import Mathlib.Tactic
import Mathlib.Algebra.Group.MinimalAxioms

/-!
Self-contained slice for Chapter 5 Section 5.4 target theorem Real.rat_between.
We copy only the minimal transitive dependency needed for parsing/typechecking.
-/

/-- From Section 4.3: definition of ε-closeness on ℚ -/
def Rat.Close (ε : ℚ) (x y : ℚ) := |x - y| ≤ ε

namespace Chapter5

/-- Definition 5.1.1 (Sequence). -/
@[ext]
structure Sequence where
  n₀ : ℤ
  seq : ℤ → ℚ
  vanish : ∀ n < n₀, seq n = 0

/-- Sequences can be thought of as functions from ℤ to ℚ. -/
instance Sequence.instCoeFun : CoeFun Sequence (fun _ ↦ ℤ → ℚ) where
  coe := fun a ↦ a.seq

/--
Functions from ℕ to ℚ can be thought of as sequences starting from 0; `ofNatFun` performs this conversion.

The `coe` attribute allows the delaborator to print `Sequence.ofNatFun f` as `↑f`, which is more concise; you may safely remove this if you prefer the more explicit notation.
-/
@[coe]
def Sequence.ofNatFun (a : ℕ → ℚ) : Sequence where
    n₀ := 0
    seq n := if n ≥ 0 then a n.toNat else 0
    vanish := by
      grind

/-- If `a : ℕ → ℚ` is used in a context where a `Sequence` is expected, automatically coerce. -/
instance : Coe (ℕ → ℚ) Sequence where
  coe := Sequence.ofNatFun

abbrev Sequence.mk' (n₀:ℤ) (a: { n // n ≥ n₀ } → ℚ) : Sequence where
  n₀ := n₀
  seq n := if h : n ≥ n₀ then a ⟨n, h⟩ else 0
  vanish := by
    grind

@[simp]
lemma Sequence.eval_coe (n:ℕ) (a: ℕ → ℚ) : (a:Sequence) n = a n := by norm_cast

@[simp]
lemma Sequence.eval_coe_at_int (n:ℤ) (a: ℕ → ℚ) : (a:Sequence) n = if n ≥ 0 then a n.toNat else 0 := by
  norm_cast

@[simp]
lemma Sequence.n0_coe (a: ℕ → ℚ) : (a:Sequence).n₀ = 0 := by norm_cast

/-- start a sequence from index n₁ (junk if n₁ < n₀) -/
abbrev Sequence.from (a:Sequence) (n₁:ℤ) : Sequence :=
  mk' (max a.n₀ n₁) (fun n ↦ a (n:ℤ))

end Chapter5

/-- Definition 5.1.3 (ε-steadiness), generalized to arbitrary starting point. -/
abbrev Rat.Steady (ε: ℚ) (a: Chapter5.Sequence) : Prop :=
  ∀ n ≥ a.n₀, ∀ m ≥ a.n₀, ε.Close (a n) (a m)

/-- Definition 5.1.6 (Eventually ε-steady). -/
abbrev Rat.EventuallySteady (ε: ℚ) (a: Chapter5.Sequence) : Prop := ∃ N ≥ a.n₀, ε.Steady (a.from N)

/-- From Section 5.2: ε-close and eventually ε-close sequences -/
abbrev Rat.CloseSeq (ε: ℚ) (a b: Chapter5.Sequence) : Prop :=
  ∀ n, n ≥ a.n₀ → n ≥ b.n₀ → ε.Close (a n) (b n)

abbrev Rat.EventuallyClose (ε: ℚ) (a b: Chapter5.Sequence) : Prop :=
  ∃ N, ε.CloseSeq (a.from N) (b.from N)

namespace Chapter5

/-- Cauchy sequences. -/
abbrev Sequence.IsCauchy (a:Sequence) : Prop := ∀ ε > (0:ℚ), ε.EventuallySteady a

/-- Definition 5.2.6 (Equivalent sequences) -/
abbrev Sequence.Equiv (a b: ℕ → ℚ) : Prop :=
  ∀ ε > (0:ℚ), ε.EventuallyClose (a:Sequence) (b:Sequence)

/-- A class of Cauchy sequences that start at zero -/
@[ext]
class CauchySequence extends Sequence where
  zero : n₀ = 0
  cauchy : toSequence.IsCauchy

/-- A sequence starting at zero that is Cauchy, can be viewed as a Cauchy sequence.-/
abbrev CauchySequence.mk' {a:ℕ → ℚ} (ha: (a:Sequence).IsCauchy) : CauchySequence where
  n₀ := 0
  seq := (a:Sequence).seq
  vanish := by aesop
  zero := rfl
  cauchy := ha

