awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.DenseSubgroup (goal)
Goal:
a b : ℝ
hb : b ≠ 0
h : ¬Dense (↑(AddSubgroup.closure {a, b}) : Set ℝ)
c : ℝ
hc : Eq (α := AddSubgroup ℝ) (AddSubgroup.closure {a, b}) (AddSubgroup.closure {c})
this : Subset (α := Set ℝ) {a, b} (range fun (x : ℤ) ↦ x • c)
⊢ ¬Irrational (a / b)
have inr : ∀ (m n : ℤ) (hb : Ne (α := ℝ) ((fun (x : ℤ) ↦ x • c) n) 0)
  (h : ¬Dense (↑(AddSubgroup.closure {(fun (x : ℤ) ↦ x • c) m, (fun (x : ℤ) ↦ x • c) n}) : Set ℝ))
  (hc :
    Eq (α := AddSubgroup ℝ) (AddSubgroup.closure {(fun (x : ℤ) ↦ x • c) m, (fun (x : ℤ) ↦ x • c) n})
      (AddSubgroup.closure {c}))
  (this : Subset (α := Set ℝ) {(fun (x : ℤ) ↦ x • c) m, (fun (x : ℤ) ↦ x • c) n} (range fun (x : ℤ) ↦ x • c)),
  ¬Irrational (HDiv.hDiv (α := ℝ) ((fun (x : ℤ) ↦ x • c) m) ((fun (x : ℤ) ↦ x • c) n))
Tactic: rcases pair_subset_iff.1 this with ⟨⟨m, rfl⟩, n, rfl⟩
Cleared hypotheses: a, b, hb, h, hc, this
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.DenseSubgroup (goal)
Goal:
a p : ℝ
⊢ (DenseRange (X := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) fun (x : ℤ) ↦
      HSMul.hSMul (β := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) x
        (↑a : ℝ ⧸ AddSubgroup.zmultiples p)) ↔
    Irrational (a / p)
have rw : (DenseRange (X := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) fun (x : ℤ) ↦
    HSMul.hSMul (β := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) x
      (↑a : ℝ ⧸ AddSubgroup.zmultiples p)) ↔
  Dense (↑(AddSubgroup.closure {a, p}) : Set ℝ)
Tactic: rw [← dense_addSubgroupClosure_pair_iff, DenseRange,
    ← QuotientAddGroup.dense_preimage_mk, ← QuotientAddGroup.coe_mk',
    ← AddSubgroup.coe_zmultiples, ← AddSubgroup.coe_comap, ← AddMonoidHom.map_zmultiples,
    AddSubgroup.comap_map_eq, QuotientAddGroup.ker_mk', AddSubgroup.zmultiples_eq_closure,
    AddSubgroup.zmultiples_eq_closure, ← AddSubgroup.closure_union, insert_eq]
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.DenseSubgroup (goal)
Goal:
a p : ℝ
rw :
  (DenseRange (X := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) fun (x : ℤ) ↦
      HSMul.hSMul (β := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) x
        (↑a : ℝ ⧸ AddSubgroup.zmultiples p)) ↔
    Dense (↑(AddSubgroup.closure {a, p}) : Set ℝ)
⊢ (DenseRange (X := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) fun (x : ℤ) ↦
      HSMul.hSMul (β := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) x
        (↑a : ℝ ⧸ AddSubgroup.zmultiples p)) ↔
    Irrational (a / p)
have rw₁ : Dense (X := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p))
    (range fun (x : ℤ) ↦
      HSMul.hSMul (β := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) x
        (↑a : ℝ ⧸ AddSubgroup.zmultiples p)) ↔
  Dense (↑(AddSubgroup.closure {a, p}) : Set ℝ)
Tactic: rw [← dense_addSubgroupClosure_pair_iff, DenseRange,
