Theory Multiseries_Expansion

(*
  File:    Multiseries_Expansion.thy
  Author:  Manuel Eberl, TU München

  Asymptotic bases and Multiseries expansions of real-valued functions.
  Contains automation to prove limits and asymptotic relationships between such functions.
*)
section ‹Multiseries expansions›
theory Multiseries_Expansion
imports
  "HOL-Library.BNF_Corec"
  "HOL-Library.Landau_Symbols"
  Lazy_Eval
  Inst_Existentials
  Eventuallize
begin

(* TODO Move *)
lemma real_powr_at_top: 
  assumes "(p::real) > 0"
  shows   "filterlim (λx. x powr p) at_top at_top"
proof (subst filterlim_cong[OF refl refl])
  show "LIM x at_top. exp (p * ln x) :> at_top"
    by (rule filterlim_compose[OF exp_at_top filterlim_tendsto_pos_mult_at_top[OF tendsto_const]])
       (simp_all add: ln_at_top assms)
  show "eventually (λx. x powr p = exp (p * ln x)) at_top"
    using eventually_gt_at_top[of 0] by eventually_elim (simp add: powr_def)
qed


subsection ‹Streams and lazy lists›

codatatype 'a msstream = MSSCons 'a "'a msstream"

primrec mssnth :: "'a msstream ⇒ nat ⇒ 'a" where
  "mssnth xs 0 = (case xs of MSSCons x xs ⇒ x)"
| "mssnth xs (Suc n) = (case xs of MSSCons x xs ⇒ mssnth xs n)"

codatatype 'a msllist = MSLNil | MSLCons 'a "'a msllist"
  for map: mslmap

fun lcoeff where
  "lcoeff MSLNil n = 0"
| "lcoeff (MSLCons x xs) 0 = x"
| "lcoeff (MSLCons x xs) (Suc n) = lcoeff xs n"

primcorec msllist_of_msstream :: "'a msstream ⇒ 'a msllist" where
  "msllist_of_msstream xs = (case xs of MSSCons x xs ⇒ MSLCons x (msllist_of_msstream xs))"
  
lemma msllist_of_msstream_MSSCons [simp]: 
  "msllist_of_msstream (MSSCons x xs) = MSLCons x (msllist_of_msstream xs)"
  by (subst msllist_of_msstream.code) simp

lemma lcoeff_llist_of_stream [simp]: "lcoeff (msllist_of_msstream xs) n = mssnth xs n"
  by (induction "msllist_of_msstream xs" n arbitrary: xs rule: lcoeff.induct;
      subst msllist_of_msstream.code) (auto split: msstream.splits)


subsection ‹Convergent power series›

subsubsection ‹Definition›

definition convergent_powser :: "real msllist ⇒ bool" where
  "convergent_powser cs ⟷ (∃R>0. ∀x. abs x < R ⟶ summable (λn. lcoeff cs n * x ^ n))"
 
lemma convergent_powser_stl:
  assumes "convergent_powser (MSLCons c cs)"
  shows   "convergent_powser cs"
proof -
  from assms obtain R where "R > 0" "⋀x. abs x < R ⟹ summable (λn. lcoeff (MSLCons c cs) n * x ^ n)"
    unfolding convergent_powser_def by blast
  hence "summable (λn. lcoeff cs n * x ^ n)" if "abs x < R" for x
    using that by (subst (asm) summable_powser_split_head [symmetric]) simp
  thus ?thesis using ‹R > 0› by (auto simp: convergent_powser_def)
qed
  
  
definition powser :: "real msllist ⇒ real ⇒ real" where
  "powser cs x = suminf (λn. lcoeff cs n * x ^ n)"

lemma isCont_powser:
  assumes "convergent_powser cs"
  shows   "isCont (powser cs) 0"
proof -
  from assms obtain R where R: "R > 0" "⋀x. abs x < R ⟹ summable (λn. lcoeff cs n * x ^ n)"
    unfolding convergent_powser_def by blast
  hence *: "summable (λn. lcoeff cs n * (R/2) ^ n)" by (intro R) simp_all
  from ‹R > 0› show ?thesis unfolding powser_def
    by (intro isCont_powser[OF *]) simp_all
qed

lemma powser_MSLNil [simp]: "powser MSLNil = (λ_. 0)"
  by (simp add: fun_eq_iff powser_def)

lemma powser_MSLCons:
  assumes "convergent_powser (MSLCons c cs)"
  shows   "eventually (λx. powser (MSLCons c cs) x = x * powser cs x + c) (nhds 0)"
proof -
  from assms obtain R where R: "R > 0" "⋀x. abs x < R ⟹ summable (λn. lcoeff (MSLCons c cs) n * x ^ n)"
    unfolding convergent_powser_def by blast
  from R have "powser (MSLCons c cs) x = x * powser cs x + c" if "abs x < R" for x
    using that unfolding powser_def by (subst powser_split_head) simp_all
  moreover have "eventually (λx. abs x < R) (nhds 0)"
    using ‹R > 0› filterlim_ident[of "nhds (0::real)"]
    by (subst (asm) tendsto_iff) (simp add: dist_real_def)
  ultimately show ?thesis by (auto elim: eventually_mono)
qed

definition convergent_powser' :: "real msllist ⇒ (real ⇒ real) ⇒ bool" where
  "convergent_powser' cs f ⟷ (∃R>0. ∀x. abs x < R ⟶ (λn. lcoeff cs n * x ^ n) sums f x)"

lemma convergent_powser'_imp_convergent_powser:
  "convergent_powser' cs f ⟹ convergent_powser cs" 
  unfolding convergent_powser_def convergent_powser'_def by (auto simp: sums_iff)

lemma convergent_powser'_eventually:
  assumes "convergent_powser' cs f"
  shows   "eventually (λx. powser cs x = f x) (nhds 0)"
proof -
  from assms obtain R where "R > 0" "⋀x. abs x < R ⟹ (λn. lcoeff cs n * x ^ n) sums f x"
    unfolding convergent_powser'_def by blast
  hence "powser cs x = f x" if "abs x < R" for x
    using that by (simp add: powser_def sums_iff)
  moreover have "eventually (λx. abs x < R) (nhds 0)"
    using ‹R > 0› filterlim_ident[of "nhds (0::real)"]
    by (subst (asm) tendsto_iff) (simp add: dist_real_def)
  ultimately show ?thesis by (auto elim: eventually_mono)
qed  


subsubsection ‹Geometric series›

primcorec mssalternate :: "'a ⇒ 'a ⇒ 'a msstream" where
  "mssalternate a b = MSSCons a (mssalternate b a)"

lemma case_mssalternate [simp]: 
  "(case mssalternate a b of MSSCons c d ⇒ f c d) = f a (mssalternate b a)"
  by (subst mssalternate.code) simp

lemma mssnth_mssalternate: "mssnth (mssalternate a b) n = (if even n then a else b)"
  by (induction n arbitrary: a b; subst mssalternate.code) simp_all
  
lemma convergent_powser'_geometric:
  "convergent_powser' (msllist_of_msstream (mssalternate 1 (-1))) (λx. inverse (1 + x))"
proof -
  have "(λn. lcoeff (msllist_of_msstream (mssalternate 1 (-1))) n * x ^ n) sums (inverse (1 + x))"
    if "abs x < 1" for x :: real
  proof -
    have "(λn. (-1) ^ n * x ^ n) sums (inverse (1 + x))"
      using that geometric_sums[of "-x"] by (simp add: field_simps power_mult_distrib [symmetric])
    also have "(λn. (-1) ^ n * x ^ n) =
                 (λn. lcoeff (msllist_of_msstream (mssalternate 1 (-1))) n * x ^ n)"
      by (auto simp add: mssnth_mssalternate fun_eq_iff)
    finally show ?thesis .
  qed
  thus ?thesis unfolding convergent_powser'_def by (force intro!: exI[of _ 1])
qed


subsubsection ‹Exponential series›

primcorec exp_series_stream_aux :: "real ⇒ real ⇒ real msstream" where
  "exp_series_stream_aux acc n = MSSCons acc (exp_series_stream_aux (acc / n) (n + 1))"
  
lemma mssnth_exp_series_stream_aux:
  "mssnth (exp_series_stream_aux (1 / fact n) (n + 1)) m = 1 / fact (n + m)"
proof (induction m arbitrary: n)
  case (0 n)
  thus ?case by (subst exp_series_stream_aux.code) simp
next
  case (Suc m n)
  from Suc.IH[of "n + 1"] show ?case
    by (subst exp_series_stream_aux.code) (simp add: algebra_simps)
qed

definition exp_series_stream :: "real msstream" where
  "exp_series_stream = exp_series_stream_aux 1 1"
  
lemma mssnth_exp_series_stream:
  "mssnth exp_series_stream n = 1 / fact n"
  unfolding exp_series_stream_def using mssnth_exp_series_stream_aux[of 0 n] by simp

lemma convergent_powser'_exp:
  "convergent_powser' (msllist_of_msstream exp_series_stream) exp"
proof -
  have "(λn. lcoeff (msllist_of_msstream exp_series_stream) n * x ^ n) sums exp x" for x :: real
    using exp_converges[of x] by (simp add: mssnth_exp_series_stream field_split_simps)
  thus ?thesis by (auto intro: exI[of _ "1::real"] simp: convergent_powser'_def)
qed
  

subsubsection ‹Logarithm series›

primcorec ln_series_stream_aux :: "bool ⇒ real ⇒ real msstream" where
  "ln_series_stream_aux b n = 
     MSSCons (if b then -inverse n else inverse n) (ln_series_stream_aux (¬b) (n+1))"

lemma mssnth_ln_series_stream_aux:
  "mssnth (ln_series_stream_aux b x) n = 
     (if b then - ((-1)^n) * inverse (x + real n) else ((-1)^n) * inverse (x + real n))"
  by (induction n arbitrary: b x; subst ln_series_stream_aux.code) simp_all

definition ln_series_stream :: "real msstream" where
  "ln_series_stream = MSSCons 0 (ln_series_stream_aux False 1)"
  
lemma mssnth_ln_series_stream:
  "mssnth ln_series_stream n = (-1) ^ Suc n / real n"
  unfolding ln_series_stream_def 
  by (cases n) (simp_all add: mssnth_ln_series_stream_aux field_simps)

lemma ln_series': 
  assumes "abs (x::real) < 1"
  shows   "(λn. - ((-x)^n) / of_nat n) sums ln (1 + x)"
proof -
  have "∀n≥1. norm (-((-x)^n)) / of_nat n ≤ norm x ^ n / 1"
    by (intro exI[of _ "1::nat"] allI impI frac_le) (simp_all add: power_abs)
  hence "∃N. ∀n≥N. norm (-((-x)^n) / of_nat n) ≤ norm x ^ n" 
    by (intro exI[of _ 1]) simp_all
  moreover from assms have "summable (λn. norm x ^ n)" 
    by (intro summable_geometric) simp_all
  ultimately have *: "summable (λn. - ((-x)^n) / of_nat n)"
    by (rule summable_comparison_test)
  moreover from assms have "0 < 1 + x" "1 + x < 2" by simp_all
  from ln_series[OF this] 
    have "ln (1 + x) = (∑n. - ((-x) ^ Suc n) / real (Suc n))"
    by (simp_all add: power_minus' mult_ac)
  hence "ln (1 + x) = (∑n. - ((-x) ^ n / real n))"
    by (subst (asm) suminf_split_head[OF *]) simp_all
  ultimately show ?thesis by (simp add: sums_iff)
qed

lemma convergent_powser'_ln:
  "convergent_powser' (msllist_of_msstream ln_series_stream) (λx. ln (1 + x))"
proof -
  have "(λn. lcoeff (msllist_of_msstream ln_series_stream)n * x ^ n) sums ln (1 + x)"
    if "abs x < 1" for x
  proof -
    from that have "(λn. - ((- x) ^ n) / real n) sums ln (1 + x)" by (rule ln_series')
    also have "(λn. - ((- x) ^ n) / real n) =
                 (λn. lcoeff (msllist_of_msstream ln_series_stream) n * x ^ n)"
      by (auto simp: fun_eq_iff mssnth_ln_series_stream power_mult_distrib [symmetric])
    finally show ?thesis .
  qed
  thus ?thesis unfolding convergent_powser'_def by (force intro!: exI[of _ 1])
qed


subsubsection ‹Generalized binomial series›

primcorec gbinomial_series_aux :: "bool ⇒ real ⇒ real ⇒ real ⇒ real msllist" where
  "gbinomial_series_aux abort x n acc =
     (if abort ∧ acc = 0 then MSLNil else 
        MSLCons acc (gbinomial_series_aux abort x (n + 1) ((x - n) / (n + 1) * acc)))"

lemma gbinomial_series_abort [simp]: "gbinomial_series_aux True x n 0 = MSLNil"
  by (subst gbinomial_series_aux.code) simp

definition gbinomial_series :: "bool ⇒ real ⇒ real msllist" where
  "gbinomial_series abort x = gbinomial_series_aux abort x 0 1"


(* TODO Move *)
lemma gbinomial_Suc_rec:
  "(x gchoose (Suc k)) = (x gchoose k) * ((x - of_nat k) / (of_nat k + 1))"
proof -
  have "((x - 1) + 1) gchoose (Suc k) = x * (x - 1 gchoose k) / (1 + of_nat k)"
    by (subst gbinomial_factors) simp
  also have "x * (x - 1 gchoose k) = (x - of_nat k) * (x gchoose k)"
    by (rule gbinomial_absorb_comp [symmetric])
  finally show ?thesis by (simp add: algebra_simps)
qed

lemma lcoeff_gbinomial_series_aux:
  "lcoeff (gbinomial_series_aux abort x m (x gchoose m)) n = (x gchoose (n + m))"
proof (induction n arbitrary: m)
  case 0
  show ?case by (subst gbinomial_series_aux.code) simp
next
  case (Suc n m)
  show ?case
  proof (cases "abort ∧ (x gchoose m) = 0")
    case True
    with gbinomial_mult_fact[of m x] obtain k where "x = real k" "k < m"
      by auto
    hence "(x gchoose Suc (n + m)) = 0"
      using gbinomial_mult_fact[of "Suc (n + m)" x]
      by (simp add: gbinomial_altdef_of_nat)
    with True show ?thesis by (subst gbinomial_series_aux.code) simp
  next
    case False
    hence "lcoeff (gbinomial_series_aux abort x m (x gchoose m)) (Suc n) = 
             lcoeff (gbinomial_series_aux abort x (Suc m) ((x gchoose m) *
             ((x - real m) / (real m + 1)))) n"
      by (subst gbinomial_series_aux.code) (auto simp: algebra_simps)
    also have "((x gchoose m) * ((x - real m) / (real m + 1))) = x gchoose (Suc m)" 
      by (rule gbinomial_Suc_rec [symmetric])
    also have "lcoeff (gbinomial_series_aux abort x (Suc m) …) n = x gchoose (n + Suc m)"
      by (rule Suc.IH)
    finally show ?thesis by simp
  qed
qed

lemma lcoeff_gbinomial_series [simp]:
  "lcoeff (gbinomial_series abort x) n = (x gchoose n)"
  using lcoeff_gbinomial_series_aux[of abort x 0 n] by (simp add: gbinomial_series_def)

lemma gbinomial_ratio_limit:
  fixes a :: "'a :: real_normed_field"
  assumes "a ∉ ℕ"
  shows "(λn. (a gchoose n) / (a gchoose Suc n)) ⇢ -1"
proof (rule Lim_transform_eventually)
  let ?f = "λn. inverse (a / of_nat (Suc n) - of_nat n / of_nat (Suc n))"
  from eventually_gt_at_top[of "0::nat"]
    show "eventually (λn. ?f n = (a gchoose n) /(a gchoose Suc n)) sequentially"
  proof eventually_elim
    fix n :: nat assume n: "n > 0"
    then obtain q where q: "n = Suc q" by (cases n) blast
    let ?P = "∏i=0..<n. a - of_nat i"
    from n have "(a gchoose n) / (a gchoose Suc n) = (of_nat (Suc n) :: 'a) *
                   (?P / (∏i=0..n. a - of_nat i))"
      by (simp add: gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost)
    also from q have "(∏i=0..n. a - of_nat i) = ?P * (a - of_nat n)"
      by (simp add: prod.atLeast0_atMost_Suc atLeastLessThanSuc_atLeastAtMost)
    also have "?P / … = (?P / ?P) / (a - of_nat n)" by (rule divide_divide_eq_left[symmetric])
    also from assms have "?P / ?P = 1" by auto
    also have "of_nat (Suc n) * (1 / (a - of_nat n)) =
                   inverse (inverse (of_nat (Suc n)) * (a - of_nat n))" by (simp add: field_simps)
    also have "inverse (of_nat (Suc n)) * (a - of_nat n) =
                 a / of_nat (Suc n) - of_nat n / of_nat (Suc n)"
      by (simp add: field_simps del: of_nat_Suc)
    finally show "?f n = (a gchoose n) / (a gchoose Suc n)" by simp
  qed

  have "(λn. norm a / (of_nat (Suc n))) ⇢ 0"
    unfolding divide_inverse
    by (intro tendsto_mult_right_zero LIMSEQ_inverse_real_of_nat)
  hence "(λn. a / of_nat (Suc n)) ⇢ 0"
    by (subst tendsto_norm_zero_iff[symmetric]) (simp add: norm_divide del: of_nat_Suc)
  hence "?f ⇢ inverse (0 - 1)"
    by (intro tendsto_inverse tendsto_diff LIMSEQ_n_over_Suc_n) simp_all
  thus "?f ⇢ -1" by simp
qed
  
lemma summable_gbinomial:
  fixes a z :: real
  assumes "norm z < 1"
  shows   "summable (λn. (a gchoose n) * z ^ n)"
proof (cases "z = 0 ∨ a ∈ ℕ")
  case False
  define r where "r = (norm z + 1) / 2"
  from assms have r: "r > norm z" "r < 1" by (simp_all add: r_def field_simps)
  from False have "abs z > 0" by auto
  from False have "a ∉ ℕ" by (auto elim!: Nats_cases)
  hence *: "(λn. norm (z / ((a gchoose n) / (a gchoose (Suc n))))) ⇢ norm (z / (-1))"
    by (intro tendsto_norm tendsto_divide tendsto_const gbinomial_ratio_limit) simp_all
  hence "∀F x in at_top. norm (z / ((a gchoose x) / (a gchoose Suc x))) > 0"
    using assms False by (intro order_tendstoD) auto
  hence nz: "∀F x in at_top. (a gchoose x) ≠ 0" by eventually_elim auto
      
  have "∀F x in at_top. norm (z / ((a gchoose x) / (a gchoose Suc x))) < r"
    using assms r by (intro order_tendstoD(2)[OF *]) simp_all
  with nz have "∀F n in at_top. norm ((a gchoose (Suc n)) * z) ≤ r * norm (a gchoose n)"
    by eventually_elim (simp add: field_simps abs_mult split: if_splits)
  hence "∀F n in at_top. norm (z ^ n) * norm ((a gchoose (Suc n)) * z) ≤
                           norm (z ^ n) * (r * norm (a gchoose n))"
    by eventually_elim (insert False, intro mult_left_mono, auto)
  hence "∀F n in at_top. norm ((a gchoose (Suc n)) * z ^ (Suc n)) ≤ 
                           r * norm ((a gchoose n) * z ^ n)"
    by (simp add: abs_mult mult_ac)
  then obtain N where N: "⋀n. n ≥ N ⟹ norm ((a gchoose (Suc n)) * z ^ (Suc n)) ≤
                                          r * norm ((a gchoose n) * z ^ n)"
    unfolding eventually_at_top_linorder by blast
  show "summable (λn. (a gchoose n) * z ^ n)"
    by (intro summable_ratio_test[OF r(2), of N] N)
next
  case True
  thus ?thesis
  proof
    assume "a ∈ ℕ"
    then obtain n where [simp]: "a = of_nat n" by (auto elim: Nats_cases)
    from eventually_gt_at_top[of n] 
      have "eventually (λn. (a gchoose n) * z ^ n = 0) at_top"
      by eventually_elim (simp add: binomial_gbinomial [symmetric])
    from summable_cong[OF this] show ?thesis by simp
  qed auto
qed

lemma gen_binomial_real:
  fixes z :: real
  assumes "norm z < 1"
  shows   "(λn. (a gchoose n) * z^n) sums (1 + z) powr a"
proof -
  define K where "K = 1 - (1 - norm z) / 2"
  from assms have K: "K > 0" "K < 1" "norm z < K"
     unfolding K_def by (auto simp: field_simps intro!: add_pos_nonneg)
  let ?f = "λn. a gchoose n" and ?f' = "diffs (λn. a gchoose n)"
  have summable_strong: "summable (λn. ?f n * z ^ n)" if "norm z < 1" for z using that
    by (intro summable_gbinomial)
  with K have summable: "summable (λn. ?f n * z ^ n)" if "norm z < K" for z using that by auto
  hence summable': "summable (λn. ?f' n * z ^ n)" if "norm z < K" for z using that
    by (intro termdiff_converges[of _ K]) simp_all

  define f f' where [abs_def]: "f z = (∑n. ?f n * z ^ n)" "f' z = (∑n. ?f' n * z ^ n)" for z
  {
    fix z :: real assume z: "norm z < K"
    from summable_mult2[OF summable'[OF z], of z]
      have summable1: "summable (λn. ?f' n * z ^ Suc n)" by (simp add: mult_ac)
    hence summable2: "summable (λn. of_nat n * ?f n * z^n)"
      unfolding diffs_def by (subst (asm) summable_Suc_iff)

    have "(1 + z) * f' z = (∑n. ?f' n * z^n) + (∑n. ?f' n * z^Suc n)"
      unfolding f_f'_def using summable' z by (simp add: algebra_simps suminf_mult)
    also have "(∑n. ?f' n * z^n) = (∑n. of_nat (Suc n) * ?f (Suc n) * z^n)"
      by (intro suminf_cong) (simp add: diffs_def)
    also have "(∑n. ?f' n * z^Suc n) = (∑n. of_nat n * ?f n * z ^ n)"
      using summable1 suminf_split_initial_segment[OF summable1] unfolding diffs_def
      by (subst suminf_split_head, subst (asm) summable_Suc_iff) simp_all
    also have "(∑n. of_nat (Suc n) * ?f (Suc n) * z^n) + (∑n. of_nat n * ?f n * z^n) =
                 (∑n. a * ?f n * z^n)"
      by (subst gbinomial_mult_1, subst suminf_add)
         (insert summable'[OF z] summable2,
          simp_all add: summable_powser_split_head algebra_simps diffs_def)
    also have "… = a * f z" unfolding f_f'_def
      by (subst suminf_mult[symmetric]) (simp_all add: summable[OF z] mult_ac)
    finally have "a * f z = (1 + z) * f' z" by simp
  } note deriv = this

  have [derivative_intros]: "(f has_field_derivative f' z) (at z)" if "norm z < of_real K" for z
    unfolding f_f'_def using K that
    by (intro termdiffs_strong[of "?f" K z] summable_strong) simp_all
  have "f 0 = (∑n. if n = 0 then 1 else 0)" unfolding f_f'_def by (intro suminf_cong) simp
  also have "… = 1" using sums_single[of 0 "λ_. 1::real"] unfolding sums_iff by simp
  finally have [simp]: "f 0 = 1" .

  have "f z * (1 + z) powr (-a) = f 0 * (1 + 0) powr (-a)"
  proof (rule DERIV_isconst3[where ?f = "λx. f x * (1 + x) powr (-a)"])
    fix z :: real assume z': "z ∈ {-K<..<K}"
    hence z: "norm z < K" using K by auto
    with K have nz: "1 + z ≠ 0" by (auto dest!: minus_unique)
    from z K have "norm z < 1" by simp
    hence "((λz. f z * (1 + z) powr (-a)) has_field_derivative
              f' z * (1 + z) powr (-a) - a * f z * (1 + z) powr (-a-1)) (at z)" using z
      by (auto intro!: derivative_eq_intros)
    also from z have "a * f z = (1 + z) * f' z" by (rule deriv)
    also have "f' z * (1 + z) powr - a - (1 + z) * f' z * (1 + z) powr (- a - 1) = 0"
      using ‹norm z < 1› by (auto simp add: field_simps powr_diff)
    finally show "((λz. f z * (1 + z) powr (-a)) has_field_derivative 0) (at z)" .
  qed (insert K, auto)
  also have "f 0 * (1 + 0) powr -a = 1" by simp
  finally have "f z = (1 + z) powr a" using K
    by (simp add: field_simps dist_real_def powr_minus)
  with summable K show ?thesis unfolding f_f'_def by (simp add: sums_iff)
qed

lemma convergent_powser'_gbinomial:
  "convergent_powser' (gbinomial_series abort p) (λx. (1 + x) powr p)"
proof -
  have "(λn. lcoeff (gbinomial_series abort p) n * x ^ n) sums (1 + x) powr p" if "abs x < 1" for x
    using that gen_binomial_real[of x p] by simp
  thus ?thesis unfolding convergent_powser'_def by (force intro!: exI[of _ 1])
qed

lemma convergent_powser'_gbinomial':
  "convergent_powser' (gbinomial_series abort (real n)) (λx. (1 + x) ^ n)"
proof -
  {
    fix x :: real
    have "(λk. if k ∈ {0..n} then real (n choose k) * x ^ k else 0) sums (x + 1) ^ n"
      using sums_If_finite_set[of "{..n}" "λk. real (n choose k) * x ^ k"]
      by (subst binomial_ring) simp
    also have "(λk. if k ∈ {0..n} then real (n choose k) * x ^ k else 0) = 
                 (λk. lcoeff (gbinomial_series abort (real n)) k * x ^ k)"
      by (simp add: fun_eq_iff binomial_gbinomial [symmetric])
    finally have "… sums (1 + x) ^ n" by (simp add: add_ac)
  }
  thus ?thesis unfolding convergent_powser'_def
    by (auto intro!: exI[of _ 1])
qed


subsubsection ‹Sine and cosine›

primcorec sin_series_stream_aux :: "bool ⇒ real ⇒ real ⇒ real msstream" where
  "sin_series_stream_aux b acc m = 
    MSSCons (if b then -inverse acc else inverse acc)
          (sin_series_stream_aux (¬b) (acc * (m + 1) * (m + 2)) (m + 2))"

lemma mssnth_sin_series_stream_aux:
  "mssnth (sin_series_stream_aux b (fact m) m) n = 
     (if b then -1 else 1) * (-1) ^ n / (fact (2 * n + m))"
proof (induction n arbitrary: b m) 
  case (0 b m)
  thus ?case by (subst sin_series_stream_aux.code) (simp add: field_simps)
next
  case (Suc n b m)
  show ?case using Suc.IH[of "¬b" "m + 2"]
    by (subst sin_series_stream_aux.code) (auto simp: field_simps)
qed

definition sin_series_stream where
  "sin_series_stream = sin_series_stream_aux False 1 1"

lemma mssnth_sin_series_stream:
  "mssnth sin_series_stream n = (-1) ^ n / fact (2 * n + 1)"
  using mssnth_sin_series_stream_aux[of False 1 n] unfolding sin_series_stream_def by simp

definition cos_series_stream where
  "cos_series_stream = sin_series_stream_aux False 1 0"

lemma mssnth_cos_series_stream:
  "mssnth cos_series_stream n = (-1) ^ n / fact (2 * n)"
  using mssnth_sin_series_stream_aux[of False 0 n] unfolding cos_series_stream_def by simp

lemma summable_pre_sin: "summable (λn. mssnth sin_series_stream n * (x::real) ^ n)"
proof -
  have *: "summable (λn. 1 / fact n * abs x ^ n)"
    using exp_converges[of "abs x"] by (simp add: sums_iff field_simps)
  {
    fix n :: nat
    have "¦x¦ ^ n / fact (2 * n + 1) ≤ ¦x¦ ^ n / fact n"
      by (intro divide_left_mono fact_mono) auto
    hence "norm (mssnth sin_series_stream n * x ^ n) ≤ 1 / fact n * abs x ^ n"
      by (simp add: mssnth_sin_series_stream abs_mult power_abs field_simps)
  }
  thus ?thesis
    by (intro summable_comparison_test[OF _ *]) (auto intro!: exI[of _ 0])
qed

lemma summable_pre_cos: "summable (λn. mssnth cos_series_stream n * (x::real) ^ n)"
proof -
  have *: "summable (λn. 1 / fact n * abs x ^ n)"
    using exp_converges[of "abs x"] by (simp add: sums_iff field_simps)
  {
    fix n :: nat
    have "¦x¦ ^ n / fact (2 * n) ≤ ¦x¦ ^ n / fact n"
      by (intro divide_left_mono fact_mono) auto
    hence "norm (mssnth cos_series_stream n * x ^ n) ≤ 1 / fact n * abs x ^ n"
      by (simp add: mssnth_cos_series_stream abs_mult power_abs field_simps)
  }
  thus ?thesis
    by (intro summable_comparison_test[OF _ *]) (auto intro!: exI[of _ 0])
qed

lemma cos_conv_pre_cos:
  "cos x = powser (msllist_of_msstream cos_series_stream) (x ^ 2)"
proof -
  have "(λn. cos_coeff (2 * n) * x ^ (2 * n)) sums cos x"
    using cos_converges[of x]
    by (subst sums_mono_reindex[of "λn. 2 * n"])
       (auto simp: strict_mono_def cos_coeff_def elim!: evenE)
  also have "(λn. cos_coeff (2 * n) * x ^ (2 * n)) =
               (λn. mssnth cos_series_stream n * (x ^ 2) ^ n)"
    by (simp add: fun_eq_iff mssnth_cos_series_stream cos_coeff_def power_mult)
  finally have sums: "(λn. mssnth cos_series_stream n * x2 ^ n) sums cos x" .
  thus ?thesis by (simp add: powser_def sums_iff)
qed

lemma sin_conv_pre_sin:
  "sin x = x * powser (msllist_of_msstream sin_series_stream) (x ^ 2)"
proof -
  have "(λn. sin_coeff (2 * n + 1) * x ^ (2 * n + 1)) sums sin x"
    using sin_converges[of x]
    by (subst sums_mono_reindex[of "λn. 2 * n + 1"])
       (auto simp: strict_mono_def sin_coeff_def elim!: oddE)
  also have "(λn. sin_coeff (2 * n + 1) * x ^ (2 * n + 1)) =
               (λn. x * mssnth sin_series_stream n * (x ^ 2) ^ n)"
    by (simp add: fun_eq_iff mssnth_sin_series_stream sin_coeff_def power_mult [symmetric] algebra_simps)
  finally have sums: "(λn. x * mssnth sin_series_stream n * x2 ^ n) sums sin x" .
  thus ?thesis using summable_pre_sin[of "x^2"] 
    by (simp add: powser_def sums_iff suminf_mult [symmetric] mult.assoc)
qed

lemma convergent_powser_sin_series_stream:
  "convergent_powser (msllist_of_msstream sin_series_stream)"
  (is "convergent_powser ?cs")
proof -
  show ?thesis using summable_pre_sin unfolding convergent_powser_def
    by (intro exI[of _ 1]) auto
qed

lemma convergent_powser_cos_series_stream:
  "convergent_powser (msllist_of_msstream cos_series_stream)"
  (is "convergent_powser ?cs")
proof -
  show ?thesis using summable_pre_cos unfolding convergent_powser_def
    by (intro exI[of _ 1]) auto
qed
  
  
subsubsection ‹Arc tangent›

definition arctan_coeffs :: "nat ⇒ 'a :: {real_normed_div_algebra, banach}" where
  "arctan_coeffs n = (if odd n then (-1) ^ (n div 2) / of_nat n else 0)"

lemma arctan_converges:
  assumes "norm x < 1"
  shows   "(λn. arctan_coeffs n * x ^ n) sums arctan x"
proof -
  have A: "(λn. diffs arctan_coeffs n * x ^ n) sums (1 / (1 + x^2))" if "norm x < 1" for x :: real
  proof -
    let ?f = "λn. (if odd n then 0 else (-1) ^ (n div 2)) * x ^ n"
    from that have "norm x ^ 2 < 1 ^ 2" by (intro power_strict_mono) simp_all
    hence "(λn. (-(x^2))^n) sums (1 / (1 - (-(x^2))))" by (intro geometric_sums) simp_all
    also have "1 - (-(x^2)) = 1 + x^2" by simp
    also have "(λn. (-(x^2))^n) = (λn. ?f (2*n))" by (auto simp: fun_eq_iff power_minus' power_mult)
    also have "range ((*) (2::nat)) = {n. even n}"
      by (auto elim!: evenE)
    hence "(λn. ?f (2*n)) sums (1 / (1 + x^2)) ⟷ ?f sums (1 / (1 + x^2))"
      by (intro sums_mono_reindex) (simp_all add: strict_mono_Suc_iff)
    also have "?f = (λn. diffs arctan_coeffs n * x ^ n)"
      by (simp add: fun_eq_iff arctan_coeffs_def diffs_def)
    finally show "(λn. diffs arctan_coeffs n * x ^ n) sums (1 / (1 + x^2))" .
  qed
  
  have B: "summable (λn. arctan_coeffs n * x ^ n)" if "norm x < 1" for x :: real
  proof (rule summable_comparison_test)
    show "∃N. ∀n≥N. norm (arctan_coeffs n * x ^ n) ≤ 1 * norm x ^ n"
      unfolding norm_mult norm_power
      by (intro exI[of _ "0::nat"] allI impI mult_right_mono)
         (simp_all add: arctan_coeffs_def field_split_simps)
    from that show "summable (λn. 1 * norm x ^ n)"
      by (intro summable_mult summable_geometric) simp_all
  qed
  
  define F :: "real ⇒ real" where "F = (λx. ∑n. arctan_coeffs n * x ^ n)"
  have [derivative_intros]:
    "(F has_field_derivative (1 / (1 + x^2))) (at x)" if "norm x < 1" for x :: real
  proof -
    define K :: real where "K = (1 + norm x) / 2"
    from that have K: "norm x < K" "K < 1" by (simp_all add: K_def field_simps)
    have "(F has_field_derivative (∑n. diffs arctan_coeffs n * x ^ n)) (at x)"
      unfolding F_def using K by (intro termdiffs_strong[where K = K] B) simp_all
    also from A[OF that] have "(∑n. diffs arctan_coeffs n * x ^ n) = 1 / (1 + x^2)"
      by (simp add: sums_iff)
    finally show ?thesis .
  qed
  from assms have "arctan x - F x = arctan 0 - F 0"
    by (intro DERIV_isconst3[of "-1" 1 _ _ "λx. arctan x - F x"])
       (auto intro!: derivative_eq_intros simp: field_simps)
  hence "F x = arctan x" by (simp add: F_def arctan_coeffs_def)
  with B[OF assms] show ?thesis by (simp add: sums_iff F_def)
qed

primcorec arctan_series_stream_aux :: "bool ⇒ real ⇒ real msstream" where
  "arctan_series_stream_aux b n = 
     MSSCons (if b then -inverse n else inverse n) (arctan_series_stream_aux (¬b) (n + 2))"

lemma mssnth_arctan_series_stream_aux: 
  "mssnth (arctan_series_stream_aux b n) m = (-1) ^ (if b then Suc m else m) / (2*m + n)"
  by (induction m arbitrary: b n; subst arctan_series_stream_aux.code)
     (auto simp: field_simps split: if_splits)

definition arctan_series_stream where
  "arctan_series_stream = arctan_series_stream_aux False 1"

lemma mssnth_arctan_series_stream:
  "mssnth arctan_series_stream n = (-1) ^ n / (2 * n + 1)"
  by (simp add: arctan_series_stream_def mssnth_arctan_series_stream_aux)

lemma summable_pre_arctan:
  assumes "norm x < 1"
  shows   "summable (λn. mssnth arctan_series_stream n * x ^ n)" (is "summable ?f")
proof (rule summable_comparison_test)
  show "summable (λn. abs x ^ n)" using assms by (intro summable_geometric) auto
  show "∃N. ∀n≥N. norm (?f n) ≤ abs x ^ n"
  proof (intro exI[of _ 0] allI impI)
    fix n :: nat
    have "norm (?f n) = ¦x¦ ^ n / (2 * real n + 1)"
      by (simp add: mssnth_arctan_series_stream abs_mult power_abs)
    also have "… ≤ ¦x¦ ^ n / 1" by (intro divide_left_mono) auto
    finally show "norm (?f n) ≤ abs x ^ n" by simp
  qed
qed

lemma arctan_conv_pre_arctan:
  assumes "norm x < 1"
  shows   "arctan x = x * powser (msllist_of_msstream arctan_series_stream) (x ^ 2)"
proof -
  from assms have "norm x * norm x < 1 * 1" by (intro mult_strict_mono) auto
  hence norm_less: "norm (x ^ 2) < 1" by (simp add: power2_eq_square)
  have "(λn. arctan_coeffs n * x ^ n) sums arctan x"
    by (intro arctan_converges assms)
  also have "?this ⟷ (λn. arctan_coeffs (2 * n + 1) * x ^ (2 * n + 1)) sums arctan x"
    by (intro sums_mono_reindex [symmetric])
       (auto simp: arctan_coeffs_def strict_mono_def elim!: oddE)
  also have "(λn. arctan_coeffs (2 * n + 1) * x ^ (2 * n + 1)) =
               (λn. x * mssnth arctan_series_stream n * (x ^ 2) ^ n)"
    by (simp add: fun_eq_iff mssnth_arctan_series_stream 
                  power_mult [symmetric] arctan_coeffs_def mult_ac)
  finally have "(λn. x * mssnth arctan_series_stream n * x2 ^ n) sums arctan x" .
  thus ?thesis using summable_pre_arctan[OF norm_less] assms
    by (simp add: powser_def sums_iff suminf_mult [symmetric] mult.assoc)
qed

lemma convergent_powser_arctan: 
  "convergent_powser (msllist_of_msstream arctan_series_stream)"
  unfolding convergent_powser_def
  using summable_pre_arctan by (intro exI[of _ 1]) auto

lemma arctan_inverse_pos: "x > 0 ⟹ arctan x = pi / 2 - arctan (inverse x)"
  using arctan_inverse[of x] by (simp add: field_simps)
    
lemma arctan_inverse_neg: "x < 0 ⟹ arctan x = -pi / 2 - arctan (inverse x)"
  using arctan_inverse[of x] by (simp add: field_simps)



subsection ‹Multiseries›

subsubsection ‹Asymptotic bases›

type_synonym basis = "(real ⇒ real) list"

lemma transp_ln_smallo_ln: "transp (λf g. (λx::real. ln (g x)) ∈ o(λx. ln (f x)))"
  by (rule transpI, erule landau_o.small.trans)

definition basis_wf :: "basis ⇒ bool" where
  "basis_wf basis ⟷ (∀f∈set basis. filterlim f at_top at_top) ∧ 
                      sorted_wrt (λf g. (λx. ln (g x)) ∈ o(λx. ln (f x))) basis"

lemma basis_wf_Nil [simp]: "basis_wf []"
  by (simp add: basis_wf_def)

lemma basis_wf_Cons: 
  "basis_wf (f # basis) ⟷ 
     filterlim f at_top at_top ∧ (∀g∈set basis. (λx. ln (g x)) ∈ o(λx. ln (f x))) ∧ basis_wf basis"
  unfolding basis_wf_def by auto

