB: "finite B"
     and f: "\<forall>x\<in> B. f x \<in> A"
   shows "setsum f B \<in> A"
   using  fB f sA
@@ -668,36 +740,39 @@
     (simp add: subspace_def sA, auto simp add: sA subspace_add)
 
 lemma subspace_linear_image:
-  assumes lf: "linear f" and sS: "subspace S"
-  shows "subspace(f ` S)"
+  assumes lf: "linear f"
+    and sS: "subspace S"
+  shows "subspace (f ` S)"
   using lf sS linear_0[OF lf]
   unfolding linear_def subspace_def
   apply (auto simp add: image_iff)
-  apply (rule_tac x="x + y" in bexI, auto)
-  apply (rule_tac x="c *\<^sub>R x" in bexI, auto)
+  apply (rule_tac x="x + y" in bexI)
+  apply auto
+  apply (rule_tac x="c *\<^sub>R x" in bexI)
+  apply auto
   done
 
 lemma subspace_linear_vimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace (f -` S)"
   by (auto simp add: subspace_def linear_def linear_0[of f])
 
-lemma subspace_linear_preimage: "linear f ==> subspace S ==> subspace {x. f x \<in> S}"
+lemma subspace_linear_preimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace {x. f x \<in> S}"
   by (auto simp add: subspace_def linear_def linear_0[of f])
 
 lemma subspace_trivial: "subspace {0}"
   by (simp add: subspace_def)
 
-lemma (in real_vector) subspace_inter: "subspace A \<Longrightarrow> subspace B ==> subspace (A \<inter> B)"
+lemma (in real_vector) subspace_inter: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<inter> B)"
   by (simp add: subspace_def)
 
-lemma subspace_Times: "\<lbrakk>subspace A; subspace B\<rbrakk> \<Longrightarrow> subspace (A \<times> B)"
+lemma subspace_Times: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<times> B)"
   unfolding subspace_def zero_prod_def by simp
 
 text {* Properties of span. *}
 
-lemma (in real_vector) span_mono: "A \<subseteq> B ==> span A \<subseteq> span B"
+lemma (in real_vector) span_mono: "A \<subseteq> B \<Longrightarrow> span A \<subseteq> span B"
   by (metis span_def hull_mono)
 
-lemma (in real_vector) subspace_span: "subspace(span S)"
+lemma (in real_vector) subspace_span: "subspace (span S)"
   unfolding span_def
   apply (rule hull_in)
   apply (simp only: subspace_def Inter_iff Int_iff subset_eq)
@@ -705,12 +780,11 @@
   done
 
 lemma (in real_vector) span_clauses:
-  "a \<in> S ==> a \<in> span S"
+  "a \<in> S \<Longrightarrow> a \<in> span S"
   "0 \<in> span S"
-  "x\<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
+  "x\<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
   "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
-  by (metis span_def hull_subset subset_eq)
-     (metis subspace_span subspace_def)+
+  by (metis span_def hull_subset subset_eq) (metis subspace_span subspace_def)+
 
 lemma span_unique:
   "S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> (\<And>T'. S \<subseteq> T' \<Longrightarrow> subspace T' \<Longrightarrow> T \<subseteq> T') \<Longrightarrow> span S = T"
@@ -722,12 +796,14 @@
 lemma (in real_vector) span_induct:
   assumes x: "x \<in> span S"
     and P: "subspace P"
-    and SP: "\<And>x. x \<in> S ==> x \<in> P"
+    and SP: "\<And>x. x \<in> S \<Longrightarrow> x \<in> P"
   shows "x \<in> P"
 proof -
-  from SP have SP': "S \<subseteq> P" by (simp add: subset_eq)
+  from SP have SP': "S \<subseteq> P"
+    by (simp add: subset_eq)
   from x hull_minimal[where S=subspace, OF SP' P, unfolded span_def[symmetric]]
-  show "x \<in> P" by (metis subset_eq)
+  show "x \<in> P"
+    by (metis subset_eq)
 qed
 
 lemma span_empty[simp]: "span {} = {0}"
@@ -742,7 +818,7 @@
 lemma dependent_single[simp]: "dependent {x} \<longleftrightarrow> x = 0"
   unfolding dependent_def by auto
 
-lemma (in real_vector) independent_mono: "independent A \<Longrightarrow> B \<subseteq> A ==> independent B"
+lemma (in real_vector) independent_mono: "independent A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> independent B"
   apply (clarsimp simp add: dependent_def span_mono)
   apply (subgoal_tac "span (B - {a}) \<le> span (A - {a})")
   apply force
@@ -760,34 +836,46 @@
   using span_induct SP P by blast
 
 inductive_set (in real_vector) span_induct_alt_help for S:: "'a set"
-  where
+where
   span_induct_alt_help_0: "0 \<in> span_induct_alt_help S"
 | span_induct_alt_help_S:
-    "x \<in> S \<Longrightarrow> z \<in> span_induct_alt_help S \<Longrightarrow> (c *\<^sub>R x + z) \<in> span_induct_alt_help S"
+    "x \<in> S \<Longrightarrow> z \<in> span_induct_alt_help S \<Longrightarrow>
+      (c *\<^sub>R x + z) \<in> span_induct_alt_help S"
 
 lemma span_induct_alt':
-  assumes h0: "h 0" and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
+  assumes h0: "h 0"
+    and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
   shows "\<forall>x \<in> span S. h x"
 proof -
-  { fix x:: "'a" assume x: "x \<in> span_induct_alt_help S"
+  {
+    fix x :: 'a
+    assume x: "x \<in> span_induct_alt_help S"
     have "h x"
       apply (rule span_induct_alt_help.induct[OF x])
       apply (rule h0)
-      apply (rule hS, assumption, assumption)
-      done }
+      apply (rule hS)
+      apply assumption
+      apply assumption
+      done
+  }
   note th0 = this
-  { fix x assume x: "x \<in> span S"
+  {
+    fix x
+    assume x: "x \<in> span S"
     have "x \<in> span_induct_alt_help S"
     proof (rule span_induct[where x=x and S=S])
-      show "x \<in> span S" using x .
+      show "x \<in> span S" by (rule x)
     next
-      fix x assume xS : "x \<in> S"
-        from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
-        show "x \<in> span_induct_alt_help S" by simp
+      fix x
+      assume xS: "x \<in> S"
+      from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
+      show "x \<in> span_induct_alt_help S"
+        by simp
     next
       have "0 \<in> span_induct_alt_help S" by (rule span_induct_alt_help_0)
       moreover
-      { fix x y
+      {
+        fix x y
         assume h: "x \<in> span_induct_alt_help S" "y \<in> span_induct_alt_help S"
         from h have "(x + y) \<in> span_induct_alt_help S"
           apply (induct rule: span_induct_alt_help.induct)
@@ -796,9 +884,11 @@
           apply (rule span_induct_alt_help_S)
           apply assumption
           apply simp
-          done }
+          done
+      }
       moreover
-      { fix c x
+      {
+        fix c x
         assume xt: "x \<in> span_induct_alt_help S"
         then have "(c *\<^sub>R x) \<in> span_induct_alt_help S"
           apply (induct rule: span_induct_alt_help.induct)
@@ -808,15 +898,17 @@
           apply assumption
           apply simp
           done }
-      ultimately
-      show "subspace (span_induct_alt_help S)"
+      ultimately show "subspace (span_induct_alt_help S)"
         unfolding subspace_def Ball_def by blast
-    qed }
+    qed
+  }
   with th0 show ?thesis by blast
 qed
 
 lemma span_induct_alt:
-  assumes h0: "h 0" and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)" and x: "x \<in> span S"
+  assumes h0: "h 0"
+    and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
+    and x: "x \<in> span S"
   shows "h x"
   using span_induct_alt'[of h S] h0 hS x by blast
 
@@ -825,35 +917,43 @@
 lemma span_span: "span (span A) = span A"
   unfolding span_def hull_hull ..
 
-lemma (in real_vector) span_superset: "x \<in> S ==> x \<in> span S" by (metis span_clauses(1))
-
-lemma (in real_vector) span_0: "0 \<in> span S" by (metis subspace_span subspace_0)
+lemma (in real_vector) span_superset: "x \<in> S \<Longrightarrow> x \<in> span S"
+  by (metis span_clauses(1))
+
+lemma (in real_vector) span_0: "0 \<in> span S"
+  by (metis subspace_span subspace_0)
 
 lemma span_inc: "S \<subseteq> span S"
   by (metis subset_eq span_superset)
 
-lemma (in real_vector) dependent_0: assumes "0\<in>A" shows "dependent A"
-  unfolding dependent_def apply(rule_tac x=0 in bexI)
-  using assms span_0 by auto
-
-lemma (in real_vector) span_add: "x \<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
+lemma (in real_vector) dependent_0:
+  assumes "0 \<in> A"
+  shows "dependent A"
+  unfolding dependent_def
+  apply (rule_tac x=0 in bexI)
+  using assms span_0
+  apply auto
+  done
+
+lemma (in real_vector) span_add: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
   by (metis subspace_add subspace_span)
 
-lemma (in real_vector) span_mul: "x \<in> span S ==> (c *\<^sub>R x) \<in> span S"
+lemma (in real_vector) span_mul: "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
   by (metis subspace_span subspace_mul)
 
-lemma span_neg: "x \<in> span S ==> - x \<in> span S"
+lemma span_neg: "x \<in> span S \<Longrightarrow> - x \<in> span S"
   by (metis subspace_neg subspace_span)
 
-lemma span_sub: "x \<in> span S \<Longrightarrow> y \<in> span S ==> x - y \<in> span S"
+lemma span_sub: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x - y \<in> span S"
   by (metis subspace_span subspace_sub)
 
-lemma (in real_vector) span_setsum: "finite A \<Longrightarrow> \<forall>x \<in> A. f x \<in> span S ==> setsum f A \<in> span S"
+lemma (in real_vector) span_setsum: "finite A \<Longrightarrow> \<forall>x \<in> A. f x \<in> span S \<Longrightarrow> setsum f A \<in> span S"
   by (rule subspace_setsum, rule subspace_span)
 
 lemma span_add_eq: "x \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
   apply (auto simp only: span_add span_sub)
-  apply (subgoal_tac "(x + y) - x \<in> span S", simp)
+  apply (subgoal_tac "(x + y) - x \<in> span S")
+  apply simp
   apply (simp only: span_add span_sub)
   done
 
@@ -871,7 +971,8 @@
   show "subspace (f ` span S)"
     using lf subspace_span by (rule subspace_linear_image)
 next
-  fix T assume "f ` S \<subseteq> T" and "subspace T"
+  fix T
+  assume "f ` S \<subseteq> T" and "subspace T"
   then show "f ` span S \<subseteq> T"
     unfolding image_subset_iff_subset_vimage
     by (intro span_minimal subspace_linear_vimage lf)
@@ -904,7 +1005,10 @@
   show "subspace (range (\<lambda>k. k *\<^sub>R x))"
     unfolding subspace_def
     by (auto intro: scaleR_add_left [symmetric])
-  fix T assume "{x} \<subseteq> T" and "subspace T" then show "range (\<lambda>k. k *\<^sub>R x) \<subseteq> T"
+next
+  fix T
+  assume "{x} \<subseteq> T" and "subspace T"
+  then show "range (\<lambda>k. k *\<^sub>R x) \<subseteq> T"
     unfolding subspace_def by auto
 qed
 
@@ -922,12 +1026,13 @@
 qed
 
 lemma span_breakdown:
-  assumes bS: "b \<in> S" and aS: "a \<in> span S"
+  assumes bS: "b \<in> S"
+    and aS: "a \<in> span S"
   shows "\<exists>k. a - k *\<^sub>R b \<in> span (S - {b})"
   using assms span_insert [of b "S - {b}"]
   by (simp add: insert_absorb)
 
-lemma span_breakdown_eq: "x \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. (x - k *\<^sub>R a) \<in> span S)"
+lemma span_breakdown_eq: "x \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. x - k *\<^sub>R a \<in> span S)"
   by (simp add: span_insert)
 
 text {* Hence some "reversal" results. *}
@@ -939,7 +1044,9 @@
 proof -
   from span_breakdown[of b "insert b S" a, OF insertI1 a]
   obtain k where k: "a - k*\<^sub>R b \<in> span (S - {b})" by auto
-  { assume k0: "k = 0"
+  show ?thesis
+  proof (cases "k = 0")
+    case True
     with k have "a \<in> span S"
       apply (simp)
       apply (rule set_rev_mp)
@@ -947,19 +1054,17 @@
       apply (rule span_mono)
       apply blast
       done
-    with na  have ?thesis by blast }
-  moreover
-  { assume k0: "k \<noteq> 0"
+    with na show ?thesis by blast
+  next
+    case False
     have eq: "b = (1/k) *\<^sub>R a - ((1/k) *\<^sub>R a - b)" by simp
-    from k0 have eq': "(1/k) *\<^sub>R (a - k*\<^sub>R b) = (1/k) *\<^sub>R a - b"
+    from False have eq': "(1/k) *\<^sub>R (a - k*\<^sub>R b) = (1/k) *\<^sub>R a - b"
       by (simp add: algebra_simps)
     from k have "(1/k) *\<^sub>R (a - k*\<^sub>R b) \<in> span (S - {b})"
       by (rule span_mul)
     then have th: "(1/k) *\<^sub>R a - b \<in> span (S - {b})"
       unfolding eq' .
-
-    from k
-    have ?thesis
+    from k show ?thesis
       apply (subst eq)
       apply (rule span_sub)
       apply (rule span_mul)
@@ -968,8 +1073,10 @@
       apply (rule set_rev_mp)
       apply (rule th)
       apply (rule span_mono)
-      using na by blast }
-  ultimately show ?thesis by blast
+      using na
+      apply blast
+      done
+  qed
 qed
 
 lemma in_span_delete:
@@ -990,7 +1097,8 @@
   unfolding span_def by (rule hull_redundant)
 
 lemma span_trans:
-  assumes x: "x \<in> span S" and y: "y \<in> span (insert x S)"
+  assumes x: "x \<in> span S"
+    and y: "y \<in> span (insert x S)"
   shows "y \<in> span S"
   using assms by (simp only: span_redundant)
 
