src/HOL/Library/Euclidean_Space.thy
 changeset 30041 9becd197a40e parent 30040 e2cd1acda1ab child 30045 b8ddd7667eed
```     1.1 --- a/src/HOL/Library/Euclidean_Space.thy	Sat Feb 21 10:58:25 2009 -0800
1.2 +++ b/src/HOL/Library/Euclidean_Space.thy	Sat Feb 21 11:18:50 2009 -0800
1.3 @@ -729,28 +729,16 @@
1.4  lemma norm_0: "norm (0::real ^ 'n) = 0"
1.5    by (rule norm_zero)
1.6
1.7 -lemma norm_pos_le: "0 <= norm (x::real^'n)"
1.8 -  by (rule norm_ge_zero)
1.9 -lemma norm_neg: " norm(-x) = norm (x:: real ^ 'n)"
1.10 -  by (rule norm_minus_cancel)
1.11 -lemma norm_sub: "norm(x - y) = norm(y - (x::real ^ 'n))"
1.12 -  by (rule norm_minus_commute)
1.13  lemma norm_mul: "norm(a *s x) = abs(a) * norm x"
1.14    by (simp add: vector_norm_def vector_component setL2_right_distrib
1.15             abs_mult cong: strong_setL2_cong)
1.16  lemma norm_eq_0_dot: "(norm x = 0) \<longleftrightarrow> (x \<bullet> x = (0::real))"
1.17    by (simp add: vector_norm_def dot_def setL2_def power2_eq_square)
1.18 -lemma norm_eq_0: "norm x = 0 \<longleftrightarrow> x = (0::real ^ 'n)"
1.19 -  by (rule norm_eq_zero)
1.20 -lemma norm_pos_lt: "0 < norm x \<longleftrightarrow> x \<noteq> (0::real ^ 'n)"
1.21 -  by (rule zero_less_norm_iff)
1.22  lemma real_vector_norm_def: "norm x = sqrt (x \<bullet> x)"
1.23    by (simp add: vector_norm_def setL2_def dot_def power2_eq_square)
1.24  lemma norm_pow_2: "norm x ^ 2 = x \<bullet> x"
1.25    by (simp add: real_vector_norm_def)
1.26 -lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_0)
1.27 -lemma norm_le_0: "norm x <= 0 \<longleftrightarrow> x = (0::real ^'n)"
1.28 -  by (rule norm_le_zero_iff)
1.29 +lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero)
1.30  lemma vector_mul_eq_0: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
1.31    by vector
1.32  lemma vector_mul_lcancel: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"
1.33 @@ -764,14 +752,14 @@
1.34  lemma norm_cauchy_schwarz: "x \<bullet> y <= norm x * norm y"
1.35  proof-
1.36    {assume "norm x = 0"
1.37 -    hence ?thesis by (simp add: norm_eq_0 dot_lzero dot_rzero norm_0)}
1.38 +    hence ?thesis by (simp add: dot_lzero dot_rzero)}
1.39    moreover
1.40    {assume "norm y = 0"
1.41 -    hence ?thesis by (simp add: norm_eq_0 dot_lzero dot_rzero norm_0)}
1.42 +    hence ?thesis by (simp add: dot_lzero dot_rzero)}
1.43    moreover
1.44    {assume h: "norm x \<noteq> 0" "norm y \<noteq> 0"
1.45      let ?z = "norm y *s x - norm x *s y"
1.46 -    from h have p: "norm x * norm y > 0" by (metis norm_pos_le le_less zero_compare_simps)
1.47 +    from h have p: "norm x * norm y > 0" by (metis norm_ge_zero le_less zero_compare_simps)
1.48      from dot_pos_le[of ?z]
1.49      have "(norm x * norm y) * (x \<bullet> y) \<le> norm x ^2 * norm y ^2"
1.50        apply (simp add: dot_rsub dot_lsub dot_lmult dot_rmult ring_simps)
1.51 @@ -782,21 +770,16 @@
1.52    ultimately show ?thesis by metis
1.53  qed
1.54
1.55 -lemma norm_abs: "abs (norm x) = norm (x::real ^'n)"
1.56 -  by (rule abs_norm_cancel) (* already declared [simp] *)
1.57 -
1.58  lemma norm_cauchy_schwarz_abs: "\<bar>x \<bullet> y\<bar> \<le> norm x * norm y"
