imports Jordan_Normal_Form_Uniqueness Perron_Frobenius_General Real_Asymp

(* Author: Thiemann *) section ‹The Jordan Blocks of the Spectral Radius are Largest› text ‹Consider a non-negative real matrix, and consider any Jordan-block of any eigenvalues whose norm is the spectral radius. We prove that there is a Jordan block of the spectral radius which has the same size or is larger.› theory Spectral_Radius_Largest_Jordan_Block imports Jordan_Normal_Form.Jordan_Normal_Form_Uniqueness Perron_Frobenius_General "HOL-Real_Asymp.Real_Asymp" begin lemma sum_root_unity: fixes x :: "'a :: {comm_ring,division_ring}" assumes "x^n = 1" shows "sum (λ i. x^i) {..< n} = (if x = 1 then of_nat n else 0)" proof (cases "x = 1 ∨ n = 0") case x: False from x obtain m where n: "n = Suc m" by (cases n, auto) have id: "{..< n} = {0..m}" unfolding n by auto show ?thesis using assms x n unfolding id sum_gp by (auto simp: divide_inverse) qed auto lemma sum_root_unity_power_pos_implies_1: assumes sumpos: "⋀ k. Re (sum (λ i. b i * x i ^ k) I) > 0" and root_unity: "⋀ i. i ∈ I ⟹ ∃ d. d ≠ 0 ∧ x i ^ d = 1" shows "1 ∈ x ` I" proof (rule ccontr) assume "¬ ?thesis" hence x: "i ∈ I ⟹ x i ≠ 1" for i by auto from sumpos[of 0] have I: "finite I" "I ≠ {}" using sum.infinite by fastforce+ have "∀ i. ∃ d. i ∈ I ⟶ d ≠ 0 ∧ x i ^ d = 1" using root_unity by auto from choice[OF this] obtain d where d: "⋀ i. i ∈ I ⟹ d i ≠ 0 ∧ x i ^ (d i) = 1" by auto define D where "D = prod d I" have D0: "0 < D" unfolding D_def by (rule prod_pos, insert d, auto) have "0 < sum (λ k. Re (sum (λ i. b i * x i ^ k) I)) {..< D}" by (rule sum_pos[OF _ _ sumpos], insert D0, auto) also have "… = Re (sum (λ k. sum (λ i. b i * x i ^ k) I) {..< D})" by auto also have "sum (λ k. sum (λ i. b i * x i ^ k) I) {..< D} = sum (λ i. sum (λ k. b i * x i ^ k) {..< D}) I" by (rule sum.swap) also have "… = sum (λ i. b i * sum (λ k. x i ^ k) {..< D}) I" by (rule sum.cong, auto simp: sum_distrib_left) also have "… = 0" proof (rule sum.neutral, intro ballI) fix i assume i: "i ∈ I" from d[OF this] x[OF this] have d: "d i ≠ 0" and rt_unity: "x i ^ d i = 1" and x: "x i ≠ 1" by auto have "∃ C. D = d i * C" unfolding D_def by (subst prod.remove[of _ i], insert i I, auto) then obtain C where D: "D = d i * C" by auto have image: "(⋀ x. f x = x) ⟹ {..< D} = f ` {..< D}" for f by auto let ?g = "(λ (a,c). a + d i * c)" have "{..< D} = ?g ` (λ j. (j mod d i, j div d i)) ` {..< d i * C}" unfolding image_image split D[symmetric] by (rule image, insert d mod_mult_div_eq, blast) also have "(λ j. (j mod d i, j div d i)) ` {..< d i * C} = {..< d i} × {..< C}" (is "?f ` ?A = ?B") proof - { fix x assume "x ∈ ?B" then obtain a c where x: "x = (a,c)" and a: "a < d i" and c: "c < C" by auto hence "a + c * d i < d i * (1 + c)" by simp also have "… ≤ d i * C" by (rule mult_le_mono2, insert c, auto) finally have "a + c * d i ∈ ?A" by auto hence "?f (a + c * d i) ∈ ?f ` ?A" by blast also have "?f (a + c * d i) = x" unfolding x using a by auto finally have "x ∈ ?f ` ?A" . } thus ?thesis using d by (auto simp: div_lt_nat) qed finally have D: "{..< D} = (λ (a,c). a + d i * c) ` ?B" by auto have inj: "inj_on ?g ?B" proof - { fix a1 a2 c1 c2 assume id: "?g (a1,c1) = ?g (a2,c2)" and *: "(a1,c1) ∈ ?B" "(a2,c2) ∈ ?B" from arg_cong[OF id, of "λ x. x div d i"] * have c: "c1 = c2" by auto from arg_cong[OF id, of "λ x. x mod d i"] * have a: "a1 = a2" by auto note a c } thus ?thesis by (smt SigmaE inj_onI) qed have "sum (λ k. x i ^ k) {..< D} = sum (λ (a,c). x i ^ (a + d i * c)) ?B" unfolding D by (subst sum.reindex, rule inj, auto intro!: sum.cong) also have "… = sum (λ (a,c). x i ^ a) ?B" by (rule sum.cong, auto simp: power_add power_mult rt_unity) also have "… = 0" unfolding sum.cartesian_product[symmetric] sum.swap[of _ "{..<C}"] by (rule sum.neutral, intro ballI, subst sum_root_unity[OF rt_unity], insert x, auto) finally show "b i * sum (λ k. x i ^ k) {..< D} = 0" by simp qed finally show False by simp qed fun j_to_jb_index :: "(nat × 'a)list ⇒ nat ⇒ nat × nat" where "j_to_jb_index ((n,a) # n_as) i = (if i < n then (0,i) else let rec = j_to_jb_index n_as (i - n) in (Suc (fst rec), snd rec))" fun jb_to_j_index :: "(nat × 'a)list ⇒ nat × nat ⇒ nat" where "jb_to_j_index n_as (0,j) = j" | "jb_to_j_index ((n,_) # n_as) (Suc i, j) = n + jb_to_j_index n_as (i,j)" lemma j_to_jb_index: assumes "i < sum_list (map fst n_as)" and "j < sum_list (map fst n_as)" and "j_to_jb_index n_as i = (bi, li)" and "j_to_jb_index n_as j = (bj, lj)" and "n_as ! bj = (n, a)" shows "((jordan_matrix n_as) ^⇩_{m}r) $$ (i,j) = (if bi = bj then ((jordan_block n a) ^⇩_{m}r) $$ (li, lj) else 0) ∧ (bi = bj ⟶ li < n ∧ lj < n ∧ bj < length n_as ∧ (n,a) ∈ set n_as)" unfolding jordan_matrix_pow using assms proof (induct n_as arbitrary: i j bi bj) case (Cons mb n_as i j bi bj) obtain m b where