Theory SEQ

(*  Title:      HOL/SEQ.thy
    Author:     Jacques D. Fleuriot, University of Cambridge
    Author:     Lawrence C Paulson
    Author:     Jeremy Avigad
    Author:     Brian Huffman

Convergence of sequences and series.
*)

header {* Sequences and Convergence *}

theory SEQ
imports Limits RComplete
begin

subsection {* Monotone sequences and subsequences *}

definition
  monoseq :: "(nat => 'a::order) => bool" where
    --{*Definition of monotonicity.
        The use of disjunction here complicates proofs considerably.
        One alternative is to add a Boolean argument to indicate the direction.
        Another is to develop the notions of increasing and decreasing first.*}
  "monoseq X = ((∀m. ∀n≥m. X m ≤ X n) | (∀m. ∀n≥m. X n ≤ X m))"

definition
  incseq :: "(nat => 'a::order) => bool" where
    --{*Increasing sequence*}
  "incseq X <-> (∀m. ∀n≥m. X m ≤ X n)"

definition
  decseq :: "(nat => 'a::order) => bool" where
    --{*Decreasing sequence*}
  "decseq X <-> (∀m. ∀n≥m. X n ≤ X m)"

definition
  subseq :: "(nat => nat) => bool" where
    --{*Definition of subsequence*}
  "subseq f <-> (∀m. ∀n>m. f m < f n)"

lemma incseq_mono: "mono f <-> incseq f"
  unfolding mono_def incseq_def by auto

lemma incseq_SucI:
  "(!!n. X n ≤ X (Suc n)) ==> incseq X"
  using lift_Suc_mono_le[of X]
  by (auto simp: incseq_def)

lemma incseqD: "!!i j. incseq f ==> i ≤ j ==> f i ≤ f j"
  by (auto simp: incseq_def)

lemma incseq_SucD: "incseq A ==> A i ≤ A (Suc i)"
  using incseqD[of A i "Suc i"] by auto

lemma incseq_Suc_iff: "incseq f <-> (∀n. f n ≤ f (Suc n))"
  by (auto intro: incseq_SucI dest: incseq_SucD)

lemma incseq_const[simp, intro]: "incseq (λx. k)"
  unfolding incseq_def by auto

lemma decseq_SucI:
  "(!!n. X (Suc n) ≤ X n) ==> decseq X"
  using order.lift_Suc_mono_le[OF dual_order, of X]
  by (auto simp: decseq_def)

lemma decseqD: "!!i j. decseq f ==> i ≤ j ==> f j ≤ f i"
  by (auto simp: decseq_def)

lemma decseq_SucD: "decseq A ==> A (Suc i) ≤ A i"
  using decseqD[of A i "Suc i"] by auto

lemma decseq_Suc_iff: "decseq f <-> (∀n. f (Suc n) ≤ f n)"
  by (auto intro: decseq_SucI dest: decseq_SucD)

lemma decseq_const[simp, intro]: "decseq (λx. k)"
  unfolding decseq_def by auto

lemma monoseq_iff: "monoseq X <-> incseq X ∨ decseq X"
  unfolding monoseq_def incseq_def decseq_def ..

lemma monoseq_Suc:
  "monoseq X <-> (∀n. X n ≤ X (Suc n)) ∨ (∀n. X (Suc n) ≤ X n)"
  unfolding monoseq_iff incseq_Suc_iff decseq_Suc_iff ..

lemma monoI1: "∀m. ∀ n ≥ m. X m ≤ X n ==> monoseq X"
by (simp add: monoseq_def)

lemma monoI2: "∀m. ∀ n ≥ m. X n ≤ X m ==> monoseq X"
by (simp add: monoseq_def)

lemma mono_SucI1: "∀n. X n ≤ X (Suc n) ==> monoseq X"
by (simp add: monoseq_Suc)

lemma mono_SucI2: "∀n. X (Suc n) ≤ X n ==> monoseq X"
by (simp add: monoseq_Suc)

lemma monoseq_minus:
  fixes a :: "nat => 'a::ordered_ab_group_add"
  assumes "monoseq a"
  shows "monoseq (λ n. - a n)"
proof (cases "∀ m. ∀ n ≥ m. a m ≤ a n")
  case True
  hence "∀ m. ∀ n ≥ m. - a n ≤ - a m" by auto
  thus ?thesis by (rule monoI2)
next
  case False
  hence "∀ m. ∀ n ≥ m. - a m ≤ - a n" using `monoseq a`[unfolded monoseq_def] by auto
  thus ?thesis by (rule monoI1)
qed

text{*Subsequence (alternative definition, (e.g. Hoskins)*}

lemma subseq_Suc_iff: "subseq f = (∀n. (f n) < (f (Suc n)))"
apply (simp add: subseq_def)
apply (auto dest!: less_imp_Suc_add)
apply (induct_tac k)
apply (auto intro: less_trans)
done

