# HG changeset patch
# User paulson <lp15@cam.ac.uk>
# Date 1529328146 -3600
# Node ID e699ca8e22b78c12686137c9e830a3540018a929
# Parent  410818a69ee382bbefcf8628f5422dcf1f5f9d35
New material in support of quaternions

diff -r 410818a69ee3 -r e699ca8e22b7 src/HOL/Analysis/Analysis.thy
--- a/src/HOL/Analysis/Analysis.thy	Mon Jun 18 11:15:46 2018 +0200
+++ b/src/HOL/Analysis/Analysis.thy	Mon Jun 18 14:22:26 2018 +0100
@@ -7,6 +7,7 @@
   Radon_Nikodym
   Fashoda_Theorem
   Determinants
+  Cross3
   Homeomorphism
   Bounded_Continuous_Function
   Function_Topology
diff -r 410818a69ee3 -r e699ca8e22b7 src/HOL/Analysis/Cross3.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/Cross3.thy	Mon Jun 18 14:22:26 2018 +0100
@@ -0,0 +1,200 @@
+theory "Cross3"
+  imports Determinants
+begin
+
+subsection\<open>Vector Cross products in real^3.                                                 \<close>
+
+definition cross3 :: "[real^3, real^3] \<Rightarrow> real^3"  (infixr "\<times>" 80)
+  where "a \<times> b \<equiv>
+    vector [a$2 * b$3 - a$3 * b$2,
+            a$3 * b$1 - a$1 * b$3,
+            a$1 * b$2 - a$2 * b$1]"
+
+subsubsection\<open> Basic lemmas.\<close>
+
+lemmas cross3_simps = cross3_def inner_vec_def sum_3 det_3 vec_eq_iff vector_def algebra_simps
+
+lemma dot_cross_self: "x \<bullet> (x \<times> y) = 0" "x \<bullet> (y \<times> x) = 0" "(x \<times> y) \<bullet> y = 0" "(y \<times> x) \<bullet> y = 0"
+  by (simp_all add: orthogonal_def cross3_simps)
+
+lemma orthogonal_cross: "orthogonal (x \<times> y) x" "orthogonal (x \<times> y) y"  
+                        "orthogonal y (x \<times> y)" "orthogonal (x \<times> y) x"
+  by (simp_all add: orthogonal_def dot_cross_self)
+
+lemma cross_zero_left [simp]: "0 \<times> x = 0" and cross_zero_right [simp]: "x \<times> 0 = 0" for x::"real^3"
+  by (simp_all add: cross3_simps)
+
+lemma cross_skew: "(x \<times> y) = -(y \<times> x)" for x::"real^3"
+  by (simp add: cross3_simps)
+
+lemma cross_refl [simp]: "x \<times> x = 0" for x::"real^3"
+  by (simp add: cross3_simps)
+
+lemma cross_add_left: "(x + y) \<times> z = (x \<times> z) + (y \<times> z)" for x::"real^3"
+  by (simp add: cross3_simps)
+
+lemma cross_add_right: "x \<times> (y + z) = (x \<times> y) + (x \<times> z)" for x::"real^3"
+  by (simp add: cross3_simps)
+
+lemma cross_mult_left: "(c *\<^sub>R x) \<times> y = c *\<^sub>R (x \<times> y)" for x::"real^3"
+  by (simp add: cross3_simps)
+
+lemma cross_mult_right: "x \<times> (c *\<^sub>R y) = c *\<^sub>R (x \<times> y)" for x::"real^3"
+  by (simp add: cross3_simps)
+
+lemma cross_minus_left [simp]: "(-x) \<times> y = - (x \<times> y)" for x::"real^3"
+  by (simp add: cross3_simps)
+
+lemma cross_minus_right [simp]: "x \<times> -y = - (x \<times> y)" for x::"real^3"
+  by (simp add: cross3_simps)
+
+lemma left_diff_distrib: "(x - y) \<times> z = x \<times> z - y \<times> z" for x::"real^3"
+  by (simp add: cross3_simps)
+
+lemma right_diff_distrib: "x \<times> (y - z) = x \<times> y - x \<times> z" for x::"real^3"
+  by (simp add: cross3_simps)
+
+lemma Jacobi: "x \<times> (y \<times> z) + y \<times> (z \<times> x) + z \<times> (x \<times> y) = 0" for x::"real^3"
+  by (simp add: cross3_simps)
+
+lemma Lagrange: "x \<times> (y \<times> z) = (x \<bullet> z) *\<^sub>R y - (x \<bullet> y) *\<^sub>R z"
