awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Algebra.IsUniformGroup.Basic (hypothesis)
Goal:
G : Type u
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : FirstCountableTopology G
N : Subgroup G
inst_4 : N.Normal
inst_5 : CompleteSpace G
this_1 : UniformSpace (G ⧸ N) := IsTopologicalGroup.rightUniformSpace (G ⧸ N)
this_2 : UniformSpace G := IsTopologicalGroup.rightUniformSpace G
this : IsCountablyGenerated.{u} (α := (G ⧸ N) × G ⧸ N) (𝓤 (G ⧸ N))
u : ℕ → Set G
hu : HasAntitoneBasis (𝓝 1) u
u_mul : ∀ (n : ℕ), u (n + 1) * u (n + 1) ⊆ u n
hv : HasBasis.{u, 1} (α := G ⧸ N) (𝓝 1) (fun (x : ℕ) ↦ True) fun (x : ℕ) ↦ mk '' u x
v_anti : Antitone.{0, u} (β := Set (G ⧸ N)) fun (x : ℕ) ↦ mk '' u x
x : ℕ → G ⧸ N
hx : CauchySeq x
key₀ :
  ∀ (i j : ℕ),
    ∃ (M : ℕ),
      j < M ∧
        ∀ (a b : ℕ), M ≤ a → M ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u i ∧ x a = (↑g' : G ⧸ N)
φ : ℕ → ℕ :=
  fun (n : ℕ) ↦
    Nat.recOn (motive := fun (x : ℕ) ↦ ℕ) n
      (Classical.choose (p := fun (x_1 : ℕ) ↦
        0 < x_1 ∧
          ∀ (a b : ℕ),
            x_1 ≤ a → x_1 ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u 0 ∧ x a = (↑g' : G ⧸ N))
        ⋯)
      fun (k yk : ℕ) ↦
      Classical.choose (p := fun (x_1 : ℕ) ↦
        yk < x_1 ∧
          ∀ (a b : ℕ),
            x_1 ≤ a → x_1 ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u (k + 1) ∧ x a = (↑g' : G ⧸ N))
        ⋯
⊢ ∃ (a : G ⧸ N), Tendsto x atTop (𝓝 a)
have hφ : ∀ (n : ℕ),
  φ n < φ (n + 1) ∧
    ∀ (a b : ℕ),
      φ (n + 1) ≤ a →
        φ (n + 1) ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u (n + 1) ∧ x a = (↑g' : G ⧸ N)
Tactic: have hφ :
    ∀ n : ℕ,
      φ n < φ (n + 1) ∧
        ∀ a b : ℕ,
          φ (n + 1) ≤ a →
            φ (n + 1) ≤ b → ∀ g : G, x b = g → ∃ g' : G, g / g' ∈ u (n + 1) ∧ x a = g' :=
    fun n => Classical.choose_spec (key₀ (n + 1) (φ n))
-- Lean 4 proof step from Mathlib.Topology.Algebra.IsUniformGroup.Basic (hypothesis)
Goal:
G : Type u
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : FirstCountableTopology G
N : Subgroup G
inst_4 : N.Normal
inst_5 : CompleteSpace G
this_1 : UniformSpace (G ⧸ N) := IsTopologicalGroup.rightUniformSpace (G ⧸ N)
this_2 : UniformSpace G := IsTopologicalGroup.rightUniformSpace G
this : IsCountablyGenerated.{u} (α := (G ⧸ N) × G ⧸ N) (𝓤 (G ⧸ N))
u : ℕ → Set G
hu : HasAntitoneBasis (𝓝 1) u
u_mul : ∀ (n : ℕ), u (n + 1) * u (n + 1) ⊆ u n
