# Theory Infinite_Products

(*File:      HOL/Analysis/Infinite_Product.thy
Author:    Manuel Eberl & LC Paulson

Basic results about convergence and absolute convergence of infinite products
and their connection to summability.
*)
section ‹Infinite Products›
theory Infinite_Products
imports Topology_Euclidean_Space Complex_Transcendental
begin

subsection✐‹tag unimportant› ‹Preliminaries›

lemma sum_le_prod:
fixes f :: "'a ⇒ 'b :: linordered_semidom"
assumes "⋀x. x ∈ A ⟹ f x ≥ 0"
shows   "sum f A ≤ (∏x∈A. 1 + f x)"
using assms
proof (induction A rule: infinite_finite_induct)
case (insert x A)
from insert.hyps have "sum f A + f x * (∏x∈A. 1) ≤ (∏x∈A. 1 + f x) + f x * (∏x∈A. 1 + f x)"
by (intro add_mono insert mult_left_mono prod_mono) (auto intro: insert.prems)
with insert.hyps show ?case by (simp add: algebra_simps)
qed simp_all

lemma prod_le_exp_sum:
fixes f :: "'a ⇒ real"
assumes "⋀x. x ∈ A ⟹ f x ≥ 0"
shows   "prod (λx. 1 + f x) A ≤ exp (sum f A)"
using assms
proof (induction A rule: infinite_finite_induct)
case (insert x A)
have "(1 + f x) * (∏x∈A. 1 + f x) ≤ exp (f x) * exp (sum f A)"
using insert.prems by (intro mult_mono insert prod_nonneg exp_ge_add_one_self) auto
qed simp_all

lemma lim_ln_1_plus_x_over_x_at_0: "(λx::real. ln (1 + x) / x) ─0→ 1"
proof (rule lhopital)
show "(λx::real. ln (1 + x)) ─0→ 0"
by (rule tendsto_eq_intros refl | simp)+
have "eventually (λx::real. x ∈ {-1/2<..<1/2}) (nhds 0)"
by (rule eventually_nhds_in_open) auto
hence *: "eventually (λx::real. x ∈ {-1/2<..<1/2}) (at 0)"
by (rule filter_leD [rotated]) (simp_all add: at_within_def)
show "eventually (λx::real. ((λx. ln (1 + x)) has_field_derivative inverse (1 + x)) (at x)) (at 0)"
using * by eventually_elim (auto intro!: derivative_eq_intros simp: field_simps)
show "eventually (λx::real. ((λx. x) has_field_derivative 1) (at x)) (at 0)"
using * by eventually_elim (auto intro!: derivative_eq_intros simp: field_simps)
show "∀⇩F x in at 0. x ≠ 0" by (auto simp: at_within_def eventually_inf_principal)
show "(λx::real. inverse (1 + x) / 1) ─0→ 1"
by (rule tendsto_eq_intros refl | simp)+
qed auto

subsection‹Definitions and basic properties›

definition✐‹tag important› raw_has_prod :: "[nat ⇒ 'a::{t2_space, comm_semiring_1}, nat, 'a] ⇒ bool"
where "raw_has_prod f M p ≡ (λn. ∏i≤n. f (i+M)) ⇢ p ∧ p ≠ 0"

text‹The nonzero and zero cases, as in \emph{Complex Analysis} by Joseph Bak and Donald J.Newman, page 241›
text✐‹tag important› ‹%whitespace›
definition✐‹tag important›
has_prod :: "(nat ⇒ 'a::{t2_space, comm_semiring_1}) ⇒ 'a ⇒ bool" (infixr "has'_prod" 80)
where "f has_prod p ≡ raw_has_prod f 0 p ∨ (∃i q. p = 0 ∧ f i = 0 ∧ raw_has_prod f (Suc i) q)"

definition✐‹tag important› convergent_prod :: "(nat ⇒ 'a :: {t2_space,comm_semiring_1}) ⇒ bool" where
"convergent_prod f ≡ ∃M p. raw_has_prod f M p"

definition✐‹tag important› prodinf :: "(nat ⇒ 'a::{t2_space, comm_semiring_1}) ⇒ 'a"
(binder "∏" 10)
where "prodinf f = (THE p. f has_prod p)"

lemmas prod_defs = raw_has_prod_def has_prod_def convergent_prod_def prodinf_def

lemma has_prod_subst[trans]: "f = g ⟹ g has_prod z ⟹ f has_prod z"
by simp

lemma has_prod_cong: "(⋀n. f n = g n) ⟹ f has_prod c ⟷ g has_prod c"
by presburger

lemma raw_has_prod_nonzero [simp]: "¬ raw_has_prod f M 0"

lemma raw_has_prod_eq_0:
fixes f :: "nat ⇒ 'a::{semidom,t2_space}"
assumes p: "raw_has_prod f m p" and i: "f i = 0" "i ≥ m"
shows "p = 0"
proof -
have eq0: "(∏k≤n. f (k+m)) = 0" if "i - m ≤ n" for n
proof -
have "∃k≤n. f (k + m) = 0"
using i that by auto
then show ?thesis
by auto
qed
have "(λn. ∏i≤n. f (i + m)) ⇢ 0"
by (rule LIMSEQ_offset [where k = "i-m"]) (simp add: eq0)
with p show ?thesis
unfolding raw_has_prod_def
using LIMSEQ_unique by blast
qed

lemma raw_has_prod_Suc:
"raw_has_prod f (Suc M) a ⟷ raw_has_prod (λn. f (Suc n)) M a"
unfolding raw_has_prod_def by auto

lemma has_prod_0_iff: "f has_prod 0 ⟷ (∃i. f i = 0 ∧ (∃p. raw_has_prod f (Suc i) p))"

lemma has_prod_unique2:
fixes f :: "nat ⇒ 'a::{semidom,t2_space}"
assumes "f has_prod a" "f has_prod b" shows "a = b"
using assms
by (auto simp: has_prod_def raw_has_prod_eq_0) (meson raw_has_prod_def sequentially_bot tendsto_unique)

lemma has_prod_unique:
fixes f :: "nat ⇒ 'a :: {semidom,t2_space}"
shows "f has_prod s ⟹ s = prodinf f"
by (simp add: has_prod_unique2 prodinf_def the_equality)

lemma has_prod_eq_0_iff:
fixes f :: "nat ⇒ 'a :: {semidom, comm_semiring_1, t2_space}"
assumes "f has_prod P"
shows   "P = 0 ⟷ 0 ∈ range f"
proof
assume "0 ∈ range f"
then obtain N where N: "f N = 0"
by auto
have "eventually (λn. n > N) at_top"
by (rule eventually_gt_at_top)
hence "eventually (λn. (∏k<n. f k) = 0) at_top"
by eventually_elim (use N in auto)
hence "(λn. ∏k<n. f k) ⇢ 0"
moreover have "(λn. ∏k<n. f k) ⇢ P"
using assms by (metis N calculation prod_defs(2) raw_has_prod_eq_0 zero_le)
ultimately show "P = 0"
using tendsto_unique by force
qed (use assms in ‹auto simp: has_prod_def›)

lemma has_prod_0D:
fixes f :: "nat ⇒ 'a :: {semidom, comm_semiring_1, t2_space}"
shows "f has_prod 0 ⟹ 0 ∈ range f"
using has_prod_eq_0_iff[of f 0] by auto

lemma has_prod_zeroI:
fixes f :: "nat ⇒ 'a :: {semidom, comm_semiring_1, t2_space}"
assumes "f has_prod P" "f n = 0"
shows   "P = 0"
using assms by (auto simp: has_prod_eq_0_iff)

lemma raw_has_prod_in_Reals:
assumes "raw_has_prod (complex_of_real ∘ z) M p"
shows "p ∈ ℝ"
using assms by (auto simp: raw_has_prod_def real_lim_sequentially)

lemma raw_has_prod_of_real_iff: "raw_has_prod (complex_of_real ∘ z) M (of_real p) ⟷ raw_has_prod z M p"
by (auto simp: raw_has_prod_def tendsto_of_real_iff simp flip: of_real_prod)

lemma convergent_prod_of_real_iff: "convergent_prod (complex_of_real ∘ z) ⟷ convergent_prod z"
by (smt (verit, best) Reals_cases convergent_prod_def raw_has_prod_in_Reals raw_has_prod_of_real_iff)