(* TODO: Move *)
lemma sorted_wrt_snoc:
  assumes trans: "transp P" and "sorted_wrt P xs" "P (last xs) y"
  shows   "sorted_wrt P (xs @ [y])"
proof (subst sorted_wrt_append, intro conjI ballI)
  fix x y' assume x: "x ∈ set xs" and y': "y' ∈ set [y]"
  from y' have [simp]: "y' = y" by simp
  show "P x y'"
  proof (cases "x = last xs")
    case False
    from x have eq: "xs = butlast xs @ [last xs]"
      by (subst append_butlast_last_id) auto
    from x and False have x': "x ∈ set (butlast xs)" by (subst (asm) eq) auto
    have "sorted_wrt P xs" by fact
    also note eq
    finally have "P x (last xs)" using x'
      by (subst (asm) sorted_wrt_append) auto
    with ‹P (last xs) y› have "P x y" using transpD[OF trans] by blast
    thus ?thesis by simp
  qed (insert assms, auto)
qed (insert assms, auto)  

lemma basis_wf_snoc:
  assumes "bs ≠ []"
  assumes "basis_wf bs" "filterlim b at_top at_top"
  assumes "(λx. ln (b x)) ∈ o(λx. ln (last bs x))"
  shows   "basis_wf (bs @ [b])"
proof -
  have "transp (λf g. (λx. ln (g x)) ∈ o(λx. ln (f x)))"
    by (auto simp: transp_def intro: landau_o.small.trans)
  thus ?thesis using assms
    by (auto simp add: basis_wf_def sorted_wrt_snoc[OF transp_ln_smallo_ln])
qed

lemma basis_wf_singleton: "basis_wf [b] ⟷ filterlim b at_top at_top"
  by (simp add: basis_wf_Cons)

lemma basis_wf_many: "basis_wf (b # b' # bs) ⟷
  filterlim b at_top at_top ∧ (λx. ln (b' x)) ∈ o(λx. ln (b x)) ∧ basis_wf (b' # bs)"
  unfolding basis_wf_def by (subst sorted_wrt2[OF transp_ln_smallo_ln]) auto


subsubsection ‹Monomials›

type_synonym monom = "real × real list"

definition eval_monom :: "monom ⇒ basis ⇒ real ⇒ real" where
  "eval_monom = (λ(c,es) basis x. c * prod_list (map (λ(b,e). b x powr e) (zip basis es)))"
  
lemma eval_monom_Nil1 [simp]: "eval_monom (c, []) basis = (λ_. c)"
  by (simp add: eval_monom_def split: prod.split)

lemma eval_monom_Nil2 [simp]: "eval_monom monom [] = (λ_. fst monom)"
  by (simp add: eval_monom_def split: prod.split)

lemma eval_monom_Cons: 
  "eval_monom (c, e # es) (b # basis) = (λx. eval_monom (c, es) basis x * b x powr e)"
  by (simp add: eval_monom_def fun_eq_iff split: prod.split)

definition inverse_monom :: "monom ⇒ monom" where
  "inverse_monom monom = (case monom of (c, es) ⇒ (inverse c, map uminus es))"

lemma length_snd_inverse_monom [simp]: 
  "length (snd (inverse_monom monom)) = length (snd monom)"
  by (simp add: inverse_monom_def split: prod.split)

lemma eval_monom_pos:
  assumes "basis_wf basis" "fst monom > 0"
  shows   "eventually (λx. eval_monom monom basis x > 0) at_top"
proof (cases monom)
  case (Pair c es)
  with assms have "c > 0" by simp
  with assms(1) show ?thesis unfolding Pair
  proof (induction es arbitrary: basis)
    case (Cons e es)
    note A = this
    thus ?case
    proof (cases basis)
      case (Cons b basis')
      with A(1)[of basis'] A(2,3) 
        have A: "∀F x in at_top. eval_monom (c, es) basis' x > 0" 
         and B: "eventually (λx. b x > 0) at_top"
        by (simp_all add: basis_wf_Cons filterlim_at_top_dense)
      thus ?thesis by eventually_elim (simp add: Cons eval_monom_Cons)
    qed simp
  qed simp
qed

lemma eval_monom_uminus: "eval_monom (-c, es) basis x = -eval_monom (c, es) basis x"
  by (simp add: eval_monom_def)

lemma eval_monom_neg:
  assumes "basis_wf basis" "fst monom < 0"
  shows   "eventually (λx. eval_monom monom basis x < 0) at_top"    
proof -
  from assms have "eventually (λx. eval_monom (-fst monom, snd monom) basis x > 0) at_top"
    by (intro eval_monom_pos) simp_all
  thus ?thesis by (simp add: eval_monom_uminus)
qed

lemma eval_monom_nonzero:
  assumes "basis_wf basis" "fst monom ≠ 0"
  shows   "eventually (λx. eval_monom monom basis x ≠ 0) at_top"
proof (cases "fst monom" "0 :: real" rule: linorder_cases)
  case greater
  with assms have "eventually (λx. eval_monom monom basis x > 0) at_top"
    by (simp add: eval_monom_pos)
  thus ?thesis by eventually_elim simp
next
  case less
  with assms have "eventually (λx. eval_monom (-fst monom, snd monom) basis x > 0) at_top"
    by (simp add: eval_monom_pos)
  thus ?thesis by eventually_elim (simp add: eval_monom_uminus)
qed (insert assms, simp_all)


subsubsection ‹Typeclass for multiseries›

class multiseries = plus + minus + times + uminus + inverse + 
  fixes is_expansion :: "'a ⇒ basis ⇒ bool"
    and expansion_level :: "'a itself ⇒ nat"
    and eval :: "'a ⇒ real ⇒ real"
    and zero_expansion :: 'a
    and const_expansion :: "real ⇒ 'a"
    and powr_expansion :: "bool ⇒ 'a ⇒ real ⇒ 'a"
    and power_expansion :: "bool ⇒ 'a ⇒ nat ⇒ 'a"
    and trimmed :: "'a ⇒ bool"
    and dominant_term :: "'a ⇒ monom"

  assumes is_expansion_length:
    "is_expansion F basis ⟹ length basis = expansion_level (TYPE('a))"
  assumes is_expansion_zero:
    "basis_wf basis ⟹ length basis = expansion_level (TYPE('a)) ⟹ 
       is_expansion zero_expansion basis"
  assumes is_expansion_const:
    "basis_wf basis ⟹ length basis = expansion_level (TYPE('a)) ⟹ 
       is_expansion (const_expansion c) basis"
  assumes is_expansion_uminus:
    "basis_wf basis ⟹ is_expansion F basis ⟹ is_expansion (-F) basis"
  assumes is_expansion_add: 
    "basis_wf basis ⟹ is_expansion F basis ⟹ is_expansion G basis ⟹ 
       is_expansion (F + G) basis"
  assumes is_expansion_minus: 
    "basis_wf basis ⟹ is_expansion F basis ⟹ is_expansion G basis ⟹ 
       is_expansion (F - G) basis"
  assumes is_expansion_mult: 
    "basis_wf basis ⟹ is_expansion F basis ⟹ is_expansion G basis ⟹ 
       is_expansion (F * G) basis"
  assumes is_expansion_inverse:
    "basis_wf basis ⟹ trimmed F ⟹ is_expansion F basis ⟹ 
       is_expansion (inverse F) basis"
  assumes is_expansion_divide:
    "basis_wf basis ⟹ trimmed G ⟹ is_expansion F basis ⟹ is_expansion G basis ⟹ 
       is_expansion (F / G) basis"
  assumes is_expansion_powr:
    "basis_wf basis ⟹ trimmed F ⟹ fst (dominant_term F) > 0 ⟹ is_expansion F basis ⟹
       is_expansion (powr_expansion abort F p) basis"
  assumes is_expansion_power:
    "basis_wf basis ⟹ trimmed F ⟹ is_expansion F basis ⟹
       is_expansion (power_expansion abort F n) basis"
  
  assumes is_expansion_imp_smallo:
    "basis_wf basis ⟹ is_expansion F basis ⟹ filterlim b at_top at_top ⟹
       (∀g∈set basis. (λx. ln (g x)) ∈ o(λx. ln (b x))) ⟹ e > 0 ⟹ eval F ∈ o(λx. b x powr e)"
  assumes is_expansion_imp_smallomega:
    "basis_wf basis ⟹ is_expansion F basis ⟹ filterlim b at_top at_top ⟹ trimmed F ⟹ 
       (∀g∈set basis. (λx. ln (g x)) ∈ o(λx. ln (b x))) ⟹ e < 0 ⟹ eval F ∈ ω(λx. b x powr e)"
  assumes trimmed_imp_eventually_sgn:
    "basis_wf basis ⟹ is_expansion F basis ⟹ trimmed F ⟹
       eventually (λx. sgn (eval F x) = sgn (fst (dominant_term F))) at_top"
  assumes trimmed_imp_eventually_nz: 
    "basis_wf basis ⟹ is_expansion F basis ⟹ trimmed F ⟹ 
       eventually (λx. eval F x ≠ 0) at_top"
  assumes trimmed_imp_dominant_term_nz: "trimmed F ⟹ fst (dominant_term F) ≠ 0"
  
  assumes dominant_term: 
    "basis_wf basis ⟹ is_expansion F basis ⟹ trimmed F ⟹
       eval F ∼[at_top] eval_monom (dominant_term F) basis"
  assumes dominant_term_bigo:
    "basis_wf basis ⟹ is_expansion F basis ⟹
       eval F ∈ O(eval_monom (1, snd (dominant_term F)) basis)"
  assumes length_dominant_term:
    "length (snd (dominant_term F)) = expansion_level TYPE('a)"
  assumes fst_dominant_term_uminus [simp]: "fst (dominant_term (- F)) = -fst (dominant_term F)"
  assumes trimmed_uminus_iff [simp]: "trimmed (-F) ⟷ trimmed F"
  
  assumes add_zero_expansion_left [simp]: "zero_expansion + F = F"
  assumes add_zero_expansion_right [simp]: "F + zero_expansion = F"
  
  assumes eval_zero [simp]: "eval zero_expansion x = 0"
  assumes eval_const [simp]: "eval (const_expansion c) x = c"
  assumes eval_uminus [simp]: "eval (-F) = (λx. -eval F x)"
  assumes eval_plus [simp]: "eval (F + G) = (λx. eval F x + eval G x)"
  assumes eval_minus [simp]: "eval (F - G) = (λx. eval F x - eval G x)"
  assumes eval_times [simp]: "eval (F * G) = (λx. eval F x * eval G x)"
  assumes eval_inverse [simp]: "eval (inverse F) = (λx. inverse (eval F x))"
  assumes eval_divide [simp]: "eval (F / G) = (λx. eval F x / eval G x)"
  assumes eval_powr [simp]: "eval (powr_expansion abort F p) = (λx. eval F x powr p)"
  assumes eval_power [simp]: "eval (power_expansion abort F n) = (λx. eval F x ^ n)"
  assumes minus_eq_plus_uminus: "F - G = F + (-G)"
  assumes times_const_expansion_1: "const_expansion 1 * F = F"
  assumes trimmed_const_expansion: "trimmed (const_expansion c) ⟷ c ≠ 0"
begin

definition trimmed_pos where
  "trimmed_pos F ⟷ trimmed F ∧ fst (dominant_term F) > 0"

definition trimmed_neg where
  "trimmed_neg F ⟷ trimmed F ∧ fst (dominant_term F) < 0"

lemma trimmed_pos_uminus: "trimmed_neg F ⟹ trimmed_pos (-F)"
  by (simp add: trimmed_neg_def trimmed_pos_def)

lemma trimmed_pos_imp_trimmed: "trimmed_pos x ⟹ trimmed x"
  by (simp add: trimmed_pos_def)

lemma trimmed_neg_imp_trimmed: "trimmed_neg x ⟹ trimmed x"
  by (simp add: trimmed_neg_def)

end


subsubsection ‹Zero-rank expansions›

instantiation real :: multiseries
begin

inductive is_expansion_real :: "real ⇒ basis ⇒ bool" where
  "is_expansion_real c []"
  
definition expansion_level_real :: "real itself ⇒ nat" where
  expansion_level_real_def [simp]: "expansion_level_real _ = 0"

definition eval_real :: "real ⇒ real ⇒ real" where
  eval_real_def [simp]: "eval_real c = (λ_. c)"

definition zero_expansion_real :: "real" where
  zero_expansion_real_def [simp]: "zero_expansion_real = 0"
  
definition const_expansion_real :: "real ⇒ real" where
  const_expansion_real_def [simp]: "const_expansion_real x = x"

definition powr_expansion_real :: "bool ⇒ real ⇒ real ⇒ real" where
  powr_expansion_real_def [simp]: "powr_expansion_real _ x p = x powr p"

definition power_expansion_real :: "bool ⇒ real ⇒ nat ⇒ real" where
  power_expansion_real_def [simp]: "power_expansion_real _ x n = x ^ n"

definition trimmed_real :: "real ⇒ bool" where
  trimmed_real_def [simp]: "trimmed_real x ⟷ x ≠ 0"

definition dominant_term_real :: "real ⇒ monom" where
  dominant_term_real_def [simp]: "dominant_term_real c = (c, [])"

instance proof
  fix basis :: basis and b :: "real ⇒ real" and e F :: real
  assume *: "basis_wf basis" "filterlim b at_top at_top" "is_expansion F basis" "0 < e"
  have "(λx. b x powr e) ∈ ω(λ_. 1)"
    by (intro smallomegaI_filterlim_at_infinity filterlim_at_top_imp_at_infinity) 
       (auto intro!: filterlim_compose[OF real_powr_at_top] * )
  with * show "eval F ∈ o(λx. b x powr e)"
    by (cases "F = 0") (auto elim!: is_expansion_real.cases simp: smallomega_iff_smallo)
next
  fix basis :: basis and b :: "real ⇒ real" and e F :: real
  assume *: "basis_wf basis" "filterlim b at_top at_top" "is_expansion F basis" 
            "e < 0" "trimmed F"
  from * have pos: "eventually (λx. b x > 0) at_top" by (simp add: filterlim_at_top_dense)
  have "(λx. b x powr -e) ∈ ω(λ_. 1)"
    by (intro smallomegaI_filterlim_at_infinity filterlim_at_top_imp_at_infinity) 
       (auto intro!: filterlim_compose[OF real_powr_at_top] * )
  with pos have "(λx. b x powr e) ∈ o(λ_. 1)" unfolding powr_minus
    by (subst (asm) landau_omega.small.inverse_eq2)
       (auto elim: eventually_mono simp: smallomega_iff_smallo)
  with * show "eval F ∈ ω(λx. b x powr e)"
    by (auto elim!: is_expansion_real.cases simp: smallomega_iff_smallo)
qed (auto intro!: is_expansion_real.intros elim!: is_expansion_real.cases)

end

lemma eval_real: "eval (c :: real) x = c" by simp


subsubsection ‹Operations on multiseries›

lemma ms_aux_cases [case_names MSLNil MSLCons]:
  fixes xs :: "('a × real) msllist"
  obtains "xs = MSLNil" | c e xs' where "xs = MSLCons (c, e) xs'"
proof (cases xs)
  case (MSLCons x xs')
  with that(2)[of "fst x" "snd x" xs'] show ?thesis by auto
qed auto

definition ms_aux_hd_exp :: "('a × real) msllist ⇒ real option" where
  "ms_aux_hd_exp xs = (case xs of MSLNil ⇒ None | MSLCons (_, e) _ ⇒ Some e)"

primrec ms_exp_gt :: "real ⇒ real option ⇒ bool" where
  "ms_exp_gt x None = True"
| "ms_exp_gt x (Some y) ⟷ x > y"

lemma ms_aux_hd_exp_MSLNil [simp]: "ms_aux_hd_exp MSLNil = None"
  by (simp add: ms_aux_hd_exp_def split: prod.split)
  
lemma ms_aux_hd_exp_MSLCons [simp]: "ms_aux_hd_exp (MSLCons x xs) = Some (snd x)"
  by (simp add: ms_aux_hd_exp_def split: prod.split)

coinductive is_expansion_aux :: 
    "('a :: multiseries × real) msllist ⇒ (real ⇒ real) ⇒ basis ⇒ bool" where
  is_expansion_MSLNil: 
    "eventually (λx. f x = 0) at_top ⟹ length basis = Suc (expansion_level TYPE('a)) ⟹
       is_expansion_aux MSLNil f basis"
| is_expansion_MSLCons: 
    "is_expansion_aux xs (λx. f x - eval C x * b x powr e) (b # basis) ⟹
       is_expansion C basis ⟹
       (⋀e'. e' > e ⟹ f ∈ o(λx. b x powr e')) ⟹ ms_exp_gt e (ms_aux_hd_exp xs) ⟹
       is_expansion_aux (MSLCons (C, e) xs) f (b # basis)"

inductive_cases is_expansion_aux_MSLCons: "is_expansion_aux (MSLCons (c, e) xs) f basis"
 
lemma is_expansion_aux_basis_nonempty: "is_expansion_aux F f basis ⟹ basis ≠ []"
  by (erule is_expansion_aux.cases) auto

lemma is_expansion_aux_expansion_level:
  assumes "is_expansion_aux (G :: ('a::multiseries × real) msllist) g basis"
  shows   "length basis = Suc (expansion_level TYPE('a))"
  using assms by (cases rule: is_expansion_aux.cases) (auto dest: is_expansion_length)  

lemma is_expansion_aux_imp_smallo:
  assumes "is_expansion_aux xs f basis" "ms_exp_gt p (ms_aux_hd_exp xs)" 
  shows   "f ∈ o(λx. hd basis x powr p)"
  using assms
proof (cases rule: is_expansion_aux.cases)
  case is_expansion_MSLNil
  show ?thesis by (simp add: landau_o.small.in_cong[OF is_expansion_MSLNil(2)])
next
  case (is_expansion_MSLCons xs C b e basis)
  with assms have "f ∈ o(λx. b x powr p)"
    by (intro is_expansion_MSLCons) (simp_all add: ms_aux_hd_exp_def)
  thus ?thesis by (simp add: is_expansion_MSLCons)
qed

lemma is_expansion_aux_imp_smallo':
  assumes "is_expansion_aux xs f basis"
  obtains e where "f ∈ o(λx. hd basis x powr e)"
proof -
  define e where "e = (case ms_aux_hd_exp xs of None ⇒ 0 | Some e ⇒ e + 1)"
  have "ms_exp_gt e (ms_aux_hd_exp xs)"
    by (auto simp add: e_def ms_aux_hd_exp_def split: msllist.splits)
  from assms this have "f ∈ o(λx. hd basis x powr e)" by (rule is_expansion_aux_imp_smallo)
  from that[OF this] show ?thesis .
qed

lemma is_expansion_aux_imp_smallo'':
  assumes "is_expansion_aux xs f basis" "ms_exp_gt e' (ms_aux_hd_exp xs)"
  obtains e where "e < e'" "f ∈ o(λx. hd basis x powr e)"
proof -
  define e where "e = (case ms_aux_hd_exp xs of None ⇒ e' - 1 | Some e ⇒ (e' + e) / 2)"
  from assms have "ms_exp_gt e (ms_aux_hd_exp xs)" "e < e'"
    by (cases xs; simp add: e_def field_simps)+
  from assms(1) this(1) have "f ∈ o(λx. hd basis x powr e)" by (rule is_expansion_aux_imp_smallo)
  from that[OF ‹e < e'› this] show ?thesis .
qed

definition trimmed_ms_aux :: "('a :: multiseries × real) msllist ⇒ bool" where
  "trimmed_ms_aux xs = (case xs of MSLNil ⇒ False | MSLCons (C, _) _ ⇒ trimmed C)"
 
lemma not_trimmed_ms_aux_MSLNil [simp]: "¬trimmed_ms_aux MSLNil"
  by (simp add: trimmed_ms_aux_def)

lemma trimmed_ms_aux_MSLCons: "trimmed_ms_aux (MSLCons x xs) = trimmed (fst x)"
  by (simp add: trimmed_ms_aux_def split: prod.split)

lemma trimmed_ms_aux_imp_nz:
  assumes "basis_wf basis" "is_expansion_aux xs f basis" "trimmed_ms_aux xs"
  shows   "eventually (λx. f x ≠ 0) at_top"
proof (cases xs rule: ms_aux_cases)
  case (MSLCons C e xs')
  from assms this obtain b basis' where *: "basis = b # basis'"
    "is_expansion_aux xs' (λx. f x - eval C x * b x powr e) (b # basis')" 
    "ms_exp_gt e (ms_aux_hd_exp xs')" "is_expansion C basis'" "⋀e'. e' > e ⟹ f ∈ o(λx. b x powr e')"
    by (auto elim!: is_expansion_aux_MSLCons)
  from assms(1,3) * have nz: "eventually (λx. eval C x ≠ 0) at_top"
    by (auto simp: trimmed_ms_aux_def MSLCons basis_wf_Cons
             intro: trimmed_imp_eventually_nz[of basis'])
  from assms(1) * have b_limit: "filterlim b at_top at_top" by (simp add: basis_wf_Cons)
  hence b_nz: "eventually (λx. b x > 0) at_top" by (simp add: filterlim_at_top_dense)
  
  from is_expansion_aux_imp_smallo''[OF *(2,3)]
  obtain e' where e': "e' < e" "(λx. f x - eval C x * b x powr e) ∈ o(λx. b x powr e')"
    unfolding list.sel by blast
  note this(2)
  also have "… = o(λx. b x powr (e' - e) * b x powr e)" by (simp add: powr_add [symmetric])
  also from assms * e' have "eval C ∈ ω(λx. b x powr (e' - e))"
    by (intro is_expansion_imp_smallomega[OF _ *(4)])
       (simp_all add: MSLCons basis_wf_Cons trimmed_ms_aux_def)
  hence "(λx. b x powr (e' - e) * b x powr e) ∈ o(λx. eval C x * b x powr e)"
    by (intro landau_o.small_big_mult is_expansion_imp_smallomega) 
       (simp_all add: smallomega_iff_smallo)
  finally have "(λx. f x - eval C x * b x powr e + eval C x * b x powr e) ∈ 
                  Θ(λx. eval C x * b x powr e)"
    by (subst bigtheta_sym, subst landau_theta.plus_absorb1) simp_all
  hence theta: "f ∈ Θ(λx. eval C x * b x powr e)" by simp
  have "eventually (λx. eval C x * b x powr e ≠ 0) at_top"
    using b_nz nz by eventually_elim simp
  thus ?thesis by (subst eventually_nonzero_bigtheta [OF theta])
qed (insert assms, simp_all add: trimmed_ms_aux_def)

lemma is_expansion_aux_imp_smallomega:
  assumes "basis_wf basis" "is_expansion_aux xs f basis" "filterlim b' at_top at_top"
          "trimmed_ms_aux xs" "∀g∈set basis. (λx. ln (g x)) ∈ o(λx. ln (b' x))" "r < 0"
  shows   "f ∈ ω(λx. b' x powr r)"
proof (cases xs rule: ms_aux_cases)
  case (MSLCons C e xs')
  from assms this obtain b basis' where *: "basis = b # basis'" "trimmed C"
    "is_expansion_aux xs' (λx. f x - eval C x * b x powr e) (b # basis')"
    "ms_exp_gt e (ms_aux_hd_exp xs')"
    "is_expansion C basis'" "⋀e'. e' > e ⟹ f ∈ o(λx. b x powr e')"
    by (auto elim!: is_expansion_aux_MSLCons simp: trimmed_ms_aux_def)
  from assms * have nz: "eventually (λx. eval C x ≠ 0) at_top"
    by (intro trimmed_imp_eventually_nz[of basis']) (simp_all add: basis_wf_Cons)
  from assms(1) * have b_limit: "filterlim b at_top at_top" by (simp add: basis_wf_Cons)
  hence b_nz: "eventually (λx. b x > 0) at_top" by (simp add: filterlim_at_top_dense)
  
  from is_expansion_aux_imp_smallo''[OF *(3,4)]
  obtain e' where e': "e' < e" "(λx. f x - eval C x * b x powr e) ∈ o(λx. b x powr e')"
    unfolding list.sel by blast
  note this(2)
  also have "… = o(λx. b x powr (e' - e) * b x powr e)" by (simp add: powr_add [symmetric])
  also from assms * e' have "eval C ∈ ω(λx. b x powr (e' - e))"
    by (intro is_expansion_imp_smallomega[OF _ *(5)])
       (simp_all add: MSLCons basis_wf_Cons trimmed_ms_aux_def)
  hence "(λx. b x powr (e' - e) * b x powr e) ∈ o(λx. eval C x * b x powr e)"
    by (intro landau_o.small_big_mult is_expansion_imp_smallomega) 
       (simp_all add: smallomega_iff_smallo)
  finally have "(λx. f x - eval C x * b x powr e + eval C x * b x powr e) ∈ 
                  Θ(λx. eval C x * b x powr e)"
    by (subst bigtheta_sym, subst landau_theta.plus_absorb1) simp_all
  hence theta: "f ∈ Θ(λx. eval C x * b x powr e)" by simp
  also from * assms e' have "(λx. eval C x * b x powr e) ∈ ω(λx. b x powr (e' - e) * b x powr e)"
    by (intro landau_omega.small_big_mult is_expansion_imp_smallomega[of basis'])
       (simp_all add: basis_wf_Cons)
  also have "… = ω(λx. b x powr e')" by (simp add: powr_add [symmetric])
  also from *(1) assms have "(λx. b x powr e') ∈ ω(λx. b' x powr r)"
    unfolding smallomega_iff_smallo by (intro ln_smallo_imp_flat') (simp_all add: basis_wf_Cons)
  finally show ?thesis .
qed (insert assms, simp_all add: trimmed_ms_aux_def)  

definition dominant_term_ms_aux :: "('a :: multiseries × real) msllist ⇒ monom" where
  "dominant_term_ms_aux xs =
     (case xs of MSLNil ⇒ (0, replicate (Suc (expansion_level TYPE('a))) 0)
               | MSLCons (C, e) _ ⇒ case dominant_term C of (c, es) ⇒ (c, e # es))"

lemma dominant_term_ms_aux_MSLCons: 
  "dominant_term_ms_aux (MSLCons (C, e) xs) = map_prod id (λes. e # es) (dominant_term C)"
  by (simp add: dominant_term_ms_aux_def case_prod_unfold map_prod_def)

lemma length_dominant_term_ms_aux [simp]:
  "length (snd (dominant_term_ms_aux (xs :: ('a :: multiseries × real) msllist))) = 
     Suc (expansion_level TYPE('a))"
proof (cases xs rule: ms_aux_cases)
  case (MSLCons C e xs')
  hence "length (snd (dominant_term_ms_aux xs)) = Suc (length (snd (dominant_term C)))"
    by (simp add: dominant_term_ms_aux_def split: prod.splits)
  also note length_dominant_term
  finally show ?thesis .
qed (simp_all add: dominant_term_ms_aux_def split: msllist.splits prod.splits)

definition const_ms_aux :: "real ⇒ ('a :: multiseries × real) msllist" where
  "const_ms_aux c = MSLCons (const_expansion c, 0) MSLNil"

definition uminus_ms_aux :: "('a :: uminus × real) msllist ⇒ ('a × real) msllist" where
  "uminus_ms_aux xs = mslmap (map_prod uminus id) xs"
  
corec (friend) plus_ms_aux :: "('a :: plus × real) msllist ⇒ ('a × real) msllist ⇒ ('a × real) msllist" where
  "plus_ms_aux xs ys =
     (case (xs, ys) of
        (MSLNil, MSLNil) ⇒ MSLNil
      | (MSLNil, MSLCons y ys) ⇒ MSLCons y ys
      | (MSLCons x xs, MSLNil) ⇒ MSLCons x xs
      | (MSLCons x xs, MSLCons y ys) ⇒
          if snd x > snd y then MSLCons x (plus_ms_aux xs (MSLCons y ys))
          else if snd x < snd y then MSLCons y (plus_ms_aux (MSLCons x xs) ys)
          else MSLCons (fst x + fst y, snd x) (plus_ms_aux xs ys))"

context
begin

friend_of_corec mslmap where
  "mslmap (f :: 'a ⇒ 'a) xs = 
     (case xs of MSLNil ⇒ MSLNil | MSLCons x xs ⇒ MSLCons (f x) (mslmap f xs))"
   by (simp split: msllist.splits, transfer_prover)

corec (friend) times_ms_aux
     :: "('a :: {plus,times} × real) msllist ⇒ ('a × real) msllist ⇒ ('a × real) msllist" where
  "times_ms_aux xs ys =
     (case (xs, ys) of
        (MSLNil, ys) ⇒ MSLNil
      | (xs, MSLNil) ⇒ MSLNil
      | (MSLCons x xs, MSLCons y ys) ⇒
           MSLCons (fst x * fst y, snd x + snd y) 
             (plus_ms_aux (mslmap (map_prod ((*) (fst x)) ((+) (snd x))) ys)
               (times_ms_aux xs (MSLCons y ys))))"

definition scale_shift_ms_aux :: "('a :: times × real) ⇒ ('a × real) msllist ⇒ ('a × real) msllist" where
  "scale_shift_ms_aux = (λ(a,b) xs. mslmap (map_prod ((*) a) ((+) b)) xs)"

lemma times_ms_aux_altdef:
  "times_ms_aux xs ys = 
     (case (xs, ys) of
        (MSLNil, ys) ⇒ MSLNil
      | (xs, MSLNil) ⇒ MSLNil
      | (MSLCons x xs, MSLCons y ys) ⇒
          MSLCons (fst x * fst y, snd x + snd y)
            (plus_ms_aux (scale_shift_ms_aux x ys) (times_ms_aux xs (MSLCons y ys))))"
  by (subst times_ms_aux.code) (simp_all add: scale_shift_ms_aux_def split: msllist.splits)
end

corec powser_ms_aux :: "real msllist ⇒ ('a :: multiseries × real) msllist ⇒ ('a × real) msllist" where
  "powser_ms_aux cs xs = (case cs of MSLNil ⇒ MSLNil | MSLCons c cs ⇒
     MSLCons (const_expansion c, 0) (times_ms_aux xs (powser_ms_aux cs xs)))"
  
definition minus_ms_aux :: "('a :: multiseries × real) msllist ⇒ _" where
  "minus_ms_aux xs ys = plus_ms_aux xs (uminus_ms_aux ys)"

lemma is_expansion_aux_coinduct [consumes 1, case_names is_expansion_aux]:
  fixes xs :: "('a :: multiseries × real) msllist"
  assumes "X xs f basis" "(⋀xs f basis. X xs f basis ⟹ 
     (xs = MSLNil ∧ (∀F x in at_top. f x = 0) ∧ length basis = Suc (expansion_level TYPE('a))) ∨
     (∃xs' b e basis' C. xs = MSLCons (C, e) xs' ∧ basis = b # basis' ∧
        (X xs' (λx. f x - eval C x * b x powr e) (b # basis')) ∧ is_expansion C basis' ∧
        (∀x>e. f ∈ o(λxa. b xa powr x)) ∧ ms_exp_gt e (ms_aux_hd_exp xs')))"
  shows "is_expansion_aux xs f basis"
using assms(1)
proof (coinduction arbitrary: xs f)
  case (is_expansion_aux xs f)
  from assms(2)[OF is_expansion_aux] show ?case by blast
qed 

lemma is_expansion_aux_cong:
  assumes "is_expansion_aux F f basis"
  assumes "eventually (λx. f x = g x) at_top"
  shows   "is_expansion_aux F g basis"
  using assms
proof (coinduction arbitrary: F f g rule: is_expansion_aux_coinduct)
  case (is_expansion_aux F f g)
  from this have ev: "eventually (λx. g x = f x) at_top" by (simp add: eq_commute)
  from ev have [simp]: "eventually (λx. g x = 0) at_top ⟷ eventually (λx. f x = 0) at_top"
    by (rule eventually_subst')
  from ev have [simp]: "(∀F x in at_top. h x = g x - h' x) ⟷
                          (∀F x in at_top. h x = f x - h' x)" for h h'
      by (rule eventually_subst')
  note [simp] = landau_o.small.in_cong[OF ev]
  from is_expansion_aux(1) show ?case
    by (cases rule: is_expansion_aux.cases) force+  
qed

lemma scale_shift_ms_aux_conv_mslmap: 
  "scale_shift_ms_aux x = mslmap (map_prod ((*) (fst x)) ((+) (snd x)))"
  by (rule ext) (simp add: scale_shift_ms_aux_def map_prod_def case_prod_unfold)

fun inverse_ms_aux :: "('a :: multiseries × real) msllist ⇒ ('a × real) msllist" where
  "inverse_ms_aux (MSLCons x xs) = 
     (let c' = inverse (fst x)
      in  scale_shift_ms_aux (c', -snd x) 
            (powser_ms_aux (msllist_of_msstream (mssalternate 1 (-1)))
              (scale_shift_ms_aux (c', -snd x) xs)))"

