awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Algebra.Ring.Compact (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
inst_3 : CompactSpace R
inst_4 : T2Space R
inst_5 : IsArtinianRing R
n : ℕ
hn : HPow.hPow (α := Ideal R) (Ideal.jacobson ⊥) n = 0
⊢ HPow.hPow (α := Ideal R) (∏ p : PrimeSpectrum R, p.asIdeal) n ≤ HPow.hPow (α := Ideal R) (Ideal.jacobson ⊥) n
have hab : LE.le.{u_1} (α := Ideal R) (∏ p : PrimeSpectrum R, p.asIdeal) (Ideal.jacobson ⊥)
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Topology.Algebra.Ring.Compact (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
inst_3 : CompactSpace R
inst_4 : T2Space R
inst_5 : IsArtinianRing R
n : ℕ
hn : HPow.hPow (α := Ideal R) (Ideal.jacobson ⊥) n = 0
⊢ LE.le.{u_1} (α := Ideal R) (∏ p : PrimeSpectrum R, p.asIdeal) (Ideal.jacobson ⊥)
have hab : ∀ (b : Ideal R),
  Membership.mem (γ := Set (Ideal R)) {I : Ideal R | I.IsMaximal} b → ∏ p : PrimeSpectrum R, p.asIdeal ≤ b
Tactic: rw [Ideal.jacobson_bot, Ring.jacobson_eq_sInf_isMaximal, le_sInf_iff]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Ring.Compact (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
inst_3 : CompactSpace R
inst_4 : T2Space R
inst_5 : IsArtinianRing R
n : ℕ
hn : HPow.hPow (α := Ideal R) (Ideal.jacobson ⊥) n = 0
⊢ Finite R
have H : HPow.hPow (α := Ideal R) (∏ p : PrimeSpectrum R, p.asIdeal) n = ⊥
Tactic: have H : (∏ p : PrimeSpectrum R, p.asIdeal) ^ n = ⊥ := by
    rw [← le_bot_iff, ← Ideal.zero_eq_bot, ← hn]
    gcongr
    rw [Ideal.jacobson_bot, Ring.jacobson_eq_sInf_isMaximal, le_sInf_iff]
    exact fun I hI ↦ Ideal.prod_le_inf.trans
      (Finset.inf_le (b := PrimeSpectrum.mk I hI.isPrime) (by simp))
-- Lean 4 proof step from Mathlib.Topology.Algebra.Ring.Compact (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
inst_3 : CompactSpace R
inst_4 : T2Space R
inst_5 : IsArtinianRing R
n : ℕ
hn : HPow.hPow (α := Ideal R) (Ideal.jacobson ⊥) n = 0
H : HPow.hPow (α := Ideal R) (∏ p : PrimeSpectrum R, p.asIdeal) n = ⊥
⊢ Finite R
have this : Finite.{u_1 + 1} (HasQuotient.Quotient R (B := Ideal R) (∏ i : PrimeSpectrum R, i.asIdeal))
Tactic: have := Ideal.finite_quotient_prod (R := R) PrimeSpectrum.asIdeal Finset.univ
    (fun _ _ ↦ IsNoetherian.noetherian _) (fun _ _ ↦ inferInstance)
-- Lean 4 proof step from Mathlib.Topology.Algebra.Ring.Compact (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
inst_3 : CompactSpace R
inst_4 : T2Space R
inst_5 : IsArtinianRing R
n : ℕ
hn : HPow.hPow (α := Ideal R) (Ideal.jacobson ⊥) n = 0
H : HPow.hPow (α := Ideal R) (∏ p : PrimeSpectrum R, p.asIdeal) n = ⊥
this : Finite.{u_1 + 1} (HasQuotient.Quotient R (B := Ideal R) (∏ i : PrimeSpectrum R, i.asIdeal))
⊢ Finite R
have this₁ : Finite.{u_1 + 1} (R ⧸ HPow.hPow (α := Ideal R) (∏ p : PrimeSpectrum R, p.asIdeal) n)
Tactic: have := Ideal.finite_quotient_pow (IsNoetherian.noetherian (∏ p : PrimeSpectrum R, p.asIdeal)) n
-- Lean 4 proof step from Mathlib.Topology.Algebra.Ring.Compact (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
inst_3 : CompactSpace R
inst_4 : T2Space R
inst_5 : IsArtinianRing R
n : ℕ
hn : HPow.hPow (α := Ideal R) (Ideal.jacobson ⊥) n = 0
H : HPow.hPow (α := Ideal R) (∏ p : PrimeSpectrum R, p.asIdeal) n = ⊥
