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

Ported from HOL Light
*)

section‹Vector Cross Products in 3 Dimensions›

theory "Cross3"
  imports Determinants Cartesian_Euclidean_Space
begin

context includes no set_product_syntax
begin ―‹locally disable syntax for set product, to avoid warnings›

definition✐‹tag important› cross3 :: "[real^3, real^3] → real^3"  (infixr ‹×› 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

open_bundle cross3_syntax
begin
notation cross3 (infixr ‹×› 80)
unbundle no set_product_syntax
end


subsection‹ Basic lemmas›

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 *_R x) × y = c *_R (x × y)" for x::"real^3"
  by (simp add: cross3_simps)

lemma  cross_mult_right: "x × (c *_R y) = c *_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) *_R y - (x ∙ y) *_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))² + (x ∙ y)² = (norm x * norm y)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]) *_R y - det(vector[w,x,y]) *_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))² = (norm x)² * (norm y)² - (x ∙ y)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)² = (x ∙ y)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, opaque_lifting) 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   ‹Preservation by rotation, or other orthogonal transformation up to sign›

lemma  cross_matrix_mult: "transpose A *v ((A *v x) × (A *v y)) = det A *_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 *_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) *_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]
           ==> (f x) × (f y) = f(x × y)"
  by (simp add: cross_orthogonal_transformation orthogonal_transformation)

subsection ‹Continuity›

lemma  continuous_cross: "[continuous F f; continuous F g] ==> continuous F (λx. (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 (λx. (f x) × (g x))"
  by (simp add: continuous_on_eq_continuous_within continuous_cross)

unbundle no cross3_syntax

end