Impressum Vector_Spaces.thy

(*  Title:      HOL/Vector_Spaces.thy
  Author: Amine Chaieb, University of Cambridge
  Author: Jose Divasón 🚫divasonm at unirioja.es>
  Author: Jesús Aransay 🚫aransay at unirioja.es>
  Author: Johannes Hölzl, VU Amsterdam
  Author: Fabian Immler, TUM
*)

section ‹Vector Spaces›

theory Vector_Spaces
  imports Modules
begin

lemma isomorphism_expand:
  "f ∘ g = id ∧ g ∘ f = id ⟷ (∀x. f (g x) = x) ∧ (∀x. g (f x) = x)"
  by (simp add: fun_eq_iff o_def id_def)

lemma left_right_inverse_eq:
  assumes fg: "f ∘ g = id"
    and gh: "g ∘ h = id"
  shows "f = h"
proof -
  have "f = f ∘ (g ∘ h)"
    unfolding gh by simp
  also have "… = (f ∘ g) ∘ h"
    by (simp add: o_assoc)
  finally show "f = h"
    unfolding fg by simp
qed

lemma ordLeq3_finite_infinite:
  assumes A: "finite A" and B: "infinite B" shows "ordLeq3 (card_of A) (card_of B)"
proof -
  have ‹ordLeq3 (card_of A) (card_of B) ∨ ordLeq3 (card_of B) (card_of A)›
    by (intro ordLeq_total card_of_Well_order)
  moreover have "¬ ordLeq3 (card_of B) (card_of A)"
    using B A card_of_ordLeq_finite[of B A] by auto
  ultimately show ?thesis by auto
qed

locale vector_space =
  fixes scale :: "'a::field ==> 'b::ab_group_add ==> 'b" (infixr ‹*s› 75)
  assumes vector_space_assms:🍋 ‹re-stating the assumptions of ‹module› instead of extending ‹module›
  allows us to rewrite in the sublocale.›
    "a *s (x + y) = a *s x + a *s y"
    "(a + b) *s x = a *s x + b *s x"
    "a *s (b *s x) = (a * b) *s x"
    "1 *s x = x"

lemma module_iff_vector_space: "module s ⟷ vector_space s"
  unfolding module_def vector_space_def ..

locale linear = vs1: vector_space s1 + vs2: vector_space s2 + module_hom s1 s2 f
  for s1 :: "'a::field ==> 'b::ab_group_add ==> 'b" (infixr ‹*a› 75)
  and s2 :: "'a::field ==> 'c::ab_group_add ==> 'c" (infixr ‹*b› 75)
  and f :: "'b ==> 'c"

lemma module_hom_iff_linear: "module_hom s1 s2 f ⟷ linear s1 s2 f"
  unfolding module_hom_def linear_def module_iff_vector_space by auto
lemmas module_hom_eq_linear = module_hom_iff_linear[abs_def, THEN meta_eq_to_obj_eq]
lemmas linear_iff_module_hom = module_hom_iff_linear[symmetric]
lemmas linear_module_homI = module_hom_iff_linear[THEN iffD1]
  and module_hom_linearI = module_hom_iff_linear[THEN iffD2]

context vector_space begin

sublocale module scale rewrites "module_hom = linear"
  by unfold_locales (fact vector_space_assms module_hom_eq_linear)+

lemmas🍋 ‹from ‹module›\›
      linear_id = module_hom_id
  and linear_ident = module_hom_ident
  and linear_scale_self = module_hom_scale_self
  and linear_scale_left = module_hom_scale_left
  and linear_uminus = module_hom_uminus

lemma linear_representation:
  assumes "independent B" "span B = UNIV"
  shows   "linear scale (*) (λv. representation B v b)"
proof (unfold_locales, goal_cases)
  case (5 x y)
  show ?case
    using assms by (subst representation_add) auto
next
  case (6 r x)
  show ?case
    using assms by (subst representation_scale) auto
qed (simp_all add: algebra_simps)

lemma linear_imp_scale:
  fixes D::"'a ==> 'b"
  assumes "linear (*) scale D"
  obtains d where "D = (λx. scale x d)"
proof -
  interpret linear "(*)" scale D by fact
  show ?thesis
    by (metis mult.commute mult.left_neutral scale that)
qed

lemma scale_eq_0_iff [simp]: "scale a x = 0 ⟷ a = 0 ∨ x = 0"
  by (metis scale_left_commute right_inverse scale_one scale_scale scale_zero_left)

lemma scale_left_imp_eq:
  assumes nonzero: "a ≠ 0"
    and scale: "scale a x = scale a y"
  shows "x = y"
proof -
  from scale have "scale a (x - y) = 0"
     by (simp add: scale_right_diff_distrib)
  with nonzero have "x - y = 0" by simp
  then show "x = y" by (simp only: right_minus_eq)
qed

lemma scale_right_imp_eq:
  assumes nonzero: "x ≠ 0"
    and scale: "scale a x = scale b x"
  shows "a = b"
proof -
  from scale have "scale (a - b) x = 0"
     by (simp add: scale_left_diff_distrib)
  with nonzero have "a - b = 0" by simp
  then show "a = b" by (simp only: right_minus_eq)
qed

lemma scale_cancel_left [simp]: "scale a x = scale a y ⟷ x = y ∨ a = 0"
  by (auto intro: scale_left_imp_eq)

lemma scale_cancel_right [simp]: "scale a x = scale b x ⟷ a = b ∨ x = 0"
  by (auto intro: scale_right_imp_eq)

lemma injective_scale: "c ≠ 0 ==> inj (λx. scale c x)"
  by (simp add: inj_on_def)

lemma dependent_def: "dependent P ⟷ (∃a ∈ P. a ∈ span (P - {a}))"
  unfolding dependent_explicit
proof safe
  fix a assume aP: "a ∈ P" and "a ∈ span (P - {a})"
  then obtain a S u
    where aP: "a ∈ P" and fS: "finite S" and SP: "S ⊆ P" "a ∉ S" and ua: "(∑v∈S. u v *s v) = a"
    unfolding span_explicit by blast
  let ?S = "insert a S"
  let ?u = "λy. if y = a then - 1 else u y"
  from fS SP have "(∑v∈?S. ?u v *s v) = 0"
    by (simp add: if_distrib[of "λr. r *s a" for a] sum.If_cases field_simps Diff_eq[symmetric] ua)
  moreover have "finite ?S" "?S ⊆ P" "a ∈ ?S" "?u a ≠ 0"
    using fS SP aP by auto
  ultimately show "∃t u. finite t ∧ t ⊆ P ∧ (∑v∈t. u v *s v) = 0 ∧ (∃v∈t. u v ≠ 0)" by fast
next
  fix S u v
  assume fS: "finite S" and SP: "S ⊆ P" and vS: "v ∈ S"
   and uv: "u v ≠ 0" and u: "(∑v∈S. u v *s v) = 0"
  let ?a = v
  let ?S = "S - {v}"
  let ?u = "λi. (- u i) / u v"
  have th0: "?a ∈ P" "finite ?S" "?S ⊆ P"
    using fS SP vS by auto
  have "(∑v∈?S. ?u v *s v) = (∑v∈S. (- (inverse (u ?a))) *s (u v *s v)) - ?u v *s v"
    using fS vS uv by (simp add: sum_diff1 field_simps)
  also have "… = ?a"
    unfolding scale_sum_right[symmetric] u using uv by simp
  finally have "(∑v∈?S. ?u v *s v) = ?a" .
  with th0 show "∃a ∈ P. a ∈ span (P - {a})"
    unfolding span_explicit by (auto intro!: bexI[where x="?a"] exI[where x="?S"] exI[where x="?u"])
qed

lemma dependent_single[simp]: "dependent {x} ⟷ x = 0"
  unfolding dependent_def by auto

lemma in_span_insert:
  assumes a: "a ∈ span (insert b S)"
    and na: "a ∉ span S"
  shows "b ∈ span (insert a S)"
proof -
  from span_breakdown[of b "insert b S" a, OF insertI1 a]
  obtain k where k: "a - k *s b ∈ span (S - {b})" by auto
  have "k ≠ 0"
  proof
    assume "k = 0"
    with k span_mono[of "S - {b}" S] have "a ∈ span S" by auto
    with na show False by blast
  qed
  then have eq: "b = (1/k) *s a - (1/k) *s (a - k *s b)"
    by (simp add: algebra_simps)

  from k have "(1/k) *s (a - k *s b) ∈ span (S - {b})"
    by (rule span_scale)
  also have "... ⊆ span (insert a S)"
    by (rule span_mono) auto
  finally show ?thesis
    using k by (subst eq) (blast intro: span_diff span_scale span_base)
qed

lemma dependent_insertD: assumes a: "a ∉ span S" and S: "dependent (insert a S)" shows "dependent S"
proof -
  have "a ∉ S" using a by (auto dest: span_base)
  obtain b where b: "b = a ∨ b ∈ S" "b ∈ span (insert a S - {b})"
    using S unfolding dependent_def by blast
  have "b ≠ a" "b ∈ S"
    using b ‹a ∉ S› a by auto
  with b have *: "b ∈ span (insert a (S - {b}))"
    by (auto simp: insert_Diff_if)
  show "dependent S"
  proof cases
    assume "b ∈ span (S - {b})" with ‹b ∈ S› show ?thesis
      by (auto simp add: dependent_def)
  next
    assume "b ∉ span (S - {b})"
    with * have "a ∈ span (insert b (S - {b}))" by (rule in_span_insert)
    with a show ?thesis
      using ‹b ∈ S› by (auto simp: insert_absorb)
  qed
qed

lemma independent_insertI: "a ∉ span S ==> independent S ==> independent (insert a S)"
  by (auto dest: dependent_insertD)

lemma independent_insert:
  "independent (insert a S) ⟷ (if a ∈ S then independent S else independent S ∧ a ∉ span S)"
proof -
  have "a ∉ S ==> a ∈ span S ==> dependent (insert a S)"
    by (auto simp: dependent_def)
  then show ?thesis
    by (auto intro: dependent_mono simp: independent_insertI)
qed

lemma maximal_independent_subset_extend:
  assumes "S ⊆ V" "independent S"
  obtains B where "S ⊆ B" "B ⊆ V" "independent B" "V ⊆ span B"
proof -
  let ?C = "{B. S ⊆ B ∧ independent B ∧ B ⊆ V}"
  have "∃M∈?C. ∀X∈?C. M ⊆ X ⟶ X = M"
  proof (rule subset_Zorn)
    fix C :: "'b set set" assume "subset.chain ?C C"
    then have C: "∧c. c ∈ C ==> c ⊆ V" "∧c. c ∈ C ==> S ⊆ c" "∧c. c ∈ C ==> independent c"
      "∧c d. c ∈ C ==> d ∈ C ==> c ⊆ d ∨ d ⊆ c"
      unfolding subset.chain_def by blast+

