Theory Expander_Graphs_Eigenvalues

section ‹Spectral Theory\label{sec:expander_eigenvalues}›

text ‹This section establishes the correspondence of the variationally defined expansion paramters
with the definitions using the spectrum of the stochastic matrix. Additionally stronger
results for the expansion parameters are derived.›

theory Expander_Graphs_Eigenvalues
  imports
    Expander_Graphs_Algebra
    Expander_Graphs_TTS
    Perron_Frobenius.HMA_Connect
    Commuting_Hermitian.Commuting_Hermitian
begin

unbundle intro_cong_syntax

hide_const Matrix_Legacy.transpose
hide_const Matrix_Legacy.row
hide_const Matrix_Legacy.mat
hide_const Matrix.mat
hide_const Matrix.row
hide_fact Matrix_Legacy.row_def
hide_fact Matrix_Legacy.mat_def
hide_fact Matrix.vec_eq_iff
hide_fact Matrix.mat_def
hide_fact Matrix.row_def
no_notation Matrix.scalar_prod  (infix "" 70)
no_notation Ordered_Semiring.max ("Maxı")

lemma mult_right_mono': "y  (0::real)  x  z  y = 0  x * y  z * y"
  by (metis mult_cancel_right mult_right_mono)

lemma poly_prod_zero:
  fixes x :: "'a :: idom"
  assumes "poly (a∈#xs. [:- a, 1:]) x = 0"
  shows "x ∈# xs"
  using assms by (induction xs, auto)

lemma poly_prod_inj_aux_1:
  fixes xs ys :: "('a :: idom) multiset"
  assumes "x ∈# xs"
  assumes "(a∈#xs. [:- a, 1:]) = (a∈#ys. [:- a, 1:])"
  shows "x ∈# ys"
proof -
  have "poly (a∈#ys. [:- a, 1:]) x = poly (a∈#xs. [:- a, 1:]) x" using assms(2) by simp
  also have "... = poly (a∈#xs - {#x#} + {#x#}. [:- a, 1:]) x"
    using assms(1) by simp
  also have "... = 0"
    by simp
  finally have "poly (a∈#ys. [:- a, 1:]) x = 0" by simp
  thus "x ∈# ys" using poly_prod_zero by blast
qed

lemma poly_prod_inj_aux_2:
  fixes xs ys :: "('a :: idom) multiset"
  assumes "x ∈# xs ∪# ys"
  assumes "(a∈#xs. [:- a, 1:]) = (a∈#ys. [:- a, 1:])"
  shows "x ∈# xs ∩# ys"
proof (cases "x ∈# xs")
  case True
  then show ?thesis using poly_prod_inj_aux_1[OF True assms(2)] by simp
next
  case False
  hence a:"x ∈# ys"
    using assms(1) by simp
  then show ?thesis
    using poly_prod_inj_aux_1[OF a assms(2)[symmetric]] by simp
qed

lemma poly_prod_inj:
  fixes xs ys :: "('a :: idom) multiset"
  assumes "(a∈#xs. [:- a, 1:]) = (a∈#ys. [:- a, 1:])"
  shows "xs = ys"
  using assms
proof (induction "size xs + size ys" arbitrary: xs ys rule:nat_less_induct)
  case 1
  show ?case
  proof (cases "xs ∪# ys = {#}")
    case True
    then show ?thesis by simp
  next
    case False
    then obtain x where "x ∈# xs ∪# ys" by auto
    hence a:"x ∈# xs ∩# ys"
      by (intro poly_prod_inj_aux_2[OF _ 1(2)])
    have b: "[:- x, 1:]  0"
      by simp
    have c: "size (xs-{#x#}) + size (ys-{#x#}) < size xs + size ys"
      using a by (simp add: add_less_le_mono size_Diff1_le size_Diff1_less)

    have "[:- x, 1:] * (a∈#xs - {#x#}. [:- a, 1:]) = (a∈#xs. [:- a, 1:])"
      using a by (subst prod_mset.insert[symmetric]) simp
    also have "... = (a∈#ys. [:- a, 1:])" using 1 by simp
    also have "... = [:- x, 1:] * (a∈#ys - {#x#}. [:- a, 1:])"
      using a by (subst prod_mset.insert[symmetric]) simp
    finally have "[:- x, 1:]*(a∈#xs-{#x#}. [:- a, 1:])=[:-x, 1:]*(a∈#ys-{#x#}. [:- a, 1:])"
      by simp
    hence "(a∈#xs-{#x#}. [:- a, 1:]) = (a∈#ys-{#x#}. [:- a, 1:])"
      using mult_left_cancel[OF b] by simp
    hence d:"xs - {#x#} = ys - {#x#}"
      using 1 c by simp
    have "xs = xs - {#x#} + {#x#}"
      using a by simp
    also have "... = ys - {#x#} + {#x#}"
      unfolding d by simp
    also have "... = ys"
      using a by simp
    finally show ?thesis by simp
  qed
qed

definition eigenvalues :: "('a::comm_ring_1)^'n^'n  'a multiset"
  where
    "eigenvalues A = (SOME as. charpoly A = (a∈#as. [:- a, 1:])  size as = CARD ('n))"

lemma char_poly_factorized_hma:
  fixes A :: "complex^'n^'n"
  shows "as. charpoly A = (aas. [:- a, 1:])  length as = CARD ('n)"
  by (transfer_hma rule:char_poly_factorized)

lemma eigvals_poly_length:
  fixes A :: "complex^'n^'n"
  shows
    "charpoly A = (a∈#eigenvalues A. [:- a, 1:])" (is "?A")
    "size (eigenvalues A) = CARD ('n)" (is "?B")
proof -
  define f where "f as = (charpoly A = (a∈#as. [:- a, 1:])  size as = CARD('n))" for as
  obtain as where as_def: "charpoly A = (aas. [:- a, 1:])" "length as = CARD('n)"
    using char_poly_factorized_hma by auto

  have "charpoly A = (aas. [:- a, 1:])"
    unfolding as_def by simp
  also have "... = (a∈#mset as. [:- a, 1:])"
    unfolding prod_mset_prod_list[symmetric] mset_map by simp
  finally have "charpoly A = (a∈#mset as. [:- a, 1:])" by simp
  moreover have "size (mset as)  = CARD('n)"
    using as_def by simp
  ultimately have "f (mset as)"
    unfolding f_def by auto
  hence "f (eigenvalues A)"
    unfolding eigenvalues_def f_def[symmetric] using someI[where x = "mset as" and P="f"] by auto
  thus ?A ?B
    unfolding f_def by auto
qed

lemma similar_matrix_eigvals:
  fixes A B :: "complex^'n^'n"
  assumes "similar_matrix A B"
  shows "eigenvalues A = eigenvalues B"
proof -
  have "(a∈#eigenvalues A. [:- a, 1:]) = (a∈#eigenvalues B. [:- a, 1:])"
    using similar_matrix_charpoly[OF assms] unfolding eigvals_poly_length(1) by simp
  thus ?thesis
    by (intro poly_prod_inj) simp
qed

definition upper_triangular_hma :: "'a::zero^'n^'n  bool"
  where "upper_triangular_hma A 
    i.  j. (to_nat j < Bij_Nat.to_nat i  A $h i $h j = 0)"

lemma for_all_reindex2:
  assumes "range f = A"
  shows "(x  A. y  A. P x y)  (x y. P (f x) (f y))"
  using assms by auto

lemma upper_triangular_hma:
  fixes A :: "('a::zero)^'n^'n"
  shows "upper_triangular (from_hmam A) = upper_triangular_hma A" (is "?L = ?R")
proof -
  have "?L  (i{0..<CARD('n)}. j{0..<CARD('n)}. j < i  A $h from_nat i $h from_nat j = 0)"
    unfolding upper_triangular_def from_hmam_def by auto
  also have "...   ((i::'n) (j::'n). to_nat j < to_nat i  A $h from_nat (to_nat i) $h from_nat (to_nat j) = 0)"
    by (intro for_all_reindex2 range_to_nat[where 'a="'n"])
  also have "...   ?R"
    unfolding upper_triangular_hma_def by auto
  finally show ?thesis by simp
qed