text{* for any sequence, there is a monotonic subsequence *}
lemma seq_monosub:
  fixes s :: "nat => 'a::linorder"
  shows "∃f. subseq f ∧ monoseq (λ n. (s (f n)))"
proof cases
  let "?P p n" = "p > n ∧ (∀m≥p. s m ≤ s p)"
  assume *: "∀n. ∃p. ?P p n"
  def f ≡ "nat_rec (SOME p. ?P p 0) (λ_ n. SOME p. ?P p n)"
  have f_0: "f 0 = (SOME p. ?P p 0)" unfolding f_def by simp
  have f_Suc: "!!i. f (Suc i) = (SOME p. ?P p (f i))" unfolding f_def nat_rec_Suc ..
  have P_0: "?P (f 0) 0" unfolding f_0 using *[rule_format] by (rule someI2_ex) auto
  have P_Suc: "!!i. ?P (f (Suc i)) (f i)" unfolding f_Suc using *[rule_format] by (rule someI2_ex) auto
  then have "subseq f" unfolding subseq_Suc_iff by auto
  moreover have "monoseq (λn. s (f n))" unfolding monoseq_Suc
  proof (intro disjI2 allI)
    fix n show "s (f (Suc n)) ≤ s (f n)"
    proof (cases n)
      case 0 with P_Suc[of 0] P_0 show ?thesis by auto
    next
      case (Suc m)
      from P_Suc[of n] Suc have "f (Suc m) ≤ f (Suc (Suc m))" by simp
      with P_Suc Suc show ?thesis by simp
    qed
  qed
  ultimately show ?thesis by auto
next
  let "?P p m" = "m < p ∧ s m < s p"
  assume "¬ (∀n. ∃p>n. (∀m≥p. s m ≤ s p))"
  then obtain N where N: "!!p. p > N ==> ∃m>p. s p < s m" by (force simp: not_le le_less)
  def f ≡ "nat_rec (SOME p. ?P p (Suc N)) (λ_ n. SOME p. ?P p n)"
  have f_0: "f 0 = (SOME p. ?P p (Suc N))" unfolding f_def by simp
  have f_Suc: "!!i. f (Suc i) = (SOME p. ?P p (f i))" unfolding f_def nat_rec_Suc ..
  have P_0: "?P (f 0) (Suc N)"
    unfolding f_0 some_eq_ex[of "λp. ?P p (Suc N)"] using N[of "Suc N"] by auto
  { fix i have "N < f i ==> ?P (f (Suc i)) (f i)"
      unfolding f_Suc some_eq_ex[of "λp. ?P p (f i)"] using N[of "f i"] . }
  note P' = this
  { fix i have "N < f i ∧ ?P (f (Suc i)) (f i)"
      by (induct i) (insert P_0 P', auto) }
  then have "subseq f" "monoseq (λx. s (f x))"
    unfolding subseq_Suc_iff monoseq_Suc by (auto simp: not_le intro: less_imp_le)
  then show ?thesis by auto
qed

lemma seq_suble: assumes sf: "subseq f" shows "n ≤ f n"
proof(induct n)
  case 0 thus ?case by simp
next
  case (Suc n)
  from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps
  have "n < f (Suc n)" by arith
  thus ?case by arith
qed

lemma eventually_subseq:
  "subseq r ==> eventually P sequentially ==> eventually (λn. P (r n)) sequentially"
  unfolding eventually_sequentially by (metis seq_suble le_trans)

lemma filterlim_subseq: "subseq f ==> filterlim f sequentially sequentially"
  unfolding filterlim_iff by (metis eventually_subseq)

lemma subseq_o: "subseq r ==> subseq s ==> subseq (r o s)"
  unfolding subseq_def by simp

lemma subseq_mono: assumes "subseq r" "m < n" shows "r m < r n"
  using assms by (auto simp: subseq_def)

lemma incseq_imp_monoseq:  "incseq X ==> monoseq X"
  by (simp add: incseq_def monoseq_def)

lemma decseq_imp_monoseq:  "decseq X ==> monoseq X"
  by (simp add: decseq_def monoseq_def)

lemma decseq_eq_incseq:
  fixes X :: "nat => 'a::ordered_ab_group_add" shows "decseq X = incseq (λn. - X n)" 
  by (simp add: decseq_def incseq_def)

lemma INT_decseq_offset:
  assumes "decseq F"
  shows "(\<Inter>i. F i) = (\<Inter>i∈{n..}. F i)"
proof safe
  fix x i assume x: "x ∈ (\<Inter>i∈{n..}. F i)"
  show "x ∈ F i"
  proof cases
    from x have "x ∈ F n" by auto
    also assume "i ≤ n" with `decseq F` have "F n ⊆ F i"
      unfolding decseq_def by simp
    finally show ?thesis .
  qed (insert x, simp)
qed auto

subsection {* Defintions of limits *}

abbreviation (in topological_space)
  LIMSEQ :: "[nat => 'a, 'a] => bool"
    ("((_)/ ----> (_))" [60, 60] 60) where
  "X ----> L ≡ (X ---> L) sequentially"

definition
  lim :: "(nat => 'a::t2_space) => 'a" where
    --{*Standard definition of limit using choice operator*}
  "lim X = (THE L. X ----> L)"

definition (in topological_space) convergent :: "(nat => 'a) => bool" where
  "convergent X = (∃L. X ----> L)"

definition
  Bseq :: "(nat => 'a::real_normed_vector) => bool" where
    --{*Standard definition for bounded sequence*}
  "Bseq X = (∃K>0.∀n. norm (X n) ≤ K)"

definition (in metric_space) Cauchy :: "(nat => 'a) => bool" where
  "Cauchy X = (∀e>0. ∃M. ∀m ≥ M. ∀n ≥ M. dist (X m) (X n) < e)"