@@ -1003,7 +1111,9 @@
   "span P = {y. \<exists>S u. finite S \<and> S \<subseteq> P \<and> setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
   (is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \<exists>S u. ?Q S u y}")
 proof -
-  { fix x assume x: "x \<in> ?E"
+  {
+    fix x
+    assume x: "x \<in> ?E"
     then obtain S u where fS: "finite S" and SP: "S\<subseteq>P" and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = x"
       by blast
     have "x \<in> span P"
@@ -1011,7 +1121,8 @@
       apply (rule span_setsum[OF fS])
       using span_mono[OF SP]
       apply (auto intro: span_superset span_mul)
-      done }
+      done
+  }
   moreover
   have "\<forall>x \<in> span P. x \<in> ?E"
   proof (rule span_induct_alt')
@@ -1022,15 +1133,20 @@
       done
   next
     fix c x y
-    assume x: "x \<in> P" and hy: "y \<in> Collect ?h"
+    assume x: "x \<in> P"
+    assume hy: "y \<in> Collect ?h"
     from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P"
       and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y" by blast
     let ?S = "insert x S"
     let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c) else u y"
-    from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P" by blast+
-    { assume xS: "x \<in> S"
+    from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P"
+      by blast+
+    have "?Q ?S ?u (c*\<^sub>R x + y)"
+    proof cases
+      assume xS: "x \<in> S"
       have S1: "S = (S - {x}) \<union> {x}"
-        and Sss:"finite (S - {x})" "finite {x}" "(S -{x}) \<inter> {x} = {}" using xS fS by auto
+        and Sss:"finite (S - {x})" "finite {x}" "(S -{x}) \<inter> {x} = {}"
+        using xS fS by auto
       have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S =(\<Sum>v\<in>S - {x}. u v *\<^sub>R v) + (u x + c) *\<^sub>R x"
         using xS
         by (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]]
@@ -1042,17 +1158,18 @@
       also have "\<dots> = c*\<^sub>R x + y"
         by (simp add: add_commute u)
       finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = c*\<^sub>R x + y" .
-    then have "?Q ?S ?u (c*\<^sub>R x + y)" using th0 by blast }
-    moreover
-    { assume xS: "x \<notin> S"
+      then show ?thesis using th0 by blast
+    next
+      assume xS: "x \<notin> S"
       have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *\<^sub>R v) = y"
         unfolding u[symmetric]
         apply (rule setsum_cong2)
-        using xS apply auto
+        using xS
+        apply auto
         done
-      have "?Q ?S ?u (c*\<^sub>R x + y)" using fS xS th0
-        by (simp add: th00 setsum_clauses add_commute cong del: if_weak_cong) }
-    ultimately have "?Q ?S ?u (c*\<^sub>R x + y)" by (cases "x \<in> S") simp_all
+      show ?thesis using fS xS th0
+        by (simp add: th00 setsum_clauses add_commute cong del: if_weak_cong)
+    qed
     then show "(c*\<^sub>R x + y) \<in> Collect ?h"
       unfolding mem_Collect_eq
       apply -
@@ -1068,15 +1185,18 @@
   "dependent P \<longleftrightarrow> (\<exists>S u. finite S \<and> S \<subseteq> P \<and> (\<exists>v\<in>S. u v \<noteq> 0 \<and> setsum (\<lambda>v. u v *\<^sub>R v) S = 0))"
   (is "?lhs = ?rhs")
 proof -
-  { assume dP: "dependent P"
+  {
+    assume dP: "dependent P"
     then obtain a S u where aP: "a \<in> P" and fS: "finite S"
       and SP: "S \<subseteq> P - {a}" and ua: "setsum (\<lambda>v. u v *\<^sub>R v) S = a"
       unfolding dependent_def span_explicit by blast
     let ?S = "insert a S"
     let ?u = "\<lambda>y. if y = a then - 1 else u y"
     let ?v = a
-    from aP SP have aS: "a \<notin> S" by blast
-    from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0" by auto
+    from aP SP have aS: "a \<notin> S"
+      by blast
+    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 *\<^sub>R v) ?S = 0"
       using fS aS
       apply (simp add: setsum_clauses field_simps)
@@ -1092,18 +1212,24 @@
       done
   }
   moreover
-  { fix S u v
+  {
+    fix S u v
     assume fS: "finite S"
-      and SP: "S \<subseteq> P" and vS: "v \<in> S" and uv: "u v \<noteq> 0"
+      and SP: "S \<subseteq> P"
+      and vS: "v \<in> S"
+      and uv: "u v \<noteq> 0"
       and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = 0"
     let ?a = v
     let ?S = "S - {v}"
     let ?u = "\<lambda>i. (- u i) / u v"
-    have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P" using fS SP vS by auto
-    have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = setsum (\<lambda>v. (- (inverse (u ?a))) *\<^sub>R (u v *\<^sub>R v)) S - ?u v *\<^sub>R v"
+    have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P"
+      using fS SP vS by auto
+    have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S =
+      setsum (\<lambda>v. (- (inverse (u ?a))) *\<^sub>R (u v *\<^sub>R v)) S - ?u v *\<^sub>R v"
       using fS vS uv by (simp add: setsum_diff1 divide_inverse field_simps)
-    also have "\<dots> = ?a" unfolding scaleR_right.setsum [symmetric] u using uv by simp
-    finally  have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = ?a" .
+    also have "\<dots> = ?a"
+      unfolding scaleR_right.setsum [symmetric] u using uv by simp
+    finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = ?a" .
     with th0 have ?lhs
       unfolding dependent_def span_explicit
       apply -
@@ -1122,61 +1248,72 @@
   shows "span S = {y. \<exists>u. setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
   (is "_ = ?rhs")
 proof -
-  { fix y
+  {
+    fix y
     assume y: "y \<in> span S"
-    from y obtain S' u where fS': "finite S'" and SS': "S' \<subseteq> S" and
-      u: "setsum (\<lambda>v. u v *\<^sub>R v) S' = y" unfolding span_explicit by blast
+    from y obtain S' u where fS': "finite S'"
+      and SS': "S' \<subseteq> S"
+      and u: "setsum (\<lambda>v. u v *\<^sub>R v) S' = y"
+      unfolding span_explicit by blast
     let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
     have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = setsum (\<lambda>v. u v *\<^sub>R v) S'"
       using SS' fS by (auto intro!: setsum_mono_zero_cong_right)
     then have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = y" by (metis u)
-    then have "y \<in> ?rhs" by auto }
+    then have "y \<in> ?rhs" by auto
+  }
   moreover
-  { fix y u
+  {
+    fix y u
     assume u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y"
-    then have "y \<in> span S" using fS unfolding span_explicit by auto }
+    then have "y \<in> span S" using fS unfolding span_explicit by auto
+  }
   ultimately show ?thesis by blast
 qed
 
 text {* This is useful for building a basis step-by-step. *}
 
 lemma independent_insert:
-  "independent(insert a S) \<longleftrightarrow>
-      (if a \<in> S then independent S
-                else independent S \<and> a \<notin> span S)" (is "?lhs \<longleftrightarrow> ?rhs")
-proof -
-  { assume aS: "a \<in> S"
-    then have ?thesis using insert_absorb[OF aS] by simp }
-  moreover
-  { assume aS: "a \<notin> S"
-    { assume i: ?lhs
-      then have ?rhs using aS
-        apply simp
-        apply (rule conjI)
-        apply (rule independent_mono)
-        apply assumption
-        apply blast
-        apply (simp add: dependent_def)
-        done }
-    moreover
-    { assume i: ?rhs
-      have ?lhs using i aS
-        apply simp
-        apply (auto simp add: dependent_def)
-        apply (case_tac "aa = a", auto)
-        apply (subgoal_tac "insert a S - {aa} = insert a (S - {aa})")
-        apply simp
-        apply (subgoal_tac "a \<in> span (insert aa (S - {aa}))")
-        apply (subgoal_tac "insert aa (S - {aa}) = S")
-        apply simp
-        apply blast
-        apply (rule in_span_insert)
-        apply assumption
-        apply blast
-        apply blast
-        done }
-    ultimately have ?thesis by blast }
-  ultimately show ?thesis by blast
+  "independent (insert a S) \<longleftrightarrow>
+    (if a \<in> S then independent S else independent S \<and> a \<notin> span S)"
+  (is "?lhs \<longleftrightarrow> ?rhs")
+proof (cases "a \<in> S")
+  case True
+  then show ?thesis
+    using insert_absorb[OF True] by simp
+next
+  case False
+  show ?thesis
+  proof
+    assume i: ?lhs
+    then show ?rhs
+      using False
+      apply simp
+      apply (rule conjI)
+      apply (rule independent_mono)
+      apply assumption
+      apply blast
+      apply (simp add: dependent_def)
+      done
+  next
+    assume i: ?rhs
+    show ?lhs
+      using i False
+      apply simp
+      apply (auto simp add: dependent_def)
+      apply (case_tac "aa = a")
+      apply auto
+      apply (subgoal_tac "insert a S - {aa} = insert a (S - {aa})")
+      apply simp
+      apply (subgoal_tac "a \<in> span (insert aa (S - {aa}))")
+      apply (subgoal_tac "insert aa (S - {aa}) = S")
+      apply simp
+      apply blast
+      apply (rule in_span_insert)
+      apply assumption
+      apply blast
+      apply blast
+      done
+  qed
 qed
 
 text {* The degenerate case of the Exchange Lemma. *}
@@ -1195,18 +1332,29 @@
   from span_mono[OF BA] span_mono[OF AsB]
   have sAB: "span A = span B" unfolding span_span by blast
 
-  { fix x assume x: "x \<in> A"
+  {
+    fix x
+    assume x: "x \<in> A"
     from iA have th0: "x \<notin> span (A - {x})"
       unfolding dependent_def using x by blast
-    from x have xsA: "x \<in> span A" by (blast intro: span_superset)
+    from x have xsA: "x \<in> span A"
+      by (blast intro: span_superset)
     have "A - {x} \<subseteq> A" by blast
-    then have th1:"span (A - {x}) \<subseteq> span A" by (metis span_mono)
-    { assume xB: "x \<notin> B"
-      from xB BA have "B \<subseteq> A -{x}" by blast
-      then have "span B \<subseteq> span (A - {x})" by (metis span_mono)
-      with th1 th0 sAB have "x \<notin> span A" by blast
-      with x have False by (metis span_superset) }
-    then have "x \<in> B" by blast }
+    then have th1: "span (A - {x}) \<subseteq> span A"
+      by (metis span_mono)
+    {
+      assume xB: "x \<notin> B"
+      from xB BA have "B \<subseteq> A - {x}"
+        by blast
+      then have "span B \<subseteq> span (A - {x})"
+        by (metis span_mono)
+      with th1 th0 sAB have "x \<notin> span A"
+        by blast
+      with x have False
+        by (metis span_superset)
+    }
+    then have "x \<in> B" by blast
+  }
   then show "A \<subseteq> B" by blast
 qed
 