this_1 : Finite.{u_1 + 1} (HasQuotient.Quotient R (B := Ideal R) (∏ i : PrimeSpectrum R, i.asIdeal))
this : Finite.{u_1 + 1} (R ⧸ HPow.hPow (α := Ideal R) (∏ p : PrimeSpectrum R, p.asIdeal) n)
⊢ Finite R
have this₁ : Finite.{u_1 + 1} (HasQuotient.Quotient R (B := Ideal R) ⊥)
Tactic: rw [H] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Topology.Algebra.Ring.Compact (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
inst_3 : CompactSpace R
inst_4 : T2Space R
inst_5 : IsLocalRing R
inst_6 : IsNoetherianRing R
I : Ideal R
⊢ IsOpen (↑I : Set R) ↔ Finite.{u_1 + 1} (R ⧸ I)
have refine : ∀ (H : Finite.{u_1 + 1} (R ⧸ I)), IsOpen (↑I : Set R)
Tactic: refine ⟨AddSubgroup.quotient_finite_of_isOpen I.toAddSubgroup, fun H ↦ ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.Ring.Compact (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
inst_3 : CompactSpace R
inst_4 : T2Space R
inst_5 : IsLocalRing R
inst_6 : IsNoetherianRing R
I : Ideal R
H : Finite.{u_1 + 1} (R ⧸ I)
⊢ IsOpen (↑I : Set R)
have obtain : ∀ (n : ℕ) (hn : HPow.hPow (α := Ideal R) (maximalIdeal R) n ≤ I), IsOpen (↑I : Set R)
Tactic: obtain ⟨n, hn⟩ := exists_maximalIdeal_pow_le_of_isArtinianRing_quotient I
-- Lean 4 proof step from Mathlib.Topology.Algebra.Ring.Compact (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
inst_3 : CompactSpace R
inst_4 : T2Space R
inst_5 : IsDedekindDomain R
I : Ideal R
hI : I ≠ ⊥
⊢ IsOpen (↑I : Set R)
have rw : IsOpen
  (↑(∏
        i ∈
          Set.Finite.toFinset (s :=
            Function.mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) sorry,
        i.maxPowDividing I) :
    Set R)
Tactic: rw [← Ideal.finprod_heightOneSpectrum_factorization hI,
    finprod_eq_finsetProd_of_mulSupport_subset _
      (s := (Ideal.hasFiniteMulSupport hI).toFinset) (by simp)]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Ring.Compact (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
inst_3 : CompactSpace R
inst_4 : T2Space R
inst_5 : IsDedekindDomain R
I : Ideal R
hI : I ≠ ⊥
⊢ IsOpen
    (↑(∏
          i ∈
            Set.Finite.toFinset (s :=
              Function.mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) ⋯,
          i.maxPowDividing I) :
      Set R)
have refine : AddSubgroup.FiniteIndex (G := R)
  (Submodule.toAddSubgroup (R := R) (module_M := Semiring.toModule)
    (∏
      i ∈
        Set.Finite.toFinset (s := Function.mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I)
          sorry,
      i.maxPowDividing I))
Tactic: refine @AddSubgroup.isOpen_of_isClosed_of_finiteIndex _ _ _ _ (Submodule.toAddSubgroup _)
    ?_ (IsNoetherianRing.isClosed_ideal _)
-- Lean 4 proof step from Mathlib.Topology.Algebra.Ring.Compact (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
inst_3 : CompactSpace R
inst_4 : T2Space R
inst_5 : IsDedekindDomain R
I : Ideal R
hI : I ≠ ⊥
⊢ AddSubgroup.FiniteIndex (G := R)
    (Submodule.toAddSubgroup (R := R) (module_M := Semiring.toModule)
      (∏
        i ∈
          Set.Finite.toFinset (s :=
            Function.mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) ⋯,
        i.maxPowDividing I))
have refine : Finite.{u_1 + 1}
  (HasQuotient.Quotient R (B := AddSubgroup R)