lemma convergent_prod_altdef:
fixes f :: "nat ⇒ 'a :: {t2_space,comm_semiring_1}"
shows "convergent_prod f ⟷ (∃M L. (∀n≥M. f n ≠ 0) ∧ (λn. ∏i≤n. f (i+M)) ⇢ L ∧ L ≠ 0)"
proof
assume "convergent_prod f"
then obtain M L where *: "(λn. ∏i≤n. f (i+M)) ⇢ L" "L ≠ 0"
by (auto simp: prod_defs)
have "f i ≠ 0" if "i ≥ M" for i
proof
assume "f i = 0"
have **: "eventually (λn. (∏i≤n. f (i+M)) = 0) sequentially"
using eventually_ge_at_top[of "i - M"]
proof eventually_elim
case (elim n)
with ‹f i = 0› and ‹i ≥ M› show ?case
by (auto intro!: bexI[of _ "i - M"] prod_zero)
qed
have "(λn. (∏i≤n. f (i+M))) ⇢ 0"
unfolding filterlim_iff
by (auto dest!: eventually_nhds_x_imp_x intro!: eventually_mono[OF **])
from tendsto_unique[OF _ this *(1)] and *(2)
show False by simp
qed
with * show "(∃M L. (∀n≥M. f n ≠ 0) ∧ (λn. ∏i≤n. f (i+M)) ⇢ L ∧ L ≠ 0)"
by blast
qed (auto simp: prod_defs)

lemma raw_has_prod_norm:
fixes a :: "'a ::real_normed_field"
assumes "raw_has_prod f M a"
shows "raw_has_prod (λn. norm (f n)) M (norm a)"
using assms by (auto simp: raw_has_prod_def prod_norm tendsto_norm)

lemma has_prod_norm:
fixes a :: "'a ::real_normed_field"
assumes f: "f has_prod a"
shows "(λn. norm (f n)) has_prod (norm a)"
using f [unfolded has_prod_def]
proof (elim disjE exE conjE)
assume f0: "raw_has_prod f 0 a"
then show "(λn. norm (f n)) has_prod norm a"
using has_prod_def raw_has_prod_norm by blast
next
fix i p
assume "a = 0" and "f i = 0" and p: "raw_has_prod f (Suc i) p"
then have "Ex (raw_has_prod (λn. norm (f n)) (Suc i))"
using raw_has_prod_norm by blast
then show ?thesis
by (metis ‹a = 0› ‹f i = 0› has_prod_0_iff norm_zero)
qed

subsection‹Absolutely convergent products›

definition✐‹tag important› abs_convergent_prod :: "(nat ⇒ _) ⇒ bool" where
"abs_convergent_prod f ⟷ convergent_prod (λi. 1 + norm (f i - 1))"

lemma abs_convergent_prodI:
assumes "convergent (λn. ∏i≤n. 1 + norm (f i - 1))"
shows   "abs_convergent_prod f"
proof -
from assms obtain L where L: "(λn. ∏i≤n. 1 + norm (f i - 1)) ⇢ L"
by (auto simp: convergent_def)
have "L ≥ 1"
proof (rule tendsto_le)
show "eventually (λn. (∏i≤n. 1 + norm (f i - 1)) ≥ 1) sequentially"
proof (intro always_eventually allI)
fix n
have "(∏i≤n. 1 + norm (f i - 1)) ≥ (∏i≤n. 1)"
by (intro prod_mono) auto
thus "(∏i≤n. 1 + norm (f i - 1)) ≥ 1" by simp
qed
qed (use L in simp_all)
hence "L ≠ 0" by auto
with L show ?thesis unfolding abs_convergent_prod_def prod_defs
by (intro exI[of _ "0::nat"] exI[of _ L]) auto
qed

lemma
fixes f :: "nat ⇒ 'a :: {topological_semigroup_mult,t2_space,idom}"
assumes "convergent_prod f"
shows   convergent_prod_imp_convergent:     "convergent (λn. ∏i≤n. f i)"
and   convergent_prod_to_zero_iff [simp]: "(λn. ∏i≤n. f i) ⇢ 0  ⟷  (∃i. f i = 0)"
proof -
from assms obtain M L
where M: "⋀n. n ≥ M ⟹ f n ≠ 0" and "(λn. ∏i≤n. f (i + M)) ⇢ L" and "L ≠ 0"
by (auto simp: convergent_prod_altdef)
note this(2)
also have "(λn. ∏i≤n. f (i + M)) = (λn. ∏i=M..M+n. f i)"
by (intro ext prod.reindex_bij_witness[of _ "λn. n - M" "λn. n + M"]) auto
finally have "(λn. (∏i<M. f i) * (∏i=M..M+n. f i)) ⇢ (∏i<M. f i) * L"
by (intro tendsto_mult tendsto_const)
also have "(λn. (∏i<M. f i) * (∏i=M..M+n. f i)) = (λn. (∏i∈{..<M}∪{M..M+n}. f i))"
by (subst prod.union_disjoint) auto
also have "(λn. {..<M} ∪ {M..M+n}) = (λn. {..n+M})" by auto
finally have lim: "(λn. prod f {..n}) ⇢ prod f {..<M} * L"
by (rule LIMSEQ_offset)
thus "convergent (λn. ∏i≤n. f i)"
by (auto simp: convergent_def)

show "(λn. ∏i≤n. f i) ⇢ 0 ⟷ (∃i. f i = 0)"
proof
assume "∃i. f i = 0"
then obtain i where "f i = 0" by auto
moreover with M have "i < M" by (cases "i < M") auto
ultimately have "(∏i<M. f i) = 0" by auto
with lim show "(λn. ∏i≤n. f i) ⇢ 0" by simp
next
assume "(λn. ∏i≤n. f i) ⇢ 0"
from tendsto_unique[OF _ this lim] and ‹L ≠ 0›
show "∃i. f i = 0" by auto
qed
qed

lemma convergent_prod_iff_nz_lim:
fixes f :: "nat ⇒ 'a :: {topological_semigroup_mult,t2_space,idom}"
assumes "⋀i. f i ≠ 0"
shows "convergent_prod f ⟷ (∃L. (λn. ∏i≤n. f i) ⇢ L ∧ L ≠ 0)"
(is "?lhs ⟷ ?rhs")
proof
assume ?lhs then show ?rhs
using assms convergentD convergent_prod_imp_convergent convergent_prod_to_zero_iff by blast
next
assume ?rhs then show ?lhs
unfolding prod_defs
by (rule_tac x=0 in exI) auto
qed

lemma✐‹tag important› convergent_prod_iff_convergent:
fixes f :: "nat ⇒ 'a :: {topological_semigroup_mult,t2_space,idom}"
assumes "⋀i. f i ≠ 0"
shows "convergent_prod f ⟷ convergent (λn. ∏i≤n. f i) ∧ lim (λn. ∏i≤n. f i) ≠ 0"
by (force simp: convergent_prod_iff_nz_lim assms convergent_def limI)

lemma bounded_imp_convergent_prod:
fixes a :: "nat ⇒ real"
assumes 1: "⋀n. a n ≥ 1" and bounded: "⋀n. (∏i≤n. a i) ≤ B"
shows "convergent_prod a"
proof -
have "bdd_above (range(λn. ∏i≤n. a i))"
by (meson bdd_aboveI2 bounded)
moreover have "incseq (λn. ∏i≤n. a i)"
unfolding mono_def by (metis 1 prod_mono2 atMost_subset_iff dual_order.trans finite_atMost zero_le_one)
ultimately obtain p where p: "(λn. ∏i≤n. a i) ⇢ p"
using LIMSEQ_incseq_SUP by blast
then have "p ≠ 0"
by (metis "1" not_one_le_zero prod_ge_1 LIMSEQ_le_const)
with 1 p show ?thesis
by (metis convergent_prod_iff_nz_lim not_one_le_zero)
qed

lemma abs_convergent_prod_altdef:
fixes f :: "nat ⇒ 'a :: {one,real_normed_vector}"
shows  "abs_convergent_prod f ⟷ convergent (λn. ∏i≤n. 1 + norm (f i - 1))"
proof
assume "abs_convergent_prod f"
thus "convergent (λn. ∏i≤n. 1 + norm (f i - 1))"
by (auto simp: abs_convergent_prod_def intro!: convergent_prod_imp_convergent)
qed (auto intro: abs_convergent_prodI)

