remove duplicated lemmas about norm
authorhuffman
Sat Feb 21 11:18:50 2009 -0800 (2009-02-21)
changeset 300419becd197a40e
parent 30040 e2cd1acda1ab
child 30043 9925ee6a5c59
remove duplicated lemmas about norm
src/HOL/Library/Determinants.thy
src/HOL/Library/Euclidean_Space.thy
     1.1 --- a/src/HOL/Library/Determinants.thy	Sat Feb 21 10:58:25 2009 -0800
     1.2 +++ b/src/HOL/Library/Determinants.thy	Sat Feb 21 11:18:50 2009 -0800
     1.3 @@ -1048,7 +1048,7 @@
     1.4    note th0 = this
     1.5    let ?g = "\<lambda>x. if x = 0 then 0 else norm x *s f (inverse (norm x) *s x)"
     1.6    {fix x:: "real ^'n" assume nx: "norm x = 1"
     1.7 -    have "?g x = f x" using nx by (simp add: norm_eq_0[symmetric])}
     1.8 +    have "?g x = f x" using nx by auto}
     1.9    hence thfg: "\<forall>x. norm x = 1 \<longrightarrow> ?g x = f x" by blast
    1.10    have g0: "?g 0 = 0" by simp
    1.11    {fix x y :: "real ^'n"
    1.12 @@ -1057,15 +1057,15 @@
    1.13      moreover
    1.14      {assume "x = 0" "y \<noteq> 0"
    1.15        then have "dist (?g x) (?g y) = dist x y" 
    1.16 -	apply (simp add: dist_def norm_neg norm_mul norm_eq_0)
    1.17 +	apply (simp add: dist_def norm_mul)
    1.18  	apply (rule f1[rule_format])
    1.19 -	by(simp add: norm_mul norm_eq_0 field_simps)}
    1.20 +	by(simp add: norm_mul field_simps)}
    1.21      moreover
    1.22      {assume "x \<noteq> 0" "y = 0"
    1.23        then have "dist (?g x) (?g y) = dist x y" 
    1.24 -	apply (simp add: dist_def norm_neg norm_mul norm_eq_0)
    1.25 +	apply (simp add: dist_def norm_mul)
    1.26  	apply (rule f1[rule_format])
    1.27 -	by(simp add: norm_mul norm_eq_0 field_simps)}
    1.28 +	by(simp add: norm_mul field_simps)}
    1.29      moreover
    1.30      {assume z: "x \<noteq> 0" "y \<noteq> 0"
    1.31        have th00: "x = norm x *s inverse (norm x) *s x" "y = norm y *s inverse (norm y) *s y" "norm x *s f (inverse (norm x) *s x) = norm x *s f (inverse (norm x) *s x)"
    1.32 @@ -1077,7 +1077,7 @@
    1.33  	"norm (f (inverse (norm x) *s x) - f (inverse (norm y) *s y)) =
    1.34  	norm (inverse (norm x) *s x - inverse (norm y) *s y)"
    1.35  	using z
    1.36 -	by (auto simp add: norm_eq_0 vector_smult_assoc field_simps norm_mul intro: f1[rule_format] fd1[rule_format, unfolded dist_def])
    1.37 +	by (auto simp add: vector_smult_assoc field_simps norm_mul intro: f1[rule_format] fd1[rule_format, unfolded dist_def])
    1.38        from z th0[OF th00] have "dist (?g x) (?g y) = dist x y" 
    1.39  	by (simp add: dist_def)}
    1.40      ultimately have "dist (?g x) (?g y) = dist x y" by blast}
    1.41 @@ -1148,4 +1148,4 @@
    1.42    by (simp add: ring_simps)
    1.43  qed
    1.44  
    1.45 -end
    1.46 \ No newline at end of file
    1.47 +end
     2.1 --- a/src/HOL/Library/Euclidean_Space.thy	Sat Feb 21 10:58:25 2009 -0800
     2.2 +++ b/src/HOL/Library/Euclidean_Space.thy	Sat Feb 21 11:18:50 2009 -0800
     2.3 @@ -729,28 +729,16 @@
     2.4  lemma norm_0: "norm (0::real ^ 'n) = 0"
     2.5    by (rule norm_zero)
     2.6  
     2.7 -lemma norm_pos_le: "0 <= norm (x::real^'n)"
     2.8 -  by (rule norm_ge_zero)
     2.9 -lemma norm_neg: " norm(-x) = norm (x:: real ^ 'n)" 
    2.10 -  by (rule norm_minus_cancel)
    2.11 -lemma norm_sub: "norm(x - y) = norm(y - (x::real ^ 'n))" 