    (Submodule.toAddSubgroup (R := R) (module_M := Semiring.toModule)
      (∏
        i ∈
          Set.Finite.toFinset (s :=
            Function.mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) sorry,
        i.maxPowDividing I)))
Tactic: refine @AddSubgroup.finiteIndex_of_finite_quotient _ _ _ ?_
-- Lean 4 proof step from Mathlib.Topology.Algebra.Ring.Compact (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
inst_3 : CompactSpace R
inst_4 : T2Space R
inst_5 : IsDedekindDomain R
I : Ideal R
hI : I ≠ ⊥
⊢ Finite.{u_1 + 1}
    (HasQuotient.Quotient R (B := AddSubgroup R)
      (Submodule.toAddSubgroup (R := R) (module_M := Semiring.toModule)
        (∏
          i ∈
            Set.Finite.toFinset (s :=
              Function.mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) ⋯,
          i.maxPowDividing I)))
have refine : ∀
  x ∈
    Set.Finite.toFinset (s := Function.mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) sorry,
  Finite.{u_1 + 1} (R ⧸ x.maxPowDividing I)
Tactic: refine Ideal.finite_quotient_prod _ _ (fun _ _ ↦ IsNoetherian.noetherian _) fun _ _ ↦ ?_
-- Lean 4 proof step from Mathlib.Topology.Algebra.Ring.Compact (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
inst_3 : CompactSpace R
inst_4 : T2Space R
inst_5 : IsDedekindDomain R
hR : ¬IsField R
I : Ideal R
⊢ IsOpen (↑I : Set R) ↔ I ≠ ⊥
have refine : IsOpen (↑I : Set R) → I ≠ ⊥
Tactic: refine ⟨?_, IsDedekindDomain.isOpen_of_ne_bot⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.Ring.Compact (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
inst_3 : CompactSpace R
inst_4 : T2Space R
inst_5 : IsDedekindDomain R
hR : ¬IsField R
H : IsOpen (↑⊥ : Set R)
⊢ False
have this : DiscreteTopology R
Tactic: have := discreteTopology_iff_isOpen_singleton_zero.mpr H
-- Lean 4 proof step from Mathlib.Topology.Algebra.Ring.Real (goal)
Goal:
⊢ (range ofNNReal).OrdConnected
have rw : OrdConnected (α := ℝ≥0∞) (Iio ∞)
Tactic: rw [range_coe']
-- Lean 4 proof step from Mathlib.Topology.Algebra.Ring.Real (goal)
Goal:
⊢ IsOpen (range ofNNReal)
have rw : IsOpen (X := ℝ≥0∞) (Iio ∞)
Tactic: rw [range_coe']
-- Lean 4 proof step from Mathlib.Topology.Algebra.Ring.Real (goal)
Goal:
α : Type u
β : Type v
γ : Type w
⊢ ContinuousAdd ℝ≥0∞
have refine : ∀ (x : ℝ≥0∞ × ℝ≥0∞), ContinuousAt (Y := ℝ≥0∞) (fun (p : ℝ≥0∞ × ℝ≥0∞) ↦ p.1 + p.2) x
Tactic: refine ⟨continuous_iff_continuousAt.2 ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.Ring.Real (goal)
Goal:
α : Type u
β : Type v
γ : Type w
b : ℝ≥0∞
a : ℝ≥0
⊢ ContinuousAt (Y := ℝ≥0∞) (fun (p : ℝ≥0∞ × ℝ≥0∞) ↦ p.1 + p.2) (some a, b)
have some : ContinuousAt (Y := ℝ≥0∞) (fun (p : ℝ≥0∞ × ℝ≥0∞) ↦ p.1 + p.2) (some a, none)
Tactic: rcases b with (_ | b)
Cleared hypotheses: b
-- Lean 4 proof step from Mathlib.Topology.Algebra.Ring.Real (goal)
Goal:
α : Type u
β : Type v
γ : Type w
b : ℝ≥0∞
a : ℝ≥0
some : ContinuousAt (Y := ℝ≥0∞) (fun (p : ℝ≥0∞ × ℝ≥0∞) ↦ p.1 + p.2) (Option.some a, none)
⊢ ContinuousAt (Y := ℝ≥0∞) (fun (p : ℝ≥0∞ × ℝ≥0∞) ↦ p.1 + p.2) (Option.some a, b)
have some₁ : ∀ (b : ℝ≥0), ContinuousAt (Y := ℝ≥0∞) (fun (p : ℝ≥0∞ × ℝ≥0∞) ↦ p.1 + p.2) (Option.some a, Option.some b)
Tactic: rcases b with (_ | b)
Cleared hypotheses: b