    ← QuotientAddGroup.dense_preimage_mk, ← QuotientAddGroup.coe_mk',
    ← AddSubgroup.coe_zmultiples, ← AddSubgroup.coe_comap, ← AddMonoidHom.map_zmultiples,
    AddSubgroup.comap_map_eq, QuotientAddGroup.ker_mk', AddSubgroup.zmultiples_eq_closure,
    AddSubgroup.zmultiples_eq_closure, ← AddSubgroup.closure_union, insert_eq]
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.DenseSubgroup (goal)
Goal:
a p : ℝ
rw :
  (DenseRange (X := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) fun (x : ℤ) ↦
      HSMul.hSMul (β := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) x
        (↑a : ℝ ⧸ AddSubgroup.zmultiples p)) ↔
    Dense (↑(AddSubgroup.closure {a, p}) : Set ℝ)
rw₁ :
  Dense (X := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p))
      (range fun (x : ℤ) ↦
        HSMul.hSMul (β := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) x
          (↑a : ℝ ⧸ AddSubgroup.zmultiples p)) ↔
    Dense (↑(AddSubgroup.closure {a, p}) : Set ℝ)
⊢ (DenseRange (X := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) fun (x : ℤ) ↦
      HSMul.hSMul (β := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) x
        (↑a : ℝ ⧸ AddSubgroup.zmultiples p)) ↔
    Irrational (a / p)
have rw₂ : Dense (X := ℝ)
    (preimage (β := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) QuotientAddGroup.mk
      (range fun (x : ℤ) ↦
        HSMul.hSMul (β := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) x
          (↑a : ℝ ⧸ AddSubgroup.zmultiples p))) ↔
  Dense (↑(AddSubgroup.closure {a, p}) : Set ℝ)
Tactic: rw [← dense_addSubgroupClosure_pair_iff, DenseRange,
    ← QuotientAddGroup.dense_preimage_mk, ← QuotientAddGroup.coe_mk',
    ← AddSubgroup.coe_zmultiples, ← AddSubgroup.coe_comap, ← AddMonoidHom.map_zmultiples,
    AddSubgroup.comap_map_eq, QuotientAddGroup.ker_mk', AddSubgroup.zmultiples_eq_closure,
    AddSubgroup.zmultiples_eq_closure, ← AddSubgroup.closure_union, insert_eq]
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.DenseSubgroup (goal)
Goal:
a p : ℝ
rw :
  (DenseRange (X := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) fun (x : ℤ) ↦
      HSMul.hSMul (β := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) x
        (↑a : ℝ ⧸ AddSubgroup.zmultiples p)) ↔
    Dense (↑(AddSubgroup.closure {a, p}) : Set ℝ)
rw₁ :
  Dense (X := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p))
      (range fun (x : ℤ) ↦
        HSMul.hSMul (β := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) x
          (↑a : ℝ ⧸ AddSubgroup.zmultiples p)) ↔
    Dense (↑(AddSubgroup.closure {a, p}) : Set ℝ)
rw₂ :
  Dense (X := ℝ)
      (preimage (β := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) QuotientAddGroup.mk
        (range fun (x : ℤ) ↦
          HSMul.hSMul (β := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) x
            (↑a : ℝ ⧸ AddSubgroup.zmultiples p))) ↔
    Dense (↑(AddSubgroup.closure {a, p}) : Set ℝ)
