Theory Linear_Algebra

(*  Title:      HOL/Analysis/Linear_Algebra.thy
    Author:     Amine Chaieb, University of Cambridge

section ‹Elementary Linear Algebra on Euclidean Spaces›

theory Linear_Algebra

lemma linear_simps:
  assumes "bounded_linear f"
    "f (a + b) = f a + f b"
    "f (a - b) = f a - f b"
    "f 0 = 0"
    "f (- a) = - f a"
    "f (s *R v) = s *R (f v)"
proof -
  interpret f: bounded_linear f by fact
  show "f (a + b) = f a + f b" by (rule f.add)
  show "f (a - b) = f a - f b" by (rule f.diff)
  show "f 0 = 0" by (rule
  show "f (- a) = - f a" by (rule f.neg)
  show "f (s *R v) = s *R (f v)" by (rule f.scale)

lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x  (UNIV::'a::finite set)}"
  using finite finite_image_set by blast

lemma substdbasis_expansion_unique:
  includes inner_syntax
  assumes d: "d  Basis"
  shows "(id. f i *R i) = (x::'a::euclidean_space) 
    (iBasis. (i  d  f i = x  i)  (i  d  x  i = 0))"
proof -
  have *: "x a b P. x * (if P then a else b) = (if P then x * a else x * b)"
    by auto
  have **: "finite d"
    by (auto intro: finite_subset[OF assms])
  have ***: "i. i  Basis  (id. f i *R i)  i = (xd. if x = i then f x else 0)"
    using d
    by (auto intro!: sum.cong simp: inner_Basis inner_sum_left)
  show ?thesis
    unfolding euclidean_eq_iff[where 'a='a] by (auto simp:[OF **] ***)

lemma independent_substdbasis: "d  Basis  independent d"
  by (rule independent_mono[OF independent_Basis])

lemma subset_translation_eq [simp]:
    fixes a :: "'a::real_vector" shows "(+) a ` s  (+) a ` t  s  t"
  by auto

lemma translate_inj_on:
  fixes A :: "'a::ab_group_add set"
  shows "inj_on (λx. a + x) A"
  unfolding inj_on_def by auto

lemma translation_assoc:
  fixes a b :: "'a::ab_group_add"
  shows "(λx. b + x) ` ((λx. a + x) ` S) = (λx. (a + b) + x) ` S"
  by auto

lemma translation_invert:
  fixes a :: "'a::ab_group_add"
  assumes "(λx. a + x) ` A = (λx. a + x) ` B"
  shows "A = B"
  using assms translation_assoc by fastforce

lemma translation_galois:
  fixes a :: "'a::ab_group_add"
  shows "T = ((λx. a + x) ` S)  S = ((λx. (- a) + x) ` T)"
  by (metis add.right_inverse group_cancel.rule0 translation_invert translation_assoc)

lemma translation_inverse_subset:
  assumes "((λx. - a + x) ` V)  (S :: 'n::ab_group_add set)"
  shows "V  ((λx. a + x) ` S)"
  by (metis assms subset_image_iff translation_galois)

subsectiontag unimportant› ‹More interesting properties of the norm›

unbundle inner_syntax

text‹Equality of vectors in terms of term(∙) products.›

lemma linear_componentwise:
  fixes f:: "'a::euclidean_space  'b::real_inner"
  assumes lf: "linear f"
  shows "(f x)  j = (iBasis. (xi) * (f ij))" (is "?lhs = ?rhs")
proof -
  interpret linear f by fact
  have "?rhs = (iBasis. (xi) *R (f i))j"
    by (simp add: inner_sum_left)
  then show ?thesis
    by (simp add: euclidean_representation sum[symmetric] scale[symmetric])

lemma vector_eq: "x = y  x  x = x  y  y  y = x  x"
  by (metis (no_types, opaque_lifting) inner_commute inner_diff_right inner_eq_zero_iff right_minus_eq)

lemma norm_triangle_half_r:
  "norm (y - x1) < e/2  norm (y - x2) < e/2  norm (x1 - x2) < e"
  using dist_triangle_half_r unfolding dist_norm[symmetric] by auto

lemma norm_triangle_half_l:
  assumes "norm (x - y) < e/2" and "norm (x' - y) < e/2"
  shows "norm (x - x') < e"
  by (metis assms dist_norm dist_triangle_half_l)

lemma abs_triangle_half_r:
  fixes y :: "'a::linordered_field"
  shows "abs (y - x1) < e/2  abs (y - x2) < e/2  abs (x1 - x2) < e"
  by linarith

lemma abs_triangle_half_l:
  fixes y :: "'a::linordered_field"
  assumes "abs (x - y) < e/2" and "abs (x' - y) < e/2"
  shows "abs (x - x') < e"
  using assms by linarith

lemma sum_clauses:
  shows "sum f {} = 0"
    and "finite S  sum f (insert x S) = (if x  S then sum f S else f x + sum f S)"
  by (auto simp add: insert_absorb)

lemma vector_eq_ldot: "(x. x  y = x  z)  y = z" and vector_eq_rdot: "(z. x  z = y  z)  x = y"
  by (metis inner_commute vector_eq)+

subsection ‹Substandard Basis›

lemma ex_card:
  assumes "n  card A"
  shows "SA. card S = n"
  by (meson assms obtain_subset_with_card_n)

lemma subspace_substandard: "subspace {x::'a::euclidean_space. (iBasis. P i  xi = 0)}"
  by (auto simp: subspace_def inner_add_left)

lemma dim_substandard:
  assumes d: "d  Basis"
  shows "dim {x::'a::euclidean_space. iBasis. i  d  xi = 0} = card d" (is "dim ?A = _")
proof (rule dim_unique)
  from d show "d  ?A"
    by (auto simp: inner_Basis)
  from d show "independent d"
    by (rule independent_mono [OF independent_Basis])
  have "x  span d" if "iBasis. i  d  x  i = 0" for x
  proof -
    have "finite d"
      by (rule finite_subset [OF d finite_Basis])
    then have "(id. (x  i) *R i)  span d"
      by (simp add: span_sum span_clauses)
    also have "(id. (x  i) *R i) = (iBasis. (x  i) *R i)"
      by (rule sum.mono_neutral_cong_left [OF finite_Basis d]) (auto simp: that)
    finally show "x  span d"
      by (simp only: euclidean_representation)
  then show "?A  span d" by auto
qed simp

subsection ‹Orthogonality›

definitiontag important› (in real_inner) "orthogonal x y  x  y = 0"

context real_inner

lemma orthogonal_self: "orthogonal x x  x = 0"
  by (simp add: orthogonal_def)

lemma orthogonal_clauses:
  "orthogonal a 0"
  "orthogonal a x  orthogonal a (c *R x)"
  "orthogonal a x  orthogonal a (- x)"
  "orthogonal a x  orthogonal a y  orthogonal a (x + y)"
  "orthogonal a x  orthogonal a y  orthogonal a (x - y)"
  "orthogonal 0 a"
  "orthogonal x a  orthogonal (c *R x) a"
  "orthogonal x a  orthogonal (- x) a"
  "orthogonal x a  orthogonal y a  orthogonal (x + y) a"
  "orthogonal x a  orthogonal y a  orthogonal (x - y) a"
  unfolding orthogonal_def inner_add inner_diff by auto


lemma orthogonal_commute: "orthogonal x y  orthogonal y x"
  by (simp add: orthogonal_def inner_commute)

lemma orthogonal_scaleR [simp]: "c  0  orthogonal (c *R x) = orthogonal x"
  by (rule ext) (simp add: orthogonal_def)

lemma pairwise_ortho_scaleR:
    "pairwise (λi j. orthogonal (f i) (g j)) B
     pairwise (λi j. orthogonal (a i *R f i) (a j *R g j)) B"
  by (auto simp: pairwise_def orthogonal_clauses)