-- Lean 4 proof step from Mathlib.Topology.Algebra.Semigroup (goal)
Goal:
M : Type u_1
inst : Nonempty M
inst_1 : Semigroup M
inst_2 : TopologicalSpace M
inst_3 : CompactSpace M
inst_4 : T2Space M
continuous_const_mul : ∀ (r : M), Continuous (Y := M) fun (x : M) ↦ x * r
S : Set (Set M) := {N : Set M | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
⊢ ∃ (m : M), m * m = m
have rsuffices : ∀ (N : Set M) (hN : Minimal (fun (x : Set M) ↦ x ∈ S) N), ∃ (m : M), m * m = m
Tactic: rsuffices ⟨N, hN⟩ : ∃ N', Minimal (· ∈ S) N'
-- Lean 4 proof step from Mathlib.Topology.Algebra.Semigroup (goal)
Goal:
M : Type u_1
inst : Nonempty M
inst_1 : Semigroup M
inst_2 : TopologicalSpace M
inst_3 : CompactSpace M
inst_4 : T2Space M
continuous_const_mul : ∀ (r : M), Continuous (Y := M) fun (x : M) ↦ x * r
S : Set (Set M) := {N : Set M | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
rsuffices : ∀ (N : Set M) (hN : Minimal (fun (x : Set M) ↦ x ∈ S) N), ∃ (m : M), m * m = m
⊢ ∃ (m : M), m * m = m
have rsuffices₁ : ∃ (N' : Set M), Minimal (fun (x : Set M) ↦ x ∈ S) N'
Tactic: rsuffices ⟨N, hN⟩ : ∃ N', Minimal (· ∈ S) N'
-- Lean 4 proof step from Mathlib.Topology.Algebra.Semigroup (goal)
Goal:
M : Type u_1
inst : Nonempty M
inst_1 : Semigroup M
inst_2 : TopologicalSpace M
inst_3 : CompactSpace M
inst_4 : T2Space M
continuous_const_mul : ∀ (r : M), Continuous (Y := M) fun (x : M) ↦ x * r
S : Set (Set M) := {N : Set M | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun (x : Set M) ↦ x ∈ S) N
⊢ ∃ (m : M), m * m = m
have obtain : ∀ (N_closed : IsClosed N) (N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N) (m : M) (hm : m ∈ N), ∃ (m : M), m * m = m
Tactic: obtain ⟨N_closed, ⟨m, hm⟩, N_mul⟩ := hN.prop
-- Lean 4 proof step from Mathlib.Topology.Algebra.Semigroup (goal)
Goal:
M : Type u_1
inst : Nonempty M
inst_1 : Semigroup M
inst_2 : TopologicalSpace M
inst_3 : CompactSpace M
inst_4 : T2Space M
continuous_const_mul : ∀ (r : M), Continuous (Y := M) fun (x : M) ↦ x * r
S : Set (Set M) := {N : Set M | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun (x : Set M) ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
⊢ ∃ (m : M), m * m = m
have h : m * m = m
Tactic: use m
-- Lean 4 proof step from Mathlib.Topology.Algebra.Semigroup (goal)
Goal:
M : Type u_1
inst : Nonempty M
inst_1 : Semigroup M
inst_2 : TopologicalSpace M
inst_3 : CompactSpace M
inst_4 : T2Space M
continuous_const_mul : ∀ (r : M), Continuous (Y := M) fun (x : M) ↦ x * r
S : Set (Set M) := {N : Set M | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun (x : Set M) ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
⊢ (fun (x : M) ↦ x * m) '' N = N
have ht : (fun (x : M) ↦ x * m) '' N ∈ S
Tactic: apply hN.eq_of_subset
-- Lean 4 proof step from Mathlib.Topology.Algebra.Semigroup (goal)
Goal:
M : Type u_1
inst : Nonempty M
inst_1 : Semigroup M
inst_2 : TopologicalSpace M
inst_3 : CompactSpace M
inst_4 : T2Space M
continuous_const_mul : ∀ (r : M), Continuous (Y := M) fun (x : M) ↦ x * r
S : Set (Set M) := {N : Set M | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun (x : Set M) ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
ht : (fun (x : M) ↦ x * m) '' N ∈ S