mb: "mb = (m,b)" by force note Cons = Cons[unfolded mb] have [simp]: "dim_col (case x of (n, a) ⇒ 1⇩_{m}n) = fst x" for x by (cases x, auto) have [simp]: "dim_row (case x of (n, a) ⇒ 1⇩_{m}n) = fst x" for x by (cases x, auto) have [simp]: "dim_col (case x of (n, a) ⇒ jordan_block n a ^⇩_{m}r) = fst x" for x by (cases x, auto) have [simp]: "dim_row (case x of (n, a) ⇒ jordan_block n a ^⇩_{m}r) = fst x" for x by (cases x, auto) consider (UL) "i < m" "j < m" | (UR) "i < m" "j ≥ m" | (LL) "i ≥ m" "j < m" | (LR) "i ≥ m" "j ≥ m" by linarith thus ?case proof cases case UL with Cons(2-) show ?thesis unfolding mb by (auto simp: Let_def) next case UR with Cons(2-) show ?thesis unfolding mb by (auto simp: Let_def dim_diag_block_mat o_def) next case LL with Cons(2-) show ?thesis unfolding mb by (auto simp: Let_def dim_diag_block_mat o_def) next case LR let ?i = "i - m" let ?j = "j - m" from LR Cons(2-) have bi: "j_to_jb_index n_as ?i = (bi - 1, li)" "bi ≠ 0" by (auto simp: Let_def) from LR Cons(2-) have bj: "j_to_jb_index n_as ?j = (bj - 1, lj)" "bj ≠ 0" by (auto simp: Let_def) from LR Cons(2-) have i: "?i < sum_list (map fst n_as)" by auto from LR Cons(2-) have j: "?j < sum_list (map fst n_as)" by auto from LR Cons(2-) bj(2) have nas: "n_as ! (bj - 1) = (n, a)" by (cases bj, auto) from bi(2) bj(2) have id: "(bi - 1 = bj - 1) = (bi = bj)" by auto note IH = Cons(1)[OF i j bi(1) bj(1) nas, unfolded id] have id: "diag_block_mat (map (λa. case a of (n, a) ⇒ jordan_block n a ^⇩_{m}r) (mb # n_as)) $$ (i, j) = diag_block_mat (map (λa. case a of (n, a) ⇒ jordan_block n a ^⇩_{m}r) n_as) $$ (?i, ?j)" using i j LR unfolding mb by (auto simp: Let_def dim_diag_block_mat o_def) show ?thesis using IH unfolding id by auto qed qed auto lemma j_to_jb_index_rev: assumes j: "j_to_jb_index n_as i = (bi, li)" and i: "i < sum_list (map fst n_as)" and k: "k ≤ li" shows "li ≤ i ∧ j_to_jb_index n_as (i - k) = (bi, li - k) ∧ ( j_to_jb_index n_as j = (bi,li - k) ⟶ j < sum_list (map fst n_as) ⟶ j = i - k)" using j i proof (induct n_as arbitrary: i bi j) case (Cons mb n_as i bi j) obtain m b where mb: "mb = (m,b)" by force note Cons = Cons[unfolded mb] show ?case proof (cases "i < m") case True thus ?thesis unfolding mb using Cons(2-) by (auto simp: Let_def) next case i_large: False let ?i = "i - m" have i: "?i < sum_list (map fst n_as)" using Cons(2-) i_large by auto from Cons(2-) i_large have j: "j_to_jb_index n_as ?i = (bi - 1, li)" and bi: "bi ≠ 0" by (auto simp: Let_def) note IH = Cons(1)[OF j i] from IH have IH1: "j_to_jb_index n_as (i - m - k) = (bi - 1, li - k)" and li: "li ≤ i - m" by auto from li have aim1: "li ≤ i" by auto from li k i_large have "i - k ≥ m" by auto hence aim2: "j_to_jb_index (mb # n_as) (i - k) = (bi, li - k)" using IH1 bi by (auto simp: mb Let_def add.commute) { assume *: "j_to_jb_index (mb # n_as) j = (bi, li - k)" "j < sum_list (map fst (mb # n_as))" from * bi have j: "j ≥ m" unfolding mb by (auto simp: Let_def split: if_splits) let ?j = "j - m" from j * have jj: "?j < sum_list (map fst n_as)" unfolding mb by auto from j * have **: "j_to_jb_index n_as (j - m) = (bi - 1, li - k)" using bi mb by (cases "j_to_jb_index n_as (j - m)", auto simp: Let_def) from IH[of ?j] jj ** have "j - m = i - m - k" by auto with j i_large k have "j = i - k" using ‹m ≤ i - k› by linarith } note aim3 = this show ?thesis using aim1 aim2 aim3 by blast qed qed auto locale spectral_radius_1_jnf_max = fixes A :: "real mat" and n m :: nat and lam :: complex and n_as assumes A: "A ∈ carrier_mat n n" and nonneg: "nonneg_mat A" and jnf: "jordan_nf (map_mat complex_of_real A) n_as" and mem: "(m, lam) ∈ set n_as" and lam1: "cmod lam = 1" and sr1: "⋀x. poly (char_poly A) x = 0 ⟹ x ≤ 1" and max_block: "⋀ k la. (k,la) ∈ set n_as ⟹ cmod la ≤ 1 ∧ (cmod la = 1 ⟶ k ≤ m)" begin lemma n_as0: "0 ∉ fst ` set n_as" using jnf[unfolded jordan_nf_def] .. lemma m0: "m ≠ 0" using mem n_as0 by force abbreviation cA where "cA ≡ map_mat complex_of_real A" abbreviation J where "J ≡ jordan_matrix n_as" lemma sim_A_J: "similar_mat cA J" using jnf[unfolded jordan_nf_def] .. definition "c = (∏ia = 0..<m-1. (of_nat m :: real) - 1 - of_nat ia)" lemma c_gt_0: "c > 0" unfolding c_def by (rule prod_pos, auto) lemma c0: "c ≠ 0" using c_gt_0 by auto lemma c_int_def: "c = (∏ia = 0..