@@ -1216,75 +1364,110 @@
   assumes f:"finite t"
     and i: "independent s"
     and sp: "s \<subseteq> span t"
-  shows "\<exists>t'. (card t' = card t) \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
+  shows "\<exists>t'. card t' = card t \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
   using f i sp
 proof (induct "card (t - s)" arbitrary: s t rule: less_induct)
   case less
   note ft = `finite t` and s = `independent s` and sp = `s \<subseteq> span t`
-  let ?P = "\<lambda>t'. (card t' = card t) \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
+  let ?P = "\<lambda>t'. card t' = card t \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
   let ?ths = "\<exists>t'. ?P t'"
-  { assume st: "s \<subseteq> t"
-    from st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t])
+  {
+    assume st: "s \<subseteq> t"
+    from st ft span_mono[OF st]
+    have ?ths
+      apply -
+      apply (rule exI[where x=t])
       apply (auto intro: span_superset)
-      done }
+      done
+  }
   moreover
-  { assume st: "t \<subseteq> s"
-    from spanning_subset_independent[OF st s sp]
-      st ft span_mono[OF st] have ?ths
-        apply -
-        apply (rule exI[where x=t])
-        apply (auto intro: span_superset)
-        done }
+  {
+    assume st: "t \<subseteq> s"
+    from spanning_subset_independent[OF st s sp] st ft span_mono[OF st]
+    have ?ths
+      apply -
+      apply (rule exI[where x=t])
+      apply (auto intro: span_superset)
+      done
+  }
   moreover
-  { assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
-    from st(2) obtain b where b: "b \<in> t" "b \<notin> s" by blast
-      from b have "t - {b} - s \<subset> t - s" by blast
-      then have cardlt: "card (t - {b} - s) < card (t - s)" using ft
-        by (auto intro: psubset_card_mono)
-      from b ft have ct0: "card t \<noteq> 0" by auto
-    { assume stb: "s \<subseteq> span(t -{b})"
-      from ft have ftb: "finite (t -{b})" by auto
+  {
+    assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
+    from st(2) obtain b where b: "b \<in> t" "b \<notin> s"
+      by blast
+    from b have "t - {b} - s \<subset> t - s"
+      by blast
+    then have cardlt: "card (t - {b} - s) < card (t - s)"
+      using ft by (auto intro: psubset_card_mono)
+    from b ft have ct0: "card t \<noteq> 0"
+      by auto
+    have ?ths
+    proof cases
+      assume stb: "s \<subseteq> span(t - {b})"
+      from ft have ftb: "finite (t -{b})"
+        by auto
       from less(1)[OF cardlt ftb s stb]
       obtain u where u: "card u = card (t-{b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u"
         and fu: "finite u" by blast
       let ?w = "insert b u"
-      have th0: "s \<subseteq> insert b u" using u by blast
-      from u(3) b have "u \<subseteq> s \<union> t" by blast
-      then have th1: "insert b u \<subseteq> s \<union> t" using u b by blast
-      have bu: "b \<notin> u" using b u by blast
-      from u(1) ft b have "card u = (card t - 1)" by auto
+      have th0: "s \<subseteq> insert b u"
+        using u by blast
+      from u(3) b have "u \<subseteq> s \<union> t"
+        by blast
+      then have th1: "insert b u \<subseteq> s \<union> t"
+        using u b by blast
+      have bu: "b \<notin> u"
+        using b u by blast
+      from u(1) ft b have "card u = (card t - 1)"
+        by auto
       then have th2: "card (insert b u) = card t"
         using card_insert_disjoint[OF fu bu] ct0 by auto
       from u(4) have "s \<subseteq> span u" .
-      also have "\<dots> \<subseteq> span (insert b u)" apply (rule span_mono) by blast
+      also have "\<dots> \<subseteq> span (insert b u)"
+        by (rule span_mono) blast
       finally have th3: "s \<subseteq> span (insert b u)" .
-      from th0 th1 th2 th3 fu have th: "?P ?w"  by blast
-      from th have ?ths by blast }
-    moreover
-    { assume stb: "\<not> s \<subseteq> span(t -{b})"
-      from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})" by blast
-      have ab: "a \<noteq> b" using a b by blast
-      have at: "a \<notin> t" using a ab span_superset[of a "t- {b}"] by auto
+      from th0 th1 th2 th3 fu have th: "?P ?w"
+        by blast
+      from th show ?thesis by blast
+    next
+      assume stb: "\<not> s \<subseteq> span(t - {b})"
+      from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})"
+        by blast
+      have ab: "a \<noteq> b"
+        using a b by blast
+      have at: "a \<notin> t"
+        using a ab span_superset[of a "t- {b}"] by auto
       have mlt: "card ((insert a (t - {b})) - s) < card (t - s)"
         using cardlt ft a b by auto
-      have ft': "finite (insert a (t - {b}))" using ft by auto
-      { fix x assume xs: "x \<in> s"
-        have t: "t \<subseteq> (insert b (insert a (t -{b})))" using b by auto
-        from b(1) have "b \<in> span t" by (simp add: span_superset)
-        have bs: "b \<in> span (insert a (t - {b}))" apply(rule in_span_delete)
-          using a sp unfolding subset_eq apply auto done
-        from xs sp have "x \<in> span t" by blast
-        with span_mono[OF t]
-        have x: "x \<in> span (insert b (insert a (t - {b})))" ..
-        from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))" . }
-      then have sp': "s \<subseteq> span (insert a (t - {b}))" by blast
-
-      from less(1)[OF mlt ft' s sp'] obtain u where
-        u: "card u = card (insert a (t -{b}))" "finite u" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t -{b})"
-          "s \<subseteq> span u" by blast
-      from u a b ft at ct0 have "?P u" by auto
-      then have ?ths by blast }
-    ultimately have ?ths by blast
+      have ft': "finite (insert a (t - {b}))"
+        using ft by auto
+      {
+        fix x
+        assume xs: "x \<in> s"
+        have t: "t \<subseteq> insert b (insert a (t - {b}))"
+          using b by auto
+        from b(1) have "b \<in> span t"
+          by (simp add: span_superset)
+        have bs: "b \<in> span (insert a (t - {b}))"
+          apply (rule in_span_delete)
+          using a sp unfolding subset_eq
+          apply auto
+          done
+        from xs sp have "x \<in> span t"
+          by blast
+        with span_mono[OF t] have x: "x \<in> span (insert b (insert a (t - {b})))" ..
+        from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))" .
+      }
+      then have sp': "s \<subseteq> span (insert a (t - {b}))"
+        by blast
+      from less(1)[OF mlt ft' s sp'] obtain u where u:
+        "card u = card (insert a (t -{b}))"
+        "finite u" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t -{b})"
+        "s \<subseteq> span u" by blast
+      from u a b ft at ct0 have "?P u"
+        by auto
+      then show ?thesis by blast
+    qed
   }
   ultimately show ?ths by blast
 qed
@@ -1292,21 +1475,24 @@
 text {* This implies corresponding size bounds. *}
 
 lemma independent_span_bound:
-  assumes f: "finite t" and i: "independent s" and sp:"s \<subseteq> span t"
+  assumes f: "finite t"
+    and i: "independent s"
+    and sp: "s \<subseteq> span t"
   shows "finite s \<and> card s \<le> card t"
   by (metis exchange_lemma[OF f i sp] finite_subset card_mono)
 
-
 lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> (UNIV::'a::finite set)}"
 proof -
-  have eq: "{f x |x. x\<in> UNIV} = f ` UNIV" by auto
+  have eq: "{f x |x. x\<in> UNIV} = f ` UNIV"
+    by auto
   show ?thesis unfolding eq
     apply (rule finite_imageI)
     apply (rule finite)
     done
 qed
 
-subsection{* Euclidean Spaces as Typeclass*}
+
+subsection {* Euclidean Spaces as Typeclass *}
 
 lemma independent_Basis: "independent Basis"
   unfolding dependent_def
@@ -1345,7 +1531,8 @@
 
 lemma setsum_norm_allsubsets_bound:
   fixes f:: "'a \<Rightarrow> 'n::euclidean_space"
-  assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
+  assumes fP: "finite P"
+    and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
   shows "(\<Sum>x\<in>P. norm (f x)) \<le> 2 * real DIM('n) * e"
 proof -
   have "(\<Sum>x\<in>P. norm (f x)) \<le> (\<Sum>x\<in>P. \<Sum>b\<in>Basis. \<bar>f x \<bullet> b\<bar>)"
@@ -1354,13 +1541,14 @@
     by (rule setsum_commute)
   also have "\<dots> \<le> of_nat (card (Basis :: 'n set)) * (2 * e)"
   proof (rule setsum_bounded)
-    fix i :: 'n assume i: "i \<in> Basis"
-    have "norm (\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le> 
+    fix i :: 'n
+    assume i: "i \<in> Basis"
+    have "norm (\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le>
       norm ((\<Sum>x\<in>P \<inter> - {x. f x \<bullet> i < 0}. f x) \<bullet> i) + norm ((\<Sum>x\<in>P \<inter> {x. f x \<bullet> i < 0}. f x) \<bullet> i)"
       by (simp add: abs_real_def setsum_cases[OF fP] setsum_negf uminus_add_conv_diff
-                    norm_triangle_ineq4 inner_setsum_left
-          del: real_norm_def)
-    also have "\<dots> \<le> e + e" unfolding real_norm_def
+            norm_triangle_ineq4 inner_setsum_left del: real_norm_def)
+    also have "\<dots> \<le> e + e"
+      unfolding real_norm_def
       by (intro add_mono norm_bound_Basis_le i fPs) auto
     finally show "(\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le> 2*e" by simp
   qed
@@ -1369,6 +1557,7 @@
   finally show ?thesis .
 qed
 
+
 subsection {* Linearity and Bilinearity continued *}
 
 lemma linear_bounded:
@@ -1377,25 +1566,32 @@
   shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
 proof -
   let ?B = "\<Sum>b\<in>Basis. norm (f b)"
-  { fix x:: "'a"
+  {
+    fix x :: 'a
     let ?g = "\<lambda>b. (x \<bullet> b) *\<^sub>R f b"
     have "norm (f x) = norm (f (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b))"
       unfolding euclidean_representation ..
     also have "\<dots> = norm (setsum ?g Basis)"
-      using linear_setsum[OF lf finite_Basis, of "\<lambda>b. (x \<bullet> b) *\<^sub>R b", unfolded o_def] linear_cmul[OF lf] by auto
+      using linear_setsum[OF lf finite_Basis, of "\<lambda>b. (x \<bullet> b) *\<^sub>R b", unfolded o_def] linear_cmul[OF lf]
+      by auto
     finally have th0: "norm (f x) = norm (setsum ?g Basis)" .
-    { fix i :: 'a assume i: "i \<in> Basis"
+    {
+      fix i :: 'a
+      assume i: "i \<in> Basis"
       from Basis_le_norm[OF i, of x]
       have "norm (?g i) \<le> norm (f i) * norm x"
         unfolding norm_scaleR
         apply (subst mult_commute)
         apply (rule mult_mono)
         apply (auto simp add: field_simps)
-        done }
+        done
+    }
     then have th: "\<forall>b\<in>Basis. norm (?g b) \<le> norm (f b) * norm x"
       by metis
     from setsum_norm_le[of _ ?g, OF th]
-    have "norm (f x) \<le> ?B * norm x" unfolding th0 setsum_left_distrib by metis}
+    have "norm (f x) \<le> ?B * norm x"
+      unfolding th0 setsum_left_distrib by metis
+  }
   then show ?thesis by blast
 qed
 
@@ -1408,7 +1604,8 @@
     B: "\<forall>x. norm (f x) \<le> B * norm x" by blast
   let ?K = "\<bar>B\<bar> + 1"
   have Kp: "?K > 0" by arith
-  { assume C: "B < 0"
+  {
+    assume C: "B < 0"
     def One \<equiv> "\<Sum>Basis ::'a"
     then have "One \<noteq> 0"
       unfolding euclidean_eq_iff[where 'a='a]
@@ -1419,14 +1616,18 @@
     with B[rule_format, of One] norm_ge_zero[of "f One"]
     have False by simp
   }
-  then have Bp: "B \<ge> 0" by (metis not_leE)
-  { fix x::"'a"
+  then have Bp: "B \<ge> 0"
+    by (metis not_leE)
+  {
+    fix x::"'a"
     have "norm (f x) \<le> ?K *  norm x"
       using B[rule_format, of x] norm_ge_zero[of x] norm_ge_zero[of "f x"] Bp
       apply (auto simp add: field_simps split add: abs_split)
       apply (erule order_trans, simp)
       done
-  } then show ?thesis using Kp by blast
+  }
+  then show ?thesis
+    using Kp by blast
 qed
 
 lemma linear_conv_bounded_linear:
@@ -1436,10 +1637,12 @@
   assume "linear f"
   show "bounded_linear f"
   proof
-    fix x y show "f (x + y) = f x + f y"
+    fix x y
+    show "f (x + y) = f x + f y"
       using `linear f` unfolding linear_def by simp
   next
-    fix r x show "f (scaleR r x) = scaleR r (f x)"
+    fix r x
+    show "f (scaleR r x) = scaleR r (f x)"
       using `linear f` unfolding linear_def by simp
   next
     have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
@@ -1450,43 +1653,43 @@
 next
   assume "bounded_linear f"
   then interpret f: bounded_linear f .
-  show "linear f"
-    by (simp add: f.add f.scaleR linear_def)
+  show "linear f" by (simp add: f.add f.scaleR linear_def)
 qed
 
 lemma bounded_linearI':
   fixes f::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
-  assumes "\<And>x y. f (x + y) = f x + f y" "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
+  assumes "\<And>x y. f (x + y) = f x + f y"
+    and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
   shows "bounded_linear f"
-  unfolding linear_conv_bounded_linear[THEN sym]
+  unfolding linear_conv_bounded_linear[symmetric]
   by (rule linearI[OF assms])
 
-
 lemma bilinear_bounded:
   fixes h:: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector"
   assumes bh: "bilinear h"
   shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
 proof (clarify intro!: exI[of _ "\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)"])
-  fix x:: "'m" and  y :: "'n"
-  have "norm (h x y) = norm (h (setsum (\<lambda>i. (x \<bullet> i) *\<^sub>R i) Basis) (setsum (\<lambda>i. (y \<bullet> i) *\<^sub>R i) Basis))" 
-    apply(subst euclidean_representation[where 'a='m])
-    apply(subst euclidean_representation[where 'a='n])
+  fix x :: 'm
+  fix y :: 'n
+  have "norm (h x y) = norm (h (setsum (\<lambda>i. (x \<bullet> i) *\<^sub>R i) Basis) (setsum (\<lambda>i. (y \<bullet> i) *\<^sub>R i) Basis))"
+    apply (subst euclidean_representation[where 'a='m])
+    apply (subst euclidean_representation[where 'a='n])
     apply rule
     done
-  also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j)) (Basis \<times> Basis))"  
+  also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j)) (Basis \<times> Basis))"
     unfolding bilinear_setsum[OF bh finite_Basis finite_Basis] ..
   finally have th: "norm (h x y) = \<dots>" .
   show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)) * norm x * norm y"
-      apply (auto simp add: setsum_left_distrib th setsum_cartesian_product)
-      apply (rule setsum_norm_le)
-      apply simp
-      apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
-        field_simps simp del: scaleR_scaleR)
-      apply (rule mult_mono)
-      apply (auto simp add: zero_le_mult_iff Basis_le_norm)
-      apply (rule mult_mono)
-      apply (auto simp add: zero_le_mult_iff Basis_le_norm)
-      done
+    apply (auto simp add: setsum_left_distrib th setsum_cartesian_product)
+    apply (rule setsum_norm_le)
+    apply simp
+    apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
+      field_simps simp del: scaleR_scaleR)
+    apply (rule mult_mono)
+    apply (auto simp add: zero_le_mult_iff Basis_le_norm)
+    apply (rule mult_mono)
+    apply (auto simp add: zero_le_mult_iff Basis_le_norm)
+    done
 qed
 
 lemma bilinear_bounded_pos:
@@ -1499,15 +1702,17 @@
   let ?K = "\<bar>B\<bar> + 1"
   have Kp: "?K > 0" by arith
   have KB: "B < ?K" by arith
-  { fix x::'a and y::'b
-    from KB Kp
-    have "B * norm x * norm y \<le> ?K * norm x * norm y"
+  {
+    fix x :: 'a
+    fix y :: 'b
+    from KB Kp have "B * norm x * norm y \<le> ?K * norm x * norm y"
       apply -
       apply (rule mult_right_mono, rule mult_right_mono)
       apply auto
       done
     then have "norm (h x y) \<le> ?K * norm x * norm y"
-      using B[rule_format, of x y] by simp }
+      using B[rule_format, of x y] by simp
+  }
   with Kp show ?thesis by blast
 qed
 
