src/HOL/Library/Inner_Product.thy
author huffman
Fri, 12 Jun 2009 16:23:07 -0700
changeset 31590 776d6a4c1327
parent 31492 5400beeddb55
child 39198 f967a16dfcdd
permissions -rw-r--r--
declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
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(* Title:      Inner_Product.thy
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   Author:     Brian Huffman
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*)
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header {* Inner Product Spaces and the Gradient Derivative *}
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theory Inner_Product
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imports Complex_Main FrechetDeriv
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begin
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subsection {* Real inner product spaces *}
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text {*
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  Temporarily relax type constraints for @{term "open"},
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  @{term dist}, and @{term norm}.
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*}
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setup {* Sign.add_const_constraint
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  (@{const_name "open"}, SOME @{typ "'a::open set \<Rightarrow> bool"}) *}
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setup {* Sign.add_const_constraint
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  (@{const_name dist}, SOME @{typ "'a::dist \<Rightarrow> 'a \<Rightarrow> real"}) *}
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setup {* Sign.add_const_constraint
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  (@{const_name norm}, SOME @{typ "'a::norm \<Rightarrow> real"}) *}
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class real_inner = real_vector + sgn_div_norm + dist_norm + open_dist +
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  fixes inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"
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  assumes inner_commute: "inner x y = inner y x"
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  and inner_add_left: "inner (x + y) z = inner x z + inner y z"
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  and inner_scaleR_left [simp]: "inner (scaleR r x) y = r * (inner x y)"
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  and inner_ge_zero [simp]: "0 \<le> inner x x"
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  and inner_eq_zero_iff [simp]: "inner x x = 0 \<longleftrightarrow> x = 0"
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  and norm_eq_sqrt_inner: "norm x = sqrt (inner x x)"
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begin
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lemma inner_zero_left [simp]: "inner 0 x = 0"
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  using inner_add_left [of 0 0 x] by simp
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lemma inner_minus_left [simp]: "inner (- x) y = - inner x y"
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  using inner_add_left [of x "- x" y] by simp
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lemma inner_diff_left: "inner (x - y) z = inner x z - inner y z"
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  by (simp add: diff_minus inner_add_left)
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text {* Transfer distributivity rules to right argument. *}
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lemma inner_add_right: "inner x (y + z) = inner x y + inner x z"
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  using inner_add_left [of y z x] by (simp only: inner_commute)
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lemma inner_scaleR_right [simp]: "inner x (scaleR r y) = r * (inner x y)"
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  using inner_scaleR_left [of r y x] by (simp only: inner_commute)
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lemma inner_zero_right [simp]: "inner x 0 = 0"
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  using inner_zero_left [of x] by (simp only: inner_commute)
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lemma inner_minus_right [simp]: "inner x (- y) = - inner x y"
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  using inner_minus_left [of y x] by (simp only: inner_commute)
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lemma inner_diff_right: "inner x (y - z) = inner x y - inner x z"
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  using inner_diff_left [of y z x] by (simp only: inner_commute)
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lemmas inner_add [algebra_simps] = inner_add_left inner_add_right
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lemmas inner_diff [algebra_simps]  = inner_diff_left inner_diff_right
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lemmas inner_scaleR = inner_scaleR_left inner_scaleR_right
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text {* Legacy theorem names *}
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lemmas inner_left_distrib = inner_add_left
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lemmas inner_right_distrib = inner_add_right
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lemmas inner_distrib = inner_left_distrib inner_right_distrib
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lemma inner_gt_zero_iff [simp]: "0 < inner x x \<longleftrightarrow> x \<noteq> 0"
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  by (simp add: order_less_le)
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lemma power2_norm_eq_inner: "(norm x)\<twosuperior> = inner x x"
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  by (simp add: norm_eq_sqrt_inner)
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lemma Cauchy_Schwarz_ineq:
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  "(inner x y)\<twosuperior> \<le> inner x x * inner y y"
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proof (cases)
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  assume "y = 0"
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  thus ?thesis by simp
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next
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  assume y: "y \<noteq> 0"
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  let ?r = "inner x y / inner y y"
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  have "0 \<le> inner (x - scaleR ?r y) (x - scaleR ?r y)"
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    by (rule inner_ge_zero)
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  also have "\<dots> = inner x x - inner y x * ?r"
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    by (simp add: inner_diff)
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  also have "\<dots> = inner x x - (inner x y)\<twosuperior> / inner y y"
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    by (simp add: power2_eq_square inner_commute)
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  finally have "0 \<le> inner x x - (inner x y)\<twosuperior> / inner y y" .