⊢ (DenseRange (X := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) fun (x : ℤ) ↦
      HSMul.hSMul (β := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) x
        (↑a : ℝ ⧸ AddSubgroup.zmultiples p)) ↔
    Irrational (a / p)
have rw₃ : Dense (X := ℝ)
    (preimage (β := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p))
      (⇑(QuotientAddGroup.mk' (AddSubgroup.zmultiples p)) : (a : ℝ) → ℝ ⧸ AddSubgroup.zmultiples p)
      (range fun (x : ℤ) ↦
        HSMul.hSMul (β := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) x
          ((QuotientAddGroup.mk' (AddSubgroup.zmultiples p) : (a : ℝ) → ℝ ⧸ AddSubgroup.zmultiples p) a :
            HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)))) ↔
  Dense (↑(AddSubgroup.closure {a, p}) : Set ℝ)
Tactic: rw [← dense_addSubgroupClosure_pair_iff, DenseRange,
    ← QuotientAddGroup.dense_preimage_mk, ← QuotientAddGroup.coe_mk',
    ← AddSubgroup.coe_zmultiples, ← AddSubgroup.coe_comap, ← AddMonoidHom.map_zmultiples,
    AddSubgroup.comap_map_eq, QuotientAddGroup.ker_mk', AddSubgroup.zmultiples_eq_closure,
    AddSubgroup.zmultiples_eq_closure, ← AddSubgroup.closure_union, insert_eq]
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.DenseSubgroup (goal)
Goal:
a p : ℝ
rw :
  (DenseRange (X := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) fun (x : ℤ) ↦
      HSMul.hSMul (β := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) x
        (↑a : ℝ ⧸ AddSubgroup.zmultiples p)) ↔
    Dense (↑(AddSubgroup.closure {a, p}) : Set ℝ)
rw₁ :
  Dense (X := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p))
      (range fun (x : ℤ) ↦
        HSMul.hSMul (β := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) x
          (↑a : ℝ ⧸ AddSubgroup.zmultiples p)) ↔
    Dense (↑(AddSubgroup.closure {a, p}) : Set ℝ)
rw₂ :
  Dense (X := ℝ)
      (preimage (β := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) QuotientAddGroup.mk
        (range fun (x : ℤ) ↦
          HSMul.hSMul (β := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) x
            (↑a : ℝ ⧸ AddSubgroup.zmultiples p))) ↔
    Dense (↑(AddSubgroup.closure {a, p}) : Set ℝ)
rw₃ :
  Dense (X := ℝ)
      (preimage (β := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p))
        (⇑(QuotientAddGroup.mk' (AddSubgroup.zmultiples p)) : (a : ℝ) → ℝ ⧸ AddSubgroup.zmultiples p)
        (range fun (x : ℤ) ↦
          HSMul.hSMul (β := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) x
            ((QuotientAddGroup.mk' (AddSubgroup.zmultiples p) : (a : ℝ) → ℝ ⧸ AddSubgroup.zmultiples p) a :
              HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)))) ↔
    Dense (↑(AddSubgroup.closure {a, p}) : Set ℝ)
⊢ (DenseRange (X := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) fun (x : ℤ) ↦
      HSMul.hSMul (β := HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p)) x
