import Mathlib
import Analysis.Section_7_2
import Analysis.Tags.Section_7_2
import Analysis.Section_7_3
import Analysis.Tags.Section_7_3
import Analysis.Section_7_4
import Analysis.Tags.Section_7_4
import Analysis.Section_8_1
import Analysis.Tags.Section_8_1

namespace Chapter7

@[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
      intro n hn
      simp [Int.not_le.mpr hn]
  }

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
    intro n hn
    simp [Int.not_le.mpr hn]

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

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

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

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

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

end Chapter7

namespace Chapter8

abbrev EqualCard (X Y : Type) : Prop := ∃ f : X → Y, Function.Bijective f

abbrev CountablyInfinite (X : Type) : Prop := EqualCard X ℕ

abbrev AtMostCountable (X : Type) : Prop := CountablyInfinite X ∨ Finite X

open Chapter7 Chapter7.Series Finset Function Filter

-- <CONTEXT>
abbrev AbsConvergent {X:Type} (f: X → ℝ) : Prop := ∃ g: ℕ → X, Bijective g ∧ (f ∘ g: Series).absConverges

theorem AbsConvergent.mk {X: Type} {f:X → ℝ} {g:ℕ → X} (h: Bijective g) (hfg: (f ∘ g:Series).absConverges) : AbsConvergent f := by
  sorry

open Classical in
noncomputable abbrev Sum {X:Type} (f: X → ℝ) : ℝ := if h: AbsConvergent f then (f ∘ h.choose:Series).sum else
  if _hX: Finite X then (∑ x ∈ @univ X (Fintype.ofFinite X), f x) else 0

theorem Sum.of_finite {X:Type} [hX:Finite X] (f:X → ℝ) : Sum f = ∑ x ∈ @Finset.univ X (Fintype.ofFinite X), f x := by
  sorry

theorem AbsConvergent.comp {X: Type} {f:X → ℝ} {g:ℕ → X} (h: Bijective g) (hf: AbsConvergent f) : (f ∘ g:Series).absConverges := by
  sorry

theorem Sum.eq {X: Type} {f:X → ℝ} {g:ℕ → X} (h: Bijective g) (hfg: (f ∘ g:Series).absConverges) : (f ∘ g:Series).convergesTo (Sum f) := by
  sorry

theorem Sum.of_comp {X Y:Type} {f:X → ℝ} (h: AbsConvergent f) {g: Y → X} (hbij: Bijective g) : AbsConvergent (f ∘ g) ∧ Sum f = Sum (f ∘ g) := by
  sorry

@[simp]
theorem Finset.Icc_eq_cast (N:ℕ) : Icc 0 (N:ℤ) = map Nat.castEmbedding (.Icc 0 N) := by
  sorry

theorem Finset.Icc_empty {N:ℤ} (h: ¬ N ≥ 0) : Icc 0 N = ∅ := by
  sorry

theorem sum_of_sum_of_AbsConvergent_nonneg {f:ℕ × ℕ → ℝ} (hf:AbsConvergent f) (hpos: ∀ n m, 0 ≤ f (n, m)) :
  (∀ n, ((fun m ↦ f (n, m)):Series).converges) ∧
  (fun n ↦ ((fun m ↦ f (n, m)):Series).sum:Series).convergesTo (Sum f) := by
  sorry

theorem sum_of_sum_of_AbsConvergent {f:ℕ × ℕ → ℝ} (hf:AbsConvergent f) :
  (∀ n, ((fun m ↦ f (n, m)):Series).absConverges) ∧
  (fun n ↦ ((fun m ↦ f (n, m)):Series).sum:Series).convergesTo (Sum f) := by
  sorry

theorem sum_of_sum_of_AbsConvergent' {f:ℕ × ℕ → ℝ} (hf:AbsConvergent f) :
  (∀ m, ((fun n ↦ f (n, m)):Series).absConverges) ∧
  (fun m ↦ ((fun n ↦ f (n, m)):Series).sum:Series).convergesTo (Sum f) := by
  sorry

theorem sum_comm {f:ℕ × ℕ → ℝ} (hf:AbsConvergent f) :
  (fun n ↦ ((fun m ↦ f (n, m)):Series).sum:Series).sum = (fun m ↦ ((fun n ↦ f (n, m)):Series).sum:Series).sum := by
  sorry

abbrev AbsConvergent' {X:Type} (f: X → ℝ) : Prop := BddAbove ( (fun A ↦ ∑ x ∈ A, |f x|) '' .univ )

theorem AbsConvergent'.of_finite {X:Type} [Finite X] (f:X → ℝ) : AbsConvergent' f := by
  sorry

theorem AbsConvergent'.of_countable {X:Type} (hX:CountablyInfinite X) {f:X → ℝ} :
  AbsConvergent' f ↔ AbsConvergent f := by
  sorry

theorem AbsConvergent'.subtype {X:Type} {f:X → ℝ} (hf: AbsConvergent' f) (A: Set X) :
  AbsConvergent' (fun x:A ↦ f x) := by
  sorry

noncomputable abbrev Sum' {X:Type} (f: X → ℝ) : ℝ := Sum (fun x : { x | f x ≠ 0 } ↦ f x)

theorem Sum'.of_finsupp {X:Type} {f:X → ℝ} {A: Finset X} (h: ∀ x ∉ A, f x = 0) : Sum' f = ∑ x ∈ A, f x := by
  sorry

theorem Sum'.of_countable_supp {X:Type} {f:X → ℝ} {A: Set X} (hA: CountablyInfinite A)
  (hfA : ∀ x ∉ A, f x = 0) (hconv: AbsConvergent' f):
  AbsConvergent' (fun x:A ↦ f x) ∧ Sum' f = Sum (fun x:A ↦ f x) := by
  sorry

theorem AbsConvergent'.iff_Summable {X:Type} (f:X → ℝ) : AbsConvergent' f ↔ Summable f := by
  sorry

theorem Filter.Eventually.int_natCast_atTop (p: ℤ → Prop) :
  (∀ᶠ n in .atTop, p n) ↔ ∀ᶠ n:ℕ in .atTop, p ↑n := by
  sorry

theorem Filter.Tendsto.int_natCast_atTop {R:Type} (f: ℤ → R) (l: Filter R) :
atTop.Tendsto f l ↔ atTop.Tendsto (f ∘ Nat.cast) l := by
  sorry

theorem Sum'.eq_tsum {X:Type} (f:X → ℝ) (h: AbsConvergent' f) :
  Sum' f = ∑' x, f x := by
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Sum'.smul {X:Type} {f:X → ℝ} (hf: AbsConvergent' f) (c: ℝ) :
  AbsConvergent' (c • f) ∧ Sum' (c • f) = c * Sum' f := by
-- </TARGET_THEOREM>

end Chapter8