    2.12 -  by (rule norm_minus_commute)
    2.13  lemma norm_mul: "norm(a *s x) = abs(a) * norm x"
    2.14    by (simp add: vector_norm_def vector_component setL2_right_distrib
    2.15             abs_mult cong: strong_setL2_cong)
    2.16  lemma norm_eq_0_dot: "(norm x = 0) \<longleftrightarrow> (x \<bullet> x = (0::real))"
    2.17    by (simp add: vector_norm_def dot_def setL2_def power2_eq_square)
    2.18 -lemma norm_eq_0: "norm x = 0 \<longleftrightarrow> x = (0::real ^ 'n)"
    2.19 -  by (rule norm_eq_zero)
    2.20 -lemma norm_pos_lt: "0 < norm x \<longleftrightarrow> x \<noteq> (0::real ^ 'n)"
    2.21 -  by (rule zero_less_norm_iff)
    2.22  lemma real_vector_norm_def: "norm x = sqrt (x \<bullet> x)"
    2.23    by (simp add: vector_norm_def setL2_def dot_def power2_eq_square)
    2.24  lemma norm_pow_2: "norm x ^ 2 = x \<bullet> x"
    2.25    by (simp add: real_vector_norm_def)
    2.26 -lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_0)
    2.27 -lemma norm_le_0: "norm x <= 0 \<longleftrightarrow> x = (0::real ^'n)"
    2.28 -  by (rule norm_le_zero_iff)
    2.29 +lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero)
    2.30  lemma vector_mul_eq_0: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
    2.31    by vector
    2.32  lemma vector_mul_lcancel: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"
    2.33 @@ -764,14 +752,14 @@
    2.34  lemma norm_cauchy_schwarz: "x \<bullet> y <= norm x * norm y"
    2.35  proof-
    2.36    {assume "norm x = 0"
    2.37 -    hence ?thesis by (simp add: norm_eq_0 dot_lzero dot_rzero norm_0)}
    2.38 +    hence ?thesis by (simp add: dot_lzero dot_rzero)}
    2.39    moreover
    2.40    {assume "norm y = 0" 
    2.41 -    hence ?thesis by (simp add: norm_eq_0 dot_lzero dot_rzero norm_0)}
    2.42 +    hence ?thesis by (simp add: dot_lzero dot_rzero)}
    2.43    moreover
    2.44    {assume h: "norm x \<noteq> 0" "norm y \<noteq> 0"
    2.45      let ?z = "norm y *s x - norm x *s y"
    2.46 -    from h have p: "norm x * norm y > 0" by (metis norm_pos_le le_less zero_compare_simps)
    2.47 +    from h have p: "norm x * norm y > 0" by (metis norm_ge_zero le_less zero_compare_simps)
    2.48      from dot_pos_le[of ?z]
    2.49      have "(norm x * norm y) * (x \<bullet> y) \<le> norm x ^2 * norm y ^2"
    2.50        apply (simp add: dot_rsub dot_lsub dot_lmult dot_rmult ring_simps)
    2.51 @@ -782,21 +770,16 @@
    2.52    ultimately show ?thesis by metis
    2.53  qed
    2.54  
    2.55 -lemma norm_abs: "abs (norm x) = norm (x::real ^'n)"
    2.56 -  by (rule abs_norm_cancel) (* already declared [simp] *)
    2.57 -
    2.58  lemma norm_cauchy_schwarz_abs: "\<bar>x \<bullet> y\<bar> \<le> norm x * norm y"
    2.59    using norm_cauchy_schwarz[of x y] norm_cauchy_schwarz[of x "-y"]
    2.60 -  by (simp add: real_abs_def dot_rneg norm_neg)
    2.61 -lemma norm_triangle: "norm(x + y) <= norm x + norm (y::real ^'n)"
    2.62 -  by (rule norm_triangle_ineq)
    2.63 +  by (simp add: real_abs_def dot_rneg)
    2.64  
    2.65  lemma norm_triangle_sub: "norm (x::real ^'n) <= norm(y) + norm(x - y)"
    2.66 -  using norm_triangle[of "y" "x - y"] by (simp add: ring_simps)
    2.67 +  using norm_triangle_ineq[of "y" "x - y"] by (simp add: ring_simps)