    show "∃U∈?C. ∀X∈C. X ⊆ U"
    proof cases
      assume "C = {}" with assms show ?thesis
        by (auto intro!: exI[of _ S])
    next
      assume "C ≠ {}"
      with C(2) have "S ⊆ ∪C"
        by auto
      moreover have "independent (∪C)"
        by (intro independent_Union_directed C)
      moreover have "∪C ⊆ V"
        using C by auto
      ultimately show ?thesis
        by auto
    qed
  qed
  then obtain B where B: "independent B" "B ⊆ V" "S ⊆ B"
    and max: "∧S. independent S ==> S ⊆ V ==> B ⊆ S ==> S = B"
    by auto
  moreover
  { assume "¬ V ⊆ span B"
    then obtain v where "v ∈ V" "v ∉ span B"
      by auto
    with B have "independent (insert v B)" by (auto intro: dependent_insertD)
    from max[OF this] ‹v ∈ V› ‹B ⊆ V›
    have "v ∈ B"
      by auto
    with ‹v ∉ span B› have False
      by (auto intro: span_base) }
  ultimately show ?thesis
    by (meson that)
qed

lemma maximal_independent_subset:
  obtains B where "B ⊆ V" "independent B" "V ⊆ span B"
  by (metis maximal_independent_subset_extend[of "{}"] empty_subsetI independent_empty)

text ‹Extends a basis from B to a basis of the entire space.›
definition extend_basis :: "'b set ==> 'b set"
  where "extend_basis B = (SOME B'. B ⊆ B' ∧ independent B' ∧ span B' = UNIV)"

lemma
  assumes B: "independent B"
  shows extend_basis_superset: "B ⊆ extend_basis B"
    and independent_extend_basis: "independent (extend_basis B)"
    and span_extend_basis[simp]: "span (extend_basis B) = UNIV"
proof -
  define p where "p B' ≡ B ⊆ B' ∧ independent B' ∧ span B' = UNIV" for B'
  obtain B' where "p B'"
    using maximal_independent_subset_extend[OF subset_UNIV B]
    by (metis top.extremum_uniqueI p_def)
  then have "p (extend_basis B)"
    unfolding extend_basis_def p_def[symmetric] by (rule someI)
  then show "B ⊆ extend_basis B" "independent (extend_basis B)" "span (extend_basis B) = UNIV"
    by (auto simp: p_def)
qed

lemma in_span_delete:
  assumes a: "a ∈ span S" and na: "a ∉ span (S - {b})"
  shows "b ∈ span (insert a (S - {b}))"
  by (metis Diff_empty Diff_insert0 a in_span_insert insert_Diff na)

lemma span_redundant: "x ∈ span S ==> span (insert x S) = span S"
  unfolding span_def by (rule hull_redundant)

lemma span_trans: "x ∈ span S ==> y ∈ span (insert x S) ==> y ∈ span S"
  by (simp only: span_redundant)

lemma span_insert_0[simp]: "span (insert 0 S) = span S"
  by (metis span_zero span_redundant)

lemma span_delete_0 [simp]: "span(S - {0}) = span S"
proof
  show "span (S - {0}) ⊆ span S"
    by (blast intro!: span_mono)
next
  have "span S ⊆ span(insert 0 (S - {0}))"
    by (blast intro!: span_mono)
  also have "... ⊆ span(S - {0})"
    using span_insert_0 by blast
  finally show "span S ⊆ span (S - {0})" .
qed

lemma span_image_scale:
  assumes "finite S" and nz: "∧x. x ∈ S ==> c x ≠ 0"
    shows "span ((λx. c x *s x) ` S) = span S"
using assms
proof (induction S arbitrary: c)
  case (empty c) show ?case by simp
next
  case (insert x F c)
  show ?case
  proof (intro set_eqI iffI)
    fix y
      assume "y ∈ span ((λx. c x *s x) ` insert x F)"
      then show "y ∈ span (insert x F)"
        using insert by (force simp: span_breakdown_eq)
  next
    fix y
      assume "y ∈ span (insert x F)"
      then show "y ∈ span ((λx. c x *s x) ` insert x F)"
        using insert
        apply (clarsimp simp: span_breakdown_eq)
        apply (rule_tac x="k / c x" in exI)
        by simp
  qed
qed

lemma exchange_lemma:
  assumes f: "finite T"
    and i: "independent S"
    and sp: "S ⊆ span T"
  shows "∃t'. card t' = card T ∧ finite t' ∧ S ⊆ t' ∧ t' ⊆ S ∪ T ∧ S ⊆ span t'"
  using f i sp
proof (induct "card (T - S)" arbitrary: S T rule: less_induct)
  case less
  note ft = ‹finite T› and S = ‹independent S› and sp = ‹S ⊆ span T›
  let ?P = "λt'. card t' = card T ∧ finite t' ∧ S ⊆ t' ∧ t' ⊆ S ∪ T ∧ S ⊆ span t'"
  show ?case
  proof (cases "S ⊆ T ∨ T ⊆ S")
    case True
    then show ?thesis
    proof
      assume "S ⊆ T" then show ?thesis
        by (metis ft Un_commute sp sup_ge1)
    next
      assume "T ⊆ S" then show ?thesis
        by (metis Un_absorb sp spanning_subset_independent[OF _ S sp] ft)
    qed
  next
    case False
    then have st: "¬ S ⊆ T" "¬ T ⊆ S"
      by auto
    from st(2) obtain b where b: "b ∈ T" "b ∉ S"
      by blast
    from b have "T - {b} - S ⊂ T - S"
      by blast
    then have cardlt: "card (T - {b} - S) < card (T - S)"
      using ft by (auto intro: psubset_card_mono)
    from b ft have ct0: "card T ≠ 0"
      by auto
    show ?thesis
    proof (cases "S ⊆ span (T - {b})")
      case True
      from ft have ftb: "finite (T - {b})"
        by auto
      from less(1)[OF cardlt ftb S True]
      obtain U where U: "card U = card (T - {b})" "S ⊆ U" "U ⊆ S ∪ (T - {b})" "S ⊆ span U"
        and fu: "finite U" by blast
      let ?w = "insert b U"
      have th0: "S ⊆ insert b U"
        using U by blast
      have th1: "insert b U ⊆ S ∪ T"
        using U b by blast
      have bu: "b ∉ U"
        using b U by blast
      from U(1) ft b have "card U = (card T - 1)"
        by auto
      then have th2: "card (insert b U) = card T"
        using card_insert_disjoint[OF fu bu] ct0 by auto
      from U(4) have "S ⊆ span U" .
      also have "… ⊆ span (insert b U)"
        by (rule span_mono) blast
      finally have th3: "S ⊆ span (insert b U)" .
      from th0 th1 th2 th3 fu have th: "?P ?w"
        by blast
      from th show ?thesis by blast
    next
      case False
      then obtain a where a: "a ∈ S" "a ∉ span (T - {b})"
        by blast
      have ab: "a ≠ b"
        using a b by blast
      have at: "a ∉ T"
        using a ab span_base[of a "T- {b}"] by auto
      have mlt: "card ((insert a (T - {b})) - S) < card (T - S)"
        using cardlt ft a b by auto
      have ft': "finite (insert a (T - {b}))"
        using ft by auto
      have sp': "S ⊆ span (insert a (T - {b}))"
      proof
        fix x
        assume xs: "x ∈ S"
        have T: "T ⊆ insert b (insert a (T - {b}))"
          using b by auto
        have bs: "b ∈ span (insert a (T - {b}))"
          by (rule in_span_delete) (use a sp in auto)
        from xs sp have "x ∈ span T"
          by blast
        with span_mono[OF T] have x: "x ∈ span (insert b (insert a (T - {b})))" ..
        from span_trans[OF bs x] show "x ∈ span (insert a (T - {b}))" .
      qed
      from less(1)[OF mlt ft' S sp'] obtain U where U:
        "card U = card (insert a (T - {b}))"
        "finite U" "S ⊆ U" "U ⊆ S ∪ insert a (T - {b})"
        "S ⊆ span U" by blast
      from U a b ft at ct0 have "?P U"
        by auto
      then show ?thesis by blast
    qed
  qed
qed

lemma independent_span_bound:
  assumes f: "finite T"
    and i: "independent S"
    and sp: "S ⊆ span T"
  shows "finite S ∧ card S ≤ card T"
  by (metis exchange_lemma[OF f i sp] finite_subset card_mono)

lemma independent_explicit_finite_subsets:
  "independent A ⟷ (∀S ⊆ A. finite S ⟶ (∀u. (∑v∈S. u v *s v) = 0 ⟶ (∀v∈S. u v = 0)))"
  unfolding dependent_explicit [of A] by (simp add: disj_not2)

lemma independent_if_scalars_zero:
  assumes fin_A: "finite A"
  and sum: "∧f x. (∑x∈A. f x *s x) = 0 ==> x ∈ A ==> f x = 0"
  shows "independent A"
proof (unfold independent_explicit_finite_subsets, clarify)
  fix S v and u :: "'b ==> 'a"
  assume S: "S ⊆ A" and v: "v ∈ S"
  let ?g = "λx. if x ∈ S then u x else 0"
  have "(∑v∈A. ?g v *s v) = (∑v∈S. u v *s v)"
    using S fin_A by (auto intro!: sum.mono_neutral_cong_right)
  also assume "(∑v∈S. u v *s v) = 0"
  finally have "?g v = 0" using v S sum by force
  thus "u v = 0" unfolding if_P[OF v] .
qed

