# Theory Cancel_Card_Constraint

(* Author: R. Thiemann *)
section ‹Elimination of CARD('n)›

text ‹In the following theory we provide a method which modifies theorems
of the form $P[CARD('n)]$ into $n != 0 \Longrightarrow P[n]$, so that they can more
easily be applied.

Known issues: there might be problems with nested meta-implications and meta-quantification.›

theory Cancel_Card_Constraint
imports
"HOL-Types_To_Sets.Types_To_Sets"
"HOL-Library.Cardinality"
begin

lemma n_zero_nonempty: "n ≠ 0 ⟹ {0 ..< n :: nat} ≠ {}" by auto

lemma type_impl_card_n: assumes "∃(Rep :: 'a ⇒ nat) Abs. type_definition Rep Abs {0 ..< n :: nat}"
shows "class.finite (TYPE('a)) ∧ CARD('a) = n"
proof -
from assms obtain rep :: "'a ⇒ nat" and abs :: "nat ⇒ 'a" where t: "type_definition rep abs {0 ..< n}" by auto
have "card (UNIV :: 'a set) = card {0 ..< n}" using t by (rule type_definition.card)
also have "… = n" by auto
finally have bn: "CARD ('a) = n" .
have "finite (abs  {0 ..< n})" by auto
also have "abs  {0 ..< n} = UNIV" using t by (rule type_definition.Abs_image)
finally have "class.finite (TYPE('a))" unfolding class.finite_def .
with bn show ?thesis by blast
qed

ML_file ‹cancel_card_constraint.ML›

(* below you find an example what the attribute cancel_card_constraint can do and how
it works internally *)

(*
(* input: some type based lemma with CARD inside, like t0 *)
consts P :: "nat ⇒ nat ⇒ bool"
axiomatization where t0: "P (CARD ('a :: finite)) (CARD('a) * m)"

(* t0 is converted into a property without the cardinality constraint via the new attribute *)
lemma t_1_to_6: "n ≠ 0 ⟹ P n (n * m)"
by (rule t0[cancel_card_constraint])

(* The internal steps are as follows. *)

(* 1st step: pull out CARD and introduce new variable n *)
lemma t1: "CARD('a :: finite) = n ⟹ P n (n * m)"
using t0[where 'a = 'a] by blast

(* 2nd step: get rid of sorts *)
lemma t2: "class.finite TYPE('a) ⟹ CARD('a) = n ⟹ P n (n * m)"
by (rule t1[internalize_sort "?'a::finite"])

(* 3rd step: restructure thm so that first two assumptions are merged into one *)
lemma t3: "class.finite TYPE('a) ∧ CARD('a) = n ⟹ P n (n * m)"
using t2 by blast

(* 4th step: choose an appropriate carrier set *)
lemma t4: "∃Rep Abs. type_definition Rep Abs {0..<n} ⟹ P n (n * m)"
by (rule t3[OF type_impl_card_n])

(* 5th step: cancel type definition *)
lemma t5: "{0..<n} ≠ {} ⟹ P n (n * m)"
by (rule t4[cancel_type_definition])

(* 6th step: simplify non-empty assumption to obtain final theorem *)
lemma t6: "n ≠ 0 ⟹ P n (n * m)"
by (rule t5[OF n_zero_nonempty])
*)

end


# File ‹cancel_card_constraint.ML›

signature CARD_ELIMINATION =
sig
val cancel_card_constraint: thm -> thm
val cancel_card_constraint_attr: attribute
end

structure Card_Elimination : CARD_ELIMINATION =
struct

(* turns A ==> B[CARD('a)] ==> C[CARD('a), CARD('b)] ==> ... ==> Z
into CARD('a) = n1 ==> CARD('b) = n2 ==> A ==> B[n1] ==> C[n1,n2] ==> ... ==> Z *)
fun extract_card_type ctxt thm = let
val prop = Thm.prop_of thm
fun add_match_terms f t = (if f t then insert (op =) t else I) #>
(case t of
t1 $t2 => add_match_terms f t1 #> add_match_terms f t2 | Abs (_,_,t1) => add_match_terms f t1 | _ => I) fun replace_terms xys t = (case AList.lookup (op =) xys t of SOME y => y | NONE => case t of t1$ t2 => replace_terms xys t1 $replace_terms xys t2 | Abs (a,b,t1) => Abs (a,b,replace_terms xys t1) | _ => t) fun find_card (Const (c,_)$ Const (t,_)) = (c = @{const_name Finite_Set.card}
andalso t = @{const_name Orderings.top_class.top})
| find_card _ = false
val card_ts = add_match_terms find_card prop []
val vars = Term.add_vars prop []
val all_tvars = Term.add_tvars prop []
val all_tfrees = map TFree (Variable.variant_frees ctxt [prop] (map (apfst fst) all_tvars))
val ns = map_index (fn (i,_) => ("n" ^ string_of_int i, @{typ nat})) card_ts
val substT_map = map fst all_tvars ~~ all_tfrees
val substT = I subst_TVars substT_map
val free_ns = Variable.variant_frees ctxt [prop] (map (apfst fst) vars @ ns)
val all_frees = map (substT o Free) free_ns
val ns = drop (length vars) all_frees
val subst = subst_vars (substT_map, map fst vars ~~ (take (length vars) all_frees))
val prop' = subst prop
val match_card_ns = add_match_terms find_card prop' [] ~~ rev ns
val prop'' = replace_terms match_card_ns prop'
val eqs = map (HOLogic.mk_eq #> HOLogic.mk_Trueprop) match_card_ns |> rev
val new_thm = Goal.prove ctxt (map fst free_ns) eqs prop'' (fn {prems = prems, context = ctxt} =>
unfold_tac ctxt (map (fn thm => @{thm sym} OF [thm]) prems)
THEN resolve_tac ctxt [thm] 1
THEN (REPEAT (assume_tac ctxt 1)))
in
(length ns, new_thm)
end

(* turns A ==> B ==> C into A /\ B ==> C *)
(* does not properly work in case of nested meta-implications and meta-quantification *)
(* TODO: fix and simplify *)
fun combine_first_prems ctxt thm = let
val prop = Thm.prop_of thm
val (p1 :: p2 :: prems) = Thm.prems_of thm
val concl = Thm.concl_of thm
val all_tvars = Term.add_tvars prop []
val all_tfrees = map TFree (Variable.variant_frees ctxt [prop] (map (apfst fst) all_tvars))
val substT_map = map fst all_tvars ~~ all_tfrees
val prop' = subst_TVars substT_map prop
val all_vars = Term.add_vars prop' []
val free_ns = Variable.variant_frees ctxt [prop'] (map (apfst fst) all_vars)
val all_frees = map Free free_ns
val subst = subst_vars (substT_map, map fst all_vars ~~ all_frees)

val p1 = subst p1
val p2 = subst p2
val prems = map subst prems
val concl = subst concl
val p = HOLogic.mk_conj (HOLogic.dest_Trueprop p1, HOLogic.dest_Trueprop p2)
|> HOLogic.mk_Trueprop
in
Goal.prove ctxt (map fst free_ns) (p :: prems) concl (fn {prems = prems, context = ctxt} =>
let
val p = hd prems
val p1 = @{thm conjunct1} OF [p]
val p2 = @{thm conjunct2} OF [p]
val prems = tl prems
in
resolve_tac ctxt [thm OF (p1 :: p2 :: prems)] 1
end)
end

(* turns CARD('a) = n1 ==> A ==> B ==> C into
A ==> B ==> n1 != 0 ==> C *)
fun eliminate_card_constraint ctxt thm = let
val v = Thm.ctyp_of ctxt (Thm.prems_of thm |> hd |> (fn t => Term.add_tvars t [] |> hd) |> TVar)
val thm = Internalize_Sort.internalize_sort v thm |> snd
val thm = combine_first_prems ctxt thm
val thm = (thm OF @{thms type_impl_card_n}) |> Local_Typedef.cancel_type_definition
val thm = thm OF @{thms n_zero_nonempty}
val thm = rotate_prems 1 thm
in
thm
end

(* turns CARD('a) = n1 ==> ... ==> CARD('z) = n_z ==> A ==> B ==> C into
A ==> B ==> n1 != 0 ==> ... ==> n_z != 0 ==> C *)
fun eliminate_card_constraints ctxt thm n =
fold (K (eliminate_card_constraint ctxt)) (replicate n true) thm

(* turns A ==> B[CARD('a)] ==> C[CARD('a), CARD('b)] ==> ... ==> Z
into A ==> B[n1] ==> C[n1,n2] ==> ... ==> n1 != 0 ==> n2 != 0 ==> Z *)
fun cancel_card_constraint thm = let
val ctxt = Proof_Context.init_global (Thm.theory_of_thm thm)
val (n,thm) = extract_card_type ctxt thm
in
eliminate_card_constraints ctxt thm n
end

val cancel_card_constraint_attr = Thm.rule_attribute [] (K cancel_card_constraint);

val _ = Context.>> (Context.map_theory (Attrib.setup @{binding cancel_card_constraint}
(Scan.succeed cancel_card_constraint_attr) "cancels carrier constraints"))
end

# Theory Bij_Nat

(*
Authors: J. Divasón, R. Thiemann, A. Yamada
*)
section ‹Connecting HMA-matrices with JNF-matrices›

text ‹The following theories provide a connection between the type-based representation of vectors
and matrices in HOL multivariate-analysis (HMA) with the set-based representation of vectors and matrices
with integer indices in the Jordan-normal-form (JNF) development.›

subsection ‹Bijections between index types of HMA and natural numbers›

text ‹At the core of HMA-connect, there has to be a translation between indices
of vectors and matrices, which are via index-types on the one hand, and natural
numbers on the other hand.

We some unspecified bijection in our application, and not the conversions
to-nat and from-nat in theory
Rank-Nullity-Theorem/Mod-Type, since our definitions
below do not enforce any further type constraints.›

theory Bij_Nat
imports
"HOL-Library.Cardinality"
"HOL-Library.Numeral_Type"
begin

lemma finite_set_to_list: "∃ xs :: 'a :: finite list. distinct xs ∧ set xs = Y"
proof -
have "finite Y" by simp
thus ?thesis
proof (induct Y rule: finite_induct)
case (insert y Y)
then obtain xs where xs: "distinct xs" "set xs = Y" by auto
show ?case
by (rule exI[of _ "y # xs"], insert xs insert(2), auto)
qed simp
qed

definition univ_list :: "'a :: finite list" where
"univ_list = (SOME xs. distinct xs ∧ set xs = UNIV)"

lemma univ_list: "distinct (univ_list :: 'a list)" "set univ_list = (UNIV :: 'a :: finite set)"
proof -
let ?xs = "univ_list :: 'a list"
have "distinct ?xs ∧ set ?xs = UNIV"
unfolding univ_list_def
by (rule someI_ex, rule finite_set_to_list)
thus "distinct ?xs" "set ?xs = UNIV" by auto
qed

definition to_nat :: "'a :: finite ⇒ nat" where
"to_nat a = (SOME i. univ_list ! i = a ∧ i < length (univ_list :: 'a list))"

definition from_nat :: "nat ⇒ 'a :: finite" where
"from_nat i = univ_list ! i"

lemma length_univ_list_card: "length (univ_list :: 'a :: finite list) = CARD('a)"
using distinct_card[of "univ_list :: 'a list", symmetric]
by (auto simp: univ_list)

lemma to_nat_ex: "∃! i. univ_list ! i = (a :: 'a :: finite) ∧ i < length (univ_list :: 'a list)"
proof -
let ?ul = "univ_list :: 'a list"
have a_in_set: "a ∈ set ?ul" unfolding univ_list by auto
from this [unfolded set_conv_nth]
obtain i where i1: "?ul ! i = a ∧ i < length ?ul" by auto
show ?thesis
proof (rule ex1I, rule i1)
fix j
assume "?ul ! j = a ∧ j < length ?ul"
moreover have "distinct ?ul" by (simp add: univ_list)
ultimately show "j = i" using i1 nth_eq_iff_index_eq by blast
qed
qed

lemma to_nat_less_card: "to_nat (a :: 'a :: finite) < CARD('a)"
proof -
let ?ul = "univ_list :: 'a list"
from to_nat_ex[of a] obtain i where
i1: "univ_list ! i = a ∧ i<length (univ_list::'a list)" by auto
show ?thesis unfolding to_nat_def
proof (rule someI2, rule i1)
fix x
assume x: "?ul ! x = a ∧ x < length ?ul"
thus "x < CARD ('a)" using x by (simp add: univ_list length_univ_list_card)
qed
qed

lemma to_nat_from_nat_id:
assumes i: "i < CARD('a :: finite)"
shows "to_nat (from_nat i :: 'a) = i"
unfolding to_nat_def from_nat_def
proof (rule some_equality, simp)
have l: "length (univ_list::'a list) = card (set (univ_list::'a list))"
by (rule distinct_card[symmetric], simp add: univ_list)
thus i2: "i < length (univ_list::'a list)"
using i unfolding univ_list by simp
fix n
assume n: "(univ_list::'a list) ! n = (univ_list::'a list) ! i ∧ n < length (univ_list::'a list)"
have d: "distinct (univ_list::'a list)" using univ_list by simp
show "n = i" using nth_eq_iff_index_eq[OF d _ i2] n by auto
qed

lemma from_nat_inj: assumes i: "i < CARD('a :: finite)"
and j: "j < CARD('a :: finite)"
and id: "(from_nat i :: 'a) = from_nat j"
shows "i = j"
proof -
from arg_cong[OF id, of to_nat]
show ?thesis using i j by (simp add: to_nat_from_nat_id)
qed

lemma from_nat_to_nat_id[simp]:
"(from_nat (to_nat a)) = (a::'a :: finite)"
proof -
have a_in_set: "a ∈ set (univ_list)" unfolding univ_list by auto
from this [unfolded set_conv_nth]
obtain i where i1: "univ_list ! i = a ∧ i<length (univ_list::'a list)" by auto
show ?thesis
unfolding to_nat_def from_nat_def
by (rule someI2, rule i1, simp)
qed

lemma to_nat_inj[simp]: assumes "to_nat a = to_nat b"
shows "a = b"
proof -
from to_nat_ex[of a] to_nat_ex[of b]
show "a = b" unfolding to_nat_def by (metis assms from_nat_to_nat_id)
qed

lemma range_to_nat: "range (to_nat :: 'a :: finite ⇒ nat) = {0 ..< CARD('a)}" (is "?l = ?r")
proof -
{
fix i
assume "i ∈ ?l"
hence "i ∈ ?r" using to_nat_less_card[where 'a = 'a] by auto
}
moreover
{
fix i
assume "i ∈ ?r"
hence "i < CARD('a)" by auto
from to_nat_from_nat_id[OF this]
have "i ∈ ?l" by (metis range_eqI)
}
ultimately show ?thesis by auto
qed

lemma inj_to_nat: "inj to_nat" by (simp add: inj_on_def)

lemma bij_to_nat: "bij_betw to_nat (UNIV :: 'a :: finite set) {0 ..< CARD('a)}"
unfolding bij_betw_def by (auto simp: range_to_nat inj_to_nat)

lemma numeral_nat: "(numeral m1 :: nat) * numeral n1 ≡ numeral (m1 * n1)"
"(numeral m1 :: nat) + numeral n1 ≡ numeral (m1 + n1)" by simp_all

lemmas card_num_simps =
card_num1 card_bit0 card_bit1
mult_num_simps
eq_num_simps
numeral_nat Suc_numeral

end


# Theory HMA_Connect

(*
Authors: J. Divasón, R. Thiemann, A. Yamada, O. Kunčar
*)

subsection ‹Transfer rules to convert theorems from JNF to HMA and vice-versa.›

theory HMA_Connect
imports
"HOL-Analysis.Determinants"
"HOL-Analysis.Cartesian_Euclidean_Space"
Bij_Nat
Cancel_Card_Constraint
"HOL-Eisbach.Eisbach"
begin

text ‹Prefer certain constants and lemmas without prefix.›

hide_const (open) Matrix.mat
hide_const (open) Matrix.row
hide_const (open) Determinant.det

lemmas mat_def = Finite_Cartesian_Product.mat_def
lemmas det_def = Determinants.det_def
lemmas row_def = Finite_Cartesian_Product.row_def

notation vec_index (infixl "$v" 90) notation vec_nth (infixl "$h" 90)

text ‹Forget that @{typ "'a mat"}, @{typ "'a Matrix.vec"}, and @{typ "'a poly"}
have been defined via lifting›

(* TODO: add to end of matrix theory, stores lifting + transfer setup *)
lifting_forget vec.lifting
lifting_forget mat.lifting

lifting_forget poly.lifting

text ‹Some notions which we did not find in the HMA-world.›
definition eigen_vector :: "'a::comm_ring_1 ^ 'n ^ 'n ⇒ 'a ^ 'n ⇒ 'a ⇒ bool" where
"eigen_vector A v ev = (v ≠ 0 ∧ A *v v = ev *s v)"

definition eigen_value :: "'a :: comm_ring_1 ^ 'n ^ 'n ⇒ 'a ⇒ bool" where
"eigen_value A k = (∃ v. eigen_vector A v k)"

definition similar_matrix_wit
:: "'a :: semiring_1 ^ 'n ^ 'n ⇒ 'a ^ 'n ^ 'n ⇒ 'a ^ 'n ^ 'n ⇒ 'a ^ 'n ^ 'n ⇒ bool" where
"similar_matrix_wit A B P Q = (P ** Q = mat 1 ∧ Q ** P = mat 1 ∧ A = P ** B ** Q)"

definition similar_matrix
:: "'a :: semiring_1 ^ 'n ^ 'n ⇒ 'a ^ 'n ^ 'n ⇒ bool" where
"similar_matrix A B = (∃ P Q. similar_matrix_wit A B P Q)"

definition spectral_radius :: "complex ^ 'n ^ 'n ⇒ real" where
"spectral_radius A = Max { norm ev | v ev. eigen_vector A v ev}"

definition Spectrum :: "'a :: field ^ 'n ^ 'n ⇒ 'a set" where
"Spectrum A = Collect (eigen_value A)"

definition vec_elements_h :: "'a ^ 'n ⇒ 'a set" where
"vec_elements_h v = range (vec_nth v)"

lemma vec_elements_h_def': "vec_elements_h v = {v $h i | i. True}" unfolding vec_elements_h_def by auto definition elements_mat_h :: "'a ^ 'nc ^ 'nr ⇒ 'a set" where "elements_mat_h A = range (λ (i,j). A$h i $h j)" lemma elements_mat_h_def': "elements_mat_h A = {A$h i $h j | i j. True}" unfolding elements_mat_h_def by auto definition map_vector :: "('a ⇒ 'b) ⇒ 'a ^'n ⇒ 'b ^'n" where "map_vector f v ≡ χ i. f (v$h i)"