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  hence "(inner x y)\<twosuperior> / inner y y \<le> inner x x"
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    by (simp add: le_diff_eq)
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  thus "(inner x y)\<twosuperior> \<le> inner x x * inner y y"
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    by (simp add: pos_divide_le_eq y)
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qed
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lemma Cauchy_Schwarz_ineq2:
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  "\<bar>inner x y\<bar> \<le> norm x * norm y"
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proof (rule power2_le_imp_le)
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  have "(inner x y)\<twosuperior> \<le> inner x x * inner y y"
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    using Cauchy_Schwarz_ineq .
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  thus "\<bar>inner x y\<bar>\<twosuperior> \<le> (norm x * norm y)\<twosuperior>"
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    by (simp add: power_mult_distrib power2_norm_eq_inner)
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  show "0 \<le> norm x * norm y"
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    unfolding norm_eq_sqrt_inner
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    by (intro mult_nonneg_nonneg real_sqrt_ge_zero inner_ge_zero)
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qed
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subclass real_normed_vector
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proof
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  fix a :: real and x y :: 'a
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  show "0 \<le> norm x"
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    unfolding norm_eq_sqrt_inner by simp
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  show "norm x = 0 \<longleftrightarrow> x = 0"
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    unfolding norm_eq_sqrt_inner by simp
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  show "norm (x + y) \<le> norm x + norm y"
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    proof (rule power2_le_imp_le)
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      have "inner x y \<le> norm x * norm y"
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        by (rule order_trans [OF abs_ge_self Cauchy_Schwarz_ineq2])
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      thus "(norm (x + y))\<twosuperior> \<le> (norm x + norm y)\<twosuperior>"
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        unfolding power2_sum power2_norm_eq_inner
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        by (simp add: inner_add inner_commute)
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      show "0 \<le> norm x + norm y"
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        unfolding norm_eq_sqrt_inner
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        by (simp add: add_nonneg_nonneg)
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    qed
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  have "sqrt (a\<twosuperior> * inner x x) = \<bar>a\<bar> * sqrt (inner x x)"
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    by (simp add: real_sqrt_mult_distrib)
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  then show "norm (a *\<^sub>R x) = \<bar>a\<bar> * norm x"
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    unfolding norm_eq_sqrt_inner
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    by (simp add: power2_eq_square mult_assoc)
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qed
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end
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text {*
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  Re-enable constraints for @{term "open"},
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  @{term dist}, and @{term norm}.