lemma bij_if_span_eq_span_bases:
  assumes B: "independent B" and C: "independent C"
    and eq: "span B = span C"
  shows "∃f. bij_betw f B C"
proof cases
  assume "finite B ∨ finite C"
  then have "finite B ∧ finite C ∧ card C = card B"
    using independent_span_bound[of B C] independent_span_bound[of C B] assms
      span_superset[of B] span_superset[of C]
    by auto
  then show ?thesis
    by (auto intro!: finite_same_card_bij)
next
  assume "¬ (finite B ∨ finite C)"
  then have "infinite B" "infinite C" by auto
  { fix B C assume B: "independent B" and C: "independent C" and "infinite B" "infinite C" and eq: "span B = span C"
    let ?R = "representation B" and ?R' = "representation C" let ?U = "λc. {v. ?R c v ≠ 0}"
    have in_span_C [simp, intro]: ‹b ∈ B ==> b ∈ span C› for b unfolding eq[symmetric] by (rule span_base)
    have in_span_B [simp, intro]: ‹c ∈ C ==> c ∈ span B› for c unfolding eq by (rule span_base)
    have ‹B ⊆ (∪c∈C. ?U c)›
    proof
      fix b assume ‹b ∈ B›
      have ‹b ∈ span C›
        using ‹b ∈ B› unfolding eq[symmetric] by (rule span_base)
      have ‹(∑v | ?R' b v ≠ 0. ∑w | ?R v w ≠ 0. (?R' b v * ?R v w) *s w) =
  (∑v | ?R' b v ≠ 0. ?R' b v *s (∑w | ?R v w ≠ 0. ?R v w *s w))›
        by (simp add: scale_sum_right)
      also have ‹… = (∑v | ?R' b v ≠ 0. ?R' b v *s v)›
        by (auto simp: sum_nonzero_representation_eq B eq span_base representation_ne_zero)
      also have ‹… = b›
        by (rule sum_nonzero_representation_eq[OF C ‹b ∈ span C›])
      finally have "?R b b = ?R (∑v | ?R' b v ≠ 0. ∑w | ?R v w ≠ 0. (?R' b v * ?R v w) *s w) b"
        by simp
      also have "... = (∑i∈{v. ?R' b v ≠ 0}. ?R (∑w | ?R i w ≠ 0. (?R' b i * ?R i w) *s w) b)"
        by (subst representation_sum[OF B]) (auto intro: span_sum span_scale span_base representation_ne_zero)
      also have "... = (∑i∈{v. ?R' b v ≠ 0}.
           ∑j ∈ {w. ?R i w ≠ 0}. ?R ((?R' b i * ?R i j ) *s  j ) b)"
        by (subst representation_sum[OF B]) (auto simp add: span_sum span_scale span_base representation_ne_zero)
      also have ‹… = (∑v | ?R' b v ≠ 0. ∑w | ?R v w ≠ 0. ?R' b v * ?R v w * ?R w b)›
        using B ‹b ∈ B› by (simp add: representation_scale[OF B] span_base representation_ne_zero)
      finally have "(∑v | ?R' b v ≠ 0. ∑w | ?R v w ≠ 0. ?R' b v * ?R v w * ?R w b) ≠ 0"
        using representation_basis[OF B ‹b ∈ B›] by auto
      then obtain v w where bv: "?R' b v ≠ 0" and vw: "?R v w ≠ 0" and "?R' b v * ?R v w * ?R w b ≠ 0"
        by (blast elim: sum.not_neutral_contains_not_neutral)
      with representation_basis[OF B, of w] vw[THEN representation_ne_zero]
      have ‹?R' b v ≠ 0› ‹?R v b ≠ 0› by (auto split: if_splits)
      then show ‹b ∈ (∪c∈C. ?U c)›
        by (auto dest: representation_ne_zero)
    qed
    then have B_eq: ‹B = (∪c∈C. ?U c)›
      by (auto intro: span_base representation_ne_zero eq)
    have "ordLeq3 (card_of B) (card_of C)"
    proof (subst B_eq, rule card_of_UNION_ordLeq_infinite[OF ‹infinite C›])
      show "ordLeq3 (card_of C) (card_of C)"
        by (intro ordLeq_refl card_of_Card_order)
      show "∀c∈C. ordLeq3 (card_of {v. ?R c v ≠ 0}) (card_of C)"
        by (intro ballI ordLeq3_finite_infinite ‹infinite C› finite_representation)
    qed }
  from this[of B C] this[of C B] B C eq ‹infinite C› ‹infinite B›
  show ?thesis by (auto simp add: ordIso_iff_ordLeq card_of_ordIso)
qed

definition dim :: "'b set ==> nat"
  where "dim V = (if ∃b. independent b ∧ span b = span V then
    card (SOME b. independent b ∧ span b = span V) else 0)"

lemma dim_eq_card:
  assumes BV: "span B = span V" and B: "independent B"
  shows "dim V = card B"
proof -
  define p where "p b ≡ independent b ∧ span b = span V" for b
  have "p (SOME B. p B)"
    using assms by (intro someI[of p B]) (auto simp: p_def)
  then have "∃f. bij_betw f B (SOME B. p B)"
    by (subst (asm) p_def, intro bij_if_span_eq_span_bases[OF B]) (simp_all add: BV)
  then have "card B = card (SOME B. p B)"
    by (auto intro: bij_betw_same_card)
  then show ?thesis
    using BV B
    by (auto simp add: dim_def p_def)
qed

lemma basis_card_eq_dim: "B ⊆ V ==> V ⊆ span B ==> independent B ==> card B = dim V"
  using dim_eq_card[of B V] span_mono[of B V] span_minimal[OF _ subspace_span, of V B] by auto

lemma basis_exists:
  obtains B where "B ⊆ V" "independent B" "V ⊆ span B" "card B = dim V"
  by (meson basis_card_eq_dim empty_subsetI independent_empty maximal_independent_subset_extend)

lemma dim_eq_card_independent: "independent B ==> dim B = card B"
  by (rule dim_eq_card[OF refl])

lemma dim_span[simp]: "dim (span S) = dim S"
  by (auto simp add: dim_def span_span)

lemma dim_span_eq_card_independent: "independent B ==> dim (span B) = card B"
  by (simp add: dim_eq_card)

lemma dim_le_card: assumes "V ⊆ span W" "finite W" shows "dim V ≤ card W"
proof -
  obtain A where "independent A" "A ⊆ V" "V ⊆ span A"
    using maximal_independent_subset[of V] by force
  with assms independent_span_bound[of W A] basis_card_eq_dim[of A V]
  show ?thesis by auto
qed

lemma span_eq_dim: "span S = span T ==> dim S = dim T"
  by (metis dim_span)

corollary dim_le_card':
  "finite s ==> dim s ≤ card s"
  by (metis basis_exists card_mono)

lemma span_card_ge_dim:
  "B ⊆ V ==> V ⊆ span B ==> finite B ==> dim V ≤ card B"
  by (simp add: dim_le_card)

lemma dim_unique:
  "B ⊆ V ==> V ⊆ span B ==> independent B ==> card B = n ==> dim V = n"
  by (metis basis_card_eq_dim)

lemma subspace_sums: "[subspace S; subspace T] ==> subspace {x + y|x y. x ∈ S ∧ y ∈ T}"
  apply (simp add: subspace_def)
  apply (intro conjI impI allI; clarsimp simp: algebra_simps)
  using add.left_neutral apply blast
   apply (metis add.assoc)
  using scale_right_distrib by blast

end

lemma linear_iff: "linear s1 s2 f ⟷
  (vector_space s1 ∧ vector_space s2 ∧ (∀x y. f (x + y) = f x + f y) ∧ (∀c x. f (s1 c x) = s2 c (f x)))"
  unfolding linear_def module_hom_iff vector_space_def module_def by auto

context begin
qualified lemma linear_compose: "linear s1 s2 f ==> linear s2 s3 g ==> linear s1 s3 (g o f)"
  unfolding module_hom_iff_linear[symmetric]
  by (rule module_hom_compose)
end

locale vector_space_pair = vs1: vector_space s1 + vs2: vector_space s2
  for s1 :: "'a::field ==> 'b::ab_group_add ==> 'b" (infixr ‹*a› 75)
  and s2 :: "'a::field ==> 'c::ab_group_add ==> 'c" (infixr ‹*b› 75)
begin

context fixes f assumes "linear s1 s2 f" begin
interpretation linear s1 s2 f by fact
lemmas🍋 ‹from locale ‹module_hom›\
  linear_0 = zero
  and linear_add = add
  and linear_scale = scale
  and linear_neg = neg
  and linear_diff = diff
  and linear_sum = sum
  and linear_inj_on_iff_eq_0 = inj_on_iff_eq_0
  and linear_inj_iff_eq_0 = inj_iff_eq_0
  and linear_subspace_image = subspace_image
  and linear_subspace_vimage = subspace_vimage
  and linear_subspace_kernel = subspace_kernel
  and linear_span_image = span_image
  and linear_dependent_inj_imageD = dependent_inj_imageD
  and linear_eq_0_on_span = eq_0_on_span
  and linear_independent_injective_image = independent_injective_image
  and linear_inj_on_span_independent_image = inj_on_span_independent_image
  and linear_inj_on_span_iff_independent_image = inj_on_span_iff_independent_image
  and linear_subspace_linear_preimage = subspace_linear_preimage
  and linear_spans_image = spans_image
  and linear_spanning_surjective_image = spanning_surjective_image
 end
 
 sublocale module_pair
  rewrites "module_hom = linear"
  by unfold_locales (fact module_hom_eq_linear)
 
 lemmas🍋 ‹from locale ‹module_pair›\
  linear_eq_on_span = module_hom_eq_on_span
  and linear_compose_scale_right = module_hom_scale
  and linear_compose_add = module_hom_add
  and linear_zero = module_hom_zero
  and linear_compose_sub = module_hom_sub
  and linear_compose_neg = module_hom_neg
  and linear_compose_scale = module_hom_compose_scale
 
 lemma linear_indep_image_lemma:
  assumes lf: "linear s1 s2 f"
  and fB: "finite B"
  and ifB: "vs2.independent (f ` B)"
  and fi: "inj_on f B"
  and xsB: "x ∈ vs1.span B"
  and fx: "f x = 0"
  shows "x = 0"
  using fB ifB fi xsB fx
 proof (induction B arbitrary: x rule: finite_induct)
  case empty
  then show ?case by auto
 next
  case (insert a b x)
  have th0: "f ` b ⊆ f ` (insert a b)"
  by (simp add: subset_insertI)
  have ifb: "vs2.independent (f ` b)"
  using vs2.independent_mono insert.prems(1) th0 by blast
  have fib: "inj_on f b"
  using insert.prems(2) by blast
  from vs1.span_breakdown[of a "insert a b", simplified, OF insert.prems(3)]
  obtain k where k: "x - k *a a ∈ vs1.span (b - {a})"
  by blast
  have "f (x - k *a a) ∈ vs2.span (f ` b)"
  unfolding linear_span_image[OF lf]
  using "insert.hyps"(2) k by auto
  then have "f x - k *b f a ∈ vs2.span (f ` b)"
  by (simp add: linear_diff linear_scale lf)
  then have th: "-k *b f a ∈ vs2.span (f ` b)"
  using insert.prems(4) by simp
  have xsb: "x ∈ vs1.span b"
  proof (cases "k = 0")
  case True
  with k have "x ∈ vs1.span (b - {a})" by simp
  then show ?thesis using vs1.span_mono[of "b - {a}" b]
  by blast
  next
  case False
  from inj_on_image_set_diff[OF insert.prems(2), of "insert a b " "{a}", symmetric]
  have "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast
  then have "f a ∉ vs2.span (f ` b)"
  using vs2.dependent_def insert.hyps(2) insert.prems(1) by fastforce
  moreover have "f a ∈ vs2.span (f ` b)"
  using False vs2.span_scale[OF th, of "- 1/ k"] by auto
  ultimately have False
  by blast
  then show ?thesis by blast
  qed
  show "x = 0"
  using ifb fib xsb insert.IH insert.prems(4) by blast
 qed
 