lemma Weierstrass_prod_ineq:
fixes f :: "'a ⇒ real"
assumes "⋀x. x ∈ A ⟹ f x ∈ {0..1}"
shows   "1 - sum f A ≤ (∏x∈A. 1 - f x)"
using assms
proof (induction A rule: infinite_finite_induct)
case (insert x A)
from insert.hyps and insert.prems
have "1 - sum f A + f x * (∏x∈A. 1 - f x) ≤ (∏x∈A. 1 - f x) + f x * (∏x∈A. 1)"
by (intro insert.IH add_mono mult_left_mono prod_mono) auto
with insert.hyps show ?case by (simp add: algebra_simps)
qed simp_all

lemma norm_prod_minus1_le_prod_minus1:
fixes f :: "nat ⇒ 'a :: {real_normed_div_algebra,comm_ring_1}"
shows "norm (prod (λn. 1 + f n) A - 1) ≤ prod (λn. 1 + norm (f n)) A - 1"
proof (induction A rule: infinite_finite_induct)
case (insert x A)
from insert.hyps have
"norm ((∏n∈insert x A. 1 + f n) - 1) =
norm ((∏n∈A. 1 + f n) - 1 + f x * (∏n∈A. 1 + f n))"
also have "… ≤ norm ((∏n∈A. 1 + f n) - 1) + norm (f x * (∏n∈A. 1 + f n))"
by (rule norm_triangle_ineq)
also have "norm (f x * (∏n∈A. 1 + f n)) = norm (f x) * (∏x∈A. norm (1 + f x))"
also have "(∏x∈A. norm (1 + f x)) ≤ (∏x∈A. norm (1::'a) + norm (f x))"
by (intro prod_mono norm_triangle_ineq ballI conjI) auto
also have "norm (1::'a) = 1" by simp
also note insert.IH
also have "(∏n∈A. 1 + norm (f n)) - 1 + norm (f x) * (∏x∈A. 1 + norm (f x)) =
(∏n∈insert x A. 1 + norm (f n)) - 1"
using insert.hyps by (simp add: algebra_simps)
finally show ?case by - (simp_all add: mult_left_mono)
qed simp_all

lemma convergent_prod_imp_ev_nonzero:
fixes f :: "nat ⇒ 'a :: {t2_space,comm_semiring_1}"
assumes "convergent_prod f"
shows   "eventually (λn. f n ≠ 0) sequentially"
using assms by (auto simp: eventually_at_top_linorder convergent_prod_altdef)

lemma convergent_prod_imp_LIMSEQ:
fixes f :: "nat ⇒ 'a :: {real_normed_field}"
assumes "convergent_prod f"
shows   "f ⇢ 1"
proof -
from assms obtain M L where L: "(λn. ∏i≤n. f (i+M)) ⇢ L" "⋀n. n ≥ M ⟹ f n ≠ 0" "L ≠ 0"
by (auto simp: convergent_prod_altdef)
hence L': "(λn. ∏i≤Suc n. f (i+M)) ⇢ L" by (subst filterlim_sequentially_Suc)
have "(λn. (∏i≤Suc n. f (i+M)) / (∏i≤n. f (i+M))) ⇢ L / L"
using L L' by (intro tendsto_divide) simp_all
also from L have "L / L = 1" by simp
also have "(λn. (∏i≤Suc n. f (i+M)) / (∏i≤n. f (i+M))) = (λn. f (n + Suc M))"
using assms L by (auto simp: fun_eq_iff atMost_Suc)
finally show ?thesis by (rule LIMSEQ_offset)
qed

lemma abs_convergent_prod_imp_summable:
fixes f :: "nat ⇒ 'a :: real_normed_div_algebra"
assumes "abs_convergent_prod f"
shows "summable (λi. norm (f i - 1))"
proof -
from assms have "convergent (λn. ∏i≤n. 1 + norm (f i - 1))"
unfolding abs_convergent_prod_def by (rule convergent_prod_imp_convergent)
then obtain L where L: "(λn. ∏i≤n. 1 + norm (f i - 1)) ⇢ L"
unfolding convergent_def by blast
have "convergent (λn. ∑i≤n. norm (f i - 1))"
proof (rule Bseq_monoseq_convergent)
have "eventually (λn. (∏i≤n. 1 + norm (f i - 1)) < L + 1) sequentially"
using L(1) by (rule order_tendstoD) simp_all
hence "∀⇩F x in sequentially. norm (∑i≤x. norm (f i - 1)) ≤ L + 1"
proof eventually_elim
case (elim n)
have "norm (∑i≤n. norm (f i - 1)) = (∑i≤n. norm (f i - 1))"
unfolding real_norm_def by (intro abs_of_nonneg sum_nonneg) simp_all
also have "… ≤ (∏i≤n. 1 + norm (f i - 1))" by (rule sum_le_prod) auto
also have "… < L + 1" by (rule elim)
finally show ?case by simp
qed
thus "Bseq (λn. ∑i≤n. norm (f i - 1))" by (rule BfunI)
next
show "monoseq (λn. ∑i≤n. norm (f i - 1))"
by (rule mono_SucI1) auto
qed
thus "summable (λi. norm (f i - 1))" by (simp add: summable_iff_convergent')
qed

lemma summable_imp_abs_convergent_prod:
fixes f :: "nat ⇒ 'a :: real_normed_div_algebra"
assumes "summable (λi. norm (f i - 1))"
shows   "abs_convergent_prod f"
proof (intro abs_convergent_prodI Bseq_monoseq_convergent)
show "monoseq (λn. ∏i≤n. 1 + norm (f i - 1))"
by (intro mono_SucI1)
(auto simp: atMost_Suc algebra_simps intro!: mult_nonneg_nonneg prod_nonneg)
next
show "Bseq (λn. ∏i≤n. 1 + norm (f i - 1))"
proof (rule Bseq_eventually_mono)
show "eventually (λn. norm (∏i≤n. 1 + norm (f i - 1)) ≤
norm (exp (∑i≤n. norm (f i - 1)))) sequentially"
by (intro always_eventually allI) (auto simp: abs_prod exp_sum intro!: prod_mono)
next
from assms have "(λn. ∑i≤n. norm (f i - 1)) ⇢ (∑i. norm (f i - 1))"
using sums_def_le by blast
hence "(λn. exp (∑i≤n. norm (f i - 1))) ⇢ exp (∑i. norm (f i - 1))"
by (rule tendsto_exp)
hence "convergent (λn. exp (∑i≤n. norm (f i - 1)))"
by (rule convergentI)
thus "Bseq (λn. exp (∑i≤n. norm (f i - 1)))"
by (rule convergent_imp_Bseq)
qed
qed

theorem abs_convergent_prod_conv_summable:
fixes f :: "nat ⇒ 'a :: real_normed_div_algebra"
shows "abs_convergent_prod f ⟷ summable (λi. norm (f i - 1))"
by (blast intro: abs_convergent_prod_imp_summable summable_imp_abs_convergent_prod)

lemma abs_convergent_prod_imp_LIMSEQ:
fixes f :: "nat ⇒ 'a :: {comm_ring_1,real_normed_div_algebra}"
assumes "abs_convergent_prod f"
shows   "f ⇢ 1"
proof -
from assms have "summable (λn. norm (f n - 1))"
by (rule abs_convergent_prod_imp_summable)
from summable_LIMSEQ_zero[OF this] have "(λn. f n - 1) ⇢ 0"
from tendsto_add[OF this tendsto_const[of 1]] show ?thesis by simp
qed

lemma abs_convergent_prod_imp_ev_nonzero:
fixes f :: "nat ⇒ 'a :: {comm_ring_1,real_normed_div_algebra}"
assumes "abs_convergent_prod f"
shows   "eventually (λn. f n ≠ 0) sequentially"
proof -
from assms have "f ⇢ 1"
by (rule abs_convergent_prod_imp_LIMSEQ)
hence "eventually (λn. dist (f n) 1 < 1) at_top"
by (auto simp: tendsto_iff)
thus ?thesis by eventually_elim auto
qed

subsection✐‹tag unimportant› ‹Ignoring initial segments›

lemma convergent_prod_offset:
assumes "convergent_prod (λn. f (n + m))"
shows   "convergent_prod f"
proof -
from assms obtain M L where "(λn. ∏k≤n. f (k + (M + m))) ⇢ L" "L ≠ 0"
thus "convergent_prod f"
unfolding prod_defs by blast
qed