hv : HasBasis.{u, 1} (α := G ⧸ N) (𝓝 1) (fun (x : ℕ) ↦ True) fun (x : ℕ) ↦ mk '' u x
v_anti : Antitone.{0, u} (β := Set (G ⧸ N)) fun (x : ℕ) ↦ mk '' u x
x : ℕ → G ⧸ N
hx : CauchySeq x
key₀ :
  ∀ (i j : ℕ),
    ∃ (M : ℕ),
      j < M ∧
        ∀ (a b : ℕ), M ≤ a → M ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u i ∧ x a = (↑g' : G ⧸ N)
φ : ℕ → ℕ :=
  fun (n : ℕ) ↦
    Nat.recOn (motive := fun (x : ℕ) ↦ ℕ) n
      (Classical.choose (p := fun (x_1 : ℕ) ↦
        0 < x_1 ∧
          ∀ (a b : ℕ),
            x_1 ≤ a → x_1 ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u 0 ∧ x a = (↑g' : G ⧸ N))
        ⋯)
      fun (k yk : ℕ) ↦
      Classical.choose (p := fun (x_1 : ℕ) ↦
        yk < x_1 ∧
          ∀ (a b : ℕ),
            x_1 ≤ a → x_1 ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u (k + 1) ∧ x a = (↑g' : G ⧸ N))
        ⋯
hφ :
  ∀ (n : ℕ),
    φ n < φ (n + 1) ∧
      ∀ (a b : ℕ),
        φ (n + 1) ≤ a →
          φ (n + 1) ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u (n + 1) ∧ x a = (↑g' : G ⧸ N)
x' : (n : ℕ) → (g : G) ×' x (φ (n + 1)) = (↑g : G ⧸ N) :=
  fun (n : ℕ) ↦
    Nat.recOn (motive := fun (x_1 : ℕ) ↦ (g : G) ×' x (φ (x_1 + 1)) = (↑g : G ⧸ N)) n
      (PSigma.mk (β := fun (g : G) ↦ x (φ (Nat.zero + 1)) = (↑g : G ⧸ N))
        (Classical.choose (p := fun (x_1 : G) ↦ (↑x_1 : G ⧸ N) = x (φ 1)) ⋯) ⋯)
      fun (k : ℕ) (hk : (g : G) ×' x (φ (k + 1)) = (↑g : G ⧸ N)) ↦
      PSigma.mk (β := fun (g : G) ↦ x (φ (k.succ + 1)) = (↑g : G ⧸ N))
        (Classical.choose (p := fun (x_1 : G) ↦
          hk.fst (β := fun (g : G) ↦ x (φ (k + 1)) = (↑g : G ⧸ N)) / x_1 ∈ u (k + 1) ∧
            x (φ (k + 1 + 1)) = (↑x_1 : G ⧸ N))
          ⋯)
        ⋯
⊢ ∃ (a : G ⧸ N), Tendsto x atTop (𝓝 a)
have hx' : ∀ (n : ℕ),
  (x' n).fst (β := fun (g : G) ↦ x (φ (n + 1)) = (↑g : G ⧸ N)) /
      PSigma.fst (β := fun (g : G) ↦ x (φ (n + 1 + 1)) = (↑g : G ⧸ N))
        (x' (n + 1) : (g : G) ×' x (φ (n + 1 + 1)) = (↑g : G ⧸ N)) ∈
    u (n + 1)
Tactic: have hx' : ∀ n : ℕ, (x' n).fst / (x' (n + 1)).fst ∈ u (n + 1) := fun n =>
    (Classical.choose_spec <| (hφ n).2 _ _ (hφ (n + 1)).1.le le_rfl (x' n).fst (x' n).snd).1
-- Lean 4 proof step from Mathlib.Topology.Algebra.IsUniformGroup.Basic (hypothesis)
Goal:
G : Type u
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : FirstCountableTopology G
N : Subgroup G
inst_4 : N.Normal
inst_5 : CompleteSpace G
this_1 : UniformSpace (G ⧸ N) := IsTopologicalGroup.rightUniformSpace (G ⧸ N)
