Quellcode-Bibliothek

^© Kompilation durch diese Firma

[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]

Datei: Cross3.thy Sprache: Isabelle

Original von: Isabelle^©

(* Title:      HOL/Analysis/Cross3.thy
   Author:     L C Paulson, University of Cambridge

Ported from HOL Light
*)

section\<open>Vector Cross Products in 3 Dimensions\<close>

theory "Cross3"
  imports Determinants Cartesian_Euclidean_Space
begin

context includes no_Set_Product_syntax
begin \<comment>\<open>locally disable syntax for set product, to avoid warnings\<close>

definition\<^marker>\<open>tag important\<close> cross3 :: "[real^3, real^3] \<Rightarrow> real^3"  (infixr "\<times>" 80)
  where "a \ b \
    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]"

end

bundle cross3_syntax begin
notation cross3 (infixr "\" 80)
no_notation Product_Type.Times (infixr "\" 80)
end

bundle no_cross3_syntax begin
no_notation cross3 (infixr "\" 80)
notation Product_Type.Times (infixr "\" 80)
end

unbundle cross3_syntax

subsection\<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 \ (x \ y) = 0" "x \ (y \ x) = 0" "(x \ y) \ y = 0" "(y \ x) \ y = 0"
  by (simp_all add: orthogonal_def cross3_simps)

lemma  orthogonal_cross: "orthogonal (x \ y) x" "orthogonal (x \ y) y"
                        "orthogonal y (x \ y)" "orthogonal (x \ y) x"
  by (simp_all add: orthogonal_def dot_cross_self)

lemma  cross_zero_left [simp]: "0 \ x = 0" and cross_zero_right [simp]: "x \ 0 = 0" for x::"real^3"
  by (simp_all add: cross3_simps)

lemma  cross_skew: "(x \ y) = -(y \ x)" for x::"real^3"
  by (simp add: cross3_simps)

lemma  cross_refl [simp]: "x \ x = 0" for x::"real^3"
  by (simp add: cross3_simps)

lemma  cross_add_left: "(x + y) \ z = (x \ z) + (y \ z)" for x::"real^3"
  by (simp add: cross3_simps)

lemma  cross_add_right: "x \ (y + z) = (x \ y) + (x \ z)" for x::"real^3"
  by (simp add: cross3_simps)

lemma  cross_mult_left: "(c *\<^sub>R x) \ y = c *\<^sub>R (x \ y)" for x::"real^3"
  by (simp add: cross3_simps)

lemma  cross_mult_right: "x \ (c *\<^sub>R y) = c *\<^sub>R (x \ y)" for x::"real^3"
  by (simp add: cross3_simps)

lemma  cross_minus_left [simp]: "(-x) \ y = - (x \ y)" for x::"real^3"
  by (simp add: cross3_simps)

lemma  cross_minus_right [simp]: "x \ -y = - (x \ y)" for x::"real^3"
  by (simp add: cross3_simps)

lemma  left_diff_distrib: "(x - y) \ z = x \ z - y \ z" for x::"real^3"
  by (simp add: cross3_simps)

lemma  right_diff_distrib: "x \ (y - z) = x \ y - x \ z" for x::"real^3"
  by (simp add: cross3_simps)

hide_fact (open) left_diff_distrib right_diff_distrib

proposition Jacobi: "x \ (y \ z) + y \ (z \ x) + z \ (x \ y) = 0" for x::"real^3"
  by (simp add: cross3_simps)

proposition Lagrange: "x \ (y \ z) = (x \ z) *\<^sub>R y - (x \ y) *\<^sub>R z"
  by (simp add: cross3_simps) (metis (full_types) exhaust_3)

proposition cross_triple: "(x \ y) \ z = (y \ z) \ x"
  by (simp add: cross3_def inner_vec_def sum_3 vec_eq_iff algebra_simps)

lemma  cross_components:
   "(x \ y)$1 = x$2 * y$3 - y$2 * x$3" "(x \ y)$2 = x$3 * y$1 - y$3 * x$1" "(x \ 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) \ (axis 2 1) = axis 3 1" "(axis 2 1) \ (axis 1 1) = -(axis 3 1)"
                   "(axis 2 1) \ (axis 3 1) = axis 1 1" "(axis 3 1) \ (axis 2 1) = -(axis 1 1)"
                   "(axis 3 1) \ (axis 1 1) = axis 2 1" "(axis 1 1) \ (axis 3 1) = -(axis 2 1)"
  using exhaust_3
  by (force simp add: axis_def cross3_simps)+

lemma  cross_basis_nonzero:
  "u \ 0 \ u \ axis 1 1 \ 0 \ u \ axis 2 1 \ 0 \ u \ axis 3 1 \ 0"
  by (clarsimp simp add: axis_def cross3_simps) (metis exhaust_3)

lemma  cross_dot_cancel:
  fixes x::"real^3"
  assumes deq: "x \ y = x \ z" and veq: "x \ y = x \ z" and x: "x \ 0"
  shows "y = z"
proof -
  have "x \ x \ 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 \ y))\<^sup>2 + (x \ 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 \ (y \ z) = det(vector[x,y,z])"
  by (simp add: cross3_simps)

lemma  cross_cross_det: "(w \ x) \ (y \ 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)

proposition  dot_cross: "(w \ x) \ (y \ z) = (w \ y) * (x \ z) - (w \ z) * (x \ y)"
  by (force simp add: cross3_simps)

proposition  norm_cross: "(norm (x \ y))\<^sup>2 = (norm x)\<^sup>2 * (norm y)\<^sup>2 - (x \ y)\<^sup>2"
  unfolding power2_norm_eq_inner power_mult_distrib
  by (simp add: cross3_simps power2_eq_square)

lemma  cross_eq_0: "x \ y = 0 \ collinear{0,x,y}"
proof -
  have "x \ y = 0 \ norm (x \ y) = 0"
    by simp
  also have "... \ (norm x * norm y)\<^sup>2 = (x \ y)\<^sup>2"
    using norm_cross [of x y] by (auto simp: power_mult_distrib)
  also have "... \ \x \ y\ = 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 "... \ collinear {0, x, y}"
    by (rule norm_cauchy_schwarz_equal)
  finally show ?thesis .
qed

lemma  cross_eq_self: "x \ y = x \ x = 0" "x \ y = y \ 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 \ y = 0 \ x \ y = 0 \ x = 0 \ 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(\)"
  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) \ (A *v y)) = det A *\<^sub>R (x \ 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) \ (A *v y) = det A *\<^sub>R (A *v (x \ 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 \ (A *v x) \ (A *v y) = A *v (x \ y)"
  by (simp add: rotation_matrix_def cross_orthogonal_matrix)

lemma  cross_rotoinversion_matrix: "rotoinversion_matrix A \ (A *v x) \ (A *v y) = - A *v (x \ 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) \ (f y) = det(matrix f) *\<^sub>R f(x \ 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:
   "\linear f; \x. norm(f x) = norm x; det(matrix f) = 1\
           \<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: "\continuous F f; continuous F g\ \ continuous F (\x. (f x) \ (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 \ real^3"
  shows "\continuous_on S f; continuous_on S g\ \ continuous_on S (\x. (f x) \ (g x))"
  by (simp add: continuous_on_eq_continuous_within continuous_cross)

unbundle no_cross3_syntax

end

¤ Dauer der Verarbeitung: 0.18 Sekunden (vorverarbeitet) ¤

Download des Quellennavigators
Download des sprechenden Kalenders
in der Quellcodebibliothek suchen

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.