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*}
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setup {* Sign.add_const_constraint
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  (@{const_name "open"}, SOME @{typ "'a::topological_space set \<Rightarrow> bool"}) *}
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setup {* Sign.add_const_constraint
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  (@{const_name dist}, SOME @{typ "'a::metric_space \<Rightarrow> 'a \<Rightarrow> real"}) *}
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setup {* Sign.add_const_constraint
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  (@{const_name norm}, SOME @{typ "'a::real_normed_vector \<Rightarrow> real"}) *}
2d91b2416de8 add extra type constraints for dist, norm
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30729
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interpretation inner:
29993
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   153
  bounded_bilinear "inner::'a::real_inner \<Rightarrow> 'a \<Rightarrow> real"
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   154
proof
84b2c432b94a new theory of real inner product spaces
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   155
  fix x y z :: 'a and r :: real
84b2c432b94a new theory of real inner product spaces
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   156
  show "inner (x + y) z = inner x z + inner y z"
31590
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   157
    by (rule inner_add_left)
29993
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parents:
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   158
  show "inner x (y + z) = inner x y + inner x z"
31590
776d6a4c1327 declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
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parents: 31492
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   159
    by (rule inner_add_right)
29993
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   160
  show "inner (scaleR r x) y = scaleR r (inner x y)"
84b2c432b94a new theory of real inner product spaces
huffman
parents:
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   161
    unfolding real_scaleR_def by (rule inner_scaleR_left)
84b2c432b94a new theory of real inner product spaces
huffman
parents:
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   162
  show "inner x (scaleR r y) = scaleR r (inner x y)"
84b2c432b94a new theory of real inner product spaces
huffman
parents:
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   163
    unfolding real_scaleR_def by (rule inner_scaleR_right)
84b2c432b94a new theory of real inner product spaces
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parents:
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   164
  show "\<exists>K. \<forall>x y::'a. norm (inner x y) \<le> norm x * norm y * K"
84b2c432b94a new theory of real inner product spaces
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parents:
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   165
  proof
84b2c432b94a new theory of real inner product spaces
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   166
    show "\<forall>x y::'a. norm (inner x y) \<le> norm x * norm y * 1"
30046
49f603f92c47 fix spelling
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   167
      by (simp add: Cauchy_Schwarz_ineq2)
29993
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  qed
84b2c432b94a new theory of real inner product spaces
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   169
qed
84b2c432b94a new theory of real inner product spaces
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   170
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   171
interpretation inner_left:
29993
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   172
  bounded_linear "\<lambda>x::'a::real_inner. inner x y"
84b2c432b94a new theory of real inner product spaces
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   173
  by (rule inner.bounded_linear_left)
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   174
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   175
interpretation inner_right:
29993
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   176
  bounded_linear "\<lambda>y::'a::real_inner. inner x y"
84b2c432b94a new theory of real inner product spaces
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   177
  by (rule inner.bounded_linear_right)
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   178
84b2c432b94a new theory of real inner product spaces
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   179
84b2c432b94a new theory of real inner product spaces
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   180
subsection {* Class instances *}
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84b2c432b94a new theory of real inner product spaces
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instantiation real :: real_inner
84b2c432b94a new theory of real inner product spaces
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   183
begin
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   184
84b2c432b94a new theory of real inner product spaces
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   185
definition inner_real_def [simp]: "inner = op *"
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   186
84b2c432b94a new theory of real inner product spaces
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instance proof
84b2c432b94a new theory of real inner product spaces
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   188
  fix x y z r :: real
84b2c432b94a new theory of real inner product spaces
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parents:
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   189
  show "inner x y = inner y x"
84b2c432b94a new theory of real inner product spaces
huffman
parents:
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   190
    unfolding inner_real_def by (rule mult_commute)
84b2c432b94a new theory of real inner product spaces