lemma abs_convergent_prod_offset:
assumes "abs_convergent_prod (λn. f (n + m))"
shows   "abs_convergent_prod f"
using assms unfolding abs_convergent_prod_def by (rule convergent_prod_offset)

lemma raw_has_prod_ignore_initial_segment:
fixes f :: "nat ⇒ 'a :: real_normed_field"
assumes "raw_has_prod f M p" "N ≥ M"
obtains q where  "raw_has_prod f N q"
proof -
have p: "(λn. ∏k≤n. f (k + M)) ⇢ p" and "p ≠ 0"
using assms by (auto simp: raw_has_prod_def)
then have nz: "⋀n. n ≥ M ⟹ f n ≠ 0"
using assms by (auto simp: raw_has_prod_eq_0)
define C where "C = (∏k<N-M. f (k + M))"
from nz have [simp]: "C ≠ 0"
by (auto simp: C_def)

from p have "(λi. ∏k≤i + (N-M). f (k + M)) ⇢ p"
by (rule LIMSEQ_ignore_initial_segment)
also have "(λi. ∏k≤i + (N-M). f (k + M)) = (λn. C * (∏k≤n. f (k + N)))"
proof (rule ext, goal_cases)
case (1 n)
have "{..n+(N-M)} = {..<(N-M)} ∪ {(N-M)..n+(N-M)}" by auto
also have "(∏k∈…. f (k + M)) = C * (∏k=(N-M)..n+(N-M). f (k + M))"
unfolding C_def by (rule prod.union_disjoint) auto
also have "(∏k=(N-M)..n+(N-M). f (k + M)) = (∏k≤n. f (k + (N-M) + M))"
by (intro ext prod.reindex_bij_witness[of _ "λk. k + (N-M)" "λk. k - (N-M)"]) auto
finally show ?case
qed
finally have "(λn. C * (∏k≤n. f (k + N)) / C) ⇢ p / C"
by (intro tendsto_divide tendsto_const) auto
hence "(λn. ∏k≤n. f (k + N)) ⇢ p / C" by simp
moreover from ‹p ≠ 0› have "p / C ≠ 0" by simp
ultimately show ?thesis
using raw_has_prod_def that by blast
qed

corollary✐‹tag unimportant› convergent_prod_ignore_initial_segment:
fixes f :: "nat ⇒ 'a :: real_normed_field"
assumes "convergent_prod f"
shows   "convergent_prod (λn. f (n + m))"
using assms
unfolding convergent_prod_def
apply clarify
apply (erule_tac N="M+m" in raw_has_prod_ignore_initial_segment)
done

corollary✐‹tag unimportant› convergent_prod_ignore_nonzero_segment:
fixes f :: "nat ⇒ 'a :: real_normed_field"
assumes f: "convergent_prod f" and nz: "⋀i. i ≥ M ⟹ f i ≠ 0"
shows "∃p. raw_has_prod f M p"
using convergent_prod_ignore_initial_segment [OF f]
by (metis convergent_LIMSEQ_iff convergent_prod_iff_convergent le_add_same_cancel2 nz prod_defs(1) zero_order(1))

corollary✐‹tag unimportant› abs_convergent_prod_ignore_initial_segment:
assumes "abs_convergent_prod f"
shows   "abs_convergent_prod (λn. f (n + m))"
using assms unfolding abs_convergent_prod_def
by (rule convergent_prod_ignore_initial_segment)

subsection‹More elementary properties›

theorem abs_convergent_prod_imp_convergent_prod:
fixes f :: "nat ⇒ 'a :: {real_normed_div_algebra,complete_space,comm_ring_1}"
assumes "abs_convergent_prod f"
shows   "convergent_prod f"
proof -
from assms have "eventually (λn. f n ≠ 0) sequentially"
by (rule abs_convergent_prod_imp_ev_nonzero)
then obtain N where N: "f n ≠ 0" if "n ≥ N" for n
by (auto simp: eventually_at_top_linorder)
let ?P = "λn. ∏i≤n. f (i + N)" and ?Q = "λn. ∏i≤n. 1 + norm (f (i + N) - 1)"

have "Cauchy ?P"
proof (rule CauchyI', goal_cases)
case (1 ε)
from assms have "abs_convergent_prod (λn. f (n + N))"
by (rule abs_convergent_prod_ignore_initial_segment)
hence "Cauchy ?Q"
unfolding abs_convergent_prod_def
by (intro convergent_Cauchy convergent_prod_imp_convergent)
from CauchyD[OF this 1] obtain M where M: "norm (?Q m - ?Q n) < ε" if "m ≥ M" "n ≥ M" for m n
by blast
show ?case
proof (rule exI[of _ M], safe, goal_cases)
case (1 m n)
have "dist (?P m) (?P n) = norm (?P n - ?P m)"
also from 1 have "{..n} = {..m} ∪ {m<..n}" by auto
hence "norm (?P n - ?P m) = norm (?P m * (∏k∈{m<..n}. f (k + N)) - ?P m)"
by (subst prod.union_disjoint [symmetric]) (auto simp: algebra_simps)
also have "… = norm (?P m * ((∏k∈{m<..n}. f (k + N)) - 1))"
also have "… = (∏k≤m. norm (f (k + N))) * norm ((∏k∈{m<..n}. f (k + N)) - 1)"
also have "… ≤ ?Q m * ((∏k∈{m<..n}. 1 + norm (f (k + N) - 1)) - 1)"
using norm_prod_minus1_le_prod_minus1[of "λk. f (k + N) - 1" "{m<..n}"]
norm_triangle_ineq[of 1 "f k - 1" for k]
by (intro mult_mono prod_mono ballI conjI norm_prod_minus1_le_prod_minus1 prod_nonneg) auto
also have "… = ?Q m * (∏k∈{m<..n}. 1 + norm (f (k + N) - 1)) - ?Q m"
also have "?Q m * (∏k∈{m<..n}. 1 + norm (f (k + N) - 1)) =
(∏k∈{..m}∪{m<..n}. 1 + norm (f (k + N) - 1))"
by (rule prod.union_disjoint [symmetric]) auto
also from 1 have "{..m}∪{m<..n} = {..n}" by auto
also have "?Q n - ?Q m ≤ norm (?Q n - ?Q m)" by simp
also from 1 have "… < ε" by (intro M) auto
finally show ?case .
qed
qed
hence conv: "convergent ?P" by (rule Cauchy_convergent)
then obtain L where L: "?P ⇢ L"
by (auto simp: convergent_def)

have "L ≠ 0"
proof
assume [simp]: "L = 0"
from tendsto_norm[OF L] have limit: "(λn. ∏k≤n. norm (f (k + N))) ⇢ 0"

from assms have "(λn. f (n + N)) ⇢ 1"
by (intro abs_convergent_prod_imp_LIMSEQ abs_convergent_prod_ignore_initial_segment)
hence "eventually (λn. norm (f (n + N) - 1) < 1) sequentially"
by (auto simp: tendsto_iff dist_norm)
then obtain M0 where M0: "norm (f (n + N) - 1) < 1" if "n ≥ M0" for n
by (auto simp: eventually_at_top_linorder)

