awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Defs (goal)
Goal:
𝕜 : Type u_1
B : Type u_2
inst : Field 𝕜
p q : 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n : ℕ
hn : 0 < n
m₁ m₂ : (↑{m : ℕ | m < n ∧ m.gcd n = 1} : Type)
h :
  HSMul.hSMul (β := HasQuotient.Quotient 𝕜 (B := AddSubgroup 𝕜) (zmultiples p)) (↑m₁ : ℕ)
      (↑(p / (↑n : 𝕜)) : 𝕜 ⧸ zmultiples p) =
    HSMul.hSMul (β := HasQuotient.Quotient 𝕜 (B := AddSubgroup 𝕜) (zmultiples p)) (↑m₂ : ℕ)
      (↑(p / (↑n : 𝕜)) : 𝕜 ⧸ zmultiples p)
⊢ (↑m₂ : ℕ) ∈ Iio (addOrderOf (G := AddCircle p) (↑(p / (↑n : 𝕜)) : 𝕜 ⧸ zmultiples p))
have refine_1 : (↑m₂ : ℕ) ∈ Iio n
Tactic: rw [addOrderOf_period_div hn]
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Defs (goal)
Goal:
𝕜 : Type u_1
B : Type u_2
inst : Field 𝕜
p q : 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n : ℕ
hn : 0 < n
u : (↑{u : AddCircle p | addOrderOf u = n} : Type u_1)
⊢ ∃ (a : (↑{m : ℕ | m < n ∧ m.gcd n = 1} : Type)),
    (fun (m : (↑{m : ℕ | m < n ∧ m.gcd n = 1} : Type)) ↦
          (Subtype.mk (p := fun (x : AddCircle p) ↦
              Membership.mem (γ := Set (AddCircle p)) {u : AddCircle p | addOrderOf u = n} x)
              (↑(HDiv.hDiv (α := 𝕜) (↑(↑m : ℕ) : 𝕜) (↑n : 𝕜) * p) : 𝕜 ⧸ zmultiples p) ⋯ :
            { x : AddCircle p // Membership.mem (γ := Set (AddCircle p)) {u : AddCircle p | addOrderOf u = n} x }))
        a =
      u
have refine_2 : ∀ (m : ℕ) (hmn : m < n) (hg : m.gcd n = 1)
  (he : (↑(HDiv.hDiv (α := 𝕜) (↑m : 𝕜) (↑n : 𝕜) * p) : 𝕜 ⧸ zmultiples p) = (↑u : AddCircle p)),
  ∃ (a : (↑{m : ℕ | m < n ∧ m.gcd n = 1} : Type)),
    (fun (m : (↑{m : ℕ | m < n ∧ m.gcd n = 1} : Type)) ↦
          (Subtype.mk (p := fun (x : AddCircle p) ↦
              Membership.mem (γ := Set (AddCircle p)) {u : AddCircle p | addOrderOf u = n} x)
              (↑(HDiv.hDiv (α := 𝕜) (↑(↑m : ℕ) : 𝕜) (↑n : 𝕜) * p) : 𝕜 ⧸ zmultiples p) sorry :
            { x : AddCircle p // Membership.mem (γ := Set (AddCircle p)) {u : AddCircle p | addOrderOf u = n} x }))
        a =
      u
Tactic: obtain ⟨m, hmn, hg, he⟩ := (addOrderOf_eq_pos_iff hn).mp u.2
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Defs (goal)
Goal:
𝕜 : Type u_1
inst : Field 𝕜
p : 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n : ℕ
⊢ Nat.card.{u_1} { u : AddCircle p // addOrderOf u = n } = n.totient
have inl : Nat.card.{u_1} { u : AddCircle p // addOrderOf u = 0 } = Nat.totient 0
Tactic: rcases n.eq_zero_or_pos with (rfl | hn)
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Defs (goal)
Goal:
𝕜 : Type u_1
inst : Field 𝕜
p : 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n : ℕ
inl : Nat.card.{u_1} { u : AddCircle p // addOrderOf u = 0 } = Nat.totient 0
⊢ Nat.card.{u_1} { u : AddCircle p // addOrderOf u = n } = n.totient
have inr : ∀ (hn : n > 0), Nat.card.{u_1} { u : AddCircle p // addOrderOf u = n } = n.totient
Tactic: rcases n.eq_zero_or_pos with (rfl | hn)
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Defs (goal)
Goal:
𝕜 : Type u_1
inst : Field 𝕜
p : 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
⊢ Nat.card.{u_1} { u : AddCircle p // addOrderOf u = 0 } = Nat.totient 0
have inl : Nat.card.{u_1} { u : AddCircle p // ¬IsOfFinAddOrder u } = 0
Tactic: simp only [Nat.totient_zero, addOrderOf_eq_zero_iff]
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Defs (goal)
Goal:
𝕜 : Type u_1
inst : Field 𝕜
p : 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
⊢ Nat.card.{u_1} { u : AddCircle p // ¬IsOfFinAddOrder u } = 0
have inl : ∀ (u : AddCircle p) (hu : ¬IsOfFinAddOrder u), Nat.card.{u_1} { u : AddCircle p // ¬IsOfFinAddOrder u } = 0
Tactic: rcases em (∃ u : AddCircle p, ¬IsOfFinAddOrder u) with (⟨u, hu⟩ | h)
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Defs (goal)