lemma from_hma_carrier:
  fixes A :: "'a^('n::finite)^('m::finite)"
  shows "from_hmam A  carrier_mat (CARD ('m)) (CARD ('n))"
  unfolding from_hmam_def by simp

definition diag_mat_hma :: "'a^'n^'n  'a multiset"
  where "diag_mat_hma A = image_mset (λi. A $h i $h i)  (mset_set UNIV)"

lemma diag_mat_hma:
  fixes A :: "'a^'n^'n"
  shows  "mset (diag_mat (from_hmam A)) = diag_mat_hma A" (is "?L = ?R")
proof -
  have "?L = {#from_hmam A $$ (i, i). i ∈# mset [0..<CARD('n)]#}"
    using from_hma_carrier[where A="A"] unfolding diag_mat_def mset_map by simp
  also have "... = {#from_hmam A $$ (i, i). i ∈# image_mset to_nat (mset_set (UNIV :: 'n set))#}"
    using range_to_nat[where 'a="'n"]
    by (intro arg_cong2[where f="image_mset"] refl) (simp add:image_mset_mset_set[OF inj_to_nat])
  also have "... = {#from_hmam A $$ (to_nat i, to_nat i). i ∈# (mset_set (UNIV :: 'n set))#}"
    by (simp add:image_mset.compositionality comp_def)
  also have "... = ?R"
    unfolding diag_mat_hma_def from_hmam_def using to_nat_less_card[where 'a="'n"]
    by (intro image_mset_cong) auto
  finally show ?thesis by simp
qed

definition adjoint_hma :: "complex^'m^'n  complex^'n^'m" where
  "adjoint_hma A = map_matrix cnj (transpose A)"

lemma adjoint_hma_eq: "adjoint_hma A $h i $h j = cnj (A $h j $h i)"
  unfolding adjoint_hma_def map_matrix_def map_vector_def transpose_def by auto

lemma adjoint_hma:
  fixes A :: "complex^('n::finite)^('m::finite)"
  shows "mat_adjoint (from_hmam A) = from_hmam (adjoint_hma A)"
proof -
  have "mat_adjoint (from_hmam A) $$ (i,j) = from_hmam (adjoint_hma A) $$ (i,j)"
    if "i < CARD('n)" "j < CARD('m)"  for i j
    using from_hma_carrier that unfolding mat_adjoint_def from_hmam_def adjoint_hma_def
      Matrix.mat_of_rows_def map_matrix_def map_vector_def transpose_def by auto
  thus ?thesis
    using from_hma_carrier
    by (intro eq_matI) auto
qed

definition cinner where "cinner v w = scalar_product v (map_vector cnj w)"

context
  includes lifting_syntax
begin

lemma cinner_hma:
  fixes x y :: "complex^'n"
  shows "cinner x y = (from_hmav x) ∙c (from_hmav y)" (is "?L = ?R")
proof -
  have "?L = (iUNIV. x $h i * cnj (y $h i))"
    unfolding cinner_def map_vector_def  scalar_product_def by simp
  also have "... = (i = 0..<CARD('n). x $h from_nat i * cnj (y $h from_nat i))"
    using to_nat_less_card to_nat_from_nat_id
    by (intro sum.reindex_bij_betw[symmetric] bij_betwI[where g="to_nat"]) auto
  also have "... = ?R"
    unfolding Matrix.scalar_prod_def from_hmav_def
    by simp
  finally show ?thesis by simp
qed

lemma cinner_hma_transfer[transfer_rule]:
  "(HMA_V ===> HMA_V ===> (=)) (∙c) cinner"
  unfolding  HMA_V_def  cinner_hma
  by (auto simp:rel_fun_def)

lemma adjoint_hma_transfer[transfer_rule]:
  "(HMA_M ===> HMA_M) (mat_adjoint) adjoint_hma"
  unfolding HMA_M_def rel_fun_def by (auto simp add:adjoint_hma)

end

lemma adjoint_adjoint_id[simp]: "adjoint_hma (adjoint_hma A ) = A"
  by (transfer) (simp add:adjoint_adjoint)

lemma adjoint_def_alter_hma:
  "cinner (A *v v) w = cinner v (adjoint_hma A *v w)"
  by (transfer_hma rule:adjoint_def_alter)

lemma cinner_0: "cinner 0 0 = 0"
  by (transfer_hma)

lemma cinner_scale_left: "cinner (a *s v) w = a * cinner v w"
  by transfer_hma

lemma cinner_scale_right: "cinner v (a *s w) = cnj a * cinner v w"
  by transfer (simp add: inner_prod_smult_right)

lemma norm_of_real:
  shows "norm (map_vector complex_of_real v) = norm v"
  unfolding norm_vec_def map_vector_def
  by (intro L2_set_cong) auto

definition unitary_hma :: "complex^'n^'n  bool"
  where "unitary_hma A  A ** adjoint_hma A = Finite_Cartesian_Product.mat 1"

definition unitarily_equiv_hma where
  "unitarily_equiv_hma A B U  (unitary_hma U  similar_matrix_wit A B U (adjoint_hma U))"

definition diagonal_mat :: "('a::zero)^('n::finite)^'n  bool" where
  "diagonal_mat A  (i. j. i  j  A $h i $h j = 0)"

lemma diagonal_mat_ex:
  assumes "diagonal_mat A"
  shows "A = diag (χ i. A $h i $h i)"
  using assms unfolding diagonal_mat_def diag_def
  by (intro iffD2[OF vec_eq_iff] allI) auto

lemma diag_diagonal_mat[simp]: "diagonal_mat (diag x)"
  unfolding diag_def diagonal_mat_def by auto

lemma diag_imp_upper_tri: "diagonal_mat A  upper_triangular_hma A"
  unfolding diagonal_mat_def upper_triangular_hma_def
  by (metis nat_neq_iff)

definition unitary_diag where
    "unitary_diag A b U  unitarily_equiv_hma A (diag b) U"

definition real_diag_decomp_hma where
  "real_diag_decomp_hma A d U  unitary_diag A d U  
  (i. d $h i  Reals)"

definition hermitian_hma :: "complex^'n^'n  bool" where
  "hermitian_hma A = (adjoint_hma A = A)"

lemma from_hma_one:
  "from_hmam (mat 1 :: (('a::{one,zero})^'n^'n)) = 1m CARD('n)"
  unfolding Finite_Cartesian_Product.mat_def from_hmam_def using from_nat_inj
  by (intro eq_matI) auto

lemma from_hma_mult:
  fixes A :: "('a :: semiring_1)^'m^'n"
  fixes B :: "'a^'k^'m::finite"
  shows "from_hmam A * from_hmam B = from_hmam (A ** B)"
  using HMA_M_mult unfolding rel_fun_def HMA_M_def by auto

lemma hermitian_hma:
  "hermitian_hma A = hermitian (from_hmam A)"
   unfolding hermitian_def adjoint_hma hermitian_hma_def by auto

lemma unitary_hma:
  fixes A :: "complex^'n^'n"
  shows  "unitary_hma A = unitary (from_hmam A)" (is "?L = ?R")
proof -
  have "?R  from_hmam A * mat_adjoint (from_hmam A) = 1m (CARD('n))"
    using from_hma_carrier
    unfolding unitary_def inverts_mat_def by simp
  also have "...  from_hmam (A ** adjoint_hma A) = from_hmam (mat 1::complex^'n^'n)"
    unfolding adjoint_hma from_hma_mult from_hma_one by simp
  also have "...  A ** adjoint_hma A = Finite_Cartesian_Product.mat 1"
    unfolding from_hmam_inj  by simp
  also have "...  ?L" unfolding unitary_hma_def by simp
  finally show ?thesis by simp
qed

lemma unitary_hmaD:
  fixes A :: "complex^'n^'n"
  assumes "unitary_hma A"
  shows "adjoint_hma A ** A = mat 1" (is "?A") "A ** adjoint_hma A = mat 1" (is "?B")
proof -
  have "mat_adjoint (from_hmam A) * from_hmam A = 1m CARD('n)"
    using assms unitary_hma by (intro unitary_simps from_hma_carrier ) auto
  thus ?A
    unfolding adjoint_hma from_hma_mult from_hma_one[symmetric] from_hmam_inj
    by simp
  show ?B
    using assms unfolding unitary_hma_def by simp
qed