@[simp]
theorem CauchySequence.coe_eq {a:ℕ → ℚ} (ha: (a:Sequence).IsCauchy) :
    (mk' ha).toSequence = (a:Sequence) := rfl

instance CauchySequence.instCoeFun : CoeFun CauchySequence (fun _ ↦ ℕ → ℚ) where
  coe a n := a.toSequence (n:ℤ)

@[simp]
theorem CauchySequence.coe_to_sequence (a: CauchySequence) :
    ((a:ℕ → ℚ):Sequence) = a.toSequence := by
  apply Sequence.ext (by simp [Sequence.n0_coe, a.zero])
  ext n; by_cases h:n ≥ 0 <;> simp_all
  rw [a.vanish]; rwa [a.zero]

@[simp]
theorem CauchySequence.coe_coe {a:ℕ → ℚ} (ha: (a:Sequence).IsCauchy) : mk' ha = a := by rfl

/-- Every constant sequence is Cauchy -/
lemma Sequence.IsCauchy.const (a:ℚ) : ((fun _:ℕ ↦ a):Sequence).IsCauchy := by
  sorry

/-- Proposition 5.3.3 / Exercise 5.3.1 -/
lemma Sequence.equiv_trans {a b c:ℕ → ℚ} (hab: Equiv a b) (hbc: Equiv b c) :
  Equiv a c := by
  sorry

/-- Proposition 5.3.3 / Exercise 5.3.1 -/
instance CauchySequence.instSetoid : Setoid CauchySequence where
  r := fun a b ↦ Sequence.Equiv a b
  iseqv := {
     refl := by
       sorry
     symm := by
       sorry
     trans := by
       sorry
  }

lemma CauchySequence.equiv_iff (a b: CauchySequence) : a ≈ b ↔ Sequence.Equiv a b := by rfl

instance CauchySequence.instZero : Zero CauchySequence where
  zero := CauchySequence.mk' (a := fun _: ℕ ↦ 0) (Sequence.IsCauchy.const (0:ℚ))

abbrev Real := Quotient CauchySequence.instSetoid

open Classical in
/--
  It is convenient in Lean to assign the "dummy" value of 0 to `LIM a` when `a` is not Cauchy.
  This requires Classical logic, because the property of being Cauchy is not computable or
  decidable.
-/
noncomputable abbrev LIM (a:ℕ → ℚ) : Real :=
  Quotient.mk _ (if h : (a:Sequence).IsCauchy then CauchySequence.mk' h else (0:CauchySequence))

/-- Definition 5.3.1 (Real numbers) -/
lemma Real.LIM_eq_LIM {a b:ℕ → ℚ} (ha: (a:Sequence).IsCauchy) (hb: (b:Sequence).IsCauchy) :
  LIM a = LIM b ↔ Sequence.Equiv a b := by
  constructor
  . intro h; replace h := Quotient.exact h
    rwa [dif_pos ha, dif_pos hb, CauchySequence.equiv_iff] at h
  intro h; apply Quotient.sound
  rwa [dif_pos ha, dif_pos hb, CauchySequence.equiv_iff]

/--Lemma 5.3.6 (Sum of Cauchy sequences is Cauchy)-/
lemma Sequence.IsCauchy.add {a b:ℕ → ℚ}  (ha: (a:Sequence).IsCauchy) (hb: (b:Sequence).IsCauchy) :
    (a + b:Sequence).IsCauchy := by
  sorry

/--Lemma 5.3.7 (Sum of equivalent sequences is equivalent)-/
lemma Sequence.add_equiv_left {a a':ℕ → ℚ} (b:ℕ → ℚ) (haa': Equiv a a') :
    Equiv (a + b) (a' + b) := by
  sorry

/--Lemma 5.3.7 (Sum of equivalent sequences is equivalent)-/
lemma Sequence.add_equiv_right {b b':ℕ → ℚ} (a:ℕ → ℚ) (hbb': Equiv b b') :
    Equiv (a + b) (a + b') := by simp_rw [add_comm]; exact add_equiv_left _ hbb'

/--Lemma 5.3.7 (Sum of equivalent sequences is equivalent)-/
lemma Sequence.add_equiv {a b a' b':ℕ → ℚ} (haa': Equiv a a')
  (hbb': Equiv b b') :
    Equiv (a + b) (a' + b') :=
  equiv_trans (add_equiv_left _ haa') (add_equiv_right _ hbb')