subsection {* Bounded Sequences *}

lemma BseqI': assumes K: "!!n. norm (X n) ≤ K" shows "Bseq X"
unfolding Bseq_def
proof (intro exI conjI allI)
  show "0 < max K 1" by simp
next
  fix n::nat
  have "norm (X n) ≤ K" by (rule K)
  thus "norm (X n) ≤ max K 1" by simp
qed

lemma BseqE: "[|Bseq X; !!K. [|0 < K; ∀n. norm (X n) ≤ K|] ==> Q|] ==> Q"
unfolding Bseq_def by auto

lemma BseqI2': assumes K: "∀n≥N. norm (X n) ≤ K" shows "Bseq X"
proof (rule BseqI')
  let ?A = "norm ` X ` {..N}"
  have 1: "finite ?A" by simp
  fix n::nat
  show "norm (X n) ≤ max K (Max ?A)"
  proof (cases rule: linorder_le_cases)
    assume "n ≥ N"
    hence "norm (X n) ≤ K" using K by simp
    thus "norm (X n) ≤ max K (Max ?A)" by simp
  next
    assume "n ≤ N"
    hence "norm (X n) ∈ ?A" by simp
    with 1 have "norm (X n) ≤ Max ?A" by (rule Max_ge)
    thus "norm (X n) ≤ max K (Max ?A)" by simp
  qed
qed

lemma Bseq_ignore_initial_segment: "Bseq X ==> Bseq (λn. X (n + k))"
unfolding Bseq_def by auto

lemma Bseq_offset: "Bseq (λn. X (n + k)) ==> Bseq X"
apply (erule BseqE)
apply (rule_tac N="k" and K="K" in BseqI2')
apply clarify
apply (drule_tac x="n - k" in spec, simp)
done

lemma Bseq_conv_Bfun: "Bseq X <-> Bfun X sequentially"
unfolding Bfun_def eventually_sequentially
apply (rule iffI)
apply (simp add: Bseq_def)
apply (auto intro: BseqI2')
done


subsection {* Limits of Sequences *}

lemma [trans]: "X=Y ==> Y ----> z ==> X ----> z"
  by simp

lemma LIMSEQ_def: "X ----> L = (∀r>0. ∃no. ∀n≥no. dist (X n) L < r)"
unfolding tendsto_iff eventually_sequentially ..

lemma LIMSEQ_iff:
  fixes L :: "'a::real_normed_vector"
  shows "(X ----> L) = (∀r>0. ∃no. ∀n ≥ no. norm (X n - L) < r)"
unfolding LIMSEQ_def dist_norm ..

lemma LIMSEQ_iff_nz: "X ----> L = (∀r>0. ∃no>0. ∀n≥no. dist (X n) L < r)"
  unfolding LIMSEQ_def by (metis Suc_leD zero_less_Suc)

lemma metric_LIMSEQ_I:
  "(!!r. 0 < r ==> ∃no. ∀n≥no. dist (X n) L < r) ==> X ----> L"
by (simp add: LIMSEQ_def)

lemma metric_LIMSEQ_D:
  "[|X ----> L; 0 < r|] ==> ∃no. ∀n≥no. dist (X n) L < r"
by (simp add: LIMSEQ_def)

lemma LIMSEQ_I:
  fixes L :: "'a::real_normed_vector"
  shows "(!!r. 0 < r ==> ∃no. ∀n≥no. norm (X n - L) < r) ==> X ----> L"
by (simp add: LIMSEQ_iff)

lemma LIMSEQ_D:
  fixes L :: "'a::real_normed_vector"
  shows "[|X ----> L; 0 < r|] ==> ∃no. ∀n≥no. norm (X n - L) < r"
by (simp add: LIMSEQ_iff)

lemma LIMSEQ_const_iff:
  fixes k l :: "'a::t2_space"
  shows "(λn. k) ----> l <-> k = l"
  using trivial_limit_sequentially by (rule tendsto_const_iff)

lemma LIMSEQ_ignore_initial_segment:
  "f ----> a ==> (λn. f (n + k)) ----> a"
apply (rule topological_tendstoI)
apply (drule (2) topological_tendstoD)
apply (simp only: eventually_sequentially)
apply (erule exE, rename_tac N)
apply (rule_tac x=N in exI)
apply simp
done

lemma LIMSEQ_offset:
  "(λn. f (n + k)) ----> a ==> f ----> a"
apply (rule topological_tendstoI)
apply (drule (2) topological_tendstoD)
apply (simp only: eventually_sequentially)
apply (erule exE, rename_tac N)
apply (rule_tac x="N + k" in exI)
apply clarify
apply (drule_tac x="n - k" in spec)
apply (simp add: le_diff_conv2)
done

lemma LIMSEQ_Suc: "f ----> l ==> (λn. f (Suc n)) ----> l"
by (drule_tac k="Suc 0" in LIMSEQ_ignore_initial_segment, simp)

lemma LIMSEQ_imp_Suc: "(λn. f (Suc n)) ----> l ==> f ----> l"
by (rule_tac k="Suc 0" in LIMSEQ_offset, simp)

lemma LIMSEQ_Suc_iff: "(λn. f (Suc n)) ----> l = f ----> l"
by (blast intro: LIMSEQ_imp_Suc LIMSEQ_Suc)

lemma LIMSEQ_linear: "[| X ----> x ; l > 0 |] ==> (λ n. X (n * l)) ----> x"
  unfolding tendsto_def eventually_sequentially
  by (metis div_le_dividend div_mult_self1_is_m le_trans nat_mult_commute)