this_2 : UniformSpace G := IsTopologicalGroup.rightUniformSpace G
this : IsCountablyGenerated.{u} (α := (G ⧸ N) × G ⧸ N) (𝓤 (G ⧸ N))
u : ℕ → Set G
hu : HasAntitoneBasis (𝓝 1) u
u_mul : ∀ (n : ℕ), u (n + 1) * u (n + 1) ⊆ u n
hv : HasBasis.{u, 1} (α := G ⧸ N) (𝓝 1) (fun (x : ℕ) ↦ True) fun (x : ℕ) ↦ mk '' u x
v_anti : Antitone.{0, u} (β := Set (G ⧸ N)) fun (x : ℕ) ↦ mk '' u x
x : ℕ → G ⧸ N
hx : CauchySeq x
key₀ :
  ∀ (i j : ℕ),
    ∃ (M : ℕ),
      j < M ∧
        ∀ (a b : ℕ), M ≤ a → M ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u i ∧ x a = (↑g' : G ⧸ N)
φ : ℕ → ℕ :=
  fun (n : ℕ) ↦
    Nat.recOn (motive := fun (x : ℕ) ↦ ℕ) n
      (Classical.choose (p := fun (x_1 : ℕ) ↦
        0 < x_1 ∧
          ∀ (a b : ℕ),
            x_1 ≤ a → x_1 ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u 0 ∧ x a = (↑g' : G ⧸ N))
        ⋯)
      fun (k yk : ℕ) ↦
      Classical.choose (p := fun (x_1 : ℕ) ↦
        yk < x_1 ∧
          ∀ (a b : ℕ),
            x_1 ≤ a → x_1 ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u (k + 1) ∧ x a = (↑g' : G ⧸ N))
        ⋯
hφ :
  ∀ (n : ℕ),
    φ n < φ (n + 1) ∧
      ∀ (a b : ℕ),
        φ (n + 1) ≤ a →
          φ (n + 1) ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u (n + 1) ∧ x a = (↑g' : G ⧸ N)
x' : (n : ℕ) → (g : G) ×' x (φ (n + 1)) = (↑g : G ⧸ N) :=
  fun (n : ℕ) ↦
    Nat.recOn (motive := fun (x_1 : ℕ) ↦ (g : G) ×' x (φ (x_1 + 1)) = (↑g : G ⧸ N)) n
      (PSigma.mk (β := fun (g : G) ↦ x (φ (Nat.zero + 1)) = (↑g : G ⧸ N))
        (Classical.choose (p := fun (x_1 : G) ↦ (↑x_1 : G ⧸ N) = x (φ 1)) ⋯) ⋯)
      fun (k : ℕ) (hk : (g : G) ×' x (φ (k + 1)) = (↑g : G ⧸ N)) ↦
      PSigma.mk (β := fun (g : G) ↦ x (φ (k.succ + 1)) = (↑g : G ⧸ N))
        (Classical.choose (p := fun (x_1 : G) ↦
          hk.fst (β := fun (g : G) ↦ x (φ (k + 1)) = (↑g : G ⧸ N)) / x_1 ∈ u (k + 1) ∧
            x (φ (k + 1 + 1)) = (↑x_1 : G ⧸ N))
          ⋯)
        ⋯
hx' :
  ∀ (n : ℕ),
    (x' n).fst (β := fun (g : G) ↦ x (φ (n + 1)) = (↑g : G ⧸ N)) /
        PSigma.fst (β := fun (g : G) ↦ x (φ (n + 1 + 1)) = (↑g : G ⧸ N))
          (x' (n + 1) : (g : G) ×' x (φ (n + 1 + 1)) = (↑g : G ⧸ N)) ∈
      u (n + 1)
⊢ CauchySeq.{u, 0} (α := G) fun (n : ℕ) ↦ (x' n).fst (β := fun (g : G) ↦ x (φ (n + 1)) = (↑g : G ⧸ N))
have h𝓤G : (𝓤 G).HasBasis (fun (x : ℕ) ↦ True) fun (i : ℕ) ↦ {x : G × G | x.2 / x.1 ∈ u i}
Tactic: have h𝓤G : (𝓤 G).HasBasis (fun _ => True) fun i => { x | x.snd / x.fst ∈ u i } := by