huffman
parents:
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   191
  show "inner (x + y) z = inner x z + inner y z"
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   192
    unfolding inner_real_def by (rule left_distrib)
84b2c432b94a new theory of real inner product spaces
huffman
parents:
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   193
  show "inner (scaleR r x) y = r * inner x y"
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   194
    unfolding inner_real_def real_scaleR_def by (rule mult_assoc)
84b2c432b94a new theory of real inner product spaces
huffman
parents:
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   195
  show "0 \<le> inner x x"
84b2c432b94a new theory of real inner product spaces
huffman
parents:
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   196
    unfolding inner_real_def by simp
84b2c432b94a new theory of real inner product spaces
huffman
parents:
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   197
  show "inner x x = 0 \<longleftrightarrow> x = 0"
84b2c432b94a new theory of real inner product spaces
huffman
parents:
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   198
    unfolding inner_real_def by simp
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   199
  show "norm x = sqrt (inner x x)"
84b2c432b94a new theory of real inner product spaces
huffman
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   200
    unfolding inner_real_def by simp
84b2c432b94a new theory of real inner product spaces
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   201
qed
84b2c432b94a new theory of real inner product spaces
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   202
84b2c432b94a new theory of real inner product spaces
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   203
end
84b2c432b94a new theory of real inner product spaces
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   204
84b2c432b94a new theory of real inner product spaces
huffman
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   205
instantiation complex :: real_inner
84b2c432b94a new theory of real inner product spaces
huffman
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   206
begin
84b2c432b94a new theory of real inner product spaces
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   207
84b2c432b94a new theory of real inner product spaces
huffman
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   208
definition inner_complex_def:
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   209
  "inner x y = Re x * Re y + Im x * Im y"
84b2c432b94a new theory of real inner product spaces
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   210
84b2c432b94a new theory of real inner product spaces
huffman
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   211
instance proof
84b2c432b94a new theory of real inner product spaces
huffman
parents:
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   212
  fix x y z :: complex and r :: real
84b2c432b94a new theory of real inner product spaces
huffman
parents:
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   213
  show "inner x y = inner y x"
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   214
    unfolding inner_complex_def by (simp add: mult_commute)
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   215
  show "inner (x + y) z = inner x z + inner y z"
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   216
    unfolding inner_complex_def by (simp add: left_distrib)
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   217
  show "inner (scaleR r x) y = r * inner x y"
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   218
    unfolding inner_complex_def by (simp add: right_distrib)
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   219
  show "0 \<le> inner x x"
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   220
    unfolding inner_complex_def by (simp add: add_nonneg_nonneg)
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   221
  show "inner x x = 0 \<longleftrightarrow> x = 0"
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   222
    unfolding inner_complex_def
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   223
    by (simp add: add_nonneg_eq_0_iff complex_Re_Im_cancel_iff)
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   224
  show "norm x = sqrt (inner x x)"
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   225
    unfolding inner_complex_def complex_norm_def
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   226
    by (simp add: power2_eq_square)
84b2c432b94a new theory of real inner product spaces
huffman
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   227
qed
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   228
84b2c432b94a new theory of real inner product spaces
huffman
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   229
end
84b2c432b94a new theory of real inner product spaces
huffman
parents:
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   230
84b2c432b94a new theory of real inner product spaces
huffman
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diff changeset
   231
84b2c432b94a new theory of real inner product spaces
huffman
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diff changeset
   232
subsection {* Gradient derivative *}
84b2c432b94a new theory of real inner product spaces
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   233
84b2c432b94a new theory of real inner product spaces
huffman
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   234
definition
84b2c432b94a new theory of real inner product spaces
huffman
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   235
  gderiv ::
84b2c432b94a new theory of real inner product spaces
huffman
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   236
    "['a::real_inner \<Rightarrow> real, 'a, 'a] \<Rightarrow> bool"
84b2c432b94a new theory of real inner product spaces
huffman
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   237
          ("(GDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
