import Mathlib

namespace Chapter7

open BigOperators

-- <CONTEXT>
@[ext]
structure Series where
  m : ℤ
  seq : ℤ → ℝ
  vanish : ∀ n < m, seq n = 0

instance Series.instCoe : Coe (ℕ → ℝ) Series where
  coe := fun a ↦ {
    m := 0
    seq n := if n ≥ 0 then a n.toNat else 0
    vanish := by grind
  }

@[simp]
theorem Series.eval_coe (a: ℕ → ℝ) (n: ℕ) : (a: Series).seq n = a n := by
  sorry

abbrev Series.mk' {m:ℤ} (a: { n // n ≥ m } → ℝ) : Series where
  m := m
  seq n := if h : n ≥ m then a ⟨n, h⟩ else 0
  vanish := by grind

theorem Series.eval_mk' {m:ℤ} (a : { n // n ≥ m } → ℝ) {n : ℤ} (h:n ≥ m) :
    (Series.mk' a).seq n = a ⟨ n, h ⟩ := by
  sorry

abbrev Series.partial (s : Series) (N:ℤ) : ℝ := ∑ n ∈ Finset.Icc s.m N, s.seq n

theorem Series.partial_succ (s : Series) {N:ℤ} (h: N ≥ s.m-1) : s.partial (N+1) = s.partial N + s.seq (N+1) := by
  sorry

theorem Series.partial_of_lt {s : Series} {N:ℤ} (h: N < s.m) : s.partial N = 0 := by
  sorry

abbrev Series.convergesTo (s : Series) (L:ℝ) : Prop := Filter.atTop.Tendsto (s.partial) (nhds L)

abbrev Series.converges (s : Series) : Prop := ∃ L, s.convergesTo L

abbrev Series.diverges (s : Series) : Prop := ¬s.converges

open Classical in
noncomputable abbrev Series.sum (s : Series) : ℝ := if h : s.converges then h.choose else 0

theorem Series.converges_of_convergesTo {s : Series} {L:ℝ} (h: s.convergesTo L) :
    s.converges := by
  sorry

theorem Series.sum_of_converges {s : Series} {L:ℝ} (h: s.convergesTo L) : s.sum = L := by
  sorry

theorem Series.convergesTo_uniq {s : Series} {L L':ℝ} (h: s.convergesTo L) (h': s.convergesTo L') :
    L = L' := by
  sorry

theorem Series.convergesTo_sum {s : Series} (h: s.converges) : s.convergesTo s.sum := by
  sorry

noncomputable abbrev Series.example_7_2_4 := mk' (m := 1) (fun n ↦ (2:ℝ)^(-n:ℤ))

theorem Series.example_7_2_4a {N:ℤ} (hN: N ≥ 1) : example_7_2_4.partial N = 1 - (2:ℝ)^(-N) := by
  sorry

theorem Series.example_7_2_4b : example_7_2_4.convergesTo 1 := by
  sorry

theorem Series.example_7_2_4c : example_7_2_4.sum = 1 := by
  sorry

noncomputable abbrev Series.example_7_2_4' := mk' (m := 1) (fun n ↦ (2:ℝ)^(n:ℤ))

theorem Series.example_7_2_4'a {N:ℤ} (hN: N ≥ 1) : example_7_2_4'.partial N = (2:ℝ)^(N+1) - 2 := by
  sorry

theorem Series.example_7_2_4'b : example_7_2_4'.diverges := by
  sorry

theorem Series.diverges_of_nodecay {s:Series} (h: ¬ Filter.atTop.Tendsto s.seq (nhds 0)) :
    s.diverges := by
  sorry

theorem Series.example_7_2_7 : ((fun _:ℕ ↦ (1:ℝ)):Series).diverges := by
  sorry

theorem Series.example_7_2_7' : ((fun n:ℕ ↦ (-1:ℝ)^n):Series).diverges := by
  sorry