lemma LIMSEQ_unique:
  fixes a b :: "'a::t2_space"
  shows "[|X ----> a; X ----> b|] ==> a = b"
  using trivial_limit_sequentially by (rule tendsto_unique)

lemma increasing_LIMSEQ:
  fixes f :: "nat => real"
  assumes inc: "!!n. f n ≤ f (Suc n)"
      and bdd: "!!n. f n ≤ l"
      and en: "!!e. 0 < e ==> ∃n. l ≤ f n + e"
  shows "f ----> l"
  unfolding LIMSEQ_def
proof safe
  fix r :: real assume "0 < r"
  with bdd en[of "r / 2"] obtain n where n: "dist (f n) l ≤ r / 2"
    by (auto simp add: field_simps dist_real_def)
  { fix N assume "n ≤ N"
    then have "dist (f N) l ≤ dist (f n) l"
      using incseq_SucI[of f] inc bdd by (auto dest!: incseqD simp: dist_real_def)
    then have "dist (f N) l < r"
      using `0 < r` n by simp }
  with `0 < r` show "∃no. ∀n≥no. dist (f n) l < r"
    by (auto simp add: LIMSEQ_def field_simps intro!: exI[of _ n])
qed

lemma Bseq_inverse_lemma:
  fixes x :: "'a::real_normed_div_algebra"
  shows "[|r ≤ norm x; 0 < r|] ==> norm (inverse x) ≤ inverse r"
apply (subst nonzero_norm_inverse, clarsimp)
apply (erule (1) le_imp_inverse_le)
done

lemma Bseq_inverse:
  fixes a :: "'a::real_normed_div_algebra"
  shows "[|X ----> a; a ≠ 0|] ==> Bseq (λn. inverse (X n))"
unfolding Bseq_conv_Bfun by (rule Bfun_inverse)

lemma LIMSEQ_diff_approach_zero:
  fixes L :: "'a::real_normed_vector"
  shows "g ----> L ==> (%x. f x - g x) ----> 0 ==> f ----> L"
  by (drule (1) tendsto_add, simp)

lemma LIMSEQ_diff_approach_zero2:
  fixes L :: "'a::real_normed_vector"
  shows "f ----> L ==> (%x. f x - g x) ----> 0 ==> g ----> L"
  by (drule (1) tendsto_diff, simp)

text{*An unbounded sequence's inverse tends to 0*}

lemma LIMSEQ_inverse_zero:
  "∀r::real. ∃N. ∀n≥N. r < X n ==> (λn. inverse (X n)) ----> 0"
  apply (rule filterlim_compose[OF tendsto_inverse_0])
  apply (simp add: filterlim_at_infinity[OF order_refl] eventually_sequentially)
  apply (metis abs_le_D1 linorder_le_cases linorder_not_le)
  done

text{*The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity*}

lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----> 0"
  by (metis filterlim_compose tendsto_inverse_0 filterlim_mono order_refl filterlim_Suc
            filterlim_compose[OF filterlim_real_sequentially] at_top_le_at_infinity)

text{*The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
infinity is now easily proved*}

lemma LIMSEQ_inverse_real_of_nat_add:
     "(%n. r + inverse(real(Suc n))) ----> r"
  using tendsto_add [OF tendsto_const LIMSEQ_inverse_real_of_nat] by auto

lemma LIMSEQ_inverse_real_of_nat_add_minus:
     "(%n. r + -inverse(real(Suc n))) ----> r"
  using tendsto_add [OF tendsto_const tendsto_minus [OF LIMSEQ_inverse_real_of_nat]]
  by auto

lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:
     "(%n. r*( 1 + -inverse(real(Suc n)))) ----> r"
  using tendsto_mult [OF tendsto_const LIMSEQ_inverse_real_of_nat_add_minus [of 1]]
  by auto

lemma LIMSEQ_le_const:
  "[|X ----> (x::real); ∃N. ∀n≥N. a ≤ X n|] ==> a ≤ x"
  using tendsto_le_const[of sequentially X x a] by (simp add: eventually_sequentially)

lemma LIMSEQ_le:
  "[|X ----> x; Y ----> y; ∃N. ∀n≥N. X n ≤ Y n|] ==> x ≤ (y::real)"
  using tendsto_le[of sequentially Y y X x] by (simp add: eventually_sequentially)

lemma LIMSEQ_le_const2:
  "[|X ----> (x::real); ∃N. ∀n≥N. X n ≤ a|] ==> x ≤ a"
  by (rule LIMSEQ_le[of X x "λn. a"]) (auto simp: tendsto_const)

subsection {* Convergence *}

lemma limI: "X ----> L ==> lim X = L"
apply (simp add: lim_def)
apply (blast intro: LIMSEQ_unique)
done

lemma convergentD: "convergent X ==> ∃L. (X ----> L)"
by (simp add: convergent_def)

lemma convergentI: "(X ----> L) ==> convergent X"
by (auto simp add: convergent_def)

lemma convergent_LIMSEQ_iff: "convergent X = (X ----> lim X)"
by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def)

lemma convergent_const: "convergent (λn. c)"
  by (rule convergentI, rule tendsto_const)

lemma convergent_add:
  fixes X Y :: "nat => 'a::real_normed_vector"
  assumes "convergent (λn. X n)"
  assumes "convergent (λn. Y n)"
  shows "convergent (λn. X n + Y n)"
  using assms unfolding convergent_def by (fast intro: tendsto_add)