    2.68  lemma norm_triangle_le: "norm(x::real ^'n) + norm y <= e ==> norm(x + y) <= e"
    2.69 -  by (metis order_trans norm_triangle)
    2.70 +  by (metis order_trans norm_triangle_ineq)
    2.71  lemma norm_triangle_lt: "norm(x::real ^'n) + norm(y) < e ==> norm(x + y) < e"
    2.72 -  by (metis basic_trans_rules(21) norm_triangle)
    2.73 +  by (metis basic_trans_rules(21) norm_triangle_ineq)
    2.74  
    2.75  lemma setsum_delta: 
    2.76    assumes fS: "finite S"
    2.77 @@ -866,13 +849,13 @@
    2.78  
    2.79  lemma norm_le_square: "norm(x) <= a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x <= a^2"
    2.80    apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
    2.81 -  using norm_pos_le[of x]
    2.82 +  using norm_ge_zero[of x]
    2.83    apply arith
    2.84    done
    2.85  
    2.86  lemma norm_ge_square: "norm(x) >= a \<longleftrightarrow> a <= 0 \<or> x \<bullet> x >= a ^ 2" 
    2.87    apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
    2.88 -  using norm_pos_le[of x]
    2.89 +  using norm_ge_zero[of x]
    2.90    apply arith
    2.91    done
    2.92  
    2.93 @@ -943,14 +926,14 @@
    2.94  lemma pth_d: "x + (0::real ^'n) == x" by (atomize (full)) vector
    2.95  
    2.96  lemma norm_imp_pos_and_ge: "norm (x::real ^ 'n) == n \<Longrightarrow> norm x \<ge> 0 \<and> n \<ge> norm x"
    2.97 -  by (atomize) (auto simp add: norm_pos_le)
    2.98 +  by (atomize) (auto simp add: norm_ge_zero)
    2.99  
   2.100  lemma real_eq_0_iff_le_ge_0: "(x::real) = 0 == x \<ge> 0 \<and> -x \<ge> 0" by arith
   2.101  
   2.102  lemma norm_pths: 
   2.103    "(x::real ^'n) = y \<longleftrightarrow> norm (x - y) \<le> 0"
   2.104    "x \<noteq> y \<longleftrightarrow> \<not> (norm (x - y) \<le> 0)"
   2.105 -  using norm_pos_le[of "x - y"] by (auto simp add: norm_0 norm_eq_0)
   2.106 +  using norm_ge_zero[of "x - y"] by auto
   2.107  
   2.108  use "normarith.ML"
   2.109  
   2.110 @@ -1070,7 +1053,7 @@
   2.111    assumes fS: "finite S"
   2.112    shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
   2.113  proof(induct rule: finite_induct[OF fS])
   2.114 -  case 1 thus ?case by (simp add: norm_zero)
   2.115 +  case 1 thus ?case by simp
   2.116  next
   2.117    case (2 x S)
   2.118    from "2.hyps" have "norm (setsum f (insert x S)) \<le> norm (f x) + norm (setsum f S)" by (simp add: norm_triangle_ineq)
   2.119 @@ -1369,7 +1352,7 @@
   2.120        by (auto simp add: setsum_component intro: abs_le_D1)
   2.121      have Pne: "setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn \<le> e"
   2.122        using i component_le_norm[OF i, of "setsum (\<lambda>x. - f x) ?Pn"]  fPs[OF PnP]
   2.123 -      by (auto simp add: setsum_negf norm_neg setsum_component vector_component intro: abs_le_D1)
   2.124 +      by (auto simp add: setsum_negf setsum_component vector_component intro: abs_le_D1)
   2.125      have "setsum (\<lambda>x. \<bar>f x $ i\<bar>) P = setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pp + setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn" 
   2.126        apply (subst thp)
   2.127        apply (rule setsum_Un_nonzero) 
   2.128 @@ -1693,7 +1676,7 @@
   2.129        unfolding norm_mul
   2.130        apply (simp only: mult_commute)
   2.131        apply (rule mult_mono)
   2.132 -      by (auto simp add: ring_simps norm_pos_le) }