lemma orthogonal_rvsum:
    "finite s; y. y  s  orthogonal x (f y)  orthogonal x (sum f s)"
  by (induction s rule: finite_induct) (auto simp: orthogonal_clauses)

lemma orthogonal_lvsum:
    "finite s; x. x  s  orthogonal (f x) y  orthogonal (sum f s) y"
  by (induction s rule: finite_induct) (auto simp: orthogonal_clauses)

lemma norm_add_Pythagorean:
  assumes "orthogonal a b"
    shows "(norm (a + b))2 = (norm a)2 + (norm b)2"
proof -
  from assms have "(a - (0 - b))  (a - (0 - b)) = a  a - (0 - b  b)"
    by (simp add: algebra_simps orthogonal_def inner_commute)
  then show ?thesis
    by (simp add: power2_norm_eq_inner)

lemma norm_sum_Pythagorean:
  assumes "finite I" "pairwise (λi j. orthogonal (f i) (f j)) I"
    shows "(norm (sum f I))2 = (iI. (norm (f i))2)"
using assms
proof (induction I rule: finite_induct)
  case empty then show ?case by simp
  case (insert x I)
  then have "orthogonal (f x) (sum f I)"
    by (metis pairwise_insert orthogonal_rvsum)
  with insert show ?case
    by (simp add: pairwise_insert norm_add_Pythagorean)

subsection  ‹Orthogonality of a transformation›

definitiontag important›  "orthogonal_transformation f  linear f  (v w. f v  f w = v  w)"

lemmatag unimportant›  orthogonal_transformation:
  "orthogonal_transformation f  linear f  (v. norm (f v) = norm v)"
  by (smt (verit, ccfv_threshold) dot_norm linear_add norm_eq_sqrt_inner orthogonal_transformation_def)

lemmatag unimportant›  orthogonal_transformation_id [simp]: "orthogonal_transformation (λx. x)"
  by (simp add: linear_iff orthogonal_transformation_def)

lemmatag unimportant›  orthogonal_orthogonal_transformation:
    "orthogonal_transformation f  orthogonal (f x) (f y)  orthogonal x y"
  by (simp add: orthogonal_def orthogonal_transformation_def)

lemmatag unimportant›  orthogonal_transformation_compose:
   "orthogonal_transformation f; orthogonal_transformation g  orthogonal_transformation(f  g)"
  by (auto simp: orthogonal_transformation_def linear_compose)

lemmatag unimportant›  orthogonal_transformation_neg:
  "orthogonal_transformation(λx. -(f x))  orthogonal_transformation f"
  by (auto simp: orthogonal_transformation_def dest: linear_compose_neg)

lemmatag unimportant›  orthogonal_transformation_scaleR: "orthogonal_transformation f  f (c *R v) = c *R f v"
  by (simp add: linear_iff orthogonal_transformation_def)

lemmatag unimportant›  orthogonal_transformation_linear:
   "orthogonal_transformation f  linear f"
  by (simp add: orthogonal_transformation_def)

lemmatag unimportant›  orthogonal_transformation_inj:
  "orthogonal_transformation f  inj f"
  unfolding orthogonal_transformation_def inj_on_def
  by (metis vector_eq)

lemmatag unimportant›  orthogonal_transformation_surj:
  "orthogonal_transformation f  surj f"
  for f :: "'a::euclidean_space  'a::euclidean_space"
  by (simp add: linear_injective_imp_surjective orthogonal_transformation_inj orthogonal_transformation_linear)

lemmatag unimportant›  orthogonal_transformation_bij:
  "orthogonal_transformation f  bij f"
  for f :: "'a::euclidean_space  'a::euclidean_space"
  by (simp add: bij_def orthogonal_transformation_inj orthogonal_transformation_surj)

lemmatag unimportant›  orthogonal_transformation_inv:
  "orthogonal_transformation f  orthogonal_transformation (inv f)"
  for f :: "'a::euclidean_space  'a::euclidean_space"
  by (metis (no_types, opaque_lifting) bijection.inv_right bijection_def inj_linear_imp_inv_linear orthogonal_transformation orthogonal_transformation_bij orthogonal_transformation_inj)

lemmatag unimportant›  orthogonal_transformation_norm:
  "orthogonal_transformation f  norm (f x) = norm x"
  by (metis orthogonal_transformation)

subsection ‹Bilinear functions›

definitiontag important›
bilinear :: "('a::real_vector  'b::real_vector  'c::real_vector)  bool" where
"bilinear f  (x. linear (λy. f x y))  (y. linear (λx. f x y))"

lemma bilinear_ladd: "bilinear h  h (x + y) z = h x z + h y z"
  by (simp add: bilinear_def linear_iff)

lemma bilinear_radd: "bilinear h  h x (y + z) = h x y + h x z"
  by (simp add: bilinear_def linear_iff)

lemma bilinear_times:
  fixes c::"'a::real_algebra" shows "bilinear (λx y::'a. x*y)"
  by (auto simp: bilinear_def distrib_left distrib_right intro!: linearI)

lemma bilinear_lmul: "bilinear h  h (c *R x) y = c *R h x y"
  by (simp add: bilinear_def linear_iff)

lemma bilinear_rmul: "bilinear h  h x (c *R y) = c *R h x y"
  by (simp add: bilinear_def linear_iff)

lemma bilinear_lneg: "bilinear h  h (- x) y = - h x y"
  by (drule bilinear_lmul [of _ "- 1"]) simp

lemma bilinear_rneg: "bilinear h  h x (- y) = - h x y"
  by (drule bilinear_rmul [of _ _ "- 1"]) simp

lemma (in ab_group_add) eq_add_iff: "x = x + y  y = 0"
  using add_left_imp_eq[of x y 0] by auto

lemma bilinear_lzero:
  assumes "bilinear h"
  shows "h 0 x = 0"
  using bilinear_ladd [OF assms, of 0 0 x] by (simp add: eq_add_iff field_simps)

lemma bilinear_rzero:
  assumes "bilinear h"
  shows "h x 0 = 0"
  using bilinear_radd [OF assms, of x 0 0 ] by (simp add: eq_add_iff field_simps)

lemma bilinear_lsub: "bilinear h  h (x - y) z = h x z - h y z"
  using bilinear_ladd [of h x "- y"] by (simp add: bilinear_lneg)

lemma bilinear_rsub: "bilinear h  h z (x - y) = h z x - h z y"
  using bilinear_radd [of h _ x "- y"] by (simp add: bilinear_rneg)

lemma bilinear_sum:
  assumes "bilinear h"
  shows "h (sum f S) (sum g T) = sum (λ(i,j). h (f i) (g j)) (S × T) "
proof -
  interpret l: linear "λx. h x y" for y using assms by (simp add: bilinear_def)
  interpret r: linear "λy. h x y" for x using assms by (simp add: bilinear_def)
  have "h (sum f S) (sum g T) = sum (λx. h (f x) (sum g T)) S"
    by (simp add: l.sum)
  also have " = sum (λx. sum (λy. h (f x) (g y)) T) S"
    by (rule sum.cong) (simp_all add: r.sum)
  finally show ?thesis
    unfolding sum.cartesian_product .