(* TODO: Move? *)
lemma inverse_prod_list: "inverse (prod_list xs :: 'a :: field) = prod_list (map inverse xs)"
  by (induction xs) simp_all

lemma eval_inverse_monom: 
  "eval_monom (inverse_monom monom) basis = (λx. inverse (eval_monom monom basis x))"
  by (rule ext) (simp add: eval_monom_def inverse_monom_def zip_map2 o_def case_prod_unfold
                   inverse_prod_list powr_minus)

fun powr_ms_aux :: "bool ⇒ ('a :: multiseries × real) msllist ⇒ real ⇒ ('a × real) msllist" where
  "powr_ms_aux abort (MSLCons x xs) p = 
      scale_shift_ms_aux (powr_expansion abort (fst x) p, snd x * p)
        (powser_ms_aux (gbinomial_series abort p)
          (scale_shift_ms_aux (inverse (fst x), -snd x) xs))"

fun power_ms_aux :: "bool ⇒ ('a :: multiseries × real) msllist ⇒ nat ⇒ ('a × real) msllist" where
  "power_ms_aux abort (MSLCons x xs) n = 
      scale_shift_ms_aux (power_expansion abort (fst x) n, snd x * real n)
        (powser_ms_aux (gbinomial_series abort (real n))
          (scale_shift_ms_aux (inverse (fst x), -snd x) xs))"

definition sin_ms_aux' :: "('a :: multiseries × real) msllist ⇒ ('a × real) msllist" where
  "sin_ms_aux' xs = times_ms_aux xs (powser_ms_aux (msllist_of_msstream sin_series_stream)
                      (times_ms_aux xs xs))"
  
definition cos_ms_aux' :: "('a :: multiseries × real) msllist ⇒ ('a × real) msllist" where
  "cos_ms_aux' xs = powser_ms_aux (msllist_of_msstream cos_series_stream) (times_ms_aux xs xs)"

subsubsection ‹Basic properties of multiseries operations›  

lemma uminus_ms_aux_MSLNil [simp]: "uminus_ms_aux MSLNil = MSLNil"
  by (simp add: uminus_ms_aux_def)
  
lemma uminus_ms_aux_MSLCons: "uminus_ms_aux (MSLCons x xs) = MSLCons (-fst x, snd x) (uminus_ms_aux xs)"
  by (simp add: uminus_ms_aux_def map_prod_def split: prod.splits)

lemma uminus_ms_aux_MSLNil_iff [simp]: "uminus_ms_aux xs = MSLNil ⟷ xs = MSLNil"
  by (simp add: uminus_ms_aux_def)
  
lemma hd_exp_uminus [simp]: "ms_aux_hd_exp (uminus_ms_aux xs) = ms_aux_hd_exp xs"
  by (simp add: uminus_ms_aux_def ms_aux_hd_exp_def split: msllist.splits prod.split)
  
lemma scale_shift_ms_aux_MSLNil_iff [simp]: "scale_shift_ms_aux x F = MSLNil ⟷ F = MSLNil"
  by (auto simp: scale_shift_ms_aux_def split: prod.split)

lemma scale_shift_ms_aux_MSLNil [simp]: "scale_shift_ms_aux x MSLNil = MSLNil"
  by (simp add: scale_shift_ms_aux_def split: prod.split)

lemma scale_shift_ms_aux_1 [simp]:
  "scale_shift_ms_aux (1 :: real, 0) xs = xs"
  by (simp add: scale_shift_ms_aux_def map_prod_def msllist.map_id)

lemma scale_shift_ms_aux_MSLCons: 
  "scale_shift_ms_aux x (MSLCons y F) = MSLCons (fst x * fst y, snd x + snd y) (scale_shift_ms_aux x F)"
  by (auto simp: scale_shift_ms_aux_def map_prod_def split: prod.split)

lemma hd_exp_basis:
  "ms_aux_hd_exp (scale_shift_ms_aux x xs) = map_option ((+) (snd x)) (ms_aux_hd_exp xs)"
  by (auto simp: ms_aux_hd_exp_def scale_shift_ms_aux_MSLCons split: msllist.split)

lemma plus_ms_aux_MSLNil_iff: "plus_ms_aux F G = MSLNil ⟷ F = MSLNil ∧ G = MSLNil"
  by (subst plus_ms_aux.code) (simp split: msllist.splits)

lemma plus_ms_aux_MSLNil [simp]: "plus_ms_aux F MSLNil = F" "plus_ms_aux MSLNil G = G"
  by (subst plus_ms_aux.code, simp split: msllist.splits)+

lemma plus_ms_aux_MSLCons: 
  "plus_ms_aux (MSLCons x xs) (MSLCons y ys) = 
     (if snd x > snd y then MSLCons x (plus_ms_aux xs (MSLCons y ys))
      else if snd x < snd y then MSLCons y (plus_ms_aux (MSLCons x xs) ys)
      else MSLCons (fst x + fst y, snd x) (plus_ms_aux xs ys))"
  by (subst plus_ms_aux.code) simp

lemma hd_exp_plus:
  "ms_aux_hd_exp (plus_ms_aux xs ys) = combine_options max (ms_aux_hd_exp xs) (ms_aux_hd_exp ys)"
  by (cases xs; cases ys)
     (simp_all add: plus_ms_aux_MSLCons ms_aux_hd_exp_def max_def split: prod.split)

lemma minus_ms_aux_MSLNil_left [simp]: "minus_ms_aux MSLNil ys = uminus_ms_aux ys"
  by (simp add: minus_ms_aux_def)

lemma minus_ms_aux_MSLNil_right [simp]: "minus_ms_aux xs MSLNil = xs"
  by (simp add: minus_ms_aux_def)

lemma times_ms_aux_MSLNil_iff: "times_ms_aux F G = MSLNil ⟷ F = MSLNil ∨ G = MSLNil"
  by (subst times_ms_aux.code) (simp split: msllist.splits)

lemma times_ms_aux_MSLNil [simp]: "times_ms_aux MSLNil G = MSLNil" "times_ms_aux F MSLNil = MSLNil"
  by (subst times_ms_aux.code, simp split: msllist.splits)+

lemma times_ms_aux_MSLCons: "times_ms_aux (MSLCons x xs) (MSLCons y ys) = 
  MSLCons (fst x * fst y, snd x + snd y) (plus_ms_aux (scale_shift_ms_aux x ys)
       (times_ms_aux xs (MSLCons y ys)))"
  by (subst times_ms_aux_altdef) simp

lemma hd_exp_times:
  "ms_aux_hd_exp (times_ms_aux xs ys) = 
     (case (ms_aux_hd_exp xs, ms_aux_hd_exp ys) of (Some x, Some y) ⇒ Some (x + y) | _ ⇒ None)"
  by (cases xs; cases ys) (simp_all add: times_ms_aux_MSLCons ms_aux_hd_exp_def split: prod.split)


subsubsection ‹Induction upto friends for multiseries›

inductive ms_closure :: 
  "(('a :: multiseries × real) msllist ⇒ (real ⇒ real) ⇒ basis ⇒ bool) ⇒
   ('a :: multiseries × real) msllist ⇒ (real ⇒ real) ⇒ basis ⇒ bool" for P where
  ms_closure_embed: 
    "P xs f basis ⟹ ms_closure P xs f basis"
| ms_closure_cong: 
    "ms_closure P xs f basis ⟹ eventually (λx. f x = g x) at_top ⟹ ms_closure P xs g basis"
| ms_closure_MSLCons:
    "ms_closure P xs (λx. f x - eval C x * b x powr e) (b # basis) ⟹
       is_expansion C basis ⟹ (⋀e'. e' > e ⟹ f ∈ o(λx. b x powr e')) ⟹
       ms_exp_gt e (ms_aux_hd_exp xs) ⟹
       ms_closure P (MSLCons (C, e) xs) f (b # basis)"
| ms_closure_add: 
    "ms_closure P xs f basis ⟹ ms_closure P ys g basis ⟹ 
       ms_closure P (plus_ms_aux xs ys) (λx. f x + g x) basis"
| ms_closure_mult: 
    "ms_closure P xs f basis ⟹ ms_closure P ys g basis ⟹ 
       ms_closure P (times_ms_aux xs ys) (λx. f x * g x) basis"
| ms_closure_basis: 
    "ms_closure P xs f (b # basis) ⟹ is_expansion C basis ⟹
       ms_closure P (scale_shift_ms_aux (C,e) xs) (λx. eval C x * b x powr e * f x) (b # basis)"
 | ms_closure_embed':
    "is_expansion_aux xs f basis ⟹ ms_closure P xs f basis"

lemma is_expansion_aux_coinduct_upto [consumes 2, case_names A B]:
  fixes xs :: "('a :: multiseries × real) msllist"
  assumes *: "X xs f basis" and basis: "basis_wf basis"
  and step: "⋀xs f basis. X xs f basis ⟹ 
    (xs = MSLNil ∧ eventually (λx. f x = 0) at_top ∧ length basis = Suc (expansion_level TYPE('a))) ∨
    (∃xs' b e basis' C. xs = MSLCons (C, e) xs' ∧ basis = b # basis' ∧
       ms_closure X xs' (λx. f x - eval C x * b x powr e) (b # basis') ∧
       is_expansion C basis' ∧ (∀e'>e. f ∈ o(λx. b x powr e')) ∧ ms_exp_gt e (ms_aux_hd_exp xs'))"
  shows "is_expansion_aux xs f basis"
proof -
  from * have "ms_closure X xs f basis" by (rule ms_closure_embed[of X xs f basis])
  thus ?thesis
  proof (coinduction arbitrary: xs f rule: is_expansion_aux_coinduct)
    case (is_expansion_aux xs f)
    from this and basis show ?case
    proof (induction rule: ms_closure.induct)
      case (ms_closure_embed xs f basis)
      from step[OF ms_closure_embed(1)] show ?case by auto
    next
      case (ms_closure_MSLCons xs f C b e basis)
      thus ?case by auto
    next
      case (ms_closure_cong xs f basis g)
      note ev = ‹∀F x in at_top. f x = g x›
      hence ev_zero_iff: "eventually (λx. f x = 0) at_top ⟷ eventually (λx. g x = 0) at_top"
        by (rule eventually_subst')
      have *: "ms_closure X xs' (λx. f x - c x * b x powr e) basis ⟷
                 ms_closure X xs' (λx. g x - c x * b x powr e) basis" for xs' c b e
        by (rule iffI; erule ms_closure.intros(2)) (insert ev, auto elim: eventually_mono)
      from ms_closure_cong show ?case
        by (simp add: ev_zero_iff * landau_o.small.in_cong[OF ev])     
    next
      case (ms_closure_embed' xs f basis)
      from this show ?case
        by (cases rule: is_expansion_aux.cases) (force intro: ms_closure.intros(7))+
    next
      
      case (ms_closure_basis xs f b basis C e)
      show ?case
      proof (cases xs rule: ms_aux_cases)
        case MSLNil
        with ms_closure_basis show ?thesis
          by (auto simp: scale_shift_ms_aux_def split: prod.split elim: eventually_mono)
      next
        case (MSLCons C' e' xs')
        with ms_closure_basis have IH:
          "ms_closure X (MSLCons (C', e') xs') f (b # basis)"
          "is_expansion C basis" "xs = MSLCons (C', e') xs'"
          "ms_closure X xs' (λx. f x - eval C' x * b x powr e') (b # basis)"
          "ms_exp_gt e' (ms_aux_hd_exp xs')"
          "is_expansion C' basis" "⋀x. x > e' ⟹ f ∈ o(λxa. b xa powr x)"
          by auto
        
        {
          fix e'' :: real assume *: "e + e' < e''"
          define d where "d = (e'' - e - e') / 2"
          from * have "d > 0" by (simp add: d_def)
          hence "(λx. eval C x * b x powr e * f x) ∈ o(λx. b x powr d * b x powr e * b x powr (e'+d))"
            using ms_closure_basis(4) IH
            by (intro landau_o.small.mult[OF landau_o.small_big_mult] IH 
              is_expansion_imp_smallo[OF _ IH(2)]) (simp_all add: basis_wf_Cons)
          also have "… = o(λx. b x powr e'')" by (simp add: d_def powr_add [symmetric])
          finally have "(λx. eval C x * b x powr e * f x) ∈ …" .
        }
        moreover have "ms_closure X xs' (λx. f x - eval C' x * b x powr e') (b # basis)"
          using ms_closure_basis IH by auto
        note ms_closure.intros(6)[OF this IH(2), of e]
        moreover have "ms_exp_gt (e + e') (ms_aux_hd_exp (scale_shift_ms_aux (C, e) xs'))"
          using ‹ms_exp_gt e' (ms_aux_hd_exp xs')›
          by (cases xs') (simp_all add: hd_exp_basis scale_shift_ms_aux_def )
        ultimately show ?thesis using IH(2,6) MSLCons ms_closure_basis(4)
          by (auto simp: scale_shift_ms_aux_MSLCons algebra_simps powr_add basis_wf_Cons
                   intro: is_expansion_mult)
      qed
      
    next
      case (ms_closure_add xs f basis ys g)
      show ?case
      proof (cases xs ys rule: ms_aux_cases[case_product ms_aux_cases])
        case MSLNil_MSLNil
        with ms_closure_add 
          have "eventually (λx. f x = 0) at_top" "eventually (λx. g x = 0) at_top"
            by simp_all
        hence "eventually (λx. f x + g x = 0) at_top" by eventually_elim simp
        with MSLNil_MSLNil ms_closure_add show ?thesis by simp
      next
        case (MSLNil_MSLCons c e xs')
        with ms_closure_add have "eventually (λx. f x = 0) at_top" by simp
        hence ev: "eventually (λx. f x + g x = g x) at_top" by eventually_elim simp
        have *: "ms_closure X xs (λx. f x + g x - y x) basis ⟷ 
                   ms_closure X xs (λx. g x - y x) basis" for y basis xs
          by (rule iffI; erule ms_closure_cong) (insert ev, simp_all)
        from MSLNil_MSLCons ms_closure_add
        show ?thesis by (simp add: * landau_o.small.in_cong[OF ev])
      next
        case (MSLCons_MSLNil c e xs')
        with ms_closure_add have "eventually (λx. g x = 0) at_top" by simp
        hence ev: "eventually (λx. f x + g x = f x) at_top" by eventually_elim simp
        have *: "ms_closure X xs (λx. f x + g x - y x) basis ⟷ 
                   ms_closure X xs (λx. f x - y x) basis" for y basis xs
          by (rule iffI; erule ms_closure_cong) (insert ev, simp_all)
        from MSLCons_MSLNil ms_closure_add
        show ?thesis by (simp add: * landau_o.small.in_cong[OF ev])          
      next
        case (MSLCons_MSLCons C1 e1 xs' C2 e2 ys')
        with ms_closure_add obtain b basis' where IH:
          "basis_wf (b # basis')" "basis = b # basis'"
          "ms_closure X (MSLCons (C1, e1) xs') f (b # basis')"
          "ms_closure X (MSLCons (C2, e2) ys') g (b # basis')"
          "xs = MSLCons (C1, e1) xs'" "ys = MSLCons (C2, e2) ys'"
          "ms_closure X xs' (λx. f x - eval C1 x * b x powr e1) (b # basis')"
          "ms_closure X ys' (λx. g x - eval C2 x * b x powr e2) (b # basis')"
          "is_expansion C1 basis'" "⋀e. e > e1 ⟹ f ∈ o(λx. b x powr e)"
          "is_expansion C2 basis'" "⋀e. e > e2 ⟹ g ∈ o(λx. b x powr e)"
          "ms_exp_gt e1 (ms_aux_hd_exp xs')" "ms_exp_gt e2 (ms_aux_hd_exp ys')"
          by blast
        have o: "(λx. f x + g x) ∈ o(λx. b x powr e)" if "e > max e1 e2" for e
          using that by (intro sum_in_smallo IH) simp_all
      
        show ?thesis
        proof (cases e1 e2 rule: linorder_cases)
          case less
          have gt: "ms_exp_gt e2 (combine_options max (Some e1) (ms_aux_hd_exp ys'))"
            using ‹ms_exp_gt e2 _› less by (cases "ms_aux_hd_exp ys'") auto
          have "ms_closure X (plus_ms_aux xs ys')
                  (λx. f x + (g x - eval C2 x * b x powr e2)) (b # basis')"
            by (rule ms_closure.intros(4)) (insert IH, simp_all)
          with less IH(2,11,14) o gt show ?thesis
            by (intro disjI2) (simp add: MSLCons_MSLCons plus_ms_aux_MSLCons algebra_simps hd_exp_plus)
        next
          case greater
          have gt: "ms_exp_gt e1 (combine_options max (ms_aux_hd_exp xs') (Some e2))"
            using ‹ms_exp_gt e1 _› greater by (cases "ms_aux_hd_exp xs'") auto
          have "ms_closure X (plus_ms_aux xs' ys)
                  (λx. (f x - eval C1 x * b x powr e1) + g x) (b # basis')"
            by (rule ms_closure.intros(4)) (insert IH, simp_all)
          with greater IH(2,9,13) o gt show ?thesis
            by (intro disjI2) (simp add:  MSLCons_MSLCons plus_ms_aux_MSLCons algebra_simps hd_exp_plus)
        next
          case equal
          have gt: "ms_exp_gt e2 (combine_options max (ms_aux_hd_exp xs')
                      (ms_aux_hd_exp ys'))"
            using ‹ms_exp_gt e1 _› ‹ms_exp_gt e2 _› equal
            by (cases "ms_aux_hd_exp xs'"; cases "ms_aux_hd_exp ys'") auto
          have "ms_closure X (plus_ms_aux xs' ys')
                  (λx. (f x - eval C1 x * b x powr e1) + (g x - eval C2 x * b x powr e2)) (b # basis')"
            by (rule ms_closure.intros(4)) (insert IH, simp_all)
          with equal IH(1,2,9,11,13,14) o gt show ?thesis
            by (intro disjI2) 
               (auto intro: is_expansion_add 
                     simp: plus_ms_aux_MSLCons basis_wf_Cons algebra_simps hd_exp_plus  MSLCons_MSLCons)
        qed
      qed
      
    next
      
      case (ms_closure_mult xs f basis ys g)
      show ?case
      proof (cases "xs = MSLNil ∨ ys = MSLNil")
        case True
        with ms_closure_mult 
          have "eventually (λx. f x = 0) at_top ∨ eventually (λx. g x = 0) at_top" by blast
        hence "eventually (λx. f x * g x = 0) at_top" by (auto elim: eventually_mono)
        with ms_closure_mult True show ?thesis by auto
      next
        case False
        with ms_closure_mult obtain C1 e1 xs' C2 e2 ys' b basis' where IH:
          "xs = MSLCons (C1, e1) xs'" "ys = MSLCons (C2, e2) ys'"
          "basis_wf (b # basis')" "basis = b # basis'"
          "ms_closure X (MSLCons (C1, e1) xs') f (b # basis')"
          "ms_closure X (MSLCons (C2, e2) ys') g (b # basis')"
          "ms_closure X xs' (λx. f x - eval C1 x * b x powr e1) (b # basis')"
          "ms_closure X ys' (λx. g x - eval C2 x * b x powr e2) (b # basis')"
          "is_expansion C1 basis'" "⋀e. e > e1 ⟹ f ∈ o(λx. b x powr e)"
          "is_expansion C2 basis'" "⋀e. e > e2 ⟹ g ∈ o(λx. b x powr e)"
          "ms_exp_gt e1 (ms_aux_hd_exp xs')" "ms_exp_gt e2 (ms_aux_hd_exp ys')"
          by blast
        have o: "(λx. f x * g x) ∈ o(λx. b x powr e)" if "e > e1 + e2" for e
        proof -
          define d where "d = (e - e1 - e2) / 2"
          from that have d: "d > 0" by (simp add: d_def)
          hence "(λx. f x * g x) ∈ o(λx. b x powr (e1 + d) * b x powr (e2 + d))"
            by (intro landau_o.small.mult IH) simp_all
          also have "… = o(λx. b x powr e)" by (simp add: d_def powr_add [symmetric])
          finally show ?thesis .
        qed

        have "ms_closure X (plus_ms_aux (scale_shift_ms_aux (C1, e1) ys') (times_ms_aux xs' ys))
                (λx. eval C1 x * b x powr e1 * (g x - eval C2 x * b x powr e2) + 
                     ((f x - eval C1 x * b x powr e1) * g x)) (b # basis')"
          (is "ms_closure _ ?zs ?f _") using ms_closure_mult IH(4)
          by (intro ms_closure.intros(4-6) IH) simp_all
        also have "?f = (λx. f x * g x - eval C1 x * eval C2 x * b x powr (e1 + e2))"
          by (intro ext) (simp add: algebra_simps powr_add)
        finally have "ms_closure X ?zs … (b # basis')" .
        moreover from ‹ms_exp_gt e1 _› ‹ms_exp_gt e2 _›
        have "ms_exp_gt (e1 + e2) (combine_options max (map_option ((+) e1) 
                (ms_aux_hd_exp ys')) (case ms_aux_hd_exp xs' of None ⇒ None | Some x ⇒
                  (case Some e2 of None ⇒ None | Some y ⇒ Some (x + y))))"
            by (cases "ms_aux_hd_exp xs'"; cases "ms_aux_hd_exp ys'")
               (simp_all add: hd_exp_times hd_exp_plus hd_exp_basis IH)
        ultimately show ?thesis using IH(1,2,3,4,9,11) o
          by (auto simp: times_ms_aux_MSLCons basis_wf_Cons hd_exp_times hd_exp_plus hd_exp_basis
                   intro: is_expansion_mult)
      qed
    qed
  qed
qed



subsubsection ‹Dominant terms›

lemma one_smallo_powr: "e > (0::real) ⟹ (λ_. 1) ∈ o(λx. x powr e)"
  by (auto simp: smallomega_iff_smallo [symmetric] real_powr_at_top 
                 smallomegaI_filterlim_at_top_norm)

lemma powr_smallo_one: "e < (0::real) ⟹ (λx. x powr e) ∈ o(λ_. 1)"
  by (intro smalloI_tendsto) (auto intro!: tendsto_neg_powr filterlim_ident)

lemma eval_monom_smallo':
  assumes "basis_wf (b # basis)" "e > 0"
  shows   "eval_monom x basis ∈ o(λx. b x powr e)"
proof (cases x)
  case (Pair c es)
  from assms show ?thesis unfolding Pair
  proof (induction es arbitrary: b basis e)
    case (Nil b basis e)
    hence b: "filterlim b at_top at_top" by (simp add: basis_wf_Cons)
    have "eval_monom (c, []) basis ∈ O(λ_. 1)" by simp
    also have "(λ_. 1) ∈ o(λx. b x powr e)"
      by (rule landau_o.small.compose[OF _ b]) (insert Nil, simp add: one_smallo_powr)
    finally show ?case .
  next
    case (Cons e' es b basis e)
    show ?case
    proof (cases basis)
      case Nil
      from Cons have b: "filterlim b at_top at_top" by (simp add: basis_wf_Cons)
      from Nil have "eval_monom (c, e' # es) basis ∈ O(λ_. 1)" by simp
      also have "(λ_. 1) ∈ o(λx. b x powr e)"
        by (rule landau_o.small.compose[OF _ b]) (insert Cons.prems, simp add: one_smallo_powr)
      finally show ?thesis .
    next
      case (Cons b' basis')
      from Cons.prems have "eval_monom (c, es) basis' ∈ o(λx. b' x powr 1)"
        by (intro Cons.IH) (simp_all add: basis_wf_Cons Cons)
      hence "(λx. eval_monom (c, es) basis' x * b' x powr e') ∈ o(λx. b' x powr 1 * b' x powr e')"
        by (rule landau_o.small_big_mult) simp_all
      also have "… = o(λx. b' x powr (1 + e'))" by (simp add: powr_add)
      also from Cons.prems have "(λx. b' x powr (1 + e')) ∈ o(λx. b x powr e)"
        by (intro ln_smallo_imp_flat) (simp_all add: basis_wf_Cons Cons)
      finally show ?thesis by (simp add: powr_add [symmetric] Cons eval_monom_Cons)
    qed
  qed
qed

lemma eval_monom_smallomega':
  assumes "basis_wf (b # basis)" "e < 0"
  shows   "eval_monom (1, es) basis ∈ ω(λx. b x powr e)"
  using assms
proof (induction es arbitrary: b basis e)
  case (Nil b basis e)
  hence b: "filterlim b at_top at_top" by (simp add: basis_wf_Cons)
  have "eval_monom (1, []) basis ∈ Ω(λ_. 1)" by simp
  also have "(λ_. 1) ∈ ω(λx. b x powr e)" unfolding smallomega_iff_smallo
    by (rule landau_o.small.compose[OF _ b]) (insert Nil, simp add: powr_smallo_one)
  finally show ?case .
next
  case (Cons e' es b basis e)
  show ?case
  proof (cases basis)
    case Nil
    from Cons have b: "filterlim b at_top at_top" by (simp add: basis_wf_Cons)
    from Nil have "eval_monom (1, e' # es) basis ∈ Ω(λ_. 1)" by simp
    also have "(λ_. 1) ∈ ω(λx. b x powr e)" unfolding smallomega_iff_smallo
      by (rule landau_o.small.compose[OF _ b]) (insert Cons.prems, simp add: powr_smallo_one)
    finally show ?thesis .
  next
    case (Cons b' basis')
    from Cons.prems have "eval_monom (1, es) basis' ∈ ω(λx. b' x powr -1)"
      by (intro Cons.IH) (simp_all add: basis_wf_Cons Cons)
    hence "(λx. eval_monom (1, es) basis' x * b' x powr e') ∈ ω(λx. b' x powr -1 * b' x powr e')"
      by (rule landau_omega.small_big_mult) simp_all
    also have "… = ω(λx. b' x powr (e' - 1))" by (simp add: powr_diff powr_minus field_simps)
    also from Cons.prems have "(λx. b' x powr (e' - 1)) ∈ ω(λx. b x powr e)"
      unfolding smallomega_iff_smallo
      by (intro ln_smallo_imp_flat') (simp_all add: basis_wf_Cons Cons)
    finally show ?thesis by (simp add: powr_add [symmetric] Cons eval_monom_Cons)
  qed
qed

lemma dominant_term_ms_aux:
  assumes basis: "basis_wf basis" and xs: "trimmed_ms_aux xs" "is_expansion_aux xs f basis"
  shows   "f ∼[at_top] eval_monom (dominant_term_ms_aux xs) basis" (is ?thesis1)
    and   "eventually (λx. sgn (f x) = sgn (fst (dominant_term_ms_aux xs))) at_top" (is ?thesis2)
proof -
  include asymp_equiv_syntax
  from xs(2,1) obtain xs' C b e basis' where xs':
    "trimmed C" "xs = MSLCons (C, e) xs'" "basis = b # basis'"
    "is_expansion_aux xs' (λx. f x - eval C x * b x powr e) (b # basis')"
    "is_expansion C basis'" "(⋀e'. e < e' ⟹ f ∈ o(λx. b x powr e'))"
    "ms_exp_gt e (ms_aux_hd_exp xs')"
    by (cases rule: is_expansion_aux.cases) (auto simp: trimmed_ms_aux_MSLCons)
  from is_expansion_aux_imp_smallo''[OF xs'(4,7)]
    obtain e' where e': "e' < e" "(λx. f x - eval C x * b x powr e) ∈ o(λx. b x powr e')"
      unfolding list.sel by blast   
  from basis xs' have "filterlim b at_top at_top" by (simp add: basis_wf_Cons)
  hence b_pos: "eventually (λx. b x > 0) at_top" by (simp add: filterlim_at_top_dense)
    
  note e'(2)
  also have "o(λx. b x powr e') = o(λx. b x powr (e' - e) * b x powr e)" 
    by (simp add: powr_add [symmetric])
  also from xs' basis e'(1) have "eval C ∈ ω(λx. b x powr (e' - e))"
    by (intro is_expansion_imp_smallomega[of basis']) (simp_all add: basis_wf_Cons)
  hence "(λx. b x powr (e' - e) * b x powr e) ∈ o(λx. eval C x * b x powr e)"
    by (intro landau_o.small_big_mult) (simp_all add: smallomega_iff_smallo)
  finally have *: "(λx. f x - eval C x * b x powr e) ∈ o(λx. eval C x * b x powr e)" .
  hence "f ∼ (λx. eval C x * b x powr e)" by (intro smallo_imp_asymp_equiv) simp
  also from basis xs' have "eval C ∼ (λx. eval_monom (dominant_term C) basis' x)"
    by (intro dominant_term) (simp_all add: basis_wf_Cons)
  also have "(λa. eval_monom (dominant_term C) basis' a * b a powr e) = 
               eval_monom (dominant_term_ms_aux xs) basis"
    by (auto simp add: dominant_term_ms_aux_def xs' eval_monom_Cons fun_eq_iff split: prod.split)
  finally show ?thesis1 by - (simp_all add: asymp_equiv_intros)
  
  have "eventually (λx. sgn (eval C x * b x powr e + (f x - eval C x * b x powr e)) = 
                          sgn (eval C x * b x powr e)) at_top"
    by (intro smallo_imp_eventually_sgn *)
  moreover have "eventually (λx. sgn (eval C x) = sgn (fst (dominant_term C))) at_top"
    using xs' basis by (intro trimmed_imp_eventually_sgn[of basis']) (simp_all add: basis_wf_Cons)
  ultimately have "eventually (λx. sgn (f x) = sgn (fst (dominant_term C))) at_top"
    using b_pos by eventually_elim (simp_all add: sgn_mult)
  thus ?thesis2 using xs'(2) by (auto simp: dominant_term_ms_aux_def split: prod.split)
qed

lemma eval_monom_smallo:
  assumes "basis ≠ []" "basis_wf basis" "length es = length basis" "e > hd es"
  shows   "eval_monom (c, es) basis ∈ o(λx. hd basis x powr e)"
proof (cases es; cases basis)
  fix b basis' e' es' assume [simp]: "basis = b # basis'" "es = e' # es'"
  from assms have "filterlim b at_top at_top" by (simp add: basis_wf_Cons)
  hence b_pos: "eventually (λx. b x > 0) at_top" 
    using filterlim_at_top_dense by blast
  have "(λx. eval_monom (1, es') basis' x * b x powr e') ∈ o(λx. b x powr (e - e') * b x powr e')"
    using assms by (intro landau_o.small_big_mult eval_monom_smallo') auto
  also have "(λx. b x powr (e - e') * b x powr e') ∈ Θ(λx. b x powr e)"
    by (intro bigthetaI_cong eventually_mono[OF b_pos]) (auto simp: powr_diff)
  finally show ?thesis by (simp add: eval_monom_def algebra_simps)
qed (insert assms, auto)

lemma eval_monom_smallomega:
  assumes "basis ≠ []" "basis_wf basis" "length es = length basis" "e < hd es"
  shows   "eval_monom (1, es) basis ∈ ω(λx. hd basis x powr e)"
proof (cases es; cases basis)
  fix b basis' e' es' assume [simp]: "basis = b # basis'" "es = e' # es'"
  from assms have "filterlim b at_top at_top" by (simp add: basis_wf_Cons)
  hence b_pos: "eventually (λx. b x > 0) at_top" 
    using filterlim_at_top_dense by blast
  have "(λx. eval_monom (1, es') basis' x * b x powr e') ∈ ω(λx. b x powr (e - e') * b x powr e')"
    using assms by (intro landau_omega.small_big_mult eval_monom_smallomega') auto
  also have "(λx. b x powr (e - e') * b x powr e') ∈ Θ(λx. b x powr e)"
    by (intro bigthetaI_cong eventually_mono[OF b_pos]) (auto simp: powr_diff)
  finally show ?thesis by (simp add: eval_monom_Cons)
qed (insert assms, auto)


subsubsection ‹Correctness of multiseries operations›

lemma drop_zero_ms_aux:
  assumes "eventually (λx. eval C x = 0) at_top"
  assumes "is_expansion_aux (MSLCons (C, e) xs) f basis"
  shows   "is_expansion_aux xs f basis"
proof (rule is_expansion_aux_cong)
  from assms(2) show "is_expansion_aux xs (λx. f x - eval C x * hd basis x powr e) basis"
    by (auto elim: is_expansion_aux_MSLCons)
  from assms(1) show "eventually (λx. f x - eval C x * hd basis x powr e = f x) at_top"
    by eventually_elim simp
qed

lemma dominant_term_ms_aux_bigo:
  assumes basis: "basis_wf basis" and xs: "is_expansion_aux xs f basis"
  shows   "f ∈ O(eval_monom (1, snd (dominant_term_ms_aux xs)) basis)" (is ?thesis1)
proof (cases xs rule: ms_aux_cases)
  case MSLNil
  with assms have "eventually (λx. f x = 0) at_top"
    by (auto elim: is_expansion_aux.cases)
  hence "f ∈ Θ(λ_. 0)" by (intro bigthetaI_cong) auto
  also have "(λ_. 0) ∈ O(eval_monom (1, snd (dominant_term_ms_aux xs)) basis)" by simp
  finally show ?thesis .
next
  case [simp]: (MSLCons C e xs')
  from xs obtain b basis' where xs':
    "basis = b # basis'"
    "is_expansion_aux xs' (λx. f x - eval C x * b x powr e) (b # basis')"
    "is_expansion C basis'" "(⋀e'. e < e' ⟹ f ∈ o(λx. b x powr e'))"
    "ms_exp_gt e (ms_aux_hd_exp xs')"
    by (cases rule: is_expansion_aux.cases) auto
  from is_expansion_aux_imp_smallo''[OF xs'(2,5)]
    obtain e' where e': "e' < e" "(λx. f x - eval C x * b x powr e) ∈ o(λx. b x powr e')"
      unfolding list.sel by blast   
  from basis xs' have "filterlim b at_top at_top" by (simp add: basis_wf_Cons)
  hence b_pos: "eventually (λx. b x > 0) at_top" by (simp add: filterlim_at_top_dense)

  let ?h = "(λx. eval_monom (1, snd (dominant_term C)) basis' x * b x powr e)"
  note e'(2)
  also have "(λx. b x powr e') ∈ Θ(λx. b x powr (e' - e) * b x powr e)"
    by (intro bigthetaI_cong eventually_mono[OF b_pos]) (auto simp: powr_diff)
  also have "(λx. b x powr (e' - e) * b x powr e) ∈ o(?h)"
    unfolding smallomega_iff_smallo [symmetric] using e'(1) basis xs'
    by (intro landau_omega.small_big_mult landau_omega.big_refl eval_monom_smallomega')
       (simp_all add: basis_wf_Cons)
  finally have "(λx. f x - eval C x * b x powr e) ∈ O(?h)" by (rule landau_o.small_imp_big)
  moreover have "(λx. eval C x * b x powr e) ∈ O(?h)"
    using basis xs' by (intro landau_o.big.mult dominant_term_bigo) (auto simp: basis_wf_Cons)
  ultimately have "(λx. f x - eval C x * b x powr e + eval C x * b x powr e) ∈ O(?h)"
    by (rule sum_in_bigo)
  hence "f ∈ O(?h)" by simp
  also have "?h = eval_monom (1, snd (dominant_term_ms_aux xs)) basis"
    using xs' by (auto simp: dominant_term_ms_aux_def case_prod_unfold eval_monom_Cons)
  finally show ?thesis .
qed

lemma const_ms_aux:
  assumes basis: "basis_wf basis" 
      and "length basis = Suc (expansion_level TYPE('a::multiseries))"
  shows   "is_expansion_aux (const_ms_aux c :: ('a × real) msllist) (λ_. c) basis"
proof -
  from assms(2) obtain b basis' where [simp]: "basis = b # basis'" by (cases basis) simp_all
  from basis have b_limit: "filterlim b at_top at_top" by (simp add: basis_wf_Cons)
  hence b_pos: "eventually (λx. b x > 0) at_top" by (simp add: filterlim_at_top_dense)

  have "(λ_. c) ∈ o(λx. b x powr e)" if "e > 0" for e
  proof -
    have "(λ_. c) ∈ O(λ_. 1)" by (cases "c = 0") simp_all  
    also from b_pos have "(λ_. 1) ∈ Θ(λx. b x powr 0)" 
      by (intro bigthetaI_cong) (auto elim: eventually_mono)
    also from that b_limit have "(λx. b x powr 0) ∈ o(λx. b x powr e)" 
      by (subst powr_smallo_iff) auto
    finally show ?thesis .
  qed
  with b_pos assms show ?thesis
    by (auto intro!: is_expansion_aux.intros simp: const_ms_aux_def is_expansion_const basis_wf_Cons 
             elim: eventually_mono)
qed

lemma uminus_ms_aux:
  assumes basis: "basis_wf basis"
  assumes F: "is_expansion_aux F f basis"
  shows   "is_expansion_aux (uminus_ms_aux F) (λx. -f x) basis"
  using F
proof (coinduction arbitrary: F f rule: is_expansion_aux_coinduct)
  case (is_expansion_aux F f)
  note IH = is_expansion_aux
  show ?case
  proof (cases F rule: ms_aux_cases)
    case MSLNil
    with IH show ?thesis by (auto simp: uminus_ms_aux_def elim: is_expansion_aux.cases)
  next
    case (MSLCons C e xs)
    with IH obtain b basis'
      where IH: "basis = b # basis'" 
                "is_expansion_aux xs (λx. f x - eval C x * b x powr e) (b # basis')"
                "is_expansion C basis'"
                "⋀e'. e < e' ⟹ f ∈ o(λx. b x powr e')" "ms_exp_gt e (ms_aux_hd_exp xs)"
      by (auto elim: is_expansion_aux_MSLCons)
    from basis IH have basis': "basis_wf basis'" by (simp add: basis_wf_Cons)
    with IH basis show ?thesis
      by (force simp: uminus_ms_aux_MSLCons MSLCons is_expansion_uminus)
  qed
qed

lemma plus_ms_aux:
  assumes "basis_wf basis" "is_expansion_aux F f basis" "is_expansion_aux G g basis"
  shows   "is_expansion_aux (plus_ms_aux F G) (λx. f x + g x) basis"
  using assms
proof (coinduction arbitrary: F f G g rule: is_expansion_aux_coinduct)
  case (is_expansion_aux F f G g)
  note IH = this
  show ?case
  proof (cases F G rule: ms_aux_cases[case_product ms_aux_cases])
    assume [simp]: "F = MSLNil" "G = MSLNil"
    with IH have *: "eventually (λx. f x = 0) at_top" "eventually (λx. g x = 0) at_top" 
                 and length: "length basis = Suc (expansion_level TYPE('a))"
      by (auto elim: is_expansion_aux.cases)
    from * have "eventually (λx. f x + g x = 0) at_top" by eventually_elim simp
    with length show ?case by simp 
  next
    assume [simp]: "F = MSLNil"
    fix C e G' assume G: "G = MSLCons (C, e) G'"
    from IH have f: "eventually (λx. f x = 0) at_top" by (auto elim: is_expansion_aux.cases)
    from f have "eventually (λx. f x + g x = g x) at_top" by eventually_elim simp
    note eq = landau_o.small.in_cong[OF this]
    from IH(3) G obtain b basis' where G':
      "basis = b # basis'"
      "is_expansion_aux G' (λx. g x - eval C x * b x powr e) (b # basis')"
      "is_expansion C basis'" "∀e'>e. g ∈ o(λx. b x powr e')" "ms_exp_gt e (ms_aux_hd_exp G')"
      by (auto elim!: is_expansion_aux_MSLCons)
    show ?thesis
      by (rule disjI2, inst_existentials G' b e basis' C F f G' "λx. g x - eval C x * b x powr e")
         (insert G' IH(1,2), simp_all add: G eq algebra_simps)
  next
    assume [simp]: "G = MSLNil"
    fix C e F' assume F: "F = MSLCons (C, e) F'"
    from IH have g: "eventually (λx. g x = 0) at_top" by (auto elim: is_expansion_aux.cases)
    hence "eventually (λx. f x + g x = f x) at_top" by eventually_elim simp
    note eq = landau_o.small.in_cong[OF this]
    from IH F obtain b basis' where F':
      "basis = b # basis'"
      "is_expansion_aux F' (λx. f x - eval C x * b x powr e) (b # basis')"
      "is_expansion C basis'" "∀e'>e. f ∈ o(λx. b x powr e')" "ms_exp_gt e (ms_aux_hd_exp F')"
      by (auto elim!: is_expansion_aux_MSLCons)
    show ?thesis
      by (rule disjI2, inst_existentials F' b e basis' C F' "λx. f x - eval C x * b x powr e" G g)
         (insert F g F' IH, simp_all add: eq algebra_simps)
  next
    fix C1 e1 F' C2 e2 G'
    assume F: "F = MSLCons (C1, e1) F'" and G: "G = MSLCons (C2, e2) G'"
    from IH F obtain b basis' where
      basis': "basis = b # basis'" and F':
      "is_expansion_aux F' (λx. f x - eval C1 x * b x powr e1) (b # basis')"
      "is_expansion C1 basis'" "⋀e'. e' > e1 ⟹ f ∈ o(λx. b x powr e')" 
      "ms_exp_gt e1 (ms_aux_hd_exp F')"
      by (auto elim!: is_expansion_aux_MSLCons)
    from IH G basis' have G':
      "is_expansion_aux G' (λx. g x - eval C2 x * b x powr e2) (b # basis')"
      "is_expansion C2 basis'" "⋀e'. e' > e2 ⟹ g ∈ o(λx. b x powr e')"
      "ms_exp_gt e2 (ms_aux_hd_exp G')"
      by (auto elim!: is_expansion_aux_MSLCons)  
    hence *: "∀x>max e1 e2. (λx. f x + g x) ∈ o(λxa. b xa powr x)"
      by (intro allI impI sum_in_smallo F' G') simp_all
      