@@ -1518,17 +1723,21 @@
   assume "bilinear h"
   show "bounded_bilinear h"
   proof
-    fix x y z show "h (x + y) z = h x z + h y z"
+    fix x y z
+    show "h (x + y) z = h x z + h y z"
       using `bilinear h` unfolding bilinear_def linear_def by simp
   next
-    fix x y z show "h x (y + z) = h x y + h x z"
+    fix x y z
+    show "h x (y + z) = h x y + h x z"
       using `bilinear h` unfolding bilinear_def linear_def by simp
   next
-    fix r x y show "h (scaleR r x) y = scaleR r (h x y)"
+    fix r x y
+    show "h (scaleR r x) y = scaleR r (h x y)"
       using `bilinear h` unfolding bilinear_def linear_def
       by simp
   next
-    fix r x y show "h x (scaleR r y) = scaleR r (h x y)"
+    fix r x y
+    show "h x (scaleR r y) = scaleR r (h x y)"
       using `bilinear h` unfolding bilinear_def linear_def
       by simp
   next
@@ -1554,13 +1763,14 @@
   using independent_span_bound[OF finite_Basis, of S] by auto
 
 lemma dependent_biggerset:
-  "(finite (S::('a::euclidean_space) set) ==> card S > DIM('a)) ==> dependent S"
+  "(finite (S::('a::euclidean_space) set) \<Longrightarrow> card S > DIM('a)) \<Longrightarrow> dependent S"
   by (metis independent_bound not_less)
 
 text {* Hence we can create a maximal independent subset. *}
 
 lemma maximal_independent_subset_extend:
-  assumes sv: "(S::('a::euclidean_space) set) \<subseteq> V"
+  fixes S :: "'a::euclidean_space set"
+  assumes sv: "S \<subseteq> V"
     and iS: "independent S"
   shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
   using sv iS
@@ -1570,15 +1780,22 @@
   let ?P = "\<lambda>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
   let ?ths = "\<exists>x. ?P x"
   let ?d = "DIM('a)"
-  { assume "V \<subseteq> span S"
-    then have ?ths  using sv i by blast }
-  moreover
-  { assume VS: "\<not> V \<subseteq> span S"
-    from VS obtain a where a: "a \<in> V" "a \<notin> span S" by blast
-    from a have aS: "a \<notin> S" by (auto simp add: span_superset)
-    have th0: "insert a S \<subseteq> V" using a sv by blast
+  show ?ths
+  proof (cases "V \<subseteq> span S")
+    case True
+    then show ?thesis
+      using sv i by blast
+  next
+    case False
+    then obtain a where a: "a \<in> V" "a \<notin> span S"
+      by blast
+    from a have aS: "a \<notin> S"
+      by (auto simp add: span_superset)
+    have th0: "insert a S \<subseteq> V"
+      using a sv by blast
     from independent_insert[of a S]  i a
-    have th1: "independent (insert a S)" by auto
+    have th1: "independent (insert a S)"
+      by auto
     have mlt: "?d - card (insert a S) < ?d - card S"
       using aS a independent_bound[OF th1] by auto
 
@@ -1586,8 +1803,8 @@
     obtain B where B: "insert a S \<subseteq> B" "B \<subseteq> V" "independent B" " V \<subseteq> span B"
       by blast
     from B have "?P B" by auto
-    then have ?ths by blast }
-  ultimately show ?ths by blast
+    then show ?thesis by blast
+  qed
 qed
 
 lemma maximal_independent_subset:
@@ -1598,7 +1815,7 @@
 
 text {* Notion of dimension. *}
 
-definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = n))"
+definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> card B = n)"
 
 lemma basis_exists:
   "\<exists>B. (B :: ('a::euclidean_space) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = dim V)"
@@ -1608,58 +1825,76 @@
 
 text {* Consequences of independence or spanning for cardinality. *}
 
-lemma independent_card_le_dim: 
-  assumes "(B::('a::euclidean_space) set) \<subseteq> V" and "independent B"
+lemma independent_card_le_dim:
+  fixes B :: "'a::euclidean_space set"
+  assumes "B \<subseteq> V"
+    and "independent B"
   shows "card B \<le> dim V"
 proof -
   from basis_exists[of V] `B \<subseteq> V`
-  obtain B' where "independent B'" and "B \<subseteq> span B'" and "card B' = dim V" by blast
+  obtain B' where "independent B'"
+    and "B \<subseteq> span B'"
+    and "card B' = dim V"
+    by blast
   with independent_span_bound[OF _ `independent B` `B \<subseteq> span B'`] independent_bound[of B']
   show ?thesis by auto
 qed
 
 lemma span_card_ge_dim:
-  "(B::('a::euclidean_space) set) \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B"
+  fixes B :: "'a::euclidean_space set"
+  shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B"
   by (metis basis_exists[of V] independent_span_bound subset_trans)
 
 lemma basis_card_eq_dim:
-  "B \<subseteq> (V:: ('a::euclidean_space) set) \<Longrightarrow> V \<subseteq> span B \<Longrightarrow>
-    independent B \<Longrightarrow> finite B \<and> card B = dim V"
+  fixes V :: "'a::euclidean_space set"
+  shows "B \<subseteq> V \<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_bound)
 
-lemma dim_unique: "(B::('a::euclidean_space) set) \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow>
-    independent B \<Longrightarrow> card B = n \<Longrightarrow> dim V = n"
+lemma dim_unique:
+  fixes B :: "'a::euclidean_space set"
+  shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> card B = n \<Longrightarrow> dim V = n"
   by (metis basis_card_eq_dim)
 
 text {* More lemmas about dimension. *}
 
-lemma dim_UNIV: "dim (UNIV :: ('a::euclidean_space) set) = DIM('a)"
+lemma dim_UNIV: "dim (UNIV :: 'a::euclidean_space set) = DIM('a)"
   using independent_Basis
   by (intro dim_unique[of Basis]) auto
 
 lemma dim_subset:
-  "(S:: ('a::euclidean_space) set) \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
+  fixes S :: "'a::euclidean_space set"
+  shows "S \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
   using basis_exists[of T] basis_exists[of S]
   by (metis independent_card_le_dim subset_trans)
 
-lemma dim_subset_UNIV: "dim (S:: ('a::euclidean_space) set) \<le> DIM('a)"
+lemma dim_subset_UNIV:
+  fixes S :: "'a::euclidean_space set"
+  shows "dim S \<le> DIM('a)"
   by (metis dim_subset subset_UNIV dim_UNIV)
 
 text {* Converses to those. *}
 
 lemma card_ge_dim_independent:
-  assumes BV:"(B::('a::euclidean_space) set) \<subseteq> V"
-    and iB:"independent B" and dVB:"dim V \<le> card B"
+  fixes B :: "'a::euclidean_space set"
+  assumes BV: "B \<subseteq> V"
+    and iB: "independent B"
+    and dVB: "dim V \<le> card B"
   shows "V \<subseteq> span B"
-proof -
-  { fix a assume aV: "a \<in> V"
-    { assume aB: "a \<notin> span B"
-      then have iaB: "independent (insert a B)" using iB aV  BV by (simp add: independent_insert)
-      from aV BV have th0: "insert a B \<subseteq> V" by blast
-      from aB have "a \<notin>B" by (auto simp add: span_superset)
-      with independent_card_le_dim[OF th0 iaB] dVB independent_bound[OF iB] have False by auto }
-    then have "a \<in> span B"  by blast }
-  then show ?thesis by blast
+proof
+  fix a
+  assume aV: "a \<in> V"
+  {
+    assume aB: "a \<notin> span B"
+    then have iaB: "independent (insert a B)"
+      using iB aV BV by (simp add: independent_insert)
+    from aV BV have th0: "insert a B \<subseteq> V"
+      by blast
+    from aB have "a \<notin>B"
+      by (auto simp add: span_superset)
+    with independent_card_le_dim[OF th0 iaB] dVB independent_bound[OF iB]
+    have False by auto
+  }
+  then show "a \<in> span B" by blast
 qed
 
 lemma card_le_dim_spanning:
@@ -1669,54 +1904,81 @@
     and dVB: "dim V \<ge> card B"
   shows "independent B"
 proof -
-  { fix a assume a: "a \<in> B" "a \<in> span (B -{a})"
-    from a fB have c0: "card B \<noteq> 0" by auto
-    from a fB have cb: "card (B -{a}) = card B - 1" by auto
-    from BV a have th0: "B -{a} \<subseteq> V" by blast
-    { fix x assume x: "x \<in> V"
-      from a have eq: "insert a (B -{a}) = B" by blast
-      from x VB have x': "x \<in> span B" by blast
+  {
+    fix a
+    assume a: "a \<in> B" "a \<in> span (B -{a})"
+    from a fB have c0: "card B \<noteq> 0"
+      by auto
+    from a fB have cb: "card (B -{a}) = card B - 1"
+      by auto
+    from BV a have th0: "B -{a} \<subseteq> V"
+      by blast
+    {
+      fix x
+      assume x: "x \<in> V"
+      from a have eq: "insert a (B -{a}) = B"
+        by blast
+      from x VB have x': "x \<in> span B"
+        by blast
       from span_trans[OF a(2), unfolded eq, OF x']
-      have "x \<in> span (B -{a})" . }
-    then have th1: "V \<subseteq> span (B -{a})" by blast
-    have th2: "finite (B -{a})" using fB by auto
+      have "x \<in> span (B -{a})" .
+    }
+    then have th1: "V \<subseteq> span (B -{a})"
+      by blast
+    have th2: "finite (B -{a})"
+      using fB by auto
     from span_card_ge_dim[OF th0 th1 th2]
     have c: "dim V \<le> card (B -{a})" .
-    from c c0 dVB cb have False by simp }
-  then show ?thesis unfolding dependent_def by blast
+    from c c0 dVB cb have False by simp
+  }
+  then show ?thesis
+    unfolding dependent_def by blast
 qed
 
-lemma card_eq_dim: "(B:: ('a::euclidean_space) set) \<subseteq> V \<Longrightarrow>
-    card B = dim V \<Longrightarrow> finite B \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
+lemma card_eq_dim:
+  fixes B :: "'a::euclidean_space set"
+  shows "B \<subseteq> V \<Longrightarrow> card B = dim V \<Longrightarrow> finite B \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
   by (metis order_eq_iff card_le_dim_spanning card_ge_dim_independent)
 
 text {* More general size bound lemmas. *}
 
 lemma independent_bound_general:
-  "independent (S:: ('a::euclidean_space) set) \<Longrightarrow> finite S \<and> card S \<le> dim S"
+  fixes S :: "'a::euclidean_space set"
+  shows "independent S \<Longrightarrow> finite S \<and> card S \<le> dim S"
   by (metis independent_card_le_dim independent_bound subset_refl)
 
 lemma dependent_biggerset_general:
-    "(finite (S:: ('a::euclidean_space) set) \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
+  fixes S :: "'a::euclidean_space set"
+  shows "(finite S \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
   using independent_bound_general[of S] by (metis linorder_not_le)
 
-lemma dim_span: "dim (span (S:: ('a::euclidean_space) set)) = dim S"
+lemma dim_span:
+  fixes S :: "'a::euclidean_space set"
+  shows "dim (span S) = dim S"
 proof -
   have th0: "dim S \<le> dim (span S)"
     by (auto simp add: subset_eq intro: dim_subset span_superset)
   from basis_exists[of S]
-  obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" by blast
-  from B have fB: "finite B" "card B = dim S" using independent_bound by blast+
-  have bSS: "B \<subseteq> span S" using B(1) by (metis subset_eq span_inc)
-  have sssB: "span S \<subseteq> span B" using span_mono[OF B(3)] by (simp add: span_span)
+  obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S"
+    by blast
+  from B have fB: "finite B" "card B = dim S"
+    using independent_bound by blast+
+  have bSS: "B \<subseteq> span S"
+    using B(1) by (metis subset_eq span_inc)
+  have sssB: "span S \<subseteq> span B"
+    using span_mono[OF B(3)] by (simp add: span_span)
   from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis
     using fB(2) by arith
 qed
 
-lemma subset_le_dim: "(S:: ('a::euclidean_space) set) \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
+lemma subset_le_dim:
+  fixes S :: "'a::euclidean_space set"
+  shows "S \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
   by (metis dim_span dim_subset)
 