Goal:
𝕜 : Type u_1
inst : Field 𝕜
p : 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
inl : ∀ (u : AddCircle p) (hu : ¬IsOfFinAddOrder u), Nat.card.{u_1} { u : AddCircle p // ¬IsOfFinAddOrder u } = 0
⊢ Nat.card.{u_1} { u : AddCircle p // ¬IsOfFinAddOrder u } = 0
have inl₁ : ∀ (h : ¬∃ (u : AddCircle p), ¬IsOfFinAddOrder u), Nat.card.{u_1} { u : AddCircle p // ¬IsOfFinAddOrder u } = 0
Tactic: rcases em (∃ u : AddCircle p, ¬IsOfFinAddOrder u) with (⟨u, hu⟩ | h)
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Defs (goal)
Goal:
𝕜 : Type u_1
inst : Field 𝕜
p : 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
u : AddCircle p
hu : ¬IsOfFinAddOrder u
⊢ Infinite.{max 1 (u_1 + 1)} { u : AddCircle p // ¬IsOfFinAddOrder u }
have rw : Set.Infinite.{u_1} (α := AddCircle p) {x : AddCircle p | ¬IsOfFinAddOrder x}
Tactic: rw [← coe_setOf, infinite_coe_iff]
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Defs (hypothesis)
Goal:
𝕜 : Type u_1
inst : Field 𝕜
p : 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
u : AddCircle p
hu : ¬IsOfFinAddOrder u
⊢ Nat.card.{u_1} { u : AddCircle p // ¬IsOfFinAddOrder u } = 0
have this : Infinite.{max 1 (u_1 + 1)} { u : AddCircle p // ¬IsOfFinAddOrder u }
Tactic: have : Infinite { u : AddCircle p // ¬IsOfFinAddOrder u } := by
        rw [← coe_setOf, infinite_coe_iff]
        exact infinite_not_isOfFinAddOrder hu
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Defs (hypothesis)
Goal:
𝕜 : Type u_1
inst : Field 𝕜
p : 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
h : ¬∃ (u : AddCircle p), ¬IsOfFinAddOrder u
⊢ Nat.card.{u_1} { u : AddCircle p // ¬IsOfFinAddOrder u } = 0
have this : IsEmpty.{max 1 (u_1 + 1)} { u : AddCircle p // ¬IsOfFinAddOrder u }
Tactic: have : IsEmpty { u : AddCircle p // ¬IsOfFinAddOrder u } := by simpa [isEmpty_subtype] using h
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Defs (goal)
Goal:
𝕜 : Type u_1
inst : Field 𝕜
p : 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n : ℕ
hn : n > 0
⊢ Nat.card.{u_1} { u : AddCircle p // addOrderOf u = n } = n.totient
have inr : Nat.card (↑{m : ℕ | m < n ∧ m.gcd n = 1} : Type) = Nat.card (↑{m : ℕ | m < n ∧ n.Coprime m} : Type)
Tactic: rw [← coe_setOf, Nat.card_congr (setAddOrderOfEquiv p hn),
      n.totient_eq_card_lt_and_coprime]
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Defs (goal)
Goal:
𝕜 : Type u_1
B : Type u_2
inst : AddCommGroup 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsOrderedAddMonoid 𝕜
p a : 𝕜
hp : Fact (0 < p)
inst_3 : Archimedean 𝕜
x : 𝕜
hx : x ∈ Icc a (a + p)
⊢ Eq (α := Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a))
    ((fun (x : 𝕋) ↦
        (Quot.mk (EndpointIdent p a)
            (Set.inclusion (s := Ico a (a + p)) (t := Icc a (a + p)) ⋯
              ((equivIco p a : (a_1 : 𝕋) → (↑(Ico a (a + p)) : Type u_1)) x)) :
          Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a)))
      ((fun (x : Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a)) ↦
          x.liftOn (r := EndpointIdent p a) (fun (x : (↑(Icc a (a + p)) : Type u_1)) ↦ (↑(↑x : 𝕜) : 𝕜 ⧸ zmultiples p))
            ⋯)
        (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) x hx))))
    (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) x hx))
have inl : ∀ (h : x ≠ a + p),
  Eq (α := Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a))
    ((fun (x : 𝕋) ↦
        (Quot.mk (EndpointIdent p a)
            (Set.inclusion (s := Ico a (a + p)) (t := Icc a (a + p)) sorry
              ((equivIco p a : (a_1 : 𝕋) → (↑(Ico a (a + p)) : Type u_1)) x)) :
          Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a)))
      ((fun (x : Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a)) ↦