    consider "e1 > e2" | "e1 < e2" | "e1 = e2" by force
    thus ?thesis
    proof cases
      assume e: "e1 > e2"
      have gt: "ms_exp_gt e1 (combine_options max (ms_aux_hd_exp F') (Some e2))"
        using ‹ms_exp_gt e1 _› e by (cases "ms_aux_hd_exp F'") simp_all
      show ?thesis
        by (rule disjI2, inst_existentials "plus_ms_aux F' G" b e1 basis' C1 F' 
                           "λx. f x - eval C1 x * b x powr e1" G g)
           (insert e F'(1,2,4) IH(1,3) basis'(1) * gt,
            simp_all add: F G plus_ms_aux_MSLCons algebra_simps hd_exp_plus)
    next
      assume e: "e1 < e2"
      have gt: "ms_exp_gt e2 (combine_options max (Some e1) (ms_aux_hd_exp G'))"
        using ‹ms_exp_gt e2 _› e by (cases "ms_aux_hd_exp G'") simp_all
      show ?thesis
        by (rule disjI2, inst_existentials "plus_ms_aux F G'" b e2 basis' C2 F f G'
                           "λx. g x - eval C2 x * b x powr e2")
           (insert e G'(1,2,4) IH(1,2) basis'(1) * gt, 
            simp_all add: F G plus_ms_aux_MSLCons algebra_simps hd_exp_plus)
    next
      assume e: "e1 = e2"
      have gt: "ms_exp_gt e2 (combine_options max (ms_aux_hd_exp F') (ms_aux_hd_exp G'))"
        using ‹ms_exp_gt e1 _› ‹ms_exp_gt e2 _› e
        by (cases "ms_aux_hd_exp F'"; cases "ms_aux_hd_exp G'") simp_all
      show ?thesis
        by (rule disjI2, inst_existentials "plus_ms_aux F' G'" b e1 basis' "C1 + C2" F' 
                         "λx. f x - eval C1 x * b x powr e1" G' "λx. g x - eval C2 x * b x powr e2")
           (insert e F'(1,2,4) G'(1,2,4) IH(1) basis'(1) * gt, 
            simp_all add: F G plus_ms_aux_MSLCons algebra_simps hd_exp_plus is_expansion_add basis_wf_Cons)
    qed
  qed
qed

lemma minus_ms_aux:
  assumes "basis_wf basis" "is_expansion_aux F f basis" "is_expansion_aux G g basis"
  shows   "is_expansion_aux (minus_ms_aux F G) (λx. f x - g x) basis"
proof -
  have "is_expansion_aux (minus_ms_aux F G) (λx. f x + (- g x)) basis"
    unfolding minus_ms_aux_def by (intro plus_ms_aux uminus_ms_aux assms)
  thus ?thesis by simp
qed

lemma scale_shift_ms_aux:
  assumes basis: "basis_wf (b # basis)"
  assumes F: "is_expansion_aux F f (b # basis)"
  assumes C: "is_expansion C basis"
  shows   "is_expansion_aux (scale_shift_ms_aux (C, e) F) (λx. eval C x * b x powr e * f x) (b # basis)"
  using F
proof (coinduction arbitrary: F f rule: is_expansion_aux_coinduct)
  case (is_expansion_aux F f)
  note IH = is_expansion_aux
  show ?case
  proof (cases F rule: ms_aux_cases)
    case MSLNil
    with IH show ?thesis 
      by (auto simp: scale_shift_ms_aux_def elim!: is_expansion_aux.cases eventually_mono)
  next
    case (MSLCons C' e' xs)
    with IH have IH: "is_expansion_aux xs (λx. f x - eval C' x * b x powr e') (b # basis)"
                     "is_expansion C' basis" "ms_exp_gt e' (ms_aux_hd_exp xs)"
                     "⋀e''. e' < e'' ⟹ f ∈ o(λx. b x powr e'')"
      by (auto elim: is_expansion_aux_MSLCons)
    
    have "(λx. eval C x * b x powr e * f x) ∈ o(λx. b x powr e'')" if "e'' > e + e'" for e''
    proof -
      define d where "d = (e'' - e - e') / 2"
      from that have d: "d > 0" by (simp add: d_def)
      have "(λx. eval C x * b x powr e * f x) ∈ o(λx. b x powr d * b x powr e * b x powr (e' + d))"
        using d basis
        by (intro landau_o.small.mult[OF landau_o.small_big_mult] 
              is_expansion_imp_smallo[OF _ C] IH) (simp_all add: basis_wf_Cons)
      also have "… = o(λx. b x powr e'')" by (simp add: d_def powr_add [symmetric])
      finally show ?thesis .
    qed
    moreover have "ms_exp_gt (e + e') (ms_aux_hd_exp (scale_shift_ms_aux (C, e) xs))"
      using IH(3) by (cases "ms_aux_hd_exp xs") (simp_all add: hd_exp_basis)
    ultimately show ?thesis using basis IH(1,2)
      by (intro disjI2, inst_existentials "scale_shift_ms_aux (C, e) xs" b "e + e'" 
                          basis "C * C'" xs "λx. f x - eval C' x * b x powr e'")
         (simp_all add: scale_shift_ms_aux_MSLCons MSLCons is_expansion_mult basis_wf_Cons
                        algebra_simps powr_add C)
  qed
qed

lemma times_ms_aux:
  assumes basis: "basis_wf basis"
  assumes F: "is_expansion_aux F f basis" and G: "is_expansion_aux G g basis"
  shows   "is_expansion_aux (times_ms_aux F G) (λx. f x * g x) basis"
  using F G
proof (coinduction arbitrary: F f G g rule: is_expansion_aux_coinduct_upto)
  case (B F f G g)
  note IH = this
  show ?case
  proof (cases "F = MSLNil ∨ G = MSLNil")
    case True
    with IH have "eventually (λx. f x = 0) at_top ∨ eventually (λx. g x = 0) at_top"
      and length: "length basis = Suc (expansion_level TYPE('a))"
      by (auto elim: is_expansion_aux.cases)
    from this(1) have "eventually (λx. f x * g x = 0) at_top" by (auto elim!: eventually_mono)
    with length True show ?thesis by auto
  next
    case False
    from False obtain C1 e1 F' where F: "F = MSLCons (C1, e1) F'" 
      by (cases F rule: ms_aux_cases) simp_all
    from False obtain C2 e2 G' where G: "G = MSLCons (C2, e2) G'" 
      by (cases G rule: ms_aux_cases) simp_all
    
    from IH(1) F obtain b basis' where
      basis': "basis = b # basis'" and F':
      "is_expansion_aux F' (λx. f x - eval C1 x * b x powr e1) (b # basis')"
      "is_expansion C1 basis'" "⋀e'. e' > e1 ⟹ f ∈ o(λx. b x powr e')"
      "ms_exp_gt e1 (ms_aux_hd_exp F')"
      by (auto elim!: is_expansion_aux_MSLCons)
    from IH(2) G basis' have G':
      "is_expansion_aux G' (λx. g x - eval C2 x * b x powr e2) (b # basis')"
      "is_expansion C2 basis'" "⋀e'. e' > e2 ⟹ g ∈ o(λx. b x powr e')"
      "ms_exp_gt e2 (ms_aux_hd_exp G')"
      by (auto elim!: is_expansion_aux_MSLCons)  
    let ?P = "(λxs'' fa'' basisa''. ∃F f G. xs'' = times_ms_aux F G ∧
            (∃g. fa'' = (λx. f x * g x) ∧ basisa'' = b # basis' ∧ is_expansion_aux F f
                  (b # basis') ∧ is_expansion_aux G g (b # basis')))"
    have "ms_closure ?P (plus_ms_aux (scale_shift_ms_aux (C1, e1) G') (times_ms_aux F' G))
            (λx. eval C1 x * b x powr e1 * (g x - eval C2 x * b x powr e2) +
                 (f x - eval C1 x * b x powr e1) * g x) (b # basis')"
      (is "ms_closure _ ?zs _ _") using IH(2) basis' basis
      by (intro ms_closure_add ms_closure_embed'[OF scale_shift_ms_aux] 
            ms_closure_mult[OF ms_closure_embed' ms_closure_embed'] F' G') simp_all
    hence "ms_closure ?P (plus_ms_aux (scale_shift_ms_aux (C1, e1) G') (times_ms_aux F' G))
            (λx. f x * g x - eval C1 x * eval C2 x * b x powr (e1 + e2)) (b # basis')"
      by (simp add: algebra_simps powr_add)
    moreover have "(λx. f x * g x) ∈ o(λx. b x powr e)" if "e > e1 + e2" for e
    proof -
      define d where "d = (e - e1 - e2) / 2"
      from that have "d > 0" by (simp add: d_def)
      hence "(λx. f x * g x) ∈ o(λx. b x powr (e1 + d) * b x powr (e2 + d))"
        by (intro landau_o.small.mult F' G') simp_all
      also have "… = o(λx. b x powr e)"
        by (simp add: d_def powr_add [symmetric])
      finally show ?thesis .
    qed
    moreover from ‹ms_exp_gt e1 _› ‹ms_exp_gt e2 _›
      have "ms_exp_gt (e1 + e2) (ms_aux_hd_exp ?zs)"
        by (cases "ms_aux_hd_exp F'"; cases "ms_aux_hd_exp G'") 
           (simp_all add: hd_exp_times hd_exp_plus hd_exp_basis G)    
    ultimately show ?thesis using F'(2) G'(2) basis' basis
      by (simp add: times_ms_aux_MSLCons basis_wf_Cons is_expansion_mult F G)
  qed
qed (insert basis, simp_all)




lemma powser_ms_aux_MSLNil_iff [simp]: "powser_ms_aux cs f = MSLNil ⟷ cs = MSLNil"
  by (subst powser_ms_aux.code) (simp split: msllist.splits)

lemma powser_ms_aux_MSLNil [simp]: "powser_ms_aux MSLNil f = MSLNil"
  by (subst powser_ms_aux.code) simp

lemma powser_ms_aux_MSLNil' [simp]: 
  "powser_ms_aux (MSLCons c cs) MSLNil = MSLCons (const_expansion c, 0) MSLNil"
  by (subst powser_ms_aux.code) simp

lemma powser_ms_aux_MSLCons: 
  "powser_ms_aux (MSLCons c cs) f = MSLCons (const_expansion c, 0) (times_ms_aux f (powser_ms_aux cs f))"
  by (subst powser_ms_aux.code) simp

lemma hd_exp_powser: "ms_aux_hd_exp (powser_ms_aux cs f) = (if cs = MSLNil then None else Some 0)"
  by (subst powser_ms_aux.code) (simp split: msllist.splits)
  
lemma powser_ms_aux:
  assumes "convergent_powser cs" and basis: "basis_wf basis"
  assumes G: "is_expansion_aux G g basis" "ms_exp_gt 0 (ms_aux_hd_exp G)"
  shows   "is_expansion_aux (powser_ms_aux cs G) (powser cs ∘ g) basis"
using assms(1)
proof (coinduction arbitrary: cs rule: is_expansion_aux_coinduct_upto)
  case (B cs)
  show ?case
  proof (cases cs)
    case MSLNil
    with G show ?thesis by (auto simp: is_expansion_aux_expansion_level)
  next
    case (MSLCons c cs')
    from is_expansion_aux_basis_nonempty[OF G(1)]
      obtain b basis' where basis': "basis = b # basis'" by (cases basis) simp_all
    from B have conv: "convergent_powser cs'" by (auto simp: MSLCons dest: convergent_powser_stl)  
    from basis basis' have b_limit: "filterlim b at_top at_top" by (simp add: basis_wf_Cons)
    hence b_pos: "eventually (λx. b x > 0) at_top" by (simp add: filterlim_at_top_dense)
    
    from G(2) have "g ∈ o(λx. hd basis x powr 0)"
      by (intro is_expansion_aux_imp_smallo[OF G(1)]) simp
    with basis' have "g ∈ o(λx. b x powr 0)" by simp
    also have "(λx. b x powr 0) ∈ Θ(λx. 1)" 
       using b_pos by (intro bigthetaI_cong) (auto elim!: eventually_mono)
    finally have g_limit: "(g ⤏ 0) at_top" by (auto dest: smalloD_tendsto)
  
    let ?P = "(λxs'' fa'' basisa''. ∃cs. xs'' = powser_ms_aux cs G ∧
                     fa'' = powser cs ∘ g ∧ basisa'' = b # basis' ∧ convergent_powser cs)"
    have "ms_closure ?P (times_ms_aux G (powser_ms_aux cs' G)) 
            (λx. g x * (powser cs' ∘ g) x) basis" using conv
      by (intro ms_closure_mult [OF ms_closure_embed' ms_closure_embed] G)
         (auto simp add: basis' o_def)
    also have "(λx. g x * (powser cs' ∘ g) x) = (λx. x * powser cs' x) ∘ g"
      by (simp add: o_def)
    finally have "ms_closure ?P (times_ms_aux G (powser_ms_aux cs' G)) 
                    (λx. (powser cs ∘ g) x - c * b x powr 0) basis" unfolding o_def
    proof (rule ms_closure_cong)
      note b_pos
      moreover have "eventually (λx. powser cs (g x) = g x * powser cs' (g x) + c) at_top"
      proof -
        from B have "eventually (λx. powser cs x = x * powser cs' x + c) (nhds 0)"
          by (simp add: MSLCons powser_MSLCons)
        from this and g_limit show ?thesis by (rule eventually_compose_filterlim)
      qed
      ultimately show "eventually (λx. g x * powser cs' (g x) = 
                         powser cs (g x) - c * b x powr 0) at_top"
        by eventually_elim simp
    qed 
    moreover from basis G have "is_expansion (const_expansion c :: 'a) basis'"
      by (intro is_expansion_const)
         (auto dest: is_expansion_aux_expansion_level simp: basis' basis_wf_Cons)
    moreover have "powser cs ∘ g ∈ o(λx. b x powr e)" if "e > 0" for e
    proof -
      have "((powser cs ∘ g) ⤏ powser cs 0) at_top" unfolding o_def
        by (intro isCont_tendsto_compose[OF _ g_limit] isCont_powser B)
      hence "powser cs ∘ g ∈ O(λ_. 1)" 
        by (intro bigoI_tendsto[where c = "powser cs 0"]) (simp_all add: o_def)
      also from b_pos have  "O(λ_. 1) = O(λx. b x powr 0)" 
        by (intro landau_o.big.cong) (auto elim: eventually_mono)
      also from that b_limit have "(λx. b x powr 0) ∈ o(λx. b x powr e)"
        by (subst powr_smallo_iff) simp_all
      finally show ?thesis .
    qed
    moreover from G have "ms_exp_gt 0 (ms_aux_hd_exp (times_ms_aux G (powser_ms_aux cs' G)))"
      by (cases "ms_aux_hd_exp G") (simp_all add: hd_exp_times MSLCons hd_exp_powser)
    ultimately show ?thesis
      by (simp add: MSLCons powser_ms_aux_MSLCons basis')
  qed
qed (insert assms, simp_all)
  
lemma powser_ms_aux':
  assumes powser: "convergent_powser' cs f" and basis: "basis_wf basis"
  assumes G: "is_expansion_aux G g basis" "ms_exp_gt 0 (ms_aux_hd_exp G)"
  shows   "is_expansion_aux (powser_ms_aux cs G) (f ∘ g) basis"
proof (rule is_expansion_aux_cong)
  from is_expansion_aux_basis_nonempty[OF G(1)] basis
    have "filterlim (hd basis) at_top at_top" by (cases basis) (simp_all add: basis_wf_Cons)
  hence pos: "eventually (λx. hd basis x > 0) at_top" by (simp add: filterlim_at_top_dense)
  from G(2) have "g ∈ o(λx. hd basis x powr 0)"
    by (intro is_expansion_aux_imp_smallo[OF G(1)]) simp
  also have "(λx. hd basis x powr 0) ∈ Θ(λx. 1)" 
     using pos by (intro bigthetaI_cong) (auto elim!: eventually_mono)
  finally have g_limit: "(g ⤏ 0) at_top" by (auto dest: smalloD_tendsto)
  
  from powser have "eventually (λx. powser cs x = f x) (nhds 0)"
    by (rule convergent_powser'_eventually)
  from this and g_limit show "eventually (λx. (powser cs ∘ g) x = (f ∘ g) x) at_top" 
    unfolding o_def by (rule eventually_compose_filterlim)
qed (rule assms powser_ms_aux convergent_powser'_imp_convergent_powser)+

lemma inverse_ms_aux:
  assumes basis: "basis_wf basis"
  assumes F: "is_expansion_aux F f basis" "trimmed_ms_aux F"
  shows   "is_expansion_aux (inverse_ms_aux F) (λx. inverse (f x)) basis"
proof (cases F rule: ms_aux_cases)
  case (MSLCons C e F')
  from F MSLCons obtain b basis' where F': "basis = b # basis'" "trimmed C"
    "is_expansion_aux F' (λx. f x - eval C x * b x powr e) (b # basis')"
    "is_expansion C basis'" "(⋀e'. e < e' ⟹ f ∈ o(λx. b x powr e'))" 
    "ms_exp_gt e (ms_aux_hd_exp F')"
    by (auto elim!: is_expansion_aux_MSLCons simp: trimmed_ms_aux_def)
  define f' where "f' = (λx. f x - eval C x * b x powr e)"
  from basis F' have b_limit: "filterlim b at_top at_top" by (simp add: basis_wf_Cons)
  hence b_pos: "eventually (λx. b x > 0) at_top" by (simp add: filterlim_at_top_dense)
  moreover from F'(6)
    have "ms_exp_gt 0 (ms_aux_hd_exp (scale_shift_ms_aux (inverse C, - e) F'))"
    by (cases "ms_aux_hd_exp F'") (simp_all add: hd_exp_basis)
  ultimately have
       "is_expansion_aux (inverse_ms_aux F) 
          (λx. eval (inverse C) x * b x powr (-e) * 
                 ((λx. inverse (1 + x)) ∘ (λx. eval (inverse C) x * b x powr (-e) * f' x)) x)
          (b # basis')" (is "is_expansion_aux _ ?h _")
    unfolding MSLCons inverse_ms_aux.simps Let_def fst_conv snd_conv f'_def eval_inverse [symmetric]
    using basis F(2) F'(1,3,4)
    by (intro scale_shift_ms_aux powser_ms_aux' is_expansion_inverse)
       (simp_all add: convergent_powser'_geometric basis_wf_Cons trimmed_ms_aux_def MSLCons)
  also have "b # basis' = basis" by (simp add: F')
  finally show ?thesis
  proof (rule is_expansion_aux_cong, goal_cases)
    case 1
    from basis F' have "eventually (λx. eval C x ≠ 0) at_top" 
      by (intro trimmed_imp_eventually_nz[of basis']) (simp_all add: basis_wf_Cons)
    with b_pos show ?case by eventually_elim (simp add: o_def field_simps powr_minus f'_def)
  qed
qed (insert assms, auto simp: trimmed_ms_aux_def)

lemma hd_exp_inverse: 
  "xs ≠ MSLNil ⟹ ms_aux_hd_exp (inverse_ms_aux xs) = map_option uminus (ms_aux_hd_exp xs)"
  by (cases xs) (auto simp: Let_def hd_exp_basis hd_exp_powser inverse_ms_aux.simps)

lemma eval_pos_if_dominant_term_pos:
  assumes "basis_wf basis" "is_expansion F basis" "trimmed F"
          "fst (dominant_term F) > 0"
  shows   "eventually (λx. eval F x > 0) at_top"
proof -
  have "eval F ∼[at_top] eval_monom (dominant_term F) basis"
    by (intro dominant_term assms)
  from trimmed_imp_eventually_sgn[OF assms(1-3)]
    have "∀F x in at_top. sgn (eval F x) = sgn (fst (dominant_term F))" .
  moreover from assms
    have "eventually (λx. eval_monom (dominant_term F) basis x > 0) at_top"
    by (intro eval_monom_pos)
  ultimately show ?thesis by eventually_elim (insert assms, simp add: sgn_if split: if_splits)
qed

lemma eval_pos_if_dominant_term_pos':
  assumes "basis_wf basis" "trimmed_ms_aux F" "is_expansion_aux F f basis" 
          "fst (dominant_term_ms_aux F) > 0"
  shows   "eventually (λx. f x > 0) at_top"
proof -
  have "f ∼[at_top] eval_monom (dominant_term_ms_aux F) basis"
    by (intro dominant_term_ms_aux assms)
  from dominant_term_ms_aux(2)[OF assms(1-3)]
    have "∀F x in at_top. sgn (f x) = sgn (fst (dominant_term_ms_aux F))" .
  moreover from assms
    have "eventually (λx. eval_monom (dominant_term_ms_aux F) basis x > 0) at_top"
    by (intro eval_monom_pos)
  ultimately show ?thesis by eventually_elim (insert assms, simp add: sgn_if split: if_splits)
qed
  
lemma eval_neg_if_dominant_term_neg':
  assumes "basis_wf basis" "trimmed_ms_aux F" "is_expansion_aux F f basis" 
          "fst (dominant_term_ms_aux F) < 0"
  shows   "eventually (λx. f x < 0) at_top"
proof -
  have "f ∼[at_top] eval_monom (dominant_term_ms_aux F) basis"
    by (intro dominant_term_ms_aux assms)
  from dominant_term_ms_aux(2)[OF assms(1-3)]
    have "∀F x in at_top. sgn (f x) = sgn (fst (dominant_term_ms_aux F))" .
  moreover from assms
  have "eventually (λx. eval_monom (dominant_term_ms_aux F) basis x < 0) at_top"
    by (intro eval_monom_neg)
  ultimately show ?thesis by eventually_elim (insert assms, simp add: sgn_if split: if_splits)
qed

lemma fst_dominant_term_ms_aux_MSLCons: 
  "fst (dominant_term_ms_aux (MSLCons x xs)) = fst (dominant_term (fst x))"
  by (auto simp: dominant_term_ms_aux_def split: prod.splits)

lemma powr_ms_aux:
  assumes basis: "basis_wf basis"
  assumes F: "is_expansion_aux F f basis" "trimmed_ms_aux F" "fst (dominant_term_ms_aux F) > 0"
  shows   "is_expansion_aux (powr_ms_aux abort F p) (λx.  f x powr p) basis"
proof (cases F rule: ms_aux_cases)
  case (MSLCons C e F')
  from F MSLCons obtain b basis' where F': "basis = b # basis'" "trimmed C" "fst (dominant_term C) > 0"
    "is_expansion_aux F' (λx. f x - eval C x * b x powr e) (b # basis')"
    "is_expansion C basis'" "(⋀e'. e < e' ⟹ f ∈ o(λx. b x powr e'))"
    "ms_exp_gt e (ms_aux_hd_exp F')"
    by (auto elim!: is_expansion_aux_MSLCons simp: trimmed_ms_aux_def dominant_term_ms_aux_def 
             split: prod.splits)
  define f' where "f' = (λx. f x - eval C x * b x powr e)"
  from basis F' have b_limit: "filterlim b at_top at_top" by (simp add: basis_wf_Cons)
  hence b_pos: "eventually (λx. b x > 0) at_top" by (simp add: filterlim_at_top_dense)
  moreover from F' have "ms_exp_gt 0 (ms_aux_hd_exp (scale_shift_ms_aux (inverse C, - e) F'))"
    by (cases "ms_aux_hd_exp F'") (simp_all add: hd_exp_basis)
  ultimately have
       "is_expansion_aux (powr_ms_aux abort F p) 
          (λx. eval (powr_expansion abort C p) x * b x powr (e * p) * 
                 ((λx. (1 + x) powr p) ∘ (λx. eval (inverse C) x * b x powr (-e) * f' x)) x)
          (b # basis')" (is "is_expansion_aux _ ?h _")
    unfolding MSLCons powr_ms_aux.simps Let_def fst_conv snd_conv f'_def eval_inverse [symmetric]
        using basis F'(1-5)
    by (intro scale_shift_ms_aux powser_ms_aux' is_expansion_inverse is_expansion_powr)
       (simp_all add: MSLCons basis_wf_Cons convergent_powser'_gbinomial)
  also have "b # basis' = basis" by (simp add: F')
  finally show ?thesis
  proof (rule is_expansion_aux_cong, goal_cases)
    case 1
    from basis F' have "eventually (λx. eval C x > 0) at_top" 
      by (intro eval_pos_if_dominant_term_pos[of basis']) (simp_all add: basis_wf_Cons)
    moreover from basis F have "eventually (λx. f x > 0) at_top"
      by (intro eval_pos_if_dominant_term_pos'[of basis F])
    ultimately show ?case using b_pos
      by eventually_elim 
         (simp add: powr_powr [symmetric] powr_minus powr_mult powr_divide f'_def field_simps)
  qed
qed (insert assms, auto simp: trimmed_ms_aux_def)

lemma power_ms_aux:
  assumes basis: "basis_wf basis"
  assumes F: "is_expansion_aux F f basis" "trimmed_ms_aux F"
  shows   "is_expansion_aux (power_ms_aux abort F n) (λx. f x ^ n) basis"
proof (cases F rule: ms_aux_cases)
  case (MSLCons C e F')
  from F MSLCons obtain b basis' where F': "basis = b # basis'" "trimmed C"
    "is_expansion_aux F' (λx. f x - eval C x * b x powr e) (b # basis')"
    "is_expansion C basis'" "(⋀e'. e < e' ⟹ f ∈ o(λx. b x powr e'))" 
    "ms_exp_gt e (ms_aux_hd_exp F')"
    by (auto elim!: is_expansion_aux_MSLCons simp: trimmed_ms_aux_def dominant_term_ms_aux_def 
             split: prod.splits)
  define f' where "f' = (λx. f x - eval C x * b x powr e)"
  from basis F' have b_limit: "filterlim b at_top at_top" by (simp add: basis_wf_Cons)
  hence b_pos: "eventually (λx. b x > 0) at_top" by (simp add: filterlim_at_top_dense)
  moreover have "ms_exp_gt 0 (ms_aux_hd_exp (scale_shift_ms_aux (inverse C, - e) F'))"
    using F'(6) by (cases "ms_aux_hd_exp F'") (simp_all add: hd_exp_basis)
  ultimately have
       "is_expansion_aux (power_ms_aux abort F n) 
          (λx. eval (power_expansion abort C n) x * b x powr (e * real n) * 
                 ((λx. (1 + x) ^ n) ∘ (λx. eval (inverse C) x * b x powr (-e) * f' x)) x)
          (b # basis')" (is "is_expansion_aux _ ?h _")
    unfolding MSLCons power_ms_aux.simps Let_def fst_conv snd_conv f'_def eval_inverse [symmetric]
        using basis F'(1-4)
    by (intro scale_shift_ms_aux powser_ms_aux' is_expansion_inverse is_expansion_power)
       (simp_all add: MSLCons basis_wf_Cons convergent_powser'_gbinomial')
  also have "b # basis' = basis" by (simp add: F')
  finally show ?thesis
  proof (rule is_expansion_aux_cong, goal_cases)
    case 1
    from F'(1,2,4) basis have "eventually (λx. eval C x ≠ 0) at_top"
      using trimmed_imp_eventually_nz[of basis' C] by (simp add: basis_wf_Cons)
    thus ?case using b_pos
      by eventually_elim
         (simp add: field_simps f'_def powr_minus powr_powr [symmetric] powr_realpow 
                    power_mult_distrib [symmetric] power_divide [symmetric]
               del: power_mult_distrib power_divide)
  qed
qed (insert assms, auto simp: trimmed_ms_aux_def)

lemma snd_dominant_term_ms_aux_MSLCons:
  "snd (dominant_term_ms_aux (MSLCons (C, e) xs)) = e # snd (dominant_term C)"
  by (simp add: dominant_term_ms_aux_def case_prod_unfold)


subsubsection ‹Type-class instantiations›

datatype 'a ms = MS "('a × real) msllist" "real ⇒ real"

instantiation ms :: (uminus) uminus
begin

primrec uminus_ms where
  "-(MS xs f) = MS (uminus_ms_aux xs) (λx. -f x)"
  
instance ..
end

instantiation ms :: (plus) plus
begin

fun plus_ms :: "'a ms ⇒ 'a ms ⇒ 'a ms" where
  "MS xs f + MS ys g = MS (plus_ms_aux xs ys) (λx. f x + g x)"

instance ..
end

instantiation ms :: ("{plus,uminus}") minus
begin

definition minus_ms :: "'a ms ⇒ 'a ms ⇒ 'a ms" where
  "F - G = F + -(G :: 'a ms)"

instance ..
end

instantiation ms :: ("{plus,times}") times
begin

fun times_ms :: "'a ms ⇒ 'a ms ⇒ 'a ms" where
  "MS xs f * MS ys g = MS (times_ms_aux xs ys) (λx. f x * g x)"

instance ..
end

instantiation ms :: (multiseries) inverse
begin

fun inverse_ms :: "'a ms ⇒ 'a ms" where
  "inverse_ms (MS xs f) = MS (inverse_ms_aux xs) (λx. inverse (f x))"

fun divide_ms :: "'a ms ⇒ 'a ms ⇒ 'a ms" where
  "divide_ms (MS xs f) (MS ys g) = MS (times_ms_aux xs (inverse_ms_aux ys)) (λx. f x / g x)"

instance ..
end


instantiation ms :: (multiseries) multiseries
begin

definition expansion_level_ms :: "'a ms itself ⇒ nat" where
  expansion_level_ms_def [simp]: "expansion_level_ms _ = Suc (expansion_level (TYPE('a)))"

definition zero_expansion_ms :: "'a ms" where
  "zero_expansion = MS MSLNil (λ_. 0)"

definition const_expansion_ms :: "real ⇒ 'a ms" where
  "const_expansion c = MS (const_ms_aux c) (λ_. c)"

primrec is_expansion_ms :: "'a ms ⇒ basis ⇒ bool" where
  "is_expansion (MS xs f) = is_expansion_aux xs f"

lemma is_expansion_ms_def': "is_expansion F = (case F of MS xs f ⇒ is_expansion_aux xs f)"
  by (simp add: is_expansion_ms_def split: ms.splits)

primrec eval_ms :: "'a ms ⇒ real ⇒ real" where
  "eval_ms (MS _ f) = f"
  
lemma eval_ms_def': "eval F = (case F of MS _ f ⇒ f)"
  by (cases F) simp_all

primrec powr_expansion_ms :: "bool ⇒ 'a ms ⇒ real ⇒ 'a ms" where
  "powr_expansion_ms abort (MS xs f) p = MS (powr_ms_aux abort xs p) (λx. f x powr p)"
  
primrec power_expansion_ms :: "bool ⇒ 'a ms ⇒ nat ⇒ 'a ms" where
  "power_expansion_ms abort (MS xs f) n = MS (power_ms_aux abort xs n) (λx. f x ^ n)"

primrec trimmed_ms :: "'a ms ⇒ bool" where
  "trimmed_ms (MS xs _) ⟷ trimmed_ms_aux xs"

primrec dominant_term_ms :: "'a ms ⇒ monom" where
  "dominant_term_ms (MS xs _) = dominant_term_ms_aux xs"

lemma length_dominant_term_ms: 
  "length (snd (dominant_term (F :: 'a ms))) = Suc (expansion_level TYPE('a))"
  by (cases F) (simp_all add: length_dominant_term)  

instance 
proof (standard, goal_cases length_basis zero const uminus plus minus times 
         inverse divide powr power smallo smallomega trimmed dominant dominant_bigo)
  case (smallo basis F b e)
  from ‹is_expansion F basis› obtain xs f where F: "F = MS xs f" "is_expansion_aux xs f basis"
    by (simp add: is_expansion_ms_def' split: ms.splits)
  from F(2) have nonempty: "basis ≠ []" by (rule is_expansion_aux_basis_nonempty)
  with smallo have filterlim_hd_basis: "filterlim (hd basis) at_top at_top"
    by (cases basis) (simp_all add: basis_wf_Cons)
  from F(2) obtain e' where "f ∈ o(λx. hd basis x powr e')"
    by (erule is_expansion_aux_imp_smallo')
  also from smallo nonempty filterlim_hd_basis have "(λx. hd basis x powr e') ∈ o(λx. b x powr e)"
    by (intro ln_smallo_imp_flat) auto
  finally show ?case by (simp add: F)
next
  case (smallomega basis F b e)
  obtain xs f where F: "F = MS xs f" by (cases F) simp_all
  from this smallomega have *: "is_expansion_aux xs f basis" by simp
  with smallomega F have "f ∈ ω(λx. b x powr e)"
    by (intro is_expansion_aux_imp_smallomega [OF _ *])
       (simp_all add: is_expansion_ms_def' split: ms.splits)
  with F show ?case by simp
next
  case (minus basis F G)
  thus ?case
    by (simp add: minus_ms_def is_expansion_ms_def' add_uminus_conv_diff [symmetric] 
                  plus_ms_aux uminus_ms_aux del: add_uminus_conv_diff split: ms.splits)
next
  case (divide basis F G)
  have "G / F = G * inverse F" by (cases F; cases G) (simp add: divide_inverse)
  with divide show ?case
    by (simp add: is_expansion_ms_def' divide_inverse times_ms_aux inverse_ms_aux split: ms.splits)
next
  fix c :: real
  show "trimmed (const_expansion c :: 'a ms) ⟷ c ≠ 0"
    by (simp add: const_expansion_ms_def trimmed_ms_aux_def const_ms_aux_def 
                  trimmed_const_expansion split: msllist.splits)
next
  fix F :: "'a ms" assume "trimmed F"
  thus "fst (dominant_term F) ≠ 0"
    by (cases F) (auto simp: dominant_term_ms_aux_def trimmed_ms_aux_MSLCons case_prod_unfold 
                    trimmed_imp_dominant_term_nz split: msllist.splits)
next
  fix F :: "'a ms"
  have "times_ms_aux (MSLCons (const_expansion 1, 0) MSLNil) xs = xs" for xs :: "('a × real) msllist"
  proof (coinduction arbitrary: xs rule: msllist.coinduct_upto)
    case Eq_real_prod_msllist
    have "map_prod ((*) (const_expansion 1 :: 'a)) ((+) (0::real)) = (λx. x)"
      by (rule ext) (simp add: map_prod_def times_const_expansion_1 case_prod_unfold)
    moreover have "mslmap … = (λx. x)" by (rule ext) (simp add: msllist.map_ident)
    ultimately have "scale_shift_ms_aux (const_expansion 1 :: 'a, 0) = (λx. x)"
      by (simp add: scale_shift_ms_aux_conv_mslmap)
    thus ?case
      by (cases xs rule: ms_aux_cases)
         (auto simp: times_ms_aux_MSLCons times_const_expansion_1
            times_ms_aux.corec.cong_refl)
  qed
  thus "const_expansion 1 * F = F"
    by (cases F) (simp add: const_expansion_ms_def const_ms_aux_def)
next
  fix F :: "'a ms"
  show "fst (dominant_term (- F)) = - fst (dominant_term F)"
       "trimmed (-F) ⟷ trimmed F"
     by (cases F; simp add: dominant_term_ms_aux_def case_prod_unfold uminus_ms_aux_MSLCons
           trimmed_ms_aux_def split: msllist.splits)+
next
  fix F :: "'a ms"
  show "zero_expansion + F = F" "F + zero_expansion = F"
    by (cases F; simp add: zero_expansion_ms_def)+
qed (auto simp: eval_ms_def' const_expansion_ms_def trimmed_ms_aux_imp_nz is_expansion_ms_def' 
                 const_ms_aux uminus_ms_aux plus_ms_aux times_ms_aux inverse_ms_aux 
                 is_expansion_aux_expansion_level dominant_term_ms_aux length_dominant_term_ms
                 minus_ms_def powr_ms_aux power_ms_aux zero_expansion_ms_def 
                 is_expansion_aux.intros(1) dominant_term_ms_aux_bigo
          split: ms.splits)

end


subsubsection ‹Changing the evaluation function of a multiseries›

definition modify_eval :: "(real ⇒ real) ⇒ 'a ms ⇒ 'a ms" where
  "modify_eval f F = (case F of MS xs _ ⇒ MS xs f)"

lemma eval_modify_eval [simp]: "eval (modify_eval f F) = f"
  by (cases F) (simp add: modify_eval_def)


subsubsection ‹Scaling expansions›

definition scale_ms :: "real ⇒ 'a :: multiseries ⇒ 'a" where
  "scale_ms c F = const_expansion c * F"

lemma scale_ms_real [simp]: "scale_ms c (c' :: real) = c * c'"
  by (simp add: scale_ms_def)

definition scale_ms_aux :: "real ⇒ ('a :: multiseries × real) msllist ⇒ ('a × real) msllist" where
  "scale_ms_aux c xs = scale_shift_ms_aux (const_expansion c, 0) xs"


lemma scale_ms_aux_eq_MSLNil_iff [simp]: "scale_ms_aux x xs = MSLNil ⟷ xs = MSLNil"
  unfolding scale_ms_aux_def by simp

lemma times_ms_aux_singleton:
  "times_ms_aux (MSLCons (c, e) MSLNil) xs = scale_shift_ms_aux (c, e) xs"
proof (coinduction arbitrary: xs rule: msllist.coinduct_strong)
  case (Eq_msllist xs)
  thus ?case
    by (cases xs rule: ms_aux_cases)
       (simp_all add: scale_shift_ms_aux_def times_ms_aux_MSLCons)
qed

lemma scale_ms [simp]: "scale_ms c (MS xs f) = MS (scale_ms_aux c xs) (λx. c * f x)"
  by (simp add: scale_ms_def scale_ms_aux_def const_expansion_ms_def const_ms_aux_def 
        times_ms_aux_singleton)