-lemma span_eq_dim: "span (S:: ('a::euclidean_space) set) = span T ==> dim S = dim T"
+lemma span_eq_dim:
+  fixes S:: "'a::euclidean_space set"
+  shows "span S = span T \<Longrightarrow> dim S = dim T"
   by (metis dim_span)
 
 lemma spans_image:
@@ -1732,12 +1994,15 @@
 proof -
   from basis_exists[of S] obtain B where
     B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" by blast
-  from B have fB: "finite B" "card B = dim S" using independent_bound by blast+
+  from B have fB: "finite B" "card B = dim S"
+    using independent_bound by blast+
   have "dim (f ` S) \<le> card (f ` B)"
     apply (rule span_card_ge_dim)
-    using lf B fB apply (auto simp add: span_linear_image spans_image subset_image_iff)
+    using lf B fB
+    apply (auto simp add: span_linear_image spans_image subset_image_iff)
     done
-  also have "\<dots> \<le> dim S" using card_image_le[OF fB(1)] fB by simp
+  also have "\<dots> \<le> dim S"
+    using card_image_le[OF fB(1)] fB by simp
   finally show ?thesis .
 qed
 
@@ -1745,12 +2010,15 @@
 
 lemma spanning_surjective_image:
   assumes us: "UNIV \<subseteq> span S"
-    and lf: "linear f" and sf: "surj f"
+    and lf: "linear f"
+    and sf: "surj f"
   shows "UNIV \<subseteq> span (f ` S)"
 proof -
-  have "UNIV \<subseteq> f ` UNIV" using sf by (auto simp add: surj_def)
-  also have " \<dots> \<subseteq> span (f ` S)" using spans_image[OF lf us] .
-finally show ?thesis .
+  have "UNIV \<subseteq> f ` UNIV"
+    using sf by (auto simp add: surj_def)
+  also have " \<dots> \<subseteq> span (f ` S)"
+    using spans_image[OF lf us] .
+  finally show ?thesis .
 qed
 
 lemma independent_injective_image:
@@ -1759,23 +2027,30 @@
     and fi: "inj f"
   shows "independent (f ` S)"
 proof -
-  { fix a
+  {
+    fix a
     assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})"
-    have eq: "f ` S - {f a} = f ` (S - {a})" using fi
-      by (auto simp add: inj_on_def)
+    have eq: "f ` S - {f a} = f ` (S - {a})"
+      using fi by (auto simp add: inj_on_def)
     from a have "f a \<in> f ` span (S -{a})"
-      unfolding eq span_linear_image[OF lf, of "S - {a}"]  by blast
-    then have "a \<in> span (S -{a})" using fi by (auto simp add: inj_on_def)
-    with a(1) iS  have False by (simp add: dependent_def) }
-  then show ?thesis unfolding dependent_def by blast
+      unfolding eq span_linear_image[OF lf, of "S - {a}"] by blast
+    then have "a \<in> span (S -{a})"
+      using fi by (auto simp add: inj_on_def)
+    with a(1) iS have False
+      by (simp add: dependent_def)
+  }
+  then show ?thesis
+    unfolding dependent_def by blast
 qed
 
 text {* Picking an orthogonal replacement for a spanning set. *}
 
-    (* FIXME : Move to some general theory ?*)
+(* FIXME : Move to some general theory ?*)
 definition "pairwise R S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y\<in> S. x\<noteq>y \<longrightarrow> R x y)"
 
-lemma vector_sub_project_orthogonal: "(b::'a::euclidean_space) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0"
+lemma vector_sub_project_orthogonal:
+  fixes b x :: "'a::euclidean_space"
+  shows "b \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0"
   unfolding inner_simps by auto
 
 lemma pairwise_orthogonal_insert:
@@ -1786,14 +2061,17 @@
   by (auto simp add: orthogonal_commute)
 
 lemma basis_orthogonal:
-  fixes B :: "('a::real_inner) set"
+  fixes B :: "'a::real_inner set"
   assumes fB: "finite B"
   shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
   (is " \<exists>C. ?P B C")
   using fB
 proof (induct rule: finite_induct)
   case empty
-  then show ?case apply (rule exI[where x="{}"]) by (auto simp add: pairwise_def)
+  then show ?case
+    apply (rule exI[where x="{}"])
+    apply (auto simp add: pairwise_def)
+    done
 next
   case (insert a B)
   note fB = `finite B` and aB = `a \<notin> B`
@@ -1802,10 +2080,12 @@
     "span C = span B" "pairwise orthogonal C" by blast
   let ?a = "a - setsum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x) C"
   let ?C = "insert ?a C"
-  from C(1) have fC: "finite ?C" by simp
+  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)
-  { fix x k
+  {
+    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"
@@ -1817,12 +2097,17 @@
       apply (rule span_mul)
       apply (rule span_superset)
       apply assumption
-      done }
+      done
+  }
   then have SC: "span ?C = span (insert a B)"
     unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto
-  { fix y assume yC: "y \<in> C"
-    then have Cy: "C = insert y (C - {y})" by blast
-    have fth: "finite (C - {y})" using C by simp
+  {
+    fix y
+    assume yC: "y \<in> C"
+    then have Cy: "C = insert y (C - {y})"
+      by blast
+    have fth: "finite (C - {y})"
+      using C by simp
     have "orthogonal ?a y"
       unfolding orthogonal_def
       unfolding inner_diff inner_setsum_left diff_eq_0_iff_eq
@@ -1831,10 +2116,12 @@
       apply (rule setsum_0')
       apply clarsimp
       apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
-      using `y \<in> C` by auto }
+      using `y \<in> C` by auto
+  }
   with `pairwise orthogonal C` have CPO: "pairwise orthogonal ?C"
     by (rule pairwise_orthogonal_insert)
-  from fC cC SC CPO have "?P (insert a B) ?C" by blast
+  from fC cC SC CPO have "?P (insert a B) ?C"
+    by blast
   then show ?case by blast
 qed
 
@@ -1843,19 +2130,29 @@
   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" by blast
-  from B have fB: "finite B" "card B = dim V" using independent_bound by auto
+    B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V"
+    by blast
+  from B have fB: "finite B" "card B = dim V"
+    using independent_bound by auto
   from basis_orthogonal[OF fB(1)] obtain C where
-    C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C" by blast
-  from C B have CSV: "C \<subseteq> span V" by (metis span_inc span_mono subset_trans)
-  from span_mono[OF B(3)]  C have SVC: "span V \<subseteq> span C" by (simp add: span_span)
+    C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C"
+    by blast
+  from C B have CSV: "C \<subseteq> span V"
+    by (metis span_inc 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 add: dim_span)
-  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)]
+  have iC: "independent C"
     by (simp add: dim_span)
-  ultimately have CdV: "card C = dim V" using C(1) by simp
-  from C B CSV CdV iC show ?thesis by auto
+  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 add: dim_span)
+  ultimately have CdV: "card C = dim V"
+    using C(1) by simp
+  from C B CSV CdV iC show ?thesis
+    by auto
 qed
 
 lemma span_eq: "span S = span T \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> span S"
@@ -1865,17 +2162,20 @@
 text {* Low-dimensional subset is in a hyperplane (weak orthogonal complement). *}
 
 lemma span_not_univ_orthogonal:
-  fixes S::"('a::euclidean_space) set"
+  fixes S :: "'a::euclidean_space set"
   assumes sU: "span S \<noteq> UNIV"
   shows "\<exists>(a::'a). a \<noteq>0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
 proof -
-  from sU obtain a where a: "a \<notin> span S" by blast
+  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"
     by blast
-  from B have fB: "finite B" "card B = dim S" using independent_bound by auto
+  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)
+  have sSB: "span S = span B"
+    by (simp add: span_span)
   let ?a = "a - setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B"
   have "setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B \<in> span S"
     unfolding sSB
@@ -1885,17 +2185,23 @@
     apply (rule span_superset)
     apply assumption
     done
-  with a have a0:"?a  \<noteq> 0" by auto
+  with a have a0:"?a  \<noteq> 0"
+    by auto
   have "\<forall>x\<in>span B. ?a \<bullet> x = 0"
   proof (rule span_induct')
     show "subspace {x. ?a \<bullet> x = 0}"
       by (auto simp add: subspace_def inner_add)
   next
-    { fix x assume x: "x \<in> B"
-      from x have B': "B = insert x (B - {x})" by blast
-      have fth: "finite (B - {x})" using fB by simp
+    {
+      fix x
+      assume x: "x \<in> B"
+      from x have B': "B = insert x (B - {x})"
+        by blast
+      have fth: "finite (B - {x})"
+        using fB by simp
       have "?a \<bullet> x = 0"
-        apply (subst B') using fB fth
+        apply (subst B')
+        using fB fth
         unfolding setsum_clauses(2)[OF fth]
         apply simp unfolding inner_simps
         apply (clarsimp simp add: inner_add inner_setsum_left)
@@ -1903,27 +2209,36 @@
         unfolding inner_commute
         apply (auto simp add: x field_simps
           intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])
-        done }
-    then show "\<forall>x \<in> B. ?a \<bullet> x = 0" by blast
+        done
+    }
+    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"])
+  with a0 show ?thesis
+    unfolding sSB by (auto intro: exI[where x="?a"])
 qed
 
 lemma span_not_univ_subset_hyperplane:
-  assumes SU: "span S \<noteq> (UNIV ::('a::euclidean_space) set)"
+  fixes S :: "'a::euclidean_space set"
+  assumes SU: "span S \<noteq> UNIV"
   shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
   using span_not_univ_orthogonal[OF SU] by auto
 
 lemma lowdim_subset_hyperplane:
-  fixes S::"('a::euclidean_space) set"
+  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 (simp add: dim_span dim_UNIV)
-    with d have False by arith }
-  then have th: "span S \<noteq> UNIV" by blast
+  {
+    assume "span S = UNIV"
+    then have "dim (span S) = dim (UNIV :: ('a) set)"
+      by simp
+    then have "dim S = DIM('a)"
+      by (simp add: dim_span dim_UNIV)
+    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
 
@@ -1945,7 +2260,9 @@
   case (2 a b x)
   have fb: "finite b" using "2.prems" by simp
   have th0: "f ` b \<subseteq> f ` (insert a b)"
-    apply (rule image_mono) by blast
+    apply (rule image_mono)
+    apply blast
+    done
   from independent_mono[ OF "2.prems"(2) th0]
   have ifb: "independent (f ` b)"  .
   have fib: "inj_on f b"
@@ -1953,23 +2270,27 @@
     apply blast
     done
   from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)]
-  obtain k where k: "x - k*\<^sub>R a \<in> span (b -{a})" by blast
+  obtain k where k: "x - k*\<^sub>R a \<in> span (b - {a})"
+    by blast
   have "f (x - k*\<^sub>R a) \<in> span (f ` b)"
     unfolding span_linear_image[OF lf]
     apply (rule imageI)
-    using k span_mono[of "b-{a}" b] apply blast
+    using k span_mono[of "b-{a}" b]
+    apply blast
     done
   then have "f x - k*\<^sub>R f a \<in> span (f ` b)"
     by (simp add: linear_sub[OF lf] linear_cmul[OF lf])
   then have th: "-k *\<^sub>R f a \<in> span (f ` b)"
     using "2.prems"(5) by simp
-  { assume k0: "k = 0"
-    from k0 k have "x \<in> span (b -{a})" by simp
-    then have "x \<in> span b" using span_mono[of "b-{a}" b]
-      by blast }
-  moreover
-  { assume k0: "k \<noteq> 0"
-    from span_mul[OF th, of "- 1/ k"] k0
+  have xsb: "x \<in> span b"
+  proof (cases "k = 0")
+    case True
+    with k have "x \<in> span (b -{a})" by simp
+    then show ?thesis using span_mono[of "b-{a}" b]
+      by blast
+  next
+    case False
+    with span_mul[OF th, of "- 1/ k"]
     have th1: "f a \<in> span (f ` b)"
       by auto
     from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric]
@@ -1979,20 +2300,21 @@
       using "2.hyps"(2)
       "2.prems"(3) by auto
     with th1 have False by blast
-    then have "x \<in> span b" by blast }
-  ultimately have xsb: "x \<in> span b" by blast
-  from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)]
-  show "x = 0" .
+    then show ?thesis by blast
+  qed
+  from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)] show "x = 0" .
 qed
 