          x.liftOn (r := EndpointIdent p a) (fun (x : (↑(Icc a (a + p)) : Type u_1)) ↦ (↑(↑x : 𝕜) : 𝕜 ⧸ zmultiples p))
            sorry)
        (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) x hx))))
    (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) x hx))
Tactic: rcases ne_or_eq x (a + p) with (h | rfl)
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Defs (goal)
Goal:
𝕜 : Type u_1
B : Type u_2
inst : AddCommGroup 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsOrderedAddMonoid 𝕜
p a : 𝕜
hp : Fact (0 < p)
inst_3 : Archimedean 𝕜
x : 𝕜
hx : x ∈ Icc a (a + p)
inl :
  ∀ (h : x ≠ a + p),
    Eq (α := Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a))
      ((fun (x : 𝕋) ↦
          (Quot.mk (EndpointIdent p a)
              (Set.inclusion (s := Ico a (a + p)) (t := Icc a (a + p)) ⋯
                ((equivIco p a : (a_1 : 𝕋) → (↑(Ico a (a + p)) : Type u_1)) x)) :
            Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a)))
        ((fun (x : Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a)) ↦
            x.liftOn (r := EndpointIdent p a) (fun (x : (↑(Icc a (a + p)) : Type u_1)) ↦ (↑(↑x : 𝕜) : 𝕜 ⧸ zmultiples p))
              ⋯)
          (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) x hx))))
      (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) x hx))
⊢ Eq (α := Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a))
    ((fun (x : 𝕋) ↦
        (Quot.mk (EndpointIdent p a)
            (Set.inclusion (s := Ico a (a + p)) (t := Icc a (a + p)) ⋯
              ((equivIco p a : (a_1 : 𝕋) → (↑(Ico a (a + p)) : Type u_1)) x)) :
          Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a)))
      ((fun (x : Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a)) ↦
          x.liftOn (r := EndpointIdent p a) (fun (x : (↑(Icc a (a + p)) : Type u_1)) ↦ (↑(↑x : 𝕜) : 𝕜 ⧸ zmultiples p))
            ⋯)
        (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) x hx))))
    (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) x hx))
have inr : ∀ (hx : a + p ∈ Icc a (a + p)),
  Eq (α := Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a))
    ((fun (x : 𝕋) ↦
        (Quot.mk (EndpointIdent p a)
            (Set.inclusion (s := Ico a (a + p)) (t := Icc a (a + p)) sorry
              ((equivIco p a : (a_1 : 𝕋) → (↑(Ico a (a + p)) : Type u_1)) x)) :
          Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a)))
      ((fun (x : Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a)) ↦
          x.liftOn (r := EndpointIdent p a) (fun (x : (↑(Icc a (a + p)) : Type u_1)) ↦ (↑(↑x : 𝕜) : 𝕜 ⧸ zmultiples p))
            sorry)
        (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) (a + p) hx))))
    (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) (a + p) hx))
Tactic: rcases ne_or_eq x (a + p) with (h | rfl)
Cleared hypotheses: x, hx
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Defs (goal)
Goal:
𝕜 : Type u_1
B : Type u_2
inst : AddCommGroup 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsOrderedAddMonoid 𝕜
p a : 𝕜
hp : Fact (0 < p)
inst_3 : Archimedean 𝕜
x : 𝕜
hx : x ∈ Icc a (a + p)
h : x ≠ a + p
⊢ Eq (α := Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a))
    ((fun (x : 𝕋) ↦
        (Quot.mk (EndpointIdent p a)
            (Set.inclusion (s := Ico a (a + p)) (t := Icc a (a + p)) ⋯
              ((equivIco p a : (a_1 : 𝕋) → (↑(Ico a (a + p)) : Type u_1)) x)) :
          Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a)))
      ((fun (x : Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a)) ↦
          x.liftOn (r := EndpointIdent p a) (fun (x : (↑(Icc a (a + p)) : Type u_1)) ↦ (↑(↑x : 𝕜) : 𝕜 ⧸ zmultiples p))
            ⋯)
        (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) x hx))))