lemma scale_ms_aux_MSLNil [simp]: "scale_ms_aux c MSLNil = MSLNil" 
  by (simp add: scale_ms_aux_def)

lemma scale_ms_aux_MSLCons: 
  "scale_ms_aux c (MSLCons (c', e) xs) = MSLCons (scale_ms c c', e) (scale_ms_aux c xs)"
  by (simp add: scale_ms_aux_def scale_shift_ms_aux_MSLCons scale_ms_def)


subsubsection ‹Additional convenience rules›

lemma trimmed_pos_real_iff [simp]: "trimmed_pos (x :: real) ⟷ x > 0"
  by (auto simp: trimmed_pos_def)

lemma trimmed_pos_ms_iff: 
  "trimmed_pos (MS xs f) ⟷ (case xs of MSLNil ⇒ False | MSLCons x xs ⇒ trimmed_pos (fst x))"
  by (auto simp add: trimmed_pos_def dominant_term_ms_aux_def trimmed_ms_aux_def
           split: msllist.splits prod.splits)

lemma not_trimmed_pos_MSLNil [simp]: "¬trimmed_pos (MS MSLNil f)"
  by (simp add: trimmed_pos_ms_iff)

lemma trimmed_pos_MSLCons [simp]: "trimmed_pos (MS (MSLCons x xs) f) = trimmed_pos (fst x)"
  by (simp add: trimmed_pos_ms_iff)

lemma drop_zero_ms:
  assumes "eventually (λx. eval C x = 0) at_top"
  assumes "is_expansion (MS (MSLCons (C, e) xs) f) basis"
  shows   "is_expansion (MS xs f) basis"
  using assms by (auto simp: is_expansion_ms_def intro: drop_zero_ms_aux)

lemma expansion_level_eq_Nil: "length [] = expansion_level TYPE(real)" by simp
lemma expansion_level_eq_Cons: 
  "length xs = expansion_level TYPE('a::multiseries) ⟹
     length (x # xs) = expansion_level TYPE('a ms)" by simp

lemma basis_wf_insert_ln: 
  assumes "basis_wf [b]"
  shows   "basis_wf [λx. ln (b x)]" (is ?thesis1)
          "basis_wf [b, λx. ln (b x)]" (is ?thesis2)
          "is_expansion (MS (MSLCons (1::real,1) MSLNil) (λx. ln (b x))) [λx. ln (b x)]" (is ?thesis3)
proof -
  have "ln ∈ o(λx. x :: real)"
    using ln_x_over_x_tendsto_0 by (intro smalloI_tendsto) auto
  with assms show ?thesis1 ?thesis2
    by (auto simp: basis_wf_Cons
          intro!: landau_o.small.compose[of _ _ _ "λx. ln (b x)"] filterlim_compose[OF ln_at_top])
  from assms have b_limit: "filterlim b at_top at_top" by (simp add: basis_wf_Cons)
  hence ev: "eventually (λx. b x > 1) at_top" by (simp add: filterlim_at_top_dense)
  have "(λx::real. x) ∈ Θ(λx. x powr 1)"
    by (intro bigthetaI_cong eventually_mono[OF eventually_gt_at_top[of 0]]) auto
  also have "(λx::real. x powr 1) ∈ o(λa. a powr e)" if "e > 1" for e
    by (subst powr_smallo_iff) (auto simp: that filterlim_ident)
  finally show ?thesis3 using assms ev
    by (auto intro!: is_expansion_aux.intros landau_o.small.compose[of _ at_top _ "λx. ln (b x)"]
                     filterlim_compose[OF ln_at_top b_limit] is_expansion_real.intros
             elim!: eventually_mono)
qed

lemma basis_wf_lift_modification:
  assumes "basis_wf (b' # b # bs)" "basis_wf (b # bs')"
  shows   "basis_wf (b' # b # bs')"
  using assms by (simp add: basis_wf_many)

lemma default_basis_wf: "basis_wf [λx. x]" 
  by (simp add: basis_wf_Cons filterlim_ident)

  
subsubsection ‹Lifting expansions›

definition is_lifting where
  "is_lifting f basis basis' ⟷ 
     (∀C. eval (f C) = eval C ∧ (is_expansion C basis ⟶ is_expansion (f C) basis') ∧
          trimmed (f C) = trimmed C ∧ fst (dominant_term (f C)) = fst (dominant_term C))"
  
lemma is_liftingD:
  assumes "is_lifting f basis basis'"
  shows   "eval (f C) = eval C" "is_expansion C basis ⟹ is_expansion (f C) basis'"
          "trimmed (f C) ⟷ trimmed C" "fst (dominant_term (f C)) = fst (dominant_term C)"
  using assms [unfolded is_lifting_def] unfolding is_lifting_def by blast+
  

definition lift_ms :: "'a :: multiseries ⇒ 'a ms" where
  "lift_ms C = MS (MSLCons (C, 0) MSLNil) (eval C)"

lemma is_expansion_lift:
  assumes "basis_wf (b # basis)" "is_expansion C basis"
  shows   "is_expansion (lift_ms C) (b # basis)"
proof -
  from assms have "filterlim b at_top at_top" by (simp add: basis_wf_Cons)
  hence b_pos: "eventually (λx. b x > 0) at_top" by (simp add: filterlim_at_top_dense)
  moreover from assms have "eval C ∈ o(λx. b x powr e)" if "e > 0" for e
    using that by (intro is_expansion_imp_smallo[OF _ assms(2)]) (simp_all add: basis_wf_Cons)
  ultimately show ?thesis using assms
    by (auto intro!: is_expansion_aux.intros simp: lift_ms_def is_expansion_length 
             elim: eventually_mono)
qed

lemma eval_lift_ms [simp]: "eval (lift_ms C) = eval C"
  by (simp add: lift_ms_def)

lemma is_lifting_lift:
  assumes "basis_wf (b # basis)"
  shows   "is_lifting lift_ms basis (b # basis)"
  using is_expansion_lift[OF assms] unfolding is_lifting_def
  by (auto simp: lift_ms_def trimmed_ms_aux_MSLCons dominant_term_ms_aux_def case_prod_unfold)


definition map_ms_aux :: 
    "('a :: multiseries ⇒ 'b :: multiseries) ⇒ 
       ('a × real) msllist ⇒ ('b × real) msllist" where
  "map_ms_aux f xs = mslmap (λ(c,e). (f c, e)) xs"

lemma map_ms_aux_MSLNil [simp]: "map_ms_aux f MSLNil = MSLNil"
  by (simp add: map_ms_aux_def)

lemma map_ms_aux_MSLCons: "map_ms_aux f (MSLCons (C, e) xs) = MSLCons (f C, e) (map_ms_aux f xs)"
  by (simp add: map_ms_aux_def)

lemma hd_exp_map [simp]: "ms_aux_hd_exp (map_ms_aux f xs) = ms_aux_hd_exp xs"
  by (simp add: ms_aux_hd_exp_def map_ms_aux_def split: msllist.splits)

lemma map_ms_altdef: "map_ms f (MS xs g) = MS (map_ms_aux f xs) g"
  by (simp add: map_ms_aux_def map_prod_def)

lemma eval_map_ms [simp]: "eval (map_ms f F) = eval F"
  by (cases F) simp_all

lemma is_expansion_map_aux:
  fixes f :: "'a :: multiseries ⇒ 'b :: multiseries"
  assumes "is_expansion_aux xs g (b # basis)"
  assumes "⋀C. is_expansion C basis ⟹ is_expansion (f C) basis'"
  assumes "length basis' = expansion_level TYPE('b)"
  assumes "⋀C. eval (f C) = eval C"
  shows   "is_expansion_aux (map_ms_aux f xs) g (b # basis')"
  using assms(1)
proof (coinduction arbitrary: xs g rule: is_expansion_aux_coinduct)
  case (is_expansion_aux xs g)
  show ?case
  proof (cases xs rule: ms_aux_cases)
    case MSLNil
    with is_expansion_aux show ?thesis
      by (auto simp: assms elim: is_expansion_aux.cases)
  next
    case (MSLCons c e xs')
    with is_expansion_aux show ?thesis
      by (auto elim!: is_expansion_aux_MSLCons simp: map_ms_aux_MSLCons assms)
  qed    
qed

lemma is_expansion_map:
  fixes f :: "'a :: multiseries ⇒ 'b :: multiseries"
    and F :: "'a ms"
  assumes "is_expansion G (b # basis)"
  assumes "⋀C. is_expansion C basis ⟹ is_expansion (f C) basis'"
  assumes "⋀C. eval (f C) = eval C"
  assumes "length basis' = expansion_level TYPE('b)"
  shows   "is_expansion (map_ms f G) (b # basis')"
  using assms by (cases G, simp only: map_ms_altdef) (auto intro!: is_expansion_map_aux)

lemma is_expansion_map_final: 
  fixes f :: "'a :: multiseries ⇒ 'b :: multiseries"
    and F :: "'a ms"
  assumes "is_lifting f basis basis'"
  assumes "is_expansion G (b # basis)"
  assumes "length basis' = expansion_level TYPE('b)"
  shows   "is_expansion (map_ms f G) (b # basis')"
  by (rule is_expansion_map[OF assms(2)]) (insert assms, simp_all add: is_lifting_def)

lemma fst_dominant_term_map_ms: 
  "is_lifting f basis basis' ⟹ fst (dominant_term (map_ms f C)) = fst (dominant_term C)"
  by (cases C)
     (simp add: dominant_term_ms_aux_def case_prod_unfold is_lifting_def split: msllist.splits)

lemma trimmed_map_ms:
  "is_lifting f basis basis' ⟹ trimmed (map_ms f C) ⟷ trimmed C"
  by (cases C) (simp add: trimmed_ms_aux_def is_lifting_def split: msllist.splits)

lemma is_lifting_map: 
  fixes f :: "'a :: multiseries ⇒ 'b :: multiseries"
    and F :: "'a ms"
  assumes "is_lifting f basis basis'"
  assumes "length basis' = expansion_level TYPE('b)"
  shows   "is_lifting (map_ms f) (b # basis) (b # basis')"
  unfolding is_lifting_def
  by (intro allI conjI impI is_expansion_map_final[OF assms(1)]) 
     (insert assms, simp_all add: is_lifting_def fst_dominant_term_map_ms[OF assms(1)] 
        trimmed_map_ms[OF assms(1)])

lemma is_lifting_id: "is_lifting (λx. x) basis basis"
  by (simp add: is_lifting_def)

lemma is_lifting_trans: 
  "is_lifting f basis basis' ⟹ is_lifting g basis' basis'' ⟹ is_lifting (λx. g (f x)) basis basis''"
  by (auto simp: is_lifting_def)

lemma is_expansion_X: "is_expansion (MS (MSLCons (1 :: real, 1) MSLNil) (λx. x)) [λx. x]"
proof -
  have "(λx::real. x) ∈ Θ(λx. x powr 1)"
    by (intro bigthetaI_cong eventually_mono[OF eventually_gt_at_top[of 0]]) auto
  also have "(λx::real. x powr 1) ∈ o(λa. a powr e)" if "e > 1" for e
    by (subst powr_smallo_iff) (auto simp: that filterlim_ident)
  finally show ?thesis  
    by (auto intro!: is_expansion_aux.intros is_expansion_real.intros 
                     eventually_mono[OF eventually_gt_at_top[of "0::real"]])
qed


inductive expands_to :: "(real ⇒ real) ⇒ 'a :: multiseries ⇒ basis ⇒ bool" 
    (infix ‹(expands'_to)› 50) where
  "is_expansion F basis ⟹ eventually (λx. eval F x = f x) at_top ⟹ (f expands_to F) basis"

lemma dominant_term_expands_to:
  assumes "basis_wf basis" "(f expands_to F) basis" "trimmed F"
  shows   "f ∼[at_top] eval_monom (dominant_term F) basis"
proof -
  from assms have "eval F ∼[at_top] f" by (intro asymp_equiv_refl_ev) (simp add: expands_to.simps)
  also from dominant_term[OF assms(1) _ assms(3)] assms(2)
    have "eval F ∼[at_top] eval_monom (dominant_term F) basis" by (simp add: expands_to.simps)
  finally show ?thesis by (subst asymp_equiv_sym) simp_all
qed

lemma expands_to_cong:
  "(f expands_to F) basis ⟹ eventually (λx. f x = g x) at_top ⟹ (g expands_to F) basis"
  by (auto simp: expands_to.simps elim: eventually_elim2)

lemma expands_to_cong':
  assumes "(f expands_to MS xs g) basis" "eventually (λx. g x = g' x) at_top"
  shows   "(f expands_to MS xs g') basis"
proof -
  have "is_expansion_aux xs g' basis"
    by (rule is_expansion_aux_cong [OF _ assms(2)]) (insert assms(1), simp add: expands_to.simps)
  with assms show ?thesis
    by (auto simp: expands_to.simps elim: eventually_elim2)
qed

lemma eval_expands_to: "(f expands_to F) basis ⟹ eventually (λx. eval F x = f x) at_top"
  by (simp add: expands_to.simps)

lemma expands_to_real_compose:
  assumes "(f expands_to (c::real)) bs"
  shows   "((λx. g (f x)) expands_to g c) bs"
  using assms by (auto simp: expands_to.simps is_expansion_real.simps elim: eventually_mono)

lemma expands_to_lift_compose:
  assumes "(f expands_to MS (MSLCons (C, e) xs) f') bs'"
  assumes "eventually (λx. f' x - h x = 0) at_top" "e = 0"
  assumes "((λx. g (h x)) expands_to G) bs" "basis_wf (b # bs)"
  shows   "((λx. g (f x)) expands_to lift_ms G) (b # bs)"
proof -
  from assms have "∀F x in at_top. f' x = f x" "∀F x in at_top. eval G x = g (h x)"
    by (auto simp: expands_to.simps)
  with assms(2) have "∀F x in at_top. eval G x = g (f x)"
    by eventually_elim simp
  with assms show ?thesis
    by (intro expands_to.intros is_expansion_lift) (auto simp: expands_to.simps)
qed

lemma expands_to_zero: 
    "basis_wf basis ⟹ length basis = expansion_level TYPE('a) ⟹
       ((λ_. 0) expands_to (zero_expansion :: 'a :: multiseries)) basis"
  by (auto simp add: expands_to.simps is_expansion_zero)
  
lemma expands_to_const: 
    "basis_wf basis ⟹ length basis = expansion_level TYPE('a) ⟹
       ((λ_. c) expands_to (const_expansion c :: 'a :: multiseries)) basis"
  by (auto simp add: expands_to.simps is_expansion_const)

lemma expands_to_X: "((λx. x) expands_to MS (MSLCons (1 :: real, 1) MSLNil) (λx. x)) [λx. x]"
  using is_expansion_X by (simp add: expands_to.simps)
    
lemma expands_to_uminus:
  "basis_wf basis ⟹ (f expands_to F) basis ⟹ ((λx. -f x) expands_to -F) basis"
  by (auto simp: expands_to.simps is_expansion_uminus)

lemma expands_to_add:
  "basis_wf basis ⟹ (f expands_to F) basis ⟹ (g expands_to G) basis ⟹
     ((λx. f x + g x) expands_to F + G) basis"
  by (auto simp: expands_to.simps is_expansion_add elim: eventually_elim2)

lemma expands_to_minus:
  "basis_wf basis ⟹ (f expands_to F) basis ⟹ (g expands_to G) basis ⟹
     ((λx. f x - g x) expands_to F - G) basis"
  by (auto simp: expands_to.simps is_expansion_minus elim: eventually_elim2)

lemma expands_to_mult:
  "basis_wf basis ⟹ (f expands_to F) basis ⟹ (g expands_to G) basis ⟹
     ((λx. f x * g x) expands_to F * G) basis"
  by (auto simp: expands_to.simps is_expansion_mult elim: eventually_elim2)

lemma expands_to_inverse:
  "trimmed F ⟹ basis_wf basis ⟹ (f expands_to F) basis ⟹
     ((λx. inverse (f x)) expands_to inverse F) basis"
  by (auto simp: expands_to.simps is_expansion_inverse)

lemma expands_to_divide:
  "trimmed G ⟹ basis_wf basis ⟹ (f expands_to F) basis ⟹ (g expands_to G) basis ⟹
     ((λx. f x / g x) expands_to F / G) basis"
  by (auto simp: expands_to.simps is_expansion_divide elim: eventually_elim2)

lemma expands_to_powr_0:
  "eventually (λx. f x = 0) at_top ⟹ g ≡ g ⟹ basis_wf bs ⟹
     length bs = expansion_level TYPE('a) ⟹
     ((λx. f x powr g x) expands_to (zero_expansion :: 'a :: multiseries)) bs"
  by (erule (1) expands_to_cong[OF expands_to_zero]) simp_all

lemma expands_to_powr_const:
  "trimmed_pos F ⟹ basis_wf basis ⟹ (f expands_to F) basis ⟹ p ≡ p ⟹
     ((λx. f x powr p) expands_to powr_expansion abort F p) basis"
  by (auto simp: expands_to.simps is_expansion_powr trimmed_pos_def elim: eventually_mono)

lemma expands_to_powr_const_0:
  "eventually (λx. f x = 0) at_top ⟹ basis_wf bs ⟹ 
     length bs = expansion_level TYPE('a :: multiseries) ⟹
     p ≡ p ⟹ ((λx. f x powr p) expands_to (zero_expansion :: 'a)) bs"
  by (rule expands_to_cong[OF expands_to_zero]) auto
  

lemma expands_to_powr:
  assumes "trimmed_pos F" "basis_wf basis'" "(f expands_to F) basis'"
  assumes "((λx. exp (ln (f x) * g x)) expands_to E) basis"
  shows   "((λx. f x powr g x) expands_to E) basis"
using assms(4)
proof (rule expands_to_cong)
  from eval_pos_if_dominant_term_pos[of basis' F] assms(1-4)
    show "eventually (λx. exp (ln (f x) * g x) = f x powr g x) at_top"
    by (auto simp: expands_to.simps trimmed_pos_def powr_def elim: eventually_elim2)
qed

lemma expands_to_ln_powr:
  assumes "trimmed_pos F" "basis_wf basis'" "(f expands_to F) basis'"
  assumes "((λx. ln (f x) * g x) expands_to E) basis"
  shows   "((λx. ln (f x powr g x)) expands_to E) basis"
using assms(4)
proof (rule expands_to_cong)
  from eval_pos_if_dominant_term_pos[of basis' F] assms(1-4)
    show "eventually (λx. ln (f x) * g x = ln (f x powr g x)) at_top"
    by (auto simp: expands_to.simps trimmed_pos_def powr_def elim: eventually_elim2)
qed
  
lemma expands_to_exp_ln:
  assumes "trimmed_pos F" "basis_wf basis" "(f expands_to F) basis"
  shows   "((λx. exp (ln (f x))) expands_to F) basis"
using assms(3)
proof (rule expands_to_cong)
  from eval_pos_if_dominant_term_pos[of basis F] assms
    show "eventually (λx. f x = exp (ln (f x))) at_top"
    by (auto simp: expands_to.simps trimmed_pos_def powr_def elim: eventually_elim2)
qed

lemma expands_to_power:
  assumes "trimmed F" "basis_wf basis" "(f expands_to F) basis"
  shows   "((λx. f x ^ n) expands_to power_expansion abort F n) basis"
  using assms by (auto simp: expands_to.simps is_expansion_power elim: eventually_mono)

lemma expands_to_power_0:
  assumes "eventually (λx. f x = 0) at_top" "basis_wf basis"
          "length basis = expansion_level TYPE('a :: multiseries)" "n ≡ n"
  shows   "((λx. f x ^ n) expands_to (const_expansion (0 ^ n) :: 'a)) basis"
  by (rule expands_to_cong[OF expands_to_const]) (insert assms, auto elim: eventually_mono)

lemma expands_to_0th_root:
  assumes  "n = 0" "basis_wf basis" "length basis = expansion_level TYPE('a :: multiseries)" "f ≡ f"
  shows   "((λx. root n (f x)) expands_to (zero_expansion :: 'a)) basis"
  by (rule expands_to_cong[OF expands_to_zero]) (insert assms, auto)

lemma expands_to_root_0:
  assumes  "n > 0" "eventually (λx. f x = 0) at_top"
           "basis_wf basis" "length basis = expansion_level TYPE('a :: multiseries)"
  shows   "((λx. root n (f x)) expands_to (zero_expansion :: 'a)) basis"
  by (rule expands_to_cong[OF expands_to_zero]) (insert assms, auto elim: eventually_mono)

lemma expands_to_root:
  assumes  "n > 0" "trimmed_pos F" "basis_wf basis" "(f expands_to F) basis"
  shows   "((λx. root n (f x)) expands_to powr_expansion False F (inverse (real n))) basis"
proof -
  have "((λx. f x powr (inverse (real n))) expands_to 
          powr_expansion False F (inverse (real n))) basis"
    using assms(2-) by (rule expands_to_powr_const)
  moreover have "eventually (λx. f x powr (inverse (real n)) = root n (f x)) at_top"
    using eval_pos_if_dominant_term_pos[of basis F] assms
    by (auto simp: trimmed_pos_def expands_to.simps root_powr_inverse field_simps
             elim: eventually_elim2)
  ultimately show ?thesis by (rule expands_to_cong)
qed

lemma expands_to_root_neg:
  assumes  "n > 0" "trimmed_neg F" "basis_wf basis" "(f expands_to F) basis"
  shows   "((λx. root n (f x)) expands_to
             -powr_expansion False (-F) (inverse (real n))) basis"
proof (rule expands_to_cong)
  show "((λx. -root n (-f x)) expands_to
          -powr_expansion False (-F) (inverse (real n))) basis"
    using assms by (intro expands_to_uminus expands_to_root trimmed_pos_uminus) auto
qed (simp_all add: real_root_minus)

lemma expands_to_sqrt:
  assumes "trimmed_pos F" "basis_wf basis" "(f expands_to F) basis"
  shows   "((λx. sqrt (f x)) expands_to powr_expansion False F (1/2)) basis"
  using expands_to_root[of 2 F basis f] assms by (simp add: sqrt_def)

lemma expands_to_exp_real:
  "(f expands_to (F :: real)) basis ⟹ ((λx. exp (f x)) expands_to exp F) basis"
  by (auto simp: expands_to.simps is_expansion_real.simps elim: eventually_mono)

lemma expands_to_exp_MSLNil:
  assumes "(f expands_to (MS MSLNil f' :: 'a :: multiseries ms)) bs" "basis_wf bs"
  shows   "((λx. exp (f x)) expands_to (const_expansion 1 :: 'a ms)) bs"
proof -
  from assms have
     "eventually (λx. f' x = 0) at_top" "eventually (λx. f' x = f x) at_top"
     and [simp]: "length bs = Suc (expansion_level(TYPE('a)))"
    by (auto simp: expands_to.simps elim: is_expansion_aux.cases)
  from this(1,2) have "eventually (λx. 1 = exp (f x)) at_top"
    by eventually_elim simp
  thus ?thesis by (auto simp: expands_to.simps intro!: is_expansion_const assms)
qed

lemma expands_to_exp_zero:
  "(g expands_to MS xs f) basis ⟹ eventually (λx. f x = 0) at_top ⟹ basis_wf basis ⟹
     length basis = expansion_level TYPE('a :: multiseries) ⟹
     ((λx. exp (g x)) expands_to (const_expansion 1 :: 'a)) basis"
  by (auto simp: expands_to.simps intro!: is_expansion_const elim: eventually_elim2)

lemma expands_to_sin_real: 
  "(f expands_to (F :: real)) basis ⟹ ((λx. sin (f x)) expands_to sin F) basis"
  by (auto simp: expands_to.simps is_expansion_real.simps elim: eventually_mono)

lemma expands_to_cos_real: 
  "(f expands_to (F :: real)) basis ⟹ ((λx. cos (f x)) expands_to cos F) basis"
  by (auto simp: expands_to.simps is_expansion_real.simps elim: eventually_mono)
   
lemma expands_to_sin_MSLNil:
  "(f expands_to MS (MSLNil:: ('a × real) msllist) g) basis ⟹ basis_wf basis ⟹ 
     ((λx. sin (f x)) expands_to MS (MSLNil :: ('a :: multiseries × real) msllist) (λx. 0)) basis"
  by (auto simp: expands_to.simps dominant_term_ms_aux_def intro!: is_expansion_aux.intros
           elim: eventually_elim2 is_expansion_aux.cases)

lemma expands_to_cos_MSLNil:
  "(f expands_to MS (MSLNil:: ('a :: multiseries × real) msllist) g) basis ⟹ basis_wf basis ⟹ 
     ((λx. cos (f x)) expands_to (const_expansion 1 :: 'a ms)) basis"
  by (auto simp: expands_to.simps dominant_term_ms_aux_def const_expansion_ms_def 
           intro!: const_ms_aux elim: is_expansion_aux.cases eventually_elim2)

lemma sin_ms_aux':
  assumes "ms_exp_gt 0 (ms_aux_hd_exp xs)" "basis_wf basis" "is_expansion_aux xs f basis"
  shows   "is_expansion_aux (sin_ms_aux' xs) (λx. sin (f x)) basis"
  unfolding sin_ms_aux'_def sin_conv_pre_sin power2_eq_square using assms
  by (intro times_ms_aux powser_ms_aux[unfolded o_def] convergent_powser_sin_series_stream)
     (auto simp: hd_exp_times add_neg_neg split: option.splits)

lemma cos_ms_aux':
  assumes "ms_exp_gt 0 (ms_aux_hd_exp xs)" "basis_wf basis" "is_expansion_aux xs f basis"
  shows   "is_expansion_aux (cos_ms_aux' xs) (λx. cos (f x)) basis"
  unfolding cos_ms_aux'_def cos_conv_pre_cos power2_eq_square using assms
  by (intro times_ms_aux powser_ms_aux[unfolded o_def] convergent_powser_cos_series_stream)
     (auto simp: hd_exp_times add_neg_neg split: option.splits)

lemma expands_to_sin_ms_neg_exp: 
  assumes "e < 0" "basis_wf basis" "(f expands_to MS (MSLCons (C, e) xs) g) basis"
  shows   "((λx. sin (f x)) expands_to MS (sin_ms_aux' (MSLCons (C, e) xs)) (λx. sin (g x))) basis"
  using assms by (auto simp: expands_to.simps snd_dominant_term_ms_aux_MSLCons o_def
                       intro!: sin_ms_aux' elim: eventually_mono)

lemma expands_to_cos_ms_neg_exp: 
  assumes "e < 0" "basis_wf basis" "(f expands_to MS (MSLCons (C, e) xs) g) basis"
  shows   "((λx. cos (f x)) expands_to MS (cos_ms_aux' (MSLCons (C, e) xs)) (λx. cos (g x))) basis"
  using assms by (auto simp: expands_to.simps snd_dominant_term_ms_aux_MSLCons o_def
                       intro!: cos_ms_aux' elim: eventually_mono)

lemma is_expansion_sin_ms_zero_exp: 
  fixes F :: "('a :: multiseries × real) msllist"
  assumes basis: "basis_wf (b # basis)" 
  assumes F: "is_expansion (MS (MSLCons (C, 0) xs) f) (b # basis)"
  assumes Sin: "((λx. sin (eval C x)) expands_to (Sin :: 'a)) basis"
  assumes Cos: "((λx. cos (eval C x)) expands_to (Cos :: 'a)) basis"
  shows   "is_expansion 
             (MS (plus_ms_aux (scale_shift_ms_aux (Sin, 0) (cos_ms_aux' xs))
                              (scale_shift_ms_aux (Cos, 0) (sin_ms_aux' xs))) 
             (λx. sin (f x))) (b # basis)" (is "is_expansion (MS ?A _) _")
proof -
  let ?g = "λx. f x - eval C x * b x powr 0"
  let ?h = "λx. eval Sin x * b x powr 0 * cos (?g x) + eval Cos x * b x powr 0 * sin (?g x)"
  from basis have "filterlim b at_top at_top" by (simp add: basis_wf_Cons)
  hence b_pos: "eventually (λx. b x > 0) at_top" by (simp add: filterlim_at_top_dense)
  from F have F': "is_expansion_aux xs ?g (b # basis)" 
                  "ms_exp_gt 0 (ms_aux_hd_exp xs)" "is_expansion C basis"
    by (auto elim!: is_expansion_aux_MSLCons) 
  have "is_expansion_aux ?A ?h (b # basis)"
    using basis Sin Cos unfolding F'(1)
    by (intro plus_ms_aux scale_shift_ms_aux cos_ms_aux' sin_ms_aux' F')
       (simp_all add: expands_to.simps)
  moreover from b_pos eval_expands_to[OF Sin] eval_expands_to[OF Cos] 
    have "eventually (λx. ?h x = sin (f x)) at_top"
    by eventually_elim (simp add: sin_add [symmetric])
  ultimately have "is_expansion_aux ?A (λx. sin (f x)) (b # basis)"
    by (rule is_expansion_aux_cong)
  thus ?thesis by simp
qed

lemma is_expansion_cos_ms_zero_exp: 
  fixes F :: "('a :: multiseries × real) msllist"
  assumes basis: "basis_wf (b # basis)" 
  assumes F: "is_expansion (MS (MSLCons (C, 0) xs) f) (b # basis)"
  assumes Sin: "((λx. sin (eval C x)) expands_to (Sin :: 'a)) basis"
  assumes Cos: "((λx. cos (eval C x)) expands_to (Cos :: 'a)) basis"
  shows   "is_expansion 
             (MS (minus_ms_aux (scale_shift_ms_aux (Cos, 0) (cos_ms_aux' xs))
                              (scale_shift_ms_aux (Sin, 0) (sin_ms_aux' xs))) 
             (λx. cos (f x))) (b # basis)" (is "is_expansion (MS ?A _) _")
proof -
  let ?g = "λx. f x - eval C x * b x powr 0"
  let ?h = "λx. eval Cos x * b x powr 0 * cos (?g x) - eval Sin x * b x powr 0 * sin (?g x)"
  from basis have "filterlim b at_top at_top" by (simp add: basis_wf_Cons)
  hence b_pos: "eventually (λx. b x > 0) at_top" by (simp add: filterlim_at_top_dense)
  from F have F': "is_expansion_aux xs ?g (b # basis)" 
                  "ms_exp_gt 0 (ms_aux_hd_exp xs)" "is_expansion C basis"
    by (auto elim!: is_expansion_aux_MSLCons) 
  have "is_expansion_aux ?A ?h (b # basis)"
    using basis Sin Cos unfolding F'(1)
    by (intro minus_ms_aux scale_shift_ms_aux cos_ms_aux' sin_ms_aux' F')
       (simp_all add: expands_to.simps)
  moreover from b_pos eval_expands_to[OF Sin] eval_expands_to[OF Cos] 
    have "eventually (λx. ?h x = cos (f x)) at_top"
    by eventually_elim (simp add: cos_add [symmetric])
  ultimately have "is_expansion_aux ?A (λx. cos (f x)) (b # basis)"
    by (rule is_expansion_aux_cong)
  thus ?thesis by simp
qed

lemma expands_to_sin_ms_zero_exp: 
  assumes e: "e = 0" and basis: "basis_wf (b # basis)" 
  assumes F: "(f expands_to MS (MSLCons (C, e) xs) g) (b # basis)"
  assumes Sin: "((λx. sin (c x)) expands_to Sin) basis"
  assumes Cos: "((λx. cos (c x)) expands_to Cos) basis"
  assumes C: "eval C = c"
  shows   "((λx. sin (f x)) expands_to 
             MS (plus_ms_aux (scale_shift_ms_aux (Sin, 0) (cos_ms_aux' xs))
                              (scale_shift_ms_aux (Cos, 0) (sin_ms_aux' xs)))
                (λx. sin (g x))) (b # basis)"
  using is_expansion_sin_ms_zero_exp[of b basis C xs g Sin Cos] assms
  by (auto simp: expands_to.simps elim: eventually_mono)
    
lemma expands_to_cos_ms_zero_exp: 
  assumes e: "e = 0" and basis: "basis_wf (b # basis)" 
  assumes F: "(f expands_to MS (MSLCons (C, e) xs) g) (b # basis)"
  assumes Sin: "((λx. sin (c x)) expands_to Sin) basis"
  assumes Cos: "((λx. cos (c x)) expands_to Cos) basis"
  assumes C: "eval C = c"
  shows   "((λx. cos (f x)) expands_to 
             MS (minus_ms_aux (scale_shift_ms_aux (Cos, 0) (cos_ms_aux' xs))
                              (scale_shift_ms_aux (Sin, 0) (sin_ms_aux' xs)))
                (λx. cos (g x))) (b # basis)"
  using is_expansion_cos_ms_zero_exp[of b basis C xs g Sin Cos] assms
  by (auto simp: expands_to.simps elim: eventually_mono)
  


lemma expands_to_sgn_zero:
  assumes "eventually (λx. f x = 0) at_top" "basis_wf basis"
          "length basis = expansion_level TYPE('a :: multiseries)"
  shows   "((λx. sgn (f x)) expands_to (zero_expansion :: 'a)) basis"
  by (rule expands_to_cong[OF expands_to_zero]) (insert assms, auto simp: sgn_eq_0_iff)

lemma expands_to_sgn_pos:
  assumes "trimmed_pos F" "(f expands_to F) basis" "basis_wf basis"
          "length basis = expansion_level TYPE('a :: multiseries)"
  shows   "((λx. sgn (f x)) expands_to (const_expansion 1 :: 'a)) basis"
proof (rule expands_to_cong[OF expands_to_const])
  from trimmed_imp_eventually_sgn[of basis F] assms
    have "eventually (λx. sgn (eval F x) = 1) at_top" 
      by (simp add: expands_to.simps trimmed_pos_def)
  moreover from assms have "eventually (λx. eval F x = f x) at_top"
    by (simp add: expands_to.simps)
  ultimately show "eventually (λx. 1 = sgn (f x)) at_top" by eventually_elim simp
qed (insert assms, auto)

lemma expands_to_sgn_neg:
  assumes "trimmed_neg F" "(f expands_to F) basis" "basis_wf basis"
          "length basis = expansion_level TYPE('a :: multiseries)"
  shows   "((λx. sgn (f x)) expands_to (const_expansion (-1) :: 'a)) basis"
proof (rule expands_to_cong[OF expands_to_const])
  from trimmed_imp_eventually_sgn[of basis F] assms
    have "eventually (λx. sgn (eval F x) = -1) at_top" 
      by (simp add: expands_to.simps trimmed_neg_def)
  moreover from assms have "eventually (λx. eval F x = f x) at_top"
    by (simp add: expands_to.simps)
  ultimately show "eventually (λx. -1 = sgn (f x)) at_top" by eventually_elim simp
qed (insert assms, auto)



lemma expands_to_abs_pos:
  assumes "trimmed_pos F" "basis_wf basis" "(f expands_to F) basis"
  shows   "((λx. abs (f x)) expands_to F) basis"
using assms(3)
proof (rule expands_to_cong)
  from trimmed_imp_eventually_sgn[of basis F] assms
    have "eventually (λx. sgn (eval F x) = 1) at_top" 
    by (simp add: expands_to.simps trimmed_pos_def)
  with assms(3) show "eventually (λx. f x = abs (f x)) at_top" 
    by (auto simp: expands_to.simps sgn_if split: if_splits elim: eventually_elim2)
qed

lemma expands_to_abs_neg:
  assumes "trimmed_neg F" "basis_wf basis" "(f expands_to F) basis"
  shows   "((λx. abs (f x)) expands_to -F) basis"
using expands_to_uminus[OF assms(2,3)]
proof (rule expands_to_cong)
  from trimmed_imp_eventually_sgn[of basis F] assms
    have "eventually (λx. sgn (eval F x) = -1) at_top" 
    by (simp add: expands_to.simps trimmed_neg_def)
  with assms(3) show "eventually (λx. -f x = abs (f x)) at_top" 
    by (auto simp: expands_to.simps sgn_if split: if_splits elim: eventually_elim2)
qed

lemma expands_to_imp_eventually_nz:
  assumes "basis_wf basis" "(f expands_to F) basis" "trimmed F"
  shows   "eventually (λx. f x ≠ 0) at_top"
  using trimmed_imp_eventually_nz[OF assms(1), of F] assms(2,3)
  by (auto simp: expands_to.simps elim: eventually_elim2)

lemma expands_to_imp_eventually_pos:
  assumes "basis_wf basis" "(f expands_to F) basis" "trimmed_pos F"
  shows   "eventually (λx. f x > 0) at_top"
  using eval_pos_if_dominant_term_pos[of basis F] assms 
  by (auto simp: expands_to.simps trimmed_pos_def elim: eventually_elim2)

lemma expands_to_imp_eventually_neg:
  assumes "basis_wf basis" "(f expands_to F) basis" "trimmed_neg F"
  shows   "eventually (λx. f x < 0) at_top"
  using eval_pos_if_dominant_term_pos[of basis "-F"] assms
  by (auto simp: expands_to.simps trimmed_neg_def is_expansion_uminus elim: eventually_elim2)

lemma expands_to_imp_eventually_gt:
  assumes "basis_wf basis" "((λx. f x - g x) expands_to F) basis" "trimmed_pos F"
  shows   "eventually (λx. f x > g x) at_top"
  using expands_to_imp_eventually_pos[OF assms] by simp

lemma expands_to_imp_eventually_lt:
  assumes "basis_wf basis" "((λx. f x - g x) expands_to F) basis" "trimmed_neg F"
  shows   "eventually (λx. f x < g x) at_top"
  using expands_to_imp_eventually_neg[OF assms] by simp

lemma eventually_diff_zero_imp_eq:
  fixes f g :: "real ⇒ real"
  assumes "eventually (λx. f x - g x = 0) at_top"
  shows   "eventually (λx. f x = g x) at_top"
  using assms by eventually_elim simp