 text {* We can extend a linear mapping from basis. *}
 
 lemma linear_independent_extend_lemma:
   fixes f :: "'a::real_vector \<Rightarrow> 'b::real_vector"
-  assumes fi: "finite B" and ib: "independent B"
-  shows "\<exists>g. (\<forall>x\<in> span B. \<forall>y\<in> span B. g (x + y) = g x + g y)
-           \<and> (\<forall>x\<in> span B. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x)
-           \<and> (\<forall>x\<in> B. g x = f x)"
+  assumes fi: "finite B"
+    and ib: "independent B"
+  shows "\<exists>g.
+    (\<forall>x\<in> span B. \<forall>y\<in> span B. g (x + y) = g x + g y) \<and>
+    (\<forall>x\<in> span B. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x) \<and>
+    (\<forall>x\<in> B. g x = f x)"
   using ib fi
 proof (induct rule: finite_induct[OF fi])
   case 1
@@ -2005,39 +2327,56 @@
     g: "\<forall>x\<in>span b. \<forall>y\<in>span b. g (x + y) = g x + g y"
     "\<forall>x\<in>span b. \<forall>c. g (c *\<^sub>R x) = c *\<^sub>R g x" "\<forall>x\<in>b. g x = f x" by blast
   let ?h = "\<lambda>z. SOME k. (z - k *\<^sub>R a) \<in> span b"
-  { fix z assume z: "z \<in> span (insert a b)"
+  {
+    fix z
+    assume z: "z \<in> span (insert a b)"
     have th0: "z - ?h z *\<^sub>R a \<in> span b"
       apply (rule someI_ex)
       unfolding span_breakdown_eq[symmetric]
-      using z .
-    { fix k assume k: "z - k *\<^sub>R a \<in> span b"
+      apply (rule z)
+      done
+    {
+      fix k
+      assume k: "z - k *\<^sub>R a \<in> span b"
       have eq: "z - ?h z *\<^sub>R a - (z - k*\<^sub>R a) = (k - ?h z) *\<^sub>R a"
         by (simp add: field_simps scaleR_left_distrib [symmetric])
-      from span_sub[OF th0 k]
-      have khz: "(k - ?h z) *\<^sub>R a \<in> span b" by (simp add: eq)
-      { assume "k \<noteq> ?h z" then have k0: "k - ?h z \<noteq> 0" by simp
+      from span_sub[OF th0 k] have khz: "(k - ?h z) *\<^sub>R a \<in> span b"
+        by (simp add: eq)
+      {
+        assume "k \<noteq> ?h z"
+        then have k0: "k - ?h z \<noteq> 0" by simp
         from k0 span_mul[OF khz, of "1 /(k - ?h z)"]
         have "a \<in> span b" by simp
         with "2.prems"(1) "2.hyps"(2) have False
-          by (auto simp add: dependent_def)}
-      then have "k = ?h z" by blast}
-    with th0 have "z - ?h z *\<^sub>R a \<in> span b \<and> (\<forall>k. z - k *\<^sub>R a \<in> span b \<longrightarrow> k = ?h z)" by blast}
+          by (auto simp add: dependent_def)
+      }
+      then have "k = ?h z" by blast
+    }
+    with th0 have "z - ?h z *\<^sub>R a \<in> span b \<and> (\<forall>k. z - k *\<^sub>R a \<in> span b \<longrightarrow> k = ?h z)"
+      by blast
+  }
   note h = this
   let ?g = "\<lambda>z. ?h z *\<^sub>R f a + g (z - ?h z *\<^sub>R a)"
-  { fix x y assume x: "x \<in> span (insert a b)" and y: "y \<in> span (insert a b)"
+  {
+    fix x y
+    assume x: "x \<in> span (insert a b)"
+      and y: "y \<in> span (insert a b)"
     have tha: "\<And>(x::'a) y a k l. (x + y) - (k + l) *\<^sub>R a = (x - k *\<^sub>R a) + (y - l *\<^sub>R a)"
       by (simp add: algebra_simps)
     have addh: "?h (x + y) = ?h x + ?h y"
       apply (rule conjunct2[OF h, rule_format, symmetric])
       apply (rule span_add[OF x y])
       unfolding tha
-      by (metis span_add x y conjunct1[OF h, rule_format])
+      apply (metis span_add x y conjunct1[OF h, rule_format])
+      done
     have "?g (x + y) = ?g x + ?g y"
       unfolding addh tha
       g(1)[rule_format,OF conjunct1[OF h, OF x] conjunct1[OF h, OF y]]
       by (simp add: scaleR_left_distrib)}
   moreover
-  { fix x:: "'a" and c:: real
+  {
+    fix x :: "'a"
+    fix c :: real
     assume x: "x \<in> span (insert a b)"
     have tha: "\<And>(x::'a) c k a. c *\<^sub>R x - (c * k) *\<^sub>R a = c *\<^sub>R (x - k *\<^sub>R a)"
       by (simp add: algebra_simps)
@@ -2048,24 +2387,29 @@
       done
     have "?g (c *\<^sub>R x) = c*\<^sub>R ?g x"
       unfolding hc tha g(2)[rule_format, OF conjunct1[OF h, OF x]]
-      by (simp add: algebra_simps) }
+      by (simp add: algebra_simps)
+  }
   moreover
-  { fix x assume x: "x \<in> (insert a b)"
-    { assume xa: "x = a"
+  {
+    fix x
+    assume x: "x \<in> insert a b"
+    {
+      assume xa: "x = a"
       have ha1: "1 = ?h a"
         apply (rule conjunct2[OF h, rule_format])
         apply (metis span_superset insertI1)
         using conjunct1[OF h, OF span_superset, OF insertI1]
         apply (auto simp add: span_0)
         done
-
       from xa ha1[symmetric] have "?g x = f x"
         apply simp
         using g(2)[rule_format, OF span_0, of 0]
         apply simp
-        done }
+        done
+    }
     moreover
-    { assume xb: "x \<in> b"
+    {
+      assume xb: "x \<in> b"
       have h0: "0 = ?h x"
         apply (rule conjunct2[OF h, rule_format])
         apply (metis  span_superset x)
@@ -2073,8 +2417,11 @@
         apply (metis span_superset xb)
         done
       have "?g x = f x"
-        by (simp add: h0[symmetric] g(3)[rule_format, OF xb]) }
-    ultimately have "?g x = f x" using x by blast }
+        by (simp add: h0[symmetric] g(3)[rule_format, OF xb])
+    }
+    ultimately have "?g x = f x"
+      using x by blast
+  }
   ultimately show ?case
     apply -
     apply (rule exI[where x="?g"])
@@ -2083,17 +2430,22 @@
 qed
 
 lemma linear_independent_extend:
-  assumes iB: "independent (B:: ('a::euclidean_space) set)"
+  fixes B :: "'a::euclidean_space set"
+  assumes iB: "independent B"
   shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)"
 proof -
   from maximal_independent_subset_extend[of B UNIV] iB
-  obtain C where C: "B \<subseteq> C" "independent C" "\<And>x. x \<in> span C" by auto
+  obtain C where C: "B \<subseteq> C" "independent C" "\<And>x. x \<in> span C"
+    by auto
 
   from C(2) independent_bound[of C] linear_independent_extend_lemma[of C f]
-  obtain g where g: "(\<forall>x\<in> span C. \<forall>y\<in> span C. g (x + y) = g x + g y)
-           \<and> (\<forall>x\<in> span C. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x)
-           \<and> (\<forall>x\<in> C. g x = f x)" by blast
-  from g show ?thesis unfolding linear_def using C
+  obtain g where g:
+    "(\<forall>x\<in> span C. \<forall>y\<in> span C. g (x + y) = g x + g y) \<and>
+     (\<forall>x\<in> span C. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x) \<and>
+     (\<forall>x\<in> C. g x = f x)" by blast
+  from g show ?thesis
+    unfolding linear_def
+    using C
     apply clarsimp
     apply blast
     done
@@ -2118,10 +2470,12 @@
     then show ?case by simp
   next
     case (2 y t)
-    from "2.prems"(1,2,5) "2.hyps"(1,2) have cst:"card s \<le> card t" by simp
-    from "2.prems"(3) [OF "2.hyps"(1) cst] obtain f where
-      f: "f ` s \<subseteq> t \<and> inj_on f s" by blast
-    from f "2.prems"(2) "2.hyps"(2) show ?case
+    from "2.prems"(1,2,5) "2.hyps"(1,2) have cst: "card s \<le> card t"
+      by simp
+    from "2.prems"(3) [OF "2.hyps"(1) cst]
+    obtain f where "f ` s \<subseteq> t" "inj_on f s"
+      by blast
+    with "2.prems"(2) "2.hyps"(2) show ?case
       apply -
       apply (rule exI[where x = "\<lambda>z. if z = x then y else f z"])
       apply (auto simp add: inj_on_def)
@@ -2135,54 +2489,74 @@
     and c: "card A = card B"
   shows "A = B"
 proof -
-  from fB AB have fA: "finite A" by (auto intro: finite_subset)
-  from fA fB have fBA: "finite (B - A)" by auto
-  have e: "A \<inter> (B - A) = {}" by blast
-  have eq: "A \<union> (B - A) = B" using AB by blast
-  from card_Un_disjoint[OF fA fBA e, unfolded eq c]
-  have "card (B - A) = 0" by arith
-  then have "B - A = {}" unfolding card_eq_0_iff using fA fB by simp
-  with AB show "A = B" by blast
+  from fB AB have fA: "finite A"
+    by (auto intro: finite_subset)
+  from fA fB have fBA: "finite (B - A)"
+    by auto
+  have e: "A \<inter> (B - A) = {}"
+    by blast
+  have eq: "A \<union> (B - A) = B"
+    using AB by blast
+  from card_Un_disjoint[OF fA fBA e, unfolded eq c] have "card (B - A) = 0"
+    by arith
+  then have "B - A = {}"
+    unfolding card_eq_0_iff using fA fB by simp
+  with AB show "A = B"
+    by blast
 qed
 
 lemma subspace_isomorphism:
-  assumes s: "subspace (S:: ('a::euclidean_space) set)"
-    and t: "subspace (T :: ('b::euclidean_space) set)"
+  fixes S :: "'a::euclidean_space set"
+    and T :: "'b::euclidean_space set"
+  assumes s: "subspace S"
+    and t: "subspace T"
     and d: "dim S = dim T"
   shows "\<exists>f. linear f \<and> f ` S = T \<and> inj_on f S"
 proof -
-  from basis_exists[of S] independent_bound obtain B where
-    B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" and fB: "finite B" by blast
-  from basis_exists[of T] independent_bound obtain C where
-    C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "card C = dim T" and fC: "finite C" by blast
-  from B(4) C(4) card_le_inj[of B C] d obtain f where
-    f: "f ` B \<subseteq> C" "inj_on f B" using `finite B` `finite C` by auto
-  from linear_independent_extend[OF B(2)] obtain g where
-    g: "linear g" "\<forall>x\<in> B. g x = f x" by blast
-  from inj_on_iff_eq_card[OF fB, of f] f(2)
-  have "card (f ` B) = card B" by simp
-  with B(4) C(4) have ceq: "card (f ` B) = card C" using d
+  from basis_exists[of S] independent_bound
+  obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" and fB: "finite B"
+    by blast
+  from basis_exists[of T] independent_bound
+  obtain C where C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "card C = dim T" and fC: "finite C"
+    by blast
+  from B(4) C(4) card_le_inj[of B C] d
+  obtain f where f: "f ` B \<subseteq> C" "inj_on f B" using `finite B` `finite C`
+    by auto
+  from linear_independent_extend[OF B(2)]
+  obtain g where g: "linear g" "\<forall>x\<in> B. g x = f x"
+    by blast
+  from inj_on_iff_eq_card[OF fB, of f] f(2) have "card (f ` B) = card B"
     by simp
-  have "g ` B = f ` B" using g(2)
-    by (auto simp add: image_iff)
+  with B(4) C(4) have ceq: "card (f ` B) = card C"
+    using d by simp
+  have "g ` B = f ` B"
+    using g(2) by (auto simp add: image_iff)
   also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] .
   finally have gBC: "g ` B = C" .
-  have gi: "inj_on g B" using f(2) g(2)
-    by (auto simp add: inj_on_def)
+  have gi: "inj_on g B"
+    using f(2) g(2) by (auto simp add: inj_on_def)
   note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
-  { fix x y assume x: "x \<in> S" and y: "y \<in> S" and gxy: "g x = g y"
-    from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B" by blast+
-    from gxy have th0: "g (x - y) = 0" by (simp add: linear_sub[OF g(1)])
-    have th1: "x - y \<in> span B" using x' y' by (metis span_sub)
-    have "x=y" using g0[OF th1 th0] by simp }
+  {
+    fix x y
+    assume x: "x \<in> S" and y: "y \<in> S" and gxy: "g x = g y"
+    from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B"
+      by blast+
+    from gxy have th0: "g (x - y) = 0"
+      by (simp add: linear_sub[OF g(1)])
+    have th1: "x - y \<in> span B"
+      using x' y' by (metis span_sub)
+    have "x = y"
+      using g0[OF th1 th0] by simp
+  }
   then have giS: "inj_on g S"
     unfolding inj_on_def by blast
-  from span_subspace[OF B(1,3) s]
-  have "g ` S = span (g ` B)" by (simp add: span_linear_image[OF g(1)])
+  from span_subspace[OF B(1,3) s] have "g ` S = span (g ` B)"
+    by (simp add: span_linear_image[OF g(1)])
   also have "\<dots> = span C" unfolding gBC ..
   also have "\<dots> = T" using span_subspace[OF C(1,3) t] .
   finally have gS: "g ` S = T" .
-  from g(1) gS giS show ?thesis by blast
+  from g(1) gS giS show ?thesis
+    by blast
 qed
 
 text {* Linear functions are equal on a subspace if they are on a spanning set. *}
@@ -2232,7 +2606,8 @@
 lemma bilinear_eq:
   assumes bf: "bilinear f"
     and bg: "bilinear g"
-    and SB: "S \<subseteq> span B" and TC: "T \<subseteq> span C"
+    and SB: "S \<subseteq> span B"
+    and TC: "T \<subseteq> span C"
     and fg: "\<forall>x\<in> B. \<forall>y\<in> C. f x y = g x y"
   shows "\<forall>x\<in>S. \<forall>y\<in>T. f x y = g x y "
 proof -
@@ -2252,11 +2627,12 @@
     apply (auto simp add: span_0 bilinear_rzero[OF bf] bilinear_rzero[OF bg] span_add Ball_def
       intro: bilinear_ladd[OF bf])
     done
-  then show ?thesis using SB TC by auto
+  then show ?thesis
+    using SB TC by auto
 qed
 