lemma unitary_hma_adjoint:
  assumes "unitary_hma A"
  shows "unitary_hma (adjoint_hma A)"
  unfolding unitary_hma_def adjoint_adjoint_id unitary_hmaD[OF assms] by simp

lemma unitarily_equiv_hma:
  fixes A :: "complex^'n^'n"
  shows  "unitarily_equiv_hma A B U =
    unitarily_equiv (from_hmam A) (from_hmam B) (from_hmam U)"
    (is "?L = ?R")
proof -
  have "?R  (unitary_hma U  similar_mat_wit (from_hmam A) (from_hmam B) (from_hmam U) (from_hmam (adjoint_hma U)))"
    unfolding Spectral_Theory_Complements.unitarily_equiv_def unitary_hma[symmetric] adjoint_hma
    by simp
  also have "...  unitary_hma U  similar_matrix_wit A B U (adjoint_hma U)"
    using HMA_similar_mat_wit unfolding rel_fun_def HMA_M_def
    by (intro arg_cong2[where f="(∧)"] refl) force
  also have "...  ?L"
    unfolding unitarily_equiv_hma_def by auto
  finally show ?thesis by simp
qed

lemma Matrix_diagonal_matD:
  assumes "Matrix.diagonal_mat A"
  assumes "i<dim_row A" "j<dim_col A"
  assumes "i  j"
  shows "A $$ (i,j) = 0"
  using assms unfolding Matrix.diagonal_mat_def by auto

lemma diagonal_mat_hma:
  fixes A :: "('a :: zero)^('n :: finite)^'n"
  shows  "diagonal_mat A = Matrix.diagonal_mat (from_hmam A)" (is "?L = ?R")
proof
  show "?L  ?R"
    unfolding diagonal_mat_def Matrix.diagonal_mat_def from_hmam_def
    using from_nat_inj  by auto
next
  assume a:"?R"

  have "A $h i $h j = 0" if "i  j" for i j
  proof -
    have "A $h i $h j = (from_hmam A) $$ (to_nat i,to_nat j)"
      unfolding from_hmam_def using to_nat_less_card[where 'a="'n"] by simp
    also have "... = 0"
      using to_nat_less_card[where 'a="'n"] to_nat_inj that
      by (intro Matrix_diagonal_matD[OF a]) auto
    finally show ?thesis by simp
  qed
  thus "?L"
    unfolding diagonal_mat_def by auto
qed

lemma unitary_diag_hma:
  fixes A :: "complex^'n^'n"
  shows "unitary_diag A d U =
    Spectral_Theory_Complements.unitary_diag (from_hmam A) (from_hmam (diag d)) (from_hmam U)"
proof -
  have "Matrix.diagonal_mat (from_hmam (diag d))"
    unfolding diagonal_mat_hma[symmetric] by simp
  thus ?thesis
    unfolding unitary_diag_def Spectral_Theory_Complements.unitary_diag_def unitarily_equiv_hma
    by auto
qed

lemma real_diag_decomp_hma:
  fixes A :: "complex^'n^'n"
  shows "real_diag_decomp_hma A d U =
    real_diag_decomp (from_hmam A) (from_hmam (diag d)) (from_hmam U)"
proof -
  have 0:"(i. d $h i  )  (i < CARD('n). from_hmam (diag d) $$ (i,i)  )"
    unfolding from_hmam_def diag_def using to_nat_less_card by fastforce
  show ?thesis
    unfolding real_diag_decomp_hma_def real_diag_decomp_def unitary_diag_hma 0
    by auto
qed

lemma diagonal_mat_diag_ex_hma:
  assumes "Matrix.diagonal_mat A" "A  carrier_mat CARD('n) CARD ('n :: finite)"
  shows "from_hmam (diag (χ (i::'n). A $$ (to_nat i,to_nat i))) = A"
  using assms from_nat_inj unfolding from_hmam_def diag_def Matrix.diagonal_mat_def
  by (intro eq_matI) (auto simp add:to_nat_from_nat_id)

theorem commuting_hermitian_family_diag_hma:
  fixes Af :: "(complex^'n^'n) set"
  assumes "finite Af"
    and "Af  {}"
    and "A. A  Af  hermitian_hma A"
    and "A B. A  Af  B Af  A ** B = B ** A"
  shows " U.  A Af. B. real_diag_decomp_hma A B U"
proof -
  have 0:"finite (from_hmam ` Af)"
    using assms(1)by (intro finite_imageI)
  have 1: "from_hmam ` Af  {}"
    using assms(2) by simp
  have 2: "A  carrier_mat (CARD ('n)) (CARD ('n))" if "A  from_hmam ` Af" for A
    using that unfolding from_hmam_def by (auto simp add:image_iff)
  have 3: "0 < CARD('n)"
    by simp
  have 4: "hermitian A" if "A  from_hmam ` Af" for A
    using hermitian_hma assms(3) that by auto
  have 5: "A * B = B * A" if "A  from_hmam ` Af" "B  from_hmam ` Af" for A B
    using that assms(4) by (auto simp add:image_iff from_hma_mult)
  have "U. A from_hmam ` Af. B. real_diag_decomp A B U"
    using commuting_hermitian_family_diag[OF 0 1 2 3 4 5] by auto
  then obtain U Bmap where U_def: "A. A  from_hmam ` Af  real_diag_decomp A (Bmap A) U"
    by metis
  define U' :: "complex^'n^'n" where "U' = to_hmam U"
  define Bmap' :: "complex^'n^'n  complex^'n"
    where "Bmap' = (λM. (χ i. (Bmap (from_hmam M)) $$ (to_nat i,to_nat i)))"

  have "real_diag_decomp_hma A (Bmap' A) U'" if "A  Af" for A
  proof -
    have rdd: "real_diag_decomp (from_hmam A) (Bmap (from_hmam A)) U"
      using U_def that by simp

    have "U  carrier_mat CARD('n) CARD('n)" "Bmap (from_hmam A)  carrier_mat CARD('n) CARD('n)"
      "Matrix.diagonal_mat (Bmap (from_hmam A))"
      using rdd unfolding real_diag_decomp_def Spectral_Theory_Complements.unitary_diag_def
        Spectral_Theory_Complements.unitarily_equiv_def similar_mat_wit_def
      by (auto simp add:Let_def)

    hence "(from_hmam (diag (Bmap' A))) = Bmap (from_hmam A)" "(from_hmam U') = U"
      unfolding Bmap'_def U'_def by (auto simp add:diagonal_mat_diag_ex_hma)
    hence "real_diag_decomp (from_hmam A) (from_hmam (diag (Bmap' A))) (from_hmam U')"
      using rdd by auto
    thus "?thesis"
      unfolding real_diag_decomp_hma by simp
  qed
  thus ?thesis
    by (intro exI[where x="U'"]) auto
qed

lemma char_poly_upper_triangular:
  fixes A :: "complex^'n^'n"
  assumes "upper_triangular_hma A"
  shows "charpoly A = (a ∈# diag_mat_hma A. [:- a, 1:])"
proof -
  have "charpoly A = char_poly (from_hmam A)"
    using HMA_char_poly unfolding rel_fun_def HMA_M_def
    by (auto simp add:eq_commute)
  also have "... = (adiag_mat (from_hmam A). [:- a, 1:])"
    using assms unfolding upper_triangular_hma[symmetric]
    by (intro char_poly_upper_triangular[where n="CARD('n)"] from_hma_carrier) auto
  also have "... = (a∈# mset (diag_mat (from_hmam A)). [:- a, 1:])"
    unfolding prod_mset_prod_list[symmetric] mset_map by simp
  also have "... = (a∈# diag_mat_hma A. [:- a, 1:])"
    unfolding diag_mat_hma by simp
  finally show "charpoly A = (a∈# diag_mat_hma A. [:- a, 1:])" by simp
qed

lemma upper_tri_eigvals:
  fixes A :: "complex^'n^'n"
  assumes "upper_triangular_hma A"
  shows "eigenvalues A = diag_mat_hma A"
proof -
  have "(a∈#eigenvalues A. [:- a, 1:]) = charpoly A"
    unfolding  eigvals_poly_length[symmetric] by simp
  also have "... = (a∈#diag_mat_hma A. [:- a, 1:])"
    by (intro char_poly_upper_triangular assms)
  finally have "(a∈#eigenvalues A. [:- a, 1:]) = (a∈#diag_mat_hma A. [:- a, 1:])"
    by simp
  thus ?thesis
    by (intro poly_prod_inj) simp
qed