definition map_matrix :: "('a ⇒ 'b) ⇒ 'a ^ 'n ^ 'm ⇒ 'b ^ 'n ^ 'm" where
"map_matrix f A ≡ χ i. map_vector f (A $h i)" definition normbound :: "'a :: real_normed_field ^ 'nc ^ 'nr ⇒ real ⇒ bool" where "normbound A b ≡ ∀ x ∈ elements_mat_h A. norm x ≤ b" lemma spectral_radius_ev_def: "spectral_radius A = Max (norm  (Collect (eigen_value A)))" unfolding spectral_radius_def eigen_value_def[abs_def] by (rule arg_cong[where f = Max], auto) lemma elements_mat: "elements_mat A = {A $$(i,j) | i j. i < dim_row A ∧ j < dim_col A}" unfolding elements_mat_def by force definition vec_elements :: "'a Matrix.vec ⇒ 'a set" where "vec_elements v = set [v i. i <- [0 ..< dim_vec v]]" lemma vec_elements: "vec_elements v = { v i | i. i < dim_vec v}" unfolding vec_elements_def by auto (* TODO: restore a bundle, for e.g., for matrix_impl *) context includes vec.lifting begin end definition from_hma⇩v :: "'a ^ 'n ⇒ 'a Matrix.vec" where "from_hma⇩v v = Matrix.vec CARD('n) (λ i. v h from_nat i)" definition from_hma⇩m :: "'a ^ 'nc ^ 'nr ⇒ 'a Matrix.mat" where "from_hma⇩m a = Matrix.mat CARD('nr) CARD('nc) (λ (i,j). a h from_nat i h from_nat j)" definition to_hma⇩v :: "'a Matrix.vec ⇒ 'a ^ 'n" where "to_hma⇩v v = (χ i. v v to_nat i)" definition to_hma⇩m :: "'a Matrix.mat ⇒ 'a ^ 'nc ^ 'nr " where "to_hma⇩m a = (χ i j. a$$ (to_nat i, to_nat j))" declare vec_lambda_eta[simp] lemma to_hma_from_hma⇩v[simp]: "to_hma⇩v (from_hma⇩v v) = v" by (auto simp: to_hma⇩v_def from_hma⇩v_def to_nat_less_card) lemma to_hma_from_hma⇩m[simp]: "to_hma⇩m (from_hma⇩m v) = v" by (auto simp: to_hma⇩m_def from_hma⇩m_def to_nat_less_card) lemma from_hma_to_hma⇩v[simp]: "v ∈ carrier_vec (CARD('n)) ⟹ from_hma⇩v (to_hma⇩v v :: 'a ^ 'n) = v" by (auto simp: to_hma⇩v_def from_hma⇩v_def to_nat_from_nat_id) lemma from_hma_to_hma⇩m[simp]: "A ∈ carrier_mat (CARD('nr)) (CARD('nc)) ⟹ from_hma⇩m (to_hma⇩m A :: 'a ^ 'nc ^ 'nr) = A" by (auto simp: to_hma⇩m_def from_hma⇩m_def to_nat_from_nat_id) lemma from_hma⇩v_inj[simp]: "from_hma⇩v x = from_hma⇩v y ⟷ x = y" by (intro iffI, insert to_hma_from_hma⇩v[of x], auto) lemma from_hma⇩m_inj[simp]: "from_hma⇩m x = from_hma⇩m y ⟷ x = y" by(intro iffI, insert to_hma_from_hma⇩m[of x], auto) definition HMA_V :: "'a Matrix.vec ⇒ 'a ^ 'n ⇒ bool" where "HMA_V = (λ v w. v = from_hma⇩v w)" definition HMA_M :: "'a Matrix.mat ⇒ 'a ^ 'nc ^ 'nr ⇒ bool" where "HMA_M = (λ a b. a = from_hma⇩m b)" definition HMA_I :: "nat ⇒ 'n :: finite ⇒ bool" where "HMA_I = (λ i a. i = to_nat a)" context includes lifting_syntax begin lemma Domainp_HMA_V [transfer_domain_rule]: "Domainp (HMA_V :: 'a Matrix.vec ⇒ 'a ^ 'n ⇒ bool) = (λ v. v ∈ carrier_vec (CARD('n )))" by(intro ext iffI, insert from_hma_to_hma⇩v[symmetric], auto simp: from_hma⇩v_def HMA_V_def) lemma Domainp_HMA_M [transfer_domain_rule]: "Domainp (HMA_M :: 'a Matrix.mat ⇒ 'a ^ 'nc ^ 'nr ⇒ bool) = (λ A. A ∈ carrier_mat CARD('nr) CARD('nc))" by (intro ext iffI, insert from_hma_to_hma⇩m[symmetric], auto simp: from_hma⇩m_def HMA_M_def) lemma Domainp_HMA_I [transfer_domain_rule]: "Domainp (HMA_I :: nat ⇒ 'n :: finite ⇒ bool) = (λ i. i < CARD('n))" (is "?l = ?r") proof (intro ext) fix i :: nat show "?l i = ?r i" unfolding HMA_I_def Domainp_iff by (auto intro: exI[of _ "from_nat i"] simp: to_nat_from_nat_id to_nat_less_card) qed lemma bi_unique_HMA_V [transfer_rule]: "bi_unique HMA_V" "left_unique HMA_V" "right_unique HMA_V" unfolding HMA_V_def bi_unique_def left_unique_def right_unique_def by auto lemma bi_unique_HMA_M [transfer_rule]: "bi_unique HMA_M" "left_unique HMA_M" "right_unique HMA_M" unfolding HMA_M_def bi_unique_def left_unique_def right_unique_def by auto lemma bi_unique_HMA_I [transfer_rule]: "bi_unique HMA_I" "left_unique HMA_I" "right_unique HMA_I" unfolding HMA_I_def bi_unique_def left_unique_def right_unique_def by auto lemma right_total_HMA_V [transfer_rule]: "right_total HMA_V" unfolding HMA_V_def right_total_def by simp lemma right_total_HMA_M [transfer_rule]: "right_total HMA_M" unfolding HMA_M_def right_total_def by simp lemma right_total_HMA_I [transfer_rule]: "right_total HMA_I" unfolding HMA_I_def right_total_def by simp lemma HMA_V_index [transfer_rule]: "(HMA_V ===> HMA_I ===> (=)) ($v) ($h)" unfolding rel_fun_def HMA_V_def HMA_I_def from_hma⇩v_def by (auto simp: to_nat_less_card) text ‹We introduce the index function to have pointwise access to HMA-matrices by a constant. Otherwise, the transfer rule with @{term "λ A i j. A$h i $h j"} instead of index is not applicable.› definition "index_hma A i j ≡ A$h i $h j" lemma HMA_M_index [transfer_rule]: "(HMA_M ===> HMA_I ===> HMA_I ===> (=)) (λ A i j. A $$(i,j)) index_hma" by (intro rel_funI, simp add: index_hma_def to_nat_less_card HMA_M_def HMA_I_def from_hma⇩m_def) lemma HMA_V_0 [transfer_rule]: "HMA_V (0⇩v CARD('n)) (0 :: 'a :: zero ^ 'n)" unfolding HMA_V_def from_hma⇩v_def by auto lemma HMA_M_0 [transfer_rule]: "HMA_M (0⇩m CARD('nr) CARD('nc)) (0 :: 'a :: zero ^ 'nc ^ 'nr )" unfolding HMA_M_def from_hma⇩m_def by auto lemma HMA_M_1[transfer_rule]: "HMA_M (1⇩m (CARD('n))) (mat 1 :: 'a::{zero,one}^'n^'n)" unfolding HMA_M_def by (auto simp add: mat_def from_hma⇩m_def from_nat_inj) lemma from_hma⇩v_add: "from_hma⇩v v + from_hma⇩v w = from_hma⇩v (v + w)" unfolding from_hma⇩v_def by auto lemma HMA_V_add [transfer_rule]: "(HMA_V ===> HMA_V ===> HMA_V) (+) (+) " unfolding rel_fun_def HMA_V_def by (auto simp: from_hma⇩v_add) lemma from_hma⇩v_diff: "from_hma⇩v v - from_hma⇩v w = from_hma⇩v (v - w)" unfolding from_hma⇩v_def by auto lemma HMA_V_diff [transfer_rule]: "(HMA_V ===> HMA_V ===> HMA_V) (-) (-)" unfolding rel_fun_def HMA_V_def by (auto simp: from_hma⇩v_diff) lemma from_hma⇩m_add: "from_hma⇩m a + from_hma⇩m b = from_hma⇩m (a + b)" unfolding from_hma⇩m_def by auto lemma HMA_M_add [transfer_rule]: "(HMA_M ===> HMA_M ===> HMA_M) (+) (+) " unfolding rel_fun_def HMA_M_def by (auto simp: from_hma⇩m_add) lemma from_hma⇩m_diff: "from_hma⇩m a - from_hma⇩m b = from_hma⇩m (a - b)" unfolding from_hma⇩m_def by auto lemma HMA_M_diff [transfer_rule]: "(HMA_M ===> HMA_M ===> HMA_M) (-) (-) " unfolding rel_fun_def HMA_M_def by (auto simp: from_hma⇩m_diff) lemma scalar_product: fixes v :: "'a :: semiring_1 ^ 'n " shows "scalar_prod (from_hma⇩v v) (from_hma⇩v w) = scalar_product v w" unfolding scalar_product_def scalar_prod_def from_hma⇩v_def dim_vec by (simp add: sum.reindex[OF inj_to_nat, unfolded range_to_nat]) lemma [simp]: "from_hma⇩m (y :: 'a ^ 'nc ^ 'nr) ∈ carrier_mat (CARD('nr)) (CARD('nc))" "dim_row (from_hma⇩m (y :: 'a ^ 'nc ^ 'nr )) = CARD('nr)" "dim_col (from_hma⇩m (y :: 'a ^ 'nc ^ 'nr )) = CARD('nc)" unfolding from_hma⇩m_def by simp_all lemma [simp]: "from_hma⇩v (y :: 'a ^ 'n) ∈ carrier_vec (CARD('n))" "dim_vec (from_hma⇩v (y :: 'a ^ 'n)) = CARD('n)" unfolding from_hma⇩v_def by simp_all declare rel_funI [intro!] lemma HMA_scalar_prod [transfer_rule]: "(HMA_V ===> HMA_V ===> (=)) scalar_prod scalar_product" by (auto simp: HMA_V_def scalar_product) lemma HMA_row [transfer_rule]: "(HMA_I ===> HMA_M ===> HMA_V) (λ i a. Matrix.row a i) row" unfolding HMA_M_def HMA_I_def HMA_V_def by (auto simp: from_hma⇩m_def from_hma⇩v_def to_nat_less_card row_def) lemma HMA_col [transfer_rule]: "(HMA_I ===> HMA_M ===> HMA_V) (λ i a. col a i) column" unfolding HMA_M_def HMA_I_def HMA_V_def by (auto simp: from_hma⇩m_def from_hma⇩v_def to_nat_less_card column_def) definition mk_mat :: "('i ⇒ 'j ⇒ 'c) ⇒ 'c^'j^'i" where "mk_mat f = (χ i j. f i j)" definition mk_vec :: "('i ⇒ 'c) ⇒ 'c^'i" where "mk_vec f = (χ i. f i)" lemma HMA_M_mk_mat[transfer_rule]: "((HMA_I ===> HMA_I ===> (=)) ===> HMA_M) (λ f. Matrix.mat (CARD('nr)) (CARD('nc)) (λ (i,j). f i j)) (mk_mat :: (('nr ⇒ 'nc ⇒ 'a) ⇒ 'a^'nc^'nr))" proof- { fix x y i j assume id: "∀ (ya :: 'nr) (yb :: 'nc). (x (to_nat ya) (to_nat yb) :: 'a) = y ya yb" and i: "i < CARD('nr)" and j: "j < CARD('nc)" from to_nat_from_nat_id[OF i] to_nat_from_nat_id[OF j] id[rule_format, of "from_nat i" "from_nat j"] have "x i j = y (from_nat i) (from_nat j)" by auto } thus ?thesis unfolding rel_fun_def mk_mat_def HMA_M_def HMA_I_def from_hma⇩m_def by auto qed lemma HMA_M_mk_vec[transfer_rule]: "((HMA_I ===> (=)) ===> HMA_V) (λ f. Matrix.vec (CARD('n)) (λ i. f i)) (mk_vec :: (('n ⇒ 'a) ⇒ 'a^'n))" proof- { fix x y i assume id: "∀ (ya :: 'n). (x (to_nat ya) :: 'a) = y ya" and i: "i < CARD('n)" from to_nat_from_nat_id[OF i] id[rule_format, of "from_nat i"] have "x i = y (from_nat i)" by auto } thus ?thesis unfolding rel_fun_def mk_vec_def HMA_V_def HMA_I_def from_hma⇩v_def by auto qed lemma mat_mult_scalar: "A ** B = mk_mat (λ i j. scalar_product (row i A) (column j B))" unfolding vec_eq_iff matrix_matrix_mult_def scalar_product_def mk_mat_def by (auto simp: row_def column_def) lemma mult_mat_vec_scalar: "A *v v = mk_vec (λ i. scalar_product (row i A) v)" unfolding vec_eq_iff matrix_vector_mult_def scalar_product_def mk_mat_def mk_vec_def by (auto simp: row_def column_def) lemma dim_row_transfer_rule: "HMA_M A (A' :: 'a ^ 'nc ^ 'nr) ⟹ (=) (dim_row A) (CARD('nr))" unfolding HMA_M_def by auto lemma dim_col_transfer_rule: "HMA_M A (A' :: 'a ^ 'nc ^ 'nr) ⟹ (=) (dim_col A) (CARD('nc))" unfolding HMA_M_def by auto lemma HMA_M_mult [transfer_rule]: "(HMA_M ===> HMA_M ===> HMA_M) ((*)) ((**))" proof - { fix A B :: "'a :: semiring_1 mat" and A' :: "'a ^ 'n ^ 'nr" and B' :: "'a ^ 'nc ^ 'n" assume 1[transfer_rule]: "HMA_M A A'" "HMA_M B B'" note [transfer_rule] = dim_row_transfer_rule[OF 1(1)] dim_col_transfer_rule[OF 1(2)] have "HMA_M (A * B) (A' ** B')" unfolding times_mat_def mat_mult_scalar by (transfer_prover_start, transfer_step+, transfer, auto) } thus ?thesis by blast qed lemma HMA_V_smult [transfer_rule]: "((=) ===> HMA_V ===> HMA_V) (⋅⇩v) ((*s))" unfolding smult_vec_def unfolding rel_fun_def HMA_V_def from_hma⇩v_def by auto lemma HMA_M_mult_vec [transfer_rule]: "(HMA_M ===> HMA_V ===> HMA_V) ((*⇩v)) ((*v))" proof - { fix A :: "'a :: semiring_1 mat" and v :: "'a Matrix.vec" and A' :: "'a ^ 'nc ^ 'nr" and v' :: "'a ^ 'nc" assume 1[transfer_rule]: "HMA_M A A'" "HMA_V v v'" note [transfer_rule] = dim_row_transfer_rule have "HMA_V (A *⇩v v) (A' *v v')" unfolding mult_mat_vec_def mult_mat_vec_scalar by (transfer_prover_start, transfer_step+, transfer, auto) } thus ?thesis by blast qed lemma HMA_det [transfer_rule]: "(HMA_M ===> (=)) Determinant.det (det :: 'a :: comm_ring_1 ^ 'n ^ 'n ⇒ 'a)" proof - { fix a :: "'a ^ 'n ^ 'n" let ?tn = "to_nat :: 'n :: finite ⇒ nat" let ?fn = "from_nat :: nat ⇒ 'n" let ?zn = "{0..< CARD('n)}" let ?U = "UNIV :: 'n set" let ?p1 = "{p. p permutes ?zn}" let ?p2 = "{p. p permutes ?U}" let ?f= "λ p i. if i ∈ ?U then ?fn (p (?tn i)) else i" let ?g = "λ p i. ?fn (p (?tn i))" have fg: "⋀ a b c. (if a ∈ ?U then b else c) = b" by auto have "?p2 = ?f  ?p1" by (rule permutes_bij', auto simp: to_nat_less_card to_nat_from_nat_id) hence id: "?p2 = ?g  ?p1" by simp have inj_g: "inj_on ?g ?p1" unfolding inj_on_def proof (intro ballI impI ext, auto) fix p q i assume p: "p permutes ?zn" and q: "q permutes ?zn" and id: "(λ i. ?fn (p (?tn i))) = (λ i. ?fn (q (?tn i)))" { fix i from permutes_in_image[OF p] have pi: "p (?tn i) < CARD('n)" by (simp add: to_nat_less_card) from permutes_in_image[OF q] have qi: "q (?tn i) < CARD('n)" by (simp add: to_nat_less_card) from fun_cong[OF id] have "?fn (p (?tn i)) = from_nat (q (?tn i))" . from arg_cong[OF this, of ?tn] have "p (?tn i) = q (?tn i)" by (simp add: to_nat_from_nat_id pi qi) } note id = this show "p i = q i" proof (cases "i < CARD('n)") case True hence "?tn (?fn i) = i" by (simp add: to_nat_from_nat_id) from id[of "?fn i", unfolded this] show ?thesis . next case False thus ?thesis using p q unfolding permutes_def by simp qed qed have mult_cong: "⋀ a b c d. a = b ⟹ c = d ⟹ a * c = b * d" by simp have "sum (λ p. signof p * (∏i∈?zn. a h ?fn i h ?fn (p i))) ?p1 = sum (λ p. of_int (sign p) * (∏i∈UNIV. a h i h p i)) ?p2" unfolding id sum.reindex[OF inj_g] proof (rule sum.cong[OF refl], unfold mem_Collect_eq o_def, rule mult_cong) fix p assume p: "p permutes ?zn" let ?q = "λ i. ?fn (p (?tn i))" from id p have q: "?q permutes ?U" by auto from p have pp: "permutation p" unfolding permutation_permutes by auto let ?ft = "λ p i. ?fn (p (?tn i))" have fin: "finite ?zn" by simp have "sign p = sign ?q ∧ p permutes ?zn" using p fin proof (induction rule: permutes_induct) case id show ?case by (auto simp: sign_id[unfolded id_def] permutes_id[unfolded id_def]) next case (swap a b p) then have ‹permutation p› by (auto intro: permutes_imp_permutation) let ?sab = "Fun.swap a b id" let ?sfab = "Fun.swap (?fn a) (?fn b) id" have p_sab: "permutation ?sab" by (rule permutation_swap_id) have p_sfab: "permutation ?sfab" by (rule permutation_swap_id) from swap(4) have IH1: "p permutes ?zn" and IH2: "sign p = sign (?ft p)" by auto have sab_perm: "?sab permutes ?zn" using swap(1-2) by (rule permutes_swap_id) from permutes_compose[OF IH1 this] have perm1: "?sab o p permutes ?zn" . from IH1 have p_p1: "p ∈ ?p1" by simp hence "?ft p ∈ ?ft  ?p1" by (rule imageI) from this[folded id] have "?ft p permutes ?U" by simp hence p_ftp: "permutation (?ft p)" unfolding permutation_permutes by auto { fix a b assume a: "a ∈ ?zn" and b: "b ∈ ?zn" hence "(?fn a = ?fn b) = (a = b)" using swap(1-2) by (auto simp: from_nat_inj) } note inj = this from inj[OF swap(1-2)] have id2: "sign ?sfab = sign ?sab" unfolding sign_swap_id by simp have id: "?ft (Fun.swap a b id ∘ p) = Fun.swap (?fn a) (?fn b) id ∘ ?ft p" proof fix c show "?ft (Fun.swap a b id ∘ p) c = (Fun.swap (?fn a) (?fn b) id ∘ ?ft p) c" proof (cases "p (?tn c) = a ∨ p (?tn c) = b") case True thus ?thesis by (cases, auto simp add: swap_id_eq) next case False hence neq: "p (?tn c) ≠ a" "p (?tn c) ≠ b" by auto have pc: "p (?tn c) ∈ ?zn" unfolding permutes_in_image[OF IH1] by (simp add: to_nat_less_card) from neq[folded inj[OF pc swap(1)] inj[OF pc swap(2)]] have "?fn (p (?tn c)) ≠ ?fn a" "?fn (p (?tn c)) ≠ ?fn b" . with neq show ?thesis by (auto simp: swap_id_eq) qed qed show ?case unfolding IH2 id sign_compose[OF p_sab ‹permutation p›] sign_compose[OF p_sfab p_ftp] id2 by (rule conjI[OF refl perm1]) qed thus "signof p = of_int (sign ?q)" unfolding signof_def sign_def by auto show "(∏i = 0..<CARD('n). a h ?fn i h ?fn (p i)) = (∏i∈UNIV. a h i h ?q i)" unfolding range_to_nat[symmetric] prod.reindex[OF inj_to_nat] by (rule prod.cong[OF refl], unfold o_def, simp) qed } thus ?thesis unfolding HMA_M_def by (auto simp: from_hma⇩m_def Determinant.det_def det_def) qed lemma HMA_mat[transfer_rule]: "((=) ===> HMA_M) (λ k. k ⋅⇩m 1⇩m CARD('n)) (Finite_Cartesian_Product.mat :: 'a::semiring_1 ⇒ 'a^'n^'n)" unfolding Finite_Cartesian_Product.mat_def[abs_def] rel_fun_def HMA_M_def by (auto simp: from_hma⇩m_def from_nat_inj) lemma HMA_mat_minus[transfer_rule]: "(HMA_M ===> HMA_M ===> HMA_M) (λ A B. A + map_mat uminus B) ((-) :: 'a :: group_add ^'nc^'nr ⇒ 'a^'nc^'nr ⇒ 'a^'nc^'nr)" unfolding rel_fun_def HMA_M_def from_hma⇩m_def by auto definition mat2matofpoly where "mat2matofpoly A = (χ i j. [: A i j :])" definition charpoly where charpoly_def: "charpoly A = det (mat (monom 1 (Suc 0)) - mat2matofpoly A)" definition erase_mat :: "'a :: zero ^ 'nc ^ 'nr ⇒ 'nr ⇒ 'nc ⇒ 'a ^ 'nc ^ 'nr" where "erase_mat A i j = (χ i'. χ j'. if i' = i ∨ j' = j then 0 else A i' j')" definition sum_UNIV_type :: "('n :: finite ⇒ 'a :: comm_monoid_add) ⇒ 'n itself ⇒ 'a" where "sum_UNIV_type f _ = sum f UNIV" definition sum_UNIV_set :: "(nat ⇒ 'a :: comm_monoid_add) ⇒ nat ⇒ 'a" where "sum_UNIV_set f n = sum f {..