<m-1. (of_nat m :: int) - 1 - of_nat ia)" using c_def by auto lemma c_int: "c ∈ ℤ" using c_int_def Ints_of_int by metis lemma c_ge_1: "c ≥ 1" using c_gt_0 unfolding c_int_def by presburger definition PP where "PP = (SOME PP. similar_mat_wit cA J (fst PP) (snd PP))" definition P where "P = fst PP" definition iP where "iP = snd PP" lemma JNF: "P ∈ carrier_mat n n" "iP ∈ carrier_mat n n" "J ∈ carrier_mat n n" "P * iP = 1⇩_{m}n" "iP * P = 1⇩_{m}n" "cA = P * J * iP" proof (atomize (full), goal_cases) case 1 have n: "n = dim_row cA" using A by auto from sim_A_J[unfolded similar_mat_def] obtain Q iQ where "similar_mat_wit cA J Q iQ" by auto hence "similar_mat_wit cA J (fst (Q,iQ)) (snd (Q,iQ))" by auto hence "similar_mat_wit cA J P iP" unfolding PP_def iP_def P_def by (rule someI) from similar_mat_witD[OF n this] show ?case by auto qed definition I :: "nat set" where "I = {i | i bi li nn la. i < n ∧ j_to_jb_index n_as i = (bi, li) ∧ n_as ! bi = (nn,la) ∧ cmod la = 1 ∧ nn = m ∧ li = nn - 1}" lemma sumlist_nf: "sum_list (map fst n_as) = n" proof - have "sum_list (map fst n_as) = dim_row (jordan_matrix n_as)" by simp also have "… = dim_row cA" using similar_matD[OF sim_A_J] by auto finally show ?thesis using A by auto qed lemma I_nonempty: "I ≠ {}" proof - from split_list[OF mem] obtain as bs where n_as: "n_as = as @ (m,lam) # bs" by auto let ?i = "sum_list (map fst as) + (m - 1)" have "j_to_jb_index n_as ?i = (length as, m - 1)" unfolding n_as by (induct as, insert m0, auto simp: Let_def) with lam1 have "?i ∈ I" unfolding I_def unfolding sumlist_nf[symmetric] n_as using m0 by auto thus ?thesis by blast qed lemma I_n: "I ⊆ {..<n}" unfolding I_def by auto lemma root_unity_cmod_1: assumes la: "la ∈ snd ` set n_as" and 1: "cmod la = 1" shows "∃ d. d ≠ 0 ∧ la ^ d = 1" proof - from la obtain k where kla: "(k,la) ∈ set n_as" by force from n_as0 kla have k0: "k ≠ 0" by force from split_list[OF kla] obtain as bs where nas: "n_as = as @ (k,la) # bs" by auto have rt: "poly (char_poly cA) la = 0" using k0 unfolding jordan_nf_char_poly[OF jnf] nas poly_prod_list prod_list_zero_iff by auto obtain ks f where decomp: "decompose_prod_root_unity (char_poly A) = (ks, f)" by force from sumlist_nf[unfolded nas] k0 have n0: "n ≠ 0" by auto note pf = perron_frobenius_for_complexity_jnf(1,7)[OF A n0 nonneg sr1 decomp, simplified] from pf(1) pf(2)[OF 1 rt] show "∃ d. d ≠ 0 ∧ la ^ d = 1" by metis qed definition d where "d = (SOME d. ∀ la. la ∈ snd ` set n_as ⟶ cmod la = 1 ⟶ d la ≠ 0 ∧ la ^ (d la) = 1)" lemma d: assumes "(k,la) ∈ set n_as" "cmod la = 1" shows "la ^ (d la) = 1 ∧ d la ≠ 0" proof - let ?P = "λ d. ∀ la. la ∈ snd ` set n_as ⟶ cmod la = 1 ⟶ d la ≠ 0 ∧ la ^ (d la) = 1" from root_unity_cmod_1 have "∀ la. ∃ d. la ∈ snd ` set n_as ⟶ cmod la = 1 ⟶ d ≠ 0 ∧ la ^ d = 1" by blast from choice[OF this] have "∃ d. ?P d" . from someI_ex[OF this] have "?P d" unfolding d_def . from this[rule_format, of la, OF _ assms(2)] assms(1) show ?thesis by force qed definition D where "D = prod_list (map (λ na. if cmod (snd na) = 1 then d (snd na) else 1) n_as)" lemma D0: "D ≠ 0" unfolding D_def by (unfold prod_list_zero_iff, insert d, force) definition K where "K off k = D * k + (m-1) + off" lemma mono_K: "strict_mono (K off)" unfolding strict_mono_def K_def using D0 by auto definition C where "C off k = (c / real (K off k) ^ (m - 1))" lemma limit_jordan_block: assumes kla: "(k, la) ∈ set n_as" and ij: "i < k" "j < k" shows "(λN. (jordan_block k la ^⇩_{m}(K off N)) $$ (i, j) * C off N) ⇢ (if i = 0 ∧ j = k - 1 ∧ cmod la = 1 ∧ k = m then la^off else 0)" proof - let ?c = "of_nat :: nat ⇒ complex" let ?r = "of_nat :: nat ⇒ real" let ?cr = complex_of_real from ij have k0: "k ≠ 0" by auto from jordan_nf_char_poly[OF jnf] have cA: "char_poly cA = (∏(n, a)←n_as. [:- a, 1:] ^ n)" . from degree_monic_char_poly[OF A] have "degree (char_poly A) = n" by auto have deg: "degree (char_poly cA) = n" using A by (simp add: degree_monic_char_poly) from this[unfolded cA] have "n = degree (∏(n, a)←n_as. [:- a, 1:] ^ n)" by auto also have "… = sum_list (map degree (map (λ(n, a). [:- a, 1:] ^ n) n_as))" by (subst degree_prod_list_eq, auto) also have "… = sum_list (map fst n_as)" by (rule arg_cong[of _ _ sum_list], auto simp: degree_linear_power) finally have sum: "sum_list (map fst n_as) = n" by auto with split_list[OF kla] k0 have n0: "n ≠ 0" by auto obtain ks f where decomp: "decompose_prod_root_unity (char_poly A) = (ks, f)" by force note pf = perron_frobenius_for_complexity_jnf[OF A n0 nonneg sr1 decomp] define ji where "ji = j - i" let ?f = "λ N. (c / (?r N)^(m-1))" let ?jb = "λ N. (jordan_block k la ^⇩_{m}N) $$ (i,j)" let ?jbc = "λ N. (jordan_block k la ^⇩_{m}N) $$ (i,j) * ?f N" define e where "e = (if i = 0 ∧ j = k - 1 ∧ cmod la = 1 ∧ k = m then la^off else 0)" let ?e1 = "λ N :: nat. (∏ia = 0..<ji. (?c N - ?c ia) / ?c (ji - ia)) * la ^ (N - ji)" let ?e2 = "λ N. (∏ia = 0..<ji. (?c N - ?c ia) / ?c (ji - ia)) * la ^ (N - ji) * (c / ((?c N^(m-1))))" define e2 where "e2 = ?e2" let ?e3 = "λ N. (((∏ia = 0..