# HG changeset patch
# User hoelzl
# Date 1475242537 -7200
# Node ID da89140186e2e42702e73912059d1ee0a6223b5e
# Parent  3b6a3632e754da4124e052257e9f95cf2779968f
HOL-Analysis: move Product_Vector and Inner_Product from Library

diff -r 3b6a3632e754 -r da89140186e2 src/HOL/Analysis/Convex_Euclidean_Space.thy
--- a/src/HOL/Analysis/Convex_Euclidean_Space.thy	Fri Sep 30 15:35:32 2016 +0200
+++ b/src/HOL/Analysis/Convex_Euclidean_Space.thy	Fri Sep 30 15:35:37 2016 +0200
@@ -11,7 +11,6 @@
 theory Convex_Euclidean_Space
 imports
   Topology_Euclidean_Space
-  "~~/src/HOL/Library/Product_Vector"
   "~~/src/HOL/Library/Set_Algebras"
 begin
 
diff -r 3b6a3632e754 -r da89140186e2 src/HOL/Analysis/Euclidean_Space.thy
--- a/src/HOL/Analysis/Euclidean_Space.thy	Fri Sep 30 15:35:32 2016 +0200
+++ b/src/HOL/Analysis/Euclidean_Space.thy	Fri Sep 30 15:35:37 2016 +0200
@@ -7,9 +7,7 @@
 
 theory Euclidean_Space
 imports
-  L2_Norm
-  "~~/src/HOL/Library/Inner_Product"
-  "~~/src/HOL/Library/Product_Vector"
+  L2_Norm Product_Vector
 begin
 
 subsection \<open>Type class of Euclidean spaces\<close>
diff -r 3b6a3632e754 -r da89140186e2 src/HOL/Analysis/Inner_Product.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/Inner_Product.thy	Fri Sep 30 15:35:37 2016 +0200
@@ -0,0 +1,402 @@
+(*  Title:      HOL/Analysis/Inner_Product.thy
+    Author:     Brian Huffman
+*)
+
+section \<open>Inner Product Spaces and the Gradient Derivative\<close>
+
+theory Inner_Product
+imports "~~/src/HOL/Complex_Main"
+begin
+
+subsection \<open>Real inner product spaces\<close>
+
+text \<open>
+  Temporarily relax type constraints for @{term "open"}, @{term "uniformity"},
+  @{term dist}, and @{term norm}.
+\<close>
+
+setup \<open>Sign.add_const_constraint
+  (@{const_name "open"}, SOME @{typ "'a::open set \<Rightarrow> bool"})\<close>
+
+setup \<open>Sign.add_const_constraint
+  (@{const_name dist}, SOME @{typ "'a::dist \<Rightarrow> 'a \<Rightarrow> real"})\<close>
+
+setup \<open>Sign.add_const_constraint
+  (@{const_name uniformity}, SOME @{typ "('a::uniformity \<times> 'a) filter"})\<close>
+
+setup \<open>Sign.add_const_constraint
+  (@{const_name norm}, SOME @{typ "'a::norm \<Rightarrow> real"})\<close>
+
+class real_inner = real_vector + sgn_div_norm + dist_norm + uniformity_dist + open_uniformity +
+  fixes inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"
+  assumes inner_commute: "inner x y = inner y x"
+  and inner_add_left: "inner (x + y) z = inner x z + inner y z"
+  and inner_scaleR_left [simp]: "inner (scaleR r x) y = r * (inner x y)"
+  and inner_ge_zero [simp]: "0 \<le> inner x x"
+  and inner_eq_zero_iff [simp]: "inner x x = 0 \<longleftrightarrow> x = 0"
+  and norm_eq_sqrt_inner: "norm x = sqrt (inner x x)"
+begin
+
+lemma inner_zero_left [simp]: "inner 0 x = 0"
+  using inner_add_left [of 0 0 x] by simp
+
+lemma inner_minus_left [simp]: "inner (- x) y = - inner x y"
+  using inner_add_left [of x "- x" y] by simp
+
+lemma inner_diff_left: "inner (x - y) z = inner x z - inner y z"
+  using inner_add_left [of x "- y" z] by simp
+
+lemma inner_setsum_left: "inner (\<Sum>x\<in>A. f x) y = (\<Sum>x\<in>A. inner (f x) y)"
+  by (cases "finite A", induct set: finite, simp_all add: inner_add_left)
+
+text \<open>Transfer distributivity rules to right argument.\<close>
+
+lemma inner_add_right: "inner x (y + z) = inner x y + inner x z"
+  using inner_add_left [of y z x] by (simp only: inner_commute)
+
+lemma inner_scaleR_right [simp]: "inner x (scaleR r y) = r * (inner x y)"
+  using inner_scaleR_left [of r y x] by (simp only: inner_commute)
+
+lemma inner_zero_right [simp]: "inner x 0 = 0"
+  using inner_zero_left [of x] by (simp only: inner_commute)
+
+lemma inner_minus_right [simp]: "inner x (- y) = - inner x y"
+  using inner_minus_left [of y x] by (simp only: inner_commute)
+
+lemma inner_diff_right: "inner x (y - z) = inner x y - inner x z"
+  using inner_diff_left [of y z x] by (simp only: inner_commute)
+
+lemma inner_setsum_right: "inner x (\<Sum>y\<in>A. f y) = (\<Sum>y\<in>A. inner x (f y))"
+  using inner_setsum_left [of f A x] by (simp only: inner_commute)
+
+lemmas inner_add [algebra_simps] = inner_add_left inner_add_right
+lemmas inner_diff [algebra_simps]  = inner_diff_left inner_diff_right
+lemmas inner_scaleR = inner_scaleR_left inner_scaleR_right
+
+text \<open>Legacy theorem names\<close>
+lemmas inner_left_distrib = inner_add_left
+lemmas inner_right_distrib = inner_add_right
+lemmas inner_distrib = inner_left_distrib inner_right_distrib
+
+lemma inner_gt_zero_iff [simp]: "0 < inner x x \<longleftrightarrow> x \<noteq> 0"
+  by (simp add: order_less_le)
+
+lemma power2_norm_eq_inner: "(norm x)\<^sup>2 = inner x x"
+  by (simp add: norm_eq_sqrt_inner)
+
+text \<open>Identities involving real multiplication and division.\<close>
+
+lemma inner_mult_left: "inner (of_real m * a) b = m * (inner a b)"
+  by (metis real_inner_class.inner_scaleR_left scaleR_conv_of_real)
+
+lemma inner_mult_right: "inner a (of_real m * b) = m * (inner a b)"
+  by (metis real_inner_class.inner_scaleR_right scaleR_conv_of_real)
+
+lemma inner_mult_left': "inner (a * of_real m) b = m * (inner a b)"
+  by (simp add: of_real_def)
+
+lemma inner_mult_right': "inner a (b * of_real m) = (inner a b) * m"
+  by (simp add: of_real_def real_inner_class.inner_scaleR_right)
+
+lemma Cauchy_Schwarz_ineq:
+  "(inner x y)\<^sup>2 \<le> inner x x * inner y y"
+proof (cases)
+  assume "y = 0"
+  thus ?thesis by simp
+next
+  assume y: "y \<noteq> 0"
+  let ?r = "inner x y / inner y y"
+  have "0 \<le> inner (x - scaleR ?r y) (x - scaleR ?r y)"
+    by (rule inner_ge_zero)
+  also have "\<dots> = inner x x - inner y x * ?r"
+    by (simp add: inner_diff)
+  also have "\<dots> = inner x x - (inner x y)\<^sup>2 / inner y y"
+    by (simp add: power2_eq_square inner_commute)
+  finally have "0 \<le> inner x x - (inner x y)\<^sup>2 / inner y y" .
+  hence "(inner x y)\<^sup>2 / inner y y \<le> inner x x"
+    by (simp add: le_diff_eq)
+  thus "(inner x y)\<^sup>2 \<le> inner x x * inner y y"
+    by (simp add: pos_divide_le_eq y)
+qed
+
+lemma Cauchy_Schwarz_ineq2:
+  "\<bar>inner x y\<bar> \<le> norm x * norm y"
+proof (rule power2_le_imp_le)
+  have "(inner x y)\<^sup>2 \<le> inner x x * inner y y"
+    using Cauchy_Schwarz_ineq .
+  thus "\<bar>inner x y\<bar>\<^sup>2 \<le> (norm x * norm y)\<^sup>2"
+    by (simp add: power_mult_distrib power2_norm_eq_inner)
+  show "0 \<le> norm x * norm y"
+    unfolding norm_eq_sqrt_inner
+    by (intro mult_nonneg_nonneg real_sqrt_ge_zero inner_ge_zero)
+qed
+
+lemma norm_cauchy_schwarz: "inner x y \<le> norm x * norm y"
+  using Cauchy_Schwarz_ineq2 [of x y] by auto
+
+subclass real_normed_vector
+proof
+  fix a :: real and x y :: 'a
+  show "norm x = 0 \<longleftrightarrow> x = 0"
+    unfolding norm_eq_sqrt_inner by simp
+  show "norm (x + y) \<le> norm x + norm y"
+    proof (rule power2_le_imp_le)
+      have "inner x y \<le> norm x * norm y"
+        by (rule norm_cauchy_schwarz)
+      thus "(norm (x + y))\<^sup>2 \<le> (norm x + norm y)\<^sup>2"
+        unfolding power2_sum power2_norm_eq_inner
+        by (simp add: inner_add inner_commute)
+      show "0 \<le> norm x + norm y"
+        unfolding norm_eq_sqrt_inner by simp
+    qed
+  have "sqrt (a\<^sup>2 * inner x x) = \<bar>a\<bar> * sqrt (inner x x)"
+    by (simp add: real_sqrt_mult_distrib)
+  then show "norm (a *\<^sub>R x) = \<bar>a\<bar> * norm x"
+    unfolding norm_eq_sqrt_inner
+    by (simp add: power2_eq_square mult.assoc)
+qed
+
+end
+
+lemma inner_divide_left:
+  fixes a :: "'a :: {real_inner,real_div_algebra}"
+  shows "inner (a / of_real m) b = (inner a b) / m"
+  by (metis (no_types) divide_inverse inner_commute inner_scaleR_right mult.left_neutral mult.right_neutral mult_scaleR_right of_real_inverse scaleR_conv_of_real times_divide_eq_left)
+
+lemma inner_divide_right:
+  fixes a :: "'a :: {real_inner,real_div_algebra}"
+  shows "inner a (b / of_real m) = (inner a b) / m"
+  by (metis inner_commute inner_divide_left)
+
+text \<open>
+  Re-enable constraints for @{term "open"}, @{term "uniformity"},
+  @{term dist}, and @{term norm}.