<n}" definition HMA_T :: "nat ⇒ 'n :: finite itself ⇒ bool" where "HMA_T n _ = (n = CARD('n))" lemma HMA_mat2matofpoly[transfer_rule]: "(HMA_M ===> HMA_M) (λx. map_mat (λa. [:a:]) x) mat2matofpoly" unfolding rel_fun_def HMA_M_def from_hma⇩m_def mat2matofpoly_def by auto lemma HMA_char_poly [transfer_rule]: "((HMA_M :: ('a:: comm_ring_1 mat ⇒ 'a^'n^'n ⇒ bool)) ===> (=)) char_poly charpoly" proof - { fix A :: "'a mat" and A' :: "'a^'n^'n" assume [transfer_rule]: "HMA_M A A'" hence [simp]: "dim_row A = CARD('n)" by (simp add: HMA_M_def) have [simp]: "monom 1 (Suc 0) = [:0, 1 :: 'a :]" by (simp add: monom_Suc) have [simp]: "map_mat uminus (map_mat (λa. [:a:]) A) = map_mat (λa. [:-a:]) A" by (rule eq_matI, auto) have "char_poly A = charpoly A'" unfolding char_poly_def[abs_def] char_poly_matrix_def charpoly_def[abs_def] by (transfer, simp) } thus ?thesis by blast qed lemma HMA_eigen_vector [transfer_rule]: "(HMA_M ===> HMA_V ===> (=)) eigenvector eigen_vector" proof - { fix A :: "'a mat" and v :: "'a Matrix.vec" and A' :: "'a ^ 'n ^ 'n" and v' :: "'a ^ 'n" and k :: 'a assume 1[transfer_rule]: "HMA_M A A'" and 2[transfer_rule]: "HMA_V v v'" hence [simp]: "dim_row A = CARD('n)" "dim_vec v = CARD('n)" by (auto simp add: HMA_V_def HMA_M_def) have [simp]: "v ∈ carrier_vec CARD('n)" using 2 unfolding HMA_V_def by simp have "eigenvector A v = eigen_vector A' v'" unfolding eigenvector_def[abs_def] eigen_vector_def[abs_def] by (transfer, simp) } thus ?thesis by blast qed lemma HMA_eigen_value [transfer_rule]: "(HMA_M ===> (=) ===> (=)) eigenvalue eigen_value" proof - { fix A :: "'a mat" and A' :: "'a ^ 'n ^ 'n" and k assume 1[transfer_rule]: "HMA_M A A'" hence [simp]: "dim_row A = CARD('n)" by (simp add: HMA_M_def) note [transfer_rule] = dim_row_transfer_rule[OF 1(1)] have "(eigenvalue A k) = (eigen_value A' k)" unfolding eigenvalue_def[abs_def] eigen_value_def[abs_def] by (transfer, auto simp add: eigenvector_def) } thus ?thesis by blast qed lemma HMA_spectral_radius [transfer_rule]: "(HMA_M ===> (=)) Spectral_Radius.spectral_radius spectral_radius" unfolding Spectral_Radius.spectral_radius_def[abs_def] spectrum_def spectral_radius_ev_def[abs_def] by transfer_prover lemma HMA_elements_mat[transfer_rule]: "((HMA_M :: ('a mat ⇒ 'a ^ 'nc ^ 'nr ⇒ bool)) ===> (=)) elements_mat elements_mat_h" proof - { fix y :: "'a ^ 'nc ^ 'nr" and i j :: nat assume i: "i < CARD('nr)" and j: "j < CARD('nc)" hence "from_hma⇩m y$$ (i, j) ∈ range (λ(i, ya). y$h i $h ya)" using to_nat_from_nat_id[OF i] to_nat_from_nat_id[OF j] by (auto simp: from_hma⇩m_def) } moreover { fix y :: "'a ^ 'nc ^ 'nr" and a b have "∃i j. y$h a $h b = from_hma⇩m y $$(i, j) ∧ i < CARD('nr) ∧ j < CARD('nc)" unfolding from_hma⇩m_def by (rule exI[of _ "Bij_Nat.to_nat a"], rule exI[of _ "Bij_Nat.to_nat b"], auto simp: to_nat_less_card) } ultimately show ?thesis unfolding elements_mat[abs_def] elements_mat_h_def[abs_def] HMA_M_def by auto qed lemma HMA_vec_elements[transfer_rule]: "((HMA_V :: ('a Matrix.vec ⇒ 'a ^ 'n ⇒ bool)) ===> (=)) vec_elements vec_elements_h" proof - { fix y :: "'a ^ 'n" and i :: nat assume i: "i < CARD('n)" hence "from_hma⇩v y i ∈ range (vec_nth y)" using to_nat_from_nat_id[OF i] by (auto simp: from_hma⇩v_def) } moreover { fix y :: "'a ^ 'n" and a have "∃i. y h a = from_hma⇩v y i ∧ i < CARD('n)" unfolding from_hma⇩v_def by (rule exI[of _ "Bij_Nat.to_nat a"], auto simp: to_nat_less_card) } ultimately show ?thesis unfolding vec_elements[abs_def] vec_elements_h_def[abs_def] rel_fun_def HMA_V_def by auto qed lemma norm_bound_elements_mat: "norm_bound A b = (∀ x ∈ elements_mat A. norm x ≤ b)" unfolding norm_bound_def elements_mat by auto lemma HMA_normbound [transfer_rule]: "((HMA_M :: 'a :: real_normed_field mat ⇒ 'a ^ 'nc ^ 'nr ⇒ bool) ===> (=) ===> (=)) norm_bound normbound" unfolding normbound_def[abs_def] norm_bound_elements_mat[abs_def] by (transfer_prover) lemma HMA_map_matrix [transfer_rule]: "((=) ===> HMA_M ===> HMA_M) map_mat map_matrix" unfolding map_vector_def map_matrix_def[abs_def] map_mat_def[abs_def] HMA_M_def from_hma⇩m_def by auto lemma HMA_transpose_matrix [transfer_rule]: "(HMA_M ===> HMA_M) transpose_mat transpose" unfolding transpose_mat_def transpose_def HMA_M_def from_hma⇩m_def by auto lemma HMA_map_vector [transfer_rule]: "((=) ===> HMA_V ===> HMA_V) map_vec map_vector" unfolding map_vector_def[abs_def] map_vec_def[abs_def] HMA_V_def from_hma⇩v_def by auto lemma HMA_similar_mat_wit [transfer_rule]: "((HMA_M :: _ ⇒ 'a :: comm_ring_1 ^ 'n ^ 'n ⇒ _) ===> HMA_M ===> HMA_M ===> HMA_M ===> (=)) similar_mat_wit similar_matrix_wit" proof (intro rel_funI, goal_cases) case (1 a A b B c C d D) note [transfer_rule] = this hence id: "dim_row a = CARD('n)" by (auto simp: HMA_M_def) have *: "(c * d = 1⇩m (dim_row a) ∧ d * c = 1⇩m (dim_row a) ∧ a = c * b * d) = (C ** D = mat 1 ∧ D ** C = mat 1 ∧ A = C ** B ** D)" unfolding id by (transfer, simp) show ?case unfolding similar_mat_wit_def Let_def similar_matrix_wit_def * using 1 by (auto simp: HMA_M_def) qed lemma HMA_similar_mat [transfer_rule]: "((HMA_M :: _ ⇒ 'a :: comm_ring_1 ^ 'n ^ 'n ⇒ _) ===> HMA_M ===> (=)) similar_mat similar_matrix" proof (intro rel_funI, goal_cases) case (1 a A b B) note [transfer_rule] = this hence id: "dim_row a = CARD('n)" by (auto simp: HMA_M_def) { fix c d assume "similar_mat_wit a b c d" hence "{c,d} ⊆ carrier_mat CARD('n) CARD('n)" unfolding similar_mat_wit_def id Let_def by auto } note * = this show ?case unfolding similar_mat_def similar_matrix_def by (transfer, insert *, blast) qed lemma HMA_spectrum[transfer_rule]: "(HMA_M ===> (=)) spectrum Spectrum" unfolding spectrum_def[abs_def] Spectrum_def[abs_def] by transfer_prover lemma HMA_M_erase_mat[transfer_rule]: "(HMA_M ===> HMA_I ===> HMA_I ===> HMA_M) mat_erase erase_mat" unfolding mat_erase_def[abs_def] erase_mat_def[abs_def] by (auto simp: HMA_M_def HMA_I_def from_hma⇩m_def to_nat_from_nat_id intro!: eq_matI) lemma HMA_M_sum_UNIV[transfer_rule]: "((HMA_I ===> (=)) ===> HMA_T ===> (=)) sum_UNIV_set sum_UNIV_type" unfolding rel_fun_def proof (clarify, rename_tac f fT n nT) fix f and fT :: "'b ⇒ 'a" and n and nT :: "'b itself" assume f: "∀x y. HMA_I x y ⟶ f x = fT y" and n: "HMA_T n nT" let ?f = "from_nat :: nat ⇒ 'b" let ?t = "to_nat :: 'b ⇒ nat" from n[unfolded HMA_T_def] have n: "n = CARD('b)" . from to_nat_from_nat_id[where 'a = 'b, folded n] have tf: "i < n ⟹ ?t (?f i) = i" for i by auto have "sum_UNIV_set f n = sum f (?t  ?f  {..<n})" unfolding sum_UNIV_set_def by (rule arg_cong[of _ _ "sum f"], insert tf, force) also have "… = sum (f ∘ ?t) (?f  {..<n})" by (rule sum.reindex, insert tf n, auto simp: inj_on_def) also have "?f  {..<n} = UNIV" using range_to_nat[where 'a = 'b, folded n] by force also have "sum (f ∘ ?t) UNIV = sum fT UNIV" proof (rule sum.cong[OF refl]) fix i :: 'b show "(f ∘ ?t) i = fT i" unfolding o_def by (rule f[rule_format], auto simp: HMA_I_def) qed also have "… = sum_UNIV_type fT nT" unfolding sum_UNIV_type_def .. finally show "sum_UNIV_set f n = sum_UNIV_type fT nT" . qed end text ‹Setup a method to easily convert theorems from JNF into HMA.› method transfer_hma uses rule = ( (fold index_hma_def)?, (* prepare matrix access for transfer *) transfer, rule rule, (unfold carrier_vec_def carrier_mat_def)?, auto) text ‹Now it becomes easy to transfer results which are not yet proven in HMA, such as:› lemma matrix_add_vect_distrib: "(A + B) *v v = A *v v + B *v v" by (transfer_hma rule: add_mult_distrib_mat_vec) lemma matrix_vector_right_distrib: "M *v (v + w) = M *v v + M *v w" by (transfer_hma rule: mult_add_distrib_mat_vec) lemma matrix_vector_right_distrib_diff: "(M :: 'a :: ring_1 ^ 'nr ^ 'nc) *v (v - w) = M *v v - M *v w" by (transfer_hma rule: mult_minus_distrib_mat_vec) lemma eigen_value_root_charpoly: "eigen_value A k ⟷ poly (charpoly (A :: 'a :: field ^ 'n ^ 'n)) k = 0" by (transfer_hma rule: eigenvalue_root_char_poly) lemma finite_spectrum: fixes A :: "'a :: field ^ 'n ^ 'n" shows "finite (Collect (eigen_value A))" by (transfer_hma rule: card_finite_spectrum(1)[unfolded spectrum_def]) lemma non_empty_spectrum: fixes A :: "complex ^ 'n ^ 'n" shows "Collect (eigen_value A) ≠ {}" by (transfer_hma rule: spectrum_non_empty[unfolded spectrum_def]) lemma charpoly_transpose: "charpoly (transpose A :: 'a :: field ^ 'n ^ 'n) = charpoly A" by (transfer_hma rule: char_poly_transpose_mat) lemma eigen_value_transpose: "eigen_value (transpose A :: 'a :: field ^ 'n ^ 'n) v = eigen_value A v" unfolding eigen_value_root_charpoly charpoly_transpose by simp lemma matrix_diff_vect_distrib: "(A - B) *v v = A *v v - B *v (v :: 'a :: ring_1 ^ 'n)" by (transfer_hma rule: minus_mult_distrib_mat_vec) lemma similar_matrix_charpoly: "similar_matrix A B ⟹ charpoly A = charpoly B" by (transfer_hma rule: char_poly_similar) lemma pderiv_char_poly_erase_mat: fixes A :: "'a :: idom ^ 'n ^ 'n" shows "monom 1 1 * pderiv (charpoly A) = sum (λ i. charpoly (erase_mat A i i)) UNIV" proof - let ?A = "from_hma⇩m A" let ?n = "CARD('n)" have tA[transfer_rule]: "HMA_M ?A A" unfolding HMA_M_def by simp have tN[transfer_rule]: "HMA_T ?n TYPE('n)" unfolding HMA_T_def by simp have A: "?A ∈ carrier_mat ?n ?n" unfolding from_hma⇩m_def by auto have id: "sum (λ i. charpoly (erase_mat A i i)) UNIV = sum_UNIV_type (λ i. charpoly (erase_mat A i i)) TYPE('n)" unfolding sum_UNIV_type_def .. show ?thesis unfolding id by (transfer, insert pderiv_char_poly_mat_erase[OF A], simp add: sum_UNIV_set_def) qed lemma degree_monic_charpoly: fixes A :: "'a :: comm_ring_1 ^ 'n ^ 'n" shows "degree (charpoly A) = CARD('n) ∧ monic (charpoly A)" proof (transfer, goal_cases) case 1 from degree_monic_char_poly[OF 1] show ?case by auto qed end  # Theory Perron_Frobenius_Aux (* Author: Thiemann *) section ‹Perron-Frobenius Theorem› subsection ‹Auxiliary Notions› text ‹We define notions like non-negative real-valued matrix, both in JNF and in HMA. These notions will be linked via HMA-connect.› theory Perron_Frobenius_Aux imports HMA_Connect begin definition real_nonneg_mat :: "complex mat ⇒ bool" where "real_nonneg_mat A ≡ ∀ a ∈ elements_mat A. a ∈ ℝ ∧ Re a ≥ 0" definition real_nonneg_vec :: "complex Matrix.vec ⇒ bool" where "real_nonneg_vec v ≡ ∀ a ∈ vec_elements v. a ∈ ℝ ∧ Re a ≥ 0" definition real_non_neg_vec :: "complex ^ 'n ⇒ bool" where "real_non_neg_vec v ≡ (∀ a ∈ vec_elements_h v. a ∈ ℝ ∧ Re a ≥ 0)" definition real_non_neg_mat :: "complex ^ 'nr ^ 'nc ⇒ bool" where "real_non_neg_mat A ≡ (∀ a ∈ elements_mat_h A. a ∈ ℝ ∧ Re a ≥ 0)" lemma real_non_neg_matD: assumes "real_non_neg_mat A" shows "A h i h j ∈ ℝ" "Re (A h i h j) ≥ 0" using assms unfolding real_non_neg_mat_def elements_mat_h_def by auto definition nonneg_mat :: "'a :: linordered_idom mat ⇒ bool" where "nonneg_mat A ≡ ∀ a ∈ elements_mat A. a ≥ 0" definition non_neg_mat :: "'a :: linordered_idom ^ 'nr ^ 'nc ⇒ bool" where "non_neg_mat A ≡ (∀ a ∈ elements_mat_h A. a ≥ 0)" context includes lifting_syntax begin lemma HMA_real_non_neg_mat [transfer_rule]: "((HMA_M :: complex mat ⇒ complex ^ 'nc ^ 'nr ⇒ bool) ===> (=)) real_nonneg_mat real_non_neg_mat" unfolding real_nonneg_mat_def[abs_def] real_non_neg_mat_def[abs_def] by transfer_prover lemma HMA_real_non_neg_vec [transfer_rule]: "((HMA_V :: complex Matrix.vec ⇒ complex ^ 'n ⇒ bool) ===> (=)) real_nonneg_vec real_non_neg_vec" unfolding real_nonneg_vec_def[abs_def] real_non_neg_vec_def[abs_def] by transfer_prover lemma HMA_non_neg_mat [transfer_rule]: "((HMA_M :: 'a :: linordered_idom mat ⇒ 'a ^ 'nc ^ 'nr ⇒ bool) ===> (=)) nonneg_mat non_neg_mat" unfolding nonneg_mat_def[abs_def] non_neg_mat_def[abs_def] by transfer_prover end primrec matpow :: "'a::semiring_1^'n^'n ⇒ nat ⇒ 'a^'n^'n" where matpow_0: "matpow A 0 = mat 1" | matpow_Suc: "matpow A (Suc n) = (matpow A n) ** A" context includes lifting_syntax begin lemma HMA_pow_mat[transfer_rule]: "((HMA_M ::'a::{semiring_1} mat ⇒ 'a^'n^'n ⇒ bool) ===> (=) ===> HMA_M) pow_mat matpow" proof - { fix A :: "'a mat" and A' :: "'a^'n^'n" and n :: nat assume [transfer_rule]: "HMA_M A A'" hence [simp]: "dim_row A = CARD('n)" unfolding HMA_M_def by simp have "HMA_M (pow_mat A n) (matpow A' n)" proof (induct n) case (Suc n) note [transfer_rule] = this show ?case by (simp, transfer_prover) qed (simp, transfer_prover) } thus ?thesis by blast qed end lemma trancl_image: "(i,j) ∈ R⇧+ ⟹ (f i, f j) ∈ (map_prod f f  R)⇧+" proof (induct rule: trancl_induct) case (step j k) from step(2) have "(f j, f k) ∈ map_prod f f  R" by auto from step(3) this show ?case by auto qed auto lemma inj_trancl_image: assumes inj: "inj f" shows "(f i, f j) ∈ (map_prod f f  R)⇧+ = ((i,j) ∈ R⇧+)" (is "?l = ?r") proof assume ?r from trancl_image[OF this] show ?l . next assume ?l from trancl_image[OF this, of "the_inv f"] show ?r unfolding image_image prod.map_comp o_def the_inv_f_f[OF inj] by auto qed lemma matrix_add_rdistrib: "((B + C) ** A) = (B ** A) + (C ** A)" by (vector matrix_matrix_mult_def sum.distrib[symmetric] field_simps) lemma norm_smult: "norm ((a :: real) *s x) = abs a * norm x" unfolding norm_vec_def by (metis norm_scaleR norm_vec_def scalar_mult_eq_scaleR) lemma nonneg_mat_mult: "nonneg_mat A ⟹ nonneg_mat B ⟹ A ∈ carrier_mat nr n ⟹ B ∈ carrier_mat n nc ⟹ nonneg_mat (A * B)" unfolding nonneg_mat_def by (auto simp: elements_mat_def scalar_prod_def intro!: sum_nonneg) lemma nonneg_mat_power: assumes "A ∈ carrier_mat n n" "nonneg_mat A" shows "nonneg_mat (A ^⇩m k)" proof (induct k) case 0 thus ?case by (auto simp: nonneg_mat_def) next case (Suc k) from nonneg_mat_mult[OF this assms(2) _ assms(1), of n] assms(1) show ?case by auto qed lemma nonneg_matD: assumes "nonneg_mat A" and "i < dim_row A" and "j < dim_col A" shows "A$$ (i,j) ≥ 0" using assms unfolding nonneg_mat_def elements_mat by auto lemma (in comm_ring_hom) similar_mat_wit_hom: assumes "similar_mat_wit A B C D" shows "similar_mat_wit (mat⇩h A) (mat⇩h B) (mat⇩h C) (mat⇩h D)" proof - obtain n where n: "n = dim_row A" by auto note * = similar_mat_witD[OF n assms] from * have [simp]: "dim_row C = n" by auto note C = *(6) note D = *(7) note id = mat_hom_mult[OF C D] mat_hom_mult[OF D C] note ** = *(1-3)[THEN arg_cong[of _ _ "mat⇩h"], unfolded id] note mult = mult_carrier_mat[of _ n n] note hom_mult = mat_hom_mult[of _ n n _ n] show ?thesis unfolding similar_mat_wit_def Let_def unfolding **(3) using **(1,2) by (auto simp: n[symmetric] hom_mult simp: *(4-) mult) qed lemma (in comm_ring_hom) similar_mat_hom: "similar_mat A B ⟹ similar_mat (mat⇩h A) (mat⇩h B)" using similar_mat_wit_hom[of A B C D for C D] by (smt similar_mat_def) lemma det_dim_1: assumes A: "A ∈ carrier_mat n n" and n: "n = 1" shows "Determinant.det A = A $$(0,0)" by (subst laplace_expansion_column[OF A[unfolded n], of 0], insert A n, auto simp: cofactor_def mat_delete_def) lemma det_dim_2: assumes A: "A ∈ carrier_mat n n" and n: "n = 2" shows "Determinant.det A = A$$ (0,0) * A $$(1,1) - A$$ (0,1) * A$$(1,0)" proof - have set: "(∑i<(2 :: nat). f i) = f 0 + f 1" for f by (subst sum.cong[of _ "{0,1}" f f], auto) show ?thesis apply (subst laplace_expansion_column[OF A[unfolded n], of 0], insert A n, auto simp: cofactor_def mat_delete_def set) apply (subst (1 2) det_dim_1, auto) done qed lemma jordan_nf_root_char_poly: fixes A :: "'a :: {semiring_no_zero_divisors, idom} mat" assumes "jordan_nf A n_as" and "(m, lam) ∈ set n_as" shows "poly (char_poly A) lam = 0" proof - from assms have m0: "m ≠ 0" unfolding jordan_nf_def by force from split_list[OF assms(2)] obtain as bs where nas: "n_as = as @ (m, lam) # bs" by auto show ?thesis using m0 unfolding jordan_nf_char_poly[OF assms(1)] nas poly_prod_list prod_list_zero_iff by (auto simp: o_def) qed lemma inverse_power_tendsto_zero: "(λx. inverse ((of_nat x :: 'a :: real_normed_div_algebra) ^ Suc d)) ⇢ 0" proof (rule filterlim_compose[OF tendsto_inverse_0], intro filterlim_at_infinity[THEN iffD2, of 0] allI impI, goal_cases) case (2 r) let ?r = "nat (ceiling r) + 1" show ?case proof (intro eventually_sequentiallyI[of ?r], unfold norm_power norm_of_nat) fix x assume r: "?r ≤ x" hence x1: "real x ≥ 1" by auto have "r ≤ real ?r" by linarith also have "… ≤ x" using r by auto also have "… ≤ real x ^ Suc d" using x1 by simp finally show "r ≤ real x ^ Suc d" . qed qed simp lemma inverse_of_nat_tendsto_zero: "(λx. inverse (of_nat x :: 'a :: real_normed_div_algebra)) ⇢ 0" using inverse_power_tendsto_zero[of 0] by auto lemma poly_times_exp_tendsto_zero: assumes b: "norm (b :: 'a :: real_normed_field) < 1" shows "(λ x. of_nat x ^ k * b ^ x) ⇢ 0" proof (cases "b = 0") case False define nla where "nla = norm b" define s where "s = sqrt nla" from b False have nla: "0 < nla" "nla < 1" unfolding nla_def by auto hence s: "0 < s" "s < 1" unfolding s_def by auto { fix x have "s^x * s^x = sqrt (nla ^ (2 * x))" unfolding s_def power_add[symmetric] unfolding real_sqrt_power[symmetric] by (rule arg_cong[of _ _ "λ x. sqrt (nla ^ x)"], simp) also have "… = nla^x" unfolding power_mult real_sqrt_power using nla by simp finally have "nla^x = s^x * s^x" by simp } note nla_s = this show "(λx. of_nat x ^ k * b ^ x) ⇢ 0" proof (rule tendsto_norm_zero_cancel, unfold norm_mult norm_power norm_of_nat nla_def[symmetric] nla_s mult.assoc[symmetric]) from poly_exp_constant_bound[OF s, of 1 k] obtain p where p: "real x ^ k * s^x ≤ p" for x by (auto simp: ac_simps) have "norm (real x ^ k * s ^ x) = real x ^ k * s^x" for x using s by auto with p have p: "norm (real x ^ k * s ^ x) ≤ p" for x by auto from s have s: "norm s < 1" by auto show "(λx. real x ^ k * s ^ x * s ^ x) ⇢ 0" by (rule lim_null_mult_left_bounded[OF _ LIMSEQ_power_zero[OF s], of _ p], insert p, auto) qed next case True show ?thesis unfolding True by (subst tendsto_cong[of _ "λ x. 0"], rule eventually_sequentiallyI[of 1], auto) qed lemma (in linorder_topology) tendsto_Min: assumes I: "I ≠ {}" and fin: "finite I" shows "(⋀i. i ∈ I ⟹ (f i ⤏ a i) F) ⟹ ((λx. Min ((λ i. f i x)  I)) ⤏ (Min (a  I) :: 'a)) F" using fin I proof (induct rule: finite_induct) case (insert i I) hence i: "(f i ⤏ a i) F" by auto show ?case proof (cases "I = {}") case True show ?thesis unfolding True using i by auto next case False have *: "Min (a  insert i I) = min (a i) (Min (a  I))" using False insert(1) by auto have **: "(λx. Min ((λi. f i x)  insert i I)) = (λx. min (f i x) (Min ((λi. f i x)  I)))" using False insert(1) by auto have IH: "((λx. Min ((λi. f i x)  I)) ⤏ Min (a  I)) F" using insert(3)[OF insert(4) False] by auto show ?thesis unfolding * ** by (auto intro!: tendsto_min i IH) qed qed simp lemma tendsto_mat_mult [tendsto_intros]: "(f ⤏ a) F ⟹ (g ⤏ b) F ⟹ ((λx. f x ** g x) ⤏ a ** b) F" for f :: "'a ⇒ 'b :: {semiring_1, real_normed_algebra} ^ 'n1 ^ 'n2" unfolding matrix_matrix_mult_def[abs_def] by (auto intro!: tendsto_intros) lemma tendsto_matpower [tendsto_intros]: "(f ⤏ a) F ⟹ ((λx. matpow (f x) n) ⤏ matpow a n) F" for f :: "'a ⇒ 'b :: {semiring_1, real_normed_algebra} ^ 'n ^ 'n" by (induct n, simp_all add: tendsto_mat_mult) lemma continuous_matpow: "continuous_on R (λ A :: 'a :: {semiring_1, real_normed_algebra_1} ^ 'n ^ 'n. matpow A n)" unfolding continuous_on_def by (auto intro!: tendsto_intros) lemma vector_smult_distrib: "(A *v ((a :: 'a :: comm_ring_1) *s x)) = a *s ((A *v x))" unfolding matrix_vector_mult_def vector_scalar_mult_def by (simp add: ac_simps sum_distrib_left) instance real :: ordered_semiring_strict by (intro_classes, auto) lemma poly_tendsto_pinfty: fixes p :: "real poly" assumes "lead_coeff p > 0" "degree p ≠ 0" shows "poly p ⇢ ∞" unfolding Lim_PInfty proof fix b show "∃N. ∀n≥N. ereal b ≤ ereal (poly p (real n))" unfolding ereal_less_eq using poly_pinfty_ge[OF assms, of b] by (meson of_nat_le_iff order_trans real_arch_simple) qed lemma div_lt_nat: "(j :: nat) < x * y ⟹ j div x < y" by (simp add: less_mult_imp_div_less mult.commute) definition diagvector :: "('n ⇒ 'a :: semiring_0) ⇒ 'a ^ 'n ^ 'n" where "diagvector x = (χ i. χ j. if i = j then x i else 0)" lemma diagvector_mult_vector[simp]: "diagvector x *v y = (χ i. x i * y$ i)"
unfolding diagvector_def matrix_vector_mult_def vec_eq_iff vec_lambda_beta
proof (rule, goal_cases)
case (1 i)
show ?case by (subst sum.remove[of _ i], auto)
qed