subsection ‹Adjoints›

definitiontag important› adjoint :: "(('a::real_inner)  ('b::real_inner))  'b  'a" where
"adjoint f = (SOME f'. x y. f x  y = x  f' y)"

lemma adjoint_unique:
  assumes "x y. inner (f x) y = inner x (g y)"
  shows "adjoint f = g"
  unfolding adjoint_def
proof (rule some_equality)
  show "x y. inner (f x) y = inner x (g y)"
    by (rule assms)
  fix h
  assume "x y. inner (f x) y = inner x (h y)"
  then show "h = g"
    by (metis assms ext vector_eq_ldot) 

text ‹TODO: The following lemmas about adjoints should hold for any
  Hilbert space (i.e. complete inner product space).
  (see 🌐‹›)

lemma adjoint_works:
  fixes f :: "'n::euclidean_space  'm::euclidean_space"
  assumes lf: "linear f"
  shows "x  adjoint f y = f x  y"
proof -
  interpret linear f by fact
  have "y. w. x. f x  y = x  w"
  proof (intro allI exI)
    fix y :: "'m" and x
    let ?w = "(iBasis. (f i  y) *R i) :: 'n"
    have "f x  y = f (iBasis. (x  i) *R i)  y"
      by (simp add: euclidean_representation)
    also have " = (iBasis. (x  i) *R f i)  y"
      by (simp add: sum scale)
    finally show "f x  y = x  ?w"
      by (simp add: inner_sum_left inner_sum_right mult.commute)
  then show ?thesis
    unfolding adjoint_def choice_iff
    by (intro someI2_ex[where Q="λf'. x  f' y = f x  y"]) auto

lemma adjoint_clauses:
  fixes f :: "'n::euclidean_space  'm::euclidean_space"
  assumes lf: "linear f"
  shows "x  adjoint f y = f x  y"
    and "adjoint f y  x = y  f x"
  by (simp_all add: adjoint_works[OF lf] inner_commute)

lemma adjoint_linear:
  fixes f :: "'n::euclidean_space  'm::euclidean_space"
  assumes lf: "linear f"
  shows "linear (adjoint f)"
  by (simp add: lf linear_iff euclidean_eq_iff[where 'a='n] euclidean_eq_iff[where 'a='m]
    adjoint_clauses[OF lf] inner_distrib)

lemma adjoint_adjoint:
  fixes f :: "'n::euclidean_space  'm::euclidean_space"
  assumes lf: "linear f"
  shows "adjoint (adjoint f) = f"
  by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])

subsectiontag unimportant› ‹Euclidean Spaces as Typeclass›

lemma independent_Basis: "independent Basis"
  by (rule independent_Basis)

lemma span_Basis [simp]: "span Basis = UNIV"
  by (rule span_Basis)

lemma in_span_Basis: "x  span Basis"
  unfolding span_Basis ..

subsectiontag unimportant› ‹Linearity and Bilinearity continued›

lemma linear_bounded:
  fixes f :: "'a::euclidean_space  'b::real_normed_vector"
  assumes lf: "linear f"
  shows "B. x. norm (f x)  B * norm x"
  interpret linear f by fact
  let ?B = "bBasis. norm (f b)"
  show "x. norm (f x)  ?B * norm x"
    fix x :: 'a
    let ?g = "λb. (x  b) *R f b"
    have "norm (f x) = norm (f (bBasis. (x  b) *R b))"
      unfolding euclidean_representation ..
    also have " = norm (sum ?g Basis)"
      by (simp add: sum scale)
    finally have th0: "norm (f x) = norm (sum ?g Basis)" .
    have th: "norm (?g i)  norm (f i) * norm x" if "i  Basis" for i
    proof -
      from Basis_le_norm[OF that, of x]
      show "norm (?g i)  norm (f i) * norm x"
        unfolding norm_scaleR by (metis mult.commute mult_left_mono norm_ge_zero)
    from sum_norm_le[of _ ?g, OF th]
    show "norm (f x)  ?B * norm x"
      by (simp add: sum_distrib_right th0)

lemma linear_conv_bounded_linear:
  fixes f :: "'a::euclidean_space  'b::real_normed_vector"
  shows "linear f  bounded_linear f"
  by (metis mult.commute bounded_linear_axioms.intro bounded_linear_def linear_bounded)

lemmas linear_linear = linear_conv_bounded_linear[symmetric]

lemma inj_linear_imp_inv_bounded_linear:
  fixes f::"'a::euclidean_space  'a"
  shows "bounded_linear f; inj f  bounded_linear (inv f)"
  by (simp add: inj_linear_imp_inv_linear linear_linear)

lemma linear_bounded_pos:
  fixes f :: "'a::euclidean_space  'b::real_normed_vector"
  assumes lf: "linear f"
 obtains B where "B > 0" "x. norm (f x)  B * norm x"
  by (metis bounded_linear.pos_bounded lf linear_linear mult.commute)

lemma linear_invertible_bounded_below_pos:
  fixes f :: "'a::real_normed_vector  'b::euclidean_space"
  assumes "linear f" "linear g" and gf: "g  f = id"
  obtains B where "B > 0" "x. B * norm x  norm(f x)"
proof -
  obtain B where "B > 0" and B: "x. norm (g x)  B * norm x"
    using linear_bounded_pos [OF linear g] by blast
  show thesis
    show "0 < 1/B"
      by (simp add: B > 0)
    show "1/B * norm x  norm (f x)" for x
      by (smt (verit, ccfv_SIG) B 0 < B gf comp_apply divide_inverse id_apply inverse_eq_divide 
              less_divide_eq mult.commute)

lemma linear_inj_bounded_below_pos:
  fixes f :: "'a::real_normed_vector  'b::euclidean_space"
  assumes "linear f" "inj f"
  obtains B where "B > 0" "x. B * norm x  norm(f x)"
  using linear_injective_left_inverse [OF assms]
    linear_invertible_bounded_below_pos assms by blast

lemma bounded_linearI':
  fixes f ::"'a::euclidean_space  'b::real_normed_vector"
  assumes "x y. f (x + y) = f x + f y"
    and "c x. f (c *R x) = c *R f x"
  shows "bounded_linear f"
  using assms linearI linear_conv_bounded_linear by blast

lemma bilinear_bounded:
  fixes h :: "'m::euclidean_space  'n::euclidean_space  'k::real_normed_vector"
  assumes bh: "bilinear h"
  shows "B. x y. norm (h x y)  B * norm x * norm y"
proof (clarify intro!: exI[of _ "iBasis. jBasis. norm (h i j)"])
  fix x :: 'm
  fix y :: 'n
  have "norm (h x y) = norm (h (sum (λi. (x  i) *R i) Basis) (sum (λi. (y  i) *R i) Basis))"
    by (simp add: euclidean_representation)
  also have " = norm (sum (λ (i,j). h ((x  i) *R i) ((y  j) *R j)) (Basis × Basis))"
    unfolding bilinear_sum[OF bh] ..
  finally have th: "norm (h x y) = " .
  have "i j. i  Basis; j  Basis
            ¦x  i¦ * (¦y  j¦ * norm (h i j))  norm x * (norm y * norm (h i j))"
    by (auto simp add: zero_le_mult_iff Basis_le_norm mult_mono)
  then show "norm (h x y)  (iBasis. jBasis. norm (h i j)) * norm x * norm y"
    unfolding sum_distrib_right th sum.cartesian_product
    by (clarsimp simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
      field_simps simp del: scaleR_scaleR intro!: sum_norm_le)

lemma bilinear_conv_bounded_bilinear:
  fixes h :: "'a::euclidean_space  'b::euclidean_space  'c::real_normed_vector"
  shows "bilinear h  bounded_bilinear h"
  assume "bilinear h"
  show "bounded_bilinear h"
    fix x y z
    show "h (x + y) z = h x z + h y z"
      using bilinear h unfolding bilinear_def linear_iff by simp
    fix x y z
    show "h x (y + z) = h x y + h x z"
      using bilinear h unfolding bilinear_def linear_iff by simp
    show "h (scaleR r x) y = scaleR r (h x y)" "h x (scaleR r y) = scaleR r (h x y)" for r x y
      using bilinear h unfolding bilinear_def linear_iff
      by simp_all
    have "B. x y. norm (h x y)  B * norm x * norm y"
      using bilinear h by (rule bilinear_bounded)
    then show "K. x y. norm (h x y)  norm x * norm y * K"
      by (simp add: ac_simps)
  assume "bounded_bilinear h"
  then interpret h: bounded_bilinear h .
  show "bilinear h"
    unfolding bilinear_def linear_conv_bounded_linear
    using h.bounded_linear_left h.bounded_linear_right by simp