        (↑a : ℝ ⧸ AddSubgroup.zmultiples p)) ↔
    Irrational (a / p)
have rw₄ : Dense (X := ℝ)
    ((⇑(QuotientAddGroup.mk' (AddSubgroup.zmultiples p)) : (a : ℝ) → ℝ ⧸ AddSubgroup.zmultiples p) ⁻¹'
      (↑(AddSubgroup.zmultiples
              ((QuotientAddGroup.mk' (AddSubgroup.zmultiples p) : (a : ℝ) → ℝ ⧸ AddSubgroup.zmultiples p) a) :
            AddSubgroup (HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (AddSubgroup.zmultiples p))) :
        Set (ℝ ⧸ AddSubgroup.zmultiples p))) ↔
  Dense (↑(AddSubgroup.closure {a, p}) : Set ℝ)
Tactic: rw [← dense_addSubgroupClosure_pair_iff, DenseRange,
    ← QuotientAddGroup.dense_preimage_mk, ← QuotientAddGroup.coe_mk',
    ← AddSubgroup.coe_zmultiples, ← AddSubgroup.coe_comap, ← AddMonoidHom.map_zmultiples,
    AddSubgroup.comap_map_eq, QuotientAddGroup.ker_mk', AddSubgroup.zmultiples_eq_closure,
    AddSubgroup.zmultiples_eq_closure, ← AddSubgroup.closure_union, insert_eq]
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.DenseSubgroup (goal)
Goal:
p : ℝ
inst : Fact (0 < p)
a : AddCircle p
⊢ (DenseRange (X := AddCircle p) fun (x : ℤ) ↦ x • a) ↔ addOrderOf a = 0
have rcases : ∀ (a : ℝ),
  (DenseRange (X := AddCircle p) fun (x : ℤ) ↦ HSMul.hSMul (β := AddCircle p) x (↑a : ℝ ⧸ AddSubgroup.zmultiples p)) ↔
    addOrderOf (G := AddCircle p) (↑a : ℝ ⧸ AddSubgroup.zmultiples p) = 0
Tactic: rcases QuotientAddGroup.mk_surjective a with ⟨a, rfl⟩
Cleared hypotheses: a
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.DenseSubgroup (goal)
Goal:
p : ℝ
inst : Fact (0 < p)
s : AddSubgroup (AddCircle p)
⊢ Dense (↑s : Set (AddCircle p)) ↔ ∀ (a : AddCircle p), addOrderOf a ≠ 0 → s ≠ AddSubgroup.zmultiples a
have mp : Dense (↑s : Set (AddCircle p)) → ∀ (a : AddCircle p), addOrderOf a ≠ 0 → s ≠ AddSubgroup.zmultiples a
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.DenseSubgroup (goal)
Goal:
p : ℝ
inst : Fact (0 < p)
s : AddSubgroup (AddCircle p)
mp : Dense (↑s : Set (AddCircle p)) → ∀ (a : AddCircle p), addOrderOf a ≠ 0 → s ≠ AddSubgroup.zmultiples a
⊢ Dense (↑s : Set (AddCircle p)) ↔ ∀ (a : AddCircle p), addOrderOf a ≠ 0 → s ≠ AddSubgroup.zmultiples a
have mpr : (∀ (a : AddCircle p), addOrderOf a ≠ 0 → s ≠ AddSubgroup.zmultiples a) → Dense (↑s : Set (AddCircle p))
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.DenseSubgroup (hypothesis)
Goal:
p : ℝ
inst : Fact (0 < p)
a : AddCircle p
ha : addOrderOf a ≠ 0
hd : Dense (↑(AddSubgroup.zmultiples a) : Set (AddCircle p))
⊢ False
have hd₁ : addOrderOf a = 0
Tactic: rw [AddSubgroup.coe_zmultiples, ← DenseRange, denseRange_zsmul_iff] at hd
Cleared hypotheses: hd
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.DenseSubgroup (goal)
Goal:
p : ℝ
inst : Fact (0 < p)
s : AddSubgroup (AddCircle p)
h : ∀ (a : AddCircle p), addOrderOf a ≠ 0 → s ≠ AddSubgroup.zmultiples a
⊢ Dense (↑s : Set (AddCircle p))