lemma diagvector_mult_left: "diagvector x ** A = (χ i j. x i * A $i$ j)" (is "?A = ?B")
unfolding vec_eq_iff
proof (intro allI)
fix i j
show "?A $h i$h j = ?B $h i$h j"
unfolding map_vector_def diagvector_def matrix_matrix_mult_def vec_lambda_beta
by (subst sum.remove[of _ i], auto)
qed

lemma diagvector_mult_right: "A ** diagvector x = (χ i j. A $i$ j * x j)" (is "?A = ?B")
unfolding vec_eq_iff
proof (intro allI)
fix i j
show "?A $h i$h j = ?B $h i$h j"
unfolding map_vector_def diagvector_def matrix_matrix_mult_def vec_lambda_beta
by (subst sum.remove[of _ j], auto)
qed

lemma diagvector_mult[simp]: "diagvector x ** diagvector y = diagvector (λ i. x i * y i)"
unfolding diagvector_mult_left unfolding diagvector_def by (auto simp: vec_eq_iff)

lemma diagvector_const[simp]: "diagvector (λ x. k) = mat k"
unfolding diagvector_def mat_def by auto

lemma diagvector_eq_mat: "diagvector x = mat a ⟷ x = (λ x. a)"
unfolding diagvector_def mat_def by (auto simp: vec_eq_iff)

lemma cmod_eq_Re: assumes "cmod x = Re x"
shows "of_real (Re x) = x"
proof (cases "Im x = 0")
case False
hence "(cmod x)^2 ≠ (Re x)^2" unfolding norm_complex_def by simp
from this[unfolded assms] show ?thesis by auto
qed (cases x, auto simp: norm_complex_def complex_of_real_def)

hide_fact (open) Matrix.vec_eq_iff

no_notation
vec_index (infixl "$" 100) lemma spectral_radius_ev: "∃ ev v. eigen_vector A v ev ∧ norm ev = spectral_radius A" proof - from non_empty_spectrum[of A] finite_spectrum[of A] have "spectral_radius A ∈ norm  (Collect (eigen_value A))" unfolding spectral_radius_ev_def by auto thus ?thesis unfolding eigen_value_def[abs_def] by auto qed lemma spectral_radius_max: assumes "eigen_value A v" shows "norm v ≤ spectral_radius A" proof - from assms have "norm v ∈ norm  (Collect (eigen_value A))" by auto from Max_ge[OF _ this, folded spectral_radius_ev_def] finite_spectrum[of A] show ?thesis by auto qed text ‹For Perron-Frobenius it is useful to use the linear norm, and not the Euclidean norm.› definition norm1 :: "'a :: real_normed_field ^ 'n ⇒ real" where "norm1 v = (∑i∈UNIV. norm (v$ i))"

lemma norm1_ge_0: "norm1 v ≥ 0" unfolding norm1_def
by (rule sum_nonneg, auto)

lemma norm1_0[simp]: "norm1 0 = 0" unfolding norm1_def by auto

lemma norm1_nonzero: assumes "v ≠ 0"
shows "norm1 v > 0"
proof -
from ‹v ≠ 0› obtain i where vi: "v $i ≠ 0" unfolding vec_eq_iff using Finite_Cartesian_Product.vec_eq_iff zero_index by force have "sum (λ i. norm (v$ i)) (UNIV - {i}) ≥ 0"
by (rule sum_nonneg, auto)
moreover have "norm (v $i) > 0" using vi by auto ultimately have "0 < norm (v$ i) + sum (λ i. norm (v $i)) (UNIV - {i})" by arith also have "… = norm1 v" unfolding norm1_def by (simp add: sum.remove) finally show "norm1 v > 0" . qed lemma norm1_0_iff[simp]: "(norm1 v = 0) = (v = 0)" using norm1_0 norm1_nonzero by (cases "v = 0", force+) lemma norm1_scaleR[simp]: "norm1 (r *⇩R v) = abs r * norm1 v" unfolding norm1_def sum_distrib_left by (rule sum.cong, auto) lemma abs_norm1[simp]: "abs (norm1 v) = norm1 v" using norm1_ge_0[of v] by arith lemma normalize_eigen_vector: assumes "eigen_vector (A :: 'a :: real_normed_field ^ 'n ^ 'n) v ev" shows "eigen_vector A ((1 / norm1 v) *⇩R v) ev" "norm1 ((1 / norm1 v) *⇩R v) = 1" proof - let ?v = "(1 / norm1 v) *⇩R v" from assms[unfolded eigen_vector_def] have nz: "v ≠ 0" and id: "A *v v = ev *s v" by auto from nz have norm1: "norm1 v ≠ 0" by auto thus "norm1 ?v = 1" by simp from norm1 nz have nz: "?v ≠ 0" by auto have "A *v ?v = (1 / norm1 v) *⇩R (A *v v)" by (auto simp: vec_eq_iff matrix_vector_mult_def real_vector.scale_sum_right) also have "A *v v = ev *s v" unfolding id .. also have "(1 / norm1 v) *⇩R (ev *s v) = ev *s ?v" by (auto simp: vec_eq_iff) finally show "eigen_vector A ?v ev" using nz unfolding eigen_vector_def by auto qed lemma norm1_cont[simp]: "isCont norm1 v" unfolding norm1_def[abs_def] by auto lemma norm1_ge_norm: "norm1 v ≥ norm v" unfolding norm1_def norm_vec_def by (rule L2_set_le_sum, auto) text ‹The following continuity lemmas have been proven with hints from Fabian Immler.› lemma tendsto_matrix_vector_mult[tendsto_intros]: "((*v) (A :: 'a :: real_normed_algebra_1 ^ 'n ^ 'k) ⤏ A *v v) (at v within S)" unfolding matrix_vector_mult_def[abs_def] by (auto intro!: tendsto_intros) lemma tendsto_matrix_matrix_mult[tendsto_intros]: "((**) (A :: 'a :: real_normed_algebra_1 ^ 'n ^ 'k) ⤏ A ** B) (at B within S)" unfolding matrix_matrix_mult_def[abs_def] by (auto intro!: tendsto_intros) lemma matrix_vect_scaleR: "(A :: 'a :: real_normed_algebra_1 ^ 'n ^ 'k) *v (a *⇩R v) = a *⇩R (A *v v)" unfolding vec_eq_iff by (auto simp: matrix_vector_mult_def scaleR_vec_def scaleR_sum_right intro!: sum.cong) lemma (in inj_semiring_hom) map_vector_0: "(map_vector hom v = 0) = (v = 0)" unfolding vec_eq_iff map_vector_def by auto lemma (in inj_semiring_hom) map_vector_inj: "(map_vector hom v = map_vector hom w) = (v = w)" unfolding vec_eq_iff map_vector_def by auto lemma (in semiring_hom) matrix_vector_mult_hom: "(map_matrix hom A) *v (map_vector hom v) = map_vector hom (A *v v)" by (transfer fixing: hom, auto simp: mult_mat_vec_hom) lemma (in semiring_hom) vector_smult_hom: "hom x *s (map_vector hom v) = map_vector hom (x *s v)" by (transfer fixing: hom, auto simp: vec_hom_smult) lemma (in inj_comm_ring_hom) eigen_vector_hom: "eigen_vector (map_matrix hom A) (map_vector hom v) (hom x) = eigen_vector A v x" unfolding eigen_vector_def matrix_vector_mult_hom vector_smult_hom map_vector_0 map_vector_inj by auto end  # Theory Perron_Frobenius (* Author: R. Thiemann *) subsection ‹Perron-Frobenius theorem via Brouwer's fixpoint theorem.› theory Perron_Frobenius imports "HOL-Analysis.Brouwer_Fixpoint" Perron_Frobenius_Aux begin text ‹We follow the textbook proof of Serre \cite[Theorem 5.2.1]{SerreMatrices}.› context fixes A :: "complex ^ 'n ^ 'n :: finite" assumes rnnA: "real_non_neg_mat A" begin private abbreviation(input) sr where "sr ≡ spectral_radius A" private definition max_v_ev :: "(complex^'n) × complex" where "max_v_ev = (SOME v_ev. eigen_vector A (fst v_ev) (snd v_ev) ∧ norm (snd v_ev) = sr)" private definition "max_v = (1 / norm1 (fst max_v_ev)) *⇩R fst max_v_ev" private definition "max_ev = snd max_v_ev" private lemma max_v_ev: "eigen_vector A max_v max_ev" "norm max_ev = sr" "norm1 max_v = 1" proof - obtain v ev where id: "max_v_ev = (v,ev)" by force from spectral_radius_ev[of A] someI_ex[of "λ v_ev. eigen_vector A (fst v_ev) (snd v_ev) ∧ norm (snd v_ev) = sr", folded max_v_ev_def, unfolded id] have v: "eigen_vector A v ev" and ev: "norm ev = sr" by auto from normalize_eigen_vector[OF v] ev show "eigen_vector A max_v max_ev" "norm max_ev = sr" "norm1 max_v = 1" unfolding max_v_def max_ev_def id by auto qed text ‹In the definition of S, we use the linear norm instead of the default euclidean norm which is defined via the type-class. The reason is that S is not convex if one uses the euclidean norm.› private definition B :: "real ^ 'n ^ 'n" where "B ≡ χ i j. Re (A$ i $j)" private definition S where "S = {v :: real ^ 'n . norm1 v = 1 ∧ (∀ i. v$ i ≥ 0) ∧
(∀ i. (B *v v) $i ≥ sr * (v$ i))}"
private definition f :: "real ^ 'n ⇒ real ^ 'n" where
"f v = (1 / norm1 (B *v v)) *⇩R (B *v v)"

private lemma closedS: "closed S"
unfolding S_def matrix_vector_mult_def[abs_def]
proof (intro closed_Collect_conj closed_Collect_all closed_Collect_le closed_Collect_eq)
show "continuous_on UNIV norm1"
qed (auto intro!: continuous_intros continuous_on_component)

private lemma boundedS: "bounded S"
proof -
{
fix v :: "real ^ 'n"
from norm1_ge_norm[of v] have "norm1 v = 1 ⟹ norm v ≤ 1" by auto
}
thus ?thesis
unfolding S_def bounded_iff
by (auto intro!: exI[of _ 1])
qed

private lemma compactS: "compact S"
using boundedS closedS

private lemmas rnn = real_non_neg_matD[OF rnnA]

lemma B_norm: "B $i$ j = norm (A $i$ j)"
using rnn[of i j]
by (cases "A $i$ j", auto simp: B_def)

lemma mult_B_mono: assumes "⋀ i. v $i ≥ w$ i"
shows "(B *v v) $i ≥ (B *v w)$ i" unfolding matrix_vector_mult_def vec_lambda_beta
by (rule sum_mono, rule mult_left_mono[OF assms], unfold B_norm, auto)

private lemma non_emptyS: "S ≠ {}"
proof -
let ?v = "(χ i. norm (max_v $i)) :: real ^ 'n" have "norm1 max_v = 1" by (rule max_v_ev(3)) hence nv: "norm1 ?v = 1" unfolding norm1_def by auto { fix i have "sr * (?v$ i) = sr * norm (max_v $i)" by auto also have "… = (norm max_ev) * norm (max_v$ i)" using max_v_ev by auto
also have "… = norm ((max_ev *s max_v) $i)" by (auto simp: norm_mult) also have "max_ev *s max_v = A *v max_v" using max_v_ev(1)[unfolded eigen_vector_def] by auto also have "norm ((A *v max_v)$ i) ≤ (B *v ?v) $i" unfolding matrix_vector_mult_def vec_lambda_beta by (rule sum_norm_le, auto simp: norm_mult B_norm) finally have "sr * (?v$ i) ≤ (B *v ?v) $i" . } note le = this have "?v ∈ S" unfolding S_def using nv le by auto thus ?thesis by blast qed private lemma convexS: "convex S" proof (rule convexI) fix v w a b assume *: "v ∈ S" "w ∈ S" "0 ≤ a" "0 ≤ b" "a + b = (1 :: real)" let ?lin = "a *⇩R v + b *⇩R w" from * have 1: "norm1 v = 1" "norm1 w = 1" unfolding S_def by auto have "norm1 ?lin = a * norm1 v + b * norm1 w" unfolding norm1_def sum_distrib_left sum.distrib[symmetric] proof (rule sum.cong) fix i :: 'n from * have "v$ i ≥ 0" "w $i ≥ 0" unfolding S_def by auto thus "norm (?lin$ i) = a * norm (v $i) + b * norm (w$ i)"
using *(3-4) by auto
qed simp
also have "… = 1" using *(5) 1 by auto
finally have norm1: "norm1 ?lin = 1" .
{
fix i
from * have "0 ≤ v $i" "sr * v$ i ≤ (B *v v) $i" unfolding S_def by auto with ‹a ≥ 0› have a: "a * (sr * v$ i) ≤ a * (B *v v) $i" by (intro mult_left_mono) from * have "0 ≤ w$ i" "sr * w $i ≤ (B *v w)$ i" unfolding S_def by auto
with ‹b ≥ 0› have b: "b * (sr * w $i) ≤ b * (B *v w)$ i" by (intro mult_left_mono)
from a b have "a * (sr * v $i) + b * (sr * w$ i) ≤ a * (B *v v) $i + b * (B *v w)$ i" by auto
} note le = this
have switch[simp]: "⋀ x y. x * a * y = a * x * y"  "⋀ x y. x * b * y = b * x * y" by auto
have [simp]: "x ∈ {v,w} ⟹ a * (r * x $h i) = r * (a * x$h i)" for a r i x by auto
show "a *⇩R v + b *⇩R w ∈ S" using * norm1 le unfolding S_def
by (auto simp: matrix_vect_scaleR matrix_vector_right_distrib ring_distribs)
qed

private abbreviation (input) r :: "real ⇒ complex" where
"r ≡ of_real"