   2.133 +      by (auto simp add: ring_simps norm_ge_zero) }
   2.134      then have th: "\<forall>i\<in> ?S. norm ((x$i) *s f (basis i :: real ^'m)) \<le> norm (f (basis i)) * norm x" by metis
   2.135      from real_setsum_norm_le[OF fS, of "\<lambda>i. (x$i) *s (f (basis i))", OF th]
   2.136      have "norm (f x) \<le> ?B * norm x" unfolding th0 setsum_left_distrib by metis}
   2.137 @@ -1710,15 +1693,15 @@
   2.138    let ?K = "\<bar>B\<bar> + 1"
   2.139    have Kp: "?K > 0" by arith
   2.140      {assume C: "B < 0"
   2.141 -      have "norm (1::real ^ 'n) > 0" by (simp add: norm_pos_lt)
   2.142 +      have "norm (1::real ^ 'n) > 0" by (simp add: zero_less_norm_iff)
   2.143        with C have "B * norm (1:: real ^ 'n) < 0"
   2.144  	by (simp add: zero_compare_simps)
   2.145 -      with B[rule_format, of 1] norm_pos_le[of "f 1"] have False by simp
   2.146 +      with B[rule_format, of 1] norm_ge_zero[of "f 1"] have False by simp
   2.147      }
   2.148      then have Bp: "B \<ge> 0" by ferrack
   2.149      {fix x::"real ^ 'n"
   2.150        have "norm (f x) \<le> ?K *  norm x"
   2.151 -      using B[rule_format, of x] norm_pos_le[of x] norm_pos_le[of "f x"] Bp
   2.152 +      using B[rule_format, of x] norm_ge_zero[of x] norm_ge_zero[of "f x"] Bp
   2.153        apply (auto simp add: ring_simps split add: abs_split)
   2.154        apply (erule order_trans, simp)
   2.155        done
   2.156 @@ -1801,9 +1784,9 @@
   2.157        apply simp
   2.158        apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh] norm_mul ring_simps)
   2.159        apply (rule mult_mono)
   2.160 -      apply (auto simp add: norm_pos_le zero_le_mult_iff component_le_norm)
   2.161 +      apply (auto simp add: norm_ge_zero zero_le_mult_iff component_le_norm)
   2.162        apply (rule mult_mono)
   2.163 -      apply (auto simp add: norm_pos_le zero_le_mult_iff component_le_norm)
   2.164 +      apply (auto simp add: norm_ge_zero zero_le_mult_iff component_le_norm)
   2.165        done}
   2.166    then show ?thesis by metis
   2.167  qed
   2.168 @@ -1823,7 +1806,7 @@
   2.169      have "B * norm x * norm y \<le> ?K * norm x * norm y"
   2.170        apply - 
   2.171        apply (rule mult_right_mono, rule mult_right_mono)
   2.172 -      by (auto simp add: norm_pos_le)
   2.173 +      by (auto simp add: norm_ge_zero)
   2.174      then have "norm (h x y) \<le> ?K * norm x * norm y"
   2.175        using B[rule_format, of x y] by simp} 
   2.176    with Kp show ?thesis by blast
   2.177 @@ -2436,21 +2419,21 @@
   2.178    moreover
   2.179    {assume H: ?lhs
   2.180      from H[rule_format, of "basis 1"] 
   2.181 -    have bp: "b \<ge> 0" using norm_pos_le[of "f (basis 1)"] dimindex_ge_1[of "UNIV:: 'n set"]
   2.182 +    have bp: "b \<ge> 0" using norm_ge_zero[of "f (basis 1)"] dimindex_ge_1[of "UNIV:: 'n set"]
   2.183        by (auto simp add: norm_basis elim: order_trans [OF norm_ge_zero])
   2.184      {fix x :: "real ^'n"
   2.185        {assume "x = 0"
   2.186 -	then have "norm (f x) \<le> b * norm x" by (simp add: linear_0[OF lf] norm_0 bp)}
   2.187 +	then have "norm (f x) \<le> b * norm x" by (simp add: linear_0[OF lf] bp)}
   2.188        moreover
   2.189        {assume x0: "x \<noteq> 0"
   2.190 -	hence n0: "norm x \<noteq> 0" by (metis norm_eq_0)