lemma cinner_self:
  fixes v :: "complex^'n"
  shows "cinner v v = norm v^2"
proof -
  have 0: "x * cnj x = complex_of_real (x  x)" for x :: complex
    unfolding inner_complex_def complex_mult_cnj by (simp add:power2_eq_square)
  thus ?thesis
    unfolding cinner_def power2_norm_eq_inner scalar_product_def inner_vec_def
      map_vector_def by simp
qed

lemma unitary_iso:
  assumes "unitary_hma U"
  shows "norm (U *v v) = norm v"
proof -
  have "norm (U *v v)^2 = cinner (U *v v) (U *v v)"
    unfolding cinner_self by simp
  also have "... = cinner v v"
    unfolding adjoint_def_alter_hma matrix_vector_mul_assoc unitary_hmaD[OF assms] by simp
  also have "... = norm v^2"
    unfolding cinner_self by simp
  finally have "complex_of_real (norm (U *v v)^2) = norm v^2" by simp
  thus ?thesis
    by (meson norm_ge_zero of_real_hom.injectivity power2_eq_iff_nonneg)
qed

lemma (in semiring_hom) mult_mat_vec_hma:
  "map_vector hom (A *v v) = map_matrix hom A *v map_vector hom v"
  using mult_mat_vec_hom by transfer auto

lemma (in semiring_hom) mat_hom_mult_hma:
  "map_matrix hom (A ** B) = map_matrix hom A ** map_matrix hom B"
  using mat_hom_mult by transfer auto

context regular_graph_tts
begin

lemma to_nat_less_n: "to_nat (x::'n) < n"
  using to_nat_less_card card_n by metis

lemma to_nat_from_nat: "x < n  to_nat (from_nat x :: 'n) = x"
  using to_nat_from_nat_id card_n by metis

lemma hermitian_A: "hermitian_hma A"
  using count_sym unfolding hermitian_hma_def adjoint_hma_def A_def map_matrix_def
    map_vector_def transpose_def by simp

lemma nonneg_A: "nonneg_mat A"
  unfolding nonneg_mat_def A_def by auto

lemma g_step_1:
  assumes "v  verts G"
  shows "g_step (λ_. 1) v = 1" (is "?L = ?R")
proof -
  have "?L = in_degree G v / d"
    unfolding g_step_def in_degree_def by simp
  also have "... = 1"
    unfolding reg(2)[OF assms] using d_gt_0 by simp
  finally show ?thesis by simp
qed

lemma markov: "markov (A :: real^'n^'n)"
proof -
  have "A *v 1 = (1::real ^'n)" (is "?L = ?R")
  proof -
    have "A *v 1 = (χ i. g_step (λ_. 1) (enum_verts i))"
      unfolding g_step_conv one_vec_def by simp
    also have "... = (χ i. 1)"
      using bij_betw_apply[OF enum_verts] by (subst g_step_1) auto
    also have "... = 1" unfolding one_vec_def by simp
    finally show ?thesis by simp
  qed
  thus ?thesis
    by (intro markov_symI nonneg_A symmetric_A)
qed

lemma nonneg_J: "nonneg_mat J"
  unfolding nonneg_mat_def J_def by auto

lemma J_eigvals: "eigenvalues J = {#1::complex#} + replicate_mset (n - 1) 0"
proof -
  define α :: "nat  real" where "α i = sqrt (i^2+i)" for i :: nat

  define q :: "nat  nat  real"
    where "q i j = (
        if i = 0 then (1/sqrt n) else (
        if j < i then ((-1) / α i) else (
        if j = i then (i / α i) else 0)))" for i j

  define Q :: "complex^'n^'n" where "Q = (χ i j. complex_of_real (q (to_nat i) (to_nat j)))"

  define D :: "complex^'n^'n" where
    "D = (χ i j. if to_nat i = 0  to_nat j = 0 then 1 else 0)"

  have 2: "[0..<n] = 0#[1..<n]"
    using n_gt_0 upt_conv_Cons by auto

  have aux0: "(k = 0..<n. q j k * q i k) = of_bool (i = j)" if 1:"i  j" "j < n"  for i j
  proof -
    consider (a) "i = j  j = 0" | (b) "i = 0  i < j" | (c) " 0 < i  i < j" | (d) "0 < i  i = j"
      using 1 by linarith
    thus ?thesis
    proof (cases)
      case a
      then show ?thesis using n_gt_0 by (simp add:q_def)
    next
      case b
      have "(k = 0..<n. q j k*q i k)=(kinsert j ({0..<j}  {j+1..<n}). q j k*q i k)"
        using that(2) by (intro sum.cong) auto
      also have "...=q j j*q i j+(k=0..<j. q j k * q i k)+(k=j+1..<n. q j k * q i k)"
        by (subst sum.insert) (auto simp add: sum.union_disjoint)
      also have "... = 0" using b unfolding q_def by simp
      finally show ?thesis using b by simp
    next
      case c
      have "(k = 0..<n. q j k*q i k)=(kinsert i ({0..<i}  {i+1..<n}). q j k*q i k)"
        using that(2) c by (intro sum.cong) auto
      also have "...=q j i*q i i+(k=0..<i. q j k * q i k)+(k=i+1..<n. q j k * q i k)"
        by (subst sum.insert) (auto simp add: sum.union_disjoint)
      also have "... =(-1) / α j * i / α i+ i * ((-1) / α j *  (-1) / α i)"
        using c unfolding q_def by simp
      also have "... = 0"
        by (simp add:algebra_simps)
      finally show ?thesis using c by simp
    next
      case d
      have "real i + real i^2 = real (i + i^2)" by simp
      also have "...  real 0"
        unfolding of_nat_eq_iff using d by simp
      finally have d_1: "real i  + real i^2  0" by simp
      have "(k = 0..<n. q j k*q i k)=(kinsert i ({0..<i}  {i+1..<n}). q j k*q i k)"
        using that(2) d by (intro sum.cong) auto
      also have "...=q j i*q i i+(k=0..<i. q j k * q i k)+(k=i+1..<n. q j k * q i k)"
        by (subst sum.insert) (auto simp add: sum.union_disjoint)
      also have "... = i/ α i * i / α i+ i * ((-1) / α i *  (-1) / α i)"
        using d that unfolding q_def by simp
      also have "... = (i^2 + i) / (α i)^2"
        by (simp add: power2_eq_square divide_simps)
      also have "... = 1"
        using d_1 unfolding α_def by (simp add:algebra_simps)
      finally show ?thesis using d by simp
    qed
  qed

  have 0:"(k = 0..<n. q j k * q i k) = of_bool (i = j)" (is "?L = ?R")  if "i < n" "j < n"  for i j
  proof -
    have "?L = (k = 0..<n. q (max i j) k * q (min i j) k)"
      by (cases "i  j") ( simp_all add:ac_simps cong:sum.cong)
    also have "... = of_bool (min i j = max i j)"
      using that by (intro aux0) auto
    also have "... = ?R"
      by (cases "i  j") auto
    finally show ?thesis by simp
  qed

  have "(kUNIV. Q $h j $h k * cnj (Q $h i $h k)) = of_bool (i=j)" (is "?L = ?R") for i j
  proof -
    have "?L = complex_of_real (k  (UNIV::'n set). q (to_nat j) (to_nat k) * q (to_nat i) (to_nat k))"
      unfolding Q_def
      by (simp add:case_prod_beta scalar_prod_def map_vector_def inner_vec_def row_def inner_complex_def)
    also have "... = complex_of_real (k=0..<n. q (to_nat j) k * q (to_nat i) k)"
      using to_nat_less_n to_nat_from_nat
      by (intro arg_cong[where f="of_real"] sum.reindex_bij_betw bij_betwI[where g="from_nat"]) (auto)
    also have "... = complex_of_real (of_bool(to_nat i = to_nat j))"
      using to_nat_less_n by (intro arg_cong[where f="of_real"] 0) auto
    also have "... = ?R"
      using to_nat_inj by auto
    finally show ?thesis by simp
  qed
  hence "Q ** adjoint_hma Q = mat 1"
    by (intro iffD2[OF vec_eq_iff]) (auto simp add:matrix_matrix_mult_def mat_def adjoint_hma_eq)
  hence unit_Q: "unitary_hma Q"
    unfolding unitary_hma_def by simp