+\<close>
+
+setup \<open>Sign.add_const_constraint
+  (@{const_name "open"}, SOME @{typ "'a::topological_space set \<Rightarrow> bool"})\<close>
+
+setup \<open>Sign.add_const_constraint
+  (@{const_name uniformity}, SOME @{typ "('a::uniform_space \<times> 'a) filter"})\<close>
+
+setup \<open>Sign.add_const_constraint
+  (@{const_name dist}, SOME @{typ "'a::metric_space \<Rightarrow> 'a \<Rightarrow> real"})\<close>
+
+setup \<open>Sign.add_const_constraint
+  (@{const_name norm}, SOME @{typ "'a::real_normed_vector \<Rightarrow> real"})\<close>
+
+lemma bounded_bilinear_inner:
+  "bounded_bilinear (inner::'a::real_inner \<Rightarrow> 'a \<Rightarrow> real)"
+proof
+  fix x y z :: 'a and r :: real
+  show "inner (x + y) z = inner x z + inner y z"
+    by (rule inner_add_left)
+  show "inner x (y + z) = inner x y + inner x z"
+    by (rule inner_add_right)
+  show "inner (scaleR r x) y = scaleR r (inner x y)"
+    unfolding real_scaleR_def by (rule inner_scaleR_left)
+  show "inner x (scaleR r y) = scaleR r (inner x y)"
+    unfolding real_scaleR_def by (rule inner_scaleR_right)
+  show "\<exists>K. \<forall>x y::'a. norm (inner x y) \<le> norm x * norm y * K"
+  proof
+    show "\<forall>x y::'a. norm (inner x y) \<le> norm x * norm y * 1"
+      by (simp add: Cauchy_Schwarz_ineq2)
+  qed
+qed
+
+lemmas tendsto_inner [tendsto_intros] =
+  bounded_bilinear.tendsto [OF bounded_bilinear_inner]
+
+lemmas isCont_inner [simp] =
+  bounded_bilinear.isCont [OF bounded_bilinear_inner]
+
+lemmas has_derivative_inner [derivative_intros] =
+  bounded_bilinear.FDERIV [OF bounded_bilinear_inner]
+
+lemmas bounded_linear_inner_left =
+  bounded_bilinear.bounded_linear_left [OF bounded_bilinear_inner]
+
+lemmas bounded_linear_inner_right =
+  bounded_bilinear.bounded_linear_right [OF bounded_bilinear_inner]
+
+lemmas bounded_linear_inner_left_comp = bounded_linear_inner_left[THEN bounded_linear_compose]
+
+lemmas bounded_linear_inner_right_comp = bounded_linear_inner_right[THEN bounded_linear_compose]
+
+lemmas has_derivative_inner_right [derivative_intros] =
+  bounded_linear.has_derivative [OF bounded_linear_inner_right]
+
+lemmas has_derivative_inner_left [derivative_intros] =
+  bounded_linear.has_derivative [OF bounded_linear_inner_left]
+
+lemma differentiable_inner [simp]:
+  "f differentiable (at x within s) \<Longrightarrow> g differentiable at x within s \<Longrightarrow> (\<lambda>x. inner (f x) (g x)) differentiable at x within s"
+  unfolding differentiable_def by (blast intro: has_derivative_inner)
+
+
+subsection \<open>Class instances\<close>
+
+instantiation real :: real_inner
+begin
+
+definition inner_real_def [simp]: "inner = op *"
+
+instance
+proof
+  fix x y z r :: real
+  show "inner x y = inner y x"
+    unfolding inner_real_def by (rule mult.commute)
+  show "inner (x + y) z = inner x z + inner y z"
+    unfolding inner_real_def by (rule distrib_right)
+  show "inner (scaleR r x) y = r * inner x y"
+    unfolding inner_real_def real_scaleR_def by (rule mult.assoc)
+  show "0 \<le> inner x x"
+    unfolding inner_real_def by simp
+  show "inner x x = 0 \<longleftrightarrow> x = 0"
+    unfolding inner_real_def by simp
+  show "norm x = sqrt (inner x x)"
+    unfolding inner_real_def by simp
+qed
+
+end
+
+lemma
+  shows real_inner_1_left[simp]: "inner 1 x = x"
+    and real_inner_1_right[simp]: "inner x 1 = x"
+  by simp_all
+
+instantiation complex :: real_inner
+begin
+
+definition inner_complex_def:
+  "inner x y = Re x * Re y + Im x * Im y"
+
+instance
+proof
+  fix x y z :: complex and r :: real
+  show "inner x y = inner y x"
+    unfolding inner_complex_def by (simp add: mult.commute)
+  show "inner (x + y) z = inner x z + inner y z"
+    unfolding inner_complex_def by (simp add: distrib_right)
+  show "inner (scaleR r x) y = r * inner x y"
+    unfolding inner_complex_def by (simp add: distrib_left)
+  show "0 \<le> inner x x"
+    unfolding inner_complex_def by simp
+  show "inner x x = 0 \<longleftrightarrow> x = 0"
+    unfolding inner_complex_def
+    by (simp add: add_nonneg_eq_0_iff complex_Re_Im_cancel_iff)
+  show "norm x = sqrt (inner x x)"
+    unfolding inner_complex_def complex_norm_def
+    by (simp add: power2_eq_square)
+qed
+
+end
+
+lemma complex_inner_1 [simp]: "inner 1 x = Re x"
+  unfolding inner_complex_def by simp
+
+lemma complex_inner_1_right [simp]: "inner x 1 = Re x"
+  unfolding inner_complex_def by simp
+
+lemma complex_inner_ii_left [simp]: "inner \<i> x = Im x"
+  unfolding inner_complex_def by simp
+
+lemma complex_inner_ii_right [simp]: "inner x \<i> = Im x"
+  unfolding inner_complex_def by simp
+
+
+subsection \<open>Gradient derivative\<close>
+
+definition
+  gderiv ::
+    "['a::real_inner \<Rightarrow> real, 'a, 'a] \<Rightarrow> bool"
+          ("(GDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
+where
+  "GDERIV f x :> D \<longleftrightarrow> FDERIV f x :> (\<lambda>h. inner h D)"
+
+lemma gderiv_deriv [simp]: "GDERIV f x :> D \<longleftrightarrow> DERIV f x :> D"
+  by (simp only: gderiv_def has_field_derivative_def inner_real_def mult_commute_abs)
+
+lemma GDERIV_DERIV_compose:
+    "\<lbrakk>GDERIV f x :> df; DERIV g (f x) :> dg\<rbrakk>
+     \<Longrightarrow> GDERIV (\<lambda>x. g (f x)) x :> scaleR dg df"
+  unfolding gderiv_def has_field_derivative_def
+  apply (drule (1) has_derivative_compose)
+  apply (simp add: ac_simps)
+  done
+
+lemma has_derivative_subst: "\<lbrakk>FDERIV f x :> df; df = d\<rbrakk> \<Longrightarrow> FDERIV f x :> d"
+  by simp
+
+lemma GDERIV_subst: "\<lbrakk>GDERIV f x :> df; df = d\<rbrakk> \<Longrightarrow> GDERIV f x :> d"
+  by simp
+
+lemma GDERIV_const: "GDERIV (\<lambda>x. k) x :> 0"
+  unfolding gderiv_def inner_zero_right by (rule has_derivative_const)
+
+lemma GDERIV_add:
+    "\<lbrakk>GDERIV f x :> df; GDERIV g x :> dg\<rbrakk>
+     \<Longrightarrow> GDERIV (\<lambda>x. f x + g x) x :> df + dg"
+  unfolding gderiv_def inner_add_right by (rule has_derivative_add)
+
+lemma GDERIV_minus:
+    "GDERIV f x :> df \<Longrightarrow> GDERIV (\<lambda>x. - f x) x :> - df"
+  unfolding gderiv_def inner_minus_right by (rule has_derivative_minus)
+
+lemma GDERIV_diff:
+    "\<lbrakk>GDERIV f x :> df; GDERIV g x :> dg\<rbrakk>
+     \<Longrightarrow> GDERIV (\<lambda>x. f x - g x) x :> df - dg"
+  unfolding gderiv_def inner_diff_right by (rule has_derivative_diff)
+
+lemma GDERIV_scaleR:
+    "\<lbrakk>DERIV f x :> df; GDERIV g x :> dg\<rbrakk>
+     \<Longrightarrow> GDERIV (\<lambda>x. scaleR (f x) (g x)) x
+      :> (scaleR (f x) dg + scaleR df (g x))"
+  unfolding gderiv_def has_field_derivative_def inner_add_right inner_scaleR_right
+  apply (rule has_derivative_subst)
+  apply (erule (1) has_derivative_scaleR)
+  apply (simp add: ac_simps)
+  done
+
+lemma GDERIV_mult:
+    "\<lbrakk>GDERIV f x :> df; GDERIV g x :> dg\<rbrakk>
+     \<Longrightarrow> GDERIV (\<lambda>x. f x * g x) x :> scaleR (f x) dg + scaleR (g x) df"
+  unfolding gderiv_def
+  apply (rule has_derivative_subst)
+  apply (erule (1) has_derivative_mult)
+  apply (simp add: inner_add ac_simps)
+  done
+
+lemma GDERIV_inverse:
+    "\<lbrakk>GDERIV f x :> df; f x \<noteq> 0\<rbrakk>
+     \<Longrightarrow> GDERIV (\<lambda>x. inverse (f x)) x :> - (inverse (f x))\<^sup>2 *\<^sub>R df"
+  apply (erule GDERIV_DERIV_compose)
+  apply (erule DERIV_inverse [folded numeral_2_eq_2])
+  done
+
+lemma GDERIV_norm:
+  assumes "x \<noteq> 0" shows "GDERIV (\<lambda>x. norm x) x :> sgn x"
+proof -
+  have 1: "FDERIV (\<lambda>x. inner x x) x :> (\<lambda>h. inner x h + inner h x)"
+    by (intro has_derivative_inner has_derivative_ident)
+  have 2: "(\<lambda>h. inner x h + inner h x) = (\<lambda>h. inner h (scaleR 2 x))"
+    by (simp add: fun_eq_iff inner_commute)