84b2c432b94a new theory of real inner product spaces
huffman
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   238
where
84b2c432b94a new theory of real inner product spaces
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   239
  "GDERIV f x :> D \<longleftrightarrow> FDERIV f x :> (\<lambda>h. inner h D)"
84b2c432b94a new theory of real inner product spaces
huffman
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diff changeset
   240
84b2c432b94a new theory of real inner product spaces
huffman
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   241
lemma deriv_fderiv: "DERIV f x :> D \<longleftrightarrow> FDERIV f x :> (\<lambda>h. h * D)"
84b2c432b94a new theory of real inner product spaces
huffman
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   242
  by (simp only: deriv_def field_fderiv_def)
84b2c432b94a new theory of real inner product spaces
huffman
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   243
84b2c432b94a new theory of real inner product spaces
huffman
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   244
lemma gderiv_deriv [simp]: "GDERIV f x :> D \<longleftrightarrow> DERIV f x :> D"
84b2c432b94a new theory of real inner product spaces
huffman
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   245
  by (simp only: gderiv_def deriv_fderiv inner_real_def)
84b2c432b94a new theory of real inner product spaces
huffman
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diff changeset
   246
84b2c432b94a new theory of real inner product spaces
huffman
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   247
lemma GDERIV_DERIV_compose:
84b2c432b94a new theory of real inner product spaces
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   248
    "\<lbrakk>GDERIV f x :> df; DERIV g (f x) :> dg\<rbrakk>
84b2c432b94a new theory of real inner product spaces
huffman
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   249
     \<Longrightarrow> GDERIV (\<lambda>x. g (f x)) x :> scaleR dg df"
84b2c432b94a new theory of real inner product spaces
huffman
parents:
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   250
  unfolding gderiv_def deriv_fderiv
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   251
  apply (drule (1) FDERIV_compose)
31590
776d6a4c1327 declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
huffman
parents: 31492
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   252
  apply (simp add: mult_ac)
29993
84b2c432b94a new theory of real inner product spaces
huffman
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   253
  done
84b2c432b94a new theory of real inner product spaces
huffman
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diff changeset
   254
84b2c432b94a new theory of real inner product spaces
huffman
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   255
lemma FDERIV_subst: "\<lbrakk>FDERIV f x :> df; df = d\<rbrakk> \<Longrightarrow> FDERIV f x :> d"
84b2c432b94a new theory of real inner product spaces
huffman
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   256
  by simp
84b2c432b94a new theory of real inner product spaces
huffman
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diff changeset
   257
84b2c432b94a new theory of real inner product spaces
huffman
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   258
lemma GDERIV_subst: "\<lbrakk>GDERIV f x :> df; df = d\<rbrakk> \<Longrightarrow> GDERIV f x :> d"
84b2c432b94a new theory of real inner product spaces
huffman
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   259
  by simp
84b2c432b94a new theory of real inner product spaces
huffman
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diff changeset
   260
84b2c432b94a new theory of real inner product spaces
huffman
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   261
lemma GDERIV_const: "GDERIV (\<lambda>x. k) x :> 0"
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   262
  unfolding gderiv_def inner_right.zero by (rule FDERIV_const)
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   263
84b2c432b94a new theory of real inner product spaces
huffman
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diff changeset
   264
lemma GDERIV_add:
84b2c432b94a new theory of real inner product spaces
huffman
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   265
    "\<lbrakk>GDERIV f x :> df; GDERIV g x :> dg\<rbrakk>
84b2c432b94a new theory of real inner product spaces
huffman
parents:
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   266
     \<Longrightarrow> GDERIV (\<lambda>x. f x + g x) x :> df + dg"
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   267
  unfolding gderiv_def inner_right.add by (rule FDERIV_add)
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   268
84b2c432b94a new theory of real inner product spaces
huffman
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   269
lemma GDERIV_minus:
84b2c432b94a new theory of real inner product spaces
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   270
    "GDERIV f x :> df \<Longrightarrow> GDERIV (\<lambda>x. - f x) x :> - df"
84b2c432b94a new theory of real inner product spaces
huffman
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diff changeset
   271
  unfolding gderiv_def inner_right.minus by (rule FDERIV_minus)
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   272
84b2c432b94a new theory of real inner product spaces
huffman
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diff changeset
   273
lemma GDERIV_diff:
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huffman
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   274
    "\<lbrakk>GDERIV f x :> df; GDERIV g x :> dg\<rbrakk>
84b2c432b94a new theory of real inner product spaces
huffman
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   275
     \<Longrightarrow> GDERIV (\<lambda>x. f x - g x) x :> df - dg"
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   276
  unfolding gderiv_def inner_right.diff by (rule FDERIV_diff)
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   277
84b2c432b94a new theory of real inner product spaces