+  by (simp add: cross3_simps) (metis (full_types) exhaust_3)
+
+lemma cross_triple: "(x \<times> y) \<bullet> z = (y \<times> z) \<bullet> x"
+  by (simp add: cross3_def inner_vec_def sum_3 vec_eq_iff algebra_simps)
+
+lemma cross_components:
+   "(x \<times> y)$1 = x$2 * y$3 - y$2 * x$3" "(x \<times> y)$2 = x$3 * y$1 - y$3 * x$1" "(x \<times> y)$3 = x$1 * y$2 - y$1 * x$2"
+  by (simp_all add: cross3_def inner_vec_def sum_3 vec_eq_iff algebra_simps)
+
+lemma cross_basis: "(axis 1 1) \<times> (axis 2 1) = axis 3 1" "(axis 2 1) \<times> (axis 1 1) = -(axis 3 1)" 
+                   "(axis 2 1) \<times> (axis 3 1) = axis 1 1" "(axis 3 1) \<times> (axis 2 1) = -(axis 1 1)" 
+                   "(axis 3 1) \<times> (axis 1 1) = axis 2 1" "(axis 1 1) \<times> (axis 3 1) = -(axis 2 1)"
+  using exhaust_3
+  by (force simp add: axis_def cross3_simps)+
+
+lemma cross_basis_nonzero:
+  "u \<noteq> 0 \<Longrightarrow> ~(u \<times> axis 1 1 = 0) \<or> ~(u \<times> axis 2 1 = 0) \<or> ~(u \<times> axis 3 1 = 0)"
+  by (clarsimp simp add: axis_def cross3_simps) (metis vector_3 exhaust_3)
+
+lemma cross_dot_cancel:
+  fixes x::"real^3"
+  assumes deq: "x \<bullet> y = x \<bullet> z" and veq: "x \<times> y = x \<times> z" and x: "x \<noteq> 0"
+  shows "y = z" 
+proof -
+  have "x \<bullet> x \<noteq> 0"
+    by (simp add: x)
+  then have "y - z = 0"
+    using veq
+    by (metis (no_types, lifting) Cross3.right_diff_distrib Lagrange deq eq_iff_diff_eq_0 inner_diff_right scale_eq_0_iff)
+  then show ?thesis
+    using eq_iff_diff_eq_0 by blast
+qed
+
+lemma norm_cross_dot: "(norm (x \<times> y))\<^sup>2 + (x \<bullet> y)\<^sup>2 = (norm x * norm y)\<^sup>2"
+  unfolding power2_norm_eq_inner power_mult_distrib
+  by (simp add: cross3_simps power2_eq_square)
+
+lemma dot_cross_det: "x \<bullet> (y \<times> z) = det(vector[x,y,z])"
+  by (simp add: cross3_simps) 
+
+lemma cross_cross_det: "(w \<times> x) \<times> (y \<times> z) = det(vector[w,x,z]) *\<^sub>R y - det(vector[w,x,y]) *\<^sub>R z"
+  using exhaust_3 by (force simp add: cross3_simps) 
+
+lemma dot_cross: "(w \<times> x) \<bullet> (y \<times> z) = (w \<bullet> y) * (x \<bullet> z) - (w \<bullet> z) * (x \<bullet> y)"
+  by (force simp add: cross3_simps) 
+
+lemma norm_cross: "(norm (x \<times> y))\<^sup>2 = (norm x)\<^sup>2 * (norm y)\<^sup>2 - (x \<bullet> y)\<^sup>2"
+  unfolding power2_norm_eq_inner power_mult_distrib
+  by (simp add: cross3_simps power2_eq_square)
+
+lemma cross_eq_0: "x \<times> y = 0 \<longleftrightarrow> collinear{0,x,y}"
+proof -
+  have "x \<times> y = 0 \<longleftrightarrow> norm (x \<times> y) = 0"
+    by simp
+  also have "... \<longleftrightarrow> (norm x * norm y)\<^sup>2 = (x \<bullet> y)\<^sup>2"
+    using norm_cross [of x y] by (auto simp: power_mult_distrib)
+  also have "... \<longleftrightarrow> \<bar>x \<bullet> y\<bar> = norm x * norm y"
+    using power2_eq_iff
+    by (metis (mono_tags, hide_lams) abs_minus abs_norm_cancel abs_power2 norm_mult power_abs real_norm_def) 
+  also have "... \<longleftrightarrow> collinear {0, x, y}"
+    by (rule norm_cauchy_schwarz_equal)
+  finally show ?thesis .
+qed
+
+lemma cross_eq_self: "x \<times> y = x \<longleftrightarrow> x = 0" "x \<times> y = y \<longleftrightarrow> y = 0"
+  apply (metis cross_zero_left dot_cross_self(1) inner_eq_zero_iff)