<ji. (?c N - ?c ia) / ?c (ji - ia))) * la ^ (N - ji)) * ?f N" { assume ij': "i ≤ j" and la0: "la ≠ 0" { fix N assume "N ≥ k" with ij ij' have ji: "j - i ≤ N" and id: "N + i - j = N - ji" unfolding ji_def by auto have "?jb N = (?c (N choose (j - i)) * la ^ (N + i - j))" unfolding jordan_block_pow using ij ij' by auto also have "… = ?e1 N" unfolding ji_def unfolding binomial_altdef_of_nat[OF ji] id ji_def proof (rule arg_cong[of _ _ "λ x. x * _"], rule prod.cong[OF refl], goal_cases) case (1 x) hence "x ≤ N" using ‹N ≥ k› ij by auto thus ?case by (simp add: of_nat_diff) qed finally have id: "?jb N = ?e1 N" . have "?jbc N = e2 N" unfolding id e2_def using c_gt_0 by (simp add: norm_mult norm_divide norm_power) } note jbc = this have e23: "?e2 N = ?e3 N" for N using c_gt_0 by auto have "(e2 o K off) ⇢ e" proof (cases "cmod la = 1 ∧ k = m ∧ i = 0 ∧ j = k - 1") case False then consider (0) "la = 0" | (small) "la ≠ 0" "cmod la < 1" | (medium) "cmod la = 1" "k < m ∨ i ≠ 0 ∨ j ≠ k - 1" using max_block[OF kla] by linarith hence main: "e2 ⇢ e" proof cases case 0 hence e0: "e = 0" unfolding e_def by auto show ?thesis unfolding e0 0 LIMSEQ_iff e2_def proof (intro exI[of _ "Suc ji"] impI allI, goal_cases) case (1 r n) thus ?case by (cases "n - ji", auto) qed next case small have e0: "e = 0 * (of_real (if m - 1 = 0 then c else 0))" using small unfolding e_def by auto show ?thesis unfolding e0 unfolding e23 e2_def proof (rule tendsto_mult[OF _ tendsto_of_real]) show "(λx. c / real x ^ (m - 1)) ⇢ (if m - 1 = 0 then c else 0)" by (cases "m - 1"; real_asymp) let ?laji = "inverse (la^ji)" let ?f = "(λx. (∏ia = 0..<ji. (?c x - ?c ia) / ?c (ji - ia)) * la ^ (x - ji))" let ?g = "λx. (∏ia = 0..<ji. (1 - ?c ia * inverse (?c x)) / ?c (ji - ia)) * (((?c x)^ji * la ^ x) * ?laji)" have fg: "∀⇩_{F}x in sequentially. ?f x = ?g x" apply (rule eventually_sequentiallyI[of "Suc ji"]) unfolding prod_pow[symmetric] prod.distrib[symmetric] mult.assoc[symmetric] unfolding prod_pow mult.assoc by (rule arg_cong2[of _ _ _ _ "(*)"], rule prod.cong, auto simp: ring_distribs, insert small, subst power_diff, auto simp: divide_inverse) have 0: "0 = (∏ia = 0..<ji. (1 - of_nat ia * 0) / of_nat (ji - ia)) * (0 * ?laji)" by simp show "?f ⇢ 0" unfolding tendsto_cong[OF fg] proof (subst 0, rule tendsto_mult[OF tendsto_prod tendsto_mult[OF _ tendsto_const]], intro tendsto_intros inverse_of_nat_tendsto_zero) show "(λx. of_nat x ^ ji * la ^ x) ⇢ 0" by (rule poly_times_exp_tendsto_zero, insert small, auto) qed auto qed next case medium with max_block[OF kla] have "k ≤ m" and 1: "⋀ x. cmod la ^ x = 1" by auto with ij medium have "ji < m - 1" unfolding ji_def by linarith then obtain d where m1: "m - 1 = Suc d + ji" using less_iff_Suc_add by auto have e0: "e = 0" using medium unfolding e_def by auto have 0: "0 = (∏ia = 0..<ji. (1 - ?c ia * 0) / ?c (ji - ia)) * (of_real c) * 0" by simp let ?e = "λ ia N. if N = 0 then 0 else (1 - ?c ia / ?c N) / ?c (ji - ia)" let ?f = "λ ia N. (1 - ?c ia * (1 / ?c N)) / ?c (ji - ia)" { fix N have "e2 N = ((∏ia = 0..<ji. (?c N - ?c ia) / ?c (ji - ia)) / ?c N ^ ji) * la ^ (N - ji) * (of_real c / ?c N ^ Suc d)" unfolding medium m1 power_add e2_def by simp also have "(∏ia = 0..<ji. (?c N - ?c ia) / ?c (ji - ia)) / ?c N ^ ji = (∏ia = 0..<ji. ?e ia N)" unfolding prod_pow[symmetric] prod_dividef[symmetric] by (cases "?c N = 0", auto simp add: field_simps) finally have "e2 N = (∏ia = 0..<ji. ?e ia N) * of_real c * inverse (?c N ^ Suc d) * la ^ (N - ji)" by (simp add: divide_inverse) also have "cmod … = cmod ((∏ia = 0..<ji. ?e ia N) * of_real c * (inverse (?c N ^ Suc d)))" unfolding norm_mult norm_power 1 by simp finally have "cmod (e2 N) = cmod ((∏ia = 0..<ji. ?e ia N) * of_real c * (inverse (?c N ^ Suc d)))" by simp } note e2 = this show ?thesis unfolding e0 apply (rule tendsto_norm_zero_cancel, unfold e2, rule tendsto_norm_zero) apply (subst (2) 0) apply (rule tendsto_mult[OF tendsto_mult[OF tendsto_prod tendsto_const] inverse_power_tendsto_zero], goal_cases) proof - case (1 i) let ?g = "λ x. (1 - ?c i * (1 / of_nat x)) / of_nat (ji - i)" have eq: "∀⇩_{F}x in sequentially. ?e i x = ?g x" by (rule eventually_sequentiallyI[of 1], auto) show "?e i ⇢ (1 - ?c i * 0) / ?c (ji - i)" unfolding tendsto_cong[OF eq] using 1 by (intro tendsto_intros lim_1_over_n, auto) qed qed show "(e2 o K off) ⇢ e" by (rule LIMSEQ_subseq_LIMSEQ[OF main mono_K]) next case True hence large: "cmod la = 1" "k = m" "i = 0" "j = k - 1" by auto hence e: "e = la^off" unfolding e_def by auto from large k0 have m0: "m ≥ 1" by auto define m1 where "m1 = m - 1" have id: "(real (m - 1) - real ia) = ?r m - 1 - ?r ia" for ia using m0 unfolding m1_def by auto let ?e4 = "λ x. (∏ia = 0..<m1. 1 - ?cr (?r ia / x))" { fix x :: nat assume x: "x ≠ 0" have "?e2 x = ((∏ia = 0..<m1. (?c x - ?c ia) / ?c (m1 - ia)) * (∏ia = 0..<m1. ?cr (real m1 - real ia))) / (∏i = 0..<m1. ?c x) * la ^ (x - (m-1))" (is "_ = ?A / ?B * ?C") unfolding m1_def ji_def large c_def prod_pow[symmetric] id by simp also have "?A = (∏ia = 0..<m1. (?cr x - ?c ia))" (is "_ = ?A") unfolding prod.distrib[symmetric] by (rule prod.cong[OF refl], subst of_nat_diff, auto) also have "?A / ?B = (∏ia = 0..