lemma bilinear_bounded_pos:
  fixes h :: "'a::euclidean_space  'b::euclidean_space  'c::real_normed_vector"
  assumes bh: "bilinear h"
  shows "B > 0. x y. norm (h x y)  B * norm x * norm y"
  by (metis mult.assoc bh bilinear_conv_bounded_bilinear bounded_bilinear.pos_bounded mult.commute)

lemma bounded_linear_imp_has_derivative: 
  "bounded_linear f  (f has_derivative f) net"
  by (auto simp add: has_derivative_def linear_diff linear_linear linear_def
      dest: bounded_linear.linear)

lemma linear_imp_has_derivative:
  fixes f :: "'a::euclidean_space  'b::real_normed_vector"
  shows "linear f  (f has_derivative f) net"
  by (simp add: bounded_linear_imp_has_derivative linear_conv_bounded_linear)

lemma bounded_linear_imp_differentiable: "bounded_linear f  f differentiable net"
  using bounded_linear_imp_has_derivative differentiable_def by blast

lemma linear_imp_differentiable:
  fixes f :: "'a::euclidean_space  'b::real_normed_vector"
  shows "linear f  f differentiable net"
  by (metis linear_imp_has_derivative differentiable_def)

lemma of_real_differentiable [simp,derivative_intros]: "of_real differentiable F"
  by (simp add: bounded_linear_imp_differentiable bounded_linear_of_real)

subsectiontag unimportant› ‹We continue›

lemma independent_bound:
  fixes S :: "'a::euclidean_space set"
  shows "independent S  finite S  card S  DIM('a)"
  by (metis dim_subset_UNIV finiteI_independent dim_span_eq_card_independent)

lemmas independent_imp_finite = finiteI_independent

corollarytag unimportant› independent_card_le:
  fixes S :: "'a::euclidean_space set"
  assumes "independent S"
  shows "card S  DIM('a)"
  using assms independent_bound by auto

lemma dependent_biggerset:
  fixes S :: "'a::euclidean_space set"
  shows "(finite S  card S > DIM('a))  dependent S"
  by (metis independent_bound not_less)

text ‹Picking an orthogonal replacement for a spanning set.›

lemma vector_sub_project_orthogonal:
  fixes b x :: "'a::euclidean_space"
  shows "b  (x - ((b  x) / (b  b)) *R b) = 0"
  unfolding inner_simps by auto

lemma pairwise_orthogonal_insert:
  assumes "pairwise orthogonal S"
    and "y. y  S  orthogonal x y"
  shows "pairwise orthogonal (insert x S)"
  using assms by (auto simp: pairwise_def orthogonal_commute)

lemma basis_orthogonal:
  fixes B :: "'a::real_inner set"
  assumes fB: "finite B"
  shows "C. finite C  card C  card B  span C = span B  pairwise orthogonal C"
  (is " C. ?P B C")
  using fB
proof (induct rule: finite_induct)
  case empty
  then show ?case
    using pairwise_empty by blast
  case (insert a B)
  note fB = finite B and aB = a  B
  from C. finite C  card C  card B  span C = span B  pairwise orthogonal C
  obtain C where C: "finite C" "card C  card B"
    "span C = span B" "pairwise orthogonal C" by blast
  let ?a = "a - sum (λx. (x  a / (x  x)) *R x) C"
  let ?C = "insert ?a C"
  from C(1) have fC: "finite ?C"
    by simp
  have cC: "card ?C  card (insert a B)"
    using C aB card_insert_if local.insert(1) by fastforce
    fix x k
    have th0: "(a::'a) b c. a - (b - c) = c + (a - b)"
      by (simp add: field_simps)
    have "x - k *R (a - (xC. (x  a / (x  x)) *R x))  span C  x - k *R a  span C"
      unfolding scaleR_right_diff_distrib th0
      by (intro span_add_eq span_scale span_sum span_base)
  then have SC: "span ?C = span (insert a B)"
    unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto
    fix y
    assume yC: "y  C"
    then have Cy: "C = insert y (C - {y})"
      by blast
    have fth: "finite (C - {y})"
      using C by simp
    have "y  0  xC - {y}. x  a * (x  y) / (x  x) = 0"
      using pairwise orthogonal C
      by (metis Cy DiffE div_0 insertCI mult_zero_right orthogonal_def pairwise_insert)
    then have "orthogonal ?a y"
      unfolding orthogonal_def
      unfolding inner_diff inner_sum_left right_minus_eq
      unfolding sum.remove [OF finite C y  C]
      by (auto simp add: sum.neutral inner_commute[of y a])
  with pairwise orthogonal C have CPO: "pairwise orthogonal ?C"
    by (rule pairwise_orthogonal_insert)
  from fC cC SC CPO have "?P (insert a B) ?C"
    by blast
  then show ?case by blast

lemma orthogonal_basis_exists:
  fixes V :: "('a::euclidean_space) set"
  shows "B. independent B  B  span V  V  span B  (card B = dim V)  pairwise orthogonal B"
proof -
  from basis_exists[of V] obtain B where
    B: "B  V" "independent B" "V  span B" "card B = dim V"
    by force
  from B have fB: "finite B" "card B = dim V"
    using independent_bound by auto
  from basis_orthogonal[OF fB(1)] obtain C where
    C: "finite C" "card C  card B" "span C = span B" "pairwise orthogonal C"
    by blast
  from C B have CSV: "C  span V"
    by (metis span_superset span_mono subset_trans)
  from span_mono[OF B(3)] C have SVC: "span V  span C"
    by (simp add: span_span)
  from C fB have "card C  dim V"
    by simp
  moreover have "dim V  card C"
    using span_card_ge_dim[OF CSV SVC C(1)]
    by simp
  ultimately have "card C = dim V"
    using C(1) by simp
  with C B CSV show ?thesis
    by (metis SVC card_eq_dim dim_span)

text ‹Low-dimensional subset is in a hyperplane (weak orthogonal complement).›

lemma span_not_univ_orthogonal:
  fixes S :: "'a::euclidean_space set"
  assumes sU: "span S  UNIV"
  shows "a::'a. a  0  (x  span S. a  x = 0)"
proof -
  from sU obtain a where a: "a  span S"
    by blast
  from orthogonal_basis_exists obtain B where
    B: "independent B" "B  span S" "S  span B" "card B = dim S" "pairwise orthogonal B"
    by blast
  from B have fB: "finite B" "card B = dim S"
    using independent_bound by auto
  have sSB: "span S = span B"
    by (simp add: B span_eq)
  let ?a = "a - sum (λb. (a  b / (b  b)) *R b) B"
  have "sum (λb. (a  b / (b  b)) *R b) B  span S"
    by (simp add: sSB span_base span_mul span_sum)
  with a have a0:"?a   0"
    by auto
  have "?a  x = 0" if "xspan B" for x
  proof (rule span_induct [OF that])
    show "subspace {x. ?a  x = 0}"
      by (auto simp add: subspace_def inner_add)
      fix x
      assume x: "x  B"
      from x have B': "B = insert x (B - {x})"
        by blast
      have fth: "finite (B - {x})"
        using fB by simp
      have "(bB - {x}. a  b * (b  x) / (b  b)) = 0" if "x  0"
        by (smt (verit) B' B(5) DiffD2 divide_eq_0_iff inner_real_def inner_zero_right insertCI orthogonal_def pairwise_insert sum.neutral)
      then have "?a  x = 0"
        apply (subst B')
        using fB fth
        unfolding sum_clauses(2)[OF fth]
        by (auto simp add: inner_add_left inner_diff_left inner_sum_left)
    then show "?a  x = 0" if "x  B" for x
      using that by blast
  with a0 sSB show ?thesis
    by blast