+  have "0 < inner x x" using \<open>x \<noteq> 0\<close> by simp
+  then have 3: "DERIV sqrt (inner x x) :> (inverse (sqrt (inner x x)) / 2)"
+    by (rule DERIV_real_sqrt)
+  have 4: "(inverse (sqrt (inner x x)) / 2) *\<^sub>R 2 *\<^sub>R x = sgn x"
+    by (simp add: sgn_div_norm norm_eq_sqrt_inner)
+  show ?thesis
+    unfolding norm_eq_sqrt_inner
+    apply (rule GDERIV_subst [OF _ 4])
+    apply (rule GDERIV_DERIV_compose [where g=sqrt and df="scaleR 2 x"])
+    apply (subst gderiv_def)
+    apply (rule has_derivative_subst [OF _ 2])
+    apply (rule 1)
+    apply (rule 3)
+    done
+qed
+
+lemmas has_derivative_norm = GDERIV_norm [unfolded gderiv_def]
+
+end
diff -r 3b6a3632e754 -r da89140186e2 src/HOL/Analysis/Product_Vector.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/Product_Vector.thy	Fri Sep 30 15:35:37 2016 +0200
@@ -0,0 +1,371 @@
+(*  Title:      HOL/Analysis/Product_Vector.thy
+    Author:     Brian Huffman
+*)
+
+section \<open>Cartesian Products as Vector Spaces\<close>
+
+theory Product_Vector
+imports
+  Inner_Product
+  "~~/src/HOL/Library/Product_plus"
+begin
+
+subsection \<open>Product is a real vector space\<close>
+
+instantiation prod :: (real_vector, real_vector) real_vector
+begin
+
+definition scaleR_prod_def:
+  "scaleR r A = (scaleR r (fst A), scaleR r (snd A))"
+
+lemma fst_scaleR [simp]: "fst (scaleR r A) = scaleR r (fst A)"
+  unfolding scaleR_prod_def by simp
+
+lemma snd_scaleR [simp]: "snd (scaleR r A) = scaleR r (snd A)"
+  unfolding scaleR_prod_def by simp
+
+lemma scaleR_Pair [simp]: "scaleR r (a, b) = (scaleR r a, scaleR r b)"
+  unfolding scaleR_prod_def by simp
+
+instance
+proof
+  fix a b :: real and x y :: "'a \<times> 'b"
+  show "scaleR a (x + y) = scaleR a x + scaleR a y"
+    by (simp add: prod_eq_iff scaleR_right_distrib)
+  show "scaleR (a + b) x = scaleR a x + scaleR b x"
+    by (simp add: prod_eq_iff scaleR_left_distrib)
+  show "scaleR a (scaleR b x) = scaleR (a * b) x"
+    by (simp add: prod_eq_iff)
+  show "scaleR 1 x = x"
+    by (simp add: prod_eq_iff)
+qed
+
+end
+
+subsection \<open>Product is a metric space\<close>
+
+(* TODO: Product of uniform spaces and compatibility with metric_spaces! *)
+
+instantiation prod :: (metric_space, metric_space) dist
+begin
+
+definition dist_prod_def[code del]:
+  "dist x y = sqrt ((dist (fst x) (fst y))\<^sup>2 + (dist (snd x) (snd y))\<^sup>2)"
+
+instance ..
+end
+
+instantiation prod :: (metric_space, metric_space) uniformity_dist
+begin
+
+definition [code del]:
+  "(uniformity :: (('a \<times> 'b) \<times> ('a \<times> 'b)) filter) =
+    (INF e:{0 <..}. principal {(x, y). dist x y < e})"
+
+instance
+  by standard (rule uniformity_prod_def)
+end
+
+declare uniformity_Abort[where 'a="'a :: metric_space \<times> 'b :: metric_space", code]
+
+instantiation prod :: (metric_space, metric_space) metric_space
+begin
+
+lemma dist_Pair_Pair: "dist (a, b) (c, d) = sqrt ((dist a c)\<^sup>2 + (dist b d)\<^sup>2)"
+  unfolding dist_prod_def by simp
+
+lemma dist_fst_le: "dist (fst x) (fst y) \<le> dist x y"
+  unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge1)
+
+lemma dist_snd_le: "dist (snd x) (snd y) \<le> dist x y"
+  unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge2)
+
+instance
+proof
+  fix x y :: "'a \<times> 'b"
+  show "dist x y = 0 \<longleftrightarrow> x = y"
+    unfolding dist_prod_def prod_eq_iff by simp
+next
+  fix x y z :: "'a \<times> 'b"
+  show "dist x y \<le> dist x z + dist y z"
+    unfolding dist_prod_def
+    by (intro order_trans [OF _ real_sqrt_sum_squares_triangle_ineq]
+        real_sqrt_le_mono add_mono power_mono dist_triangle2 zero_le_dist)
+next
+  fix S :: "('a \<times> 'b) set"
+  have *: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
+  proof
+    assume "open S" show "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S"
+    proof
+      fix x assume "x \<in> S"
+      obtain A B where "open A" "open B" "x \<in> A \<times> B" "A \<times> B \<subseteq> S"
+        using \<open>open S\<close> and \<open>x \<in> S\<close> by (rule open_prod_elim)
+      obtain r where r: "0 < r" "\<forall>y. dist y (fst x) < r \<longrightarrow> y \<in> A"
+        using \<open>open A\<close> and \<open>x \<in> A \<times> B\<close> unfolding open_dist by auto
+      obtain s where s: "0 < s" "\<forall>y. dist y (snd x) < s \<longrightarrow> y \<in> B"
+        using \<open>open B\<close> and \<open>x \<in> A \<times> B\<close> unfolding open_dist by auto
+      let ?e = "min r s"
+      have "0 < ?e \<and> (\<forall>y. dist y x < ?e \<longrightarrow> y \<in> S)"
+      proof (intro allI impI conjI)
+        show "0 < min r s" by (simp add: r(1) s(1))
+      next
+        fix y assume "dist y x < min r s"
+        hence "dist y x < r" and "dist y x < s"
+          by simp_all
+        hence "dist (fst y) (fst x) < r" and "dist (snd y) (snd x) < s"
+          by (auto intro: le_less_trans dist_fst_le dist_snd_le)
+        hence "fst y \<in> A" and "snd y \<in> B"
+          by (simp_all add: r(2) s(2))
+        hence "y \<in> A \<times> B" by (induct y, simp)
+        with \<open>A \<times> B \<subseteq> S\<close> show "y \<in> S" ..
+      qed
+      thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" ..
+    qed
+  next
+    assume *: "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" show "open S"
+    proof (rule open_prod_intro)
+      fix x assume "x \<in> S"
+      then obtain e where "0 < e" and S: "\<forall>y. dist y x < e \<longrightarrow> y \<in> S"
+        using * by fast
+      define r where "r = e / sqrt 2"
+      define s where "s = e / sqrt 2"
+      from \<open>0 < e\<close> have "0 < r" and "0 < s"
+        unfolding r_def s_def by simp_all
+      from \<open>0 < e\<close> have "e = sqrt (r\<^sup>2 + s\<^sup>2)"
+        unfolding r_def s_def by (simp add: power_divide)
+      define A where "A = {y. dist (fst x) y < r}"
+      define B where "B = {y. dist (snd x) y < s}"
+      have "open A" and "open B"
+        unfolding A_def B_def by (simp_all add: open_ball)
+      moreover have "x \<in> A \<times> B"
+        unfolding A_def B_def mem_Times_iff
+        using \<open>0 < r\<close> and \<open>0 < s\<close> by simp
+      moreover have "A \<times> B \<subseteq> S"
+      proof (clarify)
+        fix a b assume "a \<in> A" and "b \<in> B"
+        hence "dist a (fst x) < r" and "dist b (snd x) < s"
+          unfolding A_def B_def by (simp_all add: dist_commute)
+        hence "dist (a, b) x < e"
+          unfolding dist_prod_def \<open>e = sqrt (r\<^sup>2 + s\<^sup>2)\<close>
+          by (simp add: add_strict_mono power_strict_mono)
+        thus "(a, b) \<in> S"
+          by (simp add: S)
+      qed
+      ultimately show "\<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S" by fast
+    qed
+  qed
+  show "open S = (\<forall>x\<in>S. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> S)"
+    unfolding * eventually_uniformity_metric
+    by (simp del: split_paired_All add: dist_prod_def dist_commute)
+qed
+
+end
+
+declare [[code abort: "dist::('a::metric_space*'b::metric_space)\<Rightarrow>('a*'b) \<Rightarrow> real"]]
+
+lemma Cauchy_fst: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. fst (X n))"
+  unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_fst_le])
+
+lemma Cauchy_snd: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. snd (X n))"
+  unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_snd_le])
+
+lemma Cauchy_Pair:
+  assumes "Cauchy X" and "Cauchy Y"
+  shows "Cauchy (\<lambda>n. (X n, Y n))"
+proof (rule metric_CauchyI)
+  fix r :: real assume "0 < r"
+  hence "0 < r / sqrt 2" (is "0 < ?s") by simp
+  obtain M where M: "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < ?s"
+    using metric_CauchyD [OF \<open>Cauchy X\<close> \<open>0 < ?s\<close>] ..