 lemma linear_eq_on:
  assumes l: "linear s1 s2 f" "linear s1 s2 g"
  assumes x: "x ∈ vs1.span B" and eq: "∧b. b ∈ B ==> f b = g b"
  shows "f x = g x"
 proof -
  interpret d: linear s1 s2 "λx. f x - g x"
  using l by (intro linear_compose_sub) (auto simp: module_hom_iff_linear)
  have "f x - g x = 0"
  by (rule d.eq_0_on_span[OF _ x]) (auto simp: eq)
  then show ?thesis by auto
 qed
 
 definition construct :: "'b set ==> ('b ==> 'c) ==> ('b ==> 'c)"
  where "construct B g v = (∑b | vs1.representation (vs1.extend_basis B) v b ≠ 0.
  vs1.representation (vs1.extend_basis B) v b *b (if b ∈ B then g b else 0))"
 
 lemma construct_cong: "(∧b. b ∈ B ==> f b = g b) ==> construct B f = construct B g"
  unfolding construct_def by (auto intro!: sum.cong)
 
 lemma linear_construct:
  assumes B[simp]: "vs1.independent B"
  shows "linear s1 s2 (construct B f)"
  unfolding module_hom_iff_linear linear_iff
 proof safe
  have eB[simp]: "vs1.independent (vs1.extend_basis B)"
  using vs1.independent_extend_basis[OF B] .
  let ?R = "vs1.representation (vs1.extend_basis B)"
  fix c x y
  have "construct B f (x + y) =
  (∑b∈{b. ?R x b ≠ 0} ∪ {b. ?R y b ≠ 0}. ?R (x + y) b *b (if b ∈ B then f b else 0))"
  by (auto intro!: sum.mono_neutral_cong_left simp: vs1.finite_representation vs1.representation_add construct_def)
  also have "… = construct B f x + construct B f y"
  by (auto simp: construct_def vs1.representation_add vs2.scale_left_distrib sum.distrib
  intro!: arg_cong2[where f="(+)"] sum.mono_neutral_cong_right vs1.finite_representation)
  finally show "construct B f (x + y) = construct B f x + construct B f y" .
 
  show "construct B f (c *a x) = c *b construct B f x"
  by (auto simp del: vs2.scale_scale intro!: sum.mono_neutral_cong_left vs1.finite_representation
  simp add: construct_def vs2.scale_scale[symmetric] vs1.representation_scale vs2.scale_sum_right)
 qed intro_locales
 
 lemma construct_basis:
  assumes B[simp]: "vs1.independent B" and b: "b ∈ B"
  shows "construct B f b = f b"
 proof -
  have *: "vs1.representation (vs1.extend_basis B) b = (λv. if v = b then 1 else 0)"
  using vs1.extend_basis_superset[OF B] b
  by (intro vs1.representation_basis vs1.independent_extend_basis) auto
  then have "{v. vs1.representation (vs1.extend_basis B) b v ≠ 0} = {b}"
  by auto
  then show ?thesis
  unfolding construct_def by (simp add: * b)
 qed
 
 lemma construct_outside:
  assumes B: "vs1.independent B" and v: "v ∈ vs1.span (vs1.extend_basis B - B)"
  shows "construct B f v = 0"
  unfolding construct_def
 proof (clarsimp intro!: sum.neutral simp del: vs2.scale_eq_0_iff)
  fix b assume "b ∈ B"
  then have "vs1.representation (vs1.extend_basis B - B) v b = 0"
  using vs1.representation_ne_zero[of "vs1.extend_basis B - B" v b] by auto
  moreover have "vs1.representation (vs1.extend_basis B) v = vs1.representation (vs1.extend_basis B - B) v"
  using vs1.representation_extend[OF vs1.independent_extend_basis[OF B] v] by auto
  ultimately show "vs1.representation (vs1.extend_basis B) v b *b f b = 0"
  by simp
 qed
 
 lemma construct_add:
  assumes B[simp]: "vs1.independent B"
  shows "construct B (λx. f x + g x) v = construct B f v + construct B g v"
 proof (rule linear_eq_on)
  show "v ∈ vs1.span (vs1.extend_basis B)" by simp
  show "b ∈ vs1.extend_basis B ==> construct B (λx. f x + g x) b = construct B f b + construct B g b" for b
  using construct_outside[OF B vs1.span_base, of b] by (cases "b ∈ B") (auto simp: construct_basis)
 qed (intro linear_compose_add linear_construct B)+
 
 lemma construct_scale:
  assumes B[simp]: "vs1.independent B"
  shows "construct B (λx. c *b f x) v = c *b construct B f v"
 proof (rule linear_eq_on)
  show "v ∈ vs1.span (vs1.extend_basis B)" by simp
  show "b ∈ vs1.extend_basis B ==> construct B (λx. c *b f x) b = c *b construct B f b" for b
  using construct_outside[OF B vs1.span_base, of b] by (cases "b ∈ B") (auto simp: construct_basis)
 qed (intro linear_construct module_hom_scale B)+
 
 lemma construct_in_span:
  assumes B[simp]: "vs1.independent B"
  shows "construct B f v ∈ vs2.span (f ` B)"
 proof -
  interpret c: linear s1 s2 "construct B f" by (rule linear_construct) fact
  let ?R = "vs1.representation B"
  have "v ∈ vs1.span ((vs1.extend_basis B - B) ∪ B)"
  by (auto simp: Un_absorb2 vs1.extend_basis_superset)
  then obtain x y where "v = x + y" "x ∈ vs1.span (vs1.extend_basis B - B)" "y ∈ vs1.span B"
  unfolding vs1.span_Un by auto
  moreover have "construct B f (∑b | ?R y b ≠ 0. ?R y b *a b) ∈ vs2.span (f ` B)"
  by (auto simp add: c.sum c.scale construct_basis vs1.representation_ne_zero
  intro!: vs2.span_sum vs2.span_scale intro: vs2.span_base )
  ultimately show "construct B f v ∈ vs2.span (f ` B)"
  by (auto simp add: c.add construct_outside vs1.sum_nonzero_representation_eq)
 qed
 
 lemma linear_compose_sum:
  assumes lS: "∀a ∈ S. linear s1 s2 (f a)"
  shows "linear s1 s2 (λx. sum (λa. f a x) S)"
 proof (cases "finite S")
  case True
  then show ?thesis
  using lS by induct (simp_all add: linear_zero linear_compose_add)
 next
  case False
  then show ?thesis
  by (simp add: linear_zero)
 qed
 
 lemma in_span_in_range_construct:
  "x ∈ range (construct B f)" if i: "vs1.independent B" and x: "x ∈ vs2.span (f ` B)"
 proof -
  interpret linear "(*a)" "(*b)" "construct B f"
  using i by (rule linear_construct)
  obtain bb :: "('b ==> 'c) ==> ('b ==> 'c) ==> 'b set ==> 'b" where
  "∀x0 x1 x2. (∃v4. v4 ∈ x2 ∧ x1 v4 ≠ x0 v4) = (bb x0 x1 x2 ∈ x2 ∧ x1 (bb x0 x1 x2) ≠ x0 (bb x0 x1 x2))"
  by moura
  then have f2: "∀B Ba f fa. (B ≠ Ba ∨ bb fa f Ba ∈ Ba ∧ f (bb fa f Ba) ≠ fa (bb fa f Ba)) ∨ f ` B = fa ` Ba"
  by (meson image_cong)
  have "vs1.span B ⊆ vs1.span (vs1.extend_basis B)"
  by (simp add: vs1.extend_basis_superset[OF i] vs1.span_mono)
  then show "x ∈ range (construct B f)"
  using f2 x by (metis (no_types) construct_basis[OF i, of _ f]
  vs1.span_extend_basis[OF i] subsetD span_image spans_image)
 qed
 
 lemma range_construct_eq_span:
  "range (construct B f) = vs2.span (f ` B)"
  if "vs1.independent B"
  by (auto simp: that construct_in_span in_span_in_range_construct)
 
 lemma linear_independent_extend_subspace:
  🍋 ‹legacy: use 🍋‹construct› instead›
  assumes "vs1.independent B"
  shows "∃g. linear s1 s2 g ∧ (∀x∈B. g x = f x) ∧ range g = vs2.span (f`B)"
  by (rule exI[where x="construct B f"])
  (auto simp: linear_construct assms construct_basis range_construct_eq_span)
 
 lemma linear_independent_extend:
  "vs1.independent B ==> ∃g. linear s1 s2 g ∧ (∀x∈B. g x = f x)"
  using linear_independent_extend_subspace[of B f] by auto
 
 lemma linear_exists_left_inverse_on:
  assumes lf: "linear s1 s2 f"
  assumes V: "vs1.subspace V" and f: "inj_on f V"
  shows "∃g. g ` UNIV ⊆ V ∧ linear s2 s1 g ∧ (∀v∈V. g (f v) = v)"
 proof -
  interpret linear s1 s2 f by fact
  obtain B where V_eq: "V = vs1.span B" and B: "vs1.independent B"
  using vs1.maximal_independent_subset[of V] vs1.span_minimal[OF _ ‹vs1.subspace V›]
  by (metis antisym_conv)
  have f: "inj_on f (vs1.span B)"
  using f unfolding V_eq .
  show ?thesis
  proof (intro exI ballI conjI)
  interpret p: vector_space_pair s2 s1 by unfold_locales
  have fB: "vs2.independent (f ` B)"
  using independent_injective_image[OF B f] .
  let ?g = "p.construct (f ` B) (the_inv_into B f)"
  show "linear (*b) (*a) ?g"
  by (rule p.linear_construct[OF fB])
  have "?g b ∈ vs1.span (the_inv_into B f ` f ` B)" for b
  by (intro p.construct_in_span fB)
  moreover have "the_inv_into B f ` f ` B = B"
  by (auto simp: image_comp comp_def the_inv_into_f_f inj_on_subset[OF f vs1.span_superset]
  cong: image_cong)
  ultimately show "?g ` UNIV ⊆ V"
  by (auto simp: V_eq)
  have "(?g ∘ f) v = id v" if "v ∈ vs1.span B" for v
  proof (rule vector_space_pair.linear_eq_on[where x=v])
  show "vector_space_pair (*a) (*a)" by unfold_locales
  show "linear (*a) (*a) (?g ∘ f)"
  proof (rule Vector_Spaces.linear_compose[of _ "(*b)"])
  show "linear (*a) (*b) f"
  by unfold_locales
  qed fact
  show "linear (*a) (*a) id" by (rule vs1.linear_id)
  show "v ∈ vs1.span B" by fact
  show "b ∈ B ==> (p.construct (f ` B) (the_inv_into B f) ∘ f) b = id b" for b
  by (simp add: p.construct_basis fB the_inv_into_f_f inj_on_subset[OF f vs1.span_superset])
  qed
  then show "v ∈ V ==> ?g (f v) = v" for v by (auto simp: comp_def id_def V_eq)
  qed
 qed
 