      simpa [uniformity_eq_comap_nhds_one', div_eq_mul_inv] using hu.toHasBasis.comap _
-- Lean 4 proof step from Mathlib.Topology.Algebra.IsUniformGroup.Basic (goal)
Goal:
G : Type u
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : FirstCountableTopology G
N : Subgroup G
inst_4 : N.Normal
inst_5 : CompleteSpace G
this_1 : UniformSpace (G ⧸ N) := IsTopologicalGroup.rightUniformSpace (G ⧸ N)
this_2 : UniformSpace G := IsTopologicalGroup.rightUniformSpace G
this : IsCountablyGenerated.{u} (α := (G ⧸ N) × G ⧸ N) (𝓤 (G ⧸ N))
u : ℕ → Set G
hu : HasAntitoneBasis (𝓝 1) u
u_mul : ∀ (n : ℕ), u (n + 1) * u (n + 1) ⊆ u n
hv : HasBasis.{u, 1} (α := G ⧸ N) (𝓝 1) (fun (x : ℕ) ↦ True) fun (x : ℕ) ↦ mk '' u x
v_anti : Antitone.{0, u} (β := Set (G ⧸ N)) fun (x : ℕ) ↦ mk '' u x
x : ℕ → G ⧸ N
hx : CauchySeq x
key₀ :
  ∀ (i j : ℕ),
    ∃ (M : ℕ),
      j < M ∧
        ∀ (a b : ℕ), M ≤ a → M ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u i ∧ x a = (↑g' : G ⧸ N)
φ : ℕ → ℕ :=
  fun (n : ℕ) ↦
    Nat.recOn (motive := fun (x : ℕ) ↦ ℕ) n
      (Classical.choose (p := fun (x_1 : ℕ) ↦
        0 < x_1 ∧
          ∀ (a b : ℕ),
            x_1 ≤ a → x_1 ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u 0 ∧ x a = (↑g' : G ⧸ N))
        ⋯)
      fun (k yk : ℕ) ↦
      Classical.choose (p := fun (x_1 : ℕ) ↦
        yk < x_1 ∧
          ∀ (a b : ℕ),
            x_1 ≤ a → x_1 ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u (k + 1) ∧ x a = (↑g' : G ⧸ N))
        ⋯
hφ :
  ∀ (n : ℕ),
    φ n < φ (n + 1) ∧
      ∀ (a b : ℕ),
        φ (n + 1) ≤ a →
          φ (n + 1) ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u (n + 1) ∧ x a = (↑g' : G ⧸ N)
x' : (n : ℕ) → (g : G) ×' x (φ (n + 1)) = (↑g : G ⧸ N) :=
  fun (n : ℕ) ↦
    Nat.recOn (motive := fun (x_1 : ℕ) ↦ (g : G) ×' x (φ (x_1 + 1)) = (↑g : G ⧸ N)) n
      (PSigma.mk (β := fun (g : G) ↦ x (φ (Nat.zero + 1)) = (↑g : G ⧸ N))
        (Classical.choose (p := fun (x_1 : G) ↦ (↑x_1 : G ⧸ N) = x (φ 1)) ⋯) ⋯)
      fun (k : ℕ) (hk : (g : G) ×' x (φ (k + 1)) = (↑g : G ⧸ N)) ↦
      PSigma.mk (β := fun (g : G) ↦ x (φ (k.succ + 1)) = (↑g : G ⧸ N))
        (Classical.choose (p := fun (x_1 : G) ↦
          hk.fst (β := fun (g : G) ↦ x (φ (k + 1)) = (↑g : G ⧸ N)) / x_1 ∈ u (k + 1) ∧
            x (φ (k + 1 + 1)) = (↑x_1 : G ⧸ N))
          ⋯)
        ⋯
hx' :
  ∀ (n : ℕ),
    (x' n).fst (β := fun (g : G) ↦ x (φ (n + 1)) = (↑g : G ⧸ N)) /