  have "card {(k::'n). to_nat k = 0} = card {from_nat 0 :: 'n}"
    using to_nat_from_nat[where x="0"] n_gt_0
    by (intro arg_cong[where f="card"] iffD2[OF set_eq_iff]) auto
  hence 5:"card {(k::'n). to_nat k = 0} = 1" by simp
  hence 1:"adjoint_hma Q ** D = (χ i j. (if to_nat j = 0 then complex_of_real (1/sqrt n) else 0))"
    unfolding Q_def D_def by (intro iffD2[OF vec_eq_iff] allI)
     (auto simp add:adjoint_hma_eq matrix_matrix_mult_def q_def if_distrib if_distribR sum.If_cases)

  have "(adjoint_hma Q ** D ** Q) $h i $h j = J $h i $h j" (is "?L1 = ?R1") for i j
  proof -
    have "?L1 =1/((sqrt (real n)) * complex_of_real (sqrt (real n)))"
      unfolding 1 unfolding Q_def using n_gt_0 5
      by (auto simp add:matrix_matrix_mult_def q_def if_distrib if_distribR sum.If_cases)
    also have "... = 1/sqrt (real n)^2"
      unfolding of_real_divide of_real_mult power2_eq_square
      by simp
    also have "... = J $h i $h j"
      unfolding J_def card_n using n_gt_0 by simp
    finally show ?thesis by simp
  qed

  hence "adjoint_hma Q ** D ** Q = J"
    by (intro iffD2[OF vec_eq_iff] allI) auto

  hence "similar_matrix_wit J D (adjoint_hma Q) Q"
    unfolding similar_matrix_wit_def unitary_hmaD[OF unit_Q] by auto
  hence "similar_matrix J D"
    unfolding similar_matrix_def by auto
  hence "eigenvalues J = eigenvalues D"
    by (intro similar_matrix_eigvals)
  also have "... = diag_mat_hma D"
    by (intro upper_tri_eigvals diag_imp_upper_tri) (simp add:D_def diagonal_mat_def)
  also have "... = {# of_bool (to_nat i = 0). i ∈# mset_set (UNIV :: 'n set)#}"
    unfolding diag_mat_hma_def D_def of_bool_def by simp
  also have "... = {# of_bool (i = 0). i ∈# mset_set (to_nat ` (UNIV :: 'n set))#}"
    unfolding image_mset_mset_set[OF inj_to_nat, symmetric]
    by (simp add:image_mset.compositionality comp_def)
  also have "... = mset (map (λi. of_bool(i=0)) [0..<n])"
    unfolding range_to_nat card_n mset_map by simp
  also have "... = mset (1 # map (λi. 0) [1..<n])"
    unfolding 2 by (intro arg_cong[where f="mset"]) simp
  also have "... = {#1#} + replicate_mset (n-1) 0"
    using n_gt_0 by (simp add:map_replicate_const mset_repl)
  finally show ?thesis by simp
qed

lemma J_markov: "markov J"
proof -
  have "nonneg_mat J"
    unfolding J_def nonneg_mat_def by auto
  moreover have "transpose J = J"
    unfolding J_def transpose_def by auto
  moreover have "J *v 1 = (1 :: real^'n)"
    unfolding J_def by (simp add:matrix_vector_mult_def one_vec_def)
  ultimately show ?thesis
    by (intro markov_symI) auto
qed

lemma markov_complex_apply:
  assumes "markov M"
  shows "(map_matrix complex_of_real M) *v (1 :: complex^'n) = 1" (is "?L = ?R")
proof -
  have "?L = (map_matrix complex_of_real M) *v (map_vector complex_of_real 1)"
    by (intro arg_cong2[where f="(*v)"] refl) (simp add: map_vector_def one_vec_def)
  also have "... = map_vector (complex_of_real) 1"
    unfolding of_real_hom.mult_mat_vec_hma[symmetric] markov_apply[OF assms] by simp
  also have "... = ?R"
    by (simp add: map_vector_def one_vec_def)
  finally show ?thesis by simp
qed

lemma J_A_comm_real: "J ** A = A ** (J :: real^'n^'n)"
proof -
  have 0: "(kUNIV. A $h k $h i / real CARD('n)) = 1 / real CARD('n)" (is "?L = ?R") for i
  proof -
    have "?L = (1 v* A) $h i / real CARD('n)"
      unfolding vector_matrix_mult_def by (simp add:sum_divide_distrib)
    also have "... = ?R"
      unfolding markov_apply[OF markov] by simp
    finally show ?thesis by simp
  qed
  have 1: "(kUNIV. A $h i $h k / real CARD('n)) = 1 / real CARD('n)" (is "?L = ?R") for i
  proof -
    have "?L = (A *v 1) $h i / real CARD('n)"
      unfolding matrix_vector_mult_def by (simp add:sum_divide_distrib)
    also have "... = ?R"
      unfolding markov_apply[OF markov] by simp
    finally show ?thesis by simp
  qed

  show ?thesis
    unfolding J_def using 0 1
    by (intro iffD2[OF vec_eq_iff] allI) (simp add:matrix_matrix_mult_def)
qed

lemma J_A_comm: "J ** A = A ** (J :: complex^'n^'n)" (is "?L = ?R")
proof -
  have "J ** A = map_matrix complex_of_real (J ** A)"
    unfolding of_real_hom.mat_hom_mult_hma J_def A_def
    by (auto simp add:map_matrix_def map_vector_def)
  also have "... = map_matrix complex_of_real (A ** J)"
    unfolding J_A_comm_real by simp
  also have "... = map_matrix complex_of_real A ** map_matrix complex_of_real J"
    unfolding of_real_hom.mat_hom_mult_hma by simp
  also have "... = ?R"
    unfolding A_def J_def
    by (auto simp add:map_matrix_def map_vector_def)
  finally show ?thesis by simp
qed

definition γa :: "'n itself  real" where
  "γa _ = (if n > 1 then Max_mset (image_mset cmod (eigenvalues A - {#1#})) else 0)"

definition γ2 :: "'n itself  real" where
  "γ2 _ = (if n > 1 then Max_mset {# Re x. x ∈# (eigenvalues A - {#1#})#} else 0)"

lemma J_sym: "hermitian_hma J"
  unfolding J_def hermitian_hma_def
  by (intro  iffD2[OF vec_eq_iff] allI) (simp add: adjoint_hma_eq)

lemma
  shows evs_real: "set_mset (eigenvalues A::complex multiset)  " (is "?R1")
    and ev_1: "(1::complex) ∈# eigenvalues A"
    and γa_ge_0: "γa TYPE ('n)  0"
    and find_any_ev:
      "α ∈# eigenvalues A - {#1#}. v. cinner v 1 = 0  v  0  A *v v = α *s v"
    and γa_bound: "v. cinner v 1 = 0  norm (A *v v)  γa TYPE('n) * norm v"
    and γ2_bound: "(v::real^'n). v  1 = 0  v  (A *v v)  γ2 TYPE ('n) * norm v^2"
proof -
  have " U.  A {J,A}. B. real_diag_decomp_hma A B U"
    using J_sym hermitian_A J_A_comm
    by (intro commuting_hermitian_family_diag_hma) auto
  then obtain U Ad Jd
    where A_decomp: "real_diag_decomp_hma A Ad U" and K_decomp: "real_diag_decomp_hma J Jd U"
    by auto
  have J_sim: "similar_matrix_wit J (diag Jd) U (adjoint_hma U)" and
    unit_U: "unitary_hma U"
    using K_decomp unfolding real_diag_decomp_hma_def unitary_diag_def unitarily_equiv_hma_def
    by auto