 lemma linear_exists_right_inverse_on:
  assumes lf: "linear s1 s2 f"
  assumes "vs1.subspace V"
  shows "∃g. g ` UNIV ⊆ V ∧ linear s2 s1 g ∧ (∀v∈f ` V. f (g v) = v)"
 proof -
  obtain B where V_eq: "V = vs1.span B" and B: "vs1.independent B"
  using vs1.maximal_independent_subset[of V] vs1.span_minimal[OF _ ‹vs1.subspace V›]
  by (metis antisym_conv)
  obtain C where C: "vs2.independent C" and fB_C: "f ` B ⊆ vs2.span C" "C ⊆ f ` B"
  using vs2.maximal_independent_subset[of "f ` B"] by metis
  then have "∀v∈C. ∃b∈B. v = f b" by auto
  then obtain g where g: "∧v. v ∈ C ==> g v ∈ B" "∧v. v ∈ C ==> f (g v) = v" by metis
  show ?thesis
  proof (intro exI ballI conjI)
  interpret p: vector_space_pair s2 s1 by unfold_locales
  let ?g = "p.construct C g"
  show "linear (*b) (*a) ?g"
  by (rule p.linear_construct[OF C])
  have "?g v ∈ vs1.span (g ` C)" for v
  by (rule p.construct_in_span[OF C])
  also have "… ⊆ V" unfolding V_eq using g by (intro vs1.span_mono) auto
  finally show "?g ` UNIV ⊆ V" by auto
  have "(f ∘ ?g) v = id v" if v: "v ∈ f ` V" for v
  proof (rule vector_space_pair.linear_eq_on[where x=v])
  show "vector_space_pair (*b) (*b)" by unfold_locales
  show "linear (*b) (*b) (f ∘ ?g)"
  by (rule Vector_Spaces.linear_compose[of _ "(*a)"]) fact+
  show "linear (*b) (*b) id" by (rule vs2.linear_id)
  have "vs2.span (f ` B) = vs2.span C"
  using fB_C vs2.span_mono[of C "f ` B"] vs2.span_minimal[of "f`B" "vs2.span C"]
  by auto
  then show "v ∈ vs2.span C"
  using v linear_span_image[OF lf, of B] by (simp add: V_eq)
  show "(f ∘ p.construct C g) b = id b" if b: "b ∈ C" for b
  by (auto simp: p.construct_basis g C b)
  qed
  then show "v ∈ f ` V ==> f (?g v) = v" for v by (auto simp: comp_def id_def)
  qed
 qed
 
 lemma linear_inj_on_left_inverse:
  assumes lf: "linear s1 s2 f"
  assumes fi: "inj_on f (vs1.span S)"
  shows "∃g. range g ⊆ vs1.span S ∧ linear s2 s1 g ∧ (∀x∈vs1.span S. g (f x) = x)"
  using linear_exists_left_inverse_on[OF lf vs1.subspace_span fi]
  by (auto simp: linear_iff_module_hom)
 
 lemma linear_injective_left_inverse: "linear s1 s2 f ==> inj f ==> ∃g. linear s2 s1 g ∧ g ∘ f = id"
  using linear_inj_on_left_inverse[of f UNIV]
  by force
 
 lemma linear_surj_right_inverse:
  assumes lf: "linear s1 s2 f"
  assumes sf: "vs2.span T ⊆ f`vs1.span S"
  shows "∃g. range g ⊆ vs1.span S ∧ linear s2 s1 g ∧ (∀x∈vs2.span T. f (g x) = x)"
  using linear_exists_right_inverse_on[OF lf vs1.subspace_span, of S] sf
  by (force simp: linear_iff_module_hom)
 
 lemma linear_surjective_right_inverse: "linear s1 s2 f ==> surj f ==> ∃g. linear s2 s1 g ∧ f ∘ g = id"
  using linear_surj_right_inverse[of f UNIV UNIV]
  by (auto simp: fun_eq_iff)
 
 lemma finite_basis_to_basis_subspace_isomorphism:
  assumes s: "vs1.subspace S"
  and t: "vs2.subspace T"
  and d: "vs1.dim S = vs2.dim T"
  and fB: "finite B"
  and B: "B ⊆ S" "vs1.independent B" "S ⊆ vs1.span B" "card B = vs1.dim S"
  and fC: "finite C"
  and C: "C ⊆ T" "vs2.independent C" "T ⊆ vs2.span C" "card C = vs2.dim T"
  shows "∃f. linear s1 s2 f ∧ f ` B = C ∧ f ` S = T ∧ inj_on f S"
 proof -
  from B(4) C(4) card_le_inj[of B C] d obtain f where
  f: "f ` B ⊆ C" "inj_on f B" using ‹finite B› ‹finite C› by auto
  from linear_independent_extend[OF B(2)] obtain g where
  g: "linear s1 s2 g" "∀x ∈ B. g x = f x" by blast
  interpret g: linear s1 s2 g by fact
  from inj_on_iff_eq_card[OF fB, of f] f(2)
  have "card (f ` B) = card B" by simp
  with B(4) C(4) have ceq: "card (f ` B) = card C" using d
  by simp
  have "g ` B = f ` B" using g(2)
  by (auto simp add: image_iff)
  also have "… = C" using card_subset_eq[OF fC f(1) ceq] .
  finally have gBC: "g ` B = C" .
  have gi: "inj_on g B" using f(2) g(2)
  by (auto simp add: inj_on_def)
  note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
  {
  fix x y
  assume x: "x ∈ S" and y: "y ∈ S" and gxy: "g x = g y"
  from B(3) x y have x': "x ∈ vs1.span B" and y': "y ∈ vs1.span B"
  by blast+
  from gxy have th0: "g (x - y) = 0"
  by (simp add: g.diff)
  have th1: "x - y ∈ vs1.span B" using x' y'
  by (metis vs1.span_diff)
  have "x = y" using g0[OF th1 th0] by simp
  }
  then have giS: "inj_on g S" unfolding inj_on_def by blast
  from vs1.span_subspace[OF B(1,3) s]
  have "g ` S = vs2.span (g ` B)"
  by (simp add: g.span_image)
  also have "… = vs2.span C"
  unfolding gBC ..
  also have "… = T"
  using vs2.span_subspace[OF C(1,3) t] .
  finally have gS: "g ` S = T" .
  from g(1) gS giS gBC show ?thesis
  by blast
 qed
 
 end
 
 
 locale finite_dimensional_vector_space = vector_space +
  fixes Basis :: "'b set"
  assumes finite_Basis: "finite Basis"
  and independent_Basis: "independent Basis"
  and span_Basis: "span Basis = UNIV"
 begin
 
 definition "dimension = card Basis"
 
 lemma finiteI_independent: "independent B ==> finite B"
  using independent_span_bound[OF finite_Basis, of B] by (auto simp: span_Basis)
 
 lemma dim_empty [simp]: "dim {} = 0"
  by (rule dim_unique[OF order_refl]) (auto simp: dependent_def)
 
 lemma dim_insert:
  "dim (insert x S) = (if x ∈ span S then dim S else dim S + 1)"
 proof -
  show ?thesis
  proof (cases "x ∈ span S")
  case True then show ?thesis
  by (metis dim_span span_redundant)
  next
  case False
  obtain B where B: "B ⊆ span S" "independent B" "span S ⊆ span B" "card B = dim (span S)"
  using basis_exists [of "span S"] by blast
  have "dim (span (insert x S)) = Suc (dim S)"
  proof (rule dim_unique)
  show "insert x B ⊆ span (insert x S)"
  by (meson B(1) insertI1 insert_subset order_trans span_base span_mono subset_insertI)
  show "span (insert x S) ⊆ span (insert x B)"
  by (metis ‹B ⊆ span S› ‹span S ⊆ span B› span_breakdown_eq span_subspace subsetI subspace_span)
  show "independent (insert x B)"
  by (metis B(1-3) independent_insert span_subspace subspace_span False)
  show "card (insert x B) = Suc (dim S)"
  using B False finiteI_independent by force
  qed
  then show ?thesis
  by (metis False Suc_eq_plus1 dim_span)
  qed
 qed
 
 lemma dim_singleton [simp]: "dim{x} = (if x = 0 then 0 else 1)"
  by (simp add: dim_insert)
 
 proposition choose_subspace_of_subspace:
  assumes "n ≤ dim S"
  obtains T where "subspace T" "T ⊆ span S" "dim T = n"
 proof -
  have "∃T. subspace T ∧ T ⊆ span S ∧ dim T = n"
  using assms
  proof (induction n)
  case 0 then show ?case by (auto intro!: exI[where x="{0}"] span_zero)
  next
  case (Suc n)
  then obtain T where "subspace T" "T ⊆ span S" "dim T = n"
  by force
  then show ?case
  proof (cases "span S ⊆ span T")
  case True
  have "span T ⊆ span S"
  by (simp add: ‹T ⊆ span S› span_minimal)
  then have "dim S = dim T"
  by (rule span_eq_dim [OF subset_antisym [OF True]])
  then show ?thesis
  using Suc.prems ‹dim T = n› by linarith
  next
  case False
  then obtain y where y: "y ∈ S" "y ∉ T"
  by (meson span_mono subsetI)
  then have "span (insert y T) ⊆ span S"
  by (metis (no_types) ‹T ⊆ span S› subsetD insert_subset span_superset span_mono span_span)
  with ‹dim T = n› ‹subspace T› y show ?thesis
  apply (rule_tac x="span(insert y T)" in exI)
  using span_eq_iff by (fastforce simp: dim_insert)
  qed
  qed
  with that show ?thesis by blast
 qed
 
 lemma basis_subspace_exists:
  assumes "subspace S"
  obtains B where "finite B" "B ⊆ S" "independent B" "span B = S" "card B = dim S"
  by (metis assms span_subspace basis_exists finiteI_independent)
 
 lemma dim_mono: assumes "V ⊆ span W" shows "dim V ≤ dim W"
 proof -
  obtain B where "independent B" "B ⊆ W" "W ⊆ span B"
  using maximal_independent_subset[of W] by force
  with dim_le_card[of V B] assms independent_span_bound[of Basis B] basis_card_eq_dim[of B W]
  span_mono[of B W] span_minimal[OF _ subspace_span, of W B]
  show ?thesis
  by (auto simp: finite_Basis span_Basis)
 qed
 
 lemma dim_subset: "S ⊆ T ==> dim S ≤ dim T"
  using dim_mono[of S T] by (auto intro: span_base)
 
 lemma dim_eq_0 [simp]:
  "dim S = 0 ⟷ S ⊆ {0}"
  by (metis basis_exists card_eq_0_iff dim_span finiteI_independent span_empty subset_empty subset_singletonD)
 