   2.191 +	hence n0: "norm x \<noteq> 0" by (metis norm_eq_zero)
   2.192  	let ?c = "1/ norm x"
   2.193  	have "norm (?c*s x) = 1" using x0 by (simp add: n0 norm_mul)
   2.194  	with H have "norm (f(?c*s x)) \<le> b" by blast
   2.195  	hence "?c * norm (f x) \<le> b" 
   2.196  	  by (simp add: linear_cmul[OF lf] norm_mul)
   2.197  	hence "norm (f x) \<le> b * norm x" 
   2.198 -	  using n0 norm_pos_le[of x] by (auto simp add: field_simps)}
   2.199 +	  using n0 norm_ge_zero[of x] by (auto simp add: field_simps)}
   2.200        ultimately have "norm (f x) \<le> b * norm x" by blast}
   2.201      then have ?rhs by blast}
   2.202    ultimately show ?thesis by blast
   2.203 @@ -2482,12 +2465,12 @@
   2.204  qed
   2.205  
   2.206  lemma onorm_pos_le: assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" shows "0 <= onorm f"
   2.207 -  using order_trans[OF norm_pos_le onorm(1)[OF lf, of "basis 1"], unfolded norm_basis_1] by simp
   2.208 +  using order_trans[OF norm_ge_zero onorm(1)[OF lf, of "basis 1"], unfolded norm_basis_1] by simp
   2.209  
   2.210  lemma onorm_eq_0: assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" 
   2.211    shows "onorm f = 0 \<longleftrightarrow> (\<forall>x. f x = 0)"
   2.212    using onorm[OF lf]
   2.213 -  apply (auto simp add: norm_0 onorm_pos_le norm_le_0)
   2.214 +  apply (auto simp add: onorm_pos_le)
   2.215    apply atomize
   2.216    apply (erule allE[where x="0::real"])
   2.217    using onorm_pos_le[OF lf]
   2.218 @@ -2525,7 +2508,7 @@
   2.219  lemma onorm_neg_lemma: assumes lf: "linear (f::real ^'n \<Rightarrow> real^'m)"
   2.220    shows "onorm (\<lambda>x. - f x) \<le> onorm f"
   2.221    using onorm[OF linear_compose_neg[OF lf]] onorm[OF lf]
   2.222 -  unfolding norm_neg by metis
   2.223 +  unfolding norm_minus_cancel by metis
   2.224  
   2.225  lemma onorm_neg: assumes lf: "linear (f::real ^'n \<Rightarrow> real^'m)"
   2.226    shows "onorm (\<lambda>x. - f x) = onorm f"
   2.227 @@ -2537,7 +2520,7 @@
   2.228    shows "onorm (\<lambda>x. f x + g x) <= onorm f + onorm g"
   2.229    apply(rule onorm(2)[OF linear_compose_add[OF lf lg], rule_format])
   2.230    apply (rule order_trans)
   2.231 -  apply (rule norm_triangle)
   2.232 +  apply (rule norm_triangle_ineq)
   2.233    apply (simp add: distrib)
   2.234    apply (rule add_mono)
   2.235    apply (rule onorm(1)[OF lf])
   2.236 @@ -5175,10 +5158,10 @@
   2.237  lemma norm_cauchy_schwarz_eq: "(x::real ^'n) \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *s y = norm y *s x" (is "?lhs \<longleftrightarrow> ?rhs")
   2.238  proof-
   2.239    {assume h: "x = 0"
   2.240 -    hence ?thesis by (simp add: norm_0)}
   2.241 +    hence ?thesis by simp}
   2.242    moreover
   2.243    {assume h: "y = 0"
   2.244 -    hence ?thesis by (simp add: norm_0)}
   2.245 +    hence ?thesis by simp}
   2.246    moreover
   2.247    {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
   2.248      from dot_eq_0[of "norm y *s x - norm x *s y"]
   2.249 @@ -5192,7 +5175,7 @@
   2.250      also have "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)" using x y
   2.251        by (simp add: ring_simps dot_sym)
   2.252      also have "\<dots> \<longleftrightarrow> ?lhs" using x y
   2.253 -      apply (simp add: norm_eq_0)
   2.254 +      apply simp