lemma convergent_setsum:
  fixes X :: "'a => nat => 'b::real_normed_vector"
  assumes "!!i. i ∈ A ==> convergent (λn. X i n)"
  shows "convergent (λn. ∑i∈A. X i n)"
proof (cases "finite A")
  case True from this and assms show ?thesis
    by (induct A set: finite) (simp_all add: convergent_const convergent_add)
qed (simp add: convergent_const)

lemma (in bounded_linear) convergent:
  assumes "convergent (λn. X n)"
  shows "convergent (λn. f (X n))"
  using assms unfolding convergent_def by (fast intro: tendsto)

lemma (in bounded_bilinear) convergent:
  assumes "convergent (λn. X n)" and "convergent (λn. Y n)"
  shows "convergent (λn. X n ** Y n)"
  using assms unfolding convergent_def by (fast intro: tendsto)

lemma convergent_minus_iff:
  fixes X :: "nat => 'a::real_normed_vector"
  shows "convergent X <-> convergent (λn. - X n)"
apply (simp add: convergent_def)
apply (auto dest: tendsto_minus)
apply (drule tendsto_minus, auto)
done

lemma lim_le: "convergent f ==> (!!n. f n ≤ (x::real)) ==> lim f ≤ x"
  using LIMSEQ_le_const2[of f "lim f" x] by (simp add: convergent_LIMSEQ_iff)

lemma monoseq_le:
  "monoseq a ==> a ----> (x::real) ==>
    ((∀ n. a n ≤ x) ∧ (∀m. ∀n≥m. a m ≤ a n)) ∨ ((∀ n. x ≤ a n) ∧ (∀m. ∀n≥m. a n ≤ a m))"
  by (metis LIMSEQ_le_const LIMSEQ_le_const2 decseq_def incseq_def monoseq_iff)

lemma LIMSEQ_subseq_LIMSEQ:
  "[| X ----> L; subseq f |] ==> (X o f) ----> L"
  unfolding comp_def by (rule filterlim_compose[of X, OF _ filterlim_subseq])

lemma convergent_subseq_convergent:
  "[|convergent X; subseq f|] ==> convergent (X o f)"
  unfolding convergent_def by (auto intro: LIMSEQ_subseq_LIMSEQ)


subsection {* Bounded Monotonic Sequences *}

text{*Bounded Sequence*}

lemma BseqD: "Bseq X ==> ∃K. 0 < K & (∀n. norm (X n) ≤ K)"
by (simp add: Bseq_def)

lemma BseqI: "[| 0 < K; ∀n. norm (X n) ≤ K |] ==> Bseq X"
by (auto simp add: Bseq_def)

lemma lemma_NBseq_def:
  "(∃K > 0. ∀n. norm (X n) ≤ K) = (∃N. ∀n. norm (X n) ≤ real(Suc N))"
proof safe
  fix K :: real
  from reals_Archimedean2 obtain n :: nat where "K < real n" ..
  then have "K ≤ real (Suc n)" by auto
  moreover assume "∀m. norm (X m) ≤ K"
  ultimately have "∀m. norm (X m) ≤ real (Suc n)"
    by (blast intro: order_trans)
  then show "∃N. ∀n. norm (X n) ≤ real (Suc N)" ..
qed (force simp add: real_of_nat_Suc)

text{* alternative definition for Bseq *}
lemma Bseq_iff: "Bseq X = (∃N. ∀n. norm (X n) ≤ real(Suc N))"
apply (simp add: Bseq_def)
apply (simp (no_asm) add: lemma_NBseq_def)
done

lemma lemma_NBseq_def2:
     "(∃K > 0. ∀n. norm (X n) ≤ K) = (∃N. ∀n. norm (X n) < real(Suc N))"
apply (subst lemma_NBseq_def, auto)
apply (rule_tac x = "Suc N" in exI)
apply (rule_tac [2] x = N in exI)
apply (auto simp add: real_of_nat_Suc)
 prefer 2 apply (blast intro: order_less_imp_le)
apply (drule_tac x = n in spec, simp)
done

(* yet another definition for Bseq *)
lemma Bseq_iff1a: "Bseq X = (∃N. ∀n. norm (X n) < real(Suc N))"
by (simp add: Bseq_def lemma_NBseq_def2)

subsubsection{*A Few More Equivalence Theorems for Boundedness*}

text{*alternative formulation for boundedness*}
lemma Bseq_iff2: "Bseq X = (∃k > 0. ∃x. ∀n. norm (X(n) + -x) ≤ k)"
apply (unfold Bseq_def, safe)
apply (rule_tac [2] x = "k + norm x" in exI)
apply (rule_tac x = K in exI, simp)
apply (rule exI [where x = 0], auto)
apply (erule order_less_le_trans, simp)
apply (drule_tac x=n in spec, fold diff_minus)
apply (drule order_trans [OF norm_triangle_ineq2])
apply simp
done

text{*alternative formulation for boundedness*}
lemma Bseq_iff3: "Bseq X = (∃k > 0. ∃N. ∀n. norm(X(n) + -X(N)) ≤ k)"
apply safe
apply (simp add: Bseq_def, safe)
apply (rule_tac x = "K + norm (X N)" in exI)
apply auto
apply (erule order_less_le_trans, simp)
apply (rule_tac x = N in exI, safe)
apply (drule_tac x = n in spec)
apply (rule order_trans [OF norm_triangle_ineq], simp)
apply (auto simp add: Bseq_iff2)
done