 lemma dim_UNIV[simp]: "dim UNIV = card Basis"
  using dim_eq_card[of Basis UNIV] by (simp add: independent_Basis span_Basis)
 
 lemma independent_card_le_dim: assumes "B ⊆ V" and "independent B" shows "card B ≤ dim V"
  by (subst dim_eq_card[symmetric, OF refl ‹independent B›]) (rule dim_subset[OF ‹B ⊆ V›])
 
 lemma dim_subset_UNIV: "dim S ≤ dimension"
  by (metis dim_subset subset_UNIV dim_UNIV dimension_def)
 
 lemma card_ge_dim_independent:
  assumes BV: "B ⊆ V"
  and iB: "independent B"
  and dVB: "dim V ≤ card B"
  shows "V ⊆ span B"
 proof
  fix a
  assume aV: "a ∈ V"
  {
  assume aB: "a ∉ span B"
  then have iaB: "independent (insert a B)"
  using iB aV BV by (simp add: independent_insert)
  from aV BV have th0: "insert a B ⊆ V"
  by blast
  from aB have "a ∉B"
  by (auto simp add: span_base)
  with independent_card_le_dim[OF th0 iaB] dVB finiteI_independent[OF iB]
  have False by auto
  }
  then show "a ∈ span B" by blast
 qed
 
 lemma card_le_dim_spanning:
  assumes BV: "B ⊆ V"
  and VB: "V ⊆ span B"
  and fB: "finite B"
  and dVB: "dim V ≥ card B"
  shows "independent B"
 proof -
  {
  fix a
  assume a: "a ∈ B" "a ∈ span (B - {a})"
  from a fB have c0: "card B ≠ 0"
  by auto
  from a fB have cb: "card (B - {a}) = card B - 1"
  by auto
  {
  fix x
  assume x: "x ∈ V"
  from a have eq: "insert a (B - {a}) = B"
  by blast
  from x VB have x': "x ∈ span B"
  by blast
  from span_trans[OF a(2), unfolded eq, OF x']
  have "x ∈ span (B - {a})" .
  }
  then have th1: "V ⊆ span (B - {a})"
  by blast
  have th2: "finite (B - {a})"
  using fB by auto
  from dim_le_card[OF th1 th2]
  have c: "dim V ≤ card (B - {a})" .
  from c c0 dVB cb have False by simp
  }
  then show ?thesis
  unfolding dependent_def by blast
 qed
 
 lemma card_eq_dim: "B ⊆ V ==> card B = dim V ==> finite B ==> independent B ⟷ V ⊆ span B"
  by (metis order_eq_iff card_le_dim_spanning card_ge_dim_independent)
 
 lemma subspace_dim_equal:
  assumes "subspace S"
  and "subspace T"
  and "S ⊆ T"
  and "dim S ≥ dim T"
  shows "S = T"
 proof -
  obtain B where B: "B ≤ S" "independent B ∧ S ⊆ span B" "card B = dim S"
  using basis_exists[of S] by metis
  then have "span B ⊆ S"
  using span_mono[of B S] span_eq_iff[of S] assms by metis
  then have "span B = S"
  using B by auto
  have "dim S = dim T"
  using assms dim_subset[of S T] by auto
  then have "T ⊆ span B"
  using card_eq_dim[of B T] B finiteI_independent assms by auto
  then show ?thesis
  using assms ‹span B = S› by auto
 qed
 
 corollary dim_eq_span:
  shows "[S ⊆ T; dim T ≤ dim S] ==> span S = span T"
  by (simp add: span_mono subspace_dim_equal)
 
 lemma dim_psubset:
  "span S ⊂ span T ==> dim S < dim T"
  by (metis (no_types, opaque_lifting) dim_span less_le not_le subspace_dim_equal subspace_span)
 
 lemma dim_eq_full:
  shows "dim S = dimension ⟷ span S = UNIV"
  by (metis dim_eq_span dim_subset_UNIV span_Basis span_span subset_UNIV
  dim_UNIV dim_span dimension_def)
 
 lemma indep_card_eq_dim_span:
  assumes "independent B"
  shows "finite B ∧ card B = dim (span B)"
  using dim_span_eq_card_independent[OF assms] finiteI_independent[OF assms] by auto
 
 text ‹More general size bound lemmas.›
 
 lemma independent_bound_general:
  "independent S ==> finite S ∧ card S ≤ dim S"
  by (simp add: dim_eq_card_independent finiteI_independent)
 
 lemma independent_explicit:
  shows "independent B ⟷ finite B ∧ (∀c. (∑v∈B. c v *s v) = 0 ⟶ (∀v ∈ B. c v = 0))"
  using independent_bound_general
  by (fastforce simp: dependent_finite)
 
 proposition dim_sums_Int:
  assumes "subspace S" "subspace T"
  shows "dim {x + y |x y. x ∈ S ∧ y ∈ T} + dim(S ∩ T) = dim S + dim T" (is "dim ?ST + _ = _")
 proof -
  obtain B where B: "B ⊆ S ∩ T" "S ∩ T ⊆ span B"
  and indB: "independent B"
  and cardB: "card B = dim (S ∩ T)"
  using basis_exists by metis
  then obtain C D where "B ⊆ C" "C ⊆ S" "independent C" "S ⊆ span C"
  and "B ⊆ D" "D ⊆ T" "independent D" "T ⊆ span D"
  using maximal_independent_subset_extend
  by (metis Int_subset_iff ‹B ⊆ S ∩ T› indB)
  then have "finite B" "finite C" "finite D"
  by (simp_all add: finiteI_independent indB independent_bound_general)
  have Beq: "B = C ∩ D"
  proof (rule spanning_subset_independent [symmetric])
  show "independent (C ∩ D)"
  by (meson ‹independent C› independent_mono inf.cobounded1)
  qed (use B ‹C ⊆ S› ‹D ⊆ T› ‹B ⊆ C› ‹B ⊆ D› in auto)
  then have Deq: "D = B ∪ (D - C)"
  by blast
  have CUD: "C ∪ D ⊆ ?ST"
  proof (simp, intro conjI)
  show "C ⊆ ?ST"
  using span_zero span_minimal [OF _ ‹subspace T›] ‹C ⊆ S› by force
  show "D ⊆ ?ST"
  using span_zero span_minimal [OF _ ‹subspace S›] ‹D ⊆ T› by force
  qed
  have "a v = 0" if 0: "(∑v∈C. a v *s v) + (∑v∈D - C. a v *s v) = 0"
  and v: "v ∈ C ∪ (D-C)" for a v
  proof -
  have CsS: "∧x. x ∈ C ==> a x *s x ∈ S"
  using ‹C ⊆ S› ‹subspace S› subspace_scale by auto
  have eq: "(∑v∈D - C. a v *s v) = - (∑v∈C. a v *s v)"
  using that add_eq_0_iff by blast
  have "(∑v∈D - C. a v *s v) ∈ S"
  by (simp add: eq CsS ‹subspace S› subspace_neg subspace_sum)
  moreover have "(∑v∈D - C. a v *s v) ∈ T"
  apply (rule subspace_sum [OF ‹subspace T›])
  by (meson DiffD1 ‹D ⊆ T› ‹subspace T› subset_eq subspace_def)
  ultimately have "(∑v ∈ D-C. a v *s v) ∈ span B"
  using B by blast
  then obtain e where e: "(∑v∈B. e v *s v) = (∑v ∈ D-C. a v *s v)"
  using span_finite [OF ‹finite B›] by force
  have "∧c v. [(∑v∈C. c v *s v) = 0; v ∈ C] ==> c v = 0"
  using ‹finite C› ‹independent C› independentD by blast
  define cc where "cc x = (if x ∈ B then a x + e x else a x)" for x
  have [simp]: "C ∩ B = B" "D ∩ B = B" "C ∩ - B = C-D" "B ∩ (D - C) = {}"
  using ‹B ⊆ C› ‹B ⊆ D› Beq by blast+
  have f2: "(∑v∈C ∩ D. e v *s v) = (∑v∈D - C. a v *s v)"
  using Beq e by presburger
  have f3: "(∑v∈C ∪ D. a v *s v) = (∑v∈C - D. a v *s v) + (∑v∈D - C. a v *s v) + (∑v∈C ∩ D. a v *s v)"
  using ‹finite C› ‹finite D› sum.union_diff2 by blast
  have f4: "(∑v∈C ∪ (D - C). a v *s v) = (∑v∈C. a v *s v) + (∑v∈D - C. a v *s v)"
  by (meson Diff_disjoint ‹finite C› ‹finite D› finite_Diff sum.union_disjoint)
  have "(∑v∈C. cc v *s v) = 0"
  using 0 f2 f3 f4
  apply (simp add: cc_def Beq ‹finite C› sum.If_cases algebra_simps sum.distrib
  if_distrib if_distribR)
  apply (simp add: add.commute add.left_commute diff_eq)
  done
  then have "∧v. v ∈ C ==> cc v = 0"
  using independent_explicit ‹independent C› ‹finite C› by blast
  then have C0: "∧v. v ∈ C - B ==> a v = 0"
  by (simp add: cc_def Beq) meson
  then have [simp]: "(∑x∈C - B. a x *s x) = 0"
  by simp
  have "(∑x∈C. a x *s x) = (∑x∈B. a x *s x)"
  proof -
  have "C - D = C - B"
  using Beq by blast
  then show ?thesis
  using Beq ‹(∑x∈C - B. a x *s x) = 0› f3 f4 by auto
  qed
  with 0 have Dcc0: "(∑v∈D. a v *s v) = 0"
  by (subst Deq) (simp add: ‹finite B› ‹finite D› sum_Un)
  then have D0: "∧v. v ∈ D ==> a v = 0"
  using independent_explicit ‹independent D› ‹finite D› by blast
  show ?thesis
  using v C0 D0 Beq by blast
  qed
  then have "independent (C ∪ (D - C))"
  unfolding independent_explicit
  using independent_explicit
  by (simp add: independent_explicit ‹finite C› ‹finite D› sum_Un del: Un_Diff_cancel)
  then have indCUD: "independent (C ∪ D)" by simp
  have "dim (S ∩ T) = card B"
  by (rule dim_unique [OF B indB refl])
  moreover have "dim S = card C"
  by (metis ‹C ⊆ S› ‹independent C› ‹S ⊆ span C› basis_card_eq_dim)
  moreover have "dim T = card D"
  by (metis ‹D ⊆ T› ‹independent D› ‹T ⊆ span D› basis_card_eq_dim)
  moreover have "dim ?ST = card(C ∪ D)"
  proof -
  have *: "∧x y. [x ∈ S; y ∈ T] ==> x + y ∈ span (C ∪ D)"
  by (meson ‹S ⊆ span C› ‹T ⊆ span D› span_add span_mono subsetCE sup.cobounded1 sup.cobounded2)
  show ?thesis
  by (auto intro: * dim_unique [OF CUD _ indCUD refl])
  qed
  ultimately show ?thesis
  using ‹B = C ∩ D› [symmetric]
  by (simp add: ‹independent C› ‹independent D› card_Un_Int finiteI_independent)
 qed
 