lemma span_not_univ_subset_hyperplane:
  fixes S :: "'a::euclidean_space set"
  assumes SU: "span S  UNIV"
  shows " a. a 0  span S  {x. a  x = 0}"
  using span_not_univ_orthogonal[OF SU] by auto

lemma lowdim_subset_hyperplane:
  fixes S :: "'a::euclidean_space set"
  assumes d: "dim S < DIM('a)"
  shows "a::'a. a  0  span S  {x. a  x = 0}"
  using d dim_eq_full nless_le span_not_univ_subset_hyperplane by blast

lemma linear_eq_stdbasis:
  fixes f :: "'a::euclidean_space  _"
  assumes lf: "linear f"
    and lg: "linear g"
    and fg: "b. b  Basis  f b = g b"
  shows "f = g"
  using linear_eq_on_span[OF lf lg, of Basis] fg by auto

text ‹Similar results for bilinear functions.›

lemma bilinear_eq:
  assumes bf: "bilinear f"
    and bg: "bilinear g"
    and SB: "S  span B"
    and TC: "T  span C"
    and "xS" "yT"
    and fg: "x y. x  B; y C  f x y = g x y"
  shows "f x y = g x y"
proof -
  let ?P = "{x. y span C. f x y = g x y}"
  from bf bg have sp: "subspace ?P"
    unfolding bilinear_def linear_iff subspace_def bf bg
    by (auto simp add: span_zero bilinear_lzero[OF bf] bilinear_lzero[OF bg]
        span_add Ball_def
      intro: bilinear_ladd[OF bf])
  have sfg: "x. x  B  subspace {a. f x a = g x a}"
    by (auto simp: subspace_def bf bg bilinear_rzero bilinear_radd bilinear_rmul)
  have "y span C. f x y = g x y" if "x  span B" for x
    using span_induct [OF that sp] fg sfg span_induct by blast
  then show ?thesis
    using SB TC assms by auto

lemma bilinear_eq_stdbasis:
  fixes f :: "'a::euclidean_space  'b::euclidean_space  _"
  assumes bf: "bilinear f"
    and bg: "bilinear g"
    and fg: "i j. i  Basis  j  Basis  f i j = g i j"
  shows "f = g"
  using bilinear_eq[OF bf bg equalityD2[OF span_Basis] equalityD2[OF span_Basis]] fg by blast

subsection ‹Infinity norm›

definitiontag important› "infnorm (x::'a::euclidean_space) = Sup {¦x  b¦ |b. b  Basis}"

lemma infnorm_set_image:
  fixes x :: "'a::euclidean_space"
  shows "{¦x  i¦ |i. i  Basis} = (λi. ¦x  i¦) ` Basis"
  by blast

lemma infnorm_Max:
  fixes x :: "'a::euclidean_space"
  shows "infnorm x = Max ((λi. ¦x  i¦) ` Basis)"
  by (simp add: infnorm_def infnorm_set_image cSup_eq_Max)

lemma infnorm_set_lemma:
  fixes x :: "'a::euclidean_space"
  shows "finite {¦x  i¦ |i. i  Basis}"
    and "{¦x  i¦ |i. i  Basis}  {}"
  unfolding infnorm_set_image by auto

lemma infnorm_pos_le:
  fixes x :: "'a::euclidean_space"
  shows "0  infnorm x"
  by (simp add: infnorm_Max Max_ge_iff ex_in_conv)

lemma infnorm_triangle:
  fixes x :: "'a::euclidean_space"
  shows "infnorm (x + y)  infnorm x + infnorm y"
proof -
  have *: "a b c d :: real. ¦a¦  c  ¦b¦  d  ¦a + b¦  c + d"
    by simp
  show ?thesis
    by (auto simp: infnorm_Max inner_add_left intro!: *)

lemma infnorm_eq_0:
  fixes x :: "'a::euclidean_space"
  shows "infnorm x = 0  x = 0"
proof -
  have "infnorm x  0  x = 0"
    unfolding infnorm_Max by (simp add: euclidean_all_zero_iff)
  then show ?thesis
    using infnorm_pos_le[of x] by simp

lemma infnorm_0: "infnorm 0 = 0"
  by (simp add: infnorm_eq_0)

lemma infnorm_neg: "infnorm (- x) = infnorm x"
  unfolding infnorm_def by simp

lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)"
  by (metis infnorm_neg minus_diff_eq)

lemma absdiff_infnorm: "¦infnorm x - infnorm y¦  infnorm (x - y)"
  by (smt (verit, del_insts) diff_add_cancel infnorm_sub infnorm_triangle)

lemma real_abs_infnorm: "¦infnorm x¦ = infnorm x"
  using infnorm_pos_le[of x] by arith

lemma Basis_le_infnorm:
  fixes x :: "'a::euclidean_space"
  shows "b  Basis  ¦x  b¦  infnorm x"
  by (simp add: infnorm_Max)

lemma infnorm_mul: "infnorm (a *R x) = ¦a¦ * infnorm x"
  unfolding infnorm_Max
proof (safe intro!: Max_eqI)
  let ?B = "(λi. ¦x  i¦) ` Basis"
  { fix b :: 'a
    assume "b  Basis"
    then show "¦a *R x  b¦  ¦a¦ * Max ?B"
      by (simp add: abs_mult mult_left_mono)
    from Max_in[of ?B] obtain b where "b  Basis" "Max ?B = ¦x  b¦"
      by (auto simp del: Max_in)
    then show "¦a¦ * Max ((λi. ¦x  i¦) ` Basis)  (λi. ¦a *R x  i¦) ` Basis"
      by (intro image_eqI[where x=b]) (auto simp: abs_mult)
qed simp

lemma infnorm_mul_lemma: "infnorm (a *R x)  ¦a¦ * infnorm x"
  unfolding infnorm_mul ..

lemma infnorm_pos_lt: "infnorm x > 0  x  0"
  using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith

text ‹Prove that it differs only up to a bound from Euclidean norm.›

lemma infnorm_le_norm: "infnorm x  norm x"
  by (simp add: Basis_le_norm infnorm_Max)

lemma norm_le_infnorm:
  fixes x :: "'a::euclidean_space"
  shows "norm x  sqrt DIM('a) * infnorm x"
  unfolding norm_eq_sqrt_inner id_def
proof (rule real_le_lsqrt)
  show "sqrt DIM('a) * infnorm x  0"
    by (simp add: zero_le_mult_iff infnorm_pos_le)
  have "x  x  (bBasis. x  b * (x  b))"
    by (metis euclidean_inner order_refl)
  also have "  DIM('a) * ¦infnorm x¦2"
    by (rule sum_bounded_above) (metis Basis_le_infnorm abs_le_square_iff power2_eq_square real_abs_infnorm)
  also have "  (sqrt DIM('a) * infnorm x)2"
    by (simp add: power_mult_distrib)
  finally show "x  x  (sqrt DIM('a) * infnorm x)2" .