have mpr : ∀ (h : ¬Dense (↑s : Set (AddCircle p))), ∃ (a : AddCircle p), addOrderOf a ≠ 0 ∧ s = AddSubgroup.zmultiples a
Tactic: contrapose! h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.DenseSubgroup (hypothesis)
Goal:
p : ℝ
inst : Fact (0 < p)
s : AddSubgroup (AddCircle p)
h : ¬Dense (↑s : Set (AddCircle p))
⊢ ∃ (a : AddCircle p), s = AddSubgroup.zmultiples a
have h₁ : ∃ (a : ℝ), AddSubgroup.comap (QuotientAddGroup.mk' (AddSubgroup.zmultiples p)) s = AddSubgroup.zmultiples a
Tactic: rw [← QuotientAddGroup.dense_preimage_mk, ← QuotientAddGroup.coe_mk',
        ← AddSubgroup.coe_comap, xor_iff_not_iff'.1 (AddSubgroup.dense_xor_cyclic _)] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.DenseSubgroup (goal)
Goal:
p : ℝ
inst : Fact (0 < p)
s : AddSubgroup (AddCircle p)
h : ∃ (a : ℝ), AddSubgroup.comap (QuotientAddGroup.mk' (AddSubgroup.zmultiples p)) s = AddSubgroup.zmultiples a
⊢ ∃ (a : AddCircle p), s = AddSubgroup.zmultiples a
have rcases : ∀ (a : ℝ) (ha : AddSubgroup.comap (QuotientAddGroup.mk' (AddSubgroup.zmultiples p)) s = AddSubgroup.zmultiples a),
  ∃ (a : AddCircle p), s = AddSubgroup.zmultiples a
Tactic: rcases h with ⟨a, ha⟩
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.DenseSubgroup (goal)
Goal:
p : ℝ
inst : Fact (0 < p)
s : AddSubgroup (AddCircle p)
a : ℝ
ha : AddSubgroup.comap (QuotientAddGroup.mk' (AddSubgroup.zmultiples p)) s = AddSubgroup.zmultiples a
⊢ ∃ (a : AddCircle p), s = AddSubgroup.zmultiples a
have h : s = AddSubgroup.zmultiples (↑a : ℝ ⧸ AddSubgroup.zmultiples p)
Tactic: use a
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.DenseSubgroup (goal)
Goal:
p : ℝ
inst : Fact (0 < p)
s : AddSubgroup (AddCircle p)
a : ℝ
ha : AddSubgroup.comap (QuotientAddGroup.mk' (AddSubgroup.zmultiples p)) s = AddSubgroup.zmultiples a
⊢ s = AddSubgroup.zmultiples (↑a : ℝ ⧸ AddSubgroup.zmultiples p)
have h : Function.Surjective (α := ℝ) (β := AddCircle p)
  (⇑(QuotientAddGroup.mk' (AddSubgroup.zmultiples p)) : (a : ℝ) → AddCircle p)
Tactic: rw [← QuotientAddGroup.coe_mk', ← AddMonoidHom.map_zmultiples, ← ha,
        AddSubgroup.map_comap_eq_self_of_surjective]
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.DenseSubgroup (goal)
Goal:
p : ℝ
inst : Fact (0 < p)
s : AddSubgroup (AddCircle p)
h : ¬Dense (↑s : Set (AddCircle p))
⊢ ∃ (a : AddCircle p), addOrderOf a ≠ 0 ∧ s = AddSubgroup.zmultiples a
have mpr : ∀ (a : AddCircle p) (h : ¬Dense (↑(AddSubgroup.zmultiples a) : Set (AddCircle p))),
  ∃ (a_1 : AddCircle p),
    addOrderOf a_1 ≠ 0 ∧ Eq (α := AddSubgroup (AddCircle p)) (AddSubgroup.zmultiples a) (AddSubgroup.zmultiples a_1)
Tactic: obtain ⟨a, rfl⟩ : ∃ a, s = .zmultiples a := by
      rw [← QuotientAddGroup.dense_preimage_mk, ← QuotientAddGroup.coe_mk',
        ← AddSubgroup.coe_comap, xor_iff_not_iff'.1 (AddSubgroup.dense_xor_cyclic _)] at h
      rcases h with ⟨a, ha⟩
      use a
      rw [← QuotientAddGroup.coe_mk', ← AddMonoidHom.map_zmultiples, ← ha,