{
fix M assume M: "M ≥ M0"
with M0 have M: "norm (f (n + N) - 1) < 1" if "n ≥ M" for n using that by simp

have "(λn. ∏k≤n. 1 - norm (f (k+M+N) - 1)) ⇢ 0"
proof (rule tendsto_sandwich)
show "eventually (λn. (∏k≤n. 1 - norm (f (k+M+N) - 1)) ≥ 0) sequentially"
using M by (intro always_eventually prod_nonneg allI ballI) (auto intro: less_imp_le)
have "norm (1::'a) - norm (f (i + M + N) - 1) ≤ norm (f (i + M + N))" for i
using norm_triangle_ineq3[of "f (i + M + N)" 1] by simp
thus "eventually (λn. (∏k≤n. 1 - norm (f (k+M+N) - 1)) ≤ (∏k≤n. norm (f (k+M+N)))) at_top"
using M by (intro always_eventually allI prod_mono ballI conjI) (auto intro: less_imp_le)

define C where "C = (∏k<M. norm (f (k + N)))"
from N have [simp]: "C ≠ 0" by (auto simp: C_def)
from L have "(λn. norm (∏k≤n+M. f (k + N))) ⇢ 0"
by (intro LIMSEQ_ignore_initial_segment) (simp add: tendsto_norm_zero_iff)
also have "(λn. norm (∏k≤n+M. f (k + N))) = (λn. C * (∏k≤n. norm (f (k + M + N))))"
proof (rule ext, goal_cases)
case (1 n)
have "{..n+M} = {..<M} ∪ {M..n+M}" by auto
also have "norm (∏k∈…. f (k + N)) = C * norm (∏k=M..n+M. f (k + N))"
unfolding C_def by (subst prod.union_disjoint) (auto simp: norm_mult prod_norm)
also have "(∏k=M..n+M. f (k + N)) = (∏k≤n. f (k + N + M))"
by (intro prod.reindex_bij_witness[of _ "λi. i + M" "λi. i - M"]) auto
qed
finally have "(λn. C * (∏k≤n. norm (f (k + M + N))) / C) ⇢ 0 / C"
by (intro tendsto_divide tendsto_const) auto
thus "(λn. ∏k≤n. norm (f (k + M + N))) ⇢ 0" by simp
qed simp_all

have "1 - (∑i. norm (f (i + M + N) - 1)) ≤ 0"
proof (rule tendsto_le)
show "eventually (λn. 1 - (∑k≤n. norm (f (k+M+N) - 1)) ≤
(∏k≤n. 1 - norm (f (k+M+N) - 1))) at_top"
using M by (intro always_eventually allI Weierstrass_prod_ineq) (auto intro: less_imp_le)
show "(λn. ∏k≤n. 1 - norm (f (k+M+N) - 1)) ⇢ 0" by fact
show "(λn. 1 - (∑k≤n. norm (f (k + M + N) - 1)))
⇢ 1 - (∑i. norm (f (i + M + N) - 1))"
by (intro tendsto_intros summable_LIMSEQ' summable_ignore_initial_segment
abs_convergent_prod_imp_summable assms)
qed simp_all
hence "(∑i. norm (f (i + M + N) - 1)) ≥ 1" by simp
also have "… + (∑i<M. norm (f (i + N) - 1)) = (∑i. norm (f (i + N) - 1))"
by (intro suminf_split_initial_segment [symmetric] summable_ignore_initial_segment
abs_convergent_prod_imp_summable assms)
finally have "1 + (∑i<M. norm (f (i + N) - 1)) ≤ (∑i. norm (f (i + N) - 1))" by simp
} note * = this

have "1 + (∑i. norm (f (i + N) - 1)) ≤ (∑i. norm (f (i + N) - 1))"
proof (rule tendsto_le)
show "(λM. 1 + (∑i<M. norm (f (i + N) - 1))) ⇢ 1 + (∑i. norm (f (i + N) - 1))"
by (intro tendsto_intros summable_LIMSEQ summable_ignore_initial_segment
abs_convergent_prod_imp_summable assms)
show "eventually (λM. 1 + (∑i<M. norm (f (i + N) - 1)) ≤ (∑i. norm (f (i + N) - 1))) at_top"
using eventually_ge_at_top[of M0] by eventually_elim (use * in auto)
qed simp_all
thus False by simp
qed
with L show ?thesis by (auto simp: prod_defs)
qed

lemma raw_has_prod_cases:
fixes f :: "nat ⇒ 'a :: {idom,topological_semigroup_mult,t2_space}"
assumes "raw_has_prod f M p"
obtains i where "i<M" "f i = 0" | p where "raw_has_prod f 0 p"
proof -
have "(λn. ∏i≤n. f (i + M)) ⇢ p" "p ≠ 0"
using assms unfolding raw_has_prod_def by blast+
then have "(λn. prod f {..<M} * (∏i≤n. f (i + M))) ⇢ prod f {..<M} * p"
by (metis tendsto_mult_left)
moreover have "prod f {..<M} * (∏i≤n. f (i + M)) = prod f {..n+M}" for n
proof -
have "{..n+M} = {..<M} ∪ {M..n+M}"
by auto
then have "prod f {..n+M} = prod f {..<M} * prod f {M..n+M}"
by simp (subst prod.union_disjoint; force)
also have "… = prod f {..<M} * (∏i≤n. f (i + M))"
by (metis (mono_tags, lifting) add.left_neutral atMost_atLeast0 prod.shift_bounds_cl_nat_ivl)
finally show ?thesis by metis
qed
ultimately have "(λn. prod f {..n}) ⇢ prod f {..<M} * p"
by (auto intro: LIMSEQ_offset [where k=M])
then have "raw_has_prod f 0 (prod f {..<M} * p)" if "∀i<M. f i ≠ 0"
using ‹p ≠ 0› assms that by (auto simp: raw_has_prod_def)
then show thesis
using that by blast
qed

corollary convergent_prod_offset_0:
fixes f :: "nat ⇒ 'a :: {idom,topological_semigroup_mult,t2_space}"
assumes "convergent_prod f" "⋀i. f i ≠ 0"
shows "∃p. raw_has_prod f 0 p"
using assms convergent_prod_def raw_has_prod_cases by blast

lemma prodinf_eq_lim:
fixes f :: "nat ⇒ 'a :: {idom,topological_semigroup_mult,t2_space}"
assumes "convergent_prod f" "⋀i. f i ≠ 0"
shows "prodinf f = lim (λn. ∏i≤n. f i)"
using assms convergent_prod_offset_0 [OF assms]
by (simp add: prod_defs lim_def) (metis (no_types) assms(1) convergent_prod_to_zero_iff)

lemma prodinf_eq_lim':
fixes f :: "nat ⇒ 'a :: {idom,topological_semigroup_mult,t2_space}"
assumes "convergent_prod f" "⋀i. f i ≠ 0"
shows "prodinf f = lim (λn. ∏i<n. f i)"
by (metis assms prodinf_eq_lim LIMSEQ_lessThan_iff_atMost convergent_prod_iff_nz_lim limI)

lemma prodinf_eq_prod_lim:
fixes a:: "'a :: {topological_semigroup_mult,t2_space,idom}"
assumes "(λn. ∏k≤n. f k) ⇢ a" "a ≠ 0"
shows"(∏k. f k) = a"
by (metis LIMSEQ_prod_0 LIMSEQ_unique assms convergent_prod_iff_nz_lim limI prodinf_eq_lim)

lemma prodinf_eq_prod_lim':
fixes a:: "'a :: {topological_semigroup_mult,t2_space,idom}"
assumes "(λn. ∏k<n. f k) ⇢ a" "a ≠ 0"
shows"(∏k. f k) = a"
using LIMSEQ_lessThan_iff_atMost assms prodinf_eq_prod_lim by blast

lemma has_prod_one[simp, intro]: "(λn. 1) has_prod 1"
unfolding prod_defs by auto

lemma convergent_prod_one[simp, intro]: "convergent_prod (λn. 1)"
unfolding prod_defs by auto