lemma smallo_trimmed_imp_eventually_less:
  fixes f g :: "real ⇒ real"
  assumes "f ∈ o(g)" "(g expands_to G) bs" "basis_wf bs" "trimmed_pos G"
  shows   "eventually (λx. f x < g x) at_top"
proof -
  from assms(2-4) have pos: "eventually (λx. g x > 0) at_top"
    using expands_to_imp_eventually_pos by blast
  have "(1 / 2 :: real) > 0" by simp
  from landau_o.smallD[OF assms(1) this] and pos
    show ?thesis by eventually_elim (simp add: abs_if split: if_splits)
qed

lemma smallo_trimmed_imp_eventually_greater:
  fixes f g :: "real ⇒ real"
  assumes "f ∈ o(g)" "(g expands_to G) bs" "basis_wf bs" "trimmed_neg G"
  shows   "eventually (λx. f x > g x) at_top"
proof -
  from assms(2-4) have pos: "eventually (λx. g x < 0) at_top"
    using expands_to_imp_eventually_neg by blast
  have "(1 / 2 :: real) > 0" by simp
  from landau_o.smallD[OF assms(1) this] and pos
    show ?thesis by eventually_elim (simp add: abs_if split: if_splits)
qed

lemma expands_to_min_lt:
  assumes "(f expands_to F) basis" "eventually (λx. f x < g x) at_top"
  shows   "((λx. min (f x) (g x)) expands_to F) basis"
  using assms(1)
  by (rule expands_to_cong) (insert assms(2), auto simp: min_def elim: eventually_mono)

lemma expands_to_min_eq:
  assumes "(f expands_to F) basis" "eventually (λx. f x = g x) at_top"
  shows   "((λx. min (f x) (g x)) expands_to F) basis"
  using assms(1)
  by (rule expands_to_cong) (insert assms(2), auto simp: min_def elim: eventually_mono)

lemma expands_to_min_gt:
  assumes "(g expands_to G) basis" "eventually (λx. f x > g x) at_top"
  shows   "((λx. min (f x) (g x)) expands_to G) basis"
  using assms(1)
  by (rule expands_to_cong) (insert assms(2), auto simp: min_def elim: eventually_mono)
    
lemma expands_to_max_gt:
  assumes "(f expands_to F) basis" "eventually (λx. f x > g x) at_top"
  shows   "((λx. max (f x) (g x)) expands_to F) basis"
  using assms(1)
  by (rule expands_to_cong) (insert assms(2), auto simp: max_def elim: eventually_mono)
    
lemma expands_to_max_eq:
  assumes "(f expands_to F) basis" "eventually (λx. f x = g x) at_top"
  shows   "((λx. max (f x) (g x)) expands_to F) basis"
  using assms(1)
  by (rule expands_to_cong) (insert assms(2), auto simp: max_def elim: eventually_mono)

lemma expands_to_max_lt:
  assumes "(g expands_to G) basis" "eventually (λx. f x < g x) at_top"
  shows   "((λx. max (f x) (g x)) expands_to G) basis"
  using assms(1)
  by (rule expands_to_cong) (insert assms(2), auto simp: max_def elim: eventually_mono)


lemma expands_to_lift:
  "is_lifting f basis basis' ⟹ (c expands_to C) basis ⟹ (c expands_to (f C)) basis'"
  by (simp add: is_lifting_def expands_to.simps)
    
lemma expands_to_basic_powr: 
  assumes "basis_wf (b # basis)" "length basis = expansion_level TYPE('a::multiseries)"
  shows   "((λx. b x powr e) expands_to 
             MS (MSLCons (const_expansion 1 :: 'a, e) MSLNil) (λx. b x powr e)) (b # basis)"
  using assms by (auto simp: expands_to.simps basis_wf_Cons powr_smallo_iff
                       intro!: is_expansion_aux.intros is_expansion_const powr_smallo_iff)

lemma expands_to_basic_inverse: 
  assumes "basis_wf (b # basis)" "length basis = expansion_level TYPE('a::multiseries)"
  shows   "((λx. inverse (b x)) expands_to 
             MS (MSLCons (const_expansion 1 :: 'a, -1) MSLNil) (λx. b x powr -1)) (b # basis)"
proof -
  from assms have "eventually (λx. b x > 0) at_top" 
    by (simp add: basis_wf_Cons filterlim_at_top_dense)
  moreover have "(λa. inverse (a powr 1)) ∈ o(λa. a powr e')" if "e' > -1" for e' :: real
    using that by (simp add: landau_o.small.inverse_eq2 one_smallo_powr flip: powr_one' powr_add)
  ultimately show ?thesis using assms
    by (auto simp: expands_to.simps basis_wf_Cons powr_minus
             elim: eventually_mono
             intro!: is_expansion_aux.intros is_expansion_const
                     landau_o.small.compose[of _ at_top _ b])
qed

lemma expands_to_div':
  assumes "basis_wf basis" "(f expands_to F) basis" "((λx. inverse (g x)) expands_to G) basis"
  shows   "((λx. f x / g x) expands_to F * G) basis"
  using expands_to_mult[OF assms] by (simp add: field_split_simps)

lemma expands_to_basic: 
  assumes "basis_wf (b # basis)" "length basis = expansion_level TYPE('a::multiseries)"
  shows   "(b expands_to MS (MSLCons (const_expansion 1 :: 'a, 1) MSLNil) b) (b # basis)"
proof -
  from assms have "eventually (λx. b x > 0) at_top" 
    by (simp add: basis_wf_Cons filterlim_at_top_dense)
  moreover {
    fix e' :: real assume e': "e' > 1"
    have "(λx::real. x) ∈ Θ(λx. x powr 1)"
      by (intro bigthetaI_cong eventually_mono[OF eventually_gt_at_top[of 0]]) auto
    also have "(λx. x powr 1) ∈ o(λx. x powr e')"
      using e' by (subst powr_smallo_iff) (auto simp: filterlim_ident)
    finally have "(λx. x) ∈ o(λx. x powr e')" .
  }
  ultimately show ?thesis using assms
    by (auto simp: expands_to.simps basis_wf_Cons elim: eventually_mono
             intro!: is_expansion_aux.intros is_expansion_const
                     landau_o.small.compose[of _ at_top _ b])
qed

lemma expands_to_lift':
  assumes "basis_wf (b # basis)" "(f expands_to MS xs g) basis"
  shows   "(f expands_to (MS (MSLCons (MS xs g, 0) MSLNil) g)) (b # basis)"
proof -
  have "is_lifting lift_ms basis (b # basis)" by (rule is_lifting_lift) fact+
  from expands_to_lift[OF this assms(2)] show ?thesis by (simp add: lift_ms_def)
qed

lemma expands_to_lift'':
  assumes "basis_wf (b # basis)" "(f expands_to F) basis"
  shows   "(f expands_to (MS (MSLCons (F, 0) MSLNil) (eval F))) (b # basis)"
proof -
  have "is_lifting lift_ms basis (b # basis)" by (rule is_lifting_lift) fact+
  from expands_to_lift[OF this assms(2)] show ?thesis by (simp add: lift_ms_def)
qed

lemma expands_to_drop_zero:
  assumes "eventually (λx. eval C x = 0) at_top"
  assumes "(f expands_to (MS (MSLCons (C, e) xs) g)) basis"
  shows   "(f expands_to (MS xs g)) basis"
  using assms drop_zero_ms[of C e xs g basis] by (simp add: expands_to.simps)

lemma expands_to_hd'':
  assumes "(f expands_to (MS (MSLCons (C, e) xs) g)) (b # basis)"
  shows   "(eval C expands_to C) basis"
  using assms by (auto simp: expands_to.simps elim: is_expansion_aux_MSLCons)

lemma expands_to_hd:
  assumes "(f expands_to (MS (MSLCons (MS ys h, e) xs) g)) (b # basis)"
  shows   "(h expands_to MS ys h) basis"
  using assms by (auto simp: expands_to.simps elim: is_expansion_aux_MSLCons)

lemma expands_to_hd':
  assumes "(f expands_to (MS (MSLCons (c :: real, e) xs) g)) (b # basis)"
  shows   "((λ_. c) expands_to c) basis"
  using assms by (auto simp: expands_to.simps elim: is_expansion_aux_MSLCons)

lemma expands_to_trim_cong:
  assumes "(f expands_to (MS (MSLCons (C, e) xs) g)) (b # basis)"
  assumes "(eval C expands_to C') basis"
  shows   "(f expands_to (MS (MSLCons (C', e) xs) g)) (b # basis)"
proof -
  from assms(1) have "is_expansion_aux xs (λx. g x - eval C x * b x powr e) (b # basis)"
    by (auto simp: expands_to.simps elim!: is_expansion_aux_MSLCons)
  hence "is_expansion_aux xs (λx. g x - eval C' x * b x powr e) (b # basis)"
    by (rule is_expansion_aux_cong)
       (insert assms(2), auto simp: expands_to.simps elim: eventually_mono)
  with assms show ?thesis
    by (auto simp: expands_to.simps elim!: is_expansion_aux_MSLCons intro!: is_expansion_aux.intros)
qed

lemma is_expansion_aux_expands_to:
  assumes "(f expands_to MS xs g) basis"
  shows   "is_expansion_aux xs f basis"
proof -
  from assms have "is_expansion_aux xs g basis" "eventually (λx. g x = f x) at_top" 
    by (simp_all add: expands_to.simps)
  thus ?thesis by (rule is_expansion_aux_cong)
qed

lemma ln_series_stream_aux_code:
  "ln_series_stream_aux True c = MSSCons (- inverse c) (ln_series_stream_aux False (c + 1))"
  "ln_series_stream_aux False c = MSSCons (inverse c) (ln_series_stream_aux True (c + 1))"
  by (subst ln_series_stream_aux.code, simp)+

definition powser_ms_aux' where
  "powser_ms_aux' cs xs = powser_ms_aux (msllist_of_msstream cs) xs"

lemma ln_ms_aux:
  fixes C L :: "'a :: multiseries"
  assumes trimmed: "trimmed_pos C" and basis: "basis_wf (b # basis)"
  assumes F: "is_expansion_aux (MSLCons (C, e) xs) f (b # basis)"
  assumes L: "((λx. ln (eval C x) + e * ln (b x)) expands_to L) basis"
  shows   "is_expansion_aux 
             (MSLCons (L, 0) (times_ms_aux (scale_shift_ms_aux (inverse C, - e) xs)
               (powser_ms_aux' (ln_series_stream_aux False 1) 
                 (scale_shift_ms_aux (inverse C, - e) xs))))
             (λx. ln (f x)) (b # basis)"
    (is "is_expansion_aux ?G _ _")
proof -
  from F have F': 
    "is_expansion_aux xs (λx. f x - eval C x * b x powr e) (b # basis)"
    "is_expansion C basis" "∀e'>e. f ∈ o(λx. b x powr e')" "ms_exp_gt e (ms_aux_hd_exp xs)"
    by (auto elim!: is_expansion_aux_MSLCons)
  from basis have b_limit: "filterlim b at_top at_top" by (simp add: basis_wf_Cons)
  hence b_pos: "eventually (λx. b x > 1) at_top" by (simp add: filterlim_at_top_dense)
  from eval_pos_if_dominant_term_pos[of basis C] trimmed F' basis
    have C_pos: "eventually (λx. eval C x > 0) at_top"
      by (auto simp: basis_wf_Cons trimmed_pos_def)
  from eval_pos_if_dominant_term_pos'[OF basis _ F] trimmed
  have f_pos: "eventually (λx. f x > 0) at_top"
    by (simp add: trimmed_pos_def trimmed_ms_aux_def dominant_term_ms_aux_def case_prod_unfold)
  
  from F' have "ms_exp_gt 0 (map_option ((+) (-e)) (ms_aux_hd_exp xs))" 
    by (cases "ms_aux_hd_exp xs") simp_all
  hence "is_expansion_aux (powser_ms_aux' ln_series_stream (scale_shift_ms_aux (inverse C, -e) xs))
          ((λx. ln (1 + x)) ∘ (λx. eval (inverse C) x * b x powr -e * 
              (f x - eval C x * b x powr e))) (b # basis)" (is "is_expansion_aux _ ?g _")
    unfolding powser_ms_aux'_def using powser_ms_aux' basis F' trimmed
    by (intro powser_ms_aux' convergent_powser'_ln scale_shift_ms_aux is_expansion_inverse F')
       (simp_all add: trimmed_pos_def hd_exp_basis basis_wf_Cons)
  moreover have "eventually (λx. ?g x = ln (f x) - eval L x * b x powr 0) at_top"
    using b_pos C_pos f_pos eval_expands_to[OF L]
    by eventually_elim 
       (simp add: powr_minus algebra_simps ln_mult ln_inverse expands_to.simps ln_powr)
  ultimately 
    have "is_expansion_aux (powser_ms_aux' ln_series_stream (scale_shift_ms_aux (inverse C, -e) xs))
            (λx. ln (f x) - eval L x * b x powr 0) (b # basis)" 
      by (rule is_expansion_aux_cong)
  hence *: "is_expansion_aux (times_ms_aux (scale_shift_ms_aux (inverse C, - e) xs)
              (powser_ms_aux' (ln_series_stream_aux False 1) (scale_shift_ms_aux (inverse C, - e) xs)))
              (λx. ln (f x) - eval L x * b x powr 0) (b # basis)"
    unfolding ln_series_stream_def powser_ms_aux'_def powser_ms_aux_MSLCons msllist_of_msstream_MSSCons
    by (rule drop_zero_ms_aux [rotated]) simp_all
  moreover from F' have exp: "ms_exp_gt 0 (map_option ((+) (-e)) (ms_aux_hd_exp xs))"
    by (cases "ms_aux_hd_exp xs") simp_all
  moreover have "(λx. ln (f x)) ∈ o(λx. b x powr e')" if "e' > 0" for e'
  proof -
    from is_expansion_aux_imp_smallo[OF *, of e'] that exp
    have "(λx. ln (f x) - eval L x * b x powr 0) ∈ o(λx. b x powr e')" 
      by (cases "ms_aux_hd_exp xs") 
         (simp_all add: hd_exp_times hd_exp_powser hd_exp_basis powser_ms_aux'_def)
    moreover {
      from L F' basis that have "eval L ∈ o(λx. b x powr e')"
        by (intro is_expansion_imp_smallo[of basis]) (simp_all add: basis_wf_Cons expands_to.simps)
      also have "eval L ∈ o(λx. b x powr e') ⟷ (λx. eval L x * b x powr 0) ∈ o(λx. b x powr e')"
        using b_pos by (intro landau_o.small.in_cong) (auto elim: eventually_mono)
      finally have … .
    } ultimately have "(λx. ln (f x) - eval L x * b x powr 0 + eval L x * b x powr 0) ∈ 
                          o(λx. b x powr e')" by (rule sum_in_smallo)
    thus ?thesis by simp
  qed
  ultimately show "is_expansion_aux ?G (λx. ln (f x)) (b # basis)" using L
    by (intro is_expansion_aux.intros)
       (auto simp: expands_to.simps hd_exp_times hd_exp_powser hd_exp_basis
                   powser_ms_aux'_def split: option.splits)
qed

lemma expands_to_ln:
  fixes C L :: "'a :: multiseries"
  assumes trimmed: "trimmed_pos (MS (MSLCons (C, e) xs) g)" and basis: "basis_wf (b # basis)"
  assumes F: "(f expands_to MS (MSLCons (C, e) xs) g) (b # basis)"
  assumes L: "((λx. ln (h x) + e * ln (b x)) expands_to L) basis"
  and h: "eval C = h"
  shows   "((λx. ln (f x)) expands_to MS
             (MSLCons (L, 0) (times_ms_aux (scale_shift_ms_aux (inverse C, - e) xs)
               (powser_ms_aux' (ln_series_stream_aux False 1) 
                 (scale_shift_ms_aux (inverse C, - e) xs)))) (λx. ln (f x))) (b # basis)"
  using is_expansion_aux_expands_to[OF F] assms by (auto simp: expands_to.simps intro!: ln_ms_aux)

lemma trimmed_lifting:
  assumes "is_lifting f basis basis'"
  assumes "trimmed F"
  shows   "trimmed (f F)"
  using assms by (simp add: is_lifting_def)

lemma trimmed_pos_lifting:
  assumes "is_lifting f basis basis'"
  assumes "trimmed_pos F"
  shows   "trimmed_pos (f F)"
  using assms by (simp add: is_lifting_def trimmed_pos_def)

lemma expands_to_ln_aux_0:
  assumes e: "e = 0"
  assumes L1: "((λx. ln (g x)) expands_to L) basis"
  shows "((λx. ln (g x) + e * ln (b x)) expands_to L) basis"
  using assms by simp

lemma expands_to_ln_aux_1:
  assumes e: "e = 1"
  assumes basis: "basis_wf (b # basis)"
  assumes L1: "((λx. ln (g x)) expands_to L1) basis"
  assumes L2: "((λx. ln (b x)) expands_to L2) basis"
  shows "((λx. ln (g x) + e * ln (b x)) expands_to L1 + L2) basis"
  using assms by (intro expands_to_add) (simp_all add: basis_wf_Cons)

lemma expands_to_ln_eventually_1:
  assumes "basis_wf basis" "length basis = expansion_level TYPE('a)"
  assumes "eventually (λx. f x - 1 = 0) at_top"
  shows   "((λx. ln (f x)) expands_to (zero_expansion :: 'a :: multiseries)) basis"
  by (rule expands_to_cong [OF expands_to_zero]) (insert assms, auto elim: eventually_mono)  

lemma expands_to_ln_aux:
  assumes basis: "basis_wf (b # basis)"
  assumes L1: "((λx. ln (g x)) expands_to L1) basis"
  assumes L2: "((λx. ln (b x)) expands_to L2) basis"
  shows "((λx. ln (g x) + e * ln (b x)) expands_to L1 + scale_ms e L2) basis"
proof -
  from L1 have "length basis = expansion_level TYPE('a)"
    by (auto simp: expands_to.simps is_expansion_length)
  with assms show ?thesis unfolding scale_ms_def
    by (intro expands_to_add assms expands_to_mult expands_to_const) 
       (simp_all add: basis_wf_Cons)
qed

lemma expands_to_ln_to_expands_to_ln_eval:
  assumes "((λx. ln (f x) + F x) expands_to L) basis"
  shows   "((λx. ln (eval (MS xs f) x) + F x) expands_to L) basis"
  using assms by simp

lemma expands_to_ln_const:
  "((λx. ln (eval (C :: real) x)) expands_to ln C) []"
  by (simp add: expands_to.simps is_expansion_real.intros)

lemma expands_to_meta_eq_cong:
  assumes "(f expands_to F) basis" "F ≡ G"
  shows   "(f expands_to G) basis"
  using assms by simp
    
lemma expands_to_meta_eq_cong':
  assumes "(g expands_to F) basis" "f ≡ g"
  shows   "(f expands_to F) basis"
  using assms by simp


lemma uminus_ms_aux_MSLCons_eval:
  "uminus_ms_aux (MSLCons (c, e) xs) = MSLCons (-c, e) (uminus_ms_aux xs)"
  by (simp add: uminus_ms_aux_MSLCons)

lemma scale_shift_ms_aux_MSLCons_eval:
  "scale_shift_ms_aux (c, e) (MSLCons (c', e') xs) =
     MSLCons (c * c', e + e') (scale_shift_ms_aux (c, e) xs)"
  by (simp add: scale_shift_ms_aux_MSLCons)

lemma times_ms_aux_MSLCons_eval: "times_ms_aux (MSLCons (c1, e1) xs) (MSLCons (c2, e2) ys) = 
  MSLCons (c1 * c2, e1 + e2) (plus_ms_aux (scale_shift_ms_aux (c1, e1) ys) 
    (times_ms_aux xs (MSLCons (c2, e2) ys)))"
  by (simp add: times_ms_aux_MSLCons)

lemma plus_ms_aux_MSLCons_eval:
  "plus_ms_aux (MSLCons (c1, e1) xs) (MSLCons (c2, e2) ys) =
     CMP_BRANCH (COMPARE e1 e2) 
       (MSLCons (c2, e2) (plus_ms_aux (MSLCons (c1, e1) xs) ys))
       (MSLCons (c1 + c2, e1) (plus_ms_aux xs ys))
       (MSLCons (c1, e1) (plus_ms_aux xs (MSLCons (c2, e2) ys)))"
  by (simp add: CMP_BRANCH_def COMPARE_def plus_ms_aux_MSLCons)

lemma inverse_ms_aux_MSLCons: "inverse_ms_aux (MSLCons (C, e) xs) = 
   scale_shift_ms_aux (inverse C, - e)
     (powser_ms_aux' (mssalternate 1 (- 1))
       (scale_shift_ms_aux (inverse C, - e)
         xs))"
  by (simp add: Let_def inverse_ms_aux.simps powser_ms_aux'_def)

lemma powr_ms_aux_MSLCons: "powr_ms_aux abort (MSLCons (C, e) xs) p = 
  scale_shift_ms_aux (powr_expansion abort C p, e * p)
     (powser_ms_aux (gbinomial_series abort p)
       (scale_shift_ms_aux (inverse C, - e) xs))"
  by simp

lemma power_ms_aux_MSLCons: "power_ms_aux abort (MSLCons (C, e) xs) n = 
  scale_shift_ms_aux (power_expansion abort C n, e * real n)
     (powser_ms_aux (gbinomial_series abort (real n))
       (scale_shift_ms_aux (inverse C, - e) xs))"
  by simp

lemma minus_ms_aux_MSLCons_eval:
  "minus_ms_aux (MSLCons (c1, e1) xs) (MSLCons (c2, e2) ys) = 
     CMP_BRANCH (COMPARE e1 e2) 
       (MSLCons (-c2, e2) (minus_ms_aux (MSLCons (c1, e1) xs) ys))
       (MSLCons (c1 - c2, e1) (minus_ms_aux xs ys))
       (MSLCons (c1, e1) (minus_ms_aux xs (MSLCons (c2, e2) ys)))"
  by (simp add: minus_ms_aux_def plus_ms_aux_MSLCons_eval uminus_ms_aux_MSLCons minus_eq_plus_uminus)
  
lemma minus_ms_altdef: "MS xs f - MS ys g = MS (minus_ms_aux xs ys) (λx. f x - g x)"
  by (simp add: minus_ms_def minus_ms_aux_def)

lemma const_expansion_ms_eval: "const_expansion c = MS (MSLCons (const_expansion c, 0) MSLNil) (λ_. c)"
  by (simp add: const_expansion_ms_def const_ms_aux_def)
 
lemma powser_ms_aux'_MSSCons:
  "powser_ms_aux' (MSSCons c cs) xs = 
     MSLCons (const_expansion c, 0) (times_ms_aux xs (powser_ms_aux' cs xs))"
  by (simp add: powser_ms_aux'_def powser_ms_aux_MSLCons)

lemma sin_ms_aux'_altdef:
  "sin_ms_aux' xs = times_ms_aux xs (powser_ms_aux' sin_series_stream (times_ms_aux xs xs))"
  by (simp add: sin_ms_aux'_def powser_ms_aux'_def)

lemma cos_ms_aux'_altdef:
  "cos_ms_aux' xs = powser_ms_aux' cos_series_stream (times_ms_aux xs xs)"
  by (simp add: cos_ms_aux'_def powser_ms_aux'_def)

lemma sin_series_stream_aux_code:
  "sin_series_stream_aux True acc m = 
     MSSCons (-inverse acc) (sin_series_stream_aux False (acc * (m + 1) * (m + 2)) (m + 2))"
  "sin_series_stream_aux False acc m = 
     MSSCons (inverse acc) (sin_series_stream_aux True (acc * (m + 1) * (m + 2)) (m + 2))"
  by (subst sin_series_stream_aux.code; simp)+
    
lemma arctan_series_stream_aux_code:
  "arctan_series_stream_aux True n = MSSCons (-inverse n) (arctan_series_stream_aux False (n + 2))"
  "arctan_series_stream_aux False n = MSSCons (inverse n) (arctan_series_stream_aux True (n + 2))"
  by (subst arctan_series_stream_aux.code; simp)+


subsubsection ‹Composition with an asymptotic power series›

context
  fixes h :: "real ⇒ real" and F :: "real filter"
begin
  
coinductive asymp_powser :: 
    "(real ⇒ real) ⇒ real msstream ⇒ bool" where
  "asymp_powser f cs ⟹ f' ∈ O[F](λ_. 1) ⟹
     eventually (λx. f' x = c + f x * h x) F ⟹ asymp_powser f' (MSSCons c cs)"

lemma asymp_powser_coinduct [case_names asymp_powser]:
  "P x1 x2 ⟹ (⋀x1 x2.  P x1 x2 ⟹ ∃f cs c.
       x2 = MSSCons c cs ∧ asymp_powser f cs ∧
       x1 ∈ O[F](λ_. 1) ∧ (∀F x in F. x1 x = c + f x * h x)) ⟹ asymp_powser x1 x2"
  using asymp_powser.coinduct[of P x1 x2] by blast
    
lemma asymp_powser_coinduct' [case_names asymp_powser]:
  "P x1 x2 ⟹ (⋀x1 x2.  P x1 x2 ⟹ ∃f cs c.
       x2 = MSSCons c cs ∧ P f cs ∧
       x1 ∈ O[F](λ_. 1) ∧ (∀F x in F. x1 x = c + f x * h x)) ⟹ asymp_powser x1 x2"
  using asymp_powser.coinduct[of P x1 x2] by blast
  
lemma asymp_powser_transfer:
  assumes "asymp_powser f cs" "eventually (λx. f x = f' x) F"
  shows   "asymp_powser f' cs"
  using assms(1)
proof (cases rule: asymp_powser.cases)
  case (1 f cs' c)
  have "asymp_powser f' (MSSCons c cs')"
    by (rule asymp_powser.intros) 
       (insert 1 assms(2), auto elim: eventually_elim2 dest: landau_o.big.in_cong)
  thus ?thesis by (simp add: 1)
qed

lemma sum_bigo_imp_asymp_powser:
  assumes filterlim_h: "filterlim h (at 0) F"
  assumes "(⋀n. (λx. f x - (∑k<n. mssnth cs k * h x ^ k)) ∈ O[F](λx. h x ^ n))"
  shows   "asymp_powser f cs"
  using assms(2)
proof (coinduction arbitrary: f cs rule: asymp_powser_coinduct')
  case (asymp_powser f cs)
  from filterlim_h have h_nz:"eventually (λx. h x ≠ 0) F"
    by (auto simp: filterlim_at)
  show ?case
  proof (cases cs)
    case (MSSCons c cs')
    define f' where "f' = (λx. (f x - c) / h x)"  
    have "(λx. f' x - (∑k<n. mssnth cs' k * h x ^ k)) ∈ O[F](λx. h x ^ n)" for n
    proof -
      have "(λx. h x * (f' x - (∑i<n. mssnth cs' i * h x ^ i))) ∈
                Θ[F](λx. f x - c - h x * (∑i<n. mssnth cs' i * h x ^ i))"
        using h_nz by (intro bigthetaI_cong) (auto elim!: eventually_mono simp: f'_def field_simps)
      also from spec[OF asymp_powser, of "Suc n"]
        have "(λx. f x - c - h x * (∑i<n. mssnth cs' i * h x ^ i)) ∈ O[F](λx. h x * h x ^ n)"
        unfolding sum.lessThan_Suc_shift
        by (simp add: MSSCons algebra_simps sum_distrib_left sum_distrib_right)
      finally show ?thesis
        by (subst (asm) landau_o.big.mult_cancel_left) (insert h_nz, auto)
    qed
    moreover from h_nz have "∀F x in F. f x = c + f' x * h x"
      by eventually_elim (simp add: f'_def)
    ultimately show ?thesis using spec[OF asymp_powser, of 0]
      by (auto simp: MSSCons intro!: exI[of _ f'])
  qed
qed

end


lemma asymp_powser_o:
  assumes "asymp_powser h F f cs" assumes "filterlim g F G"
  shows   "asymp_powser (h ∘ g) G (f ∘ g) cs"
using assms(1)
proof (coinduction arbitrary: f cs rule: asymp_powser_coinduct')
  case (asymp_powser f cs)
  thus ?case
  proof (cases rule: asymp_powser.cases)
    case (1 f' cs' c)
    from assms(2) have "filtermap g G ≤ F" by (simp add: filterlim_def)
    moreover have "eventually (λx. f x = c + f' x * h x) F" by fact
    ultimately have "eventually (λx. f x = c + f' x * h x) (filtermap g G)" by (rule filter_leD)
    hence "eventually (λx. f (g x) = c + f' (g x) * h (g x)) G" by (simp add: eventually_filtermap)
    moreover from 1 have "f ∘ g ∈ O[G](λ_. 1)" unfolding o_def
      by (intro landau_o.big.compose[OF _ assms(2)]) auto
    ultimately show ?thesis using 1 by force
  qed
qed

lemma asymp_powser_compose:
  assumes "asymp_powser h F f cs" assumes "filterlim g F G"
  shows   "asymp_powser (λx. h (g x)) G (λx. f (g x)) cs"
  using asymp_powser_o[OF assms] by (simp add: o_def)

lemma hd_exp_powser': "ms_aux_hd_exp (powser_ms_aux' cs f) = Some 0"
  by (simp add: powser_ms_aux'_def hd_exp_powser)

lemma powser_ms_aux_asymp_powser:
  assumes "asymp_powser h at_top f cs" and basis: "basis_wf bs"
  assumes "is_expansion_aux xs h bs" "ms_exp_gt 0 (ms_aux_hd_exp xs)"
  shows   "is_expansion_aux (powser_ms_aux' cs xs) f bs"
using assms(1)
proof (coinduction arbitrary: cs f rule: is_expansion_aux_coinduct_upto)
  show "basis_wf bs" by fact
next
  case (B cs f)
  thus ?case
  proof (cases rule: asymp_powser.cases)
    case (1 f' cs' c)
    from assms(3) obtain b bs' where [simp]: "bs = b # bs'"
      by (cases bs) (auto dest: is_expansion_aux_basis_nonempty)
    from basis have b_at_top: "filterlim b at_top at_top" by (simp add: basis_wf_Cons)
    hence b_pos: "eventually (λx. b x > 0) at_top" by (auto simp: filterlim_at_top_dense)
    
    have A: "f ∈ o(λx. b x powr e)" if "e > 0" for e :: real
    proof -
      have "f ∈ O(λ_. 1)" by fact
      also have "(λ_::real. 1) ∈ o(λx. b x powr e)"
        by (rule landau_o.small.compose[OF _ b_at_top]) (insert that, simp_all add: one_smallo_powr)
      finally show ?thesis .
    qed
    have "ms_closure (λxsa'' fa'' basis''.
            ∃cs. xsa'' = powser_ms_aux' cs xs ∧ basis'' = b # bs' ∧ asymp_powser h at_top fa'' cs)
           (times_ms_aux xs (powser_ms_aux' cs' xs)) (λx. h x * f' x) bs" 
      (is "ms_closure ?Q ?ys _ _")
      by (rule ms_closure_mult[OF ms_closure_embed' ms_closure_embed])
         (insert assms 1, auto intro!: exI[of _ cs'])
    moreover have "eventually (λx. h x * f' x = f x - c * b x powr 0) at_top"
      using b_pos and ‹∀F x in at_top. f x = c + f' x * h x›
      by eventually_elim simp
    ultimately have "ms_closure ?Q ?ys (λx. f x - c * b x powr 0) bs"
      by (rule ms_closure_cong)
    with 1 assms A show ?thesis
      using is_expansion_aux_expansion_level[OF assms(3)]
      by (auto simp: powser_ms_aux'_MSSCons basis_wf_Cons hd_exp_times hd_exp_powser'
               intro!: is_expansion_const ms_closure_add ms_closure_mult split: option.splits)
  qed
qed


subsubsection ‹Arc tangent›

definition arctan_ms_aux where
  "arctan_ms_aux xs = times_ms_aux xs (powser_ms_aux' arctan_series_stream (times_ms_aux xs xs))"

lemma arctan_ms_aux_0:
  assumes "ms_exp_gt 0 (ms_aux_hd_exp xs)" "basis_wf basis" "is_expansion_aux xs f basis"
  shows   "is_expansion_aux (arctan_ms_aux xs) (λx. arctan (f x)) basis" 
proof (rule is_expansion_aux_cong)
  let ?g = "powser (msllist_of_msstream arctan_series_stream)"
  show "is_expansion_aux (arctan_ms_aux xs)
          (λx. f x * (?g ∘ (λx. f x * f x)) x) basis"
    unfolding arctan_ms_aux_def powser_ms_aux'_def using assms
    by (intro times_ms_aux powser_ms_aux powser_ms_aux convergent_powser_arctan)
       (auto simp: hd_exp_times add_neg_neg split: option.splits)
  
  have "f ∈ o(λx. hd basis x powr 0)" using assms
    by (intro is_expansion_aux_imp_smallo[OF assms(3)]) auto
  also have "(λx. hd basis x powr 0) ∈ O(λ_. 1)"
    by (intro bigoI[of _ 1]) auto
  finally have "filterlim f (nhds 0) at_top"
    by (auto dest!: smalloD_tendsto)
  from order_tendstoD(2)[OF tendsto_norm [OF this], of 1]
    have "eventually (λx. norm (f x) < 1) at_top" by simp
  thus "eventually (λx. f x * (?g ∘ (λx. f x * f x)) x = arctan (f x)) at_top"
    by eventually_elim (simp add: arctan_conv_pre_arctan power2_eq_square)
qed

lemma arctan_ms_aux_inf:
  assumes "∃e>0. ms_aux_hd_exp xs = Some e" "fst (dominant_term_ms_aux xs) > 0" "trimmed_ms_aux xs" 
          "basis_wf basis" "is_expansion_aux xs f basis"
  shows   "is_expansion_aux (minus_ms_aux (const_ms_aux (pi / 2))
             (arctan_ms_aux (inverse_ms_aux xs))) (λx. arctan (f x)) basis"
    (is "is_expansion_aux ?xs' _ _")
proof (rule is_expansion_aux_cong)
  from ‹trimmed_ms_aux xs› have [simp]: "xs ≠ MSLNil" by auto
  thus "is_expansion_aux ?xs' (λx. pi / 2 - arctan (inverse (f x))) basis" using assms
    by (intro minus_ms_aux arctan_ms_aux_0 inverse_ms_aux const_ms_aux)
       (auto simp: hd_exp_inverse is_expansion_aux_expansion_level)
next
  from assms(2-5) have "eventually (λx. f x > 0) at_top"
    by (intro eval_pos_if_dominant_term_pos') simp_all
  thus "eventually (λx. ((λx. pi/2 - arctan (inverse (f x)))) x = arctan (f x)) at_top"
    unfolding o_def by eventually_elim (subst arctan_inverse_pos, simp_all)
qed

lemma arctan_ms_aux_neg_inf:
  assumes "∃e>0. ms_aux_hd_exp xs = Some e" "fst (dominant_term_ms_aux xs) < 0" "trimmed_ms_aux xs" 
          "basis_wf basis" "is_expansion_aux xs f basis"
  shows   "is_expansion_aux (minus_ms_aux (const_ms_aux (-pi / 2))
             (arctan_ms_aux (inverse_ms_aux xs))) (λx. arctan (f x)) basis"
    (is "is_expansion_aux ?xs' _ _")
proof (rule is_expansion_aux_cong)
  from ‹trimmed_ms_aux xs› have [simp]: "xs ≠ MSLNil" by auto
  thus "is_expansion_aux ?xs' (λx. -pi / 2 - arctan (inverse (f x))) basis" using assms
    by (intro minus_ms_aux arctan_ms_aux_0 inverse_ms_aux const_ms_aux)
       (auto simp: hd_exp_inverse is_expansion_aux_expansion_level)
next
  from assms(2-5) have "eventually (λx. f x < 0) at_top"
    by (intro eval_neg_if_dominant_term_neg') simp_all
  thus "eventually (λx. ((λx. -pi/2 - arctan (inverse (f x)))) x = arctan (f x)) at_top"
    unfolding o_def by eventually_elim (subst arctan_inverse_neg, simp_all)
qed

lemma expands_to_arctan_zero:
  assumes "((λ_::real. 0::real) expands_to C) bs" "eventually (λx. f x = 0) at_top"
  shows   "((λx::real. arctan (f x)) expands_to C) bs"
  using assms by (auto simp: expands_to.simps elim: eventually_elim2)

lemma expands_to_arctan_ms_neg_exp: 
  assumes "e < 0" "basis_wf basis" "(f expands_to MS (MSLCons (C, e) xs) g) basis"
  shows   "((λx. arctan (f x)) expands_to
               MS (arctan_ms_aux (MSLCons (C, e) xs)) (λx. arctan (g x))) basis"
  using assms by (auto simp: expands_to.simps snd_dominant_term_ms_aux_MSLCons o_def
                       intro!: arctan_ms_aux_0 elim: eventually_mono)

lemma expands_to_arctan_ms_pos_exp_pos: 
  assumes "e > 0" "trimmed_pos (MS (MSLCons (C, e) xs) g)" "basis_wf basis" 
          "(f expands_to MS (MSLCons (C, e) xs) g) basis"
  shows   "((λx. arctan (f x)) expands_to MS (minus_ms_aux (const_ms_aux (pi / 2))
             (arctan_ms_aux (inverse_ms_aux (MSLCons (C, e) xs))))
               (λx. arctan (g x))) basis"
  using arctan_ms_aux_inf[of "MSLCons (C, e) xs" basis g] assms
  by (auto simp: expands_to.simps snd_dominant_term_ms_aux_MSLCons o_def
                 dominant_term_ms_aux_MSLCons trimmed_pos_def elim: eventually_mono)

lemma expands_to_arctan_ms_pos_exp_neg: 
  assumes "e > 0" "trimmed_neg (MS (MSLCons (C, e) xs) g)" "basis_wf basis" 
          "(f expands_to MS (MSLCons (C, e) xs) g) basis"
  shows   "((λx. arctan (f x)) expands_to MS (minus_ms_aux (const_ms_aux (-pi/2))
             (arctan_ms_aux (inverse_ms_aux (MSLCons (C, e) xs))))
               (λx. arctan (g x))) basis"
  using arctan_ms_aux_neg_inf[of "MSLCons (C, e) xs" basis g] assms
  by (auto simp: expands_to.simps snd_dominant_term_ms_aux_MSLCons o_def
                 dominant_term_ms_aux_MSLCons trimmed_neg_def elim: eventually_mono)                     


subsubsection ‹Exponential function›

(* TODO: Move *)
lemma ln_smallo_powr:
  assumes "(e::real) > 0"
  shows   "(λx. ln x) ∈ o(λx. x powr e)"
proof -
  have *: "ln ∈ o(λx::real. x)"
    using ln_x_over_x_tendsto_0 by (intro smalloI_tendsto) auto
  have "(λx. ln x) ∈ Θ(λx::real. ln x powr 1)"
    by (intro bigthetaI_cong eventually_mono[OF eventually_gt_at_top[of 1]]) auto
  also have "(λx::real. ln x powr 1) ∈ o(λx. x powr e)"
    by (intro ln_smallo_imp_flat filterlim_ident ln_at_top assms
              landau_o.small.compose[of _ at_top _ ln] *)
  finally show ?thesis .
qed

lemma basis_wf_insert_exp_pos:
  assumes "(f expands_to MS (MSLCons (C, e) xs) g) (b # basis)" 
          "basis_wf (b # basis)" "trimmed (MS (MSLCons (C, e) xs) g)" "e - 0 > 0"
  shows   "(λx. ln (b x)) ∈ o(f)"
proof -
  from assms have b_limit: "filterlim b at_top at_top" by (simp add: basis_wf_Cons)
  hence b_pos: "eventually (λx. b x > 0) at_top" by (simp add: filterlim_at_top_dense)

  from assms have "is_expansion_aux xs (λx. g x - eval C x * b x powr e) (b # basis)" 
                  "ms_exp_gt e (ms_aux_hd_exp xs)"
      by (auto elim!: is_expansion_aux_MSLCons simp: expands_to.simps)
  from is_expansion_aux_imp_smallo''[OF this] obtain e' where e':
    "e' < e" "(λx. g x - eval C x * b x powr e) ∈ o(λx. b x powr e')" unfolding list.sel .
  