1.59    using norm_cauchy_schwarz[of x y] norm_cauchy_schwarz[of x "-y"]
1.60 -  by (simp add: real_abs_def dot_rneg norm_neg)
1.61 -lemma norm_triangle: "norm(x + y) <= norm x + norm (y::real ^'n)"
1.62 -  by (rule norm_triangle_ineq)
1.63 +  by (simp add: real_abs_def dot_rneg)
1.64
1.65  lemma norm_triangle_sub: "norm (x::real ^'n) <= norm(y) + norm(x - y)"
1.66 -  using norm_triangle[of "y" "x - y"] by (simp add: ring_simps)
1.67 +  using norm_triangle_ineq[of "y" "x - y"] by (simp add: ring_simps)
1.68  lemma norm_triangle_le: "norm(x::real ^'n) + norm y <= e ==> norm(x + y) <= e"
1.69 -  by (metis order_trans norm_triangle)
1.70 +  by (metis order_trans norm_triangle_ineq)
1.71  lemma norm_triangle_lt: "norm(x::real ^'n) + norm(y) < e ==> norm(x + y) < e"
1.72 -  by (metis basic_trans_rules(21) norm_triangle)
1.73 +  by (metis basic_trans_rules(21) norm_triangle_ineq)
1.74
1.75  lemma setsum_delta:
1.76    assumes fS: "finite S"
1.77 @@ -866,13 +849,13 @@
1.78
1.79  lemma norm_le_square: "norm(x) <= a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x <= a^2"
1.80    apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
1.81 -  using norm_pos_le[of x]
1.82 +  using norm_ge_zero[of x]
1.83    apply arith
1.84    done
1.85
1.86  lemma norm_ge_square: "norm(x) >= a \<longleftrightarrow> a <= 0 \<or> x \<bullet> x >= a ^ 2"
1.87    apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
1.88 -  using norm_pos_le[of x]
1.89 +  using norm_ge_zero[of x]
1.90    apply arith
1.91    done
1.92
1.93 @@ -943,14 +926,14 @@
1.94  lemma pth_d: "x + (0::real ^'n) == x" by (atomize (full)) vector
1.95
1.96  lemma norm_imp_pos_and_ge: "norm (x::real ^ 'n) == n \<Longrightarrow> norm x \<ge> 0 \<and> n \<ge> norm x"
1.97 -  by (atomize) (auto simp add: norm_pos_le)
1.98 +  by (atomize) (auto simp add: norm_ge_zero)
1.99
1.100  lemma real_eq_0_iff_le_ge_0: "(x::real) = 0 == x \<ge> 0 \<and> -x \<ge> 0" by arith
1.101
1.102  lemma norm_pths:
1.103    "(x::real ^'n) = y \<longleftrightarrow> norm (x - y) \<le> 0"
1.104    "x \<noteq> y \<longleftrightarrow> \<not> (norm (x - y) \<le> 0)"
1.105 -  using norm_pos_le[of "x - y"] by (auto simp add: norm_0 norm_eq_0)
1.106 +  using norm_ge_zero[of "x - y"] by auto
1.107
1.108  use "normarith.ML"
1.109
1.110 @@ -1070,7 +1053,7 @@
1.111    assumes fS: "finite S"
1.112    shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
1.113  proof(induct rule: finite_induct[OF fS])
1.114 -  case 1 thus ?case by (simp add: norm_zero)
1.115 +  case 1 thus ?case by simp
1.116  next
1.117    case (2 x S)
1.118    from "2.hyps" have "norm (setsum f (insert x S)) \<le> norm (f x) + norm (setsum f S)" by (simp add: norm_triangle_ineq)
1.119 @@ -1369,7 +1352,7 @@
1.120        by (auto simp add: setsum_component intro: abs_le_D1)
1.121      have Pne: "setsum (\<lambda>x. \<bar>f x \$ i\<bar>) ?Pn \<le> e"
1.122        using i component_le_norm[OF i, of "setsum (\<lambda>x. - f x) ?Pn"]  fPs[OF PnP]
1.123 -      by (auto simp add: setsum_negf norm_neg setsum_component vector_component intro: abs_le_D1)
1.124 +      by (auto simp add: setsum_negf setsum_component vector_component intro: abs_le_D1)
1.125      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"