⊢ (fun (x : M) ↦ x * m) '' N = N
have hts : (fun (x : M) ↦ x * m) '' N ⊆ N
Tactic: apply hN.eq_of_subset
-- Lean 4 proof step from Mathlib.Topology.Algebra.Semigroup (goal)
Goal:
M : Type u_1
inst : Nonempty M
inst_1 : Semigroup M
inst_2 : TopologicalSpace M
inst_3 : CompactSpace M
inst_4 : T2Space M
continuous_const_mul : ∀ (r : M), Continuous (Y := M) fun (x : M) ↦ x * r
S : Set (Set M) := {N : Set M | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun (x : Set M) ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
⊢ (fun (x : M) ↦ x * m) '' N ∈ S
have ht : ∀ (m_1 : M),
  Membership.mem (γ := Set M) ((fun (x : M) ↦ x * m) '' N) m_1 →
    ∀ (m' : M),
      Membership.mem (γ := Set M) ((fun (x : M) ↦ x * m) '' N) m' →
        Membership.mem (γ := Set M) ((fun (x : M) ↦ x * m) '' N) (m_1 * m')
Tactic: refine ⟨(continuous_const_mul m).isClosedMap _ N_closed, ⟨_, ⟨m, hm, rfl⟩⟩, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.Semigroup (hypothesis)
Goal:
M : Type u_1
inst : Nonempty M
inst_1 : Semigroup M
inst_2 : TopologicalSpace M
inst_3 : CompactSpace M
inst_4 : T2Space M
continuous_const_mul : ∀ (r : M), Continuous (Y := M) fun (x : M) ↦ x * r
S : Set (Set M) := {N : Set M | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun (x : Set M) ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
⊢ m * m = m
have scaling_eq_self : (fun (x : M) ↦ x * m) '' N = N
Tactic: have scaling_eq_self : (· * m) '' N = N := by
      apply hN.eq_of_subset
      · refine ⟨(continuous_const_mul m).isClosedMap _ N_closed, ⟨_, ⟨m, hm, rfl⟩⟩, ?_⟩
        rintro _ ⟨m'', hm'', rfl⟩ _ ⟨m', hm', rfl⟩
        exact ⟨m'' * m * m', N_mul _ (N_mul _ hm'' _ hm) _ hm', mul_assoc _ _ _⟩
      · rintro _ ⟨m', hm', rfl⟩
        exact N_mul _ hm' _ hm
-- Lean 4 proof step from Mathlib.Topology.Algebra.Semigroup (goal)
Goal:
M : Type u_1
inst : Nonempty M
inst_1 : Semigroup M
inst_2 : TopologicalSpace M
inst_3 : CompactSpace M
inst_4 : T2Space M
continuous_const_mul : ∀ (r : M), Continuous (Y := M) fun (x : M) ↦ x * r
S : Set (Set M) := {N : Set M | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun (x : Set M) ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
scaling_eq_self : (fun (x : M) ↦ x * m) '' N = N
⊢ N ∩ {m' : M | m' * m = m} = N
have ht : N ∩ {m' : M | m' * m = m} ∈ S
Tactic: apply hN.eq_of_subset
-- Lean 4 proof step from Mathlib.Topology.Algebra.Semigroup (goal)
Goal:
M : Type u_1
inst : Nonempty M
inst_1 : Semigroup M
inst_2 : TopologicalSpace M
inst_3 : CompactSpace M
inst_4 : T2Space M
continuous_const_mul : ∀ (r : M), Continuous (Y := M) fun (x : M) ↦ x * r
S : Set (Set M) := {N : Set M | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun (x : Set M) ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
scaling_eq_self : (fun (x : M) ↦ x * m) '' N = N
ht : N ∩ {m' : M | m' * m = m} ∈ S
⊢ N ∩ {m' : M | m' * m = m} = N
have hts : N ∩ {m' : M | m' * m = m} ⊆ N
Tactic: apply hN.eq_of_subset
-- Lean 4 proof step from Mathlib.Topology.Algebra.Semigroup (goal)
Goal:
M : Type u_1
inst : Nonempty M
inst_1 : Semigroup M
inst_2 : TopologicalSpace M
inst_3 : CompactSpace M
inst_4 : T2Space M
continuous_const_mul : ∀ (r : M), Continuous (Y := M) fun (x : M) ↦ x * r