    (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) x hx))
have inl : ∀ (x : 𝕜) (hx : x ∈ Icc a (a + p)) (h : x ≠ a + p),
  Eq (α := Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a))
    ((fun (x : 𝕋) ↦
        (Quot.mk (EndpointIdent p a)
            (Set.inclusion (s := Ico a (a + p)) (t := Icc a (a + p)) sorry
              ((equivIco p a : (a_1 : 𝕋) → (↑(Ico a (a + p)) : Type u_1)) x)) :
          Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a)))
      ((fun (x : Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a)) ↦
          x.liftOn (r := EndpointIdent p a) (fun (x : (↑(Icc a (a + p)) : Type u_1)) ↦ (↑(↑x : 𝕜) : 𝕜 ⧸ zmultiples p))
            sorry)
        (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) x hx))))
    (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) x hx))
Tactic: revert x
Cleared hypotheses: x, hx, h
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Defs (goal)
Goal:
𝕜 : Type u_1
B : Type u_2
inst : AddCommGroup 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsOrderedAddMonoid 𝕜
p a : 𝕜
hp : Fact (0 < p)
inst_3 : Archimedean 𝕜
x : 𝕜
hx : x ∈ Icc a (a + p)
h : x ≠ a + p
⊢ Eq (α := Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a))
    (Quot.mk (EndpointIdent p a)
      (Set.inclusion (s := Ico a (a + p)) (t := Icc a (a + p)) ⋯
        ((equivIco p a : (a_1 : 𝕋) → (↑(Ico a (a + p)) : Type u_1))
          (Quot.liftOn (r := EndpointIdent p a)
            (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) x hx))
            (fun (x : (↑(Icc a (a + p)) : Type u_1)) ↦ (↑(↑x : 𝕜) : 𝕜 ⧸ zmultiples p)) ⋯))))
    (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) x hx))
have inl : Set.inclusion (s := Ico a (a + p)) (t := Icc a (a + p)) sorry
    ((equivIco p a : (a_1 : 𝕋) → (↑(Ico a (a + p)) : Type u_1))
      (Quot.liftOn (r := EndpointIdent p a)
        (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) x hx))
        (fun (x : (↑(Icc a (a + p)) : Type u_1)) ↦ (↑(↑x : 𝕜) : 𝕜 ⧸ zmultiples p)) sorry)) =
  Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) x hx
Tactic: congr
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Defs (goal)
Goal:
𝕜 : Type u_1
B : Type u_2
inst : AddCommGroup 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsOrderedAddMonoid 𝕜
p a : 𝕜
hp : Fact (0 < p)
inst_3 : Archimedean 𝕜
x : 𝕜
hx : x ∈ Icc a (a + p)
h : x ≠ a + p
⊢ Set.inclusion (s := Ico a (a + p)) (t := Icc a (a + p)) ⋯
      ((equivIco p a : (a_1 : 𝕋) → (↑(Ico a (a + p)) : Type u_1))
        (Quot.liftOn (r := EndpointIdent p a)
          (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) x hx))
          (fun (x : (↑(Icc a (a + p)) : Type u_1)) ↦ (↑(↑x : 𝕜) : 𝕜 ⧸ zmultiples p)) ⋯)) =
    Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) x hx
have inl : (↑(Set.inclusion (s := Ico a (a + p)) (t := Icc a (a + p)) sorry
        ((equivIco p a : (a_1 : 𝕋) → (↑(Ico a (a + p)) : Type u_1))
          (Quot.liftOn (r := EndpointIdent p a)
            (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) x hx))
            (fun (x : (↑(Icc a (a + p)) : Type u_1)) ↦ (↑(↑x : 𝕜) : 𝕜 ⧸ zmultiples p)) sorry))) :
    𝕜) =
  (↑(Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) x hx) : 𝕜)
Tactic: ext1
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Defs (goal)
Goal:
𝕜 : Type u_1
B : Type u_2
inst : AddCommGroup 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsOrderedAddMonoid 𝕜
p a : 𝕜
hp : Fact (0 < p)
inst_3 : Archimedean 𝕜
hx : a + p ∈ Icc a (a + p)
⊢ Eq (α := Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a))
    ((fun (x : 𝕋) ↦
        (Quot.mk (EndpointIdent p a)
            (Set.inclusion (s := Ico a (a + p)) (t := Icc a (a + p)) ⋯