+  by (metis cross_zero_right dot_cross_self(2) inner_eq_zero_iff)
+
+lemma norm_and_cross_eq_0:
+   "x \<bullet> y = 0 \<and> x \<times> y = 0 \<longleftrightarrow> x = 0 \<or> y = 0" (is "?lhs = ?rhs")
+proof 
+  assume ?lhs
+  then show ?rhs
+    by (metis cross_dot_cancel cross_zero_right inner_zero_right)
+qed auto
+
+lemma bilinear_cross: "bilinear(\<times>)"
+  apply (auto simp add: bilinear_def linear_def)
+  apply unfold_locales
+  apply (simp add: cross_add_right)
+  apply (simp add: cross_mult_right)
+  apply (simp add: cross_add_left)
+  apply (simp add: cross_mult_left)
+  done
+
+subsection\<open>Preservation by rotation, or other orthogonal transformation up to sign.\<close>
+
+lemma cross_matrix_mult: "transpose A *v ((A *v x) \<times> (A *v y)) = det A *\<^sub>R (x \<times> y)"
+  apply (simp add: vec_eq_iff   )
+  apply (simp add: vector_matrix_mult_def matrix_vector_mult_def forall_3 cross3_simps)
+  done
+
+lemma cross_orthogonal_matrix:
+  assumes "orthogonal_matrix A"
+  shows "(A *v x) \<times> (A *v y) = det A *\<^sub>R (A *v (x \<times> y))"
+proof -
+  have "mat 1 = transpose (A ** transpose A)"
+    by (metis (no_types) assms orthogonal_matrix_def transpose_mat)
+  then show ?thesis
+    by (metis (no_types) vector_matrix_mul_rid vector_transpose_matrix cross_matrix_mult matrix_vector_mul_assoc matrix_vector_mult_scaleR)
+qed
+
+lemma cross_rotation_matrix: "rotation_matrix A \<Longrightarrow> (A *v x) \<times> (A *v y) =  A *v (x \<times> y)"
+  by (simp add: rotation_matrix_def cross_orthogonal_matrix)
+
+lemma cross_rotoinversion_matrix: "rotoinversion_matrix A \<Longrightarrow> (A *v x) \<times> (A *v y) = - A *v (x \<times> y)"
+  by (simp add: rotoinversion_matrix_def cross_orthogonal_matrix scaleR_matrix_vector_assoc)
+
+lemma cross_orthogonal_transformation:
+  assumes "orthogonal_transformation f"
+  shows   "(f x) \<times> (f y) = det(matrix f) *\<^sub>R f(x \<times> y)"
+proof -
+  have orth: "orthogonal_matrix (matrix f)"
+    using assms orthogonal_transformation_matrix by blast
+  have "matrix f *v z = f z" for z
+    using assms orthogonal_transformation_matrix by force
+  with cross_orthogonal_matrix [OF orth] show ?thesis
+    by simp
+qed
+
+lemma cross_linear_image:
+   "\<lbrakk>linear f; \<And>x. norm(f x) = norm x; det(matrix f) = 1\<rbrakk>
+           \<Longrightarrow> (f x) \<times> (f y) = f(x \<times> y)"
+  by (simp add: cross_orthogonal_transformation orthogonal_transformation)
+
+subsection\<open>Continuity\<close>
+
+lemma continuous_cross: "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. (f x) \<times> (g x))"
+  apply (subst continuous_componentwise)
+  apply (clarsimp simp add: cross3_simps)
+  apply (intro continuous_intros; simp)
+  done
+
+lemma continuous_on_cross:
+  fixes f :: "'a::t2_space \<Rightarrow> real^3"
+  shows "\<lbrakk>continuous_on S f; continuous_on S g\<rbrakk> \<Longrightarrow> continuous_on S (\<lambda>x. (f x) \<times> (g x))"
+  by (simp add: continuous_on_eq_continuous_within continuous_cross)
+
+end
+
diff -r 410818a69ee3 -r e699ca8e22b7 src/HOL/Analysis/L2_Norm.thy
--- a/src/HOL/Analysis/L2_Norm.thy	Mon Jun 18 11:15:46 2018 +0200
+++ b/src/HOL/Analysis/L2_Norm.thy	Mon Jun 18 14:22:26 2018 +0100
@@ -110,12 +110,6 @@
   apply simp
   done
 