/-- Definition 5.3.4 (Addition of reals) -/
noncomputable instance Real.add_inst : Add Real where
  add := fun x y ↦
    Quotient.liftOn₂ x y (fun a b ↦ LIM (a + b)) (by
      intro a b a' b' _ _
      change LIM ((a:ℕ → ℚ) + (b:ℕ → ℚ)) = LIM ((a':ℕ → ℚ) + (b':ℕ → ℚ))
      rw [LIM_eq_LIM]
      . solve_by_elim [Sequence.add_equiv]
      all_goals apply Sequence.IsCauchy.add <;> rw [CauchySequence.coe_to_sequence] <;> convert @CauchySequence.cauchy ?_
      )

/-- Proposition 5.3.10 (Product of Cauchy sequences is Cauchy) -/
lemma Sequence.IsCauchy.mul {a b:ℕ → ℚ}  (ha: (a:Sequence).IsCauchy) (hb: (b:Sequence).IsCauchy) :
    (a * b:Sequence).IsCauchy := by
  sorry

/-- Proposition 5.3.10 (Product of equivalent sequences is equivalent) / Exercise 5.3.2 -/
lemma Sequence.mul_equiv_left {a a':ℕ → ℚ} (b:ℕ → ℚ) (hb : (b:Sequence).IsCauchy) (haa': Equiv a a') :
  Equiv (a * b) (a' * b) := by
  sorry

/--Proposition 5.3.10 (Product of equivalent sequences is equivalent) / Exercise 5.3.2 -/
lemma Sequence.mul_equiv_right {b b':ℕ → ℚ} (a:ℕ → ℚ)  (ha : (a:Sequence).IsCauchy)  (hbb': Equiv b b') :
  Equiv (a * b) (a * b') := by simp_rw [mul_comm]; exact mul_equiv_left a ha hbb'

/--Proposition 5.3.10 (Product of equivalent sequences is equivalent) / Exercise 5.3.2 -/
lemma Sequence.mul_equiv
  {a b a' b':ℕ → ℚ}
  (ha : (a:Sequence).IsCauchy)
  (hb' : (b':Sequence).IsCauchy)
  (haa': Equiv a a')
  (hbb': Equiv b b') : Equiv (a * b) (a' * b') :=
    equiv_trans (mul_equiv_right _ ha hbb') (mul_equiv_left _ hb' haa')

/-- Definition 5.3.9 (Product of reals) -/
noncomputable instance Real.mul_inst : Mul Real where
  mul := fun x y ↦
    Quotient.liftOn₂ x y (fun a b ↦ LIM (a * b)) (by
      intro a b a' b' haa' hbb'
      change LIM ((a:ℕ → ℚ) * (b:ℕ → ℚ)) = LIM ((a':ℕ → ℚ) * (b':ℕ → ℚ))
      rw [LIM_eq_LIM]
      . exact Sequence.mul_equiv (by rw [CauchySequence.coe_to_sequence]; exact a.cauchy) (by rw [CauchySequence.coe_to_sequence]; exact b'.cauchy) haa' hbb'
      all_goals apply Sequence.IsCauchy.mul <;> rw [CauchySequence.coe_to_sequence] <;> convert @CauchySequence.cauchy ?_
      )

instance Real.instRatCast : RatCast Real where
  ratCast := fun q ↦
    Quotient.mk _ (CauchySequence.mk' (a := fun _ ↦ q) (Sequence.IsCauchy.const q))

/-- OfNat instance for Real via ℚ -/
instance Real.instOfNat {n:ℕ} : OfNat Real n where
  ofNat := ((n:ℚ):Real)

noncomputable instance Real.instNeg : Neg Real where
  neg x := ((-1:ℚ):Real) * x

/-- Proposition 5.3.11 (laws of algebra) -/
noncomputable instance Real.addGroup_inst : AddGroup Real :=
  AddGroup.ofLeftAxioms (by sorry) (by sorry) (by sorry)

/-- Definition 5.4.1 (sequences bounded away from zero with sign). -/
abbrev BoundedAwayNeg (a:ℕ → ℚ) : Prop :=
  ∃ (c:ℚ), c > 0 ∧ ∀ n, a n ≤ -c

/-- Definition 5.4.1 (sequences bounded away from zero with sign). -/
abbrev Real.IsNeg (x:Real) : Prop :=
  ∃ a:ℕ → ℚ, BoundedAwayNeg a ∧ (a:Sequence).IsCauchy ∧ x = LIM a

/-- Definition 5.4.6 (Ordering of the reals) -/
instance Real.instLT : LT Real where
  lt x y := (x-y).IsNeg

/-- Proposition 5.4.14 / Exercise 5.4.5 -/
 theorem Real.rat_between {x y:Real} (hxy: x < y) : ∃ q:ℚ, x < (q:Real) ∧ (q:Real) < y := by
end Chapter5