lemma tendsto_infnorm [tendsto_intros]:
  assumes "(f  a) F"
  shows "((λx. infnorm (f x))  infnorm a) F"
proof (rule tendsto_compose [OF LIM_I assms])
  fix r :: real
  assume "r > 0"
  then show "s>0. x. x  a  norm (x - a) < s  norm (infnorm x - infnorm a) < r"
    by (metis real_norm_def le_less_trans absdiff_infnorm infnorm_le_norm)

text ‹Equality in Cauchy-Schwarz and triangle inequalities.›

lemma norm_cauchy_schwarz_eq: "x  y = norm x * norm y  norm x *R y = norm y *R x"
  (is "?lhs  ?rhs")
proof (cases "x=0")
  case True
  then show ?thesis
    by auto
  case False 
  from inner_eq_zero_iff[of "norm y *R x - norm x *R y"]
  have "?rhs 
      (norm y * (norm y * norm x * norm x - norm x * (x  y)) -
        norm x * (norm y * (y  x) - norm x * norm y * norm y) = 0)"
    using False unfolding inner_simps
    by (auto simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
  also have "  (2 * norm x * norm y * (norm x * norm y - x  y) = 0)"
    using False  by (simp add: field_simps inner_commute)
  also have "  ?lhs"
    using False by auto
  finally show ?thesis by metis

lemma norm_cauchy_schwarz_abs_eq:
  "¦x  y¦ = norm x * norm y 
    norm x *R y = norm y *R x  norm x *R y = - norm y *R x"
  using norm_cauchy_schwarz_eq [symmetric, of x y]
  using norm_cauchy_schwarz_eq [symmetric, of "-x" y] Cauchy_Schwarz_ineq2 [of x y]
  by auto

lemma norm_triangle_eq:
  fixes x y :: "'a::real_inner"
  shows "norm (x + y) = norm x + norm y  norm x *R y = norm y *R x"
proof (cases "x = 0  y = 0")
  case True
  then show ?thesis
    by force
  case False
  then have n: "norm x > 0" "norm y > 0"
    by auto
  have "norm (x + y) = norm x + norm y  (norm (x + y))2 = (norm x + norm y)2"
    by simp
  also have "  norm x *R y = norm y *R x"
    by (smt (verit, best) dot_norm inner_real_def inner_simps norm_cauchy_schwarz_eq power2_eq_square)
  finally show ?thesis .

lemma dist_triangle_eq:
  fixes x y z :: "'a::real_inner"
  shows "dist x z = dist x y + dist y z 
    norm (x - y) *R (y - z) = norm (y - z) *R (x - y)"
  by (metis (no_types, lifting) add_diff_eq diff_add_cancel dist_norm norm_triangle_eq)

subsection ‹Collinearity›

definitiontag important› collinear :: "'a::real_vector set  bool"
  where "collinear S  (u. x  S.  y  S. c. x - y = c *R u)"

lemma collinear_alt:
     "collinear S  (u v. x  S. c. x = u + c *R v)" (is "?lhs = ?rhs")
  assume ?lhs
  then show ?rhs
    unfolding collinear_def by (metis add.commute diff_add_cancel)
  assume ?rhs
  then obtain u v where *: "x. x  S  c. x = u + c *R v"
    by auto
  have "c. x - y = c *R v" if "x  S" "y  S" for x y
        by (metis *[OF x  S] *[OF y  S] scaleR_left.diff add_diff_cancel_left)
  then show ?lhs
    using collinear_def by blast

lemma collinear:
  fixes S :: "'a::{perfect_space,real_vector} set"
  shows "collinear S  (u. u  0  (x  S.  y  S. c. x - y = c *R u))"
proof -
  have "v. v  0  (xS. yS. c. x - y = c *R v)"
    if "xS. yS. c. x - y = c *R u" "u=0" for u
  proof -
    have "xS. yS. x = y"
      using that by auto
    obtain v::'a where "v  0"
      using UNIV_not_singleton [of 0] by auto
    ultimately have "xS. yS. c. x - y = c *R v"
      by auto
    then show ?thesis
      using v  0 by blast
  then show ?thesis
    by (metis collinear_def)

lemma collinear_subset: "collinear T; S  T  collinear S"
  by (meson collinear_def subsetCE)

lemma collinear_empty [iff]: "collinear {}"
  by (simp add: collinear_def)

lemma collinear_sing [iff]: "collinear {x}"
  by (simp add: collinear_def)

lemma collinear_2 [iff]: "collinear {x, y}"
  by (simp add: collinear_def) (metis minus_diff_eq scaleR_left.minus scaleR_one)

lemma collinear_lemma: "collinear {0, x, y}  x = 0  y = 0  (c. y = c *R x)"
  (is "?lhs  ?rhs")
proof (cases "x = 0  y = 0")
  case True
  then show ?thesis
    by (auto simp: insert_commute)
  case False
  show ?thesis
    assume h: "?lhs"
    then obtain u where u: " x {0,x,y}. y {0,x,y}. c. x - y = c *R u"
      unfolding collinear_def by blast
    from u[rule_format, of x 0] u[rule_format, of y 0]
    obtain cx and cy where
      cx: "x = cx *R u" and cy: "y = cy *R u"
      by auto
    from cx cy False have cx0: "cx  0" and cy0: "cy  0" by auto
    let ?d = "cy / cx"
    from cx cy cx0 have "y = ?d *R x"
      by simp
    then show ?rhs using False by blast
    assume h: "?rhs"
    then obtain c where c: "y = c *R x"
      using False by blast
    show ?lhs
      apply (simp add: collinear_def c)
      by (metis (mono_tags, lifting) scaleR_left.minus scaleR_left_diff_distrib scaleR_one)

lemma collinear_iff_Reals: "collinear {0::complex,w,z}  z/w  "
  show "z/w    collinear {0,w,z}"
    by (metis Reals_cases collinear_lemma nonzero_divide_eq_eq scaleR_conv_of_real)
qed (auto simp: collinear_lemma scaleR_conv_of_real)

lemma collinear_scaleR_iff: "collinear {0, α *R w, β *R z}  collinear {0,w,z}  α=0  β=0"
  (is "?lhs = ?rhs")
proof (cases "α=0  β=0")
  case False
  then have "(c. β *R z = (c * α) *R w) = (c. z = c *R w)"
    by (metis mult.commute scaleR_scaleR vector_fraction_eq_iff)
  then show ?thesis
    by (auto simp add: collinear_lemma)
qed (auto simp: collinear_lemma)

lemma norm_cauchy_schwarz_equal: "¦x  y¦ = norm x * norm y  collinear {0, x, y}"
proof (cases "x=0")
  case True
  then show ?thesis
    by (auto simp: insert_commute)
  case False
  then have nnz: "norm x  0"
    by auto
  show ?thesis
    assume "¦x  y¦ = norm x * norm y"
    then show "collinear {0, x, y}"
      unfolding norm_cauchy_schwarz_abs_eq collinear_lemma
      by (meson eq_vector_fraction_iff nnz)
    assume "collinear {0, x, y}"
    with False show "¦x  y¦ = norm x * norm y"
      unfolding norm_cauchy_schwarz_abs_eq collinear_lemma  by (auto simp: abs_if)

lemma norm_triangle_eq_imp_collinear:
  fixes x y :: "'a::real_inner"
  assumes "norm (x + y) = norm x + norm y"
  shows "collinear{0,x,y}"
  using assms norm_cauchy_schwarz_abs_eq norm_cauchy_schwarz_equal norm_triangle_eq 
  by blast

subsection‹Properties of special hyperplanes›

lemma subspace_hyperplane: "subspace {x. a  x = 0}"
  by (simp add: subspace_def inner_right_distrib)

lemma subspace_hyperplane2: "subspace {x. x  a = 0}"
  by (simp add: inner_commute inner_right_distrib subspace_def)

lemma special_hyperplane_span:
  fixes S :: "'n::euclidean_space set"
  assumes "k  Basis"
  shows "{x. k  x = 0} = span (Basis - {k})"
proof -
  have *: "x  span (Basis - {k})" if "k  x = 0" for x
  proof -
    have "x = (bBasis. (x  b) *R b)"
      by (simp add: euclidean_representation)
    also have " = (b  Basis - {k}. (x  b) *R b)"
      by (auto simp: sum.remove [of _ k] inner_commute assms that)
    finally have "x = (bBasis - {k}. (x  b) *R b)" .
    then show ?thesis
      by (simp add: span_finite)
  show ?thesis
    apply (rule span_subspace [symmetric])
    using assms
    apply (auto simp: inner_not_same_Basis intro: * subspace_hyperplane)

lemma dim_special_hyperplane:
  fixes k :: "'n::euclidean_space"
  shows "k  Basis  dim {x. k  x = 0} = DIM('n) - 1"
  by (metis Diff_subset card_Diff_singleton indep_card_eq_dim_span independent_substdbasis special_hyperplane_span)