+  obtain N where N: "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (Y m) (Y n) < ?s"
+    using metric_CauchyD [OF \<open>Cauchy Y\<close> \<open>0 < ?s\<close>] ..
+  have "\<forall>m\<ge>max M N. \<forall>n\<ge>max M N. dist (X m, Y m) (X n, Y n) < r"
+    using M N by (simp add: real_sqrt_sum_squares_less dist_Pair_Pair)
+  then show "\<exists>n0. \<forall>m\<ge>n0. \<forall>n\<ge>n0. dist (X m, Y m) (X n, Y n) < r" ..
+qed
+
+subsection \<open>Product is a complete metric space\<close>
+
+instance prod :: (complete_space, complete_space) complete_space
+proof
+  fix X :: "nat \<Rightarrow> 'a \<times> 'b" assume "Cauchy X"
+  have 1: "(\<lambda>n. fst (X n)) \<longlonglongrightarrow> lim (\<lambda>n. fst (X n))"
+    using Cauchy_fst [OF \<open>Cauchy X\<close>]
+    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
+  have 2: "(\<lambda>n. snd (X n)) \<longlonglongrightarrow> lim (\<lambda>n. snd (X n))"
+    using Cauchy_snd [OF \<open>Cauchy X\<close>]
+    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
+  have "X \<longlonglongrightarrow> (lim (\<lambda>n. fst (X n)), lim (\<lambda>n. snd (X n)))"
+    using tendsto_Pair [OF 1 2] by simp
+  then show "convergent X"
+    by (rule convergentI)
+qed
+
+subsection \<open>Product is a normed vector space\<close>
+
+instantiation prod :: (real_normed_vector, real_normed_vector) real_normed_vector
+begin
+
+definition norm_prod_def[code del]:
+  "norm x = sqrt ((norm (fst x))\<^sup>2 + (norm (snd x))\<^sup>2)"
+
+definition sgn_prod_def:
+  "sgn (x::'a \<times> 'b) = scaleR (inverse (norm x)) x"
+
+lemma norm_Pair: "norm (a, b) = sqrt ((norm a)\<^sup>2 + (norm b)\<^sup>2)"
+  unfolding norm_prod_def by simp
+
+instance
+proof
+  fix r :: real and x y :: "'a \<times> 'b"
+  show "norm x = 0 \<longleftrightarrow> x = 0"
+    unfolding norm_prod_def
+    by (simp add: prod_eq_iff)
+  show "norm (x + y) \<le> norm x + norm y"
+    unfolding norm_prod_def
+    apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq])
+    apply (simp add: add_mono power_mono norm_triangle_ineq)
+    done
+  show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
+    unfolding norm_prod_def
+    apply (simp add: power_mult_distrib)
+    apply (simp add: distrib_left [symmetric])
+    apply (simp add: real_sqrt_mult_distrib)
+    done
+  show "sgn x = scaleR (inverse (norm x)) x"
+    by (rule sgn_prod_def)
+  show "dist x y = norm (x - y)"
+    unfolding dist_prod_def norm_prod_def
+    by (simp add: dist_norm)
+qed
+
+end
+
+declare [[code abort: "norm::('a::real_normed_vector*'b::real_normed_vector) \<Rightarrow> real"]]
+
+instance prod :: (banach, banach) banach ..
+
+subsubsection \<open>Pair operations are linear\<close>
+
+lemma bounded_linear_fst: "bounded_linear fst"
+  using fst_add fst_scaleR
+  by (rule bounded_linear_intro [where K=1], simp add: norm_prod_def)
+
+lemma bounded_linear_snd: "bounded_linear snd"
+  using snd_add snd_scaleR
+  by (rule bounded_linear_intro [where K=1], simp add: norm_prod_def)
+
+lemmas bounded_linear_fst_comp = bounded_linear_fst[THEN bounded_linear_compose]
+
+lemmas bounded_linear_snd_comp = bounded_linear_snd[THEN bounded_linear_compose]
+
+lemma bounded_linear_Pair:
+  assumes f: "bounded_linear f"
+  assumes g: "bounded_linear g"
+  shows "bounded_linear (\<lambda>x. (f x, g x))"
+proof
+  interpret f: bounded_linear f by fact
+  interpret g: bounded_linear g by fact
+  fix x y and r :: real
+  show "(f (x + y), g (x + y)) = (f x, g x) + (f y, g y)"
+    by (simp add: f.add g.add)
+  show "(f (r *\<^sub>R x), g (r *\<^sub>R x)) = r *\<^sub>R (f x, g x)"
+    by (simp add: f.scaleR g.scaleR)
+  obtain Kf where "0 < Kf" and norm_f: "\<And>x. norm (f x) \<le> norm x * Kf"
+    using f.pos_bounded by fast
+  obtain Kg where "0 < Kg" and norm_g: "\<And>x. norm (g x) \<le> norm x * Kg"
+    using g.pos_bounded by fast
+  have "\<forall>x. norm (f x, g x) \<le> norm x * (Kf + Kg)"
+    apply (rule allI)
+    apply (simp add: norm_Pair)
+    apply (rule order_trans [OF sqrt_add_le_add_sqrt], simp, simp)
+    apply (simp add: distrib_left)
+    apply (rule add_mono [OF norm_f norm_g])
+    done
+  then show "\<exists>K. \<forall>x. norm (f x, g x) \<le> norm x * K" ..
+qed
+
+subsubsection \<open>Frechet derivatives involving pairs\<close>
+
+lemma has_derivative_Pair [derivative_intros]:
+  assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)"
+  shows "((\<lambda>x. (f x, g x)) has_derivative (\<lambda>h. (f' h, g' h))) (at x within s)"
+proof (rule has_derivativeI_sandwich[of 1])
+  show "bounded_linear (\<lambda>h. (f' h, g' h))"
+    using f g by (intro bounded_linear_Pair has_derivative_bounded_linear)
+  let ?Rf = "\<lambda>y. f y - f x - f' (y - x)"
+  let ?Rg = "\<lambda>y. g y - g x - g' (y - x)"
+  let ?R = "\<lambda>y. ((f y, g y) - (f x, g x) - (f' (y - x), g' (y - x)))"
+
+  show "((\<lambda>y. norm (?Rf y) / norm (y - x) + norm (?Rg y) / norm (y - x)) \<longlongrightarrow> 0) (at x within s)"
+    using f g by (intro tendsto_add_zero) (auto simp: has_derivative_iff_norm)
+
+  fix y :: 'a assume "y \<noteq> x"
+  show "norm (?R y) / norm (y - x) \<le> norm (?Rf y) / norm (y - x) + norm (?Rg y) / norm (y - x)"
+    unfolding add_divide_distrib [symmetric]
+    by (simp add: norm_Pair divide_right_mono order_trans [OF sqrt_add_le_add_sqrt])
+qed simp
+
+lemmas has_derivative_fst [derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_fst]
+lemmas has_derivative_snd [derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_snd]
+
+lemma has_derivative_split [derivative_intros]:
+  "((\<lambda>p. f (fst p) (snd p)) has_derivative f') F \<Longrightarrow> ((\<lambda>(a, b). f a b) has_derivative f') F"
+  unfolding split_beta' .
+
+subsection \<open>Product is an inner product space\<close>
+
+instantiation prod :: (real_inner, real_inner) real_inner
+begin
+
+definition inner_prod_def:
+  "inner x y = inner (fst x) (fst y) + inner (snd x) (snd y)"
+
+lemma inner_Pair [simp]: "inner (a, b) (c, d) = inner a c + inner b d"
+  unfolding inner_prod_def by simp
+
+instance
+proof
+  fix r :: real
+  fix x y z :: "'a::real_inner \<times> 'b::real_inner"
+  show "inner x y = inner y x"
+    unfolding inner_prod_def
+    by (simp add: inner_commute)
+  show "inner (x + y) z = inner x z + inner y z"
+    unfolding inner_prod_def
+    by (simp add: inner_add_left)
+  show "inner (scaleR r x) y = r * inner x y"
+    unfolding inner_prod_def
+    by (simp add: distrib_left)
+  show "0 \<le> inner x x"
+    unfolding inner_prod_def
+    by (intro add_nonneg_nonneg inner_ge_zero)
+  show "inner x x = 0 \<longleftrightarrow> x = 0"
+    unfolding inner_prod_def prod_eq_iff
+    by (simp add: add_nonneg_eq_0_iff)
+  show "norm x = sqrt (inner x x)"
+    unfolding norm_prod_def inner_prod_def
+    by (simp add: power2_norm_eq_inner)
+qed
+
+end
+
+lemma inner_Pair_0: "inner x (0, b) = inner (snd x) b" "inner x (a, 0) = inner (fst x) a"
+    by (cases x, simp)+
+
+lemma
+  fixes x :: "'a::real_normed_vector"
+  shows norm_Pair1 [simp]: "norm (0,x) = norm x"
+    and norm_Pair2 [simp]: "norm (x,0) = norm x"
+by (auto simp: norm_Pair)
+
+lemma norm_commute: "norm (x,y) = norm (y,x)"
+  by (simp add: norm_Pair)
+
+lemma norm_fst_le: "norm x \<le> norm (x,y)"
+  by (metis dist_fst_le fst_conv fst_zero norm_conv_dist)
+
+lemma norm_snd_le: "norm y \<le> norm (x,y)"
+  by (metis dist_snd_le snd_conv snd_zero norm_conv_dist)
+
+end
diff -r 3b6a3632e754 -r da89140186e2 src/HOL/Library/Inner_Product.thy
--- a/src/HOL/Library/Inner_Product.thy	Fri Sep 30 15:35:32 2016 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,402 +0,0 @@
-(*  Title:      HOL/Library/Inner_Product.thy
-    Author:     Brian Huffman
-*)
-
-section \<open>Inner Product Spaces and the Gradient Derivative\<close>
-
-theory Inner_Product
-imports "~~/src/HOL/Complex_Main"
-begin
-
-subsection \<open>Real inner product spaces\<close>
-
-text \<open>
-  Temporarily relax type constraints for @{term "open"}, @{term "uniformity"},
-  @{term dist}, and @{term norm}.