lemma prodinf_cong: "(⋀n. f n = g n) ⟹ prodinf f = prodinf g"
by presburger

lemma convergent_prod_cong:
fixes f g :: "nat ⇒ 'a::{field,topological_semigroup_mult,t2_space}"
assumes ev: "eventually (λx. f x = g x) sequentially" and f: "⋀i. f i ≠ 0" and g: "⋀i. g i ≠ 0"
shows "convergent_prod f = convergent_prod g"
proof -
from assms obtain N where N: "∀n≥N. f n = g n"
by (auto simp: eventually_at_top_linorder)
define C where "C = (∏k<N. f k / g k)"
with g have "C ≠ 0"
have *: "eventually (λn. prod f {..n} = C * prod g {..n}) sequentially"
using eventually_ge_at_top[of N]
proof eventually_elim
case (elim n)
then have "{..n} = {..<N} ∪ {N..n}"
by auto
also have "prod f … = prod f {..<N} * prod f {N..n}"
by (intro prod.union_disjoint) auto
also from N have "prod f {N..n} = prod g {N..n}"
by (intro prod.cong) simp_all
also have "prod f {..<N} * prod g {N..n} = C * (prod g {..<N} * prod g {N..n})"
unfolding C_def by (simp add: g prod_dividef)
also have "prod g {..<N} * prod g {N..n} = prod g ({..<N} ∪ {N..n})"
by (intro prod.union_disjoint [symmetric]) auto
also from elim have "{..<N} ∪ {N..n} = {..n}"
by auto
finally show "prod f {..n} = C * prod g {..n}" .
qed
then have cong: "convergent (λn. prod f {..n}) = convergent (λn. C * prod g {..n})"
by (rule convergent_cong)
show ?thesis
proof
assume cf: "convergent_prod f"
with f have "¬ (λn. prod f {..n}) ⇢ 0"
by simp
then have "¬ (λn. prod g {..n}) ⇢ 0"
using * ‹C ≠ 0› filterlim_cong by fastforce
then show "convergent_prod g"
by (metis convergent_mult_const_iff ‹C ≠ 0› cong cf convergent_LIMSEQ_iff convergent_prod_iff_convergent convergent_prod_imp_convergent g)
next
assume cg: "convergent_prod g"
have "∃a. C * a ≠ 0 ∧ (λn. prod g {..n}) ⇢ a"
by (metis (no_types) ‹C ≠ 0› cg convergent_prod_iff_nz_lim divide_eq_0_iff g nonzero_mult_div_cancel_right)
then show "convergent_prod f"
using "*" tendsto_mult_left filterlim_cong
by (fastforce simp add: convergent_prod_iff_nz_lim f)
qed
qed

lemma has_prod_finite:
fixes f :: "nat ⇒ 'a::{semidom,t2_space}"
assumes [simp]: "finite N"
and f: "⋀n. n ∉ N ⟹ f n = 1"
shows "f has_prod (∏n∈N. f n)"
proof -
have eq: "prod f {..n + Suc (Max N)} = prod f N" for n
proof (rule prod.mono_neutral_right)
show "N ⊆ {..n + Suc (Max N)}"
show "∀i∈{..n + Suc (Max N)} - N. f i = 1"
using f by blast
qed auto
show ?thesis
proof (cases "∀n∈N. f n ≠ 0")
case True
then have "prod f N ≠ 0"
by simp
moreover have "(λn. prod f {..n}) ⇢ prod f N"
by (rule LIMSEQ_offset[of _ "Suc (Max N)"]) (simp add: eq atLeast0LessThan del: add_Suc_right)
ultimately show ?thesis
next
case False
then obtain k where "k ∈ N" "f k = 0"
by auto
let ?Z = "{n ∈ N. f n = 0}"
have maxge: "Max ?Z ≥ n" if "f n = 0" for n
using Max_ge [of ?Z] ‹finite N› ‹f n = 0›
by (metis (mono_tags) Collect_mem_eq f finite_Collect_conjI mem_Collect_eq zero_neq_one)
let ?q = "prod f {Suc (Max ?Z)..Max N}"
have [simp]: "?q ≠ 0"
using maxge Suc_n_not_le_n le_trans by force
have eq: "(∏i≤n + Max N. f (Suc (i + Max ?Z))) = ?q" for n
proof -
have "(∏i≤n + Max N. f (Suc (i + Max ?Z))) = prod f {Suc (Max ?Z)..n + Max N + Suc (Max ?Z)}"
proof (rule prod.reindex_cong [where l = "λi. i + Suc (Max ?Z)", THEN sym])
show "{Suc (Max ?Z)..n + Max N + Suc (Max ?Z)} = (λi. i + Suc (Max ?Z))  {..n + Max N}"
using le_Suc_ex by fastforce
qed (auto simp: inj_on_def)
also have "… = ?q"
by (rule prod.mono_neutral_right)
(use Max.coboundedI [OF ‹finite N›] f in ‹force+›)
finally show ?thesis .
qed
have q: "raw_has_prod f (Suc (Max ?Z)) ?q"
show "(λn. ∏i≤n. f (Suc (i + Max ?Z))) ⇢ ?q"
by (rule LIMSEQ_offset[of _ "(Max N)"]) (simp add: eq)
qed
show ?thesis
unfolding has_prod_def
proof (intro disjI2 exI conjI)
show "prod f N = 0"
using ‹f k = 0› ‹k ∈ N› ‹finite N› prod_zero by blast
show "f (Max ?Z) = 0"
using Max_in [of ?Z] ‹finite N› ‹f k = 0› ‹k ∈ N› by auto
qed (use q in auto)
qed
qed

corollary✐‹tag unimportant› has_prod_0:
fixes f :: "nat ⇒ 'a::{semidom,t2_space}"
assumes "⋀n. f n = 1"
shows "f has_prod 1"

lemma prodinf_zero[simp]: "prodinf (λn. 1::'a::real_normed_field) = 1"
using has_prod_unique by force

lemma convergent_prod_finite:
fixes f :: "nat ⇒ 'a::{idom,t2_space}"
assumes "finite N" "⋀n. n ∉ N ⟹ f n = 1"
shows "convergent_prod f"
proof -
have "∃n p. raw_has_prod f n p"
using assms has_prod_def has_prod_finite by blast
then show ?thesis
qed

lemma has_prod_If_finite_set:
fixes f :: "nat ⇒ 'a::{idom,t2_space}"
shows "finite A ⟹ (λr. if r ∈ A then f r else 1) has_prod (∏r∈A. f r)"
using has_prod_finite[of A "(λr. if r ∈ A then f r else 1)"]
by simp

lemma has_prod_If_finite:
fixes f :: "nat ⇒ 'a::{idom,t2_space}"
shows "finite {r. P r} ⟹ (λr. if P r then f r else 1) has_prod (∏r | P r. f r)"
using has_prod_If_finite_set[of "{r. P r}"] by simp

lemma convergent_prod_If_finite_set[simp, intro]:
fixes f :: "nat ⇒ 'a::{idom,t2_space}"
shows "finite A ⟹ convergent_prod (λr. if r ∈ A then f r else 1)"

lemma convergent_prod_If_finite[simp, intro]:
fixes f :: "nat ⇒ 'a::{idom,t2_space}"
shows "finite {r. P r} ⟹ convergent_prod (λr. if P r then f r else 1)"
using convergent_prod_def has_prod_If_finite has_prod_def by fastforce

lemma has_prod_single:
fixes f :: "nat ⇒ 'a::{idom,t2_space}"
shows "(λr. if r = i then f r else 1) has_prod f i"
using has_prod_If_finite[of "λr. r = i"] by simp

text ‹The ge1 assumption can probably be weakened, at the expense of extra work›
lemma uniform_limit_prodinf:
fixes f:: "nat ⇒ real ⇒ real"
assumes "uniformly_convergent_on X (λn x. ∏k<n. f k x)"
and ge1: "⋀x k . x ∈ X ⟹ f k x ≥ 1"
shows "uniform_limit X (λn x. ∏k<n. f k x) (λx. ∏k. f k x) sequentially"
proof -
have ul: "uniform_limit X (λn x. ∏k<n. f k x) (λx. lim (λn. ∏k<n. f k x)) sequentially"
using assms uniformly_convergent_uniform_limit_iff by blast
moreover have "(∏k. f k x) = lim (λn. ∏k<n. f k x)" if "x ∈ X" for x
proof (intro prodinf_eq_lim')
have tends: "(λn. ∏k<n. f k x) ⇢ lim (λn. ∏k<n. f k x)"
using tendsto_uniform_limitI [OF ul] that by metis
moreover have "(∏k<n. f k x) ≥ 1" for n
using ge1 by (simp add: prod_ge_1 that)
ultimately have "lim (λn. ∏k<n. f k x) ≥ 1"
by (meson LIMSEQ_le_const)
then have "raw_has_prod (λk. f k x) 0 (lim (λn. ∏k<n. f k x))"
using LIMSEQ_lessThan_iff_atMost tends by (auto simp: raw_has_prod_def)
then show "convergent_prod (λk. f k x)"
unfolding convergent_prod_def by blast
show "⋀k. f k x ≠ 0"
by (smt (verit) ge1 that)
qed
ultimately show ?thesis
by (metis (mono_tags, lifting) uniform_limit_cong')
qed