        PSigma.fst (β := fun (g : G) ↦ x (φ (n + 1 + 1)) = (↑g : G ⧸ N))
          (x' (n + 1) : (g : G) ×' x (φ (n + 1 + 1)) = (↑g : G ⧸ N)) ∈
      u (n + 1)
h𝓤G : (𝓤 G).HasBasis (fun (x : ℕ) ↦ True) fun (i : ℕ) ↦ {x : G × G | x.2 / x.1 ∈ u i}
⊢ CauchySeq.{u, 0} (α := G) fun (n : ℕ) ↦ (x' n).fst (β := fun (g : G) ↦ x (φ (n + 1)) = (↑g : G ⧸ N))
have rw : ∀ (i : ℕ),
  True →
    ∃ (N_1 : ℕ),
      ∀ n ≥ N_1,
        Membership.mem (γ := SetRel G G) {x : G × G | x.2 / x.1 ∈ u i}
          ((x' n).fst (β := fun (g : G) ↦ x (φ (n + 1)) = (↑g : G ⧸ N)),
            (x' N_1).fst (β := fun (g : G) ↦ x (φ (N_1 + 1)) = (↑g : G ⧸ N)))
Tactic: rw [h𝓤G.cauchySeq_iff']
-- Lean 4 proof step from Mathlib.Topology.Algebra.IsUniformGroup.Basic (goal)
Goal:
G : Type u
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : FirstCountableTopology G
N : Subgroup G
inst_4 : N.Normal
inst_5 : CompleteSpace G
this_1 : UniformSpace (G ⧸ N) := IsTopologicalGroup.rightUniformSpace (G ⧸ N)
this_2 : UniformSpace G := IsTopologicalGroup.rightUniformSpace G
this : IsCountablyGenerated.{u} (α := (G ⧸ N) × G ⧸ N) (𝓤 (G ⧸ N))
u : ℕ → Set G
hu : HasAntitoneBasis (𝓝 1) u
u_mul : ∀ (n : ℕ), u (n + 1) * u (n + 1) ⊆ u n
hv : HasBasis.{u, 1} (α := G ⧸ N) (𝓝 1) (fun (x : ℕ) ↦ True) fun (x : ℕ) ↦ mk '' u x
v_anti : Antitone.{0, u} (β := Set (G ⧸ N)) fun (x : ℕ) ↦ mk '' u x
x : ℕ → G ⧸ N
hx : CauchySeq x
key₀ :
  ∀ (i j : ℕ),
    ∃ (M : ℕ),
      j < M ∧
        ∀ (a b : ℕ), M ≤ a → M ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u i ∧ x a = (↑g' : G ⧸ N)
φ : ℕ → ℕ :=
  fun (n : ℕ) ↦
    Nat.recOn (motive := fun (x : ℕ) ↦ ℕ) n
      (Classical.choose (p := fun (x_1 : ℕ) ↦
        0 < x_1 ∧
          ∀ (a b : ℕ),
            x_1 ≤ a → x_1 ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u 0 ∧ x a = (↑g' : G ⧸ N))
        ⋯)
      fun (k yk : ℕ) ↦
      Classical.choose (p := fun (x_1 : ℕ) ↦
        yk < x_1 ∧
          ∀ (a b : ℕ),
            x_1 ≤ a → x_1 ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u (k + 1) ∧ x a = (↑g' : G ⧸ N))
        ⋯
hφ :
  ∀ (n : ℕ),
    φ n < φ (n + 1) ∧
      ∀ (a b : ℕ),
        φ (n + 1) ≤ a →
          φ (n + 1) ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u (n + 1) ∧ x a = (↑g' : G ⧸ N)
x' : (n : ℕ) → (g : G) ×' x (φ (n + 1)) = (↑g : G ⧸ N) :=
  fun (n : ℕ) ↦
    Nat.recOn (motive := fun (x_1 : ℕ) ↦ (g : G) ×' x (φ (x_1 + 1)) = (↑g : G ⧸ N)) n
      (PSigma.mk (β := fun (g : G) ↦ x (φ (Nat.zero + 1)) = (↑g : G ⧸ N))