-\<close>
-
-setup \<open>Sign.add_const_constraint
-  (@{const_name "open"}, SOME @{typ "'a::open set \<Rightarrow> bool"})\<close>
-
-setup \<open>Sign.add_const_constraint
-  (@{const_name dist}, SOME @{typ "'a::dist \<Rightarrow> 'a \<Rightarrow> real"})\<close>
-
-setup \<open>Sign.add_const_constraint
-  (@{const_name uniformity}, SOME @{typ "('a::uniformity \<times> 'a) filter"})\<close>
-
-setup \<open>Sign.add_const_constraint
-  (@{const_name norm}, SOME @{typ "'a::norm \<Rightarrow> real"})\<close>
-
-class real_inner = real_vector + sgn_div_norm + dist_norm + uniformity_dist + open_uniformity +
-  fixes inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"
-  assumes inner_commute: "inner x y = inner y x"
-  and inner_add_left: "inner (x + y) z = inner x z + inner y z"
-  and inner_scaleR_left [simp]: "inner (scaleR r x) y = r * (inner x y)"
-  and inner_ge_zero [simp]: "0 \<le> inner x x"
-  and inner_eq_zero_iff [simp]: "inner x x = 0 \<longleftrightarrow> x = 0"
-  and norm_eq_sqrt_inner: "norm x = sqrt (inner x x)"
-begin
-
-lemma inner_zero_left [simp]: "inner 0 x = 0"
-  using inner_add_left [of 0 0 x] by simp
-
-lemma inner_minus_left [simp]: "inner (- x) y = - inner x y"
-  using inner_add_left [of x "- x" y] by simp
-
-lemma inner_diff_left: "inner (x - y) z = inner x z - inner y z"
-  using inner_add_left [of x "- y" z] by simp
-
-lemma inner_setsum_left: "inner (\<Sum>x\<in>A. f x) y = (\<Sum>x\<in>A. inner (f x) y)"
-  by (cases "finite A", induct set: finite, simp_all add: inner_add_left)
-
-text \<open>Transfer distributivity rules to right argument.\<close>
-
-lemma inner_add_right: "inner x (y + z) = inner x y + inner x z"
-  using inner_add_left [of y z x] by (simp only: inner_commute)
-
-lemma inner_scaleR_right [simp]: "inner x (scaleR r y) = r * (inner x y)"
-  using inner_scaleR_left [of r y x] by (simp only: inner_commute)
-
-lemma inner_zero_right [simp]: "inner x 0 = 0"
-  using inner_zero_left [of x] by (simp only: inner_commute)
-
-lemma inner_minus_right [simp]: "inner x (- y) = - inner x y"
-  using inner_minus_left [of y x] by (simp only: inner_commute)
-
-lemma inner_diff_right: "inner x (y - z) = inner x y - inner x z"
-  using inner_diff_left [of y z x] by (simp only: inner_commute)
-
-lemma inner_setsum_right: "inner x (\<Sum>y\<in>A. f y) = (\<Sum>y\<in>A. inner x (f y))"
-  using inner_setsum_left [of f A x] by (simp only: inner_commute)
-
-lemmas inner_add [algebra_simps] = inner_add_left inner_add_right
-lemmas inner_diff [algebra_simps]  = inner_diff_left inner_diff_right
-lemmas inner_scaleR = inner_scaleR_left inner_scaleR_right
-
-text \<open>Legacy theorem names\<close>
-lemmas inner_left_distrib = inner_add_left
-lemmas inner_right_distrib = inner_add_right
-lemmas inner_distrib = inner_left_distrib inner_right_distrib
-
-lemma inner_gt_zero_iff [simp]: "0 < inner x x \<longleftrightarrow> x \<noteq> 0"
-  by (simp add: order_less_le)
-
-lemma power2_norm_eq_inner: "(norm x)\<^sup>2 = inner x x"
-  by (simp add: norm_eq_sqrt_inner)
-
-text \<open>Identities involving real multiplication and division.\<close>
-
-lemma inner_mult_left: "inner (of_real m * a) b = m * (inner a b)"
-  by (metis real_inner_class.inner_scaleR_left scaleR_conv_of_real)
-
-lemma inner_mult_right: "inner a (of_real m * b) = m * (inner a b)"
-  by (metis real_inner_class.inner_scaleR_right scaleR_conv_of_real)
-
-lemma inner_mult_left': "inner (a * of_real m) b = m * (inner a b)"
-  by (simp add: of_real_def)
-
-lemma inner_mult_right': "inner a (b * of_real m) = (inner a b) * m"
-  by (simp add: of_real_def real_inner_class.inner_scaleR_right)
-
-lemma Cauchy_Schwarz_ineq:
-  "(inner x y)\<^sup>2 \<le> inner x x * inner y y"
-proof (cases)
-  assume "y = 0"
-  thus ?thesis by simp
-next
-  assume y: "y \<noteq> 0"
-  let ?r = "inner x y / inner y y"
-  have "0 \<le> inner (x - scaleR ?r y) (x - scaleR ?r y)"
-    by (rule inner_ge_zero)
-  also have "\<dots> = inner x x - inner y x * ?r"
-    by (simp add: inner_diff)
-  also have "\<dots> = inner x x - (inner x y)\<^sup>2 / inner y y"
-    by (simp add: power2_eq_square inner_commute)
-  finally have "0 \<le> inner x x - (inner x y)\<^sup>2 / inner y y" .
-  hence "(inner x y)\<^sup>2 / inner y y \<le> inner x x"
-    by (simp add: le_diff_eq)
-  thus "(inner x y)\<^sup>2 \<le> inner x x * inner y y"
-    by (simp add: pos_divide_le_eq y)
-qed
-
-lemma Cauchy_Schwarz_ineq2:
-  "\<bar>inner x y\<bar> \<le> norm x * norm y"
-proof (rule power2_le_imp_le)
-  have "(inner x y)\<^sup>2 \<le> inner x x * inner y y"
-    using Cauchy_Schwarz_ineq .
-  thus "\<bar>inner x y\<bar>\<^sup>2 \<le> (norm x * norm y)\<^sup>2"
-    by (simp add: power_mult_distrib power2_norm_eq_inner)
-  show "0 \<le> norm x * norm y"
-    unfolding norm_eq_sqrt_inner
-    by (intro mult_nonneg_nonneg real_sqrt_ge_zero inner_ge_zero)
-qed
-
-lemma norm_cauchy_schwarz: "inner x y \<le> norm x * norm y"
-  using Cauchy_Schwarz_ineq2 [of x y] by auto
-
-subclass real_normed_vector
-proof
-  fix a :: real and x y :: 'a
-  show "norm x = 0 \<longleftrightarrow> x = 0"
-    unfolding norm_eq_sqrt_inner by simp
-  show "norm (x + y) \<le> norm x + norm y"
-    proof (rule power2_le_imp_le)
-      have "inner x y \<le> norm x * norm y"
-        by (rule norm_cauchy_schwarz)
-      thus "(norm (x + y))\<^sup>2 \<le> (norm x + norm y)\<^sup>2"
-        unfolding power2_sum power2_norm_eq_inner
-        by (simp add: inner_add inner_commute)
-      show "0 \<le> norm x + norm y"
-        unfolding norm_eq_sqrt_inner by simp
-    qed
-  have "sqrt (a\<^sup>2 * inner x x) = \<bar>a\<bar> * sqrt (inner x x)"
-    by (simp add: real_sqrt_mult_distrib)
-  then show "norm (a *\<^sub>R x) = \<bar>a\<bar> * norm x"
-    unfolding norm_eq_sqrt_inner
-    by (simp add: power2_eq_square mult.assoc)
-qed
-
-end
-
-lemma inner_divide_left:
-  fixes a :: "'a :: {real_inner,real_div_algebra}"
-  shows "inner (a / of_real m) b = (inner a b) / m"
-  by (metis (no_types) divide_inverse inner_commute inner_scaleR_right mult.left_neutral mult.right_neutral mult_scaleR_right of_real_inverse scaleR_conv_of_real times_divide_eq_left)
-
-lemma inner_divide_right:
-  fixes a :: "'a :: {real_inner,real_div_algebra}"
-  shows "inner a (b / of_real m) = (inner a b) / m"
-  by (metis inner_commute inner_divide_left)
-
-text \<open>
-  Re-enable constraints for @{term "open"}, @{term "uniformity"},
-  @{term dist}, and @{term norm}.