-lemma sqrt_sum_squares_le_sum_abs: "sqrt (x\<^sup>2 + y\<^sup>2) \<le> \<bar>x\<bar> + \<bar>y\<bar>"
-  apply (rule power2_le_imp_le)
-  apply (simp add: power2_sum)
-  apply simp
-  done
-
 lemma L2_set_le_sum_abs: "L2_set f A \<le> (\<Sum>i\<in>A. \<bar>f i\<bar>)"
   apply (cases "finite A")
   apply (induct set: finite)
@@ -126,19 +120,6 @@
   apply simp
   done
 
-lemma L2_set_mult_ineq_lemma:
-  fixes a b c d :: real
-  shows "2 * (a * c) * (b * d) \<le> a\<^sup>2 * d\<^sup>2 + b\<^sup>2 * c\<^sup>2"
-proof -
-  have "0 \<le> (a * d - b * c)\<^sup>2" by simp
-  also have "\<dots> = a\<^sup>2 * d\<^sup>2 + b\<^sup>2 * c\<^sup>2 - 2 * (a * d) * (b * c)"
-    by (simp only: power2_diff power_mult_distrib)
-  also have "\<dots> = a\<^sup>2 * d\<^sup>2 + b\<^sup>2 * c\<^sup>2 - 2 * (a * c) * (b * d)"
-    by simp
-  finally show "2 * (a * c) * (b * d) \<le> a\<^sup>2 * d\<^sup>2 + b\<^sup>2 * c\<^sup>2"
-    by simp
-qed
-
 lemma L2_set_mult_ineq: "(\<Sum>i\<in>A. \<bar>f i\<bar> * \<bar>g i\<bar>) \<le> L2_set f A * L2_set g A"
   apply (cases "finite A")
   apply (induct set: finite)
diff -r 410818a69ee3 -r e699ca8e22b7 src/HOL/Analysis/Starlike.thy
--- a/src/HOL/Analysis/Starlike.thy	Mon Jun 18 11:15:46 2018 +0200
+++ b/src/HOL/Analysis/Starlike.thy	Mon Jun 18 14:22:26 2018 +0100
@@ -188,6 +188,14 @@
   "convex S \<longleftrightarrow> (\<forall>a\<in>S. \<forall>b\<in>S. closed_segment a b \<subseteq> S)"
   unfolding convex_alt closed_segment_def by auto
 
+lemma closed_segment_in_Reals:
+   "\<lbrakk>x \<in> closed_segment a b; a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> x \<in> Reals"
+  by (meson subsetD convex_Reals convex_contains_segment)
+
+lemma open_segment_in_Reals:
+   "\<lbrakk>x \<in> open_segment a b; a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> x \<in> Reals"
+  by (metis Diff_iff closed_segment_in_Reals open_segment_def)
+
 lemma closed_segment_subset: "\<lbrakk>x \<in> S; y \<in> S; convex S\<rbrakk> \<Longrightarrow> closed_segment x y \<subseteq> S"
   by (simp add: convex_contains_segment)
 
diff -r 410818a69ee3 -r e699ca8e22b7 src/HOL/NthRoot.thy
--- a/src/HOL/NthRoot.thy	Mon Jun 18 11:15:46 2018 +0200
+++ b/src/HOL/NthRoot.thy	Mon Jun 18 14:22:26 2018 +0100
@@ -665,30 +665,34 @@
 