        (Classical.choose (p := fun (x_1 : G) ↦ (↑x_1 : G ⧸ N) = x (φ 1)) ⋯) ⋯)
      fun (k : ℕ) (hk : (g : G) ×' x (φ (k + 1)) = (↑g : G ⧸ N)) ↦
      PSigma.mk (β := fun (g : G) ↦ x (φ (k.succ + 1)) = (↑g : G ⧸ N))
        (Classical.choose (p := fun (x_1 : G) ↦
          hk.fst (β := fun (g : G) ↦ x (φ (k + 1)) = (↑g : G ⧸ N)) / x_1 ∈ u (k + 1) ∧
            x (φ (k + 1 + 1)) = (↑x_1 : G ⧸ N))
          ⋯)
        ⋯
hx' :
  ∀ (n : ℕ),
    (x' n).fst (β := fun (g : G) ↦ x (φ (n + 1)) = (↑g : G ⧸ N)) /
        PSigma.fst (β := fun (g : G) ↦ x (φ (n + 1 + 1)) = (↑g : G ⧸ N))
          (x' (n + 1) : (g : G) ×' x (φ (n + 1 + 1)) = (↑g : G ⧸ N)) ∈
      u (n + 1)
h𝓤G : (𝓤 G).HasBasis (fun (x : ℕ) ↦ True) fun (i : ℕ) ↦ {x : G × G | x.2 / x.1 ∈ u i}
⊢ ∀ (i : ℕ),
    True →
      ∃ (N_1 : ℕ),
        ∀ n ≥ N_1,
          Membership.mem (γ := SetRel G G) {x : G × G | x.2 / x.1 ∈ u i}
            ((x' n).fst (β := fun (g : G) ↦ x (φ (n + 1)) = (↑g : G ⧸ N)),
              (x' N_1).fst (β := fun (g : G) ↦ x (φ (N_1 + 1)) = (↑g : G ⧸ N)))
have simp : ∀ (i : ℕ),
  ∃ (N_1 : ℕ),
    ∀ n ≥ N_1,
      (x' N_1).fst (β := fun (g : G) ↦ x (φ (N_1 + 1)) = (↑g : G ⧸ N)) /
          (x' n).fst (β := fun (g : G) ↦ x (φ (n + 1)) = (↑g : G ⧸ N)) ∈
        u i
Tactic: simp only [mem_setOf_eq, forall_true_left]
-- Lean 4 proof step from Mathlib.Topology.Algebra.IsUniformGroup.Basic (goal)
Goal:
G : Type u
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : FirstCountableTopology G
N : Subgroup G
inst_4 : N.Normal
inst_5 : CompleteSpace G
this_1 : UniformSpace (G ⧸ N) := IsTopologicalGroup.rightUniformSpace (G ⧸ N)
this_2 : UniformSpace G := IsTopologicalGroup.rightUniformSpace G
this : IsCountablyGenerated.{u} (α := (G ⧸ N) × G ⧸ N) (𝓤 (G ⧸ N))
u : ℕ → Set G
hu : HasAntitoneBasis (𝓝 1) u
u_mul : ∀ (n : ℕ), u (n + 1) * u (n + 1) ⊆ u n
hv : HasBasis.{u, 1} (α := G ⧸ N) (𝓝 1) (fun (x : ℕ) ↦ True) fun (x : ℕ) ↦ mk '' u x
v_anti : Antitone.{0, u} (β := Set (G ⧸ N)) fun (x : ℕ) ↦ mk '' u x
x : ℕ → G ⧸ N
hx : CauchySeq x
key₀ :
  ∀ (i j : ℕ),
    ∃ (M : ℕ),
      j < M ∧
        ∀ (a b : ℕ), M ≤ a → M ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u i ∧ x a = (↑g' : G ⧸ N)
φ : ℕ → ℕ :=
  fun (n : ℕ) ↦
    Nat.recOn (motive := fun (x : ℕ) ↦ ℕ) n
      (Classical.choose (p := fun (x_1 : ℕ) ↦
        0 < x_1 ∧
          ∀ (a b : ℕ),
            x_1 ≤ a → x_1 ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u 0 ∧ x a = (↑g' : G ⧸ N))