1.126        apply (subst thp)
1.127        apply (rule setsum_Un_nonzero)
1.128 @@ -1693,7 +1676,7 @@
1.129        unfolding norm_mul
1.130        apply (simp only: mult_commute)
1.131        apply (rule mult_mono)
1.132 -      by (auto simp add: ring_simps norm_pos_le) }
1.133 +      by (auto simp add: ring_simps norm_ge_zero) }
1.134      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
1.135      from real_setsum_norm_le[OF fS, of "\<lambda>i. (x\$i) *s (f (basis i))", OF th]
1.136      have "norm (f x) \<le> ?B * norm x" unfolding th0 setsum_left_distrib by metis}
1.137 @@ -1710,15 +1693,15 @@
1.138    let ?K = "\<bar>B\<bar> + 1"
1.139    have Kp: "?K > 0" by arith
1.140      {assume C: "B < 0"
1.141 -      have "norm (1::real ^ 'n) > 0" by (simp add: norm_pos_lt)
1.142 +      have "norm (1::real ^ 'n) > 0" by (simp add: zero_less_norm_iff)
1.143        with C have "B * norm (1:: real ^ 'n) < 0"
1.144  	by (simp add: zero_compare_simps)
1.145 -      with B[rule_format, of 1] norm_pos_le[of "f 1"] have False by simp
1.146 +      with B[rule_format, of 1] norm_ge_zero[of "f 1"] have False by simp
1.147      }
1.148      then have Bp: "B \<ge> 0" by ferrack
1.149      {fix x::"real ^ 'n"
1.150        have "norm (f x) \<le> ?K *  norm x"
1.151 -      using B[rule_format, of x] norm_pos_le[of x] norm_pos_le[of "f x"] Bp
1.152 +      using B[rule_format, of x] norm_ge_zero[of x] norm_ge_zero[of "f x"] Bp
1.153        apply (auto simp add: ring_simps split add: abs_split)
1.154        apply (erule order_trans, simp)
1.155        done
1.156 @@ -1801,9 +1784,9 @@
1.157        apply simp
1.158        apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh] norm_mul ring_simps)
1.159        apply (rule mult_mono)
1.160 -      apply (auto simp add: norm_pos_le zero_le_mult_iff component_le_norm)
1.161 +      apply (auto simp add: norm_ge_zero zero_le_mult_iff component_le_norm)
1.162        apply (rule mult_mono)
1.163 -      apply (auto simp add: norm_pos_le zero_le_mult_iff component_le_norm)
1.164 +      apply (auto simp add: norm_ge_zero zero_le_mult_iff component_le_norm)
1.165        done}
1.166    then show ?thesis by metis
1.167  qed
1.168 @@ -1823,7 +1806,7 @@
1.169      have "B * norm x * norm y \<le> ?K * norm x * norm y"
1.170        apply -
1.171        apply (rule mult_right_mono, rule mult_right_mono)
1.172 -      by (auto simp add: norm_pos_le)
1.173 +      by (auto simp add: norm_ge_zero)
1.174      then have "norm (h x y) \<le> ?K * norm x * norm y"
1.175        using B[rule_format, of x y] by simp}
1.176    with Kp show ?thesis by blast
1.177 @@ -2436,21 +2419,21 @@
1.178    moreover
1.179    {assume H: ?lhs
1.180      from H[rule_format, of "basis 1"]
1.181 -    have bp: "b \<ge> 0" using norm_pos_le[of "f (basis 1)"] dimindex_ge_1[of "UNIV:: 'n set"]
1.182 +    have bp: "b \<ge> 0" using norm_ge_zero[of "f (basis 1)"] dimindex_ge_1[of "UNIV:: 'n set"]
1.183        by (auto simp add: norm_basis elim: order_trans [OF norm_ge_zero])
1.184      {fix x :: "real ^'n"
1.185        {assume "x = 0"
1.186 -	then have "norm (f x) \<le> b * norm x" by (simp add: linear_0[OF lf] norm_0 bp)}