 lemma sqrt_sum_squares_le_sum:
   "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> sqrt (x\<^sup>2 + y\<^sup>2) \<le> x + y"
-  apply (rule power2_le_imp_le)
-  apply (simp add: power2_sum)
-  apply simp
-  done
+  by (rule power2_le_imp_le) (simp_all add: power2_sum)
+
+lemma L2_set_mult_ineq_lemma:
+  fixes a b c d :: real
+  shows "2 * (a * c) * (b * d) \<le> a\<^sup>2 * d\<^sup>2 + b\<^sup>2 * c\<^sup>2"
+proof -
+  have "0 \<le> (a * d - b * c)\<^sup>2" by simp
+  also have "\<dots> = a\<^sup>2 * d\<^sup>2 + b\<^sup>2 * c\<^sup>2 - 2 * (a * d) * (b * c)"
+    by (simp only: power2_diff power_mult_distrib)
+  also have "\<dots> = a\<^sup>2 * d\<^sup>2 + b\<^sup>2 * c\<^sup>2 - 2 * (a * c) * (b * d)"
+    by simp
+  finally show "2 * (a * c) * (b * d) \<le> a\<^sup>2 * d\<^sup>2 + b\<^sup>2 * c\<^sup>2"
+    by simp
+qed
+
+lemma sqrt_sum_squares_le_sum_abs: "sqrt (x\<^sup>2 + y\<^sup>2) \<le> \<bar>x\<bar> + \<bar>y\<bar>"
+  by (rule power2_le_imp_le) (simp_all add: power2_sum)
 
 lemma real_sqrt_sum_squares_triangle_ineq:
   "sqrt ((a + c)\<^sup>2 + (b + d)\<^sup>2) \<le> sqrt (a\<^sup>2 + b\<^sup>2) + sqrt (c\<^sup>2 + d\<^sup>2)"
-  apply (rule power2_le_imp_le)
-   apply simp
-   apply (simp add: power2_sum)
-   apply (simp only: mult.assoc distrib_left [symmetric])
-   apply (rule mult_left_mono)
-    apply (rule power2_le_imp_le)
-     apply (simp add: power2_sum power_mult_distrib)
-     apply (simp add: ring_distribs)
-     apply (subgoal_tac "0 \<le> b\<^sup>2 * c\<^sup>2 + a\<^sup>2 * d\<^sup>2 - 2 * (a * c) * (b * d)")
-      apply simp
-     apply (rule_tac b="(a * d - b * c)\<^sup>2" in ord_le_eq_trans)
-      apply (rule zero_le_power2)
-     apply (simp add: power2_diff power_mult_distrib)
-    apply simp
-   apply simp
-  apply (simp add: add_increasing)
-  done
+proof -
+  have "(a * c + b * d) \<le> (sqrt (a\<^sup>2 + b\<^sup>2) * sqrt (c\<^sup>2 + d\<^sup>2))"
+    by (rule power2_le_imp_le) (simp_all add: power2_sum power_mult_distrib ring_distribs L2_set_mult_ineq_lemma add.commute)
+  then have "(a + c)\<^sup>2 + (b + d)\<^sup>2 \<le> (sqrt (a\<^sup>2 + b\<^sup>2) + sqrt (c\<^sup>2 + d\<^sup>2))\<^sup>2"
+    by (simp add: power2_sum)
+  then show ?thesis
+    by (auto intro: power2_le_imp_le)
+qed
 
 lemma real_sqrt_sum_squares_less: "\<bar>x\<bar> < u / sqrt 2 \<Longrightarrow> \<bar>y\<bar> < u / sqrt 2 \<Longrightarrow> sqrt (x\<^sup>2 + y\<^sup>2) < u"
   apply (rule power2_less_imp_less)
diff -r 410818a69ee3 -r e699ca8e22b7 src/HOL/Real_Vector_Spaces.thy
--- a/src/HOL/Real_Vector_Spaces.thy	Mon Jun 18 11:15:46 2018 +0200
+++ b/src/HOL/Real_Vector_Spaces.thy	Mon Jun 18 14:22:26 2018 +0100
@@ -905,6 +905,11 @@
     using assms by (force intro: power_eq_imp_eq_base)
 qed
 
+lemma power_eq_1_iff:
+  fixes w :: "'a::real_normed_div_algebra"
+  shows "w ^ n = 1 \<Longrightarrow> norm w = 1 \<or> n = 0"
+  by (metis norm_one power_0_left power_eq_0_iff power_eq_imp_eq_norm power_one)
+
 lemma norm_mult_numeral1 [simp]: "norm (numeral w * a) = numeral w * norm a"
   for a b :: "'a::{real_normed_field,field}"
   by (simp add: norm_mult)