<m1. 1 - ?cr (?r ia / x))" unfolding prod_dividef[symmetric] by (rule prod.cong[OF refl], insert x, auto simp: field_simps) finally have "?e2 x = ?e4 x * ?C" . } note main = this from d[OF kla large(1)] have 1: "la ^ d la = 1" by auto from split_list[OF kla] obtain as bs where n_as: "n_as = as @ (k,la) # bs" by auto obtain C where D: "D = d la * C" unfolding D_def unfolding n_as using large by auto have "(λ x. ?e4 x * e) ⇢ (∏ia = 0..<m1. 1 - ?cr 0) * e" by (intro tendsto_intros real_tendsto_divide_at_top, auto simp: filterlim_real_sequentially) also have "(∏ia = 0..<m1. 1 - ?cr 0) = 1" unfolding e by simp finally have "(λ x. ?e4 x * e) ⇢ e" by auto from LIMSEQ_subseq_LIMSEQ[OF this mono_K] have e4: "(λ k. (?e4 o K off) k * e) ⇢ e" (is "?A ⇢ e") by (auto simp: o_def) { fix k :: nat assume k: "k ≠ 0" hence 0: "K off k ≠ 0" unfolding K_def using D0 by auto have "?e2 (K off k) = ?e4 (K off k) * la^(K off k - (m-1))" unfolding main[OF 0] .. also have "K off k - (m-1) = D * k + off" unfolding K_def by simp also have "la ^ … = e" unfolding e power_add D power_mult 1 by auto finally have "e2 (K off k) = (?e4 o K off) k * e" unfolding o_def e2_def . } note main = this have id: "(?A ⇢ e) = ((e2 o K off) ⇢ e)" by (rule tendsto_cong, unfold eventually_at_top_linorder, rule exI[of _ 1], insert main, auto) from e4[unfolded id] show ?thesis . qed also have "((e2 o K off) ⇢ e) = ((?jbc o K off) ⇢ e)" proof (rule tendsto_cong, unfold eventually_at_top_linorder, rule exI[of _ k], intro allI impI, goal_cases) case (1 n) from mono_K[of off] 1 have "K off n ≥ k" using le_trans seq_suble by blast from jbc[OF this] show ?case by (simp add: o_def) qed finally have "(?jbc o K off) ⇢ e" . } note part1 = this { assume "i > j ∨ la = 0" hence e: "e = 0" and jbn: "N ≥ k ⟹ ?jbc N = 0" for N unfolding jordan_block_pow e_def using ij by auto have "?jbc ⇢ e" unfolding e LIMSEQ_iff by (intro exI[of _ k] allI impI, subst jbn, auto) from LIMSEQ_subseq_LIMSEQ[OF this mono_K] have "(?jbc o K off) ⇢ e" . } note part2 = this from part1 part2 have "(?jbc o K off) ⇢ e" by linarith thus ?thesis unfolding e_def o_def C_def . qed definition Lam where "Lam i = snd (n_as ! fst (j_to_jb_index n_as i))" lemma cmod_Lam: "i ∈ I ⟹ cmod (Lam i) = 1" unfolding I_def Lam_def by auto lemma I_Lam: assumes i: "i ∈ I" shows "(m, Lam i) ∈ set n_as" proof - from i[unfolded I_def] obtain bi li la where i: "i < n" and jb: "j_to_jb_index n_as i = (bi, li)" and nth: "n_as ! bi = (m, la)" and "cmod la = 1 ∧ li = m - 1" by auto hence lam: "Lam i = la" unfolding Lam_def by auto from j_to_jb_index[of _ n_as, unfolded sumlist_nf, OF i i jb jb nth] lam show ?thesis by auto qed lemma limit_jordan_matrix: assumes ij: "i < n" "j < n" shows "(λN. (J ^⇩_{m}(K off N)) $$ (i, j) * C off N) ⇢ (if j ∈ I ∧ i = j - (m - 1) then (Lam j)^off else 0)" proof - obtain bi li where bi: "j_to_jb_index n_as i = (bi, li)" by force obtain bj lj where bj: "j_to_jb_index n_as j = (bj, lj)" by force define la where "la = snd (n_as ! fst (j_to_jb_index n_as j))" obtain nn where nbj: "n_as ! bj = (nn,la)" unfolding la_def bj fst_conv by (metis prod.collapse) from j_to_jb_index[OF ij[folded sumlist_nf] bi bj nbj] have eq: "bi = bj ⟹ li < nn ∧ lj < nn ∧ bj < length n_as ∧ (nn, la) ∈ set n_as" and index: "(J ^⇩_{m}r) $$ (i, j) = (if bi = bj then (jordan_block nn la ^⇩_{m}r) $$ (li, lj) else 0)" for r by auto note index_rev = j_to_jb_index_rev[OF bj, unfolded sumlist_nf, OF ij(2) le_refl] show ?thesis proof (cases "bi = bj") case False hence id: "(bi = bj) = False" by auto { assume "j ∈ I" "i = j - (m - 1)" from this[unfolded I_def] bj nbj have "i = j - lj" by auto from index_rev[folded this] bi False have False by auto } thus ?thesis unfolding index id if_False by auto next case True hence id: "(bi = bj) = True" by auto from eq[OF True] have eq: "li < nn" "lj < nn" "(nn,la) ∈ set n_as" "bj < length n_as" by auto have "(λN. (J ^⇩_{m}(K off N)) $$ (i, j) * C off N) ⇢ (if li = 0 ∧ lj = nn - 1 ∧ cmod la = 1 ∧ nn = m then la^off else 0)" unfolding index id if_True using limit_jordan_block[OF eq(3,1,2)] . also have "(li = 0 ∧ lj = nn - 1 ∧ cmod la = 1 ∧ nn = m) = (j ∈ I ∧ i = j - (m - 1))" (is "?l = ?r") proof assume ?r hence "j ∈ I" .. from this[unfolded I_def] bj nbj have *: "nn = m" "cmod la = 1" "lj = nn - 1" by auto from ‹?r› * have "i = j - lj" by auto with * have "li = 0" using index_rev bi by auto with * show ?l by auto next assume ?l hence jI: "j ∈ I" using bj nbj ij by (auto simp: I_def) from ‹?l› have "li = 0" by auto with index_rev[of i] bi ij(1) ‹?l› True have "i = j - (m - 1)" by auto with jI show ?r by auto qed finally show ?thesis unfolding la_def Lam_def . qed qed declare sumlist_nf[simp] lemma A_power_P: "cA ^⇩_{m}k * P = P * J ^⇩_{m}k" proof (induct k) case 0 show ?case using A JNF by simp next case (Suc k) have "cA ^⇩_{m}Suc k * P = cA ^⇩_{m}k * cA * P" by simp also have "… = cA ^⇩_{m}k * (P * J * iP) * P" using JNF by simp also have "… = (cA ^⇩_{m}k * P) * (J * (iP * P))" using A JNF(1-3) by (simp add: assoc_mult_mat[of _ n n _ n _ n]) also have "J * (iP * P) = J" unfolding JNF using JNF by auto finally show ?case unfolding Suc using A JNF(1-3) by (simp add: assoc_mult_mat[of _ n n _ n _ n]) qed lemma C_nonneg: "C off k ≥ 0" unfolding C_def using c_gt_0 by auto lemma P_nonzero_entry: assumes j: "j < n" shows "∃ i < n. P $$ (i,j) ≠ 0" proof (rule ccontr) assume "¬ ?thesis" hence 0: "⋀ i. i < n ⟹ P $$ (i,j) = 0" by auto have "1 = (iP * P) $$ (j,j)" using j unfolding JNF by auto also have "… = (∑i = 0..