abbrev Series.abs (s:Series) : Series := mk' (m:=s.m) (fun n ↦ |s.seq n|)

abbrev Series.absConverges (s:Series) : Prop := s.abs.converges

abbrev Series.condConverges (s:Series) : Prop := s.converges ∧ ¬ s.absConverges

theorem Series.abs_le {s:Series} (h : s.absConverges) : |s.sum| ≤ s.abs.sum := by
  sorry

theorem Series.converges_of_alternating {m:ℤ} {a: { n // n ≥ m} → ℝ} (ha: ∀ n, a n ≥ 0)
  (ha': Antitone a) :
    ((mk' (fun n ↦ (-1)^(n:ℤ) * a n)).converges ↔ Filter.atTop.Tendsto a (nhds 0)) := by
  sorry

noncomputable abbrev Series.example_7_2_13 : Series := (mk' (m:=1) (fun n ↦ (-1:ℝ)^(n:ℤ) / (n:ℤ)))

theorem Series.example_7_2_13a : example_7_2_13.converges := by
  sorry

theorem Series.example_7_2_13b : ¬ example_7_2_13.absConverges := by
  sorry

theorem Series.example_7_2_13c :  example_7_2_13.condConverges := by
  sorry

instance Series.inst_add : Add Series where
  add a b := {
    m := min a.m b.m
    seq n := a.seq n + b.seq n
    vanish n hn := by simp [a.vanish n (by omega), b.vanish n (by omega)]
  }

theorem Series.add_coe (a b: ℕ → ℝ) : (a:Series) + (b:Series) = (fun n ↦ a n + b n) := by
  sorry

theorem Series.add {s t:Series} (hs: s.converges) (ht: t.converges) :
    (s + t).converges ∧ (s+t).sum = s.sum + t.sum := by
  sorry

instance Series.inst.smul : SMul ℝ Series where
  smul c s := {
    m := s.m
    seq n := if n ≥ s.m then c * s.seq n else 0
    vanish := by grind
  }

theorem Series.smul_coe (a: ℕ → ℝ) (c: ℝ) : (c • a:Series) = (fun n ↦ c * a n) := by
  sorry

theorem Series.smul {c:ℝ} {s:Series} (hs: s.converges) :
    (c • s).converges ∧ (c • s).sum = c * s.sum := by
  sorry

instance Series.inst_sub : Sub Series where
  sub a b := {
    m := min a.m b.m
    seq n := a.seq n - b.seq n
    vanish n hn := by simp [a.vanish n (by omega), b.vanish n (by omega)]
  }

theorem Series.sub_coe (a b: ℕ → ℝ) : (a:Series) - (b:Series) = (fun n ↦ a n - b n) := by
  sorry

theorem Series.convergesTo.sub {s t:Series} {L M: ℝ} (hs: s.convergesTo L) (ht: t.convergesTo M) :
    (s - t).convergesTo (L - M) := by
  sorry

theorem Series.sub {s t:Series} (hs: s.converges) (ht: t.converges) :
    (s - t).converges ∧ (s-t).sum = s.sum - t.sum := by
  sorry

abbrev Series.from (s:Series) (m₁:ℤ) : Series := mk' (m := max s.m m₁) (fun n ↦ s.seq (n:ℤ))

theorem Series.sum_from {s:Series} (k:ℕ) (h: s.converges) :
    s.sum = ∑ n ∈ Finset.Ico s.m (s.m+k), s.seq n + (s.from (s.m+k)).sum := by
  sorry

def Series.exercise_7_2_1_convergent :
  Decidable ( (mk' (m := 1) (fun n ↦ (-1:ℝ)^(n:ℤ))).converges ) := by
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Series.telescope {a:ℕ → ℝ} (ha: Filter.atTop.Tendsto a (nhds 0)) :
    ((fun n:ℕ ↦ a n - a (n+1)):Series).convergesTo (a 0) := by
-- </TARGET_THEOREM>

end Chapter7