              ((equivIco p a : (a_1 : 𝕋) → (↑(Ico a (a + p)) : Type u_1)) x)) :
          Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a)))
      ((fun (x : Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a)) ↦
          x.liftOn (r := EndpointIdent p a) (fun (x : (↑(Icc a (a + p)) : Type u_1)) ↦ (↑(↑x : 𝕜) : 𝕜 ⧸ zmultiples p))
            ⋯)
        (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) (a + p) hx))))
    (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) (a + p) hx))
have inr : Eq (α := Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a))
  ((fun (x : 𝕋) ↦
      (Quot.mk (EndpointIdent p a)
          (Set.inclusion (s := Ico a (a + p)) (t := Icc a (a + p)) sorry
            ((equivIco p a : (a_1 : 𝕋) → (↑(Ico a (a + p)) : Type u_1)) x)) :
        Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a)))
    ((fun (x : Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a)) ↦
        x.liftOn (r := EndpointIdent p a) (fun (x : (↑(Icc a (a + p)) : Type u_1)) ↦ (↑(↑x : 𝕜) : 𝕜 ⧸ zmultiples p)) sorry)
      (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) a sorry))))
  (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) a sorry))
Tactic: rw [← Quot.sound EndpointIdent.mk]
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Defs (goal)
Goal:
𝕜 : Type u_1
B : Type u_2
inst : AddCommGroup 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsOrderedAddMonoid 𝕜
p a : 𝕜
hp : Fact (0 < p)
inst_3 : Archimedean 𝕜
hx : a + p ∈ Icc a (a + p)
⊢ Eq (α := Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a))
    (Quot.mk (EndpointIdent p a)
      (Set.inclusion (s := Ico a (a + p)) (t := Icc a (a + p)) ⋯
        ((equivIco p a : (a_1 : 𝕋) → (↑(Ico a (a + p)) : Type u_1))
          (Quot.liftOn (r := EndpointIdent p a)
            (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) a ⋯))
            (fun (x : (↑(Icc a (a + p)) : Type u_1)) ↦ (↑(↑x : 𝕜) : 𝕜 ⧸ zmultiples p)) ⋯))))
    (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) a ⋯))
have inr : Eq (α := (↑(Icc a (a + p)) : Type u_1))
  (Set.inclusion (s := Ico a (a + p)) (t := Icc a (a + p)) sorry
      ((equivIco p a : (a_1 : 𝕋) → (↑(Ico a (a + p)) : Type u_1))
        (Quot.liftOn (r := EndpointIdent p a)
          (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) a sorry))
          (fun (x : (↑(Icc a (a + p)) : Type u_1)) ↦ (↑(↑x : 𝕜) : 𝕜 ⧸ zmultiples p)) sorry)) :
    (↑(Icc a (a + p)) : Type u_1))
  (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) a sorry : { x : 𝕜 // x ∈ Icc a (a + p) })
Tactic: congr
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Defs (goal)
Goal:
𝕜 : Type u_1
B : Type u_2
inst : AddCommGroup 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsOrderedAddMonoid 𝕜
p a : 𝕜
hp : Fact (0 < p)
inst_3 : Archimedean 𝕜
hx : a + p ∈ Icc a (a + p)
⊢ Eq (α := (↑(Icc a (a + p)) : Type u_1))
    (Set.inclusion (s := Ico a (a + p)) (t := Icc a (a + p)) ⋯
        ((equivIco p a : (a_1 : 𝕋) → (↑(Ico a (a + p)) : Type u_1))
          (Quot.liftOn (r := EndpointIdent p a)
            (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) a ⋯))
            (fun (x : (↑(Icc a (a + p)) : Type u_1)) ↦ (↑(↑x : 𝕜) : 𝕜 ⧸ zmultiples p)) ⋯)) :
      (↑(Icc a (a + p)) : Type u_1))
    (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) a ⋯ : { x : 𝕜 // x ∈ Icc a (a + p) })
have inr : (↑(Set.inclusion (s := Ico a (a + p)) (t := Icc a (a + p)) sorry
        ((equivIco p a : (a_1 : 𝕋) → (↑(Ico a (a + p)) : Type u_1))
          (Quot.liftOn (r := EndpointIdent p a)
            (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) a sorry))
            (fun (x : (↑(Icc a (a + p)) : Type u_1)) ↦ (↑(↑x : 𝕜) : 𝕜 ⧸ zmultiples p)) sorry))) :
    𝕜) =