S : Set (Set M) := {N : Set M | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun (x : Set M) ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
scaling_eq_self : (fun (x : M) ↦ x * m) '' N = N
⊢ N ∩ {m' : M | m' * m = m} ∈ S
have ht : (N ∩ {m' : M | m' * m = m}).Nonempty
Tactic: refine ⟨N_closed.inter ((T1Space.t1 m).preimage (continuous_const_mul m)), ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.Semigroup (goal)
Goal:
M : Type u_1
inst : Nonempty M
inst_1 : Semigroup M
inst_2 : TopologicalSpace M
inst_3 : CompactSpace M
inst_4 : T2Space M
continuous_const_mul : ∀ (r : M), Continuous (Y := M) fun (x : M) ↦ x * r
S : Set (Set M) := {N : Set M | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun (x : Set M) ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
scaling_eq_self : (fun (x : M) ↦ x * m) '' N = N
ht : (N ∩ {m' : M | m' * m = m}).Nonempty
⊢ N ∩ {m' : M | m' * m = m} ∈ S
have ht₁ : ∀ m_1 ∈ N ∩ {m' : M | m' * m = m}, ∀ m' ∈ N ∩ {m' : M | m' * m = m}, m_1 * m' ∈ N ∩ {m' : M | m' * m = m}
Tactic: refine ⟨N_closed.inter ((T1Space.t1 m).preimage (continuous_const_mul m)), ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.Semigroup (goal)
Goal:
M : Type u_1
inst : Nonempty M
inst_1 : Semigroup M
inst_2 : TopologicalSpace M
inst_3 : CompactSpace M
inst_4 : T2Space M
continuous_const_mul : ∀ (r : M), Continuous (Y := M) fun (x : M) ↦ x * r
S : Set (Set M) := {N : Set M | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun (x : Set M) ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
scaling_eq_self : (fun (x : M) ↦ x * m) '' N = N
⊢ (N ∩ {m' : M | m' * m = m}).Nonempty
have ht : ∀ (hm : Membership.mem (γ := Set M) ((fun (x : M) ↦ x * m) '' N) m) (scaling_eq_self : (fun (x : M) ↦ x * m) '' N = N),
  (N ∩ {m' : M | m' * m = m}).Nonempty
Tactic: rwa [← scaling_eq_self] at hm
Cleared hypotheses: hm, scaling_eq_self
-- Lean 4 proof step from Mathlib.Topology.Algebra.Semigroup (goal)
Goal:
M : Type u_1
inst : Nonempty M
inst_1 : Semigroup M
inst_2 : TopologicalSpace M
inst_3 : CompactSpace M
inst_4 : T2Space M
continuous_const_mul : ∀ (r : M), Continuous (Y := M) fun (x : M) ↦ x * r
S : Set (Set M) := {N : Set M | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun (x : Set M) ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
scaling_eq_self : (fun (x : M) ↦ x * m) '' N = N
m'' : M
mem'' : m'' ∈ N
eq'' : m'' * m = m
m' : M
mem' : m' ∈ N
eq' : m' * m = m
⊢ m'' * m' ∈ N ∩ {m' : M | m' * m = m}
have ht : Membership.mem (γ := Set M) {m' : M | m' * m = m} (m'' * m')
Tactic: refine ⟨N_mul _ mem'' _ mem', ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.Semigroup (goal)
Goal:
M : Type u_1
inst : Nonempty M
inst_1 : Semigroup M
inst_2 : TopologicalSpace M
inst_3 : CompactSpace M
inst_4 : T2Space M
continuous_const_mul : ∀ (r : M), Continuous (Y := M) fun (x : M) ↦ x * r
S : Set (Set M) := {N : Set M | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun (x : Set M) ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
scaling_eq_self : (fun (x : M) ↦ x * m) '' N = N
m'' : M
mem'' : m'' ∈ N
eq'' : m'' * m = m
m' : M
mem' : m' ∈ N
eq' : m' * m = m
⊢ Membership.mem (γ := Set M) {m' : M | m' * m = m} (m'' * m')