context
fixes f :: "nat ⇒ 'a :: real_normed_field"
begin

lemma convergent_prod_imp_has_prod:
assumes "convergent_prod f"
shows "∃p. f has_prod p"
proof -
obtain M p where p: "raw_has_prod f M p"
using assms convergent_prod_def by blast
then have "p ≠ 0"
using raw_has_prod_nonzero by blast
with p have fnz: "f i ≠ 0" if "i ≥ M" for i
using raw_has_prod_eq_0 that by blast
define C where "C = (∏n<M. f n)"
show ?thesis
proof (cases "∀n≤M. f n ≠ 0")
case True
then have "C ≠ 0"
then show ?thesis
by (meson True assms convergent_prod_offset_0 fnz has_prod_def nat_le_linear)
next
case False
let ?N = "GREATEST n. f n = 0"
have 0: "f ?N = 0"
using fnz False
by (metis (mono_tags, lifting) GreatestI_ex_nat nat_le_linear)
have "f i ≠ 0" if "i > ?N" for i
by (metis (mono_tags, lifting) Greatest_le_nat fnz leD linear that)
then have "∃p. raw_has_prod f (Suc ?N) p"
using assms by (auto simp: intro!: convergent_prod_ignore_nonzero_segment)
then show ?thesis
unfolding has_prod_def using 0 by blast
qed
qed

lemma convergent_prod_has_prod [intro]:
shows "convergent_prod f ⟹ f has_prod (prodinf f)"
unfolding prodinf_def
by (metis convergent_prod_imp_has_prod has_prod_unique theI')

lemma convergent_prod_LIMSEQ:
shows "convergent_prod f ⟹ (λn. ∏i≤n. f i) ⇢ prodinf f"
by (metis convergent_LIMSEQ_iff convergent_prod_has_prod convergent_prod_imp_convergent
convergent_prod_to_zero_iff raw_has_prod_eq_0 has_prod_def prodinf_eq_lim zero_le)

theorem has_prod_iff: "f has_prod x ⟷ convergent_prod f ∧ prodinf f = x"
proof
assume "f has_prod x"
then show "convergent_prod f ∧ prodinf f = x"
apply safe
using convergent_prod_def has_prod_def apply blast
using has_prod_unique by blast
qed auto

lemma convergent_prod_has_prod_iff: "convergent_prod f ⟷ f has_prod prodinf f"
by (auto simp: has_prod_iff convergent_prod_has_prod)

lemma prodinf_finite:
assumes N: "finite N"
and f: "⋀n. n ∉ N ⟹ f n = 1"
shows "prodinf f = (∏n∈N. f n)"
using has_prod_finite[OF assms, THEN has_prod_unique] by simp

end

subsection✐‹tag unimportant› ‹Infinite products on ordered topological monoids›

context
fixes f :: "nat ⇒ 'a::{linordered_semidom,linorder_topology}"
begin

lemma has_prod_nonzero:
assumes "f has_prod a" "a ≠ 0"
shows "f k ≠ 0"
using assms by (auto simp: has_prod_def raw_has_prod_def LIMSEQ_prod_0 LIMSEQ_unique)

lemma has_prod_le:
assumes f: "f has_prod a" and g: "g has_prod b" and le: "⋀n. 0 ≤ f n ∧ f n ≤ g n"
shows "a ≤ b"
proof (cases "a=0 ∨ b=0")
case True
then show ?thesis
proof
assume [simp]: "a=0"
have "b ≥ 0"
proof (rule LIMSEQ_prod_nonneg)
show "(λn. prod g {..n}) ⇢ b"
using g by (auto simp: has_prod_def raw_has_prod_def LIMSEQ_prod_0)
qed (use le order_trans in auto)
then show ?thesis
by auto
next
assume [simp]: "b=0"
then obtain i where "g i = 0"
using g by (auto simp: prod_defs)
then have "f i = 0"
using antisym le by force
then have "a=0"
using f by (auto simp: prod_defs LIMSEQ_prod_0 LIMSEQ_unique)
then show ?thesis
by auto
qed
next
case False
then show ?thesis
using assms
unfolding has_prod_def raw_has_prod_def
by (force simp: LIMSEQ_prod_0 intro!: LIMSEQ_le prod_mono)
qed

lemma prodinf_le:
assumes f: "f has_prod a" and g: "g has_prod b" and le: "⋀n. 0 ≤ f n ∧ f n ≤ g n"
shows "prodinf f ≤ prodinf g"
using has_prod_le [OF assms] has_prod_unique f g  by blast

end

lemma prod_le_prodinf:
fixes f :: "nat ⇒ 'a::{linordered_idom,linorder_topology}"
assumes "f has_prod a" "⋀i. 0 ≤ f i" "⋀i. i≥n ⟹ 1 ≤ f i"
shows "prod f {..<n} ≤ prodinf f"
by(rule has_prod_le[OF has_prod_If_finite_set]) (use assms has_prod_unique in auto)

lemma prodinf_nonneg:
fixes f :: "nat ⇒ 'a::{linordered_idom,linorder_topology}"
assumes "f has_prod a" "⋀i. 1 ≤ f i"
shows "1 ≤ prodinf f"
using prod_le_prodinf[of f a 0] assms
by (metis order_trans prod_ge_1 zero_le_one)

lemma prodinf_le_const:
fixes f :: "nat ⇒ real"
assumes "convergent_prod f" "⋀n. n ≥ N ⟹ prod f {..<n} ≤ x"
shows "prodinf f ≤ x"
by (metis lessThan_Suc_atMost assms convergent_prod_LIMSEQ LIMSEQ_le_const2 atMost_iff lessThan_iff less_le)

lemma prodinf_eq_one_iff [simp]:
fixes f :: "nat ⇒ real"
assumes f: "convergent_prod f" and ge1: "⋀n. 1 ≤ f n"
shows "prodinf f = 1 ⟷ (∀n. f n = 1)"
proof
assume "prodinf f = 1"
then have "(λn. ∏i<n. f i) ⇢ 1"
using convergent_prod_LIMSEQ[of f] assms by (simp add: LIMSEQ_lessThan_iff_atMost)
then have "⋀i. (∏n∈{i}. f n) ≤ 1"
proof (rule LIMSEQ_le_const)
have "1 ≤ prod f n" for n
have "prod f {..<n} = 1" for n
by (metis ‹⋀n. 1 ≤ prod f n› ‹prodinf f = 1› antisym f convergent_prod_has_prod ge1 order_trans prod_le_prodinf zero_le_one)
then have "(∏n∈{i}. f n) ≤ prod f {..<n}" if "n ≥ Suc i" for i n
by (metis mult.left_neutral order_refl prod.cong prod.neutral_const prod.lessThan_Suc)
then show "∃N. ∀n≥N. (∏n∈{i}. f n) ≤ prod f {..<n}" for i
by blast
qed
with ge1 show "∀n. f n = 1"
by (auto intro!: antisym)
qed (metis prodinf_zero fun_eq_iff)

lemma prodinf_pos_iff:
fixes f :: "nat ⇒ real"
assumes "convergent_prod f" "⋀n. 1 ≤ f n"
shows "1 < prodinf f ⟷ (∃i. 1 < f i)"
using prod_le_prodinf[of f 1] prodinf_eq_one_iff
by (metis convergent_prod_has_prod assms less_le prodinf_nonneg)

lemma less_1_prodinf2:
fixes f :: "nat ⇒ real"
assumes "convergent_prod f" "⋀n. 1 ≤ f n" "1 < f i"
shows "1 < prodinf f"
proof -
have "1 < (∏n<Suc i. f n)"
using assms  by (intro less_1_prod2[where i=i]) auto
also have "… ≤ prodinf f"
by (intro prod_le_prodinf) (use assms order_trans zero_le_one in ‹blast+›)
finally show ?thesis .
qed

lemma less_1_prodinf:
fixes f :: "nat ⇒ real"
shows "⟦convergent_prod f; ⋀n. 1 < f n⟧ ⟹ 1 < prodinf f"
by (intro less_1_prodinf2[where i=1]) (auto intro: less_imp_le)

lemma prodinf_nonzero:
fixes f :: "nat ⇒ 'a :: {idom,topological_semigroup_mult,t2_space}"
assumes "convergent_prod f" "⋀i. f i ≠ 0"
shows "prodinf f ≠ 0"
by (metis assms convergent_prod_offset_0 has_prod_unique raw_has_prod_def has_prod_def)