        AddSubgroup.map_comap_eq_self_of_surjective]
      exact Quot.mk_surjective
Cleared hypotheses: s, h
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Real (goal)
Goal:
p : ℝ
inst : Fact (0 < p)
⊢ CompactSpace (AddCircle p)
have rw : IsCompact (X := AddCircle p) (QuotientAddGroup.mk '' Icc 0 (0 + p))
Tactic: rw [← isCompact_univ_iff, ← coe_image_Icc_eq p 0]
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Real (goal)
Goal:
N : ℕ
inst : NeZero N
j : ZMod N
⊢ Eq (α := UnitAddCircle) ((toAddCircle : (a : ZMod N) → UnitAddCircle) j : UnitAddCircle)
    ((↑(HDiv.hDiv (α := ℝ) (↑j.val : ℝ) (↑N : ℝ)) : HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (zmultiples 1)) :
      HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (zmultiples 1))
have rw : Eq (α := UnitAddCircle) ((toAddCircle : (a : ZMod N) → UnitAddCircle) j : UnitAddCircle)
  ((toAddCircle : (a : ZMod N) → UnitAddCircle) (↑j.val : ZMod N) : UnitAddCircle)
Tactic: rw [← toAddCircle_natCast, natCast_zmod_val]
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Real (goal)
Goal:
N : ℕ
inst : NeZero N
j : ZMod N
rw :
  Eq (α := UnitAddCircle) ((toAddCircle : (a : ZMod N) → UnitAddCircle) j : UnitAddCircle)
    ((toAddCircle : (a : ZMod N) → UnitAddCircle) (↑j.val : ZMod N) : UnitAddCircle)
⊢ Eq (α := UnitAddCircle) ((toAddCircle : (a : ZMod N) → UnitAddCircle) j : UnitAddCircle)
    ((↑(HDiv.hDiv (α := ℝ) (↑j.val : ℝ) (↑N : ℝ)) : HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (zmultiples 1)) :
      HasQuotient.Quotient ℝ (B := AddSubgroup ℝ) (zmultiples 1))
have rw₁ : Eq (α := UnitAddCircle) ((toAddCircle : (a : ZMod N) → UnitAddCircle) j : UnitAddCircle)
  ((toAddCircle : (a : ZMod N) → UnitAddCircle) j : UnitAddCircle)
Tactic: rw [← toAddCircle_natCast, natCast_zmod_val]
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Real (hypothesis)
Goal:
N : ℕ
inst : NeZero N
x y : ZMod N
hxy :
  Eq (α := UnitAddCircle) ((toAddCircle : (a : ZMod N) → UnitAddCircle) x : UnitAddCircle)
    ((toAddCircle : (a : ZMod N) → UnitAddCircle) y : UnitAddCircle)
⊢ x = y
have this : LT.lt (α := ℝ) 0 (↑N : ℝ)
Tactic: have : (0 : ℝ) < N := Nat.cast_pos.mpr (NeZero.pos _)
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Real (goal)
Goal:
N : ℕ
inst : NeZero N
x y : ZMod N
hxy :
  Eq (α := UnitAddCircle) ((toAddCircle : (a : ZMod N) → UnitAddCircle) x : UnitAddCircle)
    ((toAddCircle : (a : ZMod N) → UnitAddCircle) y : UnitAddCircle)
this : LT.lt (α := ℝ) 0 (↑N : ℝ)
⊢ x = y
have hx : ∀
  (hxy :
    Eq (α := UnitAddCircle) (↑(HDiv.hDiv (α := ℝ) (↑x.val : ℝ) (↑N : ℝ)) : ℝ ⧸ zmultiples 1)
      (↑(HDiv.hDiv (α := ℝ) (↑y.val : ℝ) (↑N : ℝ)) : ℝ ⧸ zmultiples 1))
  (this : LT.lt (α := ℝ) 0 (↑N : ℝ)), HDiv.hDiv (α := ℝ) (↑x.val : ℝ) (↑N : ℝ) ∈ Ico 0 (HAdd.hAdd (α := ℝ) 0 1)
Tactic: rwa [toAddCircle_apply, toAddCircle_apply, AddCircle.coe_eq_coe_iff_of_mem_Ico,