lemma BseqI2: "(∀n. k ≤ f n & f n ≤ (K::real)) ==> Bseq f"
apply (simp add: Bseq_def)
apply (rule_tac x = " (¦k¦ + ¦K¦) + 1" in exI, auto)
apply (drule_tac x = n in spec, arith)
done

subsubsection{*Upper Bounds and Lubs of Bounded Sequences*}

lemma Bseq_isUb:
  "!!(X::nat=>real). Bseq X ==> ∃U. isUb (UNIV::real set) {x. ∃n. X n = x} U"
by (auto intro: isUbI setleI simp add: Bseq_def abs_le_iff)

text{* Use completeness of reals (supremum property)
   to show that any bounded sequence has a least upper bound*}

lemma Bseq_isLub:
  "!!(X::nat=>real). Bseq X ==>
   ∃U. isLub (UNIV::real set) {x. ∃n. X n = x} U"
by (blast intro: reals_complete Bseq_isUb)

subsubsection{*A Bounded and Monotonic Sequence Converges*}

(* TODO: delete *)
(* FIXME: one use in NSA/HSEQ.thy *)
lemma Bmonoseq_LIMSEQ: "∀n. m ≤ n --> X n = X m ==> ∃L. (X ----> L)"
unfolding tendsto_def eventually_sequentially
apply (rule_tac x = "X m" in exI, safe)
apply (rule_tac x = m in exI, safe)
apply (drule spec, erule impE, auto)
done

text {* A monotone sequence converges to its least upper bound. *}

lemma isLub_mono_imp_LIMSEQ:
  fixes X :: "nat => real"
  assumes u: "isLub UNIV {x. ∃n. X n = x} u" (* FIXME: use 'range X' *)
  assumes X: "∀m n. m ≤ n --> X m ≤ X n"
  shows "X ----> u"
proof (rule LIMSEQ_I)
  have 1: "∀n. X n ≤ u"
    using isLubD2 [OF u] by auto
  have "∀y. (∀n. X n ≤ y) --> u ≤ y"
    using isLub_le_isUb [OF u] by (auto simp add: isUb_def setle_def)
  hence 2: "∀y<u. ∃n. y < X n"
    by (metis not_le)
  fix r :: real assume "0 < r"
  hence "u - r < u" by simp
  hence "∃m. u - r < X m" using 2 by simp
  then obtain m where "u - r < X m" ..
  with X have "∀n≥m. u - r < X n"
    by (fast intro: less_le_trans)
  hence "∃m. ∀n≥m. u - r < X n" ..
  thus "∃m. ∀n≥m. norm (X n - u) < r"
    using 1 by (simp add: diff_less_eq add_commute)
qed

text{*A standard proof of the theorem for monotone increasing sequence*}

lemma Bseq_mono_convergent:
   "Bseq X ==> ∀m. ∀n ≥ m. X m ≤ X n ==> convergent (X::nat=>real)"
  by (metis Bseq_isLub isLub_mono_imp_LIMSEQ convergentI)

lemma Bseq_minus_iff: "Bseq (%n. -(X n)) = Bseq X"
  by (simp add: Bseq_def)

text{*Main monotonicity theorem*}

lemma Bseq_monoseq_convergent: "Bseq X ==> monoseq X ==> convergent (X::nat=>real)"
  by (metis monoseq_iff incseq_def decseq_eq_incseq convergent_minus_iff Bseq_minus_iff
            Bseq_mono_convergent)

subsubsection{*Increasing and Decreasing Series*}

lemma incseq_le: "incseq X ==> X ----> L ==> X n ≤ (L::real)"
  by (metis incseq_def LIMSEQ_le_const)

lemma decseq_le: "decseq X ==> X ----> L ==> (L::real) ≤ X n"
  by (metis decseq_def LIMSEQ_le_const2)

subsection {* Cauchy Sequences *}

lemma metric_CauchyI:
  "(!!e. 0 < e ==> ∃M. ∀m≥M. ∀n≥M. dist (X m) (X n) < e) ==> Cauchy X"
  by (simp add: Cauchy_def)

lemma metric_CauchyD:
  "Cauchy X ==> 0 < e ==> ∃M. ∀m≥M. ∀n≥M. dist (X m) (X n) < e"
  by (simp add: Cauchy_def)

lemma Cauchy_iff:
  fixes X :: "nat => 'a::real_normed_vector"
  shows "Cauchy X <-> (∀e>0. ∃M. ∀m≥M. ∀n≥M. norm (X m - X n) < e)"
  unfolding Cauchy_def dist_norm ..

lemma Cauchy_iff2:
  "Cauchy X = (∀j. (∃M. ∀m ≥ M. ∀n ≥ M. ¦X m - X n¦ < inverse(real (Suc j))))"
apply (simp add: Cauchy_iff, auto)
apply (drule reals_Archimedean, safe)
apply (drule_tac x = n in spec, auto)
apply (rule_tac x = M in exI, auto)
apply (drule_tac x = m in spec, simp)
apply (drule_tac x = na in spec, auto)
done

lemma CauchyI:
  fixes X :: "nat => 'a::real_normed_vector"
  shows "(!!e. 0 < e ==> ∃M. ∀m≥M. ∀n≥M. norm (X m - X n) < e) ==> Cauchy X"
by (simp add: Cauchy_iff)