have ht : m'' * m' * m = m
Tactic: rw [Set.mem_setOf_eq, mul_assoc, eq', eq'']
-- Lean 4 proof step from Mathlib.Topology.Algebra.Semigroup (goal)
Goal:
M : Type u_1
inst : Nonempty M
inst_1 : Semigroup M
inst_2 : TopologicalSpace M
inst_3 : CompactSpace M
inst_4 : T2Space M
continuous_const_mul : ∀ (r : M), Continuous (Y := M) fun (x : M) ↦ x * r
S : Set (Set M) := {N : Set M | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun (x : Set M) ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
scaling_eq_self : (fun (x : M) ↦ x * m) '' N = N
m'' : M
mem'' : m'' ∈ N
eq'' : m'' * m = m
m' : M
mem' : m' ∈ N
eq' : m' * m = m
ht : m'' * m' * m = m
⊢ Membership.mem (γ := Set M) {m' : M | m' * m = m} (m'' * m')
have ht₁ : m'' * (m' * m) = m
Tactic: rw [Set.mem_setOf_eq, mul_assoc, eq', eq'']
-- Lean 4 proof step from Mathlib.Topology.Algebra.Semigroup (goal)
Goal:
M : Type u_1
inst : Nonempty M
inst_1 : Semigroup M
inst_2 : TopologicalSpace M
inst_3 : CompactSpace M
inst_4 : T2Space M
continuous_const_mul : ∀ (r : M), Continuous (Y := M) fun (x : M) ↦ x * r
S : Set (Set M) := {N : Set M | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun (x : Set M) ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
scaling_eq_self : (fun (x : M) ↦ x * m) '' N = N
m'' : M
mem'' : m'' ∈ N
eq'' : m'' * m = m
m' : M
mem' : m' ∈ N
eq' : m' * m = m
ht : m'' * m' * m = m
ht₁ : m'' * (m' * m) = m
ht₂ : m'' * m = m
⊢ Membership.mem (γ := Set M) {m' : M | m' * m = m} (m'' * m')
have ht₃ : m = m
Tactic: rw [Set.mem_setOf_eq, mul_assoc, eq', eq'']
-- Lean 4 proof step from Mathlib.Topology.Algebra.Semigroup (hypothesis)
Goal:
M : Type u_1
inst : Nonempty M
inst_1 : Semigroup M
inst_2 : TopologicalSpace M
inst_3 : CompactSpace M
inst_4 : T2Space M
continuous_const_mul : ∀ (r : M), Continuous (Y := M) fun (x : M) ↦ x * r
S : Set (Set M) := {N : Set M | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun (x : Set M) ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
scaling_eq_self : (fun (x : M) ↦ x * m) '' N = N
⊢ m * m = m
have absorbing_eq_self : N ∩ {m' : M | m' * m = m} = N
Tactic: have absorbing_eq_self : N ∩ { m' | m' * m = m } = N := by
      apply hN.eq_of_subset
      · refine ⟨N_closed.inter ((T1Space.t1 m).preimage (continuous_const_mul m)), ?_, ?_⟩
        · rwa [← scaling_eq_self] at hm
        · rintro m'' ⟨mem'', eq'' : _ = m⟩ m' ⟨mem', eq' : _ = m⟩
          refine ⟨N_mul _ mem'' _ mem', ?_⟩
          rw [Set.mem_setOf_eq, mul_assoc, eq', eq'']
      apply Set.inter_subset_left
-- Lean 4 proof step from Mathlib.Topology.Algebra.Semigroup (hypothesis)
Goal:
M : Type u_1
inst : Nonempty M
inst_1 : Semigroup M
inst_2 : TopologicalSpace M
inst_3 : CompactSpace M
inst_4 : T2Space M
continuous_const_mul : ∀ (r : M), Continuous (Y := M) fun (x : M) ↦ x * r
S : Set (Set M) := {N : Set M | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun (x : Set M) ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
scaling_eq_self : (fun (x : M) ↦ x * m) '' N = N
absorbing_eq_self : N ∩ {m' : M | m' * m = m} = N
⊢ m * m = m
have hm₁ : m ∈ N ∩ {m' : M | m' * m = m}
Tactic: rw [← absorbing_eq_self] at hm
Cleared hypotheses: hm, scaling_eq_self, absorbing_eq_self