  have "diag_mat_hma (diag Jd) = eigenvalues (diag Jd)"
    by (intro upper_tri_eigvals[symmetric] diag_imp_upper_tri J_sim) auto
  also have "... = eigenvalues J"
    using J_sim by (intro similar_matrix_eigvals[symmetric]) (auto simp add:similar_matrix_def)
  also have "... ={#1::complex#} + replicate_mset (n - 1) 0"
    unfolding J_eigvals by simp
  finally have 0:"diag_mat_hma (diag Jd) = {#1::complex#} + replicate_mset (n - 1) 0" by simp
  hence "1 ∈# diag_mat_hma (diag Jd)" by simp
  then obtain i where i_def:"Jd $h i = 1"
    unfolding diag_mat_hma_def diag_def by auto
  have "{# Jd $h j. j ∈# mset_set (UNIV - {i}) #} = {#Jd $h j. j ∈# mset_set UNIV - mset_set {i}#}"
    unfolding diag_mat_hma_def by (intro arg_cong2[where f="image_mset"] mset_set_Diff) auto
  also have "... = diag_mat_hma (diag Jd)  - {#1#}"
    unfolding diag_mat_hma_def diag_def by (subst image_mset_Diff) (auto simp add:i_def)
  also have "... =  replicate_mset (n - 1) 0"
    unfolding 0 by simp
  finally have "{# Jd $h j. j ∈# mset_set (UNIV - {i}) #} = replicate_mset (n - 1) 0"
    by simp
  hence "set_mset {# Jd $h j. j ∈# mset_set (UNIV - {i}) #}  {0}"
    by simp
  hence 1:"Jd $h j = 0" if "j  i" for j
    using that by auto

  define u where "u = adjoint_hma U *v 1"
  define α where "α = u $h i"

  have "U *v u = (U ** adjoint_hma U) *v 1"
    unfolding u_def by (simp add:matrix_vector_mul_assoc)
  also have "... = 1"
    unfolding unitary_hmaD[OF unit_U] by simp
  also have "...   0"
    by simp
  finally have "U *v u  0" by simp
  hence u_nz: "u  0"
    by (cases "u = 0") auto

  have "diag Jd *v u = adjoint_hma U ** U ** diag Jd ** adjoint_hma U *v 1"
    unfolding unitary_hmaD[OF unit_U] u_def by (auto simp add:matrix_vector_mul_assoc)
  also have "... = adjoint_hma U ** (U ** diag Jd ** adjoint_hma U) *v 1"
    by (simp add:matrix_mul_assoc)
  also have "... = adjoint_hma U ** J *v 1"
    using J_sim unfolding similar_matrix_wit_def by simp
  also have "... = adjoint_hma U *v (map_matrix complex_of_real J *v 1)"
    by (simp add:map_matrix_def map_vector_def J_def matrix_vector_mul_assoc)
  also have "... = u"
    unfolding u_def markov_complex_apply[OF J_markov] by simp
  finally have u_ev: "diag Jd *v u = u" by simp
  hence "Jd * u = u"
    unfolding diag_vec_mult_eq by simp
  hence "u $h j = 0" if "j  i" for j
    using 1 that unfolding times_vec_def vec_eq_iff by auto
  hence u_alt: "u = axis i α"
    unfolding α_def axis_def vec_eq_iff by auto
  hence α_nz: "α  0"
    using u_nz by (cases "α=0") auto

  have A_sim: "similar_matrix_wit A (diag Ad) U (adjoint_hma U)" and Ad_real: "i. Ad $h i  "
    using A_decomp unfolding real_diag_decomp_hma_def unitary_diag_def unitarily_equiv_hma_def
    by auto

  have "diag_mat_hma (diag Ad) = eigenvalues (diag Ad)"
    by (intro upper_tri_eigvals[symmetric] diag_imp_upper_tri A_sim) auto
  also have "... = eigenvalues A"
    using A_sim by (intro similar_matrix_eigvals[symmetric]) (auto simp add:similar_matrix_def)
  finally have 3:"diag_mat_hma (diag Ad) = eigenvalues A"
    by simp

  show ?R1
    unfolding 3[symmetric] diag_mat_hma_def diag_def using Ad_real by auto

  have "diag Ad *v u = adjoint_hma U ** U ** diag Ad ** adjoint_hma U *v 1"
    unfolding unitary_hmaD[OF unit_U] u_def by (auto simp add:matrix_vector_mul_assoc)
  also have "... = adjoint_hma U ** (U ** diag Ad ** adjoint_hma U) *v 1"
    by (simp add:matrix_mul_assoc)
  also have "... = adjoint_hma U ** A *v 1"
    using A_sim unfolding similar_matrix_wit_def by simp
  also have "... = adjoint_hma U *v (map_matrix complex_of_real A *v 1)"
    by (simp add:map_matrix_def map_vector_def A_def matrix_vector_mul_assoc)
  also have "... = u"
    unfolding u_def markov_complex_apply[OF markov] by simp
  finally have u_ev_A: "diag Ad *v u = u" by simp
  hence "Ad * u = u"
    unfolding diag_vec_mult_eq by simp
  hence 5:"Ad $h i = 1"
    using α_nz unfolding u_alt times_vec_def vec_eq_iff axis_def by force

  thus ev_1: "(1::complex) ∈# eigenvalues A"
    unfolding 3[symmetric] diag_mat_hma_def diag_def by auto

  have "eigenvalues A - {#1#} = diag_mat_hma (diag Ad) - {#1#}"
    unfolding 3 by simp
  also have "... = {#Ad $h j. j ∈# mset_set UNIV#} - {# Ad $h i #}"
    unfolding 5 diag_mat_hma_def diag_def by simp
  also have "... = {#Ad $h j. j ∈# mset_set UNIV - mset_set {i}#}"
    by (subst image_mset_Diff) auto
  also have "... = {#Ad $h j. j ∈# mset_set (UNIV - {i})#}"
    by (intro arg_cong2[where f="image_mset"] mset_set_Diff[symmetric]) auto
  finally have 4:"eigenvalues A - {#1#} = {#Ad $h j. j ∈# mset_set (UNIV - {i})#}" by simp

  have "cmod (Ad $h k)  γa TYPE ('n)" if "n > 1" "k  i" for k
    unfolding γa_def 4 using that Max_ge by auto
  moreover have "k = i" if "n = 1" for k
    using that to_nat_less_n by simp
  ultimately have norm_Ad: "norm (Ad $h k)  γa TYPE ('n)  k = i" for k
    using n_gt_0 by (cases "n = 1", auto)

  have "Re (Ad $h k)  γ2 TYPE ('n)" if "n > 1" "k  i" for k
    unfolding γ2_def 4 using that Max_ge by auto
  moreover have "k = i" if "n = 1" for k
    using that to_nat_less_n by simp
  ultimately have Re_Ad: "Re (Ad $h k)  γ2 TYPE ('n)  k = i" for k
    using n_gt_0 by (cases "n = 1", auto)

  show Λe_ge_0: "γa TYPE ('n)  0"
  proof (cases "n > 1")
    case True
    then obtain k where k_def: "k  i"
      by (metis (full_types) card_n from_nat_inj n_gt_0 one_neq_zero)
    have "0  cmod (Ad $h k)"
      by simp
    also have "...  γa TYPE ('n)"
      using norm_Ad k_def by auto
    finally show ?thesis by auto
  next
    case False
    thus ?thesis unfolding γa_def by simp
  qed

  have "v. cinner v 1 = 0  v  0  A *v v = β *s v" if β_ran: "β ∈# eigenvalues A - {#1#}" for β
  proof -
    obtain j where j_def: "β = Ad $h j" "j  i"
      using β_ran unfolding 4 by auto
    define v where "v = U *v axis j 1"

    have "A *v v = A ** U *v axis j 1"
      unfolding v_def by (simp add:matrix_vector_mul_assoc)
    also have "... = ((U ** diag Ad ** adjoint_hma U) ** U) *v axis j 1"
      using A_sim unfolding similar_matrix_wit_def by simp
    also have "... = U ** diag Ad ** (adjoint_hma U ** U) *v axis j 1"
      by (simp add:matrix_mul_assoc)
    also have "... = U ** diag Ad *v axis j 1"
      using unitary_hmaD[OF unit_U] by simp
    also have "... = U *v (Ad * axis j 1)"
      by (simp add:matrix_vector_mul_assoc[symmetric] diag_vec_mult_eq)
    also have "... = U *v (β *s axis j 1)"
      by (intro arg_cong2[where f="(*v)"] iffD2[OF vec_eq_iff]) (auto simp:j_def axis_def)
    also have "... = β *s v"
      unfolding v_def by (simp add:vector_scalar_commute)
    finally have 5:"A *v v = β *s v" by simp