  (↑(Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) a sorry : { x : 𝕜 // x ∈ Icc a (a + p) }) : 𝕜)
Tactic: ext1
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Defs (goal)
Goal:
𝕜 : Type u_1
inst : AddCommGroup 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsOrderedAddMonoid 𝕜
p a : 𝕜
hp : Fact (0 < p)
inst_3 : Archimedean 𝕜
⊢ Eq (α := 𝕜 → Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a))
    (comp (β := 𝕋) (⇑(equivIccQuot p a) : (a_1 : 𝕋) → Quot (EndpointIdent p a)) Quotient.mk'') fun (x : 𝕜) ↦
    Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) (toIocMod (p := p) ⋯ a x) ⋯)
have rw : Eq.{u_1 + 1} (α := 𝕜 → Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a))
  (fun (x : 𝕜) ↦
    Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) (toIcoMod (p := p) sorry a x) sorry))
  fun (x : 𝕜) ↦
  Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) (toIocMod (p := p) sorry a x) sorry)
Tactic: rw [equivIccQuot_comp_mk_eq_toIcoMod]
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Defs (goal)
Goal:
𝕜 : Type u_1
inst : AddCommGroup 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsOrderedAddMonoid 𝕜
p a : 𝕜
hp : Fact (0 < p)
inst_3 : Archimedean 𝕜
⊢ Eq.{u_1 + 1} (α := 𝕜 → Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a))
    (fun (x : 𝕜) ↦
      Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) (toIcoMod (p := p) ⋯ a x) ⋯))
    fun (x : 𝕜) ↦
    Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) (toIocMod (p := p) ⋯ a x) ⋯)
have h : ∀ (x : 𝕜),
  Eq (α := Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a))
    (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) (toIcoMod (p := p) sorry a x) sorry))
    (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) (toIocMod (p := p) sorry a x) sorry))
Tactic: funext x
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Defs (goal)
Goal:
𝕜 : Type u_1
inst : AddCommGroup 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsOrderedAddMonoid 𝕜
p a : 𝕜
hp : Fact (0 < p)
inst_3 : Archimedean 𝕜
x : 𝕜
⊢ Eq (α := Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a))
    (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) (toIcoMod (p := p) ⋯ a x) ⋯))
    (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) (toIocMod (p := p) ⋯ a x) ⋯))
have pos : ∀ (h : a ≡ x [PMOD p]),
  Eq (α := Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a))
    (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) (toIcoMod (p := p) sorry a x) sorry))
    (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) (toIocMod (p := p) sorry a x) sorry))
Tactic: by_cases h : a ≡ x [PMOD p]
-- Lean 4 proof step from Mathlib.Topology.Instances.AddCircle.Defs (goal)
Goal:
𝕜 : Type u_1
inst : AddCommGroup 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsOrderedAddMonoid 𝕜
p a : 𝕜
hp : Fact (0 < p)
inst_3 : Archimedean 𝕜
x : 𝕜
pos :
  ∀ (h : a ≡ x [PMOD p]),
    Eq (α := Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a))
      (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) (toIcoMod (p := p) ⋯ a x) ⋯))
      (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) (toIocMod (p := p) ⋯ a x) ⋯))
⊢ Eq (α := Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a))
    (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) (toIcoMod (p := p) ⋯ a x) ⋯))
    (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) (toIocMod (p := p) ⋯ a x) ⋯))
have neg : ∀ (h : ¬a ≡ x [PMOD p]),
  Eq (α := Quot (α := (↑(Icc a (a + p)) : Type u_1)) (EndpointIdent p a))
    (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) (toIcoMod (p := p) sorry a x) sorry))
    (Quot.mk (EndpointIdent p a) (Subtype.mk (p := fun (x : 𝕜) ↦ x ∈ Icc a (a + p)) (toIocMod (p := p) sorry a x) sorry))
Tactic: by_cases h : a ≡ x [PMOD p]