  define eps where "eps = e/2"
  have "(λx. ln (b x)) ∈ o(λx. b x powr (e - eps))" unfolding eps_def
    by (rule landau_o.small.compose[OF _ b_limit])
       (insert e'(1) ‹e - 0 > 0›, simp add: ln_smallo_powr)
  also from assms e'(1) have "eval C ∈ ω(λx. b x powr -eps)" unfolding eps_def
    by (intro is_expansion_imp_smallomega[of basis])
       (auto simp: basis_wf_Cons expands_to.simps trimmed_pos_def trimmed_ms_aux_MSLCons
             elim!: is_expansion_aux_MSLCons)
  hence "(λx. b x powr -eps * b x powr e) ∈ o(λx. eval C x * b x powr e)"
    by (intro landau_o.small_big_mult) (simp_all add: smallomega_iff_smallo)
  hence "(λx. b x powr (e - eps)) ∈ o(λx. eval C x * b x powr e)"
    by (simp add: powr_add [symmetric] field_simps)
  finally have "(λx. ln (b x)) ∈ o(λx. eval C x * b x powr e)" .
  also {
    from e' have "(λx. g x - eval C x * b x powr e) ∈ o(λx. b x powr (e' - e) * b x powr e)"
      by (simp add: powr_add [symmetric])
    also from assms e'(1) have "eval C ∈ ω(λx. b x powr (e' - e))" unfolding eps_def
      by (intro is_expansion_imp_smallomega[of basis])
         (auto simp: basis_wf_Cons expands_to.simps trimmed_pos_def trimmed_ms_aux_MSLCons
               elim!: is_expansion_aux_MSLCons)
    hence "(λx. b x powr (e' - e) * b x powr e) ∈ o(λx. eval C x * b x powr e)"
      by (intro landau_o.small_big_mult is_expansion_imp_smallo) 
         (simp_all add: smallomega_iff_smallo)
    finally have "o(λx. eval C x * b x powr e) = 
                    o(λx. g x - eval C x * b x powr e + eval C x * b x powr e)"
      by (intro landau_o.small.plus_absorb1 [symmetric]) simp_all
  }
  also have "o(λx. g x - eval C x * b x powr e + eval C x * b x powr e) = o(g)" by simp
  also from assms have "g ∈ Θ(f)" by (intro bigthetaI_cong) (simp add: expands_to.simps)
  finally show "(λx. ln (b x)) ∈ o(f)" .
qed

lemma basis_wf_insert_exp_uminus:
  "(λx. ln (b x)) ∈ o(f) ⟹ (λx. ln (b x)) ∈ o(λx. -f x :: real)"
  by simp

lemma basis_wf_insert_exp_uminus':
  "f ∈ o(λx. ln (b x)) ⟹ (λx. -f x) ∈ o(λx. ln (b x))"
  by simp
    
lemma expands_to_exp_insert_pos:
  fixes C :: "'a :: multiseries"
  assumes "(f expands_to MS (MSLCons (C, e) xs) g) (b # basis)" 
          "basis_wf (b # basis)" "trimmed_pos (MS (MSLCons (C, e) xs) g)" 
          "(λx. ln (b x)) ∈ o(f)"
  shows   "filterlim (λx. exp (f x)) at_top at_top"
          "((λx. ln (exp (f x))) expands_to MS (MSLCons (C, e) xs) g) (b # basis)"
          "((λx. exp (f x)) expands_to 
             MS (MSLCons (const_expansion 1 :: 'a ms, 1) MSLNil) (λx. exp (g x)))
             ((λx. exp (f x)) # b # basis)" (is ?thesis3)
proof -
  have "ln ∈ ω(λx::real. 1)"
    by (intro smallomegaI_filterlim_at_top_norm)
       (auto intro!: filterlim_compose[OF filterlim_abs_real] ln_at_top)
  with assms have "(λx. 1) ∈ o(λx. ln (b x))"
    by (intro landau_o.small.compose[of _ at_top _ b])
       (simp_all add: basis_wf_Cons smallomega_iff_smallo)
  also note ‹(λx. ln (b x)) ∈ o(f)›
  finally have f: "f ∈ ω(λ_. 1)" by (simp add: smallomega_iff_smallo)
  hence f: "filterlim (λx. abs (f x)) at_top at_top"
    by (auto dest: smallomegaD_filterlim_at_top_norm)
  
  define c where "c = fst (dominant_term C)"
  define es where "es = snd (dominant_term C)"
  from trimmed_imp_eventually_sgn[OF assms(2), of "MS (MSLCons (C, e) xs) g"] assms 
    have "∀F x in at_top. abs (f x) = f x"
      by (auto simp: dominant_term_ms_aux_MSLCons trimmed_pos_def expands_to.simps sgn_if 
               split: if_splits elim: eventually_elim2)
  from filterlim_cong[OF refl refl this] f 
    show *: "filterlim (λx. exp (f x)) at_top at_top" 
    by (auto intro: filterlim_compose[OF exp_at_top])
  
  {
    fix e' :: real assume e': "e' > 1"
    from assms have "(λx. exp (g x)) ∈ Θ(λx. exp (f x) powr 1)"
      by (intro bigthetaI_cong) (auto elim: eventually_mono simp: expands_to.simps)
    also from e' * have "(λx. exp (f x) powr 1) ∈ o(λx. exp (f x) powr e')"
      by (subst powr_smallo_iff) (insert *, simp_all)
    finally have "(λx. exp (g x)) ∈ o(λx. exp (f x) powr e')" .
  }
  with assms show ?thesis3
    by (auto intro!: is_expansion_aux.intros is_expansion_const 
             simp: expands_to.simps is_expansion_aux_expansion_level
             dest: is_expansion_aux_expansion_level)
qed (insert assms, simp_all)
    
lemma expands_to_exp_insert_neg:
  fixes C :: "'a :: multiseries"
  assumes "(f expands_to MS (MSLCons (C, e) xs) g) (b # basis)" 
          "basis_wf (b # basis)" "trimmed_neg (MS (MSLCons (C, e) xs) g)" 
          "(λx. ln (b x)) ∈ o(f)"
  shows   "filterlim (λx. exp (-f x)) at_top at_top" (is ?thesis1)
          "((λx. ln (exp (-f x))) expands_to MS (MSLCons (-C, e) (uminus_ms_aux xs)) (λx. - g x)) 
             (b # basis)"
          "trimmed_pos (MS (MSLCons (-C, e) (uminus_ms_aux xs)) (λx. - g x))"
          "((λx. exp (f x)) expands_to 
             MS (MSLCons (const_expansion 1 :: 'a ms, -1) MSLNil) (λx. exp (g x)))
             ((λx. exp (-f x)) # b # basis)" (is ?thesis3)
proof -
  have "ln ∈ ω(λx::real. 1)"
    by (intro smallomegaI_filterlim_at_top_norm)
       (auto intro!: filterlim_compose[OF filterlim_abs_real] ln_at_top)
  with assms have "(λx. 1) ∈ o(λx. ln (b x))"
    by (intro landau_o.small.compose[of _ at_top _ b])
       (simp_all add: basis_wf_Cons smallomega_iff_smallo)
  also note ‹(λx. ln (b x)) ∈ o(f)›
  finally have f: "f ∈ ω(λ_. 1)" by (simp add: smallomega_iff_smallo)
  hence f: "filterlim (λx. abs (f x)) at_top at_top" 
    by (auto dest: smallomegaD_filterlim_at_top_norm)
  from trimmed_imp_eventually_sgn[OF assms(2), of "MS (MSLCons (C, e) xs) g"] assms 
    have "∀F x in at_top. abs (f x) = -f x"
      by (auto simp: dominant_term_ms_aux_MSLCons trimmed_neg_def expands_to.simps sgn_if 
               split: if_splits elim: eventually_elim2)
  from filterlim_cong[OF refl refl this] f 
    show *: "filterlim (λx. exp (-f x)) at_top at_top" 
    by (auto intro: filterlim_compose[OF exp_at_top])
  
  have "((λx. -f x) expands_to -MS (MSLCons (C, e) xs) g) (b # basis)"
    by (intro expands_to_uminus assms)
  thus "((λx. ln (exp (-f x))) expands_to MS (MSLCons (-C, e) (uminus_ms_aux xs)) (λx. - g x)) 
          (b # basis)"
    by (simp add: uminus_ms_aux_MSLCons)
      
  {
    fix e' :: real assume e': "e' > -1"
    from assms have "(λx. exp (g x)) ∈ Θ(λx. exp (-f x) powr -1)"
      by (intro bigthetaI_cong) 
         (auto elim: eventually_mono simp: expands_to.simps powr_minus exp_minus)
    also from e' * have "(λx. exp (-f x) powr -1) ∈ o(λx. exp (-f x) powr e')"
      by (subst powr_smallo_iff) (insert *, simp_all)
    finally have "(λx. exp (g x)) ∈ o(λx. exp (-f x) powr e')" .
  }
  with assms show ?thesis3
    by (auto intro!: is_expansion_aux.intros is_expansion_const 
             simp: expands_to.simps is_expansion_aux_expansion_level powr_minus exp_minus
             dest: is_expansion_aux_expansion_level)
qed (insert assms, simp_all add: trimmed_pos_def trimmed_neg_def 
       trimmed_ms_aux_MSLCons dominant_term_ms_aux_MSLCons)

lemma is_expansion_aux_exp_neg:
  assumes "is_expansion_aux F f basis" "basis_wf basis" "ms_exp_gt 0 (ms_aux_hd_exp F)"
  shows   "is_expansion_aux (powser_ms_aux' exp_series_stream F) (λx. exp (f x)) basis"
  using powser_ms_aux'[OF convergent_powser'_exp assms(2,1,3)]
  by (simp add: o_def powser_ms_aux'_def)

lemma is_expansion_exp_neg:
  assumes "is_expansion (MS (MSLCons (C, e) xs) f) basis" "basis_wf basis" "e < 0"
  shows   "is_expansion (MS (powser_ms_aux' exp_series_stream (MSLCons (C, e) xs)) 
             (λx. exp (f x))) basis"
  using is_expansion_aux_exp_neg[of "MSLCons (C, e) xs" f basis] assms by simp

lemma expands_to_exp_neg:
  assumes "(f expands_to MS (MSLCons (C, e) xs) g) basis" "basis_wf basis" "e - 0 < 0"
  shows   "((λx. exp (f x)) expands_to MS (powser_ms_aux' exp_series_stream (MSLCons (C, e) xs)) 
             (λx. exp (g x))) basis"
  using assms is_expansion_exp_neg[of C e xs g basis] by (auto simp: expands_to.simps)

lemma basis_wf_insert_exp_near:
  assumes "basis_wf (b # basis)" "basis_wf ((λx. exp (f x)) # basis)" "f ∈ o(λx. ln (b x))"
  shows   "basis_wf (b # (λx. exp (f x)) # basis)"
  using assms by (simp_all add: basis_wf_Cons)


definition scale_ms_aux' where "scale_ms_aux' c F = scale_shift_ms_aux (c, 0) F"
definition shift_ms_aux where "shift_ms_aux e F = scale_shift_ms_aux (const_expansion 1, e) F"

lemma scale_ms_1 [simp]: "scale_ms 1 F = F"
  by (simp add: scale_ms_def times_const_expansion_1)

lemma scale_ms_aux_1 [simp]: "scale_ms_aux 1 xs = xs"
  by (simp add: scale_ms_aux_def scale_shift_ms_aux_def map_prod_def msllist.map_ident
        case_prod_unfold times_const_expansion_1)
    
lemma scale_ms_aux'_1 [simp]: "scale_ms_aux' (const_expansion 1) xs = xs"
  by (simp add: scale_ms_aux'_def scale_shift_ms_aux_def map_prod_def msllist.map_ident
        case_prod_unfold times_const_expansion_1)
    
lemma shift_ms_aux_0 [simp]: "shift_ms_aux 0 xs = xs"
  by (simp add: shift_ms_aux_def scale_shift_ms_aux_def map_prod_def case_prod_unfold 
        times_const_expansion_1 msllist.map_ident)
  
lemma scale_ms_aux'_MSLNil [simp]: "scale_ms_aux' C MSLNil = MSLNil"
  by (simp add: scale_ms_aux'_def)

lemma scale_ms_aux'_MSLCons [simp]: 
  "scale_ms_aux' C (MSLCons (C', e) xs) = MSLCons (C * C', e) (scale_ms_aux' C xs)"
  by (simp add: scale_ms_aux'_def scale_shift_ms_aux_MSLCons)

lemma shift_ms_aux_MSLNil [simp]: "shift_ms_aux e MSLNil = MSLNil"
  by (simp add: shift_ms_aux_def)

lemma shift_ms_aux_MSLCons [simp]: 
  "shift_ms_aux e (MSLCons (C, e') xs) = MSLCons (C, e' + e) (shift_ms_aux e xs)"
  by (simp add: shift_ms_aux_def scale_shift_ms_aux_MSLCons times_const_expansion_1)

lemma scale_ms_aux:
  fixes xs :: "('a :: multiseries × real) msllist"
  assumes "is_expansion_aux xs f basis" "basis_wf basis"
  shows   "is_expansion_aux (scale_ms_aux c xs) (λx. c * f x) basis"
proof (cases basis)
  case (Cons b basis')
  show ?thesis
  proof (rule is_expansion_aux_cong)
    show "is_expansion_aux (scale_ms_aux c xs) 
            (λx. eval (const_expansion c :: 'a) x * b x powr 0 * f x) basis"
      using assms unfolding scale_ms_aux_def Cons
      by (intro scale_shift_ms_aux is_expansion_const) 
         (auto simp: basis_wf_Cons dest: is_expansion_aux_expansion_level)
  next
    from assms have "eventually (λx. b x > 0) at_top" 
      by (simp add: basis_wf_Cons Cons filterlim_at_top_dense)
    thus "eventually (λx. eval (const_expansion c :: 'a) x * b x powr 0 * f x = c * f x) at_top"
      by eventually_elim simp
  qed
qed (insert assms, auto dest: is_expansion_aux_basis_nonempty)

lemma scale_ms_aux':
  assumes "is_expansion_aux xs f (b # basis)" "is_expansion C basis" "basis_wf (b # basis)"
  shows   "is_expansion_aux (scale_ms_aux' C xs) (λx. eval C x * f x) (b # basis)"
proof (rule is_expansion_aux_cong)
  show "is_expansion_aux (scale_ms_aux' C xs) (λx. eval C x * b x powr 0 * f x) (b # basis)"
    using assms unfolding scale_ms_aux'_def Cons by (intro scale_shift_ms_aux) simp_all
next
  from assms have "eventually (λx. b x > 0) at_top" 
    by (simp add: basis_wf_Cons filterlim_at_top_dense)
  thus "eventually (λx. eval C x * b x powr 0 * f x = eval C x * f x) at_top"
    by eventually_elim simp
qed

lemma shift_ms_aux:
  fixes xs :: "('a :: multiseries × real) msllist"
  assumes "is_expansion_aux xs f (b # basis)" "basis_wf (b # basis)"
  shows   "is_expansion_aux (shift_ms_aux e xs) (λx. b x powr e * f x) (b # basis)"
  unfolding shift_ms_aux_def 
  using scale_shift_ms_aux[OF assms(2,1) is_expansion_const[of _ 1], of e] assms
  by (auto dest!: is_expansion_aux_expansion_level simp: basis_wf_Cons)
  
lemma expands_to_exp_0:
  assumes "(f expands_to MS (MSLCons (MS ys c, e) xs) g) (b # basis)"
          "basis_wf (b # basis)" "e - 0 = 0" "((λx. exp (c x)) expands_to E) basis"
  shows   "((λx. exp (f x)) expands_to
             MS (scale_ms_aux' E (powser_ms_aux' exp_series_stream xs)) (λx. exp (g x))) (b # basis)"
          (is "(_ expands_to MS ?H _) _")
proof -
  from assms have "is_expansion_aux ?H (λx. eval E x * exp (g x - c x * b x powr 0)) (b # basis)"
    by (intro scale_ms_aux' is_expansion_aux_exp_neg)
       (auto elim!: is_expansion_aux_MSLCons simp: expands_to.simps)
  moreover {
    from ‹basis_wf (b # basis)› have "eventually (λx. b x > 0) at_top"
      by (simp add: filterlim_at_top_dense basis_wf_Cons)
    moreover from assms(4) have "eventually (λx. eval E x = exp (c x)) at_top"
      by (simp add: expands_to.simps)
    ultimately have "eventually (λx. eval E x * exp (g x - c x * b x powr 0) = exp (g x)) at_top"
      by eventually_elim (simp add: exp_diff)
  }
  ultimately have "is_expansion_aux ?H (λx. exp (g x)) (b # basis)"
    by (rule is_expansion_aux_cong)
  with assms(1) show ?thesis by (simp add: expands_to.simps)
qed

lemma expands_to_exp_0_real:
  assumes "(f expands_to MS (MSLCons (c::real, e) xs) g) [b]" "basis_wf [b]" "e - 0 = 0"
  shows   "((λx. exp (f x)) expands_to 
             MS (scale_ms_aux (exp c) (powser_ms_aux' exp_series_stream xs)) (λx. exp (g x))) [b]"
          (is "(_ expands_to MS ?H _) _")
proof -
  from assms have "is_expansion_aux ?H (λx. exp c * exp (g x - c * b x powr 0)) [b]"
    by (intro scale_ms_aux is_expansion_aux_exp_neg)
       (auto elim!: is_expansion_aux_MSLCons simp: expands_to.simps)
  moreover from ‹basis_wf [b]› have "eventually (λx. b x > 0) at_top"
    by (simp add: filterlim_at_top_dense basis_wf_Cons)
  hence "eventually (λx. exp c * exp (g x - c * b x powr 0) = exp (g x)) at_top"
    by eventually_elim (simp add: exp_diff)
  ultimately have "is_expansion_aux ?H (λx. exp (g x)) [b]"
    by (rule is_expansion_aux_cong)
  with assms(1) show ?thesis by (simp add: expands_to.simps)
qed

lemma expands_to_exp_0_pull_out1:
  assumes "(f expands_to MS (MSLCons (C, e) xs) g) (b # basis)"
  assumes "((λx. ln (b x)) expands_to L) basis" "basis_wf (b # basis)" "e - 0 = 0" "c ≡ c"
  shows   "((λx. f x - c * ln (b x)) expands_to 
             MS (MSLCons (C - scale_ms c L, e) xs) (λx. g x - c * ln (b x))) (b # basis)"
proof -
  from ‹basis_wf (b # basis)› have b_limit: "filterlim b at_top at_top" by (simp add: basis_wf_Cons)
  have "(λx. c * ln (b x)) ∈ o(λx. b x powr e')" if "e' > 0" for e'
    using that by (intro landau_o.small.compose[OF _ b_limit]) (simp add: ln_smallo_powr)
  hence *: "(λx. g x - c * ln (b x)) ∈ o(λx. b x powr e')" if "e' > 0" for e' using assms(1,4) that
    by (intro sum_in_smallo) (auto simp: expands_to.simps elim!: is_expansion_aux_MSLCons)
  
  have "is_expansion_aux xs (λx. (g x - c * b x powr 0 * eval L x) -
           eval (C - scale_ms c L) x * b x powr e) (b # basis)" 
    (is "is_expansion_aux _ ?h _") using assms
    by (auto simp: expands_to.simps algebra_simps scale_ms_def elim!: is_expansion_aux_MSLCons)
  moreover from b_limit have "eventually (λx. b x > 0) at_top"
    by (simp add: filterlim_at_top_dense)
  hence "eventually (λx. ?h x = g x - c * ln (b x) - 
           eval (C - const_expansion c * L) x * b x powr e) at_top" 
    (is "eventually (λx. _ = ?h x) _") using assms(2)
    apply (simp add: expands_to.simps scale_ms_def)
    by (smt (verit) eventually_cong)
  ultimately have **: "is_expansion_aux xs ?h (b # basis)" 
    by (rule is_expansion_aux_cong)
  from assms(1,4) have "ms_exp_gt 0 (ms_aux_hd_exp xs)" "is_expansion C basis"
    by (auto simp: expands_to.simps elim!: is_expansion_aux_MSLCons)
  moreover from assms(1) have "length basis = expansion_level TYPE('a)"
    by (auto simp: expands_to.simps dest: is_expansion_aux_expansion_level)
  ultimately have "is_expansion_aux (MSLCons (C - scale_ms c L, e) xs) 
                     (λx. g x - c * ln (b x)) (b # basis)" 
    using assms unfolding scale_ms_def
    by (intro is_expansion_aux.intros is_expansion_minus is_expansion_mult is_expansion_const * **)
       (simp_all add: basis_wf_Cons expands_to.simps)
  with assms(1) show ?thesis by (auto simp: expands_to.simps)
qed
    
lemma expands_to_exp_0_pull_out2:
  assumes "((λx. exp (f x - c * ln (b x))) expands_to MS xs g) (b # basis)"
  assumes "basis_wf (b # basis)"
  shows   "((λx. exp (f x)) expands_to MS (shift_ms_aux c xs) 
             (λx. b x powr c * g x)) (b # basis)"
proof (rule expands_to.intros)
  show "is_expansion (MS (shift_ms_aux c xs) (λx. b x powr c * g x)) (b # basis)"
    using assms unfolding expands_to.simps by (auto intro!: shift_ms_aux)
  from assms have "eventually (λx. b x > 0) at_top" 
    by (simp add: basis_wf_Cons filterlim_at_top_dense)
  with assms(1)
  show "∀F x in at_top. eval (MS (shift_ms_aux c xs) (λx. b x powr c * g x)) x = exp (f x)"
    by (auto simp: expands_to.simps exp_diff powr_def elim: eventually_elim2)
qed

lemma expands_to_exp_0_pull_out2_nat:
  assumes "((λx. exp (f x - c * ln (b x))) expands_to MS xs g) (b # basis)"
  assumes "basis_wf (b # basis)" "c = real n"
  shows   "((λx. exp (f x)) expands_to MS (shift_ms_aux c xs) 
             (λx. b x ^ n * g x)) (b # basis)"
using expands_to_exp_0_pull_out2[OF assms(1-2)]
proof (rule expands_to_cong')
  from assms(2) have "eventually (λx. b x > 0) at_top" 
    by (simp add: filterlim_at_top_dense basis_wf_Cons)
  with assms(3) show "eventually (λx. b x powr c * g x = b x ^ n * g x) at_top"
    by (auto elim!: eventually_mono simp: powr_realpow)
qed

lemma expands_to_exp_0_pull_out2_neg_nat:
  assumes "((λx. exp (f x - c * ln (b x))) expands_to MS xs g) (b # basis)"
  assumes "basis_wf (b # basis)" "c = -real n"
  shows   "((λx. exp (f x)) expands_to MS (shift_ms_aux c xs) 
             (λx. g x / b x ^ n)) (b # basis)"
using expands_to_exp_0_pull_out2[OF assms(1-2)]
proof (rule expands_to_cong')
  from assms(2) have "eventually (λx. b x > 0) at_top" 
    by (simp add: filterlim_at_top_dense basis_wf_Cons)
  with assms(3) show "eventually (λx. b x powr c * g x = g x / b x ^ n) at_top"
    by (auto elim!: eventually_mono simp: powr_realpow powr_minus field_split_simps)
qed

lemma eval_monom_collapse: "c * eval_monom (c', es) basis x = eval_monom (c * c', es) basis x"
  by (simp add: eval_monom_def)

lemma eval_monom_1_conv: "eval_monom a basis = (λx. fst a * eval_monom (1, snd a) basis x)"
  by (simp add: eval_monom_def case_prod_unfold)


subsubsection ‹Comparing exponent lists›

primrec flip_cmp_result where
  "flip_cmp_result LT = GT"
| "flip_cmp_result GT = LT"
| "flip_cmp_result EQ = EQ"

fun compare_lists :: "real list ⇒ real list ⇒ cmp_result" where
  "compare_lists [] [] = EQ"
| "compare_lists (a # as) (b # bs) = 
     (if a < b then LT else if b < a then GT else compare_lists as bs)"
| "compare_lists _ _ = undefined"

declare compare_lists.simps [simp del]
  
lemma compare_lists_Nil [simp]: "compare_lists [] [] = EQ" by (fact compare_lists.simps)

lemma compare_lists_Cons [simp]:
  "a < b ⟹ compare_lists (a # as) (b # bs) = LT"
  "a > b ⟹ compare_lists (a # as) (b # bs) = GT"
  "a = b ⟹ compare_lists (a # as) (b # bs) = compare_lists as bs"
  by (simp_all add: compare_lists.simps(2))

lemma flip_compare_lists:
  "length as = length bs ⟹ flip_cmp_result (compare_lists as bs) = compare_lists bs as"
  by (induction as bs rule: compare_lists.induct) (auto simp: compare_lists.simps(2))

lemma compare_lists_induct [consumes 1, case_names Nil Eq Neq]:
  fixes as bs :: "real list"
  assumes "length as = length bs"
  assumes "P [] []"
  assumes "⋀a as bs. P as bs ⟹ P (a # as) (a # bs)"
  assumes "⋀a as b bs. a ≠ b ⟹ P (a # as) (b # bs)"
  shows   "P as bs"
  using assms(1)
proof (induction as bs rule: list_induct2)
  case (Cons a as b bs)
  thus ?case by (cases "a < b") (auto intro: assms)
qed (simp_all add: assms)
  

lemma eval_monom_smallo_eval_monom:
  assumes "length es1 = length es2" "length es2 = length basis" "basis_wf basis"
  assumes "compare_lists es1 es2 = LT"
  shows   "eval_monom (1, es1) basis ∈ o(eval_monom (1, es2) basis)"
using assms
proof (induction es1 es2 basis rule: list_induct3)
  case (Cons e1 es1 e2 es2 b basis)
  show ?case
  proof (cases "e1 = e2")
    case True
    from Cons.prems have "eventually (λx. b x > 0) at_top" 
      by (simp add: basis_wf_Cons filterlim_at_top_dense)
    with Cons True show ?thesis
      by (auto intro: landau_o.small_big_mult simp: eval_monom_Cons basis_wf_Cons)
  next
    case False
    with Cons.prems have "e1 < e2" by (cases e1 e2 rule: linorder_cases) simp_all
    with Cons.prems have
       "(λx. eval_monom (1, es1) basis x * eval_monom (inverse_monom (1, es2)) basis x *
           b x powr e1) ∈ o(λx. b x powr ((e2 - e1)/2) * b x powr ((e2 - e1)/2) * b x powr e1)"
      (is "?f ∈ _") by (intro landau_o.small_big_mult[OF _ landau_o.big_refl] landau_o.small.mult 
                          eval_monom_smallo') simp_all
    also have "… = o(λx. b x powr e2)" by (simp add: powr_add [symmetric])
    also have "?f = (λx. eval_monom (1, e1 # es1) (b # basis) x / eval_monom (1, es2) basis x)"
      by (simp add: eval_inverse_monom field_simps eval_monom_Cons)
    also have "… ∈ o(λx. b x powr e2) ⟷ 
                 eval_monom (1, e1 # es1) (b # basis) ∈ o(eval_monom (1, e2 # es2) (b # basis))"
      using Cons.prems 
      by (subst landau_o.small.divide_eq2)
         (simp_all add: eval_monom_Cons mult_ac eval_monom_nonzero basis_wf_Cons)
    finally show ?thesis .
  qed
qed simp_all

lemma eval_monom_eq:
  assumes "length es1 = length es2" "length es2 = length basis" "basis_wf basis"
  assumes "compare_lists es1 es2 = EQ"
  shows   "eval_monom (1, es1) basis = eval_monom (1, es2) basis"
  using assms
  by (induction es1 es2 basis rule: list_induct3)
     (auto simp: basis_wf_Cons eval_monom_Cons compare_lists.simps(2) split: if_splits)

definition compare_expansions :: "'a :: multiseries ⇒ 'a ⇒ cmp_result × real × real" where
  "compare_expansions F G =
     (case compare_lists (snd (dominant_term F)) (snd (dominant_term G)) of
        LT ⇒ (LT, 0, 0)
      | GT ⇒ (GT, 0, 0)
      | EQ ⇒ (EQ, fst (dominant_term F),  fst (dominant_term G)))"

lemma compare_expansions_real:
  "compare_expansions (c1 :: real) c2 = (EQ, c1, c2)"
  by (simp add: compare_expansions_def)

lemma compare_expansions_MSLCons_eval:
  "compare_expansions (MS (MSLCons (C1, e1) xs) f) (MS (MSLCons (C2, e2) ys) g) =
     CMP_BRANCH (COMPARE e1 e2) (LT, 0, 0) (compare_expansions C1 C2) (GT, 0, 0)"
  by (simp add: compare_expansions_def dominant_term_ms_aux_def case_prod_unfold 
        COMPARE_def CMP_BRANCH_def split: cmp_result.splits)

lemma compare_expansions_LT_I:
  assumes "e1 - e2 < 0"
  shows   "compare_expansions (MS (MSLCons (C1, e1) xs) f) (MS (MSLCons (C2, e2) ys) g) = (LT, 0, 0)"
  using assms by (simp add: compare_expansions_MSLCons_eval CMP_BRANCH_def COMPARE_def)

lemma compare_expansions_GT_I:
  assumes "e1 - e2 > 0"
  shows   "compare_expansions (MS (MSLCons (C1, e1) xs) f) (MS (MSLCons (C2, e2) ys) g) = (GT, 0, 0)"
  using assms by (simp add: compare_expansions_MSLCons_eval CMP_BRANCH_def COMPARE_def)

lemma compare_expansions_same_exp:
  assumes "e1 - e2 = 0" "compare_expansions C1 C2 = res"
  shows   "compare_expansions (MS (MSLCons (C1, e1) xs) f) (MS (MSLCons (C2, e2) ys) g) = res"
  using assms by (simp add: compare_expansions_MSLCons_eval CMP_BRANCH_def COMPARE_def)
  

lemma compare_expansions_GT_flip:
  "length (snd (dominant_term F)) = length (snd (dominant_term G)) ⟹
     compare_expansions F G = (GT, c) ⟷ compare_expansions G F = (LT, c)"
  using flip_compare_lists[of "snd (dominant_term F)" "snd (dominant_term G)"]
  by (auto simp: compare_expansions_def split: cmp_result.splits)

lemma compare_expansions_LT:
  assumes "compare_expansions F G = (LT, c, c')" "trimmed G"
          "(f expands_to F) basis" "(g expands_to G) basis" "basis_wf basis"
  shows   "f ∈ o(g)"
proof -
  from assms have "f ∈ Θ(eval F)"
    by (auto simp: expands_to.simps eq_commute intro!: bigthetaI_cong)
  also have "eval F ∈ O(eval_monom (1, snd (dominant_term F)) basis)"
    using assms by (intro dominant_term_bigo) (auto simp: expands_to.simps)
  also have "eval_monom (1, snd (dominant_term F)) basis ∈ 
               o(eval_monom (1, snd (dominant_term G)) basis)" using assms 
    by (intro eval_monom_smallo_eval_monom)
       (auto simp: length_dominant_term expands_to.simps is_expansion_length compare_expansions_def
             split: cmp_result.splits)
  also have "eval_monom (1, snd (dominant_term G)) basis ∈ Θ(eval_monom (dominant_term G) basis)"
    by (subst (2) eval_monom_1_conv, subst landau_theta.cmult)
       (insert assms, simp_all add: trimmed_imp_dominant_term_nz)    
  also have "eval_monom (dominant_term G) basis ∈ Θ(g)"
    by (intro asymp_equiv_imp_bigtheta[OF asymp_equiv_symI] dominant_term_expands_to
              asymp_equiv_imp_bigtheta assms)
  finally show ?thesis .
qed

lemma compare_expansions_GT:
  assumes "compare_expansions F G = (GT, c, c')" "trimmed F"
          "(f expands_to F) basis" "(g expands_to G) basis" "basis_wf basis"
  shows   "g ∈ o(f)"
  by (rule compare_expansions_LT[OF _ assms(2,4,3)])
     (insert assms, simp_all add: compare_expansions_GT_flip length_dominant_term)

lemma compare_expansions_EQ:
  assumes "compare_expansions F G = (EQ, c, c')" "trimmed F" "trimmed G"
          "(f expands_to F) basis" "(g expands_to G) basis" "basis_wf basis"
  shows   "(λx. c' * f x) ∼[at_top] (λx. c * g x)"
proof -
  from assms(1) have c: "c = fst (dominant_term F)" "c' = fst (dominant_term G)"
    by (auto simp: compare_expansions_def split: cmp_result.splits)
  have "(λx. c' * f x) ∼[at_top] (λx. c' * (c * eval_monom (1, snd (dominant_term F)) basis x))" 
    unfolding c by (rule asymp_equiv_mult, rule asymp_equiv_refl, subst eval_monom_collapse)
                   (auto intro: dominant_term_expands_to assms)
  also have "eval_monom (1, snd (dominant_term F)) basis =
               eval_monom (1, snd (dominant_term G)) basis" using assms
    by (intro eval_monom_eq)
       (simp_all add: compare_expansions_def length_dominant_term is_expansion_length 
          expands_to.simps split: cmp_result.splits)
  also have "(λx. c' * (c * … x)) = (λx. c * (c' * … x))" by (simp add: mult_ac)
  also have "… ∼[at_top] (λx. c * g x)"
    unfolding c by (rule asymp_equiv_mult, rule asymp_equiv_refl, subst eval_monom_collapse, 
                       rule asymp_equiv_symI)  (auto intro: dominant_term_expands_to assms)
  finally show ?thesis by (simp add: asymp_equiv_sym)
qed

lemma compare_expansions_EQ_imp_bigo:
  assumes "compare_expansions F G = (EQ, c, c')" "trimmed G"
          "(f expands_to F) basis" "(g expands_to G) basis" "basis_wf basis"
  shows   "f ∈ O(g)"
proof -
  from assms(1) have c: "c = fst (dominant_term F)" "c' = fst (dominant_term G)"
    by (auto simp: compare_expansions_def split: cmp_result.splits)
  from assms(3) have [simp]: "expansion_level TYPE('a) = length basis"
    by (auto simp: expands_to.simps is_expansion_length)

  have "f ∈ Θ(eval F)"
    using assms by (intro bigthetaI_cong) (auto simp: expands_to.simps eq_commute)
  also have "eval F ∈ O(eval_monom (1, snd (dominant_term F)) basis)"
    using assms by (intro dominant_term_bigo assms) (auto simp: expands_to.simps)
  also have "eval_monom (1, snd (dominant_term F)) basis =
               eval_monom (1, snd (dominant_term G)) basis" using assms
    by (intro eval_monom_eq) (auto simp: compare_expansions_def case_prod_unfold
                                length_dominant_term split: cmp_result.splits)
  also have "… ∈ O(eval_monom (dominant_term G) basis)" using assms(2)
    by (auto simp add: eval_monom_def case_prod_unfold dest: trimmed_imp_dominant_term_nz)
  also have "eval_monom (dominant_term G) basis ∈ Θ(eval G)"
    by (rule asymp_equiv_imp_bigtheta, rule asymp_equiv_symI, rule dominant_term)
       (insert assms, auto simp: expands_to.simps)
  also have "eval G ∈ Θ(g)"
    using assms by (intro bigthetaI_cong) (auto simp: expands_to.simps)
  finally show ?thesis .
qed

lemma compare_expansions_EQ_same:
  assumes "compare_expansions F G = (EQ, c, c')" "trimmed F" "trimmed G"
          "(f expands_to F) basis" "(g expands_to G) basis" "basis_wf basis"
          "c = c'"
  shows   "f ∼[at_top] g"
proof -
  from assms have [simp]: "c' ≠ 0" 
    by (auto simp: compare_expansions_def trimmed_imp_dominant_term_nz split: cmp_result.splits)
  have "(λx. inverse c * (c' * f x)) ∼[at_top] (λx. inverse c * (c * g x))"
    by (rule asymp_equiv_mult[OF asymp_equiv_refl]) (rule compare_expansions_EQ[OF assms(1-6)])
  with assms(7) show ?thesis by (simp add: field_split_simps)
qed
  