lemma less_0_prodinf:
fixes f :: "nat ⇒ real"
assumes f: "convergent_prod f" and 0: "⋀i. f i > 0"
shows "0 < prodinf f"
proof -
have "prodinf f ≠ 0"
by (metis assms less_irrefl prodinf_nonzero)
moreover have "0 < (∏n<i. f n)" for i
then have "prodinf f ≥ 0"
using convergent_prod_LIMSEQ [OF f] LIMSEQ_prod_nonneg 0 less_le by blast
ultimately show ?thesis
by auto
qed

lemma prod_less_prodinf2:
fixes f :: "nat ⇒ real"
assumes f: "convergent_prod f" and 1: "⋀m. m≥n ⟹ 1 ≤ f m" and 0: "⋀m. 0 < f m" and i: "n ≤ i" "1 < f i"
shows "prod f {..<n} < prodinf f"
proof -
have "prod f {..<n} ≤ prod f {..<i}"
by (rule prod_mono2) (use assms less_le in auto)
then have "prod f {..<n} < f i * prod f {..<i}"
using mult_less_le_imp_less[of 1 "f i" "prod f {..<n}" "prod f {..<i}"] assms
moreover have "prod f {..<Suc i} ≤ prodinf f"
using prod_le_prodinf[of f _ "Suc i"]
by (meson "0" "1" Suc_leD convergent_prod_has_prod f ‹n ≤ i› le_trans less_eq_real_def)
ultimately show ?thesis
by (metis le_less_trans mult.commute not_le prod.lessThan_Suc)
qed

lemma prod_less_prodinf:
fixes f :: "nat ⇒ real"
assumes f: "convergent_prod f" and 1: "⋀m. m≥n ⟹ 1 < f m" and 0: "⋀m. 0 < f m"
shows "prod f {..<n} < prodinf f"
by (meson "0" "1" f le_less prod_less_prodinf2)

lemma raw_has_prodI_bounded:
fixes f :: "nat ⇒ real"
assumes pos: "⋀n. 1 ≤ f n"
and le: "⋀n. (∏i<n. f i) ≤ x"
shows "∃p. raw_has_prod f 0 p"
proof (rule exI LIMSEQ_incseq_SUP conjI)+
show "bdd_above (range (λn. prod f {..n}))"
by (metis bdd_aboveI2 le lessThan_Suc_atMost)
then have "(SUP i. prod f {..i}) > 0"
by (metis UNIV_I cSUP_upper less_le_trans pos prod_pos zero_less_one)
then show "(SUP i. prod f {..i}) ≠ 0"
by auto
show "incseq (λn. prod f {..n})"
using pos order_trans [OF zero_le_one] by (auto simp: mono_def intro!: prod_mono2)
qed

lemma convergent_prodI_nonneg_bounded:
fixes f :: "nat ⇒ real"
assumes "⋀n. 1 ≤ f n" "⋀n. (∏i<n. f i) ≤ x"
shows "convergent_prod f"
using convergent_prod_def raw_has_prodI_bounded [OF assms] by blast

subsection✐‹tag unimportant› ‹Infinite products on topological spaces›

context
fixes f g :: "nat ⇒ 'a::{t2_space,topological_semigroup_mult,idom}"
begin

lemma raw_has_prod_mult: "⟦raw_has_prod f M a; raw_has_prod g M b⟧ ⟹ raw_has_prod (λn. f n * g n) M (a * b)"
by (force simp add: prod.distrib tendsto_mult raw_has_prod_def)

lemma has_prod_mult_nz: "⟦f has_prod a; g has_prod b; a ≠ 0; b ≠ 0⟧ ⟹ (λn. f n * g n) has_prod (a * b)"

end

context
fixes f g :: "nat ⇒ 'a::real_normed_field"
begin

lemma has_prod_mult:
assumes f: "f has_prod a" and g: "g has_prod b"
shows "(λn. f n * g n) has_prod (a * b)"
using f [unfolded has_prod_def]
proof (elim disjE exE conjE)
assume f0: "raw_has_prod f 0 a"
show ?thesis
using g [unfolded has_prod_def]
proof (elim disjE exE conjE)
assume g0: "raw_has_prod g 0 b"
with f0 show ?thesis
by (force simp add: has_prod_def prod.distrib tendsto_mult raw_has_prod_def)
next
fix j q
assume "b = 0" and "g j = 0" and q: "raw_has_prod g (Suc j) q"
obtain p where p: "raw_has_prod f (Suc j) p"
using f0 raw_has_prod_ignore_initial_segment by blast
then have "Ex (raw_has_prod (λn. f n * g n) (Suc j))"
using q raw_has_prod_mult by blast
then show ?thesis
using ‹b = 0› ‹g j = 0› has_prod_0_iff by fastforce
qed
next
fix i p
assume "a = 0" and "f i = 0" and p: "raw_has_prod f (Suc i) p"
show ?thesis
using g [unfolded has_prod_def]
proof (elim disjE exE conjE)
assume g0: "raw_has_prod g 0 b"
obtain q where q: "raw_has_prod g (Suc i) q"
using g0 raw_has_prod_ignore_initial_segment by blast
then have "Ex (raw_has_prod (λn. f n * g n) (Suc i))"
using raw_has_prod_mult p by blast
then show ?thesis
using ‹a = 0› ‹f i = 0› has_prod_0_iff by fastforce
next
fix j q
assume "b = 0" and "g j = 0" and q: "raw_has_prod g (Suc j) q"
obtain p' where p': "raw_has_prod f (Suc (max i j)) p'"
by (metis raw_has_prod_ignore_initial_segment max_Suc_Suc max_def p)
moreover
obtain q' where q': "raw_has_prod g (Suc (max i j)) q'"
by (metis raw_has_prod_ignore_initial_segment max.cobounded2 max_Suc_Suc q)
ultimately show ?thesis
using ‹b = 0› by (simp add: has_prod_def) (metis ‹f i = 0› ‹g j = 0› raw_has_prod_mult max_def)
qed
qed

lemma convergent_prod_mult:
assumes f: "convergent_prod f" and g: "convergent_prod g"
shows "convergent_prod (λn. f n * g n)"
unfolding convergent_prod_def
proof -
obtain M p N q where p: "raw_has_prod f M p" and q: "raw_has_prod g N q"
using convergent_prod_def f g by blast+
then obtain p' q' where p': "raw_has_prod f (max M N) p'" and q': "raw_has_prod g (max M N) q'"
by (meson raw_has_prod_ignore_initial_segment max.cobounded1 max.cobounded2)
then show "∃M p. raw_has_prod (λn. f n * g n) M p"
using raw_has_prod_mult by blast
qed

lemma prodinf_mult: "convergent_prod f ⟹ convergent_prod g ⟹ prodinf f * prodinf g = (∏n. f n * g n)"
by (intro has_prod_unique has_prod_mult convergent_prod_has_prod)

end

context
fixes f :: "'i ⇒ nat ⇒ 'a::real_normed_field"
and I :: "'i set"
begin

lemma has_prod_prod: "(⋀i. i ∈ I ⟹ (f i) has_prod (x i)) ⟹ (λn. ∏i∈I. f i n) has_prod (∏i∈I. x i)"
by (induct I rule: infinite_finite_induct) (auto intro!: has_prod_mult)

lemma prodinf_prod: "(⋀i. i ∈ I ⟹ convergent_prod (f i)) ⟹ (∏n. ∏i∈I. f i n) = (∏i∈I. ∏n. f i n)"
using has_prod_unique[OF has_prod_prod, OF convergent_prod_has_prod] by simp

lemma convergent_prod_prod: "(⋀i. i ∈ I ⟹ convergent_prod (f i)) ⟹ convergent_prod (λn. ∏i∈I. f i n)"
using convergent_prod_has_prod_iff has_prod_prod prodinf_prod by force

end

subsection✐‹tag unimportant› ‹Infinite summability on real normed fields›

context
fixes f :: "nat ⇒ 'a::real_normed_field"
begin

lemma raw_has_prod_Suc_iff: "raw_has_prod f M (a * f M) ⟷ raw_has_prod (λn. f (Suc n)) M a ∧ f M ≠ 0"
proof -
have "raw_has_prod f M (a * f M) ⟷ (λi. ∏j≤Suc i. f (j+M)) ⇢ a * f M ∧ a * f M ≠ 0"
by (subst filterlim_sequentially_Suc) (simp add: raw_has_prod_def)
also have "… ⟷ (λi. (∏j≤i. f (Suc j + M)) * f M) `