    have "cinner v 1 = cinner (axis j 1) (adjoint_hma U *v 1)"
      unfolding v_def adjoint_def_alter_hma by simp
    also have "... = cinner (axis j 1) (axis i α)"
      unfolding u_def[symmetric] u_alt by simp
    also have " ... = 0"
      using j_def(2) unfolding cinner_def axis_def scalar_product_def  map_vector_def
      by (auto simp:if_distrib if_distribR sum.If_cases)
    finally have 6:"cinner v 1  = 0 "
      by simp

    have "cinner v v = cinner (axis j 1) (adjoint_hma U *v (U *v (axis j 1)))"
      unfolding v_def adjoint_def_alter_hma by simp
    also have "... = cinner (axis j 1) (axis j 1)"
      unfolding matrix_vector_mul_assoc unitary_hmaD[OF unit_U] by simp
    also have "... = 1"
      unfolding cinner_def axis_def scalar_product_def  map_vector_def
      by (auto simp:if_distrib if_distribR sum.If_cases)
    finally have "cinner v v = 1"
      by simp
    hence 7:"v  0"
      by (cases "v=0") (auto simp add:cinner_0)

    show ?thesis
      by (intro exI[where x="v"] conjI 6 7 5)
  qed

  thus "α ∈# eigenvalues A - {#1#}. v. cinner v 1 = 0  v  0  A *v v = α *s v"
    by simp

  have "norm (A *v v)  γa TYPE('n) * norm v" if "cinner v 1 = 0" for v
  proof -
    define w where "w= adjoint_hma U *v v"

    have "w $h i = cinner w (axis i 1)"
      unfolding cinner_def axis_def scalar_product_def map_vector_def
      by (auto simp:if_distrib if_distribR sum.If_cases)
    also have "... = cinner v (U *v axis i 1)"
      unfolding w_def adjoint_def_alter_hma by simp
    also have "... = cinner v ((1 / α) *s (U *v u))"
      unfolding vector_scalar_commute[symmetric] u_alt using α_nz
      by (intro_cong "[σ2 cinner, σ2 (*v)]") (auto simp add:axis_def vec_eq_iff)
    also have "... = cinner v ((1 / α) *s 1)"
      unfolding u_def matrix_vector_mul_assoc unitary_hmaD[OF unit_U] by simp
    also have "... = 0"
      unfolding cinner_scale_right that by simp
    finally have w_orth: "w $h i = 0" by simp

    have "norm (A *v v) = norm (U *v (diag Ad *v w))"
      using A_sim  unfolding matrix_vector_mul_assoc similar_matrix_wit_def w_def
      by (simp add:matrix_mul_assoc)
    also have "... = norm (diag Ad *v w)"
      unfolding unitary_iso[OF unit_U] by simp
    also have "... = norm (Ad * w)"
      unfolding diag_vec_mult_eq by simp
    also have "... = sqrt (iUNIV. (cmod (Ad $h i) * cmod (w $h i))2)"
      unfolding norm_vec_def L2_set_def times_vec_def by (simp add:norm_mult)
    also have "...  sqrt (iUNIV. ((γa TYPE('n)) * cmod (w $h i))^2)"
      using w_orth norm_Ad
      by (intro iffD2[OF real_sqrt_le_iff] sum_mono power_mono mult_right_mono') auto
    also have "... = ¦γa TYPE('n)¦ * sqrt (iUNIV. (cmod (w $h i))2)"
      by (simp add:power_mult_distrib sum_distrib_left[symmetric] real_sqrt_mult)
    also have "... = ¦γa TYPE('n)¦ * norm w"
      unfolding norm_vec_def L2_set_def by simp
    also have "... = γa TYPE('n) * norm w"
      using Λe_ge_0 by simp
    also have "... = γa TYPE('n) * norm v"
      unfolding w_def unitary_iso[OF unitary_hma_adjoint[OF unit_U]] by simp
    finally show "norm (A *v v)  γa TYPE('n) * norm v"
      by simp
  qed

  thus "v. cinner v 1 = 0  norm (A *v v)  γa TYPE('n) * norm v" by auto

  have "v  (A *v v)  γ2 TYPE ('n) * norm v^2" if "v  1 = 0" for v :: "real^'n"
  proof -
    define v' where "v' = map_vector complex_of_real v"
    define w where "w= adjoint_hma U *v v'"

    have "w $h i = cinner w (axis i 1)"
      unfolding cinner_def axis_def scalar_product_def map_vector_def
      by (auto simp:if_distrib if_distribR sum.If_cases)
    also have "... = cinner v' (U *v axis i 1)"
      unfolding w_def adjoint_def_alter_hma by simp
    also have "... = cinner v' ((1 / α) *s (U *v u))"
      unfolding vector_scalar_commute[symmetric] u_alt using α_nz
      by (intro_cong "[σ2 cinner, σ2 (*v)]") (auto simp add:axis_def vec_eq_iff)
    also have "... = cinner v' ((1 / α) *s 1)"
      unfolding u_def matrix_vector_mul_assoc unitary_hmaD[OF unit_U] by simp
    also have "... = cnj (1 / α) * cinner v' 1"
      unfolding cinner_scale_right by simp
    also have "... = cnj (1 / α) * complex_of_real (v  1)"
      unfolding cinner_def scalar_product_def map_vector_def inner_vec_def v'_def
      by (intro arg_cong2[where f="(*)"] refl) (simp)
    also have "... = 0"
      unfolding that by simp
    finally have w_orth: "w $h i = 0" by simp

    have "complex_of_real (norm v^2) = complex_of_real (v  v)"
      by (simp add: power2_norm_eq_inner)
    also have "... = cinner v' v'"
      unfolding v'_def cinner_def scalar_product_def inner_vec_def map_vector_def by simp
    also have "... = norm v'^2"
      unfolding cinner_self by simp
    also have "... = norm w^2"
      unfolding w_def unitary_iso[OF unitary_hma_adjoint[OF unit_U]] by simp
    also have "... = cinner w w"
      unfolding cinner_self by simp
    also have "... = (jUNIV. complex_of_real (cmod (w $h j)^2))"
      unfolding cinner_def scalar_product_def map_vector_def
      cmod_power2 complex_mult_cnj[symmetric] by simp
    also have "... = complex_of_real (jUNIV.  (cmod (w $h j)^2))"
      by simp
    finally have "complex_of_real (norm v^2) = complex_of_real (jUNIV.  (cmod (w $h j)^2))"
      by simp
    hence norm_v: "norm v^2 = (jUNIV.  (cmod (w $h j)^2))"
      using of_real_hom.injectivity by blast

    have "complex_of_real (v  (A *v v)) = cinner v' (map_vector of_real (A *v v))"
      unfolding v'_def cinner_def scalar_product_def inner_vec_def map_vector_def
      by simp
    also have "... = cinner v' (map_matrix of_real A *v v')"
      unfolding v'_def of_real_hom.mult_mat_vec_hma by simp
    also have "... = cinner v' (A *v v')"
      unfolding map_matrix_def map_vector_def A_def by auto
    also have "... = cinner v' (U ** diag Ad ** adjoint_hma U *v v')"
      using A_sim unfolding similar_matrix_wit_def by simp
    also have "... = cinner (adjoint_hma U *v v') (diag Ad ** adjoint_hma U *v v')"
      unfolding adjoint_def_alter_hma adjoint_adjoint adjoint_adjoint_id
      by (simp add:matrix_vector_mul_assoc matrix_mul_assoc)
    also have "... = cinner w (diag Ad *v w)"
      unfolding w_def by (simp add:matrix_vector_mul_assoc)
    also have "... = cinner w (Ad * w)"
      unfolding diag_vec_mult_eq by simp
    also have "... = (jUNIV. cnj (Ad $h j) * cmod (w $h j)^2)"
      unfolding cinner_def map_vector_def scalar_product_def cmod_power2 complex_mult_cnj[symmetric]
      by (simp add:algebra_simps)
    also have "... = (jUNIV. Ad $h j * cmod (w $h j)^2)"
      using Ad_real by (intro sum.cong refl arg_cong2[where f="(*)"] iffD1[OF Reals_cnj_iff]) auto
    also have "... = (jUNIV. complex_of_real (Re (Ad $h j) * cmod (w $h j)^2))"
      using Ad_real by (intro sum.cong refl) simp
    also have "... = complex_of_real (j UNIV. Re (Ad $h j) * cmod (w $h j)^2)"
      by simp
    finally have "complex_of_real (v(A *v v)) = of_real(jUNIV. Re (Ad $h j) * cmod (w $h j)^2)"
      by simp
    hence "v(A *v v) = (jUNIV. Re (Ad $h j) * cmod (w $h j)^2)"
      using of_real_hom.injectivity by blast
    also have "...  (jUNIV. γ2 TYPE ('n) * cmod (w $h j)^2)"
      using w_orth Re_Ad by (intro sum_mono mult_right_mono') auto
    also have "... = γ2 TYPE ('n) * (jUNIV. cmod (w $h j)^2)"
      by (simp add:sum_distrib_left)
    also have "... = γ2 TYPE ('n) * norm v^2"
      unfolding norm_v by simp
    finally show ?thesis by simp
  qed