    div_left_inj' this.ne', Nat.cast_inj, (val_injective N).eq_iff] at hxy
Cleared hypotheses: hxy, this
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Real (goal)
Goal:
N : ℕ
inst : NeZero N
x y : ZMod N
hxy :
  Eq (α := UnitAddCircle) ((toAddCircle : (a : ZMod N) → UnitAddCircle) x : UnitAddCircle)
    ((toAddCircle : (a : ZMod N) → UnitAddCircle) y : UnitAddCircle)
this : LT.lt (α := ℝ) 0 (↑N : ℝ)
hx :
  ∀
    (hxy :
      Eq (α := UnitAddCircle) (↑(HDiv.hDiv (α := ℝ) (↑x.val : ℝ) (↑N : ℝ)) : ℝ ⧸ zmultiples 1)
        (↑(HDiv.hDiv (α := ℝ) (↑y.val : ℝ) (↑N : ℝ)) : ℝ ⧸ zmultiples 1))
    (this : LT.lt (α := ℝ) 0 (↑N : ℝ)), HDiv.hDiv (α := ℝ) (↑x.val : ℝ) (↑N : ℝ) ∈ Ico 0 (HAdd.hAdd (α := ℝ) 0 1)
⊢ x = y
have hy : ∀
  (hxy :
    Eq (α := UnitAddCircle) (↑(HDiv.hDiv (α := ℝ) (↑x.val : ℝ) (↑N : ℝ)) : ℝ ⧸ zmultiples 1)
      (↑(HDiv.hDiv (α := ℝ) (↑y.val : ℝ) (↑N : ℝ)) : ℝ ⧸ zmultiples 1))
  (this : LT.lt (α := ℝ) 0 (↑N : ℝ)), HDiv.hDiv (α := ℝ) (↑y.val : ℝ) (↑N : ℝ) ∈ Ico 0 (HAdd.hAdd (α := ℝ) 0 1)
Tactic: rwa [toAddCircle_apply, toAddCircle_apply, AddCircle.coe_eq_coe_iff_of_mem_Ico,
    div_left_inj' this.ne', Nat.cast_inj, (val_injective N).eq_iff] at hxy
Cleared hypotheses: hxy, this
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Real (goal)
Goal:
N : ℕ
inst : NeZero N
x y : ZMod N
hxy :
  Eq (α := UnitAddCircle) ((toAddCircle : (a : ZMod N) → UnitAddCircle) x : UnitAddCircle)
    ((toAddCircle : (a : ZMod N) → UnitAddCircle) y : UnitAddCircle)
this : LT.lt (α := ℝ) 0 (↑N : ℝ)
hx :
  ∀
    (hxy :
      Eq (α := UnitAddCircle) (↑(HDiv.hDiv (α := ℝ) (↑x.val : ℝ) (↑N : ℝ)) : ℝ ⧸ zmultiples 1)
        (↑(HDiv.hDiv (α := ℝ) (↑y.val : ℝ) (↑N : ℝ)) : ℝ ⧸ zmultiples 1))
    (this : LT.lt (α := ℝ) 0 (↑N : ℝ)), HDiv.hDiv (α := ℝ) (↑x.val : ℝ) (↑N : ℝ) ∈ Ico 0 (HAdd.hAdd (α := ℝ) 0 1)
hy :
  ∀
    (hxy :
      Eq (α := UnitAddCircle) (↑(HDiv.hDiv (α := ℝ) (↑x.val : ℝ) (↑N : ℝ)) : ℝ ⧸ zmultiples 1)
        (↑(HDiv.hDiv (α := ℝ) (↑y.val : ℝ) (↑N : ℝ)) : ℝ ⧸ zmultiples 1))
    (this : LT.lt (α := ℝ) 0 (↑N : ℝ)), HDiv.hDiv (α := ℝ) (↑y.val : ℝ) (↑N : ℝ) ∈ Ico 0 (HAdd.hAdd (α := ℝ) 0 1)
⊢ x = y
have rwa : (hxy :
    Eq (α := UnitAddCircle) (↑(HDiv.hDiv (α := ℝ) (↑x.val : ℝ) (↑N : ℝ)) : ℝ ⧸ zmultiples 1)
      (↑(HDiv.hDiv (α := ℝ) (↑y.val : ℝ) (↑N : ℝ)) : ℝ ⧸ zmultiples 1)) →
  (this : LT.lt (α := ℝ) 0 (↑N : ℝ)) → ℝ
Tactic: rwa [toAddCircle_apply, toAddCircle_apply, AddCircle.coe_eq_coe_iff_of_mem_Ico,
    div_left_inj' this.ne', Nat.cast_inj, (val_injective N).eq_iff] at hxy
Cleared hypotheses: hxy, this
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
⊢ Membership.mem (γ := Set ℝ) (Set.Icc 0 1) (1 / 4) ∧ Membership.mem (γ := Set ℝ) (Set.Icc 0 1) (3 / 4)
have zero : 0 ≤ 1 / 4
Tactic: refine ⟨⟨ ?_, ?_⟩, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
zero : 0 ≤ 1 / 4