private abbreviation rv :: "real ^'n ⇒ complex ^'n" where
"rv v ≡ χ i. r (v $i)" private lemma rv_0: "(rv v = 0) = (v = 0)" by (simp add: of_real_hom.map_vector_0 map_vector_def vec_eq_iff) private lemma rv_mult: "A *v rv v = rv (B *v v)" proof - have "map_matrix r B = A" using rnnA unfolding map_matrix_def B_def real_non_neg_mat_def map_vector_def elements_mat_h_def by vector thus ?thesis using of_real_hom.matrix_vector_mult_hom[of B, where 'a = complex] unfolding map_vector_def by auto qed context assumes zero_no_ev: "⋀ v. v ∈ S ⟹ A *v rv v ≠ 0" begin private lemma normB_S: assumes v: "v ∈ S" shows "norm1 (B *v v) ≠ 0" proof - from zero_no_ev[OF v, unfolded rv_mult rv_0] show ?thesis by auto qed private lemma image_f: "f  S ⊆ S" proof - { fix v assume v: "v ∈ S" hence norm: "norm1 v = 1" and ge: "⋀ i. v$ i ≥ 0" "⋀ i. sr * v $i ≤ (B *v v)$ i" unfolding S_def by auto
from normB_S[OF v] have normB: "norm1 (B *v v) > 0" using norm1_nonzero by auto
have fv: "f v = (1 / norm1 (B *v v)) *⇩R (B *v v)" unfolding f_def by auto
from normB have Bv0: "B *v v ≠ 0" unfolding norm1_0_iff[symmetric] by linarith
have norm: "norm1 (f v) = 1" unfolding fv using normB Bv0 by simp
define c where "c = (1 / norm1 (B *v v))"
have c: "c > 0" unfolding c_def using normB by auto
{
fix i
have 1: "f v $i ≥ 0" unfolding fv c_def[symmetric] using c ge by (auto simp: matrix_vector_mult_def sum_distrib_left B_norm intro!: sum_nonneg) have id1: "⋀ i. (B *v f v)$ i = c * ((B *v (B *v v)) $i)" unfolding f_def c_def matrix_vect_scaleR by simp have id3: "⋀ i. sr * f v$ i = c * ((B *v (sr *⇩R v)) $i)" unfolding f_def c_def[symmetric] matrix_vect_scaleR by auto have 2: "sr * f v$ i ≤ (B *v f v) $i" unfolding id1 id3 unfolding mult_le_cancel_iff2[OF ‹c > 0›] by (rule mult_B_mono, insert ge(2), auto) note 1 2 } with norm have "f v ∈ S" unfolding S_def by auto } thus ?thesis by blast qed private lemma cont_f: "continuous_on S f" unfolding f_def[abs_def] continuous_on using normB_S unfolding norm1_def by (auto intro!: tendsto_eq_intros) qualified lemma perron_frobenius_positive_ev: "∃ v. eigen_vector A v (r sr) ∧ real_non_neg_vec v" proof - from brouwer[OF compactS convexS non_emptyS cont_f image_f] obtain v where v: "v ∈ S" and fv: "f v = v" by auto define ev where "ev = norm1 (B *v v)" from normB_S[OF v] have "ev ≠ 0" unfolding ev_def by auto with norm1_ge_0[of "B *v v", folded ev_def] have norm: "ev > 0" by auto from arg_cong[OF fv[unfolded f_def], of "λ (w :: real ^ 'n). ev *⇩R w"] norm have ev: "B *v v = ev *s v" unfolding ev_def[symmetric] scalar_mult_eq_scaleR by simp with v[unfolded S_def] have ge: "⋀ i. sr * v$ i ≤ ev * v $i" by auto have "A *v rv v = rv (B *v v)" unfolding rv_mult .. also have "… = ev *s rv v" unfolding ev vec_eq_iff by (simp add: scaleR_conv_of_real scaleR_vec_def) finally have ev: "A *v rv v = ev *s rv v" . from v have v0: "v ≠ 0" unfolding S_def by auto hence "rv v ≠ 0" unfolding rv_0 . with ev have ev: "eigen_vector A (rv v) ev" unfolding eigen_vector_def by auto hence "eigen_value A ev" unfolding eigen_value_def by auto from spectral_radius_max[OF this] have le: "norm (r ev) ≤ sr" . from v0 obtain i where "v$ i ≠ 0" unfolding vec_eq_iff by auto
from v have "v $i ≥ 0" unfolding S_def by auto with ‹v$ i ≠ 0› have "v $i > 0" by auto with ge[of i] have ge: "sr ≤ ev" by auto with le have sr: "r sr = ev" by auto from v have *: "real_non_neg_vec (rv v)" unfolding S_def real_non_neg_vec_def vec_elements_h_def by auto show ?thesis unfolding sr by (rule exI[of _ "rv v"], insert * ev norm, auto) qed end qualified lemma perron_frobenius_both: "∃ v. eigen_vector A v (r sr) ∧ real_non_neg_vec v" proof (cases "∀ v ∈ S. A *v rv v ≠ 0") case True show ?thesis by (rule Perron_Frobenius.perron_frobenius_positive_ev[OF rnnA], insert True, auto) next case False then obtain v where v: "v ∈ S" and A0: "A *v rv v = 0" by auto hence id: "A *v rv v = 0 *s rv v" and v0: "v ≠ 0" unfolding S_def by auto from v0 have "rv v ≠ 0" unfolding rv_0 . with id have ev: "eigen_vector A (rv v) 0" unfolding eigen_vector_def by auto hence "eigen_value A 0" unfolding eigen_value_def .. from spectral_radius_max[OF this] have 0: "0 ≤ sr" by auto from v[unfolded S_def] have ge: "⋀ i. sr * v$ i ≤ (B *v v) $i" by auto from v[unfolded S_def] have rnn: "real_non_neg_vec (rv v)" unfolding real_non_neg_vec_def vec_elements_h_def by auto from v0 obtain i where "v$ i ≠ 0" unfolding vec_eq_iff by auto
from v have "v $i ≥ 0" unfolding S_def by auto with ‹v$ i ≠ 0› have vi: "v $i > 0" by auto from rv_mult[of v, unfolded A0] have "rv (B *v v) = 0" by simp hence "B *v v = 0" unfolding rv_0 . from ge[of i, unfolded this] vi have ge: "sr ≤ 0" by (simp add: mult_le_0_iff) with ‹0 ≤ sr› have "sr = 0" by auto show ?thesis unfolding ‹sr = 0› using rnn ev by auto qed end text ‹Perron Frobenius: The largest complex eigenvalue of a real-valued non-negative matrix is a real one, and it has a real-valued non-negative eigenvector.› lemma perron_frobenius: assumes "real_non_neg_mat A" shows "∃v. eigen_vector A v (of_real (spectral_radius A)) ∧ real_non_neg_vec v" by (rule Perron_Frobenius.perron_frobenius_both[OF assms]) text ‹And a version which ignores the eigenvector.› lemma perron_frobenius_eigen_value: assumes "real_non_neg_mat A" shows "eigen_value A (of_real (spectral_radius A))" using perron_frobenius[OF assms] unfolding eigen_value_def by blast end  # Theory Roots_Unity (* author: Thiemann *) section ‹Roots of Unity› theory Roots_Unity imports Polynomial_Factorization.Order_Polynomial "HOL-Computational_Algebra.Fundamental_Theorem_Algebra" Polynomial_Interpolation.Ring_Hom_Poly begin lemma cis_mult_cmod_id: "cis (arg x) * of_real (cmod x) = x" using rcis_cmod_arg[unfolded rcis_def] by (simp add: ac_simps) lemma rcis_mult_cis[simp]: "rcis n a * cis b = rcis n (a + b)" unfolding cis_rcis_eq rcis_mult by simp lemma rcis_div_cis[simp]: "rcis n a / cis b = rcis n (a - b)" unfolding cis_rcis_eq rcis_divide by simp lemma cis_plus_2pi[simp]: "cis (x + 2 * pi) = cis x" by (auto simp: complex_eq_iff) lemma cis_plus_2pi_neq_1: assumes x: "0 < x" "x < 2 * pi" shows "cis x ≠ 1" proof - from x have "cos x ≠ 1" by (smt cos_2pi_minus cos_monotone_0_pi cos_zero) thus ?thesis by (auto simp: complex_eq_iff) qed lemma cis_times_2pi[simp]: "cis (of_nat n * 2 * pi) = 1" proof (induct n) case (Suc n) have "of_nat (Suc n) * 2 * pi = of_nat n * 2 * pi + 2 * pi" by (simp add: distrib_right) also have "cis … = 1" unfolding cis_plus_2pi Suc .. finally show ?case . qed simp declare cis_pi[simp] lemma cis_pi_2[simp]: "cis (pi / 2) = 𝗂" by (auto simp: complex_eq_iff) lemma cis_add_pi[simp]: "cis (pi + x) = - cis x" by (auto simp: complex_eq_iff) lemma cis_3_pi_2[simp]: "cis (pi * 3 / 2) = - 𝗂" proof - have "cis (pi * 3 / 2) = cis (pi + pi / 2)" by (rule arg_cong[of _ _ cis], simp) also have "… = - 𝗂" unfolding cis_add_pi by simp finally show ?thesis . qed lemma rcis_plus_2pi[simp]: "rcis y (x + 2 * pi) = rcis y x" unfolding rcis_def by simp lemma rcis_times_2pi[simp]: "rcis r (of_nat n * 2 * pi) = of_real r" unfolding rcis_def cis_times_2pi by simp lemma arg_rcis_cis: assumes n: "n > 0" shows "arg (rcis n x) = arg (cis x)" using arg_bounded arg_unique cis_arg complex_mod_rcis n rcis_def sgn_eq by auto lemma arg_eqD: assumes "arg (cis x) = arg (cis y)" "-pi < x" "x ≤ pi" "-pi < y" "y ≤ pi" shows "x = y" using assms(1) unfolding arg_unique[OF sgn_cis assms(2-3)] arg_unique[OF sgn_cis assms(4-5)] . lemma rcis_inj_on: assumes r: "r ≠ 0" shows "inj_on (rcis r) {0 ..< 2 * pi}" proof (rule inj_onI, goal_cases) case (1 x y) from arg_cong[OF 1(3), of "λ x. x / r"] have "cis x = cis y" using r by (simp add: rcis_def) from arg_cong[OF this, of "λ x. inverse x"] have "cis (-x) = cis (-y)" by simp from arg_cong[OF this, of uminus] have *: "cis (-x + pi) = cis (-y + pi)" by (auto simp: complex_eq_iff) have "- x + pi = - y + pi" by (rule arg_eqD[OF arg_cong[OF *, of arg]], insert 1(1-2), auto) thus ?case by simp qed lemma cis_inj_on: "inj_on cis {0 ..< 2 * pi}" using rcis_inj_on[of 1] unfolding rcis_def by auto definition root_unity :: "nat ⇒ 'a :: comm_ring_1 poly" where "root_unity n = monom 1 n - 1" lemma poly_root_unity: "poly (root_unity n) x = 0 ⟷ x^n = 1" unfolding root_unity_def by (simp add: poly_monom) lemma degree_root_unity[simp]: "degree (root_unity n) = n" (is "degree ?p = _") proof - have p: "?p = monom 1 n + (-1)" unfolding root_unity_def by auto show ?thesis proof (cases n) case 0 thus ?thesis unfolding p by simp next case (Suc m) show ?thesis unfolding p unfolding Suc by (subst degree_add_eq_left, auto simp: degree_monom_eq) qed qed lemma zero_root_unit[simp]: "root_unity n = 0 ⟷ n = 0" (is "?p = 0 ⟷ _") proof (cases "n = 0") case True thus ?thesis unfolding root_unity_def by simp next case False from degree_root_unity[of n] False have "degree ?p ≠ 0" by auto hence "?p ≠ 0" by fastforce thus ?thesis using False by auto qed definition prod_root_unity :: "nat list ⇒ 'a :: idom poly" where "prod_root_unity ns = prod_list (map root_unity ns)" lemma poly_prod_root_unity: "poly (prod_root_unity ns) x = 0 ⟷ (∃k∈set ns. x ^ k = 1)" unfolding prod_root_unity_def by (simp add: poly_prod_list prod_list_zero_iff o_def image_def poly_root_unity) lemma degree_prod_root_unity[simp]: "0 ∉ set ns ⟹ degree (prod_root_unity ns) = sum_list ns" unfolding prod_root_unity_def by (subst degree_prod_list_eq, auto simp: o_def) lemma zero_prod_root_unit[simp]: "prod_root_unity ns = 0 ⟷ 0 ∈ set ns" unfolding prod_root_unity_def prod_list_zero_iff by auto lemma roots_of_unity: assumes n: "n ≠ 0" shows "(λ i. (cis (of_nat i * 2 * pi / n)))  {0 ..< n} = { x :: complex. x ^ n = 1}" (is "?prod = ?Roots") "{x. poly (root_unity n) x = 0} = { x :: complex. x ^ n = 1}" "card { x :: complex. x ^ n = 1} = n" proof (atomize(full), goal_cases) case 1 let ?one = "1 :: complex" let ?p = "monom ?one n - 1" have degM: "degree (monom ?one n) = n" by (rule degree_monom_eq, simp) have "degree ?p = degree (monom ?one n + (-1))" by simp also have "… = degree (monom ?one n)" by (rule degree_add_eq_left, insert n, simp add: degM) finally have degp: "degree ?p = n" unfolding degM . with n have p: "?p ≠ 0" by auto have roots: "?Roots = {x. poly ?p x = 0}" unfolding poly_diff poly_monom by simp also have "finite …" by (rule poly_roots_finite[OF p]) finally have fin: "finite ?Roots" . have sub: "?prod ⊆ ?Roots" proof fix x assume "x ∈ ?prod" then obtain i where x: "x = cis (real i * 2 * pi / n)" by auto have "x ^ n = cis (real i * 2 * pi)" unfolding x DeMoivre using n by simp also have "… = 1" by simp finally show "x ∈ ?Roots" by auto qed have Rn: "card ?Roots ≤ n" unfolding roots by (rule poly_roots_degree[of ?p, unfolded degp, OF p]) have "… = card {0 ..< n}" by simp also have "… = card ?prod" proof (rule card_image[symmetric], rule inj_onI, goal_cases) case (1 x y) { fix m assume "m < n" hence "real m < real n" by simp from mult_strict_right_mono[OF this, of "2 * pi / real n"] n have "real m * 2 * pi / real n < real n * 2 * pi / real n" by simp hence "real m * 2 * pi / real n < 2 * pi" using n by simp } note [simp] = this have 0: "(1 :: real) ≠ 0" using n by auto have "real x * 2 * pi / real n = real y * 2 * pi / real n" by (rule inj_onD[OF rcis_inj_on 1(3)[unfolded cis_rcis_eq]], insert 1(1-2), auto) with n show "x = y" by auto qed finally have cn: "card ?prod = n" .. with Rn have "card ?prod ≥ card ?Roots" by auto with card_mono[OF fin sub] have card: "card ?prod = card ?Roots" by auto have "?prod = ?Roots" by (rule card_subset_eq[OF fin sub card]) from this roots[symmetric] cn[unfolded this] show ?case unfolding root_unity_def by blast qed lemma poly_roots_dvd: fixes p :: "'a :: field poly" assumes "p ≠ 0" and "degree p = n" and "card {x. poly p x = 0} ≥ n" and "{x. poly p x = 0} ⊆ {x. poly q x = 0}" shows "p dvd q" proof - from poly_roots_degree[OF assms(1)] assms(2-3) have "card {x. poly p x = 0} = n" by auto from assms(1-2) this assms(4) show ?thesis proof (induct n arbitrary: p q) case (0 p q) from is_unit_iff_degree[OF 0(1)] 0(2) show ?case by blast next case (Suc n p q) let ?P = "{x. poly p x = 0}" let ?Q = "{x. poly q x = 0}" from Suc(4-5) card_gt_0_iff[of ?P] obtain x where x: "poly p x = 0" "poly q x = 0" and fin: "finite ?P" by auto define r where "r = [:-x, 1:]" from x[unfolded poly_eq_0_iff_dvd r_def[symmetric]] obtain p' q' where p: "p = r * p'" and q: "q = r * q'" unfolding dvd_def by auto from Suc(2) have "degree p = degree r + degree p'" unfolding p by (subst degree_mult_eq, auto) with Suc(3) have deg: "degree p' = n" unfolding r_def by auto from Suc(2) p have p'0: "p' ≠ 0" by auto let ?P' = "{x. poly p' x = 0}" let ?Q' = "{x. poly q' x = 0}" have P: "?P = insert x ?P'" unfolding p poly_mult unfolding r_def by auto have Q: "?Q = insert x ?Q'" unfolding q poly_mult unfolding r_def by auto { assume "x ∈ ?P'" hence "?P = ?P'" unfolding P by auto from arg_cong[OF this, of card, unfolded Suc(4)] deg have False using poly_roots_degree[OF p'0] by auto } note xp' = this hence xP': "x ∉ ?P'" by auto have "card ?P = Suc (card ?P')" unfolding P by (rule card_insert_disjoint[OF _ xP'], insert fin[unfolded P], auto) with Suc(4) have card: "card ?P' = n" by auto from Suc(5)[unfolded P Q] xP' have "?P' ⊆ ?Q'" by auto from Suc(1)[OF p'0 deg card this] have IH: "p' dvd q'" . show ?case unfolding p q using IH by simp qed qed lemma root_unity_decomp: assumes n: "n ≠ 0" shows "root_unity n = prod_list (map (λ i. [:-cis (of_nat i * 2 * pi / n), 1:]) [0 ..< n])" (is "?u = ?p") proof - have deg: "degree ?u = n" by simp note main = roots_of_unity[OF n] have dvd: "?u dvd ?p" proof (rule poly_roots_dvd[OF _ deg]) show "n ≤ card {x. poly ?u x = 0}" using main by auto show "?u ≠ 0" using n by auto show "{x. poly ?u x = 0} ⊆ {x. poly ?p x = 0}" unfolding main(2) main(1)[symmetric] poly_prod_list prod_list_zero_iff by auto qed have deg': "degree ?p = n" by (subst degree_prod_list_eq, auto simp: o_def sum_list_triv) have mon: "monic ?u" using deg unfolding root_unity_def using n by auto have mon': "monic ?p" by (rule monic_prod_list, auto) from dvd[unfolded dvd_def] obtain f where puf: "?p = ?u * f" by auto have "degree ?p = degree ?u + degree f" using mon' n unfolding puf by (subst degree_mult_eq, auto) with deg deg' have "degree f = 0" by auto from degree0_coeffs[OF this] obtain a where f: "f = [:a:]" by blast from arg_cong[OF puf, of lead_coeff] mon mon' have "a = 1" unfolding puf f by (cases "a = 0", auto) with f have f: "f = 1" by auto with puf show ?thesis by auto qed lemma order_monic_linear: "order x [:y,1:] = (if y + x = 0 then 1 else 0)" proof (cases "y + x = 0") case True hence "poly [:y,1:] x = 0" by simp from this[unfolded order_root] have "order x [:y,1:] ≠ 0" by auto moreover from order_degree[of "[:y,1:]" x] have "order x [:y,1:] ≤ 1" by auto ultimately show ?thesis unfolding True by auto next case False hence "poly [:y,1:] x ≠ 0" by auto from order_0I[OF this] False show ?thesis by auto qed lemma order_root_unity: fixes x :: complex assumes n: "n ≠ 0" shows "order x (root_unity n) = (if x^n = 1 then 1 else 0)" (is "order _ ?u = _") proof (cases "x^n = 1") case False with roots_of_unity(2)[OF n] have "poly ?u x ≠ 0" by auto from False order_0I[OF this] show ?thesis by auto next case True let ?phi = "λ i :: nat. i * 2 * pi / n" from True roots_of_unity(1)[OF n] obtain i where i: "i < n" and x: "x = cis (?phi i)" by force from i have n_split: "[0 ..< n] = [0 ..< i] @ i # [Suc i ..