1.187 +	then have "norm (f x) \<le> b * norm x" by (simp add: linear_0[OF lf] bp)}
1.188        moreover
1.189        {assume x0: "x \<noteq> 0"
1.190 -	hence n0: "norm x \<noteq> 0" by (metis norm_eq_0)
1.191 +	hence n0: "norm x \<noteq> 0" by (metis norm_eq_zero)
1.192  	let ?c = "1/ norm x"
1.193  	have "norm (?c*s x) = 1" using x0 by (simp add: n0 norm_mul)
1.194  	with H have "norm (f(?c*s x)) \<le> b" by blast
1.195  	hence "?c * norm (f x) \<le> b"
1.196  	  by (simp add: linear_cmul[OF lf] norm_mul)
1.197  	hence "norm (f x) \<le> b * norm x"
1.198 -	  using n0 norm_pos_le[of x] by (auto simp add: field_simps)}
1.199 +	  using n0 norm_ge_zero[of x] by (auto simp add: field_simps)}
1.200        ultimately have "norm (f x) \<le> b * norm x" by blast}
1.201      then have ?rhs by blast}
1.202    ultimately show ?thesis by blast
1.203 @@ -2482,12 +2465,12 @@
1.204  qed
1.205
1.206  lemma onorm_pos_le: assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" shows "0 <= onorm f"
1.207 -  using order_trans[OF norm_pos_le onorm(1)[OF lf, of "basis 1"], unfolded norm_basis_1] by simp
1.208 +  using order_trans[OF norm_ge_zero onorm(1)[OF lf, of "basis 1"], unfolded norm_basis_1] by simp
1.209
1.210  lemma onorm_eq_0: assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)"
1.211    shows "onorm f = 0 \<longleftrightarrow> (\<forall>x. f x = 0)"
1.212    using onorm[OF lf]
1.213 -  apply (auto simp add: norm_0 onorm_pos_le norm_le_0)
1.214 +  apply (auto simp add: onorm_pos_le)
1.215    apply atomize
1.216    apply (erule allE[where x="0::real"])
1.217    using onorm_pos_le[OF lf]
1.218 @@ -2525,7 +2508,7 @@
1.219  lemma onorm_neg_lemma: assumes lf: "linear (f::real ^'n \<Rightarrow> real^'m)"
1.220    shows "onorm (\<lambda>x. - f x) \<le> onorm f"
1.221    using onorm[OF linear_compose_neg[OF lf]] onorm[OF lf]
1.222 -  unfolding norm_neg by metis
1.223 +  unfolding norm_minus_cancel by metis
1.224
1.225  lemma onorm_neg: assumes lf: "linear (f::real ^'n \<Rightarrow> real^'m)"
1.226    shows "onorm (\<lambda>x. - f x) = onorm f"
1.227 @@ -2537,7 +2520,7 @@
1.228    shows "onorm (\<lambda>x. f x + g x) <= onorm f + onorm g"
1.229    apply(rule onorm(2)[OF linear_compose_add[OF lf lg], rule_format])
1.230    apply (rule order_trans)
1.231 -  apply (rule norm_triangle)
1.232 +  apply (rule norm_triangle_ineq)
1.233    apply (simp add: distrib)
1.234    apply (rule add_mono)
1.235    apply (rule onorm(1)[OF lf])
1.236 @@ -5175,10 +5158,10 @@
1.237  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")
1.238  proof-
1.239    {assume h: "x = 0"
1.240 -    hence ?thesis by (simp add: norm_0)}
1.241 +    hence ?thesis by simp}
1.242    moreover
1.243    {assume h: "y = 0"
1.244 -    hence ?thesis by (simp add: norm_0)}
1.245 +    hence ?thesis by simp}
1.246    moreover
1.247    {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
1.248      from dot_eq_0[of "norm y *s x - norm x *s y"]
1.249 @@ -5192,7 +5175,7 @@
1.250      also have "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)" using x y
1.251        by (simp add: ring_simps dot_sym)
1.252      also have "\<dots> \<longleftrightarrow> ?lhs" using x y