 lemma bilinear_eq_stdbasis:
-  fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _"
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _"
   assumes bf: "bilinear f"
     and bg: "bilinear g"
     and fg: "\<forall>i\<in>Basis. \<forall>j\<in>Basis. f i j = g i j"
@@ -2266,50 +2642,53 @@
 text {* Detailed theorems about left and right invertibility in general case. *}
 
 lemma linear_injective_left_inverse:
-  fixes f::"'a::euclidean_space => 'b::euclidean_space"
+  fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes lf: "linear f" and fi: "inj f"
   shows "\<exists>g. linear g \<and> g o f = id"
 proof -
   from linear_independent_extend[OF independent_injective_image, OF independent_Basis, OF lf fi]
-  obtain h:: "'b => 'a" where
-    h: "linear h" "\<forall>x \<in> f ` Basis. h x = inv f x" by blast
+  obtain h:: "'b \<Rightarrow> 'a" where h: "linear h" "\<forall>x \<in> f ` Basis. h x = inv f x"
+    by blast
   from h(2) have th: "\<forall>i\<in>Basis. (h \<circ> f) i = id i"
     using inv_o_cancel[OF fi, unfolded fun_eq_iff id_def o_def]
     by auto
-
   from linear_eq_stdbasis[OF linear_compose[OF lf h(1)] linear_id th]
   have "h o f = id" .
-  then show ?thesis using h(1) by blast
+  then show ?thesis
+    using h(1) by blast
 qed
 
 lemma linear_surjective_right_inverse:
-  fixes f::"'a::euclidean_space => 'b::euclidean_space"
-  assumes lf: "linear f" and sf: "surj f"
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes lf: "linear f"
+    and sf: "surj f"
   shows "\<exists>g. linear g \<and> f o g = id"
 proof -
   from linear_independent_extend[OF independent_Basis[where 'a='b],of "inv f"]
-  obtain h:: "'b \<Rightarrow> 'a" where
-    h: "linear h" "\<forall>x\<in>Basis. h x = inv f x" by blast
-  from h(2)
-  have th: "\<forall>i\<in>Basis. (f o h) i = id i"
+  obtain h:: "'b \<Rightarrow> 'a" where h: "linear h" "\<forall>x\<in>Basis. h x = inv f x"
+    by blast
+  from h(2) have th: "\<forall>i\<in>Basis. (f o h) i = id i"
     using sf by (auto simp add: surj_iff_all)
   from linear_eq_stdbasis[OF linear_compose[OF h(1) lf] linear_id th]
   have "f o h = id" .
-  then show ?thesis using h(1) by blast
+  then show ?thesis
+    using h(1) by blast
 qed
 
 text {* An injective map @{typ "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"} is also surjective. *}
 
 lemma linear_injective_imp_surjective:
-  fixes f::"'a::euclidean_space => 'a::euclidean_space"
-  assumes lf: "linear f" and fi: "inj f"
+  fixes f::"'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
+  assumes lf: "linear f"
+    and fi: "inj f"
   shows "surj f"
 proof -
   let ?U = "UNIV :: 'a set"
   from basis_exists[of ?U] obtain B
     where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "card B = dim ?U"
     by blast
-  from B(4) have d: "dim ?U = card B" by simp
+  from B(4) have d: "dim ?U = card B"
+    by simp
   have th: "?U \<subseteq> span (f ` B)"
     apply (rule card_ge_dim_independent)
     apply blast
@@ -2333,51 +2712,66 @@
     and fT: "finite T"
     and c: "card S = card T"
     and ST: "f ` S \<subseteq> T"
-  shows "(\<forall>y \<in> T. \<exists>x \<in> S. f x = y) \<longleftrightarrow> inj_on f S" (is "?lhs \<longleftrightarrow> ?rhs")
-proof -
-  { assume h: "?lhs"
-    { fix x y
-      assume x: "x \<in> S" and y: "y \<in> S" and f: "f x = f y"
-      from x fS have S0: "card S \<noteq> 0" by auto
-      { assume xy: "x \<noteq> y"
-        have th: "card S \<le> card (f ` (S - {y}))"
-          unfolding c
-          apply (rule card_mono)
-          apply (rule finite_imageI)
-          using fS apply simp
-          using h xy x y f unfolding subset_eq image_iff
-          apply auto
-          apply (case_tac "xa = f x")
-          apply (rule bexI[where x=x])
-          apply auto
-          done
-        also have " \<dots> \<le> card (S -{y})"
-          apply (rule card_image_le)
-          using fS by simp
-        also have "\<dots> \<le> card S - 1" using y fS by simp
-        finally have False  using S0 by arith }
-      then have "x = y" by blast}
-    then have ?rhs unfolding inj_on_def by blast}
-  moreover
-  { assume h: ?rhs
-    have "f ` S = T"
-      apply (rule card_subset_eq[OF fT ST])
-      unfolding card_image[OF h] using c .
-    then have ?lhs by blast}
-  ultimately show ?thesis by blast
+  shows "(\<forall>y \<in> T. \<exists>x \<in> S. f x = y) \<longleftrightarrow> inj_on f S"
+  (is "?lhs \<longleftrightarrow> ?rhs")
+proof
+  assume h: "?lhs"
+  {
+    fix x y
+    assume x: "x \<in> S"
+    assume y: "y \<in> S"
+    assume f: "f x = f y"
+    from x fS have S0: "card S \<noteq> 0"
+      by auto
+    have "x = y"
+    proof (rule ccontr)
+      assume xy: "x \<noteq> y"
+      have th: "card S \<le> card (f ` (S - {y}))"
+        unfolding c
+        apply (rule card_mono)
+        apply (rule finite_imageI)
+        using fS apply simp
+        using h xy x y f unfolding subset_eq image_iff
+        apply auto
+        apply (case_tac "xa = f x")
+        apply (rule bexI[where x=x])
+        apply auto
+        done
+      also have " \<dots> \<le> card (S -{y})"
+        apply (rule card_image_le)
+        using fS by simp
+      also have "\<dots> \<le> card S - 1" using y fS by simp
+      finally show False using S0 by arith
+    qed
+  }
+  then show ?rhs
+    unfolding inj_on_def by blast
+next
+  assume h: ?rhs
+  have "f ` S = T"
+    apply (rule card_subset_eq[OF fT ST])
+    unfolding card_image[OF h]
+    apply (rule c)
+    done
+  then show ?lhs by blast
 qed
 
 lemma linear_surjective_imp_injective:
-  fixes f::"'a::euclidean_space => 'a::euclidean_space"
-  assumes lf: "linear f" and sf: "surj f"
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
+  assumes lf: "linear f"
+    and sf: "surj f"
   shows "inj f"
 proof -
   let ?U = "UNIV :: 'a set"
   from basis_exists[of ?U] obtain B
     where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" and d: "card B = dim ?U"
     by blast
-  { fix x assume x: "x \<in> span B" and fx: "f x = 0"
-    from B(2) have fB: "finite B" using independent_bound by auto
+  {
+    fix x
+    assume x: "x \<in> span B"
+    assume fx: "f x = 0"
+    from B(2) have fB: "finite B"
+      using independent_bound by auto
     have fBi: "independent (f ` B)"
       apply (rule card_le_dim_spanning[of "f ` B" ?U])
       apply blast
@@ -2394,81 +2788,98 @@
       apply blast
       unfolding span_linear_image[OF lf]
       apply (rule subset_trans[where B = "f ` UNIV"])
-      using sf unfolding surj_def apply blast
+      using sf unfolding surj_def
+      apply blast
       apply (rule image_mono)
       apply (rule B(3))
       apply (metis finite_imageI fB)
       done
-
     moreover have "card (f ` B) \<le> card B"
       by (rule card_image_le, rule fB)
-    ultimately have th1: "card B = card (f ` B)" unfolding d by arith
+    ultimately have th1: "card B = card (f ` B)"
+      unfolding d by arith
     have fiB: "inj_on f B"
       unfolding surjective_iff_injective_gen[OF fB finite_imageI[OF fB] th1 subset_refl, symmetric]
       by blast
     from linear_indep_image_lemma[OF lf fB fBi fiB x] fx
-    have "x = 0" by blast}
-  note th = this
-  from th show ?thesis unfolding linear_injective_0[OF lf]
-    using B(3) by blast
+    have "x = 0" by blast
+  }
+  then show ?thesis
+    unfolding linear_injective_0[OF lf]
+    using B(3)
+    by blast
 qed
 
 text {* Hence either is enough for isomorphism. *}
 
 lemma left_right_inverse_eq:
-  assumes fg: "f o g = id" and gh: "g o h = id"
+  assumes fg: "f \<circ> g = id"
+    and gh: "g \<circ> h = id"
   shows "f = h"
 proof -
-  have "f = f o (g o h)" unfolding gh by simp
-  also have "\<dots> = (f o g) o h" by (simp add: o_assoc)
-  finally show "f = h" unfolding fg by simp
+  have "f = f \<circ> (g \<circ> h)"
+    unfolding gh by simp
+  also have "\<dots> = (f \<circ> g) \<circ> h"
+    by (simp add: o_assoc)
+  finally show "f = h"
+    unfolding fg by simp
 qed
 
 lemma isomorphism_expand:
-  "f o g = id \<and> g o f = id \<longleftrightarrow> (\<forall>x. f(g x) = x) \<and> (\<forall>x. g(f x) = x)"
+  "f \<circ> g = id \<and> g \<circ> f = id \<longleftrightarrow> (\<forall>x. f (g x) = x) \<and> (\<forall>x. g (f x) = x)"
   by (simp add: fun_eq_iff o_def id_def)
 
 lemma linear_injective_isomorphism:
-  fixes f::"'a::euclidean_space => 'a::euclidean_space"
-  assumes lf: "linear f" and fi: "inj f"
+  fixes f::"'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
+  assumes lf: "linear f"
+    and fi: "inj f"
   shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
   unfolding isomorphism_expand[symmetric]
   using linear_surjective_right_inverse[OF lf linear_injective_imp_surjective[OF lf fi]]
     linear_injective_left_inverse[OF lf fi]
   by (metis left_right_inverse_eq)
 
-lemma linear_surjective_isomorphism: fixes f::"'a::euclidean_space => 'a::euclidean_space"
-  assumes lf: "linear f" and sf: "surj f"
+lemma linear_surjective_isomorphism:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
+  assumes lf: "linear f"
+    and sf: "surj f"
   shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
   unfolding isomorphism_expand[symmetric]
   using linear_surjective_right_inverse[OF lf sf]
     linear_injective_left_inverse[OF lf linear_surjective_imp_injective[OF lf sf]]
   by (metis left_right_inverse_eq)
 
-text {* Left and right inverses are the same for @{typ "'a::euclidean_space => 'a::euclidean_space"}. *}
+text {* Left and right inverses are the same for
+  @{typ "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"}. *}
 
 lemma linear_inverse_left:
-  fixes f::"'a::euclidean_space => 'a::euclidean_space"
-  assumes lf: "linear f" and lf': "linear f'"
-  shows "f o f' = id \<longleftrightarrow> f' o f = id"
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
+  assumes lf: "linear f"
+    and lf': "linear f'"
+  shows "f \<circ> f' = id \<longleftrightarrow> f' \<circ> f = id"
 proof -
-  { fix f f':: "'a => 'a"
-    assume lf: "linear f" "linear f'" and f: "f o f' = id"
+  {
+    fix f f':: "'a \<Rightarrow> 'a"
+    assume lf: "linear f" "linear f'"
+    assume f: "f \<circ> f' = id"
     from f have sf: "surj f"
       apply (auto simp add: o_def id_def surj_def)
       apply metis
       done
     from linear_surjective_isomorphism[OF lf(1) sf] lf f
-    have "f' o f = id" unfolding fun_eq_iff o_def id_def
-      by metis }
-  then show ?thesis using lf lf' by metis
+    have "f' \<circ> f = id"
+      unfolding fun_eq_iff o_def id_def by metis
+  }
+  then show ?thesis
+    using lf lf' by metis
 qed
 
 text {* Moreover, a one-sided inverse is automatically linear. *}
 
 lemma left_inverse_linear:
-  fixes f::"'a::euclidean_space => 'a::euclidean_space"
-  assumes lf: "linear f" and gf: "g o f = id"
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
+  assumes lf: "linear f"
+    and gf: "g \<circ> f = id"
   shows "linear g"
 proof -
   from gf have fi: "inj f"
@@ -2476,8 +2887,8 @@
     apply metis
     done
   from linear_injective_isomorphism[OF lf fi]
-  obtain h:: "'a \<Rightarrow> 'a" where
-    h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" by blast
+  obtain h :: "'a \<Rightarrow> 'a" where h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x"
+    by blast
   have "h = g"
     apply (rule ext) using gf h(2,3)
     apply (simp add: o_def id_def fun_eq_iff)
@@ -2495,22 +2906,26 @@
   by auto
 
 lemma infnorm_set_image:
-  "{ abs ((x::'a::euclidean_space) \<bullet> i) |i. i \<in> Basis} = (\<lambda>i. abs(x \<bullet> i)) ` Basis"
+  "{abs ((x::'a::euclidean_space) \<bullet> i) |i. i \<in> Basis} = (\<lambda>i. abs(x \<bullet> i)) ` Basis"
   by blast
 
 lemma infnorm_Max: "infnorm (x::'a::euclidean_space) = Max ((\<lambda>i. abs(x \<bullet> i)) ` Basis)"
   by (simp add: infnorm_def infnorm_set_image cSup_eq_Max)
 
 lemma infnorm_set_lemma:
-  shows "finite {abs((x::'a::euclidean_space) \<bullet> i) |i. i \<in> Basis}"
-  and "{abs(x \<bullet> i) |i. i \<in> Basis} \<noteq> {}"
+  "finite {abs((x::'a::euclidean_space) \<bullet> i) |i. i \<in> Basis}"
+  "{abs(x \<bullet> i) |i. i \<in> Basis} \<noteq> {}"
   unfolding infnorm_set_image
   by auto
 