< n]" by (metis le_Suc_ex less_imp_le_nat not_le_imp_less not_less0 upt_add_eq_append upt_conv_Cons) { fix j assume j: "j < n ∨ j < i" and eq: "cis (?phi i) = cis (?phi j)" from inj_onD[OF cis_inj_on eq] i j n have "i = j" by (auto simp: field_simps) } note inj = this have "order x ?u = 1" unfolding root_unity_decomp[OF n] unfolding x n_split using inj by (subst order_prod_list, force, fastforce simp: order_monic_linear) with True show ?thesis by auto qed lemma order_prod_root_unity: assumes 0: "0 ∉ set ks" shows "order (x :: complex) (prod_root_unity ks) = length (filter (λ k. x^k = 1) ks)" proof - have "order x (prod_root_unity ks) = (∑k←ks. order x (root_unity k))" unfolding prod_root_unity_def by (subst order_prod_list, insert 0, auto simp: o_def) also have "… = (∑k←ks. (if x^k = 1 then 1 else 0))" by (rule arg_cong, rule map_cong, insert 0, force, intro order_root_unity, metis) also have "… = length (filter (λ k. x^k = 1) ks)" by (subst sum_list_map_filter'[symmetric], simp add: sum_list_triv) finally show ?thesis . qed lemma root_unity_witness: fixes xs :: "complex list" assumes "prod_list (map (λ x. [:-x,1:]) xs) = monom 1 n - 1" shows "x^n = 1 ⟷ x ∈ set xs" proof - from assms have n0: "n ≠ 0" by (cases "n = 0", auto simp: prod_list_zero_iff) have "x ∈ set xs ⟷ poly (prod_list (map (λ x. [:-x,1:]) xs)) x = 0" unfolding poly_prod_list prod_list_zero_iff by auto also have "… ⟷ x^n = 1" using roots_of_unity(2)[OF n0] unfolding assms root_unity_def by auto finally show ?thesis by auto qed lemma root_unity_explicit: fixes x :: complex shows "(x ^ 1 = 1) ⟷ x = 1" "(x ^ 2 = 1) ⟷ (x ∈ {1, -1})" "(x ^ 3 = 1) ⟷ (x ∈ {1, Complex (-1/2) (sqrt 3 / 2), Complex (-1/2) (- sqrt 3 / 2)})" "(x ^ 4 = 1) ⟷ (x ∈ {1, -1, 𝗂, - 𝗂})" proof - show "(x ^ 1 = 1) ⟷ x = 1" by (subst root_unity_witness[of "[1]"], code_simp, auto) show "(x ^ 2 = 1) ⟷ (x ∈ {1, -1})" by (subst root_unity_witness[of "[1,-1]"], code_simp, auto) show "(x ^ 4 = 1) ⟷ (x ∈ {1, -1, 𝗂, - 𝗂})" by (subst root_unity_witness[of "[1,-1, 𝗂, - 𝗂]"], code_simp, auto) have 3: "3 = Suc (Suc (Suc 0))" "1 = [:1:]" by auto show "(x ^ 3 = 1) ⟷ (x ∈ {1, Complex (-1/2) (sqrt 3 / 2), Complex (-1/2) (- sqrt 3 / 2)})" by (subst root_unity_witness[of "[1, Complex (-1/2) (sqrt 3 / 2), Complex (-1/2) (- sqrt 3 / 2)]"], auto simp: 3 monom_altdef complex_mult complex_eq_iff) qed definition primitive_root_unity :: "nat ⇒ 'a :: power ⇒ bool" where "primitive_root_unity k x = (k ≠ 0 ∧ x^k = 1 ∧ (∀ k' < k. k' ≠ 0 ⟶ x^k' ≠ 1))" lemma primitive_root_unityD: assumes "primitive_root_unity k x" shows "k ≠ 0" "x^k = 1" "k' ≠ 0 ⟹ x^k' = 1 ⟹ k ≤ k'" proof - note * = assms[unfolded primitive_root_unity_def] from * have **: "k' < k ⟹ k' ≠ 0 ⟹ x ^ k' ≠ 1" by auto show "k ≠ 0" "x^k = 1" using * by auto show "k' ≠ 0 ⟹ x^k' = 1 ⟹ k ≤ k'" using ** by force qed lemma primitive_root_unity_exists: assumes "k ≠ 0" "x ^ k = 1" shows "∃ k'. k' ≤ k ∧ primitive_root_unity k' x" proof - let ?P = "λ k. x ^ k = 1 ∧ k ≠ 0" define k' where "k' = (LEAST k. ?P k)" from assms have Pk: "∃ k. ?P k" by auto from LeastI_ex[OF Pk, folded k'_def] have "k' ≠ 0" "x ^ k' = 1" by auto with not_less_Least[of _ ?P, folded k'_def] have "primitive_root_unity k' x" unfolding primitive_root_unity_def by auto with primitive_root_unityD(3)[OF this assms] show ?thesis by auto qed lemma primitive_root_unity_dvd: fixes x :: "complex" assumes k: "primitive_root_unity k x" shows "x ^ n = 1 ⟷ k dvd n" proof assume "k dvd n" then obtain j where n: "n = k * j" unfolding dvd_def by auto have "x ^ n = (x ^ k) ^ j" unfolding n power_mult by simp also have "… = 1" unfolding primitive_root_unityD[OF k] by simp finally show "x ^ n = 1" . next assume n: "x ^ n = 1" note k = primitive_root_unityD[OF k] show "k dvd n" proof (cases "n = 0") case n0: False from k(3)[OF n0] n have nk: "n ≥ k" by force from roots_of_unity[OF k(1)] k(2) obtain i :: nat where xk: "x = cis (i * 2 * pi / k)" and ik: "i < k" by force from roots_of_unity[OF n0] n obtain j :: nat where xn: "x = cis (j * 2 * pi / n)" and jn: "j < n" by force have cop: "coprime i k" proof (rule gcd_eq_1_imp_coprime) from k(1) have "gcd i k ≠ 0" by auto from gcd_coprime_exists[OF this] this obtain i' k' g where *: "i = i' * g" "k = k' * g" "g ≠ 0" and g: "g = gcd i k" by blast from *(2) k(1) have k': "k' ≠ 0" by auto have "x = cis (i * 2 * pi / k)" by fact also have "i * 2 * pi / k = i' * 2 * pi / k'" unfolding * using *(3) by auto finally have "x ^ k' = 1" by (simp add: DeMoivre k') with k(3)[OF k'] have "k' ≥ k" by linarith moreover with * k(1) have "g = 1" by auto then show "gcd i k = 1" by (simp add: g) qed from inj_onD[OF cis_inj_on xk[unfolded xn]] n0 k(1) ik jn have "j * real k = i * real n" by (auto simp: field_simps) hence "real (j * k) = real (i * n)" by simp hence eq: "j * k = i * n" by linarith with cop show "k dvd n" by (metis coprime_commute coprime_dvd_mult_right_iff dvd_triv_right) qed auto qed lemma primitive_root_unity_simple_computation: "primitive_root_unity k x = (if k = 0 then False else x ^ k = 1 ∧ (∀ i ∈ {1 ..< k}. x ^ i ≠ 1))" unfolding primitive_root_unity_def by auto lemma primitive_root_unity_explicit: fixes x :: complex shows "primitive_root_unity 1 x ⟷ x = 1" "primitive_root_unity 2 x ⟷ x = -1" "primitive_root_unity 3 x ⟷ (x ∈ {Complex (-1/2) (sqrt 3 / 2), Complex (-1/2) (- sqrt 3 / 2)})" "primitive_root_unity 4 x ⟷ (x ∈ {𝗂, - 𝗂})" proof (atomize(full), goal_cases) case 1 { fix P :: "nat ⇒ bool" have *: "{1 ..< 2 :: nat} = {1}" "{1 ..< 3 :: nat} = {1,2}" "{1 ..< 4 :: nat} = {1,2,3}" by code_simp+ have "(∀i∈ {1 ..< 2}. P i) = P 1" "(∀i∈ {1 ..< 3}. P i) ⟷ P 1 ∧ P 2" "(∀i∈ {1 ..< 4}. P i) ⟷ P 1 ∧ P 2 ∧ P 3" unfolding * by auto } note * = this show ?case unfolding primitive_root_unity_simple_computation root_unity_explicit * by (auto simp: complex_eq_iff) qed function decompose_prod_root_unity_main :: "'a :: field poly ⇒ nat ⇒ nat list × 'a poly" where "decompose_prod_root_unity_main p k = ( if k = 0 then ([], p) else let q = root_unity k in if q dvd p then if p = 0 then ([],0) else map_prod (Cons k) id (decompose_prod_root_unity_main (p div q) k) else decompose_prod_root_unity_main p (k - 1))" by pat_completeness auto termination by (relation "measure (λ (p,k). degree p + k)", auto simp: degree_div_less) declare decompose_prod_root_unity_main.simps[simp del] lemma decompose_prod_root_unity_main: fixes p :: "complex poly" assumes p: "p = prod_root_unity ks * f" and d: "decompose_prod_root_unity_main p k = (ks',g)" and f: "⋀ x. cmod x = 1 ⟹ poly f x ≠ 0" and k: "⋀ k'. k' > k ⟹ ¬ root_unity k' dvd p" shows "p = prod_root_unity ks' * f ∧ f = g ∧ set ks = set ks'" using d p k proof (induct p k arbitrary: ks ks' rule: decompose_prod_root_unity_main.induct) case (1 p k ks ks') note p = 1(4) note k = 1(5) from k[of "Suc k"] have p0: "p ≠ 0" by auto hence "p = 0 ⟷ False" by auto note d = 1(3)[unfolded decompose_prod_root_unity_main.simps[of p k] this if_False Let_def] from p0[unfolded p] have ks0: "0 ∉ set ks" by simp from f[of 1] have f0: "f ≠ 0" by auto note IH = 1(1)[OF _ refl _ p0] 1(2)[OF _ refl] show ?case proof (cases "k = 0") case True with p k[unfolded this, of "hd ks"] p0 have "ks = []" by (cases ks, auto simp: prod_root_unity_def) with d p True show ?thesis by (auto simp: prod_root_unity_def) next case k0: False note IH = IH[OF k0] from k0 have "k = 0 ⟷ False" by auto note d = d[unfolded this if_False] let ?u = "root_unity k :: complex poly" show ?thesis proof (cases "?u dvd p") case True note IH = IH(1)[OF True] let ?call = "decompose_prod_root_unity_main (p div ?u) k" from True d obtain Ks where rec: "?call = (Ks,g)" and ks': "ks' = (k # Ks)" by (cases ?call, auto) from True have "?u dvd p ⟷ True" by simp note d = d[unfolded this if_True rec] let ?x = "cis (2 * pi / k)" have rt: "poly ?u ?x = 0" unfolding poly_root_unity using cis_times_2pi[of 1] by (simp add: DeMoivre) with True have "poly p ?x = 0" unfolding dvd_def by auto from this[unfolded p] f[of ?x] rt have "poly (prod_root_unity ks) ?x = 0" unfolding poly_root_unity by auto from this[unfolded poly_prod_root_unity] ks0 obtain k' where k': "k' ∈ set ks" and rt: "?x ^ k' = 1" and k'0: "k' ≠ 0" by auto let ?u' = "root_unity k' :: complex poly" from k' rt k'0 have rtk': "poly ?u' ?x = 0" unfolding poly_root_unity by auto { let ?phi = " k' * (2 * pi / k)" assume "k' < k" hence "0 < ?phi" "?phi < 2 * pi" using k0 k'0 by (auto simp: field_simps) from cis_plus_2pi_neq_1[OF this] rtk' have False unfolding poly_root_unity DeMoivre .. } hence kk': "k ≤ k'" by presburger { assume "k' > k" from k[OF this, unfolded p] have "¬ ?u' dvd prod_root_unity ks" using dvd_mult2 by auto with k' have False unfolding prod_root_unity_def using prod_list_dvd[of ?u' "map root_unity ks"] by auto } with kk' have kk': "k' = k" by presburger with k' have "k ∈ set ks" by auto from split_list[OF this] obtain ks1 ks2 where ks: "ks = ks1 @ k # ks2" by auto hence "p div ?u = (?u * (prod_root_unity (ks1 @ ks2) * f)) div ?u" by (simp add: ac_simps p prod_root_unity_def) also have "… = prod_root_unity (ks1 @ ks2) * f" by (rule nonzero_mult_div_cancel_left, insert k0, auto) finally have id: "p div ?u = prod_root_unity (ks1 @ ks2) * f" . from d have ks': "ks' = k # Ks" by auto have "k < k' ⟹ ¬ root_unity k' dvd p div ?u" for k' using k[of k'] True by (metis dvd_div_mult_self dvd_mult2) from IH[OF rec id this] have id: "p div root_unity k = prod_root_unity Ks * f" and *: "f = g ∧ set (ks1 @ ks2) = set Ks" by auto from arg_cong[OF id, of "λ x. x * ?u"] True have "p = prod_root_unity Ks * f * root_unity k" by auto thus ?thesis using * unfolding ks ks' by (auto simp: prod_root_unity_def) next case False from d False have "decompose_prod_root_unity_main p (k - 1) = (ks',g)" by auto note IH = IH(2)[OF False this p] have k: "k - 1 < k' ⟹ ¬ root_unity k' dvd p" for k' using False k[of k'] k0 by (cases "k' = k", auto) show ?thesis by (rule IH, insert False k, auto) qed qed qed definition "decompose_prod_root_unity p = decompose_prod_root_unity_main p (degree p)" lemma decompose_prod_root_unity: fixes p :: "complex poly" assumes p: "p = prod_root_unity ks * f" and d: "decompose_prod_root_unity p = (ks',g)" and f: "⋀ x. cmod x = 1 ⟹ poly f x ≠ 0" and p0: "p ≠ 0" shows "p = prod_root_unity ks' * f ∧ f = g ∧ set ks = set ks'" proof (rule decompose_prod_root_unity_main[OF p d[unfolded decompose_prod_root_unity_def] f]) fix k assume deg: "degree p < k" hence "degree p < degree (root_unity k)" by simp with p0 show "¬ root_unity k dvd p" by (simp add: poly_divides_conv0) qed lemma (in comm_ring_hom) hom_root_unity: "map_poly hom (root_unity n) = root_unity n" proof - interpret p: map_poly_comm_ring_hom hom .. show ?thesis unfolding root_unity_def by (simp add: hom_distribs) qed lemma (in idom_hom) hom_prod_root_unity: "map_poly hom (prod_root_unity n) = prod_root_unity n" proof - interpret p: map_poly_comm_ring_hom hom .. show ?thesis unfolding prod_root_unity_def p.hom_prod_list map_map o_def hom_root_unity .. qed lemma (in field_hom) hom_decompose_prod_root_unity_main: "decompose_prod_root_unity_main (map_poly hom p) k = map_prod id (map_poly hom) (decompose_prod_root_unity_main p k)" proof (induct p k rule: decompose_prod_root_unity_main.induct) case (1 p k) let ?h = "map_poly hom" let ?p = "?h p" let ?u = "root_unity k :: 'a poly" let ?u' = "root_unity k :: 'b poly" interpret p: map_poly_inj_idom_divide_hom hom .. have u': "?u' = ?h ?u" unfolding hom_root_unity .. note simp = decompose_prod_root_unity_main.simps let ?rec1 = "decompose_prod_root_unity_main (p div ?u) k" have 0: "?p = 0 ⟷ p = 0" by simp show ?case unfolding simp[of ?p k] simp[of p k] if_distrib[of "map_prod id ?h"] Let_def u' unfolding 0 p.hom_div[symmetric] p.hom_dvd_iff by (rule if_cong[OF refl], force, rule if_cong[OF refl if_cong[OF refl]], force, (subst 1(1), auto, cases ?rec1, auto)[1], (subst 1(2), auto)) qed lemma (in field_hom) hom_decompose_prod_root_unity: "decompose_prod_root_unity (map_poly hom p) = map_prod id (map_poly hom) (decompose_prod_root_unity p)" unfolding decompose_prod_root_unity_def by (subst hom_decompose_prod_root_unity_main, simp) end  # Theory Perron_Frobenius_Irreducible (* Author: Thiemann *) subsection ‹The Perron Frobenius Theorem for Irreducible Matrices› theory Perron_Frobenius_Irreducible imports Perron_Frobenius Roots_Unity Rank_Nullity_Theorem.Miscellaneous (* for scalar-matrix-multiplication, this import is incompatible with field_simps, ac_simps *) begin lifting_forget vec.lifting lifting_forget mat.lifting lifting_forget poly.lifting lemma charpoly_of_real: "charpoly (map_matrix complex_of_real A) = map_poly of_real (charpoly A)" by (transfer_hma rule: of_real_hom.char_poly_hom) context includes lifting_syntax begin lemma HMA_M_smult[transfer_rule]: "((=) ===> HMA_M ===> HMA_M) (⋅⇩m) ((*k))" unfolding smult_mat_def unfolding rel_fun_def HMA_M_def from_hma⇩m_def by (auto simp: matrix_scalar_mult_def) end lemma order_charpoly_smult: fixes A :: "complex ^ 'n ^ 'n" assumes k: "k ≠ 0" shows "order x (charpoly (k *k A)) = order (x / k) (charpoly A)" by (transfer fixing: k, rule order_char_poly_smult[OF _ k]) (* use field, since the *k-lemmas have been stated for fields *) lemma smult_eigen_vector: fixes a :: "'a :: field" assumes "eigen_vector A v x" shows "eigen_vector (a *k A) v (a * x)" proof - from assms[unfolded eigen_vector_def] have v: "v ≠ 0" and id: "A *v v = x *s v" by auto from arg_cong[OF id, of "(*s) a"] have id: "(a *k A) *v v = (a * x) *s v" unfolding scalar_matrix_vector_assoc by simp thus "eigen_vector (a *k A) v (a * x)" using v unfolding eigen_vector_def by auto qed lemma smult_eigen_value: fixes a :: "'a :: field" assumes "eigen_value A x" shows "eigen_value (a *k A) (a * x)" using assms smult_eigen_vector[of A _ x a] unfolding eigen_value_def by blast locale fixed_mat = fixes A :: "'a :: zero ^ 'n ^ 'n" begin definition G :: "'n rel" where "G = { (i,j). A$ i $j ≠ 0}" definition irreducible :: bool where "irreducible = (UNIV ⊆ G^+)" end lemma G_transpose: "fixed_mat.G (transpose A) = ((fixed_mat.G A))^-1" unfolding fixed_mat.G_def by (force simp: transpose_def) lemma G_transpose_trancl: "(fixed_mat.G (transpose A))^+ = ((fixed_mat.G A)^+)^-1" unfolding G_transpose trancl_converse by auto locale pf_nonneg_mat = fixed_mat A for A :: "'a :: linordered_idom ^ 'n ^ 'n" + assumes non_neg_mat: "non_neg_mat A" begin lemma nonneg: "A$ i $j ≥ 0" using non_neg_mat unfolding non_neg_mat_def elements_mat_h_def by auto lemma nonneg_matpow: "matpow A n$ i $j ≥ 0" by (induct n arbitrary: i j, insert nonneg, auto intro!: sum_nonneg simp: matrix_matrix_mult_def mat_def) lemma G_relpow_matpow_pos: "(i,j) ∈ G ^^ n ⟹ matpow A n$ i $j > 0" proof (induct n arbitrary: i j) case (0 i) thus ?case by (auto simp: mat_def) next case (Suc n i j) from Suc(2) have "(i,j) ∈ G ^^ n O G" by (simp add: relpow_commute) then obtain k where ik: "A$ k $j ≠ 0" and kj: "(i, k) ∈ G ^^ n" by (auto simp: G_def) from ik nonneg[of k j] have ik: "A$ k $j > 0" by auto from Suc(1)[OF kj] have IH: "matpow A n$h i $h k > 0" . thus ?case using ik by (auto simp: nonneg_matpow nonneg matrix_matrix_mult_def intro!: sum_pos2[of _ k] mult_nonneg_nonneg) qed lemma matpow_mono: assumes B: "⋀ i j. B$ i $j ≥ A$ i $j" shows "matpow B n$ i $j ≥ matpow A n$ i $j" proof (induct n arbitrary: i j) case (Suc n i j) thus ?case using B nonneg_matpow[of n] nonneg by (auto simp: matrix_matrix_mult_def intro!: sum_mono mult_mono') qed simp lemma matpow_sum_one_mono: "matpow (A + mat 1) (n + k)$ i $j ≥ matpow (A + mat 1) n$ i $j" proof (induct k) case (Suc k) have "(matpow (A + mat 1) (n + k) ** A)$h i $h j ≥ 0" unfolding matrix_matrix_mult_def using order.trans[OF nonneg_matpow matpow_mono[of "A + mat 1" "n + k"]] by (auto intro!: sum_nonneg mult_nonneg_nonneg nonneg simp: mat_def) thus ?case using Suc by (simp add: matrix_add_ldistrib matrix_mul_rid) qed simp lemma G_relpow_matpow_pos_ge: assumes "(i,j) ∈ G ^^ m" "n ≥ m" shows "matpow (A + mat 1) n$ i $j > 0" proof - from assms(2) obtain k where n: "n = m + k" using le_Suc_ex by blast have "0 < matpow A m$ i $j" by (rule G_relpow_matpow_pos[OF assms(1)]) also have "… ≤ matpow (A + mat 1) m$ i $j" by (rule matpow_mono, auto simp: mat_def) also have "… ≤ matpow (A + mat 1) n$ i $j" unfolding n using matpow_sum_one_mono . finally show ?thesis . qed end locale perron_frobenius = pf_nonneg_mat A for A :: "real ^ 'n ^ 'n" + assumes irr: irreducible begin definition N where "N = (SOME N. ∀ ij. ∃ n ≤ N. ij ∈ G ^^ n)" lemma N: "∃ n ≤ N. ij ∈ G ^^ n" proof - { fix ij have "ij ∈ G^+" using irr[unfolded irreducible_def] by auto from this[unfolded trancl_power] have "∃ n. ij ∈ G ^^ n" by blast } hence "∀ ij. ∃ n. ij ∈ G ^^ n" by auto from choice[OF this] obtain f where f: "⋀ ij. ij ∈ G ^^ (f ij)" by auto define N where N: "N = Max (range f)" { fix ij from f[of ij] have "ij ∈ G ^^ f ij" . moreover have "f ij ≤ N" unfolding N by (rule Max_ge, auto) ultimately have "∃ n ≤ N. ij ∈ G ^^ n" by blast } note main = this let ?P = "λ N. ∀ ij. ∃ n ≤ N. ij ∈ G ^^ n" from main have "?P N" by blast from someI[of ?P, OF this, folded N_def] show ?thesis by blast qed lemma irreducible_matpow_pos: assumes irreducible shows "matpow (A + mat 1) N$ i $j > 0" proof - from N obtain n where n: "n ≤ N" and reach: "(i,j) ∈ G ^^ n" by auto show ?thesis by (rule G_relpow_matpow_pos_ge[OF reach n]) qed lemma pf_transpose: "perron_frobenius (transpose A)" proof show "fixed_mat.irreducible (transpose A)" unfolding fixed_mat.irreducible_def G_transpose_trancl using irr[unfolded irreducible_def] by auto qed (insert nonneg, auto simp: transpose_def non_neg_mat_def elements_mat_h_def) abbreviation le_vec :: "real ^ 'n ⇒ real ^ 'n ⇒ bool" where "le_vec x y ≡ (∀ i. x$ i ≤ y $i)" abbreviation lt_vec :: "real ^ 'n ⇒ real ^ 'n ⇒ bool" where "lt_vec x y ≡ (∀ i. x$ i < y $i)" definition "A1n = matpow (A + mat 1) N" lemmas A1n_pos = irreducible_matpow_pos[OF irr, folded A1n_def] definition r :: "real ^ 'n ⇒ real" where "r x = Min { (A *v x)$ j / x $j | j. x$ j ≠ 0 }"