<n. iP $$ (j, i) * P $$ (i, j))" using j JNF(1-2) by (auto simp: scalar_prod_def) also have "… = 0" by (rule sum.neutral, insert 0, auto) finally show False by auto qed definition i where "i = (SOME i. i ∈ I)" lemma i: "i ∈ I" unfolding i_def using I_nonempty some_in_eq by blast lemma i_n: "i < n" using i unfolding I_def by auto definition "j = (SOME j. j < n ∧ P $$ (j, i - (m - 1)) ≠ 0)" lemma j: "j < n" "P $$ (j, i - (m - 1)) ≠ 0" proof - from i_n have lt: "i - (m - 1) < n" by auto show "j < n" "P $$ (j, i - (m - 1)) ≠ 0" unfolding j_def using someI_ex[OF P_nonzero_entry[OF lt]] by auto qed definition "B = cmod (P $$ (j, i - (m - 1))) / 2" lemma B: "0 < B" unfolding B_def using j by auto definition "w = P *⇩_{v}unit_vec n i" lemma w: "w ∈ carrier_vec n" using JNF unfolding w_def by auto definition "v = map_vec cmod w" lemma v: "v ∈ carrier_vec n" unfolding v_def using w by auto lemma main_step: "∃ a. ∀ l. 0 < Re (∑i∈I. a i * Lam i ^ l)" proof - let ?c = "complex_of_real" let ?cv = "map_vec ?c" let ?cm = "map_mat ?c" let ?v = "?cv v" define u where "u = iP *⇩_{v}?v" define cc where "cc = (λ i. ((∑k = 0..<n. (if k = i - (m - 1) then P $$ (j, k) else 0)) * u $v i))" define a where "a = (λ i. P $$ (j, i - (m - 1)) * u $v i)" have u: "u ∈ carrier_vec n" unfolding u_def using JNF(2) v by auto { fix off from i i_n have iI: "i ∈ I" and i: "i < n" by auto let ?exp = "λ k. sum (λ ii. P $$ (j, ii) * (J ^⇩_{m}k) $$ (ii,i)) {..<n}" define M where "M = (λ k. cmod (?exp (K off k) * C off k))" let ?i = "i - (m - 1)" define G where "G = (λ k. (A ^⇩_{m}K off k *⇩_{v}v) $v j * C off k)" { fix kk define k where "k = K off kk" define cAk where "cAk = cA ^⇩_{m}k" have cAk: "cAk ∈ carrier_mat n n" unfolding cAk_def using A by auto have "((A ^⇩_{m}k) *⇩_{v}v) $ j = ((map_mat cmod cAk) *⇩_{v}map_vec cmod w) $ j" unfolding v_def[symmetric] cAk_def by (rule arg_cong[of _ _ "λ x. (x *⇩_{v}v) $ j"], unfold of_real_hom.mat_hom_pow[OF A, symmetric], insert nonneg_mat_power[OF A nonneg, of k], insert i j, auto simp: nonneg_mat_def elements_mat_def) also have "… ≥ cmod ((cAk *⇩_{v}w) $ j)" by (subst (1 2) index_mult_mat_vec, insert j cAk w, auto simp: scalar_prod_def intro!: sum_norm_le norm_mult_ineq) also have "cAk *⇩_{v}w = (cAk * P) *⇩_{v}unit_vec n i" unfolding w_def i_def using JNF cAk by simp also have "… = P *⇩_{v}(J ^⇩_{m}k *⇩_{v}unit_vec n i)" unfolding cAk_def A_power_P using JNF by (subst assoc_mult_mat_vec[of _ n n _ n], auto) also have "J ^⇩_{m}k *⇩_{v}unit_vec n i = col (J ^⇩_{m}k) i" by (rule eq_vecI, insert i, auto) also have "(P *⇩_{v}(col (J ^⇩_{m}k) i)) $ j = Matrix.row P j ∙ col (J ^⇩_{m}k) i" by (subst index_mult_mat_vec, insert j JNF, auto) also have "… = sum (λ ii. P $$ (j, ii) * (J ^⇩_{m}k) $$ (ii,i)) {..<n}" unfolding scalar_prod_def by (rule sum.cong, insert i j JNF(1), auto) finally have "(A ^⇩_{m}k *⇩_{v}v) $v j ≥ cmod (?exp k)" . from mult_right_mono[OF this C_nonneg] have "(A ^⇩_{m}k *⇩_{v}v) $v j * C off kk ≥ cmod (?exp k * C off kk)" unfolding norm_mult using C_nonneg by auto } hence ge: "(A ^⇩_{m}K off k *⇩_{v}v) $v j * C off k ≥ M k" for k unfolding M_def by auto from i have mem: "i - (m - 1) ∈ {..<n}" by auto have "(λ k. ?exp (K off k) * C off k) ⇢ (∑ii<n. P $$ (j, ii) * (if i ∈ I ∧ ii = i - (m - 1) then Lam i ^ off else 0))" (is "_ ⇢ ?sum") unfolding sum_distrib_right mult.assoc by (rule tendsto_sum, rule tendsto_mult, force, rule limit_jordan_matrix[OF _ i], auto) also have "?sum = P $$ (j, i - (m - 1)) * Lam i ^ off" by (subst sum.remove[OF _ mem], force, subst sum.neutral, insert iI, auto) finally have tend1: "(λ k. ?exp (K off k) * C off k) ⇢ P $$ (j, i - (m - 1)) * Lam i ^ off" . have tend2: "M ⇢ cmod (P $$ (j, i - (m - 1)) * Lam i ^ off)" unfolding M_def by (rule tendsto_norm, rule tend1) { from j have 0: "P $$ (j, i - (m - 1)) ≠ 0" by auto define E where "E = cmod (P $$ (j, i - (m - 1)) * Lam i ^ off)" from cmod_Lam[OF iI] 0 have E: "E / 2 > 0" unfolding E_def by auto from tend2[folded E_def] have tend2: "M ⇢ E" . from ge have ge: "G k ≥ M k" for k unfolding G_def . from tend2[unfolded LIMSEQ_iff, rule_format, OF