import Mathlib.Tactic

/-! 
Self-contained slice for Chapter 5 (reals via Cauchy sequences) sufficient to typecheck
Chapter5.Real.not_zero_pos. We copy only the necessary pieces from Sections 5.1–5.4 and 4.3.
-/

open Lean Elab Tactic

-- Minimal stand-in for the project-specific `grind` tactic used in proof fields.
elab "grind" : tactic => do
  let t ← `(tactic| intro n; intro hn; try simp [hn.le]; try simp [not_le.mpr hn])
  evalTactic t

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

namespace Chapter5

/-! Section 5.1: Sequences and Cauchy sequences over ℚ -/

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

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

/-- A convenient constructor for sequences starting at an arbitrary `n₀`. -/
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

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

/-- Convert a function `ℕ → ℚ` into a `Sequence` starting at 0 (as a coercion). -/
instance : Coe (ℕ → ℚ) Sequence where
  coe a :=
    { n₀ := 0
      seq := fun n ↦ if n ≥ 0 then a n.toNat else 0
      vanish := by
        intro n hn
        have : ¬ n ≥ 0 := by exact not_le.mpr hn
        simp [this] }

/-- Definition 5.1.3 (ε-steadiness) for general starting points. -/
abbrev Rat.Steady (ε : ℚ) (a : Sequence) : Prop :=
  ∀ n ≥ a.n₀, ∀ m ≥ a.n₀, Rat.Close ε (a n) (a m)

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

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

end Chapter5

/-! Section 5.2: Eventually ε-close sequences and equivalence of sequences -/

abbrev Rat.CloseSeq (ε : ℚ) (a b : Chapter5.Sequence) : Prop :=
  ∀ n, n ≥ a.n₀ → n ≥ b.n₀ → Rat.Close ε (a n) (b n)

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

namespace Chapter5

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

end Chapter5

/-! Section 5.3: The construction of the real numbers -/

namespace Chapter5

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

/-- View a `CauchySequence` as a function `ℕ → ℚ`. -/
instance CauchySequence.instCoeFun : CoeFun CauchySequence (fun _ ↦ ℕ → ℚ) where
  coe a n := a.toSequence (n : ℤ)

/-- A sequence starting at zero that is Cauchy, 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

/-- Setoid on Cauchy sequences given by `Sequence.Equiv`. -/
instance CauchySequence.instSetoid : Setoid CauchySequence where
  r := fun a b ↦ Sequence.Equiv a b
  iseqv :=
    { refl := by sorry
      symm := by sorry
      trans := by sorry }

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

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

/-- The real numbers: reals are Cauchy sequences modulo equivalence. -/
abbrev Real := Quotient CauchySequence.instSetoid

open Classical in
/-- Formal limit of a sequence (dummy value 0 if not Cauchy). -/
noncomputable abbrev LIM (a : ℕ → ℚ) : Real :=
  Quotient.mk _ (if h : (a : Sequence).IsCauchy then CauchySequence.mk' h else (0 : CauchySequence))

/-- Rationals embed into reals via constant Cauchy sequences. -/
instance Real.instRatCast : RatCast Real where
  ratCast := fun q ↦ Quotient.mk _ (CauchySequence.mk' (a := fun _ ↦ q) (Sequence.IsCauchy.const q))

/-- Natural numbers embed into reals via `ℕ ⊆ ℚ ⊆ Real`. -/
instance Real.instOfNat {n : ℕ} : OfNat Real n where ofNat := ((n : ℚ) : Real)

end Chapter5

/-! Section 5.4: Sign on the reals -/

namespace Chapter5

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

/-- Definition 5.4.1 (positive reals via Cauchy sequences bounded away from 0 on the right). -/
abbrev Real.IsPos (x : Real) : Prop :=
  ∃ a : ℕ → ℚ, BoundedAwayPos a ∧ (a : Sequence).IsCauchy ∧ x = LIM a

/-- Unfolding lemma for `IsPos`. -/
lemma Real.isPos_def (x : Real) : IsPos x ↔ ∃ a : ℕ → ℚ, BoundedAwayPos a ∧ (a : Sequence).IsCauchy ∧ x = LIM a := by
  rfl

/-- Proposition 5.4.4 (basic properties of positive reals) / Exercise 5.4.1 -/
 theorem Real.not_zero_pos (x:Real) : ¬(x = 0 ∧ x.IsPos) := by
end Chapter5