-\<close>
-
-setup \<open>Sign.add_const_constraint
-  (@{const_name "open"}, SOME @{typ "'a::topological_space set \<Rightarrow> bool"})\<close>
-
-setup \<open>Sign.add_const_constraint
-  (@{const_name uniformity}, SOME @{typ "('a::uniform_space \<times> 'a) filter"})\<close>
-
-setup \<open>Sign.add_const_constraint
-  (@{const_name dist}, SOME @{typ "'a::metric_space \<Rightarrow> 'a \<Rightarrow> real"})\<close>
-
-setup \<open>Sign.add_const_constraint
-  (@{const_name norm}, SOME @{typ "'a::real_normed_vector \<Rightarrow> real"})\<close>
-
-lemma bounded_bilinear_inner:
-  "bounded_bilinear (inner::'a::real_inner \<Rightarrow> 'a \<Rightarrow> real)"
-proof
-  fix x y z :: 'a and r :: real
-  show "inner (x + y) z = inner x z + inner y z"
-    by (rule inner_add_left)
-  show "inner x (y + z) = inner x y + inner x z"
-    by (rule inner_add_right)
-  show "inner (scaleR r x) y = scaleR r (inner x y)"
-    unfolding real_scaleR_def by (rule inner_scaleR_left)
-  show "inner x (scaleR r y) = scaleR r (inner x y)"
-    unfolding real_scaleR_def by (rule inner_scaleR_right)
-  show "\<exists>K. \<forall>x y::'a. norm (inner x y) \<le> norm x * norm y * K"
-  proof
-    show "\<forall>x y::'a. norm (inner x y) \<le> norm x * norm y * 1"
-      by (simp add: Cauchy_Schwarz_ineq2)
-  qed
-qed
-
-lemmas tendsto_inner [tendsto_intros] =
-  bounded_bilinear.tendsto [OF bounded_bilinear_inner]
-
-lemmas isCont_inner [simp] =
-  bounded_bilinear.isCont [OF bounded_bilinear_inner]
-
-lemmas has_derivative_inner [derivative_intros] =
-  bounded_bilinear.FDERIV [OF bounded_bilinear_inner]
-
-lemmas bounded_linear_inner_left =
-  bounded_bilinear.bounded_linear_left [OF bounded_bilinear_inner]
-
-lemmas bounded_linear_inner_right =
-  bounded_bilinear.bounded_linear_right [OF bounded_bilinear_inner]
-
-lemmas bounded_linear_inner_left_comp = bounded_linear_inner_left[THEN bounded_linear_compose]
-
-lemmas bounded_linear_inner_right_comp = bounded_linear_inner_right[THEN bounded_linear_compose]
-
-lemmas has_derivative_inner_right [derivative_intros] =
-  bounded_linear.has_derivative [OF bounded_linear_inner_right]
-
-lemmas has_derivative_inner_left [derivative_intros] =
-  bounded_linear.has_derivative [OF bounded_linear_inner_left]
-
-lemma differentiable_inner [simp]:
-  "f differentiable (at x within s) \<Longrightarrow> g differentiable at x within s \<Longrightarrow> (\<lambda>x. inner (f x) (g x)) differentiable at x within s"
-  unfolding differentiable_def by (blast intro: has_derivative_inner)
-
-
-subsection \<open>Class instances\<close>
-
-instantiation real :: real_inner
-begin
-
-definition inner_real_def [simp]: "inner = op *"
-
-instance
-proof
-  fix x y z r :: real
-  show "inner x y = inner y x"
-    unfolding inner_real_def by (rule mult.commute)
-  show "inner (x + y) z = inner x z + inner y z"
-    unfolding inner_real_def by (rule distrib_right)
-  show "inner (scaleR r x) y = r * inner x y"
-    unfolding inner_real_def real_scaleR_def by (rule mult.assoc)
-  show "0 \<le> inner x x"
-    unfolding inner_real_def by simp
-  show "inner x x = 0 \<longleftrightarrow> x = 0"
-    unfolding inner_real_def by simp
-  show "norm x = sqrt (inner x x)"
-    unfolding inner_real_def by simp
-qed
-
-end
-
-lemma
-  shows real_inner_1_left[simp]: "inner 1 x = x"
-    and real_inner_1_right[simp]: "inner x 1 = x"
-  by simp_all
-
-instantiation complex :: real_inner
-begin
-
-definition inner_complex_def:
-  "inner x y = Re x * Re y + Im x * Im y"
-
-instance
-proof
-  fix x y z :: complex and r :: real
-  show "inner x y = inner y x"
-    unfolding inner_complex_def by (simp add: mult.commute)
-  show "inner (x + y) z = inner x z + inner y z"
-    unfolding inner_complex_def by (simp add: distrib_right)
-  show "inner (scaleR r x) y = r * inner x y"
-    unfolding inner_complex_def by (simp add: distrib_left)
-  show "0 \<le> inner x x"
-    unfolding inner_complex_def by simp
-  show "inner x x = 0 \<longleftrightarrow> x = 0"
-    unfolding inner_complex_def
-    by (simp add: add_nonneg_eq_0_iff complex_Re_Im_cancel_iff)
-  show "norm x = sqrt (inner x x)"
-    unfolding inner_complex_def complex_norm_def
-    by (simp add: power2_eq_square)
-qed
-
-end
-
-lemma complex_inner_1 [simp]: "inner 1 x = Re x"
-  unfolding inner_complex_def by simp
-
-lemma complex_inner_1_right [simp]: "inner x 1 = Re x"
-  unfolding inner_complex_def by simp
-
-lemma complex_inner_ii_left [simp]: "inner \<i> x = Im x"
-  unfolding inner_complex_def by simp
-
-lemma complex_inner_ii_right [simp]: "inner x \<i> = Im x"
-  unfolding inner_complex_def by simp
-
-
-subsection \<open>Gradient derivative\<close>
-
-definition
-  gderiv ::
-    "['a::real_inner \<Rightarrow> real, 'a, 'a] \<Rightarrow> bool"
-          ("(GDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
-where
-  "GDERIV f x :> D \<longleftrightarrow> FDERIV f x :> (\<lambda>h. inner h D)"
-
-lemma gderiv_deriv [simp]: "GDERIV f x :> D \<longleftrightarrow> DERIV f x :> D"
-  by (simp only: gderiv_def has_field_derivative_def inner_real_def mult_commute_abs)
-
-lemma GDERIV_DERIV_compose:
-    "\<lbrakk>GDERIV f x :> df; DERIV g (f x) :> dg\<rbrakk>
-     \<Longrightarrow> GDERIV (\<lambda>x. g (f x)) x :> scaleR dg df"
-  unfolding gderiv_def has_field_derivative_def
-  apply (drule (1) has_derivative_compose)
-  apply (simp add: ac_simps)
-  done
-
-lemma has_derivative_subst: "\<lbrakk>FDERIV f x :> df; df = d\<rbrakk> \<Longrightarrow> FDERIV f x :> d"
-  by simp
-
-lemma GDERIV_subst: "\<lbrakk>GDERIV f x :> df; df = d\<rbrakk> \<Longrightarrow> GDERIV f x :> d"
-  by simp
-
-lemma GDERIV_const: "GDERIV (\<lambda>x. k) x :> 0"
-  unfolding gderiv_def inner_zero_right by (rule has_derivative_const)
-
-lemma GDERIV_add:
-    "\<lbrakk>GDERIV f x :> df; GDERIV g x :> dg\<rbrakk>
-     \<Longrightarrow> GDERIV (\<lambda>x. f x + g x) x :> df + dg"
-  unfolding gderiv_def inner_add_right by (rule has_derivative_add)
-
-lemma GDERIV_minus:
-    "GDERIV f x :> df \<Longrightarrow> GDERIV (\<lambda>x. - f x) x :> - df"
-  unfolding gderiv_def inner_minus_right by (rule has_derivative_minus)
-
-lemma GDERIV_diff:
-    "\<lbrakk>GDERIV f x :> df; GDERIV g x :> dg\<rbrakk>
-     \<Longrightarrow> GDERIV (\<lambda>x. f x - g x) x :> df - dg"
-  unfolding gderiv_def inner_diff_right by (rule has_derivative_diff)
-
-lemma GDERIV_scaleR:
-    "\<lbrakk>DERIV f x :> df; GDERIV g x :> dg\<rbrakk>
-     \<Longrightarrow> GDERIV (\<lambda>x. scaleR (f x) (g x)) x
-      :> (scaleR (f x) dg + scaleR df (g x))"
-  unfolding gderiv_def has_field_derivative_def inner_add_right inner_scaleR_right
-  apply (rule has_derivative_subst)
-  apply (erule (1) has_derivative_scaleR)
-  apply (simp add: ac_simps)
-  done
-
-lemma GDERIV_mult:
-    "\<lbrakk>GDERIV f x :> df; GDERIV g x :> dg\<rbrakk>
-     \<Longrightarrow> GDERIV (\<lambda>x. f x * g x) x :> scaleR (f x) dg + scaleR (g x) df"
-  unfolding gderiv_def
-  apply (rule has_derivative_subst)
-  apply (erule (1) has_derivative_mult)
-  apply (simp add: inner_add ac_simps)
-  done
-
-lemma GDERIV_inverse:
-    "\<lbrakk>GDERIV f x :> df; f x \<noteq> 0\<rbrakk>
-     \<Longrightarrow> GDERIV (\<lambda>x. inverse (f x)) x :> - (inverse (f x))\<^sup>2 *\<^sub>R df"
-  apply (erule GDERIV_DERIV_compose)
-  apply (erule DERIV_inverse [folded numeral_2_eq_2])
-  done
-
-lemma GDERIV_norm:
-  assumes "x \<noteq> 0" shows "GDERIV (\<lambda>x. norm x) x :> sgn x"
-proof -
-  have 1: "FDERIV (\<lambda>x. inner x x) x :> (\<lambda>h. inner x h + inner h x)"
-    by (intro has_derivative_inner has_derivative_ident)
-  have 2: "(\<lambda>h. inner x h + inner h x) = (\<lambda>h. inner h (scaleR 2 x))"
-    by (simp add: fun_eq_iff inner_commute)
-  have "0 < inner x x" using \<open>x \<noteq> 0\<close> by simp
-  then have 3: "DERIV sqrt (inner x x) :> (inverse (sqrt (inner x x)) / 2)"
-    by (rule DERIV_real_sqrt)
-  have 4: "(inverse (sqrt (inner x x)) / 2) *\<^sub>R 2 *\<^sub>R x = sgn x"
-    by (simp add: sgn_div_norm norm_eq_sqrt_inner)
-  show ?thesis
-    unfolding norm_eq_sqrt_inner
-    apply (rule GDERIV_subst [OF _ 4])
-    apply (rule GDERIV_DERIV_compose [where g=sqrt and df="scaleR 2 x"])
-    apply (subst gderiv_def)