E] obtain k' where diff: "⋀ k. k ≥ k' ⟹ norm (M k - E) < E / 2" by auto { fix k assume "k ≥ k'" from diff[OF this] have norm: "norm (M k - E) < E / 2" . have "M k ≥ 0" unfolding M_def by auto with E norm have "M k ≥ E / 2" by (smt real_norm_def field_sum_of_halves) with ge[of k] E have "G k ≥ E / 2" by auto also have "E / 2 = B" unfolding E_def B_def j norm_mult norm_power cmod_Lam[OF iI] by auto finally have "G k ≥ B" . } hence "∃ k'. ∀ k. k ≥ k' ⟶ G k ≥ B" by auto } hence Bound: "∃k'. ∀k≥k'. B ≤ G k" by auto { fix kk define k where "k = K off kk" have "((A ^⇩_{m}k) *⇩_{v}v) $ j * C off kk = Re (?c (((A ^⇩_{m}k) *⇩_{v}v) $ j * C off kk))" by simp also have "?c (((A ^⇩_{m}k) *⇩_{v}v) $ j * C off kk) = ?cv ((A ^⇩_{m}k) *⇩_{v}v) $ j * ?c (C off kk)" using j A by simp also have "?cv ((A ^⇩_{m}k) *⇩_{v}v) = (?cm (A ^⇩_{m}k) *⇩_{v}?v)" using A by (subst of_real_hom.mult_mat_vec_hom[OF _ v], auto) also have "… = (cA ^⇩_{m}k *⇩_{v}?v)" by (simp add: of_real_hom.mat_hom_pow[OF A]) also have "… = (cA ^⇩_{m}k *⇩_{v}((P * iP) *⇩_{v}?v))" unfolding JNF using v by auto also have "… = (cA ^⇩_{m}k *⇩_{v}(P *⇩_{v}u))" unfolding u_def by (subst assoc_mult_mat_vec, insert JNF v, auto) also have "… = (P * J ^⇩_{m}k *⇩_{v}u)" unfolding A_power_P[symmetric] by (subst assoc_mult_mat_vec, insert u JNF(1) A, auto) also have "… = (P *⇩_{v}(J ^⇩_{m}k *⇩_{v}u))" by (rule assoc_mult_mat_vec, insert u JNF(1) A, auto) finally have "(A ^⇩_{m}k *⇩_{v}v) $v j * C off kk = Re ((P *⇩_{v}(J ^⇩_{m}k *⇩_{v}u)) $ j * C off kk)" by simp also have "… = Re (∑i = 0..<n. P $$ (j, i) * (∑ia = 0..< n. (J ^⇩_{m}k) $$ (i, ia) * u $v ia * C off kk))" by (subst index_mult_mat_vec, insert JNF(1) j u, auto simp: scalar_prod_def sum_distrib_right[symmetric] mult.assoc[symmetric]) finally have "(A ^⇩_{m}k *⇩_{v}v) $v j * C off kk = Re (∑i = 0..<n. P $$ (j, i) * (∑ia = 0..<n. (J ^⇩_{m}k) $$ (i, ia) * C off kk * u $v ia))" unfolding k_def by (simp only: ac_simps) } note A_to_u = this define F where "F = (∑ia∈I. a ia * Lam ia ^ off)" have "G ⇢ Re (∑i = 0..<n. P $$ (j, i) * (∑ia = 0..<n. (if ia ∈ I ∧ i = ia - (m - 1) then (Lam ia)^off else 0) * u $v ia))" unfolding A_to_u G_def by (rule tendsto_Re[OF tendsto_sum[OF tendsto_mult[OF _ tendsto_sum[OF tendsto_mult[OF limit_jordan_matrix]]]]], auto) also have "(∑i = 0..<n. P $$ (j, i) * (∑ia = 0..<n. (if ia ∈ I ∧ i = ia - (m - 1) then (Lam ia)^off else 0) * u $v ia)) = (∑i = 0..<n. (∑ia ∈ I. (if ia ∈ I ∧ i = ia - (m - 1) then P $$ (j, i) else 0) * ((Lam ia)^off * u $v ia)))" by (rule sum.cong[OF refl], unfold sum_distrib_left, subst (2) sum.mono_neutral_left[of "{0..<n}"], insert I_n, auto intro!: sum.cong) also have "… = (∑ia ∈ I. (∑i = 0..<n. (if i = ia - (m - 1) then P $$ (j, i) else 0)) * ((Lam ia)^off * u $v ia))" unfolding sum.swap[of _ I] sum_distrib_right by (rule sum.cong[OF refl], auto) also have "… = (∑ia ∈ I. cc ia * (Lam ia)^off)" unfolding cc_def by (rule sum.cong[OF refl], simp) also have "… = F" unfolding cc_def a_def F_def by (rule sum.cong[OF refl], insert I_n, auto) finally have tend3: "G ⇢ Re F" . from this[unfolded LIMSEQ_iff, rule_format, of "B / 2"] B obtain kk where kk: "⋀ k. k ≥ kk ⟹ norm (G k - Re F) < B / 2" by auto from Bound obtain kk' where kk': "⋀ k. k ≥ kk' ⟹ B ≤ G k" by auto define k where "k = max kk kk'" with kk kk' have 1: "norm (G k - Re F) < B / 2" "B ≤ G k" by auto with B have "Re F > 0" by (smt real_norm_def field_sum_of_halves) } thus ?thesis by blast qed lemma main_theorem: "(m, 1) ∈ set n_as" proof - from main_step obtain a where pos: "⋀ l. 0 < Re (∑i∈I. a i * Lam i ^ l)" by auto have "1 ∈ Lam ` I" proof (rule sum_root_unity_power_pos_implies_1[of a Lam I, OF pos]) fix i assume "i ∈ I" from d[OF I_Lam[OF this] cmod_Lam[OF this]] show "∃d. d ≠ 0 ∧ Lam i ^ d = 1" by auto qed then obtain i where i: "i ∈ I" and "Lam i = 1" by auto with I_Lam[OF i] show ?thesis by auto qed end lemma nonneg_sr_1_largest_jb: assumes nonneg: "nonneg_mat A" and jnf: "jordan_nf (map_mat complex_of_real A) n_as" and mem: "(m, lam) ∈ set n_as" and lam1: "cmod lam = 1" and sr1: "⋀x. poly (char_poly A) x = 0 ⟹ x ≤ 1" shows "∃ M. M ≥ m ∧ (M,1) ∈ set n_as" proof - note jnf' = jnf[unfolded jordan_nf_def] from jnf' similar_matD[OF jnf'[THEN conjunct2]] obtain n where A: "A ∈ carrier_mat n n" and n_as0: "0 ∉ fst ` set n_as" by auto let ?M = "{ m. ∃ lam. (m,lam) ∈ set n_as ∧ cmod lam = 1}" have m: "m ∈ ?M" using mem lam1 by auto have fin: "finite ?M" by (rule finite_subset[OF _ finite_set[of "map fst n_as"]], force) define M where "M = Max ?M" have "M ∈ ?M" using fin m unfolding M_def using Max_in by blast then obtain Lam where M: "(M,Lam) ∈ set n_as" "cmod Lam = 1" by auto from m fin have mM: "m ≤ M" unfolding M_def by simp interpret spectral_radius_1_jnf_max A n M Lam proof (unfold_locales, rule A, rule nonneg, rule jnf, rule M, rule M, rule