huffman
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diff changeset
   278
lemma GDERIV_scaleR:
84b2c432b94a new theory of real inner product spaces
huffman
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   279
    "\<lbrakk>DERIV f x :> df; GDERIV g x :> dg\<rbrakk>
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huffman
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diff changeset
   280
     \<Longrightarrow> GDERIV (\<lambda>x. scaleR (f x) (g x)) x
84b2c432b94a new theory of real inner product spaces
huffman
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   281
      :> (scaleR (f x) dg + scaleR df (g x))"
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   282
  unfolding gderiv_def deriv_fderiv inner_right.add inner_right.scaleR
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   283
  apply (rule FDERIV_subst)
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   284
  apply (erule (1) scaleR.FDERIV)
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   285
  apply (simp add: mult_ac)
84b2c432b94a new theory of real inner product spaces
huffman
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diff changeset
   286
  done
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   287
84b2c432b94a new theory of real inner product spaces
huffman
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   288
lemma GDERIV_mult:
84b2c432b94a new theory of real inner product spaces
huffman
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   289
    "\<lbrakk>GDERIV f x :> df; GDERIV g x :> dg\<rbrakk>
84b2c432b94a new theory of real inner product spaces
huffman
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diff changeset
   290
     \<Longrightarrow> GDERIV (\<lambda>x. f x * g x) x :> scaleR (f x) dg + scaleR (g x) df"
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   291
  unfolding gderiv_def
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   292
  apply (rule FDERIV_subst)
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   293
  apply (erule (1) FDERIV_mult)
31590
776d6a4c1327 declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
huffman
parents: 31492
diff changeset
   294
  apply (simp add: inner_add mult_ac)
29993
84b2c432b94a new theory of real inner product spaces
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diff changeset
   295
  done
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   296
84b2c432b94a new theory of real inner product spaces
huffman
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diff changeset
   297
lemma GDERIV_inverse:
84b2c432b94a new theory of real inner product spaces
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diff changeset
   298
    "\<lbrakk>GDERIV f x :> df; f x \<noteq> 0\<rbrakk>
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   299
     \<Longrightarrow> GDERIV (\<lambda>x. inverse (f x)) x :> - (inverse (f x))\<twosuperior> *\<^sub>R df"
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   300
  apply (erule GDERIV_DERIV_compose)
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   301
  apply (erule DERIV_inverse [folded numeral_2_eq_2])
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   302
  done
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   303
84b2c432b94a new theory of real inner product spaces
huffman
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diff changeset
   304
lemma GDERIV_norm:
84b2c432b94a new theory of real inner product spaces
huffman
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diff changeset
   305
  assumes "x \<noteq> 0" shows "GDERIV (\<lambda>x. norm x) x :> sgn x"
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   306
proof -
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   307
  have 1: "FDERIV (\<lambda>x. inner x x) x :> (\<lambda>h. inner x h + inner h x)"
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   308
    by (intro inner.FDERIV FDERIV_ident)
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   309
  have 2: "(\<lambda>h. inner x h + inner h x) = (\<lambda>h. inner h (scaleR 2 x))"
31590
776d6a4c1327 declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
huffman
parents: 31492
diff changeset
   310
    by (simp add: expand_fun_eq inner_commute)
29993
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   311
  have "0 < inner x x" using `x \<noteq> 0` by simp
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   312
  then have 3: "DERIV sqrt (inner x x) :> (inverse (sqrt (inner x x)) / 2)"
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   313
    by (rule DERIV_real_sqrt)
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   314
  have 4: "(inverse (sqrt (inner x x)) / 2) *\<^sub>R 2 *\<^sub>R x = sgn x"
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   315
    by (simp add: sgn_div_norm norm_eq_sqrt_inner)
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   316
  show ?thesis
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   317
    unfolding norm_eq_sqrt_inner
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   318
    apply (rule GDERIV_subst [OF _ 4])
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   319
    apply (rule GDERIV_DERIV_compose [where g=sqrt and df="scaleR 2 x"])
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   320
    apply (subst gderiv_def)
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   321
    apply (rule FDERIV_subst [OF _ 2])
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   322
    apply (rule 1)
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   323
    apply (rule 3)
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   324
    done
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   325
qed
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   326
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   327
lemmas FDERIV_norm = GDERIV_norm [unfolded gderiv_def]
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   328
84b2c432b94a new theory of real inner product spaces
huffman
parents:
diff changeset
   329
end