-    apply (rule has_derivative_subst [OF _ 2])
-    apply (rule 1)
-    apply (rule 3)
-    done
-qed
-
-lemmas has_derivative_norm = GDERIV_norm [unfolded gderiv_def]
-
-end
diff -r 3b6a3632e754 -r da89140186e2 src/HOL/Library/Library.thy
--- a/src/HOL/Library/Library.thy	Fri Sep 30 15:35:32 2016 +0200
+++ b/src/HOL/Library/Library.thy	Fri Sep 30 15:35:37 2016 +0200
@@ -37,7 +37,6 @@
   Groups_Big_Fun
   Indicator_Function
   Infinite_Set
-  Inner_Product
   IArray
   Lattice_Algebras
   Lattice_Syntax
@@ -62,7 +61,7 @@
   Polynomial
   Polynomial_FPS
   Preorder
-  Product_Vector
+  Product_plus
   Quadratic_Discriminant
   Quotient_List
   Quotient_Option
diff -r 3b6a3632e754 -r da89140186e2 src/HOL/Library/Product_Vector.thy
--- a/src/HOL/Library/Product_Vector.thy	Fri Sep 30 15:35:32 2016 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,369 +0,0 @@
-(*  Title:      HOL/Library/Product_Vector.thy
-    Author:     Brian Huffman
-*)
-
-section \<open>Cartesian Products as Vector Spaces\<close>
-
-theory Product_Vector
-imports Inner_Product Product_plus
-begin
-
-subsection \<open>Product is a real vector space\<close>
-
-instantiation prod :: (real_vector, real_vector) real_vector
-begin
-
-definition scaleR_prod_def:
-  "scaleR r A = (scaleR r (fst A), scaleR r (snd A))"
-
-lemma fst_scaleR [simp]: "fst (scaleR r A) = scaleR r (fst A)"
-  unfolding scaleR_prod_def by simp
-
-lemma snd_scaleR [simp]: "snd (scaleR r A) = scaleR r (snd A)"
-  unfolding scaleR_prod_def by simp
-
-lemma scaleR_Pair [simp]: "scaleR r (a, b) = (scaleR r a, scaleR r b)"
-  unfolding scaleR_prod_def by simp
-
-instance
-proof
-  fix a b :: real and x y :: "'a \<times> 'b"
-  show "scaleR a (x + y) = scaleR a x + scaleR a y"
-    by (simp add: prod_eq_iff scaleR_right_distrib)
-  show "scaleR (a + b) x = scaleR a x + scaleR b x"
-    by (simp add: prod_eq_iff scaleR_left_distrib)
-  show "scaleR a (scaleR b x) = scaleR (a * b) x"
-    by (simp add: prod_eq_iff)
-  show "scaleR 1 x = x"
-    by (simp add: prod_eq_iff)
-qed
-
-end
-
-subsection \<open>Product is a metric space\<close>
-
-(* TODO: Product of uniform spaces and compatibility with metric_spaces! *)
-
-instantiation prod :: (metric_space, metric_space) dist
-begin
-
-definition dist_prod_def[code del]:
-  "dist x y = sqrt ((dist (fst x) (fst y))\<^sup>2 + (dist (snd x) (snd y))\<^sup>2)"
-
-instance ..
-end
-
-instantiation prod :: (metric_space, metric_space) uniformity_dist
-begin
-
-definition [code del]:
-  "(uniformity :: (('a \<times> 'b) \<times> ('a \<times> 'b)) filter) =
-    (INF e:{0 <..}. principal {(x, y). dist x y < e})"
-
-instance
-  by standard (rule uniformity_prod_def)
-end
-
-declare uniformity_Abort[where 'a="'a :: metric_space \<times> 'b :: metric_space", code]
-
-instantiation prod :: (metric_space, metric_space) metric_space
-begin
-
-lemma dist_Pair_Pair: "dist (a, b) (c, d) = sqrt ((dist a c)\<^sup>2 + (dist b d)\<^sup>2)"
-  unfolding dist_prod_def by simp
-
-lemma dist_fst_le: "dist (fst x) (fst y) \<le> dist x y"
-  unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge1)
-
-lemma dist_snd_le: "dist (snd x) (snd y) \<le> dist x y"
-  unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge2)
-
-instance
-proof
-  fix x y :: "'a \<times> 'b"
-  show "dist x y = 0 \<longleftrightarrow> x = y"
-    unfolding dist_prod_def prod_eq_iff by simp
-next
-  fix x y z :: "'a \<times> 'b"
-  show "dist x y \<le> dist x z + dist y z"
-    unfolding dist_prod_def
-    by (intro order_trans [OF _ real_sqrt_sum_squares_triangle_ineq]
-        real_sqrt_le_mono add_mono power_mono dist_triangle2 zero_le_dist)
-next
-  fix S :: "('a \<times> 'b) set"
-  have *: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
-  proof
-    assume "open S" show "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S"
-    proof
-      fix x assume "x \<in> S"
-      obtain A B where "open A" "open B" "x \<in> A \<times> B" "A \<times> B \<subseteq> S"
-        using \<open>open S\<close> and \<open>x \<in> S\<close> by (rule open_prod_elim)
-      obtain r where r: "0 < r" "\<forall>y. dist y (fst x) < r \<longrightarrow> y \<in> A"
-        using \<open>open A\<close> and \<open>x \<in> A \<times> B\<close> unfolding open_dist by auto
-      obtain s where s: "0 < s" "\<forall>y. dist y (snd x) < s \<longrightarrow> y \<in> B"
-        using \<open>open B\<close> and \<open>x \<in> A \<times> B\<close> unfolding open_dist by auto
-      let ?e = "min r s"
-      have "0 < ?e \<and> (\<forall>y. dist y x < ?e \<longrightarrow> y \<in> S)"
-      proof (intro allI impI conjI)
-        show "0 < min r s" by (simp add: r(1) s(1))
-      next
-        fix y assume "dist y x < min r s"
-        hence "dist y x < r" and "dist y x < s"
-          by simp_all
-        hence "dist (fst y) (fst x) < r" and "dist (snd y) (snd x) < s"
-          by (auto intro: le_less_trans dist_fst_le dist_snd_le)
-        hence "fst y \<in> A" and "snd y \<in> B"
-          by (simp_all add: r(2) s(2))
-        hence "y \<in> A \<times> B" by (induct y, simp)
-        with \<open>A \<times> B \<subseteq> S\<close> show "y \<in> S" ..
-      qed
-      thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" ..
-    qed
-  next
-    assume *: "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" show "open S"
-    proof (rule open_prod_intro)
-      fix x assume "x \<in> S"
-      then obtain e where "0 < e" and S: "\<forall>y. dist y x < e \<longrightarrow> y \<in> S"
-        using * by fast
-      define r where "r = e / sqrt 2"
-      define s where "s = e / sqrt 2"
-      from \<open>0 < e\<close> have "0 < r" and "0 < s"
-        unfolding r_def s_def by simp_all
-      from \<open>0 < e\<close> have "e = sqrt (r\<^sup>2 + s\<^sup>2)"
-        unfolding r_def s_def by (simp add: power_divide)
-      define A where "A = {y. dist (fst x) y < r}"
-      define B where "B = {y. dist (snd x) y < s}"
-      have "open A" and "open B"
-        unfolding A_def B_def by (simp_all add: open_ball)
-      moreover have "x \<in> A \<times> B"
-        unfolding A_def B_def mem_Times_iff
-        using \<open>0 < r\<close> and \<open>0 < s\<close> by simp
-      moreover have "A \<times> B \<subseteq> S"
-      proof (clarify)
-        fix a b assume "a \<in> A" and "b \<in> B"
-        hence "dist a (fst x) < r" and "dist b (snd x) < s"
-          unfolding A_def B_def by (simp_all add: dist_commute)
-        hence "dist (a, b) x < e"
-          unfolding dist_prod_def \<open>e = sqrt (r\<^sup>2 + s\<^sup>2)\<close>
-          by (simp add: add_strict_mono power_strict_mono)
-        thus "(a, b) \<in> S"
-          by (simp add: S)
-      qed
-      ultimately show "\<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S" by fast
-    qed
-  qed
-  show "open S = (\<forall>x\<in>S. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> S)"
-    unfolding * eventually_uniformity_metric
-    by (simp del: split_paired_All add: dist_prod_def dist_commute)
-qed
-
-end
-
-declare [[code abort: "dist::('a::metric_space*'b::metric_space)\<Rightarrow>('a*'b) \<Rightarrow> real"]]
-
-lemma Cauchy_fst: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. fst (X n))"
-  unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_fst_le])
-
-lemma Cauchy_snd: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. snd (X n))"
-  unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_snd_le])
-
-lemma Cauchy_Pair:
-  assumes "Cauchy X" and "Cauchy Y"
-  shows "Cauchy (\<lambda>n. (X n, Y n))"
-proof (rule metric_CauchyI)
-  fix r :: real assume "0 < r"
-  hence "0 < r / sqrt 2" (is "0 < ?s") by simp
-  obtain M where M: "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < ?s"
-    using metric_CauchyD [OF \<open>Cauchy X\<close> \<open>0 < ?s\<close>] ..
-  obtain N where N: "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (Y m) (Y n) < ?s"
-    using metric_CauchyD [OF \<open>Cauchy Y\<close> \<open>0 < ?s\<close>] ..
-  have "\<forall>m\<ge>max M N. \<forall>n\<ge>max M N. dist (X m, Y m) (X n, Y n) < r"
-    using M N by (simp add: real_sqrt_sum_squares_less dist_Pair_Pair)
-  then show "\<exists>n0. \<forall>m\<ge>n0. \<forall>n\<ge>n0. dist (X m, Y m) (X n, Y n) < r" ..