lemma CauchyD:
  fixes X :: "nat => 'a::real_normed_vector"
  shows "[|Cauchy X; 0 < e|] ==> ∃M. ∀m≥M. ∀n≥M. norm (X m - X n) < e"
by (simp add: Cauchy_iff)

lemma Cauchy_subseq_Cauchy:
  "[| Cauchy X; subseq f |] ==> Cauchy (X o f)"
apply (auto simp add: Cauchy_def)
apply (drule_tac x=e in spec, clarify)
apply (rule_tac x=M in exI, clarify)
apply (blast intro: le_trans [OF _ seq_suble] dest!: spec)
done

subsubsection {* Cauchy Sequences are Bounded *}

text{*A Cauchy sequence is bounded -- this is the standard
  proof mechanization rather than the nonstandard proof*}

lemma lemmaCauchy: "∀n ≥ M. norm (X M - X n) < (1::real)
          ==>  ∀n ≥ M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
apply (clarify, drule spec, drule (1) mp)
apply (simp only: norm_minus_commute)
apply (drule order_le_less_trans [OF norm_triangle_ineq2])
apply simp
done

lemma Cauchy_Bseq: "Cauchy X ==> Bseq X"
apply (simp add: Cauchy_iff)
apply (drule spec, drule mp, rule zero_less_one, safe)
apply (drule_tac x="M" in spec, simp)
apply (drule lemmaCauchy)
apply (rule_tac k="M" in Bseq_offset)
apply (simp add: Bseq_def)
apply (rule_tac x="1 + norm (X M)" in exI)
apply (rule conjI, rule order_less_le_trans [OF zero_less_one], simp)
apply (simp add: order_less_imp_le)
done

subsubsection {* Cauchy Sequences are Convergent *}

class complete_space = metric_space +
  assumes Cauchy_convergent: "Cauchy X ==> convergent X"

class banach = real_normed_vector + complete_space

theorem LIMSEQ_imp_Cauchy:
  assumes X: "X ----> a" shows "Cauchy X"
proof (rule metric_CauchyI)
  fix e::real assume "0 < e"
  hence "0 < e/2" by simp
  with X have "∃N. ∀n≥N. dist (X n) a < e/2" by (rule metric_LIMSEQ_D)
  then obtain N where N: "∀n≥N. dist (X n) a < e/2" ..
  show "∃N. ∀m≥N. ∀n≥N. dist (X m) (X n) < e"
  proof (intro exI allI impI)
    fix m assume "N ≤ m"
    hence m: "dist (X m) a < e/2" using N by fast
    fix n assume "N ≤ n"
    hence n: "dist (X n) a < e/2" using N by fast
    have "dist (X m) (X n) ≤ dist (X m) a + dist (X n) a"
      by (rule dist_triangle2)
    also from m n have "… < e" by simp
    finally show "dist (X m) (X n) < e" .
  qed
qed

lemma convergent_Cauchy: "convergent X ==> Cauchy X"
unfolding convergent_def
by (erule exE, erule LIMSEQ_imp_Cauchy)

lemma Cauchy_convergent_iff:
  fixes X :: "nat => 'a::complete_space"
  shows "Cauchy X = convergent X"
by (fast intro: Cauchy_convergent convergent_Cauchy)

text {*
Proof that Cauchy sequences converge based on the one from
http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html
*}

text {*
  If sequence @{term "X"} is Cauchy, then its limit is the lub of
  @{term "{r::real. ∃N. ∀n≥N. r < X n}"}
*}

lemma isUb_UNIV_I: "(!!y. y ∈ S ==> y ≤ u) ==> isUb UNIV S u"
by (simp add: isUbI setleI)

lemma real_Cauchy_convergent:
  fixes X :: "nat => real"
  assumes X: "Cauchy X"
  shows "convergent X"
proof -
  def S ≡ "{x::real. ∃N. ∀n≥N. x < X n}"
  then have mem_S: "!!N x. ∀n≥N. x < X n ==> x ∈ S" by auto

  { fix N x assume N: "∀n≥N. X n < x"
  have "isUb UNIV S x"
  proof (rule isUb_UNIV_I)
  fix y::real assume "y ∈ S"
  hence "∃M. ∀n≥M. y < X n"
    by (simp add: S_def)
  then obtain M where "∀n≥M. y < X n" ..
  hence "y < X (max M N)" by simp
  also have "… < x" using N by simp
  finally show "y ≤ x"
    by (rule order_less_imp_le)
  qed }
  note bound_isUb = this 

  have "∃u. isLub UNIV S u"
  proof (rule reals_complete)
  obtain N where "∀m≥N. ∀n≥N. norm (X m - X n) < 1"
    using CauchyD [OF X zero_less_one] by auto
  hence N: "∀n≥N. norm (X n - X N) < 1" by simp
  show "∃x. x ∈ S"
  proof
    from N have "∀n≥N. X N - 1 < X n"
      by (simp add: abs_diff_less_iff)
    thus "X N - 1 ∈ S" by (rule mem_S)
  qed
  show "∃u. isUb UNIV S u"
  proof
    from N have "∀n≥N. X n < X N + 1"
      by (simp add: abs_diff_less_iff)
    thus "isUb UNIV S (X N + 1)"
      by (rule bound_isUb)
  qed
  qed
  then obtain x where x: "isLub UNIV S x" ..
  have "X ----> x"
  proof (rule LIMSEQ_I)
  fix r::real assume "0 < r"
  hence r: "0 < r/2" by simp
  obtain N where "∀n≥N. ∀m≥N. norm (X n - X m) < r/2"
    using CauchyD [OF X r] by auto
  hence "∀n≥N. norm (X n - X N) < r/2" by simp
  hence N: "∀n≥N. X N - r/2 < X n ∧ X n < X N + r/2"
    by (simp only: real_norm_def abs_diff_less_iff)