definition X :: "(real ^ 'n )set" where
"X = { x . le_vec 0 x ∧ x ≠ 0 }"

lemma nonneg_Ax: "x ∈ X ⟹ le_vec 0 (A *v x)"
unfolding X_def using nonneg
by (auto simp: matrix_vector_mult_def intro!: sum_nonneg)

lemma A_nonzero_fixed_i: "∃ j. A $i$ j ≠ 0"
proof -
from irr[unfolded irreducible_def] have "(i,i) ∈ G^+" by auto
then obtain j where "(i,j) ∈ G" by (metis converse_tranclE)
hence Aij: "A $i$ j ≠ 0" unfolding G_def by auto
thus ?thesis ..
qed

lemma A_nonzero_fixed_j: "∃ i. A $i$ j ≠ 0"
proof -
from irr[unfolded irreducible_def] have "(j,j) ∈ G^+" by auto
then obtain i where "(i,j) ∈ G" by (cases, auto)
hence Aij: "A $i$ j ≠ 0" unfolding G_def by auto
thus ?thesis ..
qed

lemma Ax_pos: assumes x: "lt_vec 0 x"
shows "lt_vec 0 (A *v x)"
proof
fix i
from A_nonzero_fixed_i[of i] obtain j where "A $i$ j ≠ 0" by auto
with nonneg[of i j] have A: "A $i$ j > 0" by simp
from x have "x $j ≥ 0" for j by (auto simp: order.strict_iff_order) note nonneg = mult_nonneg_nonneg[OF nonneg[of i] this] have "(A *v x)$ i = (∑j∈UNIV. A $i$ j * x $j)" unfolding matrix_vector_mult_def by simp also have "… = A$ i $j * x$ j + (∑j∈UNIV - {j}. A $i$ j * x $j)" by (subst sum.remove, auto) also have "… > 0 + 0" by (rule add_less_le_mono, insert A x[rule_format] nonneg, auto intro!: sum_nonneg mult_pos_pos) finally show "0$ i < (A *v x) $i" by simp qed lemma nonzero_Ax: assumes x: "x ∈ X" shows "A *v x ≠ 0" proof assume 0: "A *v x = 0" from x[unfolded X_def] have x: "le_vec 0 x" "x ≠ 0" by auto from x(2) obtain j where xj: "x$ j ≠ 0"
by (metis vec_eq_iff zero_index)
from A_nonzero_fixed_j[of j]  obtain i where Aij: "A $i$ j ≠ 0" by auto
from arg_cong[OF 0, of "λ v. v $i", unfolded matrix_vector_mult_def] have "0 = (∑ k ∈ UNIV. A$h i $h k * x$h k)" by auto
also have "… = A $h i$h j * x $h j + (∑ k ∈ UNIV - {j}. A$h i $h k * x$h k)"
by (subst sum.remove[of _ j], auto)
also have "… > 0 + 0"
by (rule add_less_le_mono, insert nonneg[of i] Aij x(1) xj,
auto intro!: sum_nonneg mult_pos_pos simp: dual_order.not_eq_order_implies_strict)
finally show False by simp
qed

lemma r_witness: assumes x: "x ∈ X"
shows "∃ j. x $j > 0 ∧ r x = (A *v x)$ j / x $j" proof - from x[unfolded X_def] have x: "le_vec 0 x" "x ≠ 0" by auto let ?A = "{ (A *v x)$ j / x $j | j. x$ j ≠ 0 }"
from x(2) obtain j where "x $j ≠ 0" by (metis vec_eq_iff zero_index) hence empty: "?A ≠ {}" by auto from Min_in[OF _ this, folded r_def] obtain j where "x$ j ≠ 0" and rx: "r x = (A *v x) $j / x$ j" by auto
with x have "x $j > 0" by (auto simp: dual_order.not_eq_order_implies_strict) with rx show ?thesis by auto qed lemma rx_nonneg: assumes x: "x ∈ X" shows "r x ≥ 0" proof - from x[unfolded X_def] have x: "le_vec 0 x" "x ≠ 0" by auto let ?A = "{ (A *v x)$ j / x $j | j. x$ j ≠ 0 }"
from r_witness[OF ‹x ∈ X›]
have empty: "?A ≠ {}" by force
show ?thesis unfolding r_def X_def
proof (subst Min_ge_iff, force, use empty in force, intro ballI)
fix y
assume "y ∈ ?A"
then obtain j where "y = (A *v x) $j / x$ j" and "x $j ≠ 0" by auto from nonneg_Ax[OF ‹x ∈ X›] this x show "0 ≤ y" by simp qed qed lemma rx_pos: assumes x: "lt_vec 0 x" shows "r x > 0" proof - from Ax_pos[OF x] have lt: "lt_vec 0 (A *v x)" . from x have x': "x ∈ X" unfolding X_def order.strict_iff_order by auto let ?A = "{ (A *v x)$ j / x $j | j. x$ j ≠ 0 }"
from r_witness[OF ‹x ∈ X›]
have empty: "?A ≠ {}" by force
show ?thesis unfolding r_def X_def
proof (subst Min_gr_iff, force, use empty in force, intro ballI)
fix y
assume "y ∈ ?A"
then obtain j where "y = (A *v x) $j / x$ j" and "x $j ≠ 0" by auto from lt this x show "0 < y" by simp qed qed lemma rx_le_Ax: assumes x: "x ∈ X" shows "le_vec (r x *s x) (A *v x)" proof (intro allI) fix i show "(r x *s x)$h i ≤ (A *v x) $h i" proof (cases "x$ i = 0")
case True
with nonneg_Ax[OF x] show ?thesis by auto
next
case False
with x[unfolded X_def] have pos: "x $i > 0" by (auto simp: dual_order.not_eq_order_implies_strict) from False have "(A *v x)$h i / x $i ∈ { (A *v x)$ j / x $j | j. x$ j ≠ 0 }" by auto
hence "(A *v x) $h i / x$ i ≥ r x" unfolding r_def by simp
hence "x $i * r x ≤ x$ i * ((A *v x) $h i / x$ i)" unfolding mult_le_cancel_left_pos[OF pos] .
also have "… = (A *v x) $h i" using pos by simp finally show ?thesis by (simp add: ac_simps) qed qed lemma rho_le_x_Ax_imp_rho_le_rx: assumes x: "x ∈ X" and ρ: "le_vec (ρ *s x) (A *v x)" shows "ρ ≤ r x" proof - from r_witness[OF x] obtain j where rx: "r x = (A *v x)$ j / x $j" and xj: "x$ j > 0" "x $j ≥ 0" by auto from divide_right_mono[OF ρ[rule_format, of j] xj(2)] show ?thesis unfolding rx using xj by simp qed lemma rx_Max: assumes x: "x ∈ X" shows "r x = Sup { ρ . le_vec (ρ *s x) (A *v x) }" (is "_ = Sup ?S") proof - have "r x ∈ ?S" using rx_le_Ax[OF x] by auto moreover { fix y assume "y ∈ ?S" hence y: "le_vec (y *s x) (A *v x)" by auto from rho_le_x_Ax_imp_rho_le_rx[OF x this] have "y ≤ r x" . } ultimately show ?thesis by (metis (mono_tags, lifting) cSup_eq_maximum) qed lemma r_smult: assumes x: "x ∈ X" and a: "a > 0" shows "r (a *s x) = r x" unfolding r_def by (rule arg_cong[of _ _ Min], unfold vector_smult_distrib, insert a, simp) definition "X1 = (X ∩ {x. norm x = 1})" lemma bounded_X1: "bounded X1" unfolding bounded_iff X1_def by auto lemma closed_X1: "closed X1" proof - have X1: "X1 = {x. le_vec 0 x ∧ norm x = 1}" unfolding X1_def X_def by auto show ?thesis unfolding X1 by (intro closed_Collect_conj closed_Collect_all closed_Collect_le closed_Collect_eq, auto intro: continuous_intros) qed lemma compact_X1: "compact X1" using bounded_X1 closed_X1 by (simp add: compact_eq_bounded_closed) definition "pow_A_1 x = A1n *v x" lemma continuous_pow_A_1: "continuous_on R pow_A_1" unfolding pow_A_1_def continuous_on by (auto intro: tendsto_intros) definition "Y = pow_A_1  X1" lemma compact_Y: "compact Y" unfolding Y_def using compact_X1 continuous_pow_A_1[of X1] by (metis compact_continuous_image) lemma Y_pos_main: assumes y: "y ∈ pow_A_1  X" shows "y$ i > 0"
proof -
from y obtain x where x: "x ∈ X" and y: "y = pow_A_1 x" unfolding Y_def X1_def by auto
from r_witness[OF x] obtain j where xj: "x $j > 0" by auto from x[unfolded X_def] have xi: "x$ i ≥ 0" for i by auto
have nonneg: "0 ≤ A1n $i$ k * x $k" for k using A1n_pos[of i k] xi[of k] by auto have "y$ i = (∑j∈UNIV. A1n $i$ j * x $j)" unfolding y pow_A_1_def matrix_vector_mult_def by simp also have "… = A1n$ i $j * x$ j + (∑j∈UNIV - {j}. A1n $i$ j * x $j)" by (subst sum.remove, auto) also have "… > 0 + 0" by (rule add_less_le_mono, insert xj A1n_pos nonneg, auto intro!: sum_nonneg mult_pos_pos simp: dual_order.not_eq_order_implies_strict) finally show ?thesis by simp qed lemma Y_pos: assumes y: "y ∈ Y" shows "y$ i > 0"
using Y_pos_main[of y i] y unfolding Y_def X1_def by auto

lemma Y_nonzero: assumes y: "y ∈ Y"
shows "y $i ≠ 0" using Y_pos[OF y, of i] by auto definition r' :: "real ^ 'n ⇒ real" where "r' x = Min (range (λ j. (A *v x)$ j / x $j))" lemma r'_r: assumes x: "x ∈ Y" shows "r' x = r x" unfolding r'_def r_def proof (rule arg_cong[of _ _ Min]) have "range (λj. (A *v x)$ j / x $j) ⊆ {(A *v x)$ j / x $j |j. x$ j ≠ 0}" (is "?L ⊆ ?R")
proof
fix y
assume "y ∈ ?L"
then obtain j where "y = (A *v x) $j / x$ j" by auto
with Y_pos[OF x, of j] show "y ∈ ?R" by auto
qed
moreover have "?L ⊇ ?R" by auto
ultimately show "?L = ?R" by blast
qed

lemma continuous_Y_r: "continuous_on Y r"
proof -
have *: "(∀ y ∈ Y. P y (r y)) = (∀ y ∈ Y. P y (r' y))" for P using r'_r by auto
have "continuous_on Y r = continuous_on Y r'"
by (rule continuous_on_cong[OF refl r'_r[symmetric]])
also have …
unfolding continuous_on r'_def using Y_nonzero
by (auto intro!: tendsto_Min tendsto_intros)
finally show ?thesis .
qed

lemma X1_nonempty: "X1 ≠ {}"
proof -
define x where "x = ((χ i. if i = undefined then 1 else 0) :: real ^ 'n)"
{
assume "x = 0"
from arg_cong[OF this, of "λ x. x $undefined"] have False unfolding x_def by auto } hence x: "x ≠ 0" by auto moreover have "le_vec 0 x" unfolding x_def by auto moreover have "norm x = 1" unfolding norm_vec_def L2_set_def by (auto, subst sum.remove[of _ undefined], auto simp: x_def) ultimately show ?thesis unfolding X1_def X_def by auto qed lemma Y_nonempty: "Y ≠ {}" unfolding Y_def using X1_nonempty by auto definition z where "z = (SOME z. z ∈ Y ∧ (∀ y ∈ Y. r y ≤ r z))" abbreviation "sr ≡ r z" lemma z: "z ∈ Y" and sr_max_Y: "⋀ y. y ∈ Y ⟹ r y ≤ sr" proof - let ?P = "λ z. z ∈ Y ∧ (∀ y ∈ Y. r y ≤ r z)" from continuous_attains_sup[OF compact_Y Y_nonempty continuous_Y_r] obtain y where "?P y" by blast from someI[of ?P, OF this, folded z_def] show "z ∈ Y" "⋀ y. y ∈ Y ⟹ r y ≤ r z" by blast+ qed lemma Y_subset_X: "Y ⊆ X" proof fix y assume "y ∈ Y" from Y_pos[OF this] show "y ∈ X" unfolding X_def by (auto simp: order.strict_iff_order) qed lemma zX: "z ∈ X" using z(1) Y_subset_X by auto lemma le_vec_mono_left: assumes B: "⋀ i j. B$ i $j ≥ 0" and "le_vec x y" shows "le_vec (B *v x) (B *v y)" proof (intro allI) fix i show "(B *v x)$ i ≤ (B *v y) $i" unfolding matrix_vector_mult_def using B[of i] assms(2) by (auto intro!: sum_mono mult_left_mono) qed lemma matpow_1_commute: "matpow (A + mat 1) n ** A = A ** matpow (A + mat 1) n" by (induct n, auto simp: matrix_add_rdistrib matrix_add_ldistrib matrix_mul_rid matrix_mul_lid matrix_mul_assoc[symmetric]) lemma A1n_commute: "A1n ** A = A ** A1n" unfolding A1n_def by (rule matpow_1_commute) lemma le_vec_pow_A_1: assumes le: "le_vec (rho *s x) (A *v x)" shows "le_vec (rho *s pow_A_1 x) (A *v pow_A_1 x)" proof - have "A1n$ i $j ≥ 0" for i j using A1n_pos[of i j] by auto from le_vec_mono_left[OF this le] have "le_vec (A1n *v (rho *s x)) (A1n *v (A *v x))" . also have "A1n *v (A *v x) = (A1n ** A) *v x" by (simp add: matrix_vector_mul_assoc) also have "… = A *v (A1n *v x)" unfolding A1n_commute by (simp add: matrix_vector_mul_assoc) also have "… = A *v (pow_A_1 x)" unfolding pow_A_1_def .. also have "A1n *v (rho *s x) = rho *s (A1n *v x)" unfolding vector_smult_distrib .. also have "… = rho *s pow_A_1 x" unfolding pow_A_1_def .. finally show "le_vec (rho *s pow_A_1 x) (A *v pow_A_1 x)" . qed lemma r_pow_A_1: assumes x: "x ∈ X" shows "r x ≤ r (pow_A_1 x)" proof - let ?y = "pow_A_1 x" have "?y ∈ pow_A_1  X" using x by auto from Y_pos_main[OF this] have y: "?y ∈ X" unfolding X_def by (auto simp: order.strict_iff_order) let ?A = "{ρ. le_vec (ρ *s x) (A *v x)}" let ?B = "{ρ. le_vec (ρ *s pow_A_1 x) (A *v pow_A_1 x)}" show ?thesis unfolding rx_Max[OF x] rx_Max[OF y] proof (rule cSup_mono) show "bdd_above ?B" using rho_le_x_Ax_imp_rho_le_rx[OF y] by fast show "?A ≠ {}" using rx_le_Ax[OF x] by auto fix rho assume "rho ∈ ?A" hence "le_vec (rho *s x) (A *v x)" by auto from le_vec_pow_A_1[OF this] have "rho ∈ ?B" by auto thus "∃ rho' ∈ ?B. rho ≤ rho'" by auto qed qed lemma sr_max: assumes x: "x ∈ X" shows "r x ≤ sr" proof - let ?n = "norm x" define x' where "x' = inverse ?n *s x" from x[unfolded X_def] have x0: "x ≠ 0" by auto hence n: "?n > 0" by auto have x': "x' ∈ X1" "x' ∈ X" using x n unfolding X1_def X_def x'_def by (auto simp: norm_smult) have id: "r x = r x'" unfolding x'_def by (rule sym, rule r_smult[OF x], insert n, auto) define y where "y = pow_A_1 x'" from x' have y: "y ∈ Y" unfolding Y_def y_def by auto note id also have "r x' ≤ r y" using r_pow_A_1[OF x'(2)] unfolding y_def . also have "… ≤ r z" using sr_max_Y[OF y] . finally show "r x ≤ r z" . qed lemma z_pos: "z$ i > 0"
using Y_pos[OF z(1)] by auto

lemma sr_pos: "sr > 0"
by (rule rx_pos, insert z_pos, auto)

context fixes u
assumes u: "u ∈ X" and ru: "r u = sr"
begin

lemma sr_imp_eigen_vector_main: "sr *s u = A *v u"
proof (rule ccontr)
assume *: "sr *s u ≠ A *v u"
let ?x = "A *v u - sr *s u"
from * have 0: "?x ≠ 0" by auto
let ?y = "pow_A_1 u"
have "le_vec (sr *s u) (A *v u)" using rx_le_Ax[OF u] unfolding ru .
hence le: "le_vec 0 ?x" by auto
from 0 le have x: "?x ∈ X" unfolding X_def by auto
have y_pos: "lt_vec 0 ?y" using Y_pos_main[of ?y] u by auto
hence y: "?y ∈ X" unfolding X_def by (auto simp: order.strict_iff_order)
from Y_pos_main[of "pow_A_1 ?x"] x
have "lt_vec 0 (pow_A_1 ?x)" by auto
hence lt: "lt_vec (sr *s ?y) (A *v ?y)" unfolding pow_A_1_def matrix_vector_right_distrib_diff
matrix_vector_mul_assoc A1n_commute vector_smult_distrib by simp
let ?f = "(λ i. (A *v ?y - sr *s ?y) $i / ?y$ i)"
let ?U = "UNIV :: 'n set"
define eps where "eps = Min (?f  ?U)"
have U: "finite (?f  ?U)" "?f  ?U ≠ {}" by auto
have eps: "eps > 0" unfolding eps_def Min_gr_iff[OF U]
using lt sr_pos y_pos by auto
have le: "le_vec ((sr + eps) *s ?y) (A *v ?y)"
proof
fix i
have "((sr + eps) *s ?y) $i = sr * ?y$ i + eps * ?y $i" by (simp add: comm_semiring_class.distrib) also have "… ≤ sr * ?y$ i + ?f i * ?y $i" proof (rule add_left_mono[OF mult_right_mono]) show "0 ≤ ?y$ i" using y_pos[rule_format, of i] by auto
show "eps ≤ ?f i" unfolding eps_def by (rule Min_le, auto)
qed
also have "… = (A *v ?y) $i" using sr_pos y_pos[rule_format, of i] by simp finally show "((sr + eps) *s ?y)$ i ≤ (A *v ?y) $i" . qed from rho_le_x_Ax_imp_rho_le_rx[OF y le] have "r ?y ≥ sr + eps" . with sr_max[OF y] eps show False by auto qed lemma sr_imp_eigen_vector: "eigen_vector A u sr" unfolding eigen_vector_def sr_imp_eigen_vector_main using u unfolding X_def by auto lemma sr_u_pos: "lt_vec 0 u" proof - let ?y = "pow_A_1 u" define n where "n = N" define c where "c = (sr + 1)^N" have c: "c > 0" using sr_pos unfolding c_def by auto have "lt_vec 0 ?y" using Y_pos_main[of ?y] u by auto also have "?y = A1n *v u" unfolding pow_A_1_def .. also have "… = c *s u" unfolding c_def A1n_def n_def[symmetric] proof (induct n) case (Suc n) then show ?case by (simp add: matrix_vector_mul_assoc[symmetric] algebra_simps vec.scale sr_imp_eigen_vector_main[symmetric]) qed auto finally have lt: "lt_vec 0 (c *s u)" . have "0 < u$ i" for i using lt[rule_format, of i] c by simp (metis zero_less_mult_pos)
thus "lt_vec 0 u" by simp
qed
end

lemma eigen_vector_z_sr: "eigen_vector A z sr"
using sr_imp_eigen_vector[OF zX refl] by auto

lemma eigen_value_sr: "eigen_value A sr"
using eigen_vector_z_sr unfolding eigen_value_def by auto

abbreviation "c ≡ complex_of_real"
abbreviation "cA ≡ map_matrix c A"
abbreviation "norm_v ≡ map_vector (norm :: complex ⇒ real)"

lemma norm_v_ge_0: "le_vec 0 (norm_v v)" by (auto simp: map_vector_def)
lemma norm_v_eq_0: "norm_v v = 0 ⟷ v = 0" by (auto simp: map_vector_def vec_eq_iff)

lemma cA_index: "cA $i$ j = c (A $i$ j)"
unfolding map_matrix_def map_vector_def by simp

lemma norm_cA[simp]: "norm (cA $i$ j) = A $i$ j"
using nonneg[of i j] by (simp add: cA_index)

context fixes α v
assumes ev: "eigen_vector cA v α"
begin

lemma evD: "α *s v = cA *v v" "v ≠ 0"
using ev[unfolded eigen_vector_def] by auto

lemma ev_alpha_norm_v: "norm_v (α *s v) = (norm α *s norm_v v)"
by (auto simp: map_vector_def norm_mult vec_eq_iff)

lemma ev_A_norm_v: "norm_v (cA *v v) $j ≤ (A *v norm_v v)$ j"
proof -
have "norm_v (cA *v v) $j = norm (∑i∈UNIV. cA$ j $i * v$ i)"
unfolding map_vector_def by (simp add: matrix_vector_mult_def)
also have "… ≤ (∑i∈UNIV. norm (cA $j$ i * v $i))" by (rule norm_sum) also have "… = (∑i∈UNIV. A$ j $i * norm_v v$ i)"
by (rule sum.cong[OF refl], auto simp: norm_mult map_vector_def)
also have "… = (A *v norm_v v) $j" by (simp add: matrix_vector_mult_def) finally show ?thesis . qed lemma ev_le_vec: "le_vec (norm α *s norm_v v) (A *v norm_v v)" using arg_cong[OF evD(1), of norm_v, unfolded ev_alpha_norm_v] ev_A_norm_v by auto lemma norm_v_X: "norm_v v ∈ X" using norm_v_ge_0[of v] evD(2) norm_v_eq_0[of v] unfolding X_def by auto lemma ev_inequalities: "norm α ≤ r (norm_v v)" "r (norm_v v) ≤ sr" proof - have v: "norm_v v ∈ X" by (rule norm_v_X) from rho_le_x_Ax_imp_rho_le_rx[OF v ev_le_vec] show "norm α ≤ r (norm_v v)" . from sr_max[OF v] show "r (norm_v v) ≤ sr" . qed lemma eigen_vector_norm_sr: "norm α ≤ sr" using ev_inequalities by auto end lemma eigen_value_norm_sr: assumes "eigen_value cA α" shows "norm α ≤ sr" using eigen_vector_norm_sr[of _ α] assms unfolding eigen_value_def by auto lemma le_vec_trans: "le_vec x y ⟹ le_vec y u ⟹ le_vec x u" using order.trans[of "x$ i" "y $i" "u$ i" for i] by auto