proposition dim_hyperplane:
  fixes a :: "'a::euclidean_space"
  assumes "a  0"
    shows "dim {x. a  x = 0} = DIM('a) - 1"
proof -
  have span0: "span {x. a  x = 0} = {x. a  x = 0}"
    by (rule span_unique) (auto simp: subspace_hyperplane)
  then obtain B where "independent B"
              and Bsub: "B  {x. a  x = 0}"
              and subspB: "{x. a  x = 0}  span B"
              and card0: "(card B = dim {x. a  x = 0})"
              and ortho: "pairwise orthogonal B"
    using orthogonal_basis_exists by metis
  with assms have "a  span B"
    by (metis (mono_tags, lifting) span_eq inner_eq_zero_iff mem_Collect_eq span0)
  then have ind: "independent (insert a B)"
    by (simp add: independent B independent_insert)
  have "finite B"
    using independent B independent_bound by blast
  have "UNIV  span (insert a B)"
  proof fix y::'a
    obtain r z where "y = r *R a + z" "a  z = 0"
      by (metis add.commute diff_add_cancel vector_sub_project_orthogonal)
    then show "y  span (insert a B)"
      by (metis (mono_tags, lifting) Bsub add_diff_cancel_left'
          mem_Collect_eq span0 span_breakdown_eq span_eq subspB)
  then have "DIM('a) = dim(insert a B)"
    by (metis independent_Basis span_Basis dim_eq_card top.extremum_uniqueI)
  then show ?thesis
    by (metis One_nat_def a  span B finite B card0 card_insert_disjoint 
        diff_Suc_Suc diff_zero dim_eq_card_independent ind span_base)

lemma lowdim_eq_hyperplane:
  fixes S :: "'a::euclidean_space set"
  assumes "dim S = DIM('a) - 1"
  obtains a where "a  0" and "span S = {x. a  x = 0}"
proof -
  obtain b where b: "b  0" "span S  {a. b  a = 0}"
    by (metis DIM_positive assms diff_less zero_less_one lowdim_subset_hyperplane)
  then show ?thesis
    by (metis assms dim_hyperplane dim_span dim_subset subspace_dim_equal subspace_hyperplane subspace_span that)

lemma dim_eq_hyperplane:
  fixes S :: "'n::euclidean_space set"
  shows "dim S = DIM('n) - 1  (a. a  0  span S = {x. a  x = 0})"
by (metis One_nat_def dim_hyperplane dim_span lowdim_eq_hyperplane)

subsection‹ Orthogonal bases and Gram-Schmidt process›

lemma pairwise_orthogonal_independent:
  assumes "pairwise orthogonal S" and "0  S"
    shows "independent S"
proof -
  have 0: "x y. x  y; x  S; y  S  x  y = 0"
    using assms by (simp add: pairwise_def orthogonal_def)
  have "False" if "a  S" and a: "a  span (S - {a})" for a
  proof -
    obtain T U where "T  S - {a}" "a = (vT. U v *R v)"
      using a by (force simp: span_explicit)
    then have "a  a = a  (vT. U v *R v)"
      by simp
    also have " = 0"
      apply (simp add: inner_sum_right)
      by (smt (verit) "0" DiffE T  S - {a} in_mono insertCI mult_not_zero sum.neutral that(1))
    finally show ?thesis
      using 0  S a  S by auto
  then show ?thesis
    by (force simp: dependent_def)

lemma pairwise_orthogonal_imp_finite:
  fixes S :: "'a::euclidean_space set"
  assumes "pairwise orthogonal S"
    shows "finite S"
  by (metis Set.set_insert assms finite_insert independent_bound pairwise_insert 

lemma subspace_orthogonal_to_vector: "subspace {y. orthogonal x y}"
  by (simp add: subspace_def orthogonal_clauses)

lemma subspace_orthogonal_to_vectors: "subspace {y. x  S. orthogonal x y}"
  by (simp add: subspace_def orthogonal_clauses)

lemma orthogonal_to_span:
  assumes a: "a  span S" and x: "y. y  S  orthogonal x y"
    shows "orthogonal x a"
  by (metis a orthogonal_clauses(1,2,4)
      span_induct_alt x)

proposition Gram_Schmidt_step:
  fixes S :: "'a::euclidean_space set"
  assumes S: "pairwise orthogonal S" and x: "x  span S"
    shows "orthogonal x (a - (bS. (b  a / (b  b)) *R b))"
proof -
  have "finite S"
    by (simp add: S pairwise_orthogonal_imp_finite)
  have "orthogonal (a - (bS. (b  a / (b  b)) *R b)) x"
       if "x  S" for x
  proof -
    have "a  x = (yS. if y = x then y  a else 0)"
      by (simp add: finite S inner_commute that)
    also have " =  (bS. b  a * (b  x) / (b  b))"
      apply (rule sum.cong [OF refl], simp)
      by (meson S orthogonal_def pairwise_def that)
   finally show ?thesis
     by (simp add: orthogonal_def algebra_simps inner_sum_left)
  then show ?thesis
    using orthogonal_to_span orthogonal_commute x by blast

lemma orthogonal_extension_aux:
  fixes S :: "'a::euclidean_space set"
  assumes "finite T" "finite S" "pairwise orthogonal S"
    shows "U. pairwise orthogonal (S  U)  span (S  U) = span (S  T)"
using assms
proof (induction arbitrary: S)
  case empty then show ?case
    by simp (metis sup_bot_right)
  case (insert a T)
  have 0: "x y. x  y; x  S; y  S  x  y = 0"
    using insert by (simp add: pairwise_def orthogonal_def)
  define a' where "a' = a - (bS. (b  a / (b  b)) *R b)"
  obtain U where orthU: "pairwise orthogonal (S  insert a' U)"
             and spanU: "span (insert a' S  U) = span (insert a' S  T)"
    by (rule exE [OF insert.IH [of "insert a' S"]])
      (auto simp: Gram_Schmidt_step a'_def insert.prems orthogonal_commute
        pairwise_orthogonal_insert span_clauses)
  have orthS: "x. x  S  a'  x = 0"
    using Gram_Schmidt_step a'_def insert.prems orthogonal_commute orthogonal_def span_base by blast
  have "span (S  insert a' U) = span (insert a' (S  T))"
    using spanU by simp
  also have " = span (insert a (S  T))"
    by (simp add: a'_def span_neg span_sum span_base span_mul eq_span_insert_eq)
  also have " = span (S  insert a T)"
    by simp
  finally show ?case
    using orthU by blast

proposition orthogonal_extension:
  fixes S :: "'a::euclidean_space set"
  assumes S: "pairwise orthogonal S"
  obtains U where "pairwise orthogonal (S  U)" "span (S  U) = span (S  T)"
proof -
  obtain B where "finite B" "span B = span T"
    using basis_subspace_exists [of "span T"] subspace_span by metis
  with orthogonal_extension_aux [of B S]
  obtain U where "pairwise orthogonal (S  U)" "span (S  U) = span (S  B)"
    using assms pairwise_orthogonal_imp_finite by auto
  with span B = span T show ?thesis
    by (rule_tac U=U in that) (auto simp: span_Un)

corollarytag unimportant› orthogonal_extension_strong:
  fixes S :: "'a::euclidean_space set"
  assumes S: "pairwise orthogonal S"
  obtains U where "U  (insert 0 S) = {}" "pairwise orthogonal (S  U)"
                  "span (S  U) = span (S  T)"
proof -
  obtain U where U: "pairwise orthogonal (S  U)" "span (S  U) = span (S  T)"
    using orthogonal_extension assms by blast
  moreover have "pairwise orthogonal (S  (U - insert 0 S))"
    by (smt (verit, best) Un_Diff_Int Un_iff U pairwise_def)
  ultimately show ?thesis
    by (metis Diff_disjoint Un_Diff_cancel Un_insert_left inf_commute span_insert_0 that)