1.253 -      apply (simp add: norm_eq_0)
1.254 +      apply simp
1.255        by metis
1.256      finally have ?thesis by blast}
1.257    ultimately show ?thesis by blast
1.258 @@ -5203,14 +5186,14 @@
1.259  proof-
1.260    have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> abs x = a \<longleftrightarrow> x = a \<or> x = - a" by arith
1.261    have "?rhs \<longleftrightarrow> norm x *s y = norm y *s x \<or> norm (- x) *s y = norm y *s (- x)"
1.262 -    apply (simp add: norm_neg) by vector
1.263 +    apply simp by vector
1.264    also have "\<dots> \<longleftrightarrow>(x \<bullet> y = norm x * norm y \<or>
1.265       (-x) \<bullet> y = norm x * norm y)"
1.266      unfolding norm_cauchy_schwarz_eq[symmetric]
1.267 -    unfolding norm_neg
1.268 +    unfolding norm_minus_cancel
1.269        norm_mul by blast
1.270    also have "\<dots> \<longleftrightarrow> ?lhs"
1.271 -    unfolding th[OF mult_nonneg_nonneg, OF norm_pos_le[of x] norm_pos_le[of y]] dot_lneg
1.272 +    unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] dot_lneg
1.273      by arith
1.274    finally show ?thesis ..
1.275  qed
1.276 @@ -5218,17 +5201,17 @@
1.277  lemma norm_triangle_eq: "norm(x + y) = norm x + norm y \<longleftrightarrow> norm x *s y = norm y *s x"
1.278  proof-
1.279    {assume x: "x =0 \<or> y =0"
1.280 -    hence ?thesis by (cases "x=0", simp_all add: norm_0)}
1.281 +    hence ?thesis by (cases "x=0", simp_all)}
1.282    moreover
1.283    {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
1.284      hence "norm x \<noteq> 0" "norm y \<noteq> 0"
1.285 -      by (simp_all add: norm_eq_0)
1.286 +      by simp_all
1.287      hence n: "norm x > 0" "norm y > 0"
1.288 -      using norm_pos_le[of x] norm_pos_le[of y]
1.289 +      using norm_ge_zero[of x] norm_ge_zero[of y]
1.290        by arith+
1.291      have th: "\<And>(a::real) b c. a + b + c \<noteq> 0 ==> (a = b + c \<longleftrightarrow> a^2 = (b + c)^2)" by algebra
1.292      have "norm(x + y) = norm x + norm y \<longleftrightarrow> norm(x + y)^ 2 = (norm x + norm y) ^2"
1.293 -      apply (rule th) using n norm_pos_le[of "x + y"]
1.294 +      apply (rule th) using n norm_ge_zero[of "x + y"]
1.295        by arith
1.296      also have "\<dots> \<longleftrightarrow> norm x *s y = norm y *s x"
1.297        unfolding norm_cauchy_schwarz_eq[symmetric]
1.298 @@ -5298,8 +5281,8 @@
1.299
1.300  lemma norm_cauchy_schwarz_equal: "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow> collinear {(0::real^'n),x,y}"
1.301  unfolding norm_cauchy_schwarz_abs_eq
1.302 -apply (cases "x=0", simp_all add: collinear_2 norm_0)
1.303 -apply (cases "y=0", simp_all add: collinear_2 norm_0 insert_commute)
1.304 +apply (cases "x=0", simp_all add: collinear_2)
1.305 +apply (cases "y=0", simp_all add: collinear_2 insert_commute)
1.306  unfolding collinear_lemma
1.307  apply simp
1.308  apply (subgoal_tac "norm x \<noteq> 0")
1.309 @@ -5324,8 +5307,8 @@
1.310  apply (simp add: ring_simps)
1.311  apply (case_tac "c <= 0", simp add: ring_simps)
1.312  apply (simp add: ring_simps)
1.313 -apply (simp add: norm_eq_0)
1.314 -apply (simp add: norm_eq_0)
1.315 +apply simp
1.316 +apply simp
1.317  done
1.318
1.319  end
```