 lemma dependent_biggerset_general:
  "(finite S ==> card S > dim S) ==> dependent S"
  using independent_bound_general[of S] by (metis linorder_not_le)
 
 lemma subset_le_dim:
  "S ⊆ span T ==> dim S ≤ dim T"
  by (metis dim_span dim_subset)
 
 lemma linear_inj_imp_surj:
  assumes lf: "linear scale scale f"
  and fi: "inj f"
  shows "surj f"
 proof -
  interpret lf: linear scale scale f by fact
  from basis_exists[of UNIV] obtain B
  where B: "B ⊆ UNIV" "independent B" "UNIV ⊆ span B" "card B = dim UNIV"
  by blast
  from B(4) have d: "dim UNIV = card B"
  by simp
  have "UNIV ⊆ span (f ` B)"
  proof (rule card_ge_dim_independent)
  show "independent (f ` B)"
  by (simp add: B(2) fi lf.independent_inj_image)
  have "card (f ` B) = dim UNIV"
  by (metis B(1) card_image d fi inj_on_subset)
  then show "dim UNIV ≤ card (f ` B)"
  by simp
  qed blast
  then show ?thesis
  unfolding lf.span_image surj_def
  using B(3) by blast
 qed
 
 end
 
 locale finite_dimensional_vector_space_pair_1 =
  vs1: finite_dimensional_vector_space s1 B1 + vs2: vector_space s2
  for s1 :: "'a::field ==> 'b::ab_group_add ==> 'b" (infixr ‹*a› 75)
  and B1 :: "'b set"
  and s2 :: "'a::field ==> 'c::ab_group_add ==> 'c" (infixr ‹*b› 75)
 begin
 
 sublocale vector_space_pair s1 s2 by unfold_locales
 
 lemma dim_image_eq:
  assumes lf: "linear s1 s2 f"
  and fi: "inj_on f (vs1.span S)"
  shows "vs2.dim (f ` S) = vs1.dim S"
 proof -
  interpret lf: linear by fact
  obtain B where B: "B ⊆ S" "vs1.independent B" "S ⊆ vs1.span B" "card B = vs1.dim S"
  using vs1.basis_exists[of S] by auto
  then have "vs1.span S = vs1.span B"
  using vs1.span_mono[of B S] vs1.span_mono[of S "vs1.span B"] vs1.span_span[of B] by auto
  moreover have "card (f ` B) = card B"
  using assms card_image[of f B] inj_on_subset[of f "vs1.span S" B] B vs1.span_superset by auto
  moreover have "(f ` B) ⊆ (f ` S)"
  using B by auto
  ultimately show ?thesis
  by (metis B(2) B(4) fi lf.dependent_inj_imageD lf.span_image vs2.dim_eq_card_independent vs2.dim_span)
 qed
 
 lemma dim_image_le:
  assumes lf: "linear s1 s2 f"
  shows "vs2.dim (f ` S) ≤ vs1.dim (S)"
 proof -
  from vs1.basis_exists[of S] obtain B where
  B: "B ⊆ S" "vs1.independent B" "S ⊆ vs1.span B" "card B = vs1.dim S" by blast
  from B have fB: "finite B" "card B = vs1.dim S"
  using vs1.independent_bound_general by blast+
  have "vs2.dim (f ` S) ≤ card (f ` B)"
  apply (rule vs2.span_card_ge_dim)
  using lf B fB
  apply (auto simp add: module_hom.span_image module_hom.spans_image subset_image_iff
  linear_iff_module_hom)
  done
  also have "… ≤ vs1.dim S"
  using card_image_le[OF fB(1)] fB by simp
  finally show ?thesis .
 qed
 
 end
 
 locale finite_dimensional_vector_space_pair =
  vs1: finite_dimensional_vector_space s1 B1 + vs2: finite_dimensional_vector_space s2 B2
  for s1 :: "'a::field ==> 'b::ab_group_add ==> 'b" (infixr ‹*a› 75)
  and B1 :: "'b set"
  and s2 :: "'a::field ==> 'c::ab_group_add ==> 'c" (infixr ‹*b› 75)
  and B2 :: "'c set"
 begin
 
 sublocale finite_dimensional_vector_space_pair_1 ..
 
 lemma linear_surjective_imp_injective:
  assumes lf: "linear s1 s2 f" and sf: "surj f" and eq: "vs2.dim UNIV = vs1.dim UNIV"
  shows "inj f"
 proof -
  interpret linear s1 s2 f by fact
  have *: "card (f ` B1) ≤ vs2.dim UNIV"
  using vs1.finite_Basis vs1.dim_eq_card[of B1 UNIV] sf
  by (auto simp: vs1.span_Basis vs1.independent_Basis eq
  simp del: vs2.dim_UNIV
  intro!: card_image_le)
  have indep_fB: "vs2.independent (f ` B1)"
  using vs1.finite_Basis vs1.dim_eq_card[of B1 UNIV] sf *
  by (intro vs2.card_le_dim_spanning[of "f ` B1" UNIV]) (auto simp: span_image vs1.span_Basis )
  have "vs2.dim UNIV ≤ card (f ` B1)"
  unfolding eq sf[symmetric] vs2.dim_span_eq_card_independent[symmetric, OF indep_fB]
  vs2.dim_span
  by (intro vs2.dim_mono) (auto simp: span_image vs1.span_Basis)
  with * have "card (f ` B1) = vs2.dim UNIV" by auto
  also have "... = card B1"
  unfolding eq vs1.dim_UNIV ..
  finally have "inj_on f B1"
  by (subst inj_on_iff_eq_card[OF vs1.finite_Basis])
  then show "inj f"
  using inj_on_span_iff_independent_image[OF indep_fB] vs1.span_Basis by auto
 qed
 
 lemma linear_injective_imp_surjective:
  assumes lf: "linear s1 s2 f" and sf: "inj f" and eq: "vs2.dim UNIV = vs1.dim UNIV"
  shows "surj f"
 proof -
  interpret linear s1 s2 f by fact
  have *: False if b: "b ∉ vs2.span (f ` B1)" for b
  proof -
  have *: "vs2.independent (f ` B1)"
  using vs1.independent_Basis by (intro independent_injective_image inj_on_subset[OF sf]) auto
  have **: "vs2.independent (insert b (f ` B1))"
  using b * by (rule vs2.independent_insertI)
 
  have "b ∉ f ` B1" using vs2.span_base[of b "f ` B1"] b by auto
  then have "Suc (card B1) = card (insert b (f ` B1))"
  using sf[THEN inj_on_subset, of B1] by (subst card.insert_remove) (auto intro: vs1.finite_Basis simp: card_image)
  also have "… = vs2.dim (insert b (f ` B1))"
  using vs2.dim_eq_card_independent[OF **] by simp
  also have "vs2.dim (insert b (f ` B1)) ≤ vs2.dim B2"
  by (rule vs2.dim_mono) (auto simp: vs2.span_Basis)
  also have "… = card B1"
  using vs1.dim_span[of B1] vs2.dim_span[of B2] unfolding vs1.span_Basis vs2.span_Basis eq
  vs1.dim_eq_card_independent[OF vs1.independent_Basis] by simp
  finally show False by simp
  qed
  have "f ` UNIV = f ` vs1.span B1" unfolding vs1.span_Basis ..
  also have "… = vs2.span (f ` B1)" unfolding span_image ..
  also have "… = UNIV" using * by blast
  finally show ?thesis .
 qed
 
 lemma linear_injective_isomorphism:
  assumes lf: "linear s1 s2 f"
  and fi: "inj f"
  and dims: "vs2.dim UNIV = vs1.dim UNIV"
  shows "∃f'. linear s2 s1 f' ∧ (∀x. f' (f x) = x) ∧ (∀x. f (f' x) = x)"
  unfolding isomorphism_expand[symmetric]
  using linear_injective_imp_surjective[OF lf fi dims]
  using fi left_right_inverse_eq lf linear_injective_left_inverse linear_surjective_right_inverse by blast
 
 lemma linear_surjective_isomorphism:
  assumes lf: "linear s1 s2 f"
  and sf: "surj f"
  and dims: "vs2.dim UNIV = vs1.dim UNIV"
  shows "∃f'. linear s2 s1 f' ∧ (∀x. f' (f x) = x) ∧ (∀x. f (f' x) = x)"
  using linear_surjective_imp_injective[OF lf sf dims] sf
  linear_exists_right_inverse_on[OF lf vs1.subspace_UNIV]
  linear_exists_left_inverse_on[OF lf vs1.subspace_UNIV]
  dims lf linear_injective_isomorphism by auto
 
 lemma basis_to_basis_subspace_isomorphism:
  assumes s: "vs1.subspace S"
  and t: "vs2.subspace T"
  and d: "vs1.dim S = vs2.dim T"
  and B: "B ⊆ S" "vs1.independent B" "S ⊆ vs1.span B" "card B = vs1.dim S"
  and C: "C ⊆ T" "vs2.independent C" "T ⊆ vs2.span C" "card C = vs2.dim T"
  shows "∃f. linear s1 s2 f ∧ f ` B = C ∧ f ` S = T ∧ inj_on f S"
 proof -
  from B have fB: "finite B"
  by (simp add: vs1.finiteI_independent)
  from C have fC: "finite C"
  by (simp add: vs2.finiteI_independent)
  from finite_basis_to_basis_subspace_isomorphism[OF s t d fB B fC C] show ?thesis .
 qed
 
 lemma basis_change_exists':
  assumes "vs1.independent B" "vs2.independent B'"
  assumes "vs1.span B = UNIV" "vs2.span B' = UNIV" "vs1.dimension = vs2.dimension"
  shows "∃g. linear s1 s2 g ∧ bij g ∧ bij_betw g B B'"
 proof -
  have "finite B" "finite B'"
  using assms by (auto intro: vs1.finiteI_independent vs2.finiteI_independent)
  moreover from assms have "card B = card B'"
  by (metis vs1.dim_span vs2.dim_span vs1.dim_UNIV vs2.dim_UNIV vs1.dimension_def vs2.dimension_def
  vs1.dim_eq_card_independent vs2.dim_eq_card_independent)
  ultimately obtain h where h: "bij_betw h B B'"
  using ‹card B = card B'› by (meson bij_betw_iff_card)
 
  define g where "g = construct B h"
  interpret g: Vector_Spaces.linear s1 s2 g
  unfolding g_def by (rule linear_construct) fact
 
  have "bij_betw g B B' ⟷ bij_betw h B B'"
  using assms(1) by (intro bij_betw_cong) (auto simp: g_def construct_basis)
  hence "bij_betw g B B'"
  using h by simp
  moreover from this have "bij g"
  using assms(3-5)
  by (metis bij_betw_imp_surj_on bij_def g.linear_axioms local.linear_span_image
  local.linear_surjective_imp_injective vs1.dim_UNIV vs1.dimension_def
  vs2.dim_UNIV vs2.dimension_def)
  ultimately show ?thesis
  using g.linear_axioms by blast
 qed
 
 lemma basis_change_exists:
  assumes "vs1.dimension = vs2.dimension"
  shows "∃g. linear s1 s2 g ∧ bij g ∧ bij_betw g B1 B2"
  using basis_change_exists'[OF vs1.independent_Basis vs2.independent_Basis
  vs1.span_Basis vs2.span_Basis assms] .
 