subsection‹Decomposing a vector into parts in orthogonal subspaces›

text‹existence of orthonormal basis for a subspace.›

lemma orthogonal_spanningset_subspace:
  fixes S :: "'a :: euclidean_space set"
  assumes "subspace S"
  obtains B where "B  S" "pairwise orthogonal B" "span B = S"
  by (metis assms basis_orthogonal basis_subspace_exists span_eq)

lemma orthogonal_basis_subspace:
  fixes S :: "'a :: euclidean_space set"
  assumes "subspace S"
  obtains B where "0  B" "B  S" "pairwise orthogonal B" "independent B"
                  "card B = dim S" "span B = S"
  by (metis assms dependent_zero orthogonal_basis_exists span_eq span_eq_iff)

proposition orthonormal_basis_subspace:
  fixes S :: "'a :: euclidean_space set"
  assumes "subspace S"
  obtains B where "B  S" "pairwise orthogonal B"
              and "x. x  B  norm x = 1"
              and "independent B" "card B = dim S" "span B = S"
proof -
  obtain B where "0  B" "B  S"
             and orth: "pairwise orthogonal B"
             and "independent B" "card B = dim S" "span B = S"
    by (blast intro: orthogonal_basis_subspace [OF assms])
  have 1: "(λx. x /R norm x) ` B  S"
    using span B = S span_superset span_mul by fastforce
  have 2: "pairwise orthogonal ((λx. x /R norm x) ` B)"
    using orth by (force simp: pairwise_def orthogonal_clauses)
  have 3: "x. x  (λx. x /R norm x) ` B  norm x = 1"
    by (metis (no_types, lifting) 0  B image_iff norm_sgn sgn_div_norm)
  have 4: "independent ((λx. x /R norm x) ` B)"
    by (metis "2" "3" norm_zero pairwise_orthogonal_independent zero_neq_one)
  have "inj_on (λx. x /R norm x) B"
    fix x y
    assume "x  B" "y  B" "x /R norm x = y /R norm y"
    moreover have "i. i  B  norm (i /R norm i) = 1"
      using 3 by blast
    ultimately show "x = y"
      by (metis norm_eq_1 orth orthogonal_clauses(7) orthogonal_commute orthogonal_def pairwise_def zero_neq_one)
  then have 5: "card ((λx. x /R norm x) ` B) = dim S"
    by (metis card B = dim S card_image)
  have 6: "span ((λx. x /R norm x) ` B) = S"
    by (metis "1" "4" "5" assms card_eq_dim independent_imp_finite span_subspace)
  show ?thesis
    by (rule that [OF 1 2 3 4 5 6])

propositiontag unimportant› orthogonal_to_subspace_exists_gen:
  fixes S :: "'a :: euclidean_space set"
  assumes "span S  span T"
  obtains x where "x  0" "x  span T" "y. y  span S  orthogonal x y"
proof -
  obtain B where "B  span S" and orthB: "pairwise orthogonal B"
             and "x. x  B  norm x = 1"
             and "independent B" "card B = dim S" "span B = span S"
    by (metis dim_span orthonormal_basis_subspace subspace_span)
  with assms obtain u where spanBT: "span B  span T" and "u  span B" "u  span T"
    by auto
  obtain C where orthBC: "pairwise orthogonal (B  C)" and spanBC: "span (B  C) = span (B  {u})"
    by (blast intro: orthogonal_extension [OF orthB])
  show thesis
  proof (cases "C  insert 0 B")
    case True
    then have "C  span B"
      using span_eq
      by (metis span_insert_0 subset_trans)
    moreover have "u  span (B  C)"
      using span (B  C) = span (B  {u}) span_superset by force
    ultimately show ?thesis
      using True u  span B
      by (metis Un_insert_left span_insert_0 sup.orderE)
    case False
    then obtain x where "x  C" "x  0" "x  B"
      by blast
    then have "x  span T"
      by (smt (verit, ccfv_SIG) Set.set_insert  u  span T empty_subsetI insert_subset 
          le_sup_iff spanBC spanBT span_mono span_span span_superset subset_trans)
    moreover have "orthogonal x y" if "y  span B" for y
      using that
    proof (rule span_induct)
      show "subspace {a. orthogonal x a}"
        by (simp add: subspace_orthogonal_to_vector)
      show "b. b  B  orthogonal x b"
        by (metis Un_iff x  C x  B orthBC pairwise_def)
    ultimately show ?thesis
      using x  0 that span B = span S by auto

corollarytag unimportant› orthogonal_to_subspace_exists:
  fixes S :: "'a :: euclidean_space set"
  assumes "dim S < DIM('a)"
  obtains x where "x  0" "y. y  span S  orthogonal x y"
proof -
  have "span S  UNIV"
    by (metis assms dim_eq_full order_less_imp_not_less top.not_eq_extremum)
  with orthogonal_to_subspace_exists_gen [of S UNIV] that show ?thesis
    by (auto)

corollarytag unimportant› orthogonal_to_vector_exists:
  fixes x :: "'a :: euclidean_space"
  assumes "2  DIM('a)"
  obtains y where "y  0" "orthogonal x y"
proof -
  have "dim {x} < DIM('a)"
    using assms by auto
  then show thesis
    by (rule orthogonal_to_subspace_exists) (simp add: orthogonal_commute span_base that)

propositiontag unimportant› orthogonal_subspace_decomp_exists:
  fixes S :: "'a :: euclidean_space set"
  obtains y z
  where "y  span S"
    and "w. w  span S  orthogonal z w"
    and "x = y + z"
proof -
  obtain T where "0  T" "T  span S" "pairwise orthogonal T" "independent T"
    "card T = dim (span S)" "span T = span S"
    using orthogonal_basis_subspace subspace_span by blast
  let ?a = "bT. (b  x / (b  b)) *R b"
  have orth: "orthogonal (x - ?a) w" if "w  span S" for w
    by (simp add: Gram_Schmidt_step pairwise orthogonal T span T = span S
        orthogonal_commute that)
  with that[of ?a "x-?a"] T  span S show ?thesis
    by (simp add: span_mul span_sum subsetD)

lemma orthogonal_subspace_decomp_unique:
  fixes S :: "'a :: euclidean_space set"
  assumes "x + y = x' + y'"
      and ST: "x  span S" "x'  span S" "y  span T" "y'  span T"
      and orth: "a b. a  S; b  T  orthogonal a b"
  shows "x = x'  y = y'"
proof -
  have "x + y - y' = x'"
    by (simp add: assms)
  moreover have "a b. a  span S; b  span T  orthogonal a b"
    by (meson orth orthogonal_commute orthogonal_to_span)
  ultimately have "0 = x' - x"
    using assms
    by (metis add.commute add_diff_cancel_right' diff_right_commute orthogonal_self span_diff)
  with assms show ?thesis by auto

lemma vector_in_orthogonal_spanningset:
  fixes a :: "'a::euclidean_space"
  obtains S where "a  S" "pairwise orthogonal S" "span S = UNIV"
  by (metis UnI1 Un_UNIV_right insertI1 orthogonal_extension pairwise_singleton span_UNIV)

lemma vector_in_orthogonal_basis:
  fixes a :: "'a::euclidean_space"
  assumes "a  0"
  obtains S where "a  S" "0  S" "pairwise orthogonal S" "independent S" "finite S"
                  "span S = UNIV" "card S = DIM('a)"
proof -
  obtain S where S: "a  S" "pairwise orthogonal S" "span S = UNIV"
    using vector_in_orthogonal_spanningset .
  show thesis
    show "pairwise orthogonal (S - {0})"
      using pairwise_mono S(2) by blast
    show "independent (S - {0})"
      by (simp add: pairwise orthogonal (S - {0}) pairwise_orthogonal_independent)
    show "finite (S - {0})"
      using independent (S - {0}) independent_imp_finite by blast
    show "card (S - {0}) = DIM(