lemma compare_expansions_EQ_imp_bigtheta:
  assumes "compare_expansions F G = (EQ, c, c')" "trimmed F" "trimmed G"
          "(f expands_to F) basis" "(g expands_to G) basis" "basis_wf basis"
  shows   "f ∈ Θ(g)"
proof -
  from assms have "(λx. c' * f x) ∈ Θ(λx. c * g x)"
    by (intro asymp_equiv_imp_bigtheta compare_expansions_EQ)
  moreover from assms have "c ≠ 0" "c' ≠ 0"
    by (auto simp: compare_expansions_def trimmed_imp_dominant_term_nz split: cmp_result.splits)
  ultimately show ?thesis by simp
qed

lemma expands_to_MSLCons_0_asymp_equiv_hd:
  assumes "(f expands_to (MS (MSLCons (C, 0) xs) g)) basis" "trimmed (MS (MSLCons (C, 0) xs) f)"
          "basis_wf basis"
  shows   "f ∼[at_top] eval C"
proof -
  from assms(1) is_expansion_aux_basis_nonempty obtain b basis' where [simp]: "basis = b # basis'"
    by (cases basis) (auto simp: expands_to.simps)
  from assms have b_pos: "eventually (λx. b x > 0) at_top" 
    by (simp add: filterlim_at_top_dense basis_wf_Cons)
  from assms have "f ∼[at_top] eval_monom (dominant_term (MS (MSLCons (C, 0) xs) g)) basis" 
    by (intro dominant_term_expands_to) simp_all
  also have "… = (λx. eval_monom (dominant_term C) basis' x * b x powr 0)"
    by (simp_all add: dominant_term_ms_aux_def case_prod_unfold eval_monom_Cons)
  also have "eventually (λx. … x = eval_monom (dominant_term C) basis' x) at_top"
    using b_pos by eventually_elim simp
  also from assms have "eval_monom (dominant_term C) basis' ∼[at_top] eval C"
    by (intro asymp_equiv_symI [OF dominant_term_expands_to] 
          expands_to_hd''[of f C 0 xs g b basis']) (auto simp: trimmed_ms_aux_MSLCons basis_wf_Cons)
  finally show ?thesis .
qed
    
  
lemma compare_expansions_LT':
  assumes "compare_expansions (MS ys h) G ≡ (LT, c, c')" "trimmed G"
          "(f expands_to (MS (MSLCons (MS ys h, e) xs) f')) (b # basis)" "(g expands_to G) basis"
          "e = 0" "basis_wf (b # basis)"
  shows   "h ∈ o(g)"   
proof -
  from assms show ?thesis
    by (intro compare_expansions_LT[OF _ assms(2) _ assms(4)])
       (auto intro: expands_to_hd'' simp: trimmed_ms_aux_MSLCons basis_wf_Cons expands_to.simps 
             elim!: is_expansion_aux_MSLCons)
qed

lemma compare_expansions_GT':
  assumes "compare_expansions C G ≡ (GT, c, c')"
          "trimmed (MS (MSLCons (C, e) xs) f')"
          "(f expands_to (MS (MSLCons (C, e) xs) f')) (b # basis)" "(g expands_to G) basis"
          "e = 0" "basis_wf (b # basis)"
  shows   "g ∈ o(f)"
proof -
  from assms have "g ∈ o(eval C)"
    by (intro compare_expansions_GT[of C G c c' _ basis])
       (auto intro: expands_to_hd'' simp: trimmed_ms_aux_MSLCons basis_wf_Cons)
  also from assms have "f ∈ Θ(eval C)"
    by (intro asymp_equiv_imp_bigtheta expands_to_MSLCons_0_asymp_equiv_hd[of f C xs f' "b#basis"])
       auto
  finally show ?thesis .
qed

lemma compare_expansions_EQ':
  assumes "compare_expansions C G = (EQ, c, c')" 
          "trimmed (MS (MSLCons (C, e) xs) f')" "trimmed G"
          "(f expands_to (MS (MSLCons (C, e) xs) f')) (b # basis)" "(g expands_to G) basis"
          "e = 0" "basis_wf (b # basis)"
  shows   "f ∼[at_top] (λx. c / c' * g x)" 
proof -
  from assms have [simp]: "c' ≠ 0" 
    by (auto simp: compare_expansions_def trimmed_imp_dominant_term_nz split: cmp_result.splits)
  from assms have "f ∼[at_top] eval C"
    by (intro expands_to_MSLCons_0_asymp_equiv_hd[of f C xs f' "b#basis"]) auto
  also from assms(2,4,7) have *: "(λx. c' * eval C x) ∼[at_top] (λx. c * g x)"
    by (intro compare_expansions_EQ[OF assms(1) _ assms(3) _ assms(5)])
       (auto intro: expands_to_hd'' simp: trimmed_ms_aux_MSLCons basis_wf_Cons)
  have "(λx. inverse c' * (c' * eval C x)) ∼[at_top] (λx. inverse c' * (c * g x))"
    by (rule asymp_equiv_mult) (insert *, simp_all)
  hence "eval C ∼[at_top] (λx. c / c' * g x)" by (simp add: field_split_simps)
  finally show ?thesis .
qed

lemma expands_to_insert_ln: 
  assumes "basis_wf [b]"
  shows   "((λx. ln (b x)) expands_to MS (MSLCons (1::real,1) MSLNil) (λx. ln (b x))) [λx. ln (b x)]"
proof -
  from assms have b_limit: "filterlim b at_top at_top" by (simp add: basis_wf_Cons)
  hence b: "eventually (λx. b x > 1) at_top" by (simp add: filterlim_at_top_dense)
  have "(λx::real. x) ∈ Θ(λx. x powr 1)"
    by (intro bigthetaI_cong eventually_mono[OF eventually_gt_at_top[of 0]]) auto
  also have "(λx::real. x powr 1) ∈ o(λa. a powr e)" if "e > 1" for e
    by (subst powr_smallo_iff) (auto simp: that filterlim_ident)
  finally show ?thesis using b assms
    by (auto intro!: is_expansion_aux.intros landau_o.small.compose[of _ at_top _ "λx. ln (b x)"]
                     filterlim_compose[OF ln_at_top b_limit] is_expansion_real.intros
             elim!: eventually_mono simp: expands_to.simps)
qed

lemma trimmed_pos_insert_ln:
  assumes "basis_wf [b]"
  shows   "trimmed_pos (MS (MSLCons (1::real, 1) MSLNil) (λx. ln (b x)))"
  by (simp_all add: trimmed_ms_aux_def)


primrec compare_list_0 where
  "compare_list_0 [] = EQ"
| "compare_list_0 (c # cs) = CMP_BRANCH (COMPARE c 0) LT (compare_list_0 cs) GT"


lemma compare_reals_diff_sgnD:
  "a - (b :: real) < 0 ⟹ a < b" "a - b = 0 ⟹ a = b" "a - b > 0 ⟹ a > b"
  by simp_all

lemma expands_to_real_imp_filterlim:
  assumes "(f expands_to (c :: real)) basis"
  shows   "(f ⤏ c) at_top"
  using assms by (auto elim!: expands_to.cases simp: eq_commute[of c] tendsto_eventually)

lemma expands_to_MSLNil_imp_filterlim:
  assumes "(f expands_to MS MSLNil f') basis"
  shows   "(f ⤏ 0) at_top"
proof -
  from assms have "eventually (λx. f' x = 0) at_top" "eventually (λx. f' x = f x) at_top"
    by (auto elim!: expands_to.cases is_expansion_aux.cases[of MSLNil])
  hence "eventually (λx. f x = 0) at_top" by eventually_elim auto
  thus ?thesis by (simp add: tendsto_eventually)
qed

lemma expands_to_neg_exponent_imp_filterlim:
  assumes "(f expands_to MS (MSLCons (C, e) xs) f') basis" "basis_wf basis" "e < 0"
  shows   "(f ⤏ 0) at_top"
proof -
  let ?F = "MS (MSLCons (C, e) xs) f'"
  let ?es = "snd (dominant_term ?F)"
  from assms have exp: "is_expansion ?F basis"
    by (simp add: expands_to.simps)
  from assms have "f ∈ Θ(f')" by (intro bigthetaI_cong) (simp add: expands_to.simps eq_commute)
  also from dominant_term_bigo[OF assms(2) exp]
    have "f' ∈ O(eval_monom (1, ?es) basis)" by simp
  also have "(eval_monom (1, ?es) basis) ∈ o(λx. hd basis x powr 0)" using exp assms
    by (intro eval_monom_smallo)
       (auto simp: is_expansion_aux_basis_nonempty dominant_term_ms_aux_def
                   case_prod_unfold length_dominant_term is_expansion_aux_expansion_level)
  also have "(λx. hd basis x powr 0) ∈ O(λ_. 1)"
    by (intro landau_o.bigI[of 1]) auto
  finally show ?thesis by (auto dest!: smalloD_tendsto)
qed

lemma expands_to_pos_exponent_imp_filterlim:
  assumes "(f expands_to MS (MSLCons (C, e) xs) f') basis" "trimmed (MS (MSLCons (C, e) xs) f')"
          "basis_wf basis" "e > 0"
  shows   "filterlim f at_infinity at_top"
proof -
  let ?F = "MS (MSLCons (C, e) xs) f'"
  let ?es = "snd (dominant_term ?F)"
  from assms have exp: "is_expansion ?F basis"
    by (simp add: expands_to.simps)
  with assms have "filterlim (hd basis) at_top at_top"
    using is_expansion_aux_basis_nonempty[of "MSLCons (C, e) xs" f' basis]
    by (cases basis) (simp_all add: basis_wf_Cons)
  hence b_pos: "eventually (λx. hd basis x > 0) at_top" using filterlim_at_top_dense by blast

  from assms have "f ∈ Θ(f')" by (intro bigthetaI_cong) (simp add: expands_to.simps eq_commute)
  also from dominant_term[OF assms(3) exp assms(2)]
    have "f' ∈ Θ(eval_monom (dominant_term ?F) basis)" by (simp add: asymp_equiv_imp_bigtheta)
  also have "(eval_monom (dominant_term ?F) basis) ∈ Θ(eval_monom (1, ?es) basis)"
    using assms by (simp add: eval_monom_def case_prod_unfold dominant_term_ms_aux_def 
                              trimmed_imp_dominant_term_nz trimmed_ms_aux_def)
  also have "eval_monom (1, ?es) basis ∈ ω(λx. hd basis x powr 0)" using exp assms
    by (intro eval_monom_smallomega)
       (auto simp: is_expansion_aux_basis_nonempty dominant_term_ms_aux_def
                   case_prod_unfold length_dominant_term is_expansion_aux_expansion_level)
  also have "(λx. hd basis x powr 0) ∈ Θ(λ_. 1)"
    by (intro bigthetaI_cong eventually_mono[OF b_pos]) auto
  finally show ?thesis by (auto dest!: smallomegaD_filterlim_at_infinity)
qed

lemma expands_to_zero_exponent_imp_filterlim:
  assumes "(f expands_to MS (MSLCons (C, e) xs) f') basis"
          "basis_wf basis" "e = 0"
  shows   "filterlim (eval C) at_infinity at_top ⟹ filterlim f at_infinity at_top"
    and   "filterlim (eval C) (nhds L) at_top ⟹ filterlim f (nhds L) at_top"
proof -
  from assms obtain b basis' where *:
    "basis = b # basis'" "is_expansion C basis'" "ms_exp_gt 0 (ms_aux_hd_exp xs)"
    "is_expansion_aux xs (λx. f' x - eval C x * b x powr 0) basis"
    by (auto elim!: expands_to.cases is_expansion_aux_MSLCons)

  from *(1) assms have "filterlim b at_top at_top" by (simp add: basis_wf_Cons)
  hence b_pos: "eventually (λx. b x > 0) at_top" using filterlim_at_top_dense by blast

  from assms(1) have "eventually (λx. f' x = f x) at_top" by (simp add: expands_to.simps)
  hence "eventually (λx. f' x - eval C x * b x powr 0 = f x - eval C x) at_top"
    using b_pos by eventually_elim auto
  from *(4) and this have "is_expansion_aux xs (λx. f x - eval C x) basis"
    by (rule is_expansion_aux_cong)
  hence "(λx. f x - eval C x) ∈ o(λx. hd basis x powr 0)"
    by (rule is_expansion_aux_imp_smallo) fact
  also have "(λx. hd basis x powr 0) ∈ Θ(λ_. 1)"
    by (intro bigthetaI_cong eventually_mono[OF b_pos]) (auto simp: *(1))
  finally have lim: "filterlim (λx. f x - eval C x) (nhds 0) at_top"
    by (auto dest: smalloD_tendsto)

  show "filterlim f at_infinity at_top" if "filterlim (eval C) at_infinity at_top"
    using tendsto_add_filterlim_at_infinity[OF lim that] by simp
  show "filterlim f (nhds L) at_top" if "filterlim (eval C) (nhds L) at_top"
    using tendsto_add[OF lim that] by simp
qed

lemma expands_to_lift_function:
  assumes "eventually (λx. f x - eval c x = 0) at_top"
          "((λx. g (eval (c :: 'a :: multiseries) x)) expands_to G) bs'"
  shows   "((λx. g (f x)) expands_to G) bs'"
  by (rule expands_to_cong[OF assms(2)]) (insert assms, auto elim: eventually_mono) 

lemma drop_zero_ms':
  fixes c f
  assumes "c = (0::real)" "(f expands_to MS (MSLCons (c, e) xs) g) basis"
  shows   "(f expands_to MS xs g) basis"
  using assms drop_zero_ms[of c e xs g basis] by (simp add: expands_to.simps)

lemma trimmed_realI: "c ≠ (0::real) ⟹ trimmed c"
  by simp

lemma trimmed_pos_realI: "c > (0::real) ⟹ trimmed_pos c"
  by simp

lemma trimmed_neg_realI: "c < (0::real) ⟹ trimmed_neg c"
  by (simp add: trimmed_neg_def)

lemma trimmed_pos_hd_coeff: "trimmed_pos (MS (MSLCons (C, e) xs) f) ⟹ trimmed_pos C"
  by simp

lemma lift_trimmed: "trimmed C ⟹ trimmed (MS (MSLCons (C, e) xs) f)"
  by (auto simp: trimmed_ms_aux_def)

lemma lift_trimmed_pos: "trimmed_pos C ⟹ trimmed_pos (MS (MSLCons (C, e) xs) f)"
  by simp

lemma lift_trimmed_neg: "trimmed_neg C ⟹ trimmed_neg (MS (MSLCons (C, e) xs) f)"
  by (simp add: trimmed_neg_def fst_dominant_term_ms_aux_MSLCons trimmed_ms_aux_MSLCons)

lemma lift_trimmed_pos':
  "trimmed_pos C ⟹ MS (MSLCons (C, e) xs) f ≡ MS (MSLCons (C, e) xs) f ⟹
     trimmed_pos (MS (MSLCons (C, e) xs) f)"
  by simp

lemma lift_trimmed_neg':
  "trimmed_neg C ⟹ MS (MSLCons (C, e) xs) f ≡ MS (MSLCons (C, e) xs) f ⟹
     trimmed_neg (MS (MSLCons (C, e) xs) f)"
  by (simp add: lift_trimmed_neg)

lemma trimmed_eq_cong: "trimmed C ⟹ C ≡ C' ⟹ trimmed C'"
  and trimmed_pos_eq_cong: "trimmed_pos C ⟹ C ≡ C' ⟹ trimmed_pos C'"
  and trimmed_neg_eq_cong: "trimmed_neg C ⟹ C ≡ C' ⟹ trimmed_neg C'"
  by simp_all

lemma trimmed_hd: "trimmed (MS (MSLCons (C, e) xs) f) ⟹ trimmed C"
  by (simp add: trimmed_ms_aux_MSLCons)

lemma trim_lift_eq:
  assumes "A ≡ MS (MSLCons (C, e) xs) f" "C ≡ D"
  shows   "A ≡ MS (MSLCons (D, e) xs) f" 
  using assms by simp

lemma basis_wf_manyI: 
  "filterlim b' at_top at_top ⟹ (λx. ln (b x)) ∈ o(λx. ln (b' x)) ⟹
     basis_wf (b # basis) ⟹ basis_wf (b' # b # basis)"
  by (simp_all add: basis_wf_many)

lemma ln_smallo_ln_exp: "(λx. ln (b x)) ∈ o(f) ⟹ (λx. ln (b x)) ∈ o(λx. ln (exp (f x :: real)))"
  by simp


subsection ‹Reification and other technical details›

text ‹
  The following is used by the automation in order to avoid writing terms like $x^2$ or $x^{-2}$
  as term‹λx::real. x powr 2› etc.\ but as the more agreeable term‹λx::real. x ^ 2› or
  term‹λx::real. inverse (x ^ 2)›.
›

lemma intyness_0: "0 ≡ real 0"
  and intyness_1: "1 ≡ real 1"
  and intyness_numeral: "num ≡ num ⟹ numeral num ≡ real (numeral num)"
  and intyness_uminus:  "x ≡ real n ⟹ -x ≡ -real n"
  and intyness_of_nat:  "n ≡ n ⟹ real n ≡ real n"
  by simp_all
    
lemma intyness_simps:
  "real a + real b = real (a + b)"
  "real a * real b = real (a * b)"
  "real a ^ n = real (a ^ n)"
  "1 = real 1" "0 = real 0" "numeral num = real (numeral num)" 
  by simp_all

lemma odd_Numeral1: "odd (Numeral1)"
  by simp

lemma even_addI:
  "even a ⟹ even b ⟹ even (a + b)"
  "odd a ⟹ odd b ⟹ even (a + b)"
  by auto

lemma odd_addI:
  "even a ⟹ odd b ⟹ odd (a + b)"
  "odd a ⟹ even b ⟹ odd (a + b)"
  by auto

lemma even_diffI:
  "even a ⟹ even b ⟹ even (a - b :: int)"
  "odd a ⟹ odd b ⟹ even (a - b)"
  by auto

lemma odd_diffI:
  "even a ⟹ odd b ⟹ odd (a - b :: int)"
  "odd a ⟹ even b ⟹ odd (a - b)"
  by auto

lemma even_multI: "even a ⟹ even (a * b)" "even b ⟹ even (a * b)"
  by auto

lemma odd_multI: "odd a ⟹ odd b ⟹ odd (a * b)"
  by auto

lemma even_uminusI: "even a ⟹ even (-a :: int)"
  and odd_uminusI:  "odd a ⟹ odd (-a :: int)"
  by auto

lemma odd_powerI: "odd a ⟹ odd (a ^ n)"
  by auto


text ‹
  The following theorem collection is used to pre-process functions for multiseries expansion.
›
named_theorems real_asymp_reify_simps

lemmas [real_asymp_reify_simps] =
  sinh_field_def cosh_field_def tanh_real_altdef arsinh_def arcosh_def artanh_def


text ‹
  This is needed in order to handle things like term‹λn. f n ^ n›.
›
definition powr_nat :: "real ⇒ real ⇒ real" where 
  "powr_nat x y = 
     (if y = 0 then 1
      else if x < 0 then cos (pi * y) * (-x) powr y else x powr y)"
  
lemma powr_nat_of_nat: "powr_nat x (of_nat n) = x ^ n"
  by (cases "x > 0") (auto simp: powr_nat_def powr_realpow not_less power_mult_distrib [symmetric])

lemma powr_nat_conv_powr: "x > 0 ⟹ powr_nat x y = x powr y"
  by (simp_all add: powr_nat_def)

lemma reify_power: "x ^ n ≡ powr_nat x (of_nat n)"
  by (simp add: powr_nat_of_nat)


lemma sqrt_conv_root [real_asymp_reify_simps]: "sqrt x = root 2 x"
  by (simp add: sqrt_def)

lemma tan_conv_sin_cos [real_asymp_reify_simps]: "tan x = sin x / cos x"
  by (simp add: tan_def)

definition rfloor :: "real ⇒ real" where "rfloor x = real_of_int (floor x)"
definition rceil :: "real ⇒ real" where "rceil x = real_of_int (ceiling x)"
definition rnatmod :: "real ⇒ real ⇒ real"
  where "rnatmod x y = real (nat ⌊x⌋ mod nat ⌊y⌋)"
definition rintmod :: "real ⇒ real ⇒ real"
  where "rintmod x y = real_of_int (⌊x⌋ mod ⌊y⌋)"

lemmas [real_asymp_reify_simps] =
  ln_exp log_def rfloor_def [symmetric] rceil_def [symmetric]

lemma expands_to_powr_nat_0_0:
  assumes "eventually (λx. f x = 0) at_top" "eventually (λx. g x = 0) at_top"
          "basis_wf basis" "length basis = expansion_level TYPE('a :: multiseries)"
  shows   "((λx. powr_nat (f x) (g x)) expands_to (const_expansion 1 :: 'a)) basis"
proof (rule expands_to_cong [OF expands_to_const])
  from assms(1,2) show "eventually (λx. 1 = powr_nat (f x) (g x)) at_top"
    by eventually_elim (simp add: powr_nat_def)
qed fact+

lemma expands_to_powr_nat_0:
  assumes "eventually (λx. f x = 0) at_top" "(g expands_to G) basis" "trimmed G"
          "basis_wf basis" "length basis = expansion_level TYPE('a :: multiseries)"
  shows   "((λx. powr_nat (f x) (g x)) expands_to (zero_expansion :: 'a)) basis"
proof (rule expands_to_cong [OF expands_to_zero])
  from assms have "eventually (λx. g x ≠ 0) at_top"
    using expands_to_imp_eventually_nz[of basis g G] by auto
  with assms(1) show "eventually (λx. 0 = powr_nat (f x) (g x)) at_top"
    by eventually_elim (simp add: powr_nat_def)
qed (insert assms, auto elim!: eventually_mono simp: powr_nat_def)

lemma expands_to_powr_nat:
  assumes "trimmed_pos F" "basis_wf basis'" "(f expands_to F) basis'"
  assumes "((λx. exp (ln (f x) * g x)) expands_to E) basis"
  shows   "((λx. powr_nat (f x) (g x)) expands_to E) basis"
using assms(4)
proof (rule expands_to_cong)
  from eval_pos_if_dominant_term_pos[of basis' F] assms(1-4)
    show "eventually (λx. exp (ln (f x) * g x) = powr_nat (f x) (g x)) at_top"
    by (auto simp: expands_to.simps trimmed_pos_def powr_def powr_nat_def elim: eventually_elim2)
qed


subsection ‹Some technical lemmas›

lemma landau_meta_eq_cong: "f ∈ L(g) ⟹ f' ≡ f ⟹ g' ≡ g ⟹ f' ∈ L(g')"
  and asymp_equiv_meta_eq_cong: "f ∼[at_top] g ⟹ f' ≡ f ⟹ g' ≡ g ⟹ f' ∼[at_top] g'"
  by simp_all
    
lemma filterlim_mono': "filterlim f F G ⟹ F ≤ F' ⟹ filterlim f F' G"
  by (erule (1) filterlim_mono) simp_all

lemma at_within_le_nhds: "at x within A ≤ nhds x"
  by (simp add: at_within_def)

lemma at_within_le_at: "at x within A ≤ at x"
  by (rule at_le) simp_all

lemma at_right_to_top': "at_right c = filtermap (λx::real. c + inverse x) at_top"
  by (subst at_right_to_0, subst at_right_to_top) (simp add: filtermap_filtermap add_ac)

lemma at_left_to_top': "at_left c = filtermap (λx::real. c - inverse x) at_top"
  by (subst at_left_minus, subst at_right_to_0, subst at_right_to_top) 
     (simp add: filtermap_filtermap add_ac)

lemma at_left_to_top: "at_left 0 = filtermap (λx::real. - inverse x) at_top"
  by (simp add: at_left_to_top')

lemma filterlim_conv_filtermap:
  "G = filtermap g G' ⟹
     PROP (Trueprop (filterlim f F G)) ≡ PROP (Trueprop (filterlim (λx. f (g x)) F G'))"
  by (simp add: filterlim_filtermap)

lemma eventually_conv_filtermap:
  "G = filtermap g G' ⟹ 
     PROP (Trueprop (eventually P G)) ≡ PROP (Trueprop (eventually (λx. P (g x)) G'))"
  by (simp add: eventually_filtermap)

lemma eventually_lt_imp_eventually_le:
  "eventually (λx. f x < (g x :: real)) F ⟹ eventually (λx. f x ≤ g x) F"
  by (erule eventually_mono) simp
    
lemma smallo_imp_smallomega: "f ∈ o[F](g) ⟹ g ∈ ω[F](f)"
  by (simp add: smallomega_iff_smallo)

lemma bigo_imp_bigomega: "f ∈ O[F](g) ⟹ g ∈ Ω[F](f)"
  by (simp add: bigomega_iff_bigo)

context
  fixes L L' :: "real filter ⇒ (real ⇒ _) ⇒ _" and Lr :: "real filter ⇒ (real ⇒ real) ⇒ _"
  assumes LS: "landau_symbol L L' Lr"
begin
  
interpretation landau_symbol L L' Lr by (fact LS)

lemma landau_symbol_at_top_imp_at_bot:
  "(λx. f (-x)) ∈ L' at_top (λx. g (-x)) ⟹ f ∈ L at_bot g"
  by (simp add: in_filtermap_iff at_bot_mirror)

lemma landau_symbol_at_top_imp_at_right_0:
  "(λx. f (inverse x)) ∈ L' at_top (λx. g (inverse x)) ⟹ f ∈ L (at_right 0) g"
  by (simp add: in_filtermap_iff at_right_to_top')
    
lemma landau_symbol_at_top_imp_at_left_0:
  "(λx. f ( -inverse x)) ∈ L' at_top (λx. g (-inverse x)) ⟹ f ∈ L (at_left 0) g"
  by (simp add: in_filtermap_iff at_left_to_top')    

lemma landau_symbol_at_top_imp_at_right:
  "(λx. f (a + inverse x)) ∈ L' at_top (λx. g (a + inverse x)) ⟹ f ∈ L (at_right a) g"
  by (simp add: in_filtermap_iff at_right_to_top')
    
lemma landau_symbol_at_top_imp_at_left:
  "(λx. f (a - inverse x)) ∈ L' at_top (λx. g (a - inverse x)) ⟹ f ∈ L (at_left a) g"
  by (simp add: in_filtermap_iff at_left_to_top')
    
lemma landau_symbol_at_top_imp_at:
  "(λx. f (a - inverse x)) ∈ L' at_top (λx. g (a - inverse x)) ⟹
   (λx. f (a + inverse x)) ∈ L' at_top (λx. g (a + inverse x)) ⟹
   f ∈ L (at a) g"
  by (simp add: sup at_eq_sup_left_right 
        landau_symbol_at_top_imp_at_left landau_symbol_at_top_imp_at_right)
      
lemma landau_symbol_at_top_imp_at_0:
  "(λx. f (-inverse x)) ∈ L' at_top (λx. g (-inverse x)) ⟹
   (λx. f (inverse x)) ∈ L' at_top (λx. g (inverse x)) ⟹
   f ∈ L (at 0) g"
  by (rule landau_symbol_at_top_imp_at) simp_all

end


context
  fixes f g :: "real ⇒ real"
begin

lemma asymp_equiv_at_top_imp_at_bot:
  "(λx. f (- x)) ∼[at_top] (λx. g (-x)) ⟹ f ∼[at_bot] g"
  by (simp add: asymp_equiv_def filterlim_at_bot_mirror)

lemma asymp_equiv_at_top_imp_at_right_0:
  "(λx. f (inverse x)) ∼[at_top] (λx. g (inverse x)) ⟹ f ∼[at_right 0] g"
  by (simp add: at_right_to_top asymp_equiv_filtermap_iff)

lemma asymp_equiv_at_top_imp_at_left_0:
  "(λx. f (-inverse x)) ∼[at_top] (λx. g (-inverse x)) ⟹ f ∼[at_left 0] g"
  by (simp add: at_left_to_top asymp_equiv_filtermap_iff)

lemma asymp_equiv_at_top_imp_at_right:
  "(λx. f (a + inverse x)) ∼[at_top] (λx. g (a + inverse x)) ⟹ f ∼[at_right a] g"
  by (simp add: at_right_to_top' asymp_equiv_filtermap_iff)

lemma asymp_equiv_at_top_imp_at_left:
  "(λx. f (a - inverse x)) ∼[at_top] (λx. g (a - inverse x)) ⟹ f ∼[at_left a] g"
  by (simp add: at_left_to_top' asymp_equiv_filtermap_iff)

lemma asymp_equiv_at_top_imp_at:
  "(λx. f (a - inverse x)) ∼[at_top] (λx. g (a - inverse x)) ⟹
   (λx. f (a + inverse x)) ∼[at_top] (λx. g (a + inverse x)) ⟹ f ∼[at a] g"
  unfolding asymp_equiv_def
  using asymp_equiv_at_top_imp_at_left[of a] asymp_equiv_at_top_imp_at_right[of a]
  by (intro filterlim_split_at) (auto simp: asymp_equiv_def)

lemma asymp_equiv_at_top_imp_at_0:
  "(λx. f (-inverse x)) ∼[at_top] (λx. g (-inverse x)) ⟹
   (λx. f (inverse x)) ∼[at_top] (λx. g (inverse x)) ⟹ f ∼[at 0] g"
  by (rule asymp_equiv_at_top_imp_at) auto

end


lemmas eval_simps =
  eval_const eval_plus eval_minus eval_uminus eval_times eval_inverse eval_divide
  eval_map_ms eval_lift_ms eval_real_def eval_ms.simps

lemma real_eqI: "a - b = 0 ⟹ a = (b::real)"
  by simp

lemma lift_ms_simps:
  "lift_ms (MS xs f) = MS (MSLCons (MS xs f, 0) MSLNil) f"
  "lift_ms (c :: real) = MS (MSLCons (c, 0) MSLNil) (λ_. c)"
  by (simp_all add: lift_ms_def)

lemmas landau_reduce_to_top = 
  landau_symbols [THEN landau_symbol_at_top_imp_at_bot]
  landau_symbols [THEN landau_symbol_at_top_imp_at_left_0]
  landau_symbols [THEN landau_symbol_at_top_imp_at_right_0]
  landau_symbols [THEN landau_symbol_at_top_imp_at_left]
  landau_symbols [THEN landau_symbol_at_top_imp_at_right]
  asymp_equiv_at_top_imp_at_bot
  asymp_equiv_at_top_imp_at_left_0
  asymp_equiv_at_top_imp_at_right_0
  asymp_equiv_at_top_imp_at_left
  asymp_equiv_at_top_imp_at_right

lemmas landau_at_top_imp_at = 
  landau_symbols [THEN landau_symbol_at_top_imp_at_0]
  landau_symbols [THEN landau_symbol_at_top_imp_at]
  asymp_equiv_at_top_imp_at_0
  asymp_equiv_at_top_imp_at


lemma nhds_leI: "x = y ⟹ nhds x ≤ nhds y"
  by simp
    
lemma of_nat_diff_real: "of_nat (a - b) = max (0::real) (of_nat a - of_nat b)"
  and of_nat_max_real: "of_nat (max a b) = max (of_nat a) (of_nat b :: real)"
  and of_nat_min_real: "of_nat (min a b) = min (of_nat a) (of_nat b :: real)"
  and of_int_max_real: "of_int (max c d) = max (of_int c) (of_int d :: real)"
  and of_int_min_real: "of_int (min c d) = min (of_int c) (of_int d :: real)"
  by simp_all

named_theorems real_asymp_nat_reify

lemmas [real_asymp_nat_reify] = 
  of_nat_add of_nat_mult of_nat_diff_real of_nat_max_real of_nat_min_real of_nat_power 
  of_nat_Suc of_nat_numeral

lemma of_nat_div_real [real_asymp_nat_reify]:
  "of_nat (a div b) = max 0 (rfloor (of_nat a / of_nat b))"
  by (simp add: rfloor_def floor_divide_of_nat_eq)

lemma of_nat_mod_real [real_asymp_nat_reify]: "of_nat (a mod b) = rnatmod (of_nat a) (of_nat b)"
  by (simp add: rnatmod_def)

lemma real_nat [real_asymp_nat_reify]: "real (nat a) = max 0 (of_int a)"
  by simp


named_theorems real_asymp_int_reify

lemmas [real_asymp_int_reify] = 
  of_int_add of_int_mult of_int_diff of_int_minus of_int_max_real of_int_min_real
  of_int_power of_int_of_nat_eq of_int_numeral

lemma of_int_div_real [real_asymp_int_reify]:
  "of_int (a div b) = rfloor (of_int a / of_int b)"
  by (simp add: rfloor_def floor_divide_of_int_eq)

named_theorems real_asymp_preproc

lemmas [real_asymp_preproc] =
  of_nat_add of_nat_mult of_nat_diff_real of_nat_max_real of_nat_min_real of_nat_power 
  of_nat_Suc of_nat_numeral
  of_int_add of_int_mult of_int_diff of_int_minus of_int_max_real of_int_min_real
  of_int_power of_int_of_nat_eq of_int_numeral real_nat

lemma of_int_mod_real [real_asymp_int_reify]: "of_int (a mod b) = rintmod (of_int a) (of_int b)"
  by (simp add: rintmod_def)


lemma filterlim_of_nat_at_top:
  "filterlim f F at_top ⟹ filterlim (λx. f (of_nat x :: real)) F at_top"
  by (erule filterlim_compose[OF _filterlim_real_sequentially])

lemma asymp_equiv_real_nat_transfer:
  "f ∼[at_top] g ⟹ (λx. f (of_nat x :: real)) ∼[at_top] (λx. g (of_nat x))"
  unfolding asymp_equiv_def by (rule filterlim_of_nat_at_top)

lemma eventually_nat_real:
  assumes "eventually P (at_top :: real filter)"
  shows   "eventually (λx. P (real x)) (at_top :: nat filter)"
  using assms filterlim_real_sequentially
  unfolding filterlim_def le_filter_def eventually_filtermap by auto

lemmas real_asymp_nat_intros =
  filterlim_of_nat_at_top eventually_nat_real smallo_real_nat_transfer bigo_real_nat_transfer
  smallomega_real_nat_transfer bigomega_real_nat_transfer bigtheta_real_nat_transfer
  asymp_equiv_real_nat_transfer


lemma filterlim_of_int_at_top:
  "filterlim f F at_top ⟹ filterlim (λx. f (of_int x :: real)) F at_top"
  by (erule filterlim_compose[OF _ filterlim_real_of_int_at_top])

(* TODO Move *)
lemma filterlim_real_of_int_at_bot [tendsto_intros]:
  "filterlim real_of_int at_bot at_bot"
  unfolding filterlim_at_bot
proof
  fix C :: real
  show "eventually (λn. real_of_int n ≤ C) at_bot"
    using eventually_le_at_bot[of "⌊C⌋"] by eventually_elim linarith
qed

lemma filterlim_of_int_at_bot:
  "filterlim f F at_bot ⟹ filterlim (λx. f (of_int x :: real)) F at_bot"
  by (erule filterlim_compose[OF _ filterlim_real_of_int_at_bot])

lemma asymp_equiv_real_int_transfer_at_top:
  "f ∼[at_top] g ⟹ (λx. f (of_int x :: real)) ∼[at_top] (λx. g (of_int x))"
  unfolding asymp_equiv_def by (rule filterlim_of_int_at_top)

lemma asymp_equiv_real_int_transfer_at_bot:
  "f ∼[at_bot] g ⟹ (λx. f (of_int x :: real)) ∼[at_bot] (λx. g (of_int x))"
  unfolding asymp_equiv_def by (rule filterlim_of_int_at_bot)

lemma eventually_int_real_at_top:
  assumes "eventually P (at_top :: real filter)"
  shows   "eventually (λx. P (of_int x)) (at_top :: int filter)"
  using assms filterlim_of_int_at_top filterlim_iff filterlim_real_of_int_at_top by blast

lemma eventually_int_real_at_bot:
  assumes "eventually P (at_bot :: real filter)"
  shows   "eventually (λx. P (of_int x)) (at_bot :: int filter)"
  using assms filterlim_of_int_at_bot filterlim_iff filterlim_real_of_int_at_bot by blast


lemmas real_asymp_int_intros =
  filterlim_of_int_at_bot filterlim_of_int_at_top
  landau_symbols[THEN landau_symbol.compose[OF _ _ filterlim_real_of_int_at_top]]
  landau_symbols[THEN landau_symbol.compose[OF _ _ filterlim_real_of_int_at_bot]]
  asymp_equiv_real_int_transfer_at_top asymp_equiv_real_int_transfer_at_bot

(* TODO: Move? *)
lemma tendsto_discrete_iff:
  "filterlim f (nhds (c :: 'a :: discrete_topology)) F ⟷ eventually (λx. f x = c) F"
  by (auto simp: nhds_discrete filterlim_principal)

lemma tendsto_of_nat_iff:
  "filterlim (λn. of_nat (f n)) (nhds (of_nat c :: 'a :: real_normed_div_algebra)) F ⟷
     eventually (λn. f n = c) F"
proof
  assume "((λn. of_nat (f n)) ⤏ (of_nat c :: 'a)) F"
  hence "eventually (λn. ¦real (f n) - real c¦ < 1/2) F"
    by (force simp: tendsto_iff dist_of_nat dest: spec[of _ "1/2"])
  thus "eventually (λn. f n = c) F"
    by eventually_elim (simp add: abs_if split: if_splits)
next
  assume "eventually (λn. f n = c) F"
  hence "eventually (λn. of_nat (f n) = (of_nat c :: 'a)) F"
    by eventually_elim simp