 end
 
 context finite_dimensional_vector_space begin
 
 lemma linear_surj_imp_inj:
  assumes lf: "linear scale scale f" and sf: "surj f"
  shows "inj f"
 proof -
  interpret finite_dimensional_vector_space_pair scale Basis scale Basis by unfold_locales
  let ?U = "UNIV :: 'b set"
  from basis_exists[of ?U] obtain B
  where B: "B ⊆ ?U" "independent B" "?U ⊆ span B" and d: "card B = dim ?U"
  by blast
  {
  fix x
  assume x: "x ∈ span B" and fx: "f x = 0"
  from B(2) have fB: "finite B"
  using finiteI_independent by auto
  have Uspan: "UNIV ⊆ span (f ` B)"
  by (simp add: B(3) lf linear_spanning_surjective_image sf)
  have fBi: "independent (f ` B)"
  proof (rule card_le_dim_spanning)
  show "card (f ` B) ≤ dim ?U"
  using card_image_le d fB by fastforce
  qed (use fB Uspan in auto)
  have th0: "dim ?U ≤ card (f ` B)"
  by (rule span_card_ge_dim) (use Uspan fB in auto)
  moreover have "card (f ` B) ≤ card B"
  by (rule card_image_le, rule fB)
  ultimately have th1: "card B = card (f ` B)"
  unfolding d by arith
  have fiB: "inj_on f B"
  by (simp add: eq_card_imp_inj_on fB th1)
  from linear_indep_image_lemma[OF lf fB fBi fiB x] fx
  have "x = 0" by blast
  }
  then show ?thesis
  unfolding linear_inj_iff_eq_0[OF lf] using B(3) by blast
 qed
 
 lemma linear_inverse_left:
  assumes lf: "linear scale scale f"
  and lf': "linear scale scale f'"
  shows "f ∘ f' = id ⟷ f' ∘ f = id"
 proof -
  {
  fix f f':: "'b ==> 'b"
  assume lf: "linear scale scale f" "linear scale scale f'"
  assume f: "f ∘ f' = id"
  from f have sf: "surj f"
  by (auto simp add: o_def id_def surj_def) metis
  interpret finite_dimensional_vector_space_pair scale Basis scale Basis by unfold_locales
  from linear_surjective_isomorphism[OF lf(1) sf] lf f
  have "f' ∘ f = id"
  unfolding fun_eq_iff o_def id_def by metis
  }
  then show ?thesis
  using lf lf' by metis
 qed
 
 lemma left_inverse_linear:
  assumes lf: "linear scale scale f"
  and gf: "g ∘ f = id"
  shows "linear scale scale g"
 proof -
  from gf have fi: "inj f"
  by (auto simp add: inj_on_def o_def id_def fun_eq_iff) metis
  interpret finite_dimensional_vector_space_pair scale Basis scale Basis by unfold_locales
  from linear_injective_isomorphism[OF lf fi]
  obtain h :: "'b ==> 'b" where "linear scale scale h" and h: "∀x. h (f x) = x" "∀x. f (h x) = x"
  by blast
  have "h = g"
  by (metis gf h isomorphism_expand left_right_inverse_eq)
  with ‹linear scale scale h› show ?thesis by blast
 qed
 
 lemma inj_linear_imp_inv_linear:
  assumes "linear scale scale f" "inj f" shows "linear scale scale (inv f)"
  using assms inj_iff left_inverse_linear by blast
 
 lemma right_inverse_linear:
  assumes lf: "linear scale scale f"
  and gf: "f ∘ g = id"
  shows "linear scale scale g"
 proof -
  from gf have fi: "surj f"
  by (auto simp add: surj_def o_def id_def) metis
  interpret finite_dimensional_vector_space_pair scale Basis scale Basis by unfold_locales
  from linear_surjective_isomorphism[OF lf fi]
  obtain h:: "'b ==> 'b" where h: "linear scale scale h" "∀x. h (f x) = x" "∀x. f (h x) = x"
  by blast
  then have "h = g"
  by (metis gf isomorphism_expand left_right_inverse_eq)
  with h(1) show ?thesis by blast
 qed
 
 lemma linear_independent_extend_inj:
  assumes "independent B" "independent (f ` B)" "inj_on f B"
  shows "∃g. linear scale scale g ∧ inj g ∧ (∀x∈B. g x = f x)"
 proof -
  obtain B' where B': "span B' = UNIV" "B ⊆ B'" "independent B'"
  using assms(1) by (meson extend_basis_superset independent_extend_basis span_extend_basis)
  obtain B'' where B'': "span B'' = UNIV" "f ` B ⊆ B''" "independent B''"
  using assms(2) by (meson extend_basis_superset independent_extend_basis span_extend_basis)
  have "card B' = card B''"
  using B' B'' by (metis local.dim_eq_card)
  hence "card (B' - B) = card (B'' - f ` B)"
  using finiteI_independent[OF assms(1)] B' B''
  by (subst (1 2) card_Diff_subset) (auto simp: card_image[OF ‹inj_on f B›])
  hence "∃h. bij_betw h (B' - B) (B'' - f ` B)"
  using finiteI_independent[of B'] finiteI_independent[of B''] B' B''
  by (intro finite_same_card_bij) auto
  then obtain h where h: "bij_betw h (B' - B) (B'' - f ` B)"
  by blast
 
  have [simp, intro]: "finite B'"
  using finiteI_independent[of B'] B' by auto
  define r where "r = representation B'"
  define g where "g = (λv. ∑b∈B'. scale (r v b) (if b ∈ B then f b else h b))"
  interpret pair: vector_space_pair scale scale ..
  interpret g: Vector_Spaces.linear scale scale g
  unfolding g_def r_def
  by (intro pair.module_hom_sum pair.linear_compose_scale linear_representation B' conjI module_axioms)
 
  have g_B: "g b = f b" if "b ∈ B" for b
  proof -
  from that have "g b = (∑b'∈{b}. f b)"
  unfolding g_def using B'
  by (intro sum.mono_neutral_cong_right) (auto simp: r_def representation_basis)
  thus "g b = f b"
  by simp
  qed
 
  have g_not_B: "g b = h b" if "b ∈ B' - B" for b
  proof -
  from that have "g b = (∑b'∈{b}. h b)"
  unfolding g_def using B'
  by (intro sum.mono_neutral_cong_right) (auto simp: r_def representation_basis)
  thus "g b = h b"
  by simp
  qed
 
  show ?thesis
  proof (rule exI[of _ g]; safe)
  have "B'' ⊆ range g"
  proof
  fix b assume b: "b ∈ B''"
  thus "b ∈ range g"
  proof (cases "b ∈ f ` B")
  case True
  then obtain b' where b': "b' ∈ B" "f b' = b"
  by blast
  thus ?thesis
  by (auto simp: image_def g_B intro!: exI[of _ b'])
  next
  case False
  with b have "b ∈ B'' - f ` B"
  by blast
  then obtain b' where "h b' = b" "b' ∈ B' - B"
  using h by (metis bij_betw_iff_bijections)
  thus ?thesis
  by (intro rev_image_eqI[of b']) (simp_all add: g_not_B)
  qed
  qed
  hence "span B'' ≤ span (range g)"
  by (intro span_mono)
  also have "span B'' = UNIV"
  using B'' by simp
  finally show "inj g"
  by (simp add: g.linear_axioms g.span_image local.linear_surj_imp_inj set_eq_subset)
  qed (use g_B g.linear_axioms in auto)
 qed
 
 end
 
 context finite_dimensional_vector_space_pair begin
 
 lemma subspace_isomorphism:
  assumes s: "vs1.subspace S"
  and t: "vs2.subspace T"
  and d: "vs1.dim S = vs2.dim T"
  shows "∃f. linear s1 s2 f ∧ f ` S = T ∧ inj_on f S"
 proof -
  from vs1.basis_exists[of S] vs1.finiteI_independent
  obtain B where B: "B ⊆ S" "vs1.independent B" "S ⊆ vs1.span B" "card B = vs1.dim S" and fB: "finite B"
  by metis
  from vs2.basis_exists[of T] vs2.finiteI_independent
  obtain C where C: "C ⊆ T" "vs2.independent C" "T ⊆ vs2.span C" "card C = vs2.dim T" and fC: "finite C"
  by metis
  from B(4) C(4) card_le_inj[of B C] d
  obtain f where f: "f ` B ⊆ C" "inj_on f B" using ‹finite B› ‹finite C›
  by auto
  from linear_independent_extend[OF B(2)]
  obtain g where g: "linear s1 s2 g" "∀x∈ B. g x = f x"
  by blast
  interpret g: linear s1 s2 g by fact
  from inj_on_iff_eq_card[OF fB, of f] f(2) have "card (f ` B) = card B"
  by simp
  with B(4) C(4) have ceq: "card (f ` B) = card C"
  using d by simp
  have "g ` B = f ` B"
  using g(2) by (auto simp add: image_iff)
  also have "… = C" using card_subset_eq[OF fC f(1) ceq] .
  finally have gBC: "g ` B = C" .
  have gi: "inj_on g B"
  using f(2) g(2) by (auto simp add: inj_on_def)
  note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
  {
  fix x y
  assume x: "x ∈ S" and y: "y ∈ S" and gxy: "g x = g y"
  from B(3) x y have x': "x ∈ vs1.span B" and y': "y ∈ vs1.span B"
  by blast+
  from gxy have th0: "g (x - y) = 0"
  by (simp add: linear_diff g)
  have th1: "x - y ∈ vs1.span B"
  using x' y' by (metis vs1.span_diff)
  have "x = y"
  using g0[OF th1 th0] by simp
  }
  then have giS: "inj_on g S"
  unfolding inj_on_def by blast
  from vs1.span_subspace[OF B(1,3) s] have "g ` S = vs2.span (g ` B)"
  by (simp add: module_hom.span_image[OF g(1)[unfolded linear_iff_module_hom]])
  also have "… = vs2.span C" unfolding gBC ..
  also have "… = T" using vs2.span_subspace[OF C(1,3) t] .
  finally have gS: "g ` S = T" .
  from g(1) gS giS show ?thesis
  by blast
 qed
 
 end
 
 hide_const (open) linear
 
 end