⊢ Membership.mem (γ := Set ℝ) (Set.Icc 0 1) (1 / 4) ∧ Membership.mem (γ := Set ℝ) (Set.Icc 0 1) (3 / 4)
have zero₁ : 1 / 4 ≤ 1
Tactic: refine ⟨⟨ ?_, ?_⟩, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
zero : 0 ≤ 1 / 4
zero₁ : 1 / 4 ≤ 1
⊢ Membership.mem (γ := Set ℝ) (Set.Icc 0 1) (1 / 4) ∧ Membership.mem (γ := Set ℝ) (Set.Icc 0 1) (3 / 4)
have zero₂ : 0 ≤ 3 / 4
Tactic: refine ⟨⟨ ?_, ?_⟩, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
zero : 0 ≤ 1 / 4
zero₁ : 1 / 4 ≤ 1
zero₂ : 0 ≤ 3 / 4
⊢ Membership.mem (γ := Set ℝ) (Set.Icc 0 1) (1 / 4) ∧ Membership.mem (γ := Set ℝ) (Set.Icc 0 1) (3 / 4)
have zero₃ : 3 / 4 ≤ 1
Tactic: refine ⟨⟨ ?_, ?_⟩, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
n : ℕ
ih : 1 / 4 ∈ preCantorSet n ∧ 3 / 4 ∈ preCantorSet n
⊢ 1 / 4 ∈ preCantorSet (n + 1) ∧ 3 / 4 ∈ preCantorSet (n + 1)
have succ : 1 / 4 ∈ preCantorSet (n + 1)
Tactic: apply And.intro
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
n : ℕ
ih : 1 / 4 ∈ preCantorSet n ∧ 3 / 4 ∈ preCantorSet n
succ : 1 / 4 ∈ preCantorSet (n + 1)
⊢ 1 / 4 ∈ preCantorSet (n + 1) ∧ 3 / 4 ∈ preCantorSet (n + 1)
have succ₁ : 3 / 4 ∈ preCantorSet (n + 1)
Tactic: apply And.intro
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
n : ℕ
⊢ 1 / 4 ∈ preCantorSet n ∧ 3 / 4 ∈ preCantorSet n
have zero : 1 / 4 ∈ preCantorSet 0 ∧ 3 / 4 ∈ preCantorSet 0
Tactic: induction n with
  | zero =>
    simp only [preCantorSet_zero]
    refine ⟨⟨ ?_, ?_⟩, ?_, ?_⟩ <;> norm_num
  | succ n ih =>
    apply And.intro
    · -- goal: 1 / 4 ∈ preCantorSet (n + 1)
      -- follows by the inductive hypothesis, since 3 / 4 ∈ preCantorSet n
      exact Or.inl ⟨3 / 4, ih.2, by norm_num⟩
    · -- goal: 3 / 4 ∈ preCantorSet (n + 1)
      -- follows by the inductive hypothesis, since 1 / 4 ∈ preCantorSet n
      exact Or.inr ⟨1 / 4, ih.1, by norm_num⟩
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
n : ℕ
zero : 1 / 4 ∈ preCantorSet 0 ∧ 3 / 4 ∈ preCantorSet 0
⊢ 1 / 4 ∈ preCantorSet n ∧ 3 / 4 ∈ preCantorSet n
have succ : ∀ (n : ℕ) (ih : 1 / 4 ∈ preCantorSet n ∧ 3 / 4 ∈ preCantorSet n),
  1 / 4 ∈ preCantorSet (n + 1) ∧ 3 / 4 ∈ preCantorSet (n + 1)
Tactic: induction n with
  | zero =>
    simp only [preCantorSet_zero]
    refine ⟨⟨ ?_, ?_⟩, ?_, ?_⟩ <;> norm_num
  | succ n ih =>
    apply And.intro
    · -- goal: 1 / 4 ∈ preCantorSet (n + 1)
      -- follows by the inductive hypothesis, since 3 / 4 ∈ preCantorSet n
      exact Or.inl ⟨3 / 4, ih.2, by norm_num⟩
    · -- goal: 3 / 4 ∈ preCantorSet (n + 1)
      -- follows by the inductive hypothesis, since 1 / 4 ∈ preCantorSet n
      exact Or.inr ⟨1 / 4, ih.1, by norm_num⟩
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.Topology.Instances.CantorSet (goal)
Goal:
n : ℕ
⊢ 0 ∈ preCantorSet n
have zero : 0 ∈ preCantorSet 0
Tactic: induction n with
  | zero =>
    simp [preCantorSet]
  | succ n ih =>
    exact Or.inl ⟨0, ih, by simp only [zero_div]⟩
Cleared hypotheses: n