-qed
-
-subsection \<open>Product is a complete metric space\<close>
-
-instance prod :: (complete_space, complete_space) complete_space
-proof
-  fix X :: "nat \<Rightarrow> 'a \<times> 'b" assume "Cauchy X"
-  have 1: "(\<lambda>n. fst (X n)) \<longlonglongrightarrow> lim (\<lambda>n. fst (X n))"
-    using Cauchy_fst [OF \<open>Cauchy X\<close>]
-    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
-  have 2: "(\<lambda>n. snd (X n)) \<longlonglongrightarrow> lim (\<lambda>n. snd (X n))"
-    using Cauchy_snd [OF \<open>Cauchy X\<close>]
-    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
-  have "X \<longlonglongrightarrow> (lim (\<lambda>n. fst (X n)), lim (\<lambda>n. snd (X n)))"
-    using tendsto_Pair [OF 1 2] by simp
-  then show "convergent X"
-    by (rule convergentI)
-qed
-
-subsection \<open>Product is a normed vector space\<close>
-
-instantiation prod :: (real_normed_vector, real_normed_vector) real_normed_vector
-begin
-
-definition norm_prod_def[code del]:
-  "norm x = sqrt ((norm (fst x))\<^sup>2 + (norm (snd x))\<^sup>2)"
-
-definition sgn_prod_def:
-  "sgn (x::'a \<times> 'b) = scaleR (inverse (norm x)) x"
-
-lemma norm_Pair: "norm (a, b) = sqrt ((norm a)\<^sup>2 + (norm b)\<^sup>2)"
-  unfolding norm_prod_def by simp
-
-instance
-proof
-  fix r :: real and x y :: "'a \<times> 'b"
-  show "norm x = 0 \<longleftrightarrow> x = 0"
-    unfolding norm_prod_def
-    by (simp add: prod_eq_iff)
-  show "norm (x + y) \<le> norm x + norm y"
-    unfolding norm_prod_def
-    apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq])
-    apply (simp add: add_mono power_mono norm_triangle_ineq)
-    done
-  show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
-    unfolding norm_prod_def
-    apply (simp add: power_mult_distrib)
-    apply (simp add: distrib_left [symmetric])
-    apply (simp add: real_sqrt_mult_distrib)
-    done
-  show "sgn x = scaleR (inverse (norm x)) x"
-    by (rule sgn_prod_def)
-  show "dist x y = norm (x - y)"
-    unfolding dist_prod_def norm_prod_def
-    by (simp add: dist_norm)
-qed
-
-end
-
-declare [[code abort: "norm::('a::real_normed_vector*'b::real_normed_vector) \<Rightarrow> real"]]
-
-instance prod :: (banach, banach) banach ..
-
-subsubsection \<open>Pair operations are linear\<close>
-
-lemma bounded_linear_fst: "bounded_linear fst"
-  using fst_add fst_scaleR
-  by (rule bounded_linear_intro [where K=1], simp add: norm_prod_def)
-
-lemma bounded_linear_snd: "bounded_linear snd"
-  using snd_add snd_scaleR
-  by (rule bounded_linear_intro [where K=1], simp add: norm_prod_def)
-
-lemmas bounded_linear_fst_comp = bounded_linear_fst[THEN bounded_linear_compose]
-
-lemmas bounded_linear_snd_comp = bounded_linear_snd[THEN bounded_linear_compose]
-
-lemma bounded_linear_Pair:
-  assumes f: "bounded_linear f"
-  assumes g: "bounded_linear g"
-  shows "bounded_linear (\<lambda>x. (f x, g x))"
-proof
-  interpret f: bounded_linear f by fact
-  interpret g: bounded_linear g by fact
-  fix x y and r :: real
-  show "(f (x + y), g (x + y)) = (f x, g x) + (f y, g y)"
-    by (simp add: f.add g.add)
-  show "(f (r *\<^sub>R x), g (r *\<^sub>R x)) = r *\<^sub>R (f x, g x)"
-    by (simp add: f.scaleR g.scaleR)
-  obtain Kf where "0 < Kf" and norm_f: "\<And>x. norm (f x) \<le> norm x * Kf"
-    using f.pos_bounded by fast
-  obtain Kg where "0 < Kg" and norm_g: "\<And>x. norm (g x) \<le> norm x * Kg"
-    using g.pos_bounded by fast
-  have "\<forall>x. norm (f x, g x) \<le> norm x * (Kf + Kg)"
-    apply (rule allI)
-    apply (simp add: norm_Pair)
-    apply (rule order_trans [OF sqrt_add_le_add_sqrt], simp, simp)
-    apply (simp add: distrib_left)
-    apply (rule add_mono [OF norm_f norm_g])
-    done
-  then show "\<exists>K. \<forall>x. norm (f x, g x) \<le> norm x * K" ..
-qed
-
-subsubsection \<open>Frechet derivatives involving pairs\<close>
-
-lemma has_derivative_Pair [derivative_intros]:
-  assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)"
-  shows "((\<lambda>x. (f x, g x)) has_derivative (\<lambda>h. (f' h, g' h))) (at x within s)"
-proof (rule has_derivativeI_sandwich[of 1])
-  show "bounded_linear (\<lambda>h. (f' h, g' h))"
-    using f g by (intro bounded_linear_Pair has_derivative_bounded_linear)
-  let ?Rf = "\<lambda>y. f y - f x - f' (y - x)"
-  let ?Rg = "\<lambda>y. g y - g x - g' (y - x)"
-  let ?R = "\<lambda>y. ((f y, g y) - (f x, g x) - (f' (y - x), g' (y - x)))"
-
-  show "((\<lambda>y. norm (?Rf y) / norm (y - x) + norm (?Rg y) / norm (y - x)) \<longlongrightarrow> 0) (at x within s)"
-    using f g by (intro tendsto_add_zero) (auto simp: has_derivative_iff_norm)
-
-  fix y :: 'a assume "y \<noteq> x"
-  show "norm (?R y) / norm (y - x) \<le> norm (?Rf y) / norm (y - x) + norm (?Rg y) / norm (y - x)"
-    unfolding add_divide_distrib [symmetric]
-    by (simp add: norm_Pair divide_right_mono order_trans [OF sqrt_add_le_add_sqrt])
-qed simp
-
-lemmas has_derivative_fst [derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_fst]
-lemmas has_derivative_snd [derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_snd]
-
-lemma has_derivative_split [derivative_intros]:
-  "((\<lambda>p. f (fst p) (snd p)) has_derivative f') F \<Longrightarrow> ((\<lambda>(a, b). f a b) has_derivative f') F"
-  unfolding split_beta' .
-
-subsection \<open>Product is an inner product space\<close>
-
-instantiation prod :: (real_inner, real_inner) real_inner
-begin
-
-definition inner_prod_def:
-  "inner x y = inner (fst x) (fst y) + inner (snd x) (snd y)"
-
-lemma inner_Pair [simp]: "inner (a, b) (c, d) = inner a c + inner b d"
-  unfolding inner_prod_def by simp
-
-instance
-proof
-  fix r :: real
-  fix x y z :: "'a::real_inner \<times> 'b::real_inner"
-  show "inner x y = inner y x"
-    unfolding inner_prod_def
-    by (simp add: inner_commute)
-  show "inner (x + y) z = inner x z + inner y z"
-    unfolding inner_prod_def
-    by (simp add: inner_add_left)
-  show "inner (scaleR r x) y = r * inner x y"
-    unfolding inner_prod_def
-    by (simp add: distrib_left)
-  show "0 \<le> inner x x"
-    unfolding inner_prod_def
-    by (intro add_nonneg_nonneg inner_ge_zero)
-  show "inner x x = 0 \<longleftrightarrow> x = 0"
-    unfolding inner_prod_def prod_eq_iff
-    by (simp add: add_nonneg_eq_0_iff)
-  show "norm x = sqrt (inner x x)"
-    unfolding norm_prod_def inner_prod_def
-    by (simp add: power2_norm_eq_inner)
-qed
-
-end
-
-lemma inner_Pair_0: "inner x (0, b) = inner (snd x) b" "inner x (a, 0) = inner (fst x) a"
-    by (cases x, simp)+
-
-lemma
-  fixes x :: "'a::real_normed_vector"
-  shows norm_Pair1 [simp]: "norm (0,x) = norm x"
-    and norm_Pair2 [simp]: "norm (x,0) = norm x"
-by (auto simp: norm_Pair)
-
-lemma norm_commute: "norm (x,y) = norm (y,x)"
-  by (simp add: norm_Pair)
-
-lemma norm_fst_le: "norm x \<le> norm (x,y)"
-  by (metis dist_fst_le fst_conv fst_zero norm_conv_dist)
-
-lemma norm_snd_le: "norm y \<le> norm (x,y)"
-  by (metis dist_snd_le snd_conv snd_zero norm_conv_dist)
-
-end
diff -r 3b6a3632e754 -r da89140186e2 src/HOL/Mirabelle/ex/Ex.thy
--- a/src/HOL/Mirabelle/ex/Ex.thy	Fri Sep 30 15:35:32 2016 +0200
+++ b/src/HOL/Mirabelle/ex/Ex.thy	Fri Sep 30 15:35:37 2016 +0200
@@ -3,7 +3,7 @@
 
 ML \<open>
   val rc = Isabelle_System.bash
-    "cd \"$ISABELLE_HOME/src/HOL/Library\"; isabelle mirabelle arith Inner_Product.thy";
+    "cd \"$ISABELLE_HOME/src/HOL/Analysis\"; isabelle mirabelle arith Inner_Product.thy";
   if rc <> 0 then error ("Mirabelle example failed: " ^ string_of_int rc)
   else ();
 \<close> \<comment> "some arbitrary small test case"