-lemma infnorm_pos_le: "0 \<le> infnorm (x::'a::euclidean_space)"
+lemma infnorm_pos_le:
+  fixes x :: "'a::euclidean_space"
+  shows "0 \<le> infnorm x"
   by (simp add: infnorm_Max Max_ge_iff ex_in_conv)
 
-lemma infnorm_triangle: "infnorm ((x::'a::euclidean_space) + y) \<le> infnorm x + infnorm y"
+lemma infnorm_triangle:
+  fixes x :: "'a::euclidean_space"
+  shows "infnorm (x + y) \<le> infnorm x + infnorm y"
 proof -
   have *: "\<And>a b c d :: real. \<bar>a\<bar> \<le> c \<Longrightarrow> \<bar>b\<bar> \<le> d \<Longrightarrow> \<bar>a + b\<bar> \<le> c + d"
     by simp
@@ -2518,7 +2933,9 @@
     by (auto simp: infnorm_Max inner_add_left intro!: *)
 qed
 
-lemma infnorm_eq_0: "infnorm x = 0 \<longleftrightarrow> (x::_::euclidean_space) = 0"
+lemma infnorm_eq_0:
+  fixes x :: "'a::euclidean_space"
+  shows "infnorm x = 0 \<longleftrightarrow> x = 0"
 proof -
   have "infnorm x \<le> 0 \<longleftrightarrow> x = 0"
     unfolding infnorm_Max by (simp add: euclidean_all_zero_iff)
@@ -2539,41 +2956,47 @@
 lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)"
 proof -
   have "y - x = - (x - y)" by simp
-  then show ?thesis  by (metis infnorm_neg)
+  then show ?thesis
+    by (metis infnorm_neg)
 qed
 
-lemma real_abs_sub_infnorm: "\<bar> infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
+lemma real_abs_sub_infnorm: "\<bar>infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
 proof -
-  have th: "\<And>(nx::real) n ny. nx <= n + ny \<Longrightarrow> ny <= n + nx ==> \<bar>nx - ny\<bar> <= n"
+  have th: "\<And>(nx::real) n ny. nx \<le> n + ny \<Longrightarrow> ny <= n + nx \<Longrightarrow> \<bar>nx - ny\<bar> \<le> n"
     by arith
   from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
   have ths: "infnorm x \<le> infnorm (x - y) + infnorm y"
     "infnorm y \<le> infnorm (x - y) + infnorm x"
     by (simp_all add: field_simps infnorm_neg)
-  from th[OF ths]  show ?thesis .
+  from th[OF ths] show ?thesis .
 qed
 
-lemma real_abs_infnorm: " \<bar>infnorm x\<bar> = infnorm x"
+lemma real_abs_infnorm: "\<bar>infnorm x\<bar> = infnorm x"
   using infnorm_pos_le[of x] by arith
 
 lemma Basis_le_infnorm:
-  "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> infnorm (x::'a::euclidean_space)"
+  fixes x :: "'a::euclidean_space"
+  shows "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> infnorm x"
   by (simp add: infnorm_Max)
 
 lemma infnorm_mul: "infnorm(a *\<^sub>R x) = abs a * infnorm x"
   unfolding infnorm_Max
 proof (safe intro!: Max_eqI)
   let ?B = "(\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis"
-  show "\<And>b :: 'a. b \<in> Basis \<Longrightarrow> \<bar>a *\<^sub>R x \<bullet> b\<bar> \<le> \<bar>a\<bar> * Max ?B"
-    by (simp add: abs_mult mult_left_mono)
-
-  from Max_in[of ?B] obtain b where "b \<in> Basis" "Max ?B = \<bar>x \<bullet> b\<bar>"
-    by (auto simp del: Max_in)
-  then show "\<bar>a\<bar> * Max ((\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis) \<in> (\<lambda>i. \<bar>a *\<^sub>R x \<bullet> i\<bar>) ` Basis"
-    by (intro image_eqI[where x=b]) (auto simp: abs_mult)
+  {
+    fix b :: 'a
+    assume "b \<in> Basis"
+    then show "\<bar>a *\<^sub>R x \<bullet> b\<bar> \<le> \<bar>a\<bar> * Max ?B"
+      by (simp add: abs_mult mult_left_mono)
+  next
+    from Max_in[of ?B] obtain b where "b \<in> Basis" "Max ?B = \<bar>x \<bullet> b\<bar>"
+      by (auto simp del: Max_in)
+    then show "\<bar>a\<bar> * Max ((\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis) \<in> (\<lambda>i. \<bar>a *\<^sub>R x \<bullet> i\<bar>) ` Basis"
+      by (intro image_eqI[where x=b]) (auto simp: abs_mult)
+  }
 qed simp
 
-lemma infnorm_mul_lemma: "infnorm(a *\<^sub>R x) \<le> \<bar>a\<bar> * infnorm x"
+lemma infnorm_mul_lemma: "infnorm (a *\<^sub>R x) \<le> \<bar>a\<bar> * infnorm x"
   unfolding infnorm_mul ..
 
 lemma infnorm_pos_lt: "infnorm x > 0 \<longleftrightarrow> x \<noteq> 0"
@@ -2591,7 +3014,8 @@
 lemma norm_le_infnorm: "norm x \<le> sqrt DIM('a) * infnorm(x::'a::euclidean_space)"
 proof -
   let ?d = "DIM('a)"
-  have "real ?d \<ge> 0" by simp
+  have "real ?d \<ge> 0"
+    by simp
   then have d2: "(sqrt (real ?d))\<^sup>2 = real ?d"
     by (auto intro: real_sqrt_pow2)
   have th: "sqrt (real ?d) * infnorm x \<ge> 0"
@@ -2608,29 +3032,37 @@
     apply (auto simp: infnorm_Max)
     done
   from real_le_lsqrt[OF inner_ge_zero th th1]
-  show ?thesis unfolding norm_eq_sqrt_inner id_def .
+  show ?thesis
+    unfolding norm_eq_sqrt_inner id_def .
 qed
 
 lemma tendsto_infnorm [tendsto_intros]:
   assumes "(f ---> a) F"
   shows "((\<lambda>x. infnorm (f x)) ---> infnorm a) F"
 proof (rule tendsto_compose [OF LIM_I assms])
-  fix r :: real assume "0 < r"
+  fix r :: real
+  assume "r > 0"
   then show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (infnorm x - infnorm a) < r"
     by (metis real_norm_def le_less_trans real_abs_sub_infnorm infnorm_le_norm)
 qed
 
 text {* Equality in Cauchy-Schwarz and triangle inequalities. *}
 
-lemma norm_cauchy_schwarz_eq: "x \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x" (is "?lhs \<longleftrightarrow> ?rhs")
+lemma norm_cauchy_schwarz_eq: "x \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
+  (is "?lhs \<longleftrightarrow> ?rhs")
 proof -
-  { assume h: "x = 0"
-    then have ?thesis by simp }
+  {
+    assume h: "x = 0"
+    then have ?thesis by simp
+  }
   moreover
-  { assume h: "y = 0"
-    then have ?thesis by simp }
+  {
+    assume h: "y = 0"
+    then have ?thesis by simp
+  }
   moreover
-  { assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
+  {
+    assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
     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)) -
@@ -2648,49 +3080,58 @@
       apply simp
       apply metis
       done
-    finally have ?thesis by blast }
+    finally have ?thesis by blast
+  }
   ultimately show ?thesis by blast
 qed
 
 lemma norm_cauchy_schwarz_abs_eq:
   "abs(x \<bullet> y) = 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")
+    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> abs x = a \<longleftrightarrow> x = a \<or> x = - a" by arith
+  have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> abs x = 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)"
+  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
+    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
 
 lemma norm_triangle_eq:
   fixes x y :: "'a::real_inner"
-  shows "norm(x + y) = norm x + norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
+  shows "norm (x + y) = norm x + norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
 proof -
-  { assume x: "x = 0 \<or> y = 0"
-    then have ?thesis by (cases "x = 0") simp_all }
+  {
+    assume x: "x = 0 \<or> y = 0"
+    then have ?thesis
+      by (cases "x = 0") simp_all
+  }
   moreover
-  { assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
+  {
+    assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
     then have "norm x \<noteq> 0" "norm y \<noteq> 0"
       by simp_all
     then have n: "norm x > 0" "norm y > 0"
       using norm_ge_zero[of x] norm_ge_zero[of y] by arith+
-    have th: "\<And>(a::real) b c. a + b + c \<noteq> 0 ==> (a = b + c \<longleftrightarrow> a\<^sup>2 = (b + c)\<^sup>2)"
+    have th: "\<And>(a::real) b c. a + b + c \<noteq> 0 \<Longrightarrow> a = b + c \<longleftrightarrow> a\<^sup>2 = (b + c)\<^sup>2"
       by algebra
     have "norm (x + y) = norm x + norm y \<longleftrightarrow> (norm (x + y))\<^sup>2 = (norm x + norm y)\<^sup>2"
-      apply (rule th) using n norm_ge_zero[of "x + y"]
+      apply (rule th)
+      using n norm_ge_zero[of "x + y"]
       apply arith
       done
     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)
-    finally have ?thesis .}
+    finally have ?thesis .
+  }
   ultimately show ?thesis by blast
 qed
 
@@ -2700,7 +3141,8 @@
 definition collinear :: "'a::real_vector set \<Rightarrow> bool"
   where "collinear S \<longleftrightarrow> (\<exists>u. \<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *\<^sub>R u)"
 
-lemma collinear_empty:  "collinear {}" by (simp add: collinear_def)
+lemma collinear_empty: "collinear {}"
+  by (simp add: collinear_def)
 
 lemma collinear_sing: "collinear {x}"
   by (simp add: collinear_def)
@@ -2713,14 +3155,20 @@
   apply (rule exI[where x="- 1"], simp)
   done
 
-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")
+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")
 proof -
-  { assume "x=0 \<or> y = 0"
-    then have ?thesis by (cases "x = 0") (simp_all add: collinear_2 insert_commute) }
+  {
+    assume "x = 0 \<or> y = 0"
+    then have ?thesis
+      by (cases "x = 0") (simp_all add: collinear_2 insert_commute)
+  }
   moreover
-  { assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
-    { assume h: "?lhs"
+  {
+    assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
+    have ?thesis
+    proof
+      assume h: "?lhs"
       then obtain u where u: "\<forall> x\<in> {0,x,y}. \<forall>y\<in> {0,x,y}. \<exists>c. x - y = c *\<^sub>R u"
         unfolding collinear_def by blast
       from u[rule_format, of x 0] u[rule_format, of y 0]
@@ -2732,11 +3180,13 @@
       let ?d = "cy / cx"
       from cx cy cx0 have "y = ?d *\<^sub>R x"
         by simp
-      then have ?rhs using x y by blast }
-    moreover
-    { assume h: "?rhs"
-      then obtain c where c: "y = c *\<^sub>R x" using x y by blast
-      have ?lhs unfolding collinear_def c
+      then show ?rhs using x y by blast
+    next
+      assume h: "?rhs"
+      then obtain c where c: "y = c *\<^sub>R x"
+        using x y by blast
+      show ?lhs
+        unfolding collinear_def c
         apply (rule exI[where x=x])
         apply auto
         apply (rule exI[where x="- 1"], simp)
@@ -2744,12 +3194,13 @@
         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 }
-    ultimately have ?thesis by blast }
+        done
+    qed
+  }
   ultimately show ?thesis by blast
 qed
 
-lemma norm_cauchy_schwarz_equal: "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow> collinear {0,x,y}"
+lemma norm_cauchy_schwarz_equal: "abs (x \<bullet> y) = norm x * norm y \<longleftrightarrow> collinear {0, x, y}"
   unfolding norm_cauchy_schwarz_abs_eq
   apply (cases "x=0", simp_all add: collinear_2)
   apply (cases "y=0", simp_all add: collinear_2 insert_commute)
@@ -2773,9 +3224,9 @@
   unfolding scaleR_scaleR
   unfolding norm_scaleR
   apply (subgoal_tac "norm x * c = \<bar>c\<bar> * norm x \<or> norm x * c = - \<bar>c\<bar> * norm x")
-  apply (case_tac "c <= 0", simp add: field_simps)
+  apply (case_tac "c \<le> 0", simp add: field_simps)
   apply (simp add: field_simps)
-  apply (case_tac "c <= 0", simp add: field_simps)
+  apply (case_tac "c \<le> 0", simp add: field_simps)
   apply (simp add: field_simps)
   apply simp
   apply simp
@@ -2801,11 +3252,12 @@
     fast intro: order_trans)
 
 lemma atLeastAtMost_singleton_euclidean[simp]:
-  fixes a :: "'a::ordered_euclidean_space" shows "{a .. a} = {a}"
+  fixes a :: "'a::ordered_euclidean_space"
+  shows "{a .. a} = {a}"
   by (force simp: eucl_le[where 'a='a] euclidean_eq_iff[where 'a='a])
 
 instance real :: ordered_euclidean_space
-  by default (auto simp add: Basis_real_def)
+  by default auto
 
 instantiation prod :: (ordered_euclidean_space, ordered_euclidean_space) ordered_euclidean_space
 begin