sr1) fix k la assume kla: "(k, la) ∈ set n_as" with fin have 1: "cmod la = 1 ⟶ k ≤ M" unfolding M_def using Max_ge by blast obtain ks f where decomp: "decompose_prod_root_unity (char_poly A) = (ks, f)" by force from n_as0 kla have k0: "k ≠ 0" by force let ?cA = "map_mat complex_of_real A" from split_list[OF kla] obtain as bs where nas: "n_as = as @ (k,la) # bs" by auto have rt: "poly (char_poly ?cA) la = 0" using k0 unfolding jordan_nf_char_poly[OF jnf] nas poly_prod_list prod_list_zero_iff by auto have sumlist_nf: "sum_list (map fst n_as) = n" proof - have "sum_list (map fst n_as) = dim_row (jordan_matrix n_as)" by simp also have "… = dim_row ?cA" using similar_matD[OF jnf'[THEN conjunct2]] by auto finally show ?thesis using A by auto qed from this[unfolded nas] k0 have n0: "n ≠ 0" by auto from perron_frobenius_for_complexity_jnf(4)[OF A n0 nonneg sr1 decomp rt] have "cmod la ≤ 1" . with 1 show "cmod la ≤ 1 ∧ (cmod la = 1 ⟶ k ≤ M)" by auto qed from main_theorem show ?thesis using mM by auto qed hide_const(open) spectral_radius lemma (in ring_hom) hom_smult_mat: "mat⇩_{h}(a ⋅⇩_{m}A) = hom a ⋅⇩_{m}mat⇩_{h}A" by (rule eq_matI, auto simp: hom_mult) lemma root_char_poly_smult: fixes A :: "complex mat" assumes A: "A ∈ carrier_mat n n" and k: "k ≠ 0" shows "(poly (char_poly (k ⋅⇩_{m}A)) x = 0) = (poly (char_poly A) (x / k) = 0)" using order_char_poly_smult[OF A k, of x] by (metis A degree_0 degree_monic_char_poly monic_degree_0 order_root smult_carrier_mat) theorem real_nonneg_mat_spectral_radius_largest_jordan_block: assumes "real_nonneg_mat A" and "jordan_nf A n_as" and "(m, lam) ∈ set n_as" and "cmod lam = spectral_radius A" shows "∃ M ≥ m. (M, of_real (spectral_radius A)) ∈ set n_as" proof - from similar_matD[OF assms(2)[unfolded jordan_nf_def, THEN conjunct2]] obtain n where A: "A ∈ carrier_mat n n" by auto let ?c = complex_of_real define B where "B = map_mat Re A" have B: "B ∈ carrier_mat n n" unfolding B_def using A by auto have AB: "A = map_mat ?c B" unfolding B_def using assms(1) by (auto simp: real_nonneg_mat_def elements_mat_def) have nonneg: "nonneg_mat B" using assms(1) unfolding AB by (auto simp: real_nonneg_mat_def elements_mat_def nonneg_mat_def) let ?sr = "spectral_radius A" show ?thesis proof (cases "?sr = 0") case False define isr where "isr = inverse ?sr" let ?nas = "map (λ(n, a). (n, ?c isr * a)) n_as" from False have isr0: "isr ≠ 0" unfolding isr_def by auto hence cisr0: "?c isr ≠ 0" by auto from False assms(4) have isr_pos: "isr > 0" unfolding isr_def by (smt norm_ge_zero positive_imp_inverse_positive) define C where "C = isr ⋅⇩_{m}B" have C: "C ∈ carrier_mat n n" using B unfolding C_def by auto have BC: "B = ?sr ⋅⇩_{m}C" using isr0 unfolding C_def isr_def by auto have nonneg: "nonneg_mat C" unfolding C_def using isr_pos nonneg unfolding nonneg_mat_def elements_mat_def by auto from jordan_nf_smult[OF assms(2)[unfolded AB] cisr0] have jnf: "jordan_nf (map_mat ?c C) ?nas" unfolding C_def by (auto simp: of_real_hom.hom_smult_mat) from assms(3) have mem: "(m, ?c isr * lam) ∈ set ?nas" by auto have 1: "cmod (?c isr * lam) = 1" using False isr_pos unfolding isr_def norm_mult assms(4) by (smt mult.commute norm_of_real right_inverse) { fix x have B': "map_mat ?c B ∈ carrier_mat n n" using B by auto assume "poly (char_poly C) x = 0" hence "poly (char_poly (map_mat ?c C)) (?c x) = 0" unfolding of_real_hom.char_poly_hom[OF C] by auto hence "poly (char_poly A) (?c x / ?c isr) = 0" unfolding C_def of_real_hom.hom_smult_mat AB unfolding root_char_poly_smult[OF B' cisr0] . hence "eigenvalue A (?c x / ?c isr)" unfolding eigenvalue_root_char_poly[OF A] . hence mem: "cmod (?c x / ?c isr) ∈ cmod ` spectrum A" unfolding spectrum_def by auto from Max_ge[OF finite_imageI this] have "cmod (?c x / ?c isr) ≤ ?sr" unfolding Spectral_Radius.spectral_radius_def using A card_finite_spectrum(1) by blast hence "cmod (?c x) ≤ 1" using isr0 isr_pos unfolding isr_def by (auto simp: field_simps norm_divide norm_mult) hence "x ≤ 1" by auto } note sr = this from nonneg_sr_1_largest_jb[OF nonneg jnf mem 1 sr] obtain M where M: "M ≥ m" "(M,1) ∈ set ?nas" by blast from M(2) obtain a where mem: "(M,a) ∈ set n_as" and "1 = ?c isr * a" by auto from this(2) have a: "a = ?c ?sr" using isr0 unfolding isr_def by (auto simp: field_simps) show ?thesis by (intro exI[of _ M], insert mem a M(1), auto) next case True from jordan_nf_root_char_poly[OF assms(2,3)] have "eigenvalue A lam" unfolding eigenvalue_root_char_poly[OF A] . hence "cmod lam ∈ cmod ` spectrum A" unfolding spectrum_def by auto from Max_ge[OF finite_imageI this] have "cmod lam ≤ ?sr" unfolding Spectral_Radius.spectral_radius_def using A card_finite_spectrum(1) by blast from this[unfolded True] have lam0: "lam = 0" by auto show ?thesis unfolding True using assms(3)[unfolded lam0] by auto qed qed end