lemma eigen_vector_z_sr_c: "eigen_vector cA (map_vector c z) (c sr)"
unfolding of_real_hom.eigen_vector_hom by (rule eigen_vector_z_sr)

lemma eigen_value_sr_c: "eigen_value cA (c sr)"
using eigen_vector_z_sr_c unfolding eigen_value_def by auto

definition "w = perron_frobenius.z (transpose A)"

lemma w: "transpose A *v w = sr *s w" "lt_vec 0 w" "perron_frobenius.sr (transpose A) = sr"
proof -
interpret t: perron_frobenius "transpose A"
by (rule pf_transpose)
from eigen_vector_z_sr_c t.eigen_vector_z_sr_c
have ev: "eigen_value cA (c sr)" "eigen_value t.cA (c t.sr)"
unfolding eigen_value_def by auto
{
fix x
have "eigen_value (t.cA) x = eigen_value (transpose cA) x"
unfolding map_matrix_def map_vector_def transpose_def
by (auto simp: vec_eq_iff)
also have "… = eigen_value cA x" by (rule eigen_value_transpose)
finally have "eigen_value (t.cA) x = eigen_value cA x" .
} note ev_id = this
with ev have ev: "eigen_value t.cA (c sr)" "eigen_value cA (c t.sr)" by auto
from eigen_value_norm_sr[OF ev(2)] t.eigen_value_norm_sr[OF ev(1)]
show id: "t.sr = sr" by auto
from t.eigen_vector_z_sr[unfolded id, folded w_def] show "transpose A *v w = sr *s w"
unfolding eigen_vector_def by auto
from t.z_pos[folded w_def] show "lt_vec 0 w" by auto
qed

lemma c_cmod_id: "a ∈ ℝ ⟹ Re a ≥ 0 ⟹ c (cmod a) = a" by (auto simp: Reals_def)

lemma pos_rowvector_mult_0: assumes lt: "lt_vec 0 x"
and 0: "(rowvector x :: real ^ 'n ^ 'n) *v y = 0" (is "?x *v _ = 0") and le: "le_vec 0 y"
shows "y = 0"
proof -
{
fix i
assume "y $i ≠ 0" with le have yi: "y$ i > 0" by (auto simp: order.strict_iff_order)
have "0 = (?x *v y) $i" unfolding 0 by simp also have "… = (∑j∈UNIV. x$ j * y $j)" unfolding rowvector_def matrix_vector_mult_def by simp also have "… > 0" by (rule sum_pos2[of _ i], insert yi lt le, auto intro!: mult_nonneg_nonneg simp: order.strict_iff_order) finally have False by simp } thus ?thesis by (auto simp: vec_eq_iff) qed lemma pos_matrix_mult_0: assumes le: "⋀ i j. B$ i $j ≥ 0" and lt: "lt_vec 0 x" and 0: "B *v x = 0" shows "B = 0" proof - { fix i j assume "B$ i $j ≠ 0" with le have gt: "B$ i $j > 0" by (auto simp: order.strict_iff_order) have "0 = (B *v x)$ i" unfolding 0 by simp
also have "… = (∑j∈UNIV. B $i$ j * x $j)" unfolding matrix_vector_mult_def by simp also have "… > 0" by (rule sum_pos2[of _ j], insert gt lt le, auto intro!: mult_nonneg_nonneg simp: order.strict_iff_order) finally have False by simp } thus "B = 0" unfolding vec_eq_iff by auto qed lemma eigen_value_smaller_matrix: assumes B: "⋀ i j. 0 ≤ B$ i $j ∧ B$ i $j ≤ A$ i $j" and AB: "A ≠ B" and ev: "eigen_value (map_matrix c B) sigma" shows "cmod sigma < sr" proof - let ?B = "map_matrix c B" let ?sr = "spectral_radius ?B" define σ where "σ = ?sr" have "real_non_neg_mat ?B" unfolding real_non_neg_mat_def elements_mat_h_def by (auto simp: map_matrix_def map_vector_def B) from perron_frobenius[OF this, folded σ_def] obtain x where ev_sr: "eigen_vector ?B x (c σ)" and rnn: "real_non_neg_vec x" by auto define y where "y = norm_v x" from rnn have xy: "x = map_vector c y" unfolding real_non_neg_vec_def vec_elements_h_def y_def by (auto simp: map_vector_def vec_eq_iff c_cmod_id) from spectral_radius_max[OF ev, folded σ_def] have sigma_sigma: "cmod sigma ≤ σ" . from ev_sr[unfolded xy of_real_hom.eigen_vector_hom] have ev_B: "eigen_vector B y σ" . from ev_B[unfolded eigen_vector_def] have ev_B': "B *v y = σ *s y" by auto have ypos: "y$ i ≥ 0" for i unfolding y_def by (auto simp: map_vector_def)
from ev_B this have y: "y ∈ X" unfolding eigen_vector_def X_def by auto

have BA: "(B *v y) $i ≤ (A *v y)$ i" for i
unfolding matrix_vector_mult_def vec_lambda_beta
by (rule sum_mono, rule mult_right_mono, insert B ypos, auto)
hence le_vec: "le_vec (σ *s y) (A *v y)" unfolding ev_B' by auto
from rho_le_x_Ax_imp_rho_le_rx[OF y le_vec]
have "σ ≤ r y" by auto
also have "… ≤ sr" using y by (rule sr_max)
finally have sig_le_sr: "σ ≤ sr" .
{
assume "σ = sr"
hence r_sr: "r y = sr" and sr_sig: "sr = σ" using ‹σ ≤ r y› ‹r y ≤ sr› by auto
from sr_u_pos[OF y r_sr] have pos: "lt_vec 0 y" .
from sr_imp_eigen_vector[OF y r_sr] have ev': "eigen_vector A y sr" .
have "(A - B) *v y = A *v y - B *v y" unfolding matrix_vector_mult_def
by (auto simp: vec_eq_iff field_simps sum_subtractf)
also have "A *v y = sr *s y" using ev'[unfolded eigen_vector_def] by auto
also have "B *v y = sr *s y" unfolding ev_B' sr_sig ..
finally have id: "(A - B) *v y = 0" by simp
from pos_matrix_mult_0[OF _ pos id] assms(1-2) have False by auto
}
with sig_le_sr sigma_sigma show ?thesis by argo
qed

lemma charpoly_erase_mat_sr: "0 < poly (charpoly (erase_mat A i i)) sr"
proof -
let ?A = "erase_mat A i i"
let ?pos = "poly (charpoly ?A) sr"
{
from A_nonzero_fixed_j[of i] obtain k where "A $k$ i ≠ 0" by auto
assume "A = ?A"
hence "A $k$ i = ?A $k$ i" by simp
also have "?A $k$ i = 0" by (auto simp: erase_mat_def)
also have "A $k$ i ≠ 0" by fact
finally have False by simp
}
hence AA: "A ≠ ?A" by auto
have le: "0 ≤ ?A $i$ j ∧ ?A $i$ j ≤ A $i$ j" for i j
by (auto simp: erase_mat_def nonneg)
note ev_small = eigen_value_smaller_matrix[OF le AA]
{
fix rho :: real
assume "eigen_value ?A rho"
hence ev: "eigen_value (map_matrix c ?A) (c rho)"
unfolding eigen_value_def using of_real_hom.eigen_vector_hom[of ?A _ rho] by auto
from ev_small[OF this] have "abs rho < sr" by auto
} note ev_small_real = this
have pos0: "?pos ≠ 0"
using ev_small_real[of sr] by (auto simp: eigen_value_root_charpoly)
{
define p where "p = charpoly ?A"
assume pos: "?pos < 0"
hence neg: "poly p sr < 0" unfolding p_def by auto
from degree_monic_charpoly[of ?A] have mon: "monic p" and deg: "degree p ≠ 0" unfolding p_def by auto
let ?f = "poly p"
have cont: "continuous_on {a..b} ?f" for a b by (auto intro: continuous_intros)
from pos have le: "?f sr ≤ 0" by (auto simp: p_def)
from mon have lc: "lead_coeff p > 0" by auto
from poly_pinfty_ge[OF this deg, of 0] obtain z where lez: "⋀ x. z ≤ x ⟹ 0 ≤ ?f x" by auto
define y where "y = max z sr"
have yr: "y ≥ sr" and "y ≥ z" unfolding y_def by auto
from lez[OF this(2)] have y0: "?f y ≥ 0" .
from IVT'[of ?f, OF le y0 yr cont] obtain x where ge: "x ≥ sr" and rt: "?f x = 0"
unfolding p_def by auto
hence "eigen_value ?A x" unfolding p_def by (simp add: eigen_value_root_charpoly)
from ev_small_real[OF this] ge have False by auto
}
with pos0 show ?thesis by argo
qed

lemma multiplicity_sr_1: "order sr (charpoly A) = 1"
proof -
{
assume "poly (pderiv (charpoly A)) sr = 0"
hence "0 = poly (monom 1 1 * pderiv (charpoly A)) sr" by simp
also have "… = sum (λ i. poly (charpoly (erase_mat A i i)) sr) UNIV"
unfolding pderiv_char_poly_erase_mat poly_sum ..
also have "… > 0"
by (rule sum_pos, (force simp: charpoly_erase_mat_sr)+)
finally have False by simp
}
hence nZ: "poly (pderiv (charpoly A)) sr ≠ 0" and nZ': "pderiv (charpoly A) ≠ 0" by auto
from eigen_vector_z_sr have "eigen_value A sr" unfolding eigen_value_def ..
from this[unfolded eigen_value_root_charpoly]
have "poly (charpoly A) sr = 0" .
hence "order sr (charpoly A) ≠ 0" unfolding order_root using nZ' by auto
from order_pderiv[OF nZ' this] order_0I[OF nZ]
show ?thesis by simp
qed

proof -
from eigen_vector_z_sr_c have "eigen_value cA (c sr)"
unfolding eigen_value_def by auto
have sr: "sr ≤ spectral_radius cA" by auto
show ?thesis by force
qed

lemma le_vec_A_mu: assumes y: "y ∈ X" and le: "le_vec (A *v y) (mu *s y)"
shows "sr ≤ mu" "lt_vec 0 y"
"mu = sr ∨ A *v y = mu *s y ⟹ mu = sr ∧ A *v y = mu *s y"
proof -
let ?w = "rowvector w"
let ?w' = "columnvector w"
have "?w ** A = transpose (transpose (?w ** A))"
unfolding transpose_transpose by simp
also have "transpose (?w ** A) = transpose A ** transpose ?w"
by (rule matrix_transpose_mul)
also have "transpose ?w = columnvector w" by (rule transpose_rowvector)
also have "transpose A ** … = columnvector (transpose A *v w)"
unfolding dot_rowvector_columnvector[symmetric] ..
also have "transpose A *v w = sr *s w" unfolding w by simp
also have "transpose (columnvector …) = rowvector (sr *s w)"
unfolding transpose_def columnvector_def rowvector_def vector_scalar_mult_def by auto
finally have 1: "?w ** A = rowvector (sr *s w)" .
have "sr *s (?w *v y) = ?w ** A *v y" unfolding 1
by (auto simp: rowvector_def vector_scalar_mult_def matrix_vector_mult_def vec_eq_iff
sum_distrib_left mult.assoc)
also have "… = ?w *v (A *v y)" by (simp add: matrix_vector_mul_assoc)
finally have eq1: "sr *s (rowvector w *v y) = rowvector w *v (A *v y)" .
have "le_vec (rowvector w *v (A *v y)) (?w *v (mu *s y))"
by (rule le_vec_mono_left[OF _ le], insert w(2), auto simp: rowvector_def order.strict_iff_order)
also have "?w *v (mu *s y) = mu *s (?w *v y)" by (simp add: algebra_simps vec.scale)
finally have le1: "le_vec (rowvector w *v (A *v y)) (mu *s (?w *v y))" .
from le1[unfolded eq1[symmetric]]
have 2: "le_vec (sr *s (?w *v y)) (mu *s (?w *v y))" .
{
from y obtain i where yi: "y $i > 0" and y: "⋀ j. y$ j ≥ 0" unfolding X_def
by (auto simp: order.strict_iff_order vec_eq_iff)
from w(2) have wi: "w $i > 0" and w: "⋀ j. w$ j ≥ 0"
by (auto simp: order.strict_iff_order)
have "(?w *v y) $i > 0" using yi y wi w by (auto simp: matrix_vector_mult_def rowvector_def intro!: sum_pos2[of _ i] mult_nonneg_nonneg) moreover from 2[rule_format, of i] have "sr * (?w *v y)$ i ≤ mu * (?w *v y) $i" by simp ultimately have "sr ≤ mu" by simp } thus *: "sr ≤ mu" . define cc where "cc = (mu + 1)^ N" define n where "n = N" from * sr_pos have mu: "mu ≥ 0" "mu > 0" by auto hence cc: "cc > 0" unfolding cc_def by simp from y have "pow_A_1 y ∈ pow_A_1  X" by auto from Y_pos_main[OF this] have lt: "0 < (A1n *v y)$ i" for i by (simp add: pow_A_1_def)
have le: "le_vec (A1n *v y) (cc *s y)" unfolding cc_def A1n_def n_def[symmetric]
proof (induct n)
case (Suc n)
let ?An = "matpow (A + mat 1) n"
let ?mu = "(mu + 1)"
have id': "matpow (A + mat 1) (Suc n) *v y = A *v (?An *v y) + ?An *v y" (is "?a = ?b + ?c")
matrix_vector_mul_assoc[symmetric])
have "le_vec ?b (?mu^n *s (A *v y))"
using le_vec_mono_left[OF nonneg Suc] by (simp add: algebra_simps vec.scale)
moreover have "le_vec (?mu^n *s (A *v y)) (?mu^n *s (mu *s y))"
using le mu by auto
moreover have id: "?mu^n *s (mu *s y) = (?mu^n * mu) *s y" by simp
from le_vec_trans[OF calculation[unfolded id]]
have le1: "le_vec ?b ((?mu^n * mu) *s y)" .
from Suc have le2: "le_vec ?c ((mu + 1) ^ n *s y)" .
have le: "le_vec ?a ((?mu^n * mu) *s y + ?mu^n *s y)"
unfolding id' using add_mono[OF le1[rule_format] le2[rule_format]] by auto
have id'': "(?mu^n * mu) *s y + ?mu^n *s y = ?mu^Suc n *s y" by (simp add: algebra_simps)
show ?case using le unfolding id'' .
have lt: "0 < cc * y $i" for i using lt[of i] le[rule_format, of i] by auto have "y$ i > 0" for i using lt[of i] cc by (rule zero_less_mult_pos)
thus "lt_vec 0 y" by auto
assume **: "mu = sr ∨ A *v y = mu *s y"
{
assume "A *v y = mu *s y"
with y have "eigen_vector A y mu" unfolding X_def eigen_vector_def by auto
hence "eigen_vector cA (map_vector c y) (c mu)" unfolding of_real_hom.eigen_vector_hom .
from eigen_vector_norm_sr[OF this] * have "mu = sr" by auto
}
with ** have mu_sr: "mu = sr" by auto
from eq1[folded vector_smult_distrib]
have 0: "?w *v (sr *s y - A *v y) = 0"
unfolding matrix_vector_right_distrib_diff by simp
have le0: "le_vec 0 (sr *s y - A *v y)" using assms(2)[unfolded mu_sr] by auto
have "sr *s y - A *v y = 0" using pos_rowvector_mult_0[OF w(2) 0 le0] .
hence ev_y: "A *v y = sr *s y" by auto
show "mu = sr ∧ A *v y = mu *s y" using ev_y mu_sr by auto
qed

lemma nonnegative_eigenvector_has_ev_sr: assumes "eigen_vector A v mu" and le: "le_vec 0 v"
shows "mu = sr"
proof -
from assms(1)[unfolded eigen_vector_def] have v: "v ≠ 0" and ev: "A *v v = mu *s v" by auto
from le v have v: "v ∈ X" unfolding X_def by auto
from ev have "le_vec (A *v v) (mu *s v)" by auto
from le_vec_A_mu[OF v this] ev show ?thesis by auto
qed

lemma similar_matrix_rotation: assumes ev: "eigen_value cA α" and α: "cmod α = sr"
shows "similar_matrix (cis (arg α) *k cA) cA"
proof -
from ev obtain y where ev: "eigen_vector cA y α" unfolding eigen_value_def by auto
let ?y = "norm_v y"
note maps = map_vector_def map_matrix_def
define yp where "yp = norm_v y"
let ?yp = "map_vector c yp"
have yp: "yp ∈ X" unfolding yp_def by (rule norm_v_X[OF ev])
from ev[unfolded eigen_vector_def] have ev_y: "cA *v y = α *s y" by auto
from ev_le_vec[OF ev, unfolded α, folded yp_def]
have 1: "le_vec (sr *s yp) (A *v yp)" by simp
from rho_le_x_Ax_imp_rho_le_rx[OF yp 1] have "sr ≤ r yp" by auto
with ev_inequalities[OF ev, folded yp_def]
have 2: "r yp = sr" by auto
have ev_yp: "A *v yp = sr *s yp"
and pos_yp: "lt_vec 0 yp"
using sr_imp_eigen_vector_main[OF yp 2] sr_u_pos[OF yp 2] by auto
define D where "D = diagvector (λ j. cis (arg (y $j)))" define inv_D where "inv_D = diagvector (λ j. cis (- arg (y$ j)))"
have DD: "inv_D ** D = mat 1" "D ** inv_D = mat 1" unfolding D_def inv_D_def
by (auto simp add: diagvector_eq_mat cis_mult)
{
fix i
have "(D *v ?yp) $i = cis (arg (y$ i)) * c (cmod (y $i))" unfolding D_def yp_def by (simp add: maps) also have "… = y$ i" by (simp add: cis_mult_cmod_id)
also note calculation
}
hence y_D_yp: "y = D *v ?yp" by (auto simp: vec_eq_iff)
define φ where "φ = arg α"
let ?φ = "cis (- φ)"
have [simp]: "cis (- φ) * rcis sr φ = sr" unfolding cis_rcis_eq rcis_mult by simp
have α: "α = rcis sr φ" unfolding φ_def α[symmetric] rcis_cmod_arg ..
define F where "F = ?φ *k (inv_D ** cA ** D)"
have "cA *v (D *v ?yp) = α *s y" unfolding y_D_yp[symmetric] ev_y by simp
also have "inv_D *v … = α *s ?yp"
unfolding vector_smult_distrib y_D_yp matrix_vector_mul_assoc DD matrix_vector_mul_lid ..
also have "?φ *s … = sr *s ?yp" unfolding α by simp
also have "… = map_vector c (sr *s yp)" unfolding vec_eq_iff by (auto simp: maps)
also have "… = cA *v ?yp" unfolding ev_yp[symmetric] by (auto simp: maps matrix_vector_mult_def)
finally have F: "F *v ?yp = cA *v ?yp" unfolding F_def matrix_scalar_vector_ac[symmetric]
unfolding matrix_vector_mul_assoc[symmetric] vector_smult_distrib .
have prod: "inv_D ** cA ** D = (χ i j. cis (- arg (y $i)) * cA$ i $j * cis (arg (y$ j)))"
unfolding inv_D_def D_def diagvector_mult_right diagvector_mult_left by simp
{
fix i j
have "cmod (F $i$ j) = cmod (?φ * cA $h i$h j * (cis (- arg (y $h i)) * cis (arg (y$h j))))"
unfolding F_def prod vec_lambda_beta matrix_scalar_mult_def
by (simp only: ac_simps)
also have "… = A $i$ j" unfolding cis_mult unfolding norm_mult by simp
also note calculation
}
hence FA: "map_matrix norm F = A" unfolding maps by auto
let ?F = "map_matrix c (map_matrix norm F)"
let ?G = "?F - F"
let ?Re = "map_matrix Re"
from F[folded FA] have 0: "?G *v ?yp = 0" unfolding matrix_diff_vect_distrib by simp
have "?Re ?G *v yp = map_vector Re (?G *v ?yp)"
unfolding maps matrix_vector_mult_def vec_lambda_beta Re_sum by auto
also have "… = 0" unfolding 0 by (simp add: vec_eq_iff maps)
finally have 0: "?Re ?G *v yp = 0" .
have "?Re ?G = 0"
by (rule pos_matrix_mult_0[OF _ pos_yp 0], auto simp: maps complex_Re_le_cmod)
hence "?F = F" by (auto simp: maps vec_eq_iff cmod_eq_Re)
with FA have AF: "cA = F" by simp
from arg_cong[OF this, of "λ A. cis φ *k A"]
have sim: "cis φ *k cA = inv_D ** cA ** D" unfolding F_def matrix.scale_scale cis_mult
by simp
have "similar_matrix (cis φ *k cA) cA" unfolding similar_matrix_def similar_matrix_wit_def
sim
by (rule exI[of _ inv_D