   2.255        by metis
   2.256      finally have ?thesis by blast}
   2.257    ultimately show ?thesis by blast
   2.258 @@ -5203,14 +5186,14 @@
   2.259  proof-
   2.260    have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> abs x = a \<longleftrightarrow> x = a \<or> x = - a" by arith
   2.261    have "?rhs \<longleftrightarrow> norm x *s y = norm y *s x \<or> norm (- x) *s y = norm y *s (- x)"
   2.262 -    apply (simp add: norm_neg) by vector
   2.263 +    apply simp by vector
   2.264    also have "\<dots> \<longleftrightarrow>(x \<bullet> y = norm x * norm y \<or>
   2.265       (-x) \<bullet> y = norm x * norm y)"
   2.266      unfolding norm_cauchy_schwarz_eq[symmetric]
   2.267 -    unfolding norm_neg
   2.268 +    unfolding norm_minus_cancel
   2.269        norm_mul by blast
   2.270    also have "\<dots> \<longleftrightarrow> ?lhs"
   2.271 -    unfolding th[OF mult_nonneg_nonneg, OF norm_pos_le[of x] norm_pos_le[of y]] dot_lneg
   2.272 +    unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] dot_lneg
   2.273      by arith
   2.274    finally show ?thesis ..
   2.275  qed
   2.276 @@ -5218,17 +5201,17 @@
   2.277  lemma norm_triangle_eq: "norm(x + y) = norm x + norm y \<longleftrightarrow> norm x *s y = norm y *s x"
   2.278  proof-
   2.279    {assume x: "x =0 \<or> y =0"
   2.280 -    hence ?thesis by (cases "x=0", simp_all add: norm_0)}
   2.281 +    hence ?thesis by (cases "x=0", simp_all)}
   2.282    moreover
   2.283    {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
   2.284      hence "norm x \<noteq> 0" "norm y \<noteq> 0"
   2.285 -      by (simp_all add: norm_eq_0)
   2.286 +      by simp_all
   2.287      hence n: "norm x > 0" "norm y > 0" 
   2.288 -      using norm_pos_le[of x] norm_pos_le[of y]
   2.289 +      using norm_ge_zero[of x] norm_ge_zero[of y]
   2.290        by arith+
   2.291      have th: "\<And>(a::real) b c. a + b + c \<noteq> 0 ==> (a = b + c \<longleftrightarrow> a^2 = (b + c)^2)" by algebra
   2.292      have "norm(x + y) = norm x + norm y \<longleftrightarrow> norm(x + y)^ 2 = (norm x + norm y) ^2"
   2.293 -      apply (rule th) using n norm_pos_le[of "x + y"]
   2.294 +      apply (rule th) using n norm_ge_zero[of "x + y"]
   2.295        by arith
   2.296      also have "\<dots> \<longleftrightarrow> norm x *s y = norm y *s x"
   2.297        unfolding norm_cauchy_schwarz_eq[symmetric]
   2.298 @@ -5298,8 +5281,8 @@
   2.299  
   2.300  lemma norm_cauchy_schwarz_equal: "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow> collinear {(0::real^'n),x,y}"
   2.301  unfolding norm_cauchy_schwarz_abs_eq
   2.302 -apply (cases "x=0", simp_all add: collinear_2 norm_0)
   2.303 -apply (cases "y=0", simp_all add: collinear_2 norm_0 insert_commute)
   2.304 +apply (cases "x=0", simp_all add: collinear_2)
   2.305 +apply (cases "y=0", simp_all add: collinear_2 insert_commute)
   2.306  unfolding collinear_lemma
   2.307  apply simp
   2.308  apply (subgoal_tac "norm x \<noteq> 0")
   2.309 @@ -5324,8 +5307,8 @@
   2.310  apply (simp add: ring_simps)
   2.311  apply (case_tac "c <= 0", simp add: ring_simps)
   2.312  apply (simp add: ring_simps)
   2.313 -apply (simp add: norm_eq_0)
   2.314 -apply (simp add: norm_eq_0)
   2.315 +apply simp
   2.316 +apply simp
   2.317  done
   2.318  
   2.319  end