  thus "(v::real^'n). v  1 = 0  v  (A *v v)  γ2 TYPE ('n) * norm v^2"
    by auto
qed

lemma find_any_real_ev:
  assumes "complex_of_real α ∈# eigenvalues A - {#1#}"
  shows "v. v  1 = 0  v  0  A *v v = α *s v"
proof -
  obtain w where w_def: "cinner w 1 = 0" "w  0" "A *v w = α *s w"
    using find_any_ev assms by auto

  have "w = 0" if "map_vector Re (1 *s w) = 0" "map_vector Re (𝗂 *s w) = 0"
    using that by (simp add:vec_eq_iff map_vector_def complex_eq_iff)
  then obtain c where c_def: "map_vector Re (c *s w)  0"
    using w_def(2) by blast

  define u where "u = c *s w"

  define v where "v = map_vector Re u"

  hence "v  1 = Re (cinner u 1)"
    unfolding cinner_def inner_vec_def scalar_product_def map_vector_def by simp
  also have "... = 0"
    unfolding u_def cinner_scale_left w_def(1) by simp
  finally have 1:"v  1 = 0" by simp

  have "A *v v = (χ i. jUNIV. A $h i $h j * Re (u $h j))"
    unfolding matrix_vector_mult_def v_def map_vector_def by simp
  also have "... = (χ i. jUNIV. Re ( of_real (A $h i $h j) * u $h j))"
    by simp
  also have "... = (χ i. Re (jUNIV. A $h i $h j * u $h j))"
    unfolding A_def by simp
  also have "... = map_vector Re (A *v u)"
    unfolding map_vector_def matrix_vector_mult_def by simp
  also have "... = map_vector Re (of_real α *s u)"
    unfolding u_def vector_scalar_commute w_def(3)
    by (simp add:ac_simps)
  also have "... = α *s v"
    unfolding v_def by (simp add:vec_eq_iff map_vector_def)
  finally have 2: "A *v v = α *s v" by simp

  have 3:"v  0"
    unfolding v_def u_def using c_def by simp

  show ?thesis
    by (intro exI[where x="v"] conjI 1 2 3)
qed

lemma size_evs:
  "size (eigenvalues A - {#1::complex#}) = n-1"
proof -
  have "size (eigenvalues A :: complex multiset) = n"
    using eigvals_poly_length card_n[symmetric] by auto
  thus "size (eigenvalues A - {#(1::complex)#}) = n -1"
    using ev_1 by (simp add: size_Diff_singleton)
qed

lemma find_γ2:
  assumes "n > 1"
  shows "γa TYPE('n) ∈# image_mset cmod (eigenvalues A - {#1::complex#})"
proof -
  have "set_mset (eigenvalues A - {#(1::complex)#})  {}"
    using assms size_evs by auto
  hence 2: "cmod ` set_mset (eigenvalues A - {#1#})  {}"
    by simp
  have "γa TYPE('n)  set_mset (image_mset cmod (eigenvalues A - {#1#}))"
    unfolding γa_def using assms 2 Max_in by auto
  thus "γa TYPE('n) ∈# image_mset cmod (eigenvalues A - {#1#})"
    by simp
qed

lemma γ2_real_ev:
  assumes "n > 1"
  shows "v. (α. abs α=γa TYPE('n)  v  1=0  v  0  A *v v = α *s v)"
proof -
  obtain α where α_def: "cmod α = γa TYPE('n)" "α ∈# eigenvalues A - {#1#}"
    using find_γ2[OF assms] by auto
  have "α  "
    using in_diffD[OF α_def(2)] evs_real by auto
  then obtain β where β_def: "α = of_real β"
    using Reals_cases by auto

  have 0:"complex_of_real β ∈# eigenvalues A-{#1#}"
    using α_def unfolding β_def by auto

  have 1: "¦β¦ = γa TYPE('n)"
    using α_def unfolding β_def by simp
  show ?thesis
    using find_any_real_ev[OF 0] 1 by auto
qed

lemma γa_real_bound:
  fixes v :: "real^'n"
  assumes "v  1 = 0"
  shows "norm (A *v v)  γa TYPE('n) * norm v"
proof -
  define w where "w = map_vector complex_of_real v"

  have "cinner w 1 = v  1"
    unfolding w_def cinner_def map_vector_def scalar_product_def inner_vec_def
    by simp
  also have "... = 0" using assms by simp
  finally have 0: "cinner w 1 = 0" by simp
  have "norm (A *v v) = norm (map_matrix complex_of_real A *v (map_vector complex_of_real v))"
    unfolding norm_of_real of_real_hom.mult_mat_vec_hma[symmetric] by simp
  also have "... = norm (A *v w)"
    unfolding w_def A_def map_matrix_def map_vector_def by simp
  also have "...  γa TYPE('n) * norm w"
    using γa_bound 0 by auto
  also have "... = γa TYPE('n) * norm v"
    unfolding w_def norm_of_real by simp
  finally show ?thesis by simp
qed

lemma Λe_eq_Λ: "Λa = γa TYPE('n)"
proof -
  have "¦g_inner f (g_step f)¦  γa TYPE('n) * (g_norm f)2"
    (is "?L  ?R") if "g_inner f (λ_. 1) = 0" for f
  proof -
    define v where "v = (χ i. f (enum_verts i))"
    have 0: " v  1 = 0"
      using that unfolding g_inner_conv one_vec_def v_def by auto
    have "?L = ¦v  (A *v v)¦"
      unfolding g_inner_conv g_step_conv v_def by simp
    also have "...  (norm v * norm (A *v v))"
      by (intro Cauchy_Schwarz_ineq2)
    also have "...  (norm v * (γa TYPE('n) * norm v))"
      by (intro mult_left_mono γa_real_bound 0) auto
    also have "... = ?R"
      unfolding g_norm_conv v_def by (simp add:algebra_simps power2_eq_square)
    finally show ?thesis by simp
  qed
  hence "Λa  γa TYPE('n)"
    using γa_ge_0 by (intro expander_intro_1) auto

  moreover have "Λa  γa TYPE('n)"
  proof (cases "n > 1")
    case True
    then obtain v α where v_def: "abs α = γa TYPE('n)" "A *v v =α *s v" "v  0" "v  1 = 0"
      using γ2_real_ev by auto
    define f where "f x = v $h enum_verts_inv x" for x
    have v_alt: "v = (χ i. f (enum_verts i))"
      unfolding f_def Rep_inverse by simp

    have "g_inner f (λ_. 1) = v  1"
      unfolding g_inner_conv v_alt one_vec_def by simp
    also have "... = 0" using v_def by simp
    finally have 2:"g_inner f (λ_. 1) = 0" by simp

    have "γa TYPE('n) * g_norm f^2 = γa TYPE('n) * norm v^2"
      unfolding g_norm_conv v_alt by simp
    also have "... = γa TYPE('n) * ¦v  v¦"
      by (simp add: power2_norm_eq_inner)
    also have "... = ¦v  (α *s v)¦