  from N have "∀n≥N. X N - r/2 < X n" by fast
  hence "X N - r/2 ∈ S" by (rule mem_S)
  hence 1: "X N - r/2 ≤ x" using x isLub_isUb isUbD by fast

  from N have "∀n≥N. X n < X N + r/2" by fast
  hence "isUb UNIV S (X N + r/2)" by (rule bound_isUb)
  hence 2: "x ≤ X N + r/2" using x isLub_le_isUb by fast

  show "∃N. ∀n≥N. norm (X n - x) < r"
  proof (intro exI allI impI)
    fix n assume n: "N ≤ n"
    from N n have "X n < X N + r/2" and "X N - r/2 < X n" by simp+
    thus "norm (X n - x) < r" using 1 2
      by (simp add: abs_diff_less_iff)
  qed
  qed
  then show ?thesis unfolding convergent_def by auto
qed

instance real :: banach
  by intro_classes (rule real_Cauchy_convergent)


subsection {* Power Sequences *}

text{*The sequence @{term "x^n"} tends to 0 if @{term "0≤x"} and @{term
"x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
  also fact that bounded and monotonic sequence converges.*}

lemma Bseq_realpow: "[| 0 ≤ (x::real); x ≤ 1 |] ==> Bseq (%n. x ^ n)"
apply (simp add: Bseq_def)
apply (rule_tac x = 1 in exI)
apply (simp add: power_abs)
apply (auto dest: power_mono)
done

lemma monoseq_realpow: fixes x :: real shows "[| 0 ≤ x; x ≤ 1 |] ==> monoseq (%n. x ^ n)"
apply (clarify intro!: mono_SucI2)
apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
done

lemma convergent_realpow:
  "[| 0 ≤ (x::real); x ≤ 1 |] ==> convergent (%n. x ^ n)"
by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)

lemma LIMSEQ_inverse_realpow_zero: "1 < (x::real) ==> (λn. inverse (x ^ n)) ----> 0"
  by (rule filterlim_compose[OF tendsto_inverse_0 filterlim_realpow_sequentially_gt1]) simp

lemma LIMSEQ_realpow_zero:
  "[|0 ≤ (x::real); x < 1|] ==> (λn. x ^ n) ----> 0"
proof cases
  assume "0 ≤ x" and "x ≠ 0"
  hence x0: "0 < x" by simp
  assume x1: "x < 1"
  from x0 x1 have "1 < inverse x"
    by (rule one_less_inverse)
  hence "(λn. inverse (inverse x ^ n)) ----> 0"
    by (rule LIMSEQ_inverse_realpow_zero)
  thus ?thesis by (simp add: power_inverse)
qed (rule LIMSEQ_imp_Suc, simp add: tendsto_const)

lemma LIMSEQ_power_zero:
  fixes x :: "'a::{real_normed_algebra_1}"
  shows "norm x < 1 ==> (λn. x ^ n) ----> 0"
apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
apply (simp only: tendsto_Zfun_iff, erule Zfun_le)
apply (simp add: power_abs norm_power_ineq)
done

lemma LIMSEQ_divide_realpow_zero: "1 < x ==> (λn. a / (x ^ n) :: real) ----> 0"
  by (rule tendsto_divide_0 [OF tendsto_const filterlim_realpow_sequentially_gt1]) simp

text{*Limit of @{term "c^n"} for @{term"¦c¦ < 1"}*}

lemma LIMSEQ_rabs_realpow_zero: "¦c¦ < 1 ==> (λn. ¦c¦ ^ n :: real) ----> 0"
  by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])

lemma LIMSEQ_rabs_realpow_zero2: "¦c¦ < 1 ==> (λn. c ^ n :: real) ----> 0"
  by (rule LIMSEQ_power_zero) simp

lemma tendsto_at_topI_sequentially:
  fixes f :: "real => real"
  assumes mono: "mono f"
  assumes limseq: "(λn. f (real n)) ----> y"
  shows "(f ---> y) at_top"
proof (rule tendstoI)
  fix e :: real assume "0 < e"
  with limseq obtain N :: nat where N: "!!n. N ≤ n ==> ¦f (real n) - y¦ < e"
    by (auto simp: LIMSEQ_def dist_real_def)
  { fix x :: real
    from ex_le_of_nat[of x] guess n ..
    note monoD[OF mono this]
    also have "f (real_of_nat n) ≤ y"
      by (rule LIMSEQ_le_const[OF limseq])
         (auto intro: exI[of _ n] monoD[OF mono] simp: real_eq_of_nat[symmetric])
    finally have "f x ≤ y" . }
  note le = this
  have "eventually (λx. real N ≤ x) at_top"
    by (rule eventually_ge_at_top)
  then show "eventually (λx. dist (f x) y < e) at_top"
  proof eventually_elim
    fix x assume N': "real N ≤ x"
    with N[of N] le have "y - f (real N) < e" by auto
    moreover note monoD[OF mono N']
    ultimately show "dist (f x) y < e"
      using le[of x] by (auto simp: dist_real_def field_simps)
  qed
qed

end