        ⋯)
      fun (k yk : ℕ) ↦
      Classical.choose (p := fun (x_1 : ℕ) ↦
        yk < x_1 ∧
          ∀ (a b : ℕ),
            x_1 ≤ a → x_1 ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u (k + 1) ∧ x a = (↑g' : G ⧸ N))
        ⋯
hφ :
  ∀ (n : ℕ),
    φ n < φ (n + 1) ∧
      ∀ (a b : ℕ),
        φ (n + 1) ≤ a →
          φ (n + 1) ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u (n + 1) ∧ x a = (↑g' : G ⧸ N)
x' : (n : ℕ) → (g : G) ×' x (φ (n + 1)) = (↑g : G ⧸ N) :=
  fun (n : ℕ) ↦
    Nat.recOn (motive := fun (x_1 : ℕ) ↦ (g : G) ×' x (φ (x_1 + 1)) = (↑g : G ⧸ N)) n
      (PSigma.mk (β := fun (g : G) ↦ x (φ (Nat.zero + 1)) = (↑g : G ⧸ N))
        (Classical.choose (p := fun (x_1 : G) ↦ (↑x_1 : G ⧸ N) = x (φ 1)) ⋯) ⋯)
      fun (k : ℕ) (hk : (g : G) ×' x (φ (k + 1)) = (↑g : G ⧸ N)) ↦
      PSigma.mk (β := fun (g : G) ↦ x (φ (k.succ + 1)) = (↑g : G ⧸ N))
        (Classical.choose (p := fun (x_1 : G) ↦
          hk.fst (β := fun (g : G) ↦ x (φ (k + 1)) = (↑g : G ⧸ N)) / x_1 ∈ u (k + 1) ∧
            x (φ (k + 1 + 1)) = (↑x_1 : G ⧸ N))
          ⋯)
        ⋯
hx' :
  ∀ (n : ℕ),
    (x' n).fst (β := fun (g : G) ↦ x (φ (n + 1)) = (↑g : G ⧸ N)) /
        PSigma.fst (β := fun (g : G) ↦ x (φ (n + 1 + 1)) = (↑g : G ⧸ N))
          (x' (n + 1) : (g : G) ×' x (φ (n + 1 + 1)) = (↑g : G ⧸ N)) ∈
      u (n + 1)
x'_cauchy : CauchySeq.{u, 0} (α := G) fun (n : ℕ) ↦ (x' n).fst (β := fun (g : G) ↦ x (φ (n + 1)) = (↑g : G ⧸ N))
⊢ ∃ (a : G ⧸ N), Tendsto x atTop (𝓝 a)
have rcases : ∀ (x₀ : G) (hx₀ : Tendsto (fun (n : ℕ) ↦ (x' n).fst (β := fun (g : G) ↦ x (φ (n + 1)) = (↑g : G ⧸ N))) atTop (𝓝 x₀)),
  ∃ (a : G ⧸ N), Tendsto x atTop (𝓝 a)
Tactic: rcases cauchySeq_tendsto_of_complete x'_cauchy with ⟨x₀, hx₀⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.IsUniformGroup.Basic (goal)
Goal:
G : Type u
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : FirstCountableTopology G
N : Subgroup G
inst_4 : N.Normal
inst_5 : CompleteSpace G
this_1 : UniformSpace (G ⧸ N) := IsTopologicalGroup.rightUniformSpace (G ⧸ N)
this_2 : UniformSpace G := IsTopologicalGroup.rightUniformSpace G
this : IsCountablyGenerated.{u} (α := (G ⧸ N) × G ⧸ N) (𝓤 (G ⧸ N))
u : ℕ → Set G
hu : HasAntitoneBasis (𝓝 1) u
u_mul : ∀ (n : ℕ), u (n + 1) * u (n + 1) ⊆ u n
hv : HasBasis.{u, 1} (α := G ⧸ N) (𝓝 1) (fun (x : ℕ) ↦ True) fun (x : ℕ) ↦ mk '' u x
v_anti : Antitone.{0, u} (β := Set (G ⧸ N)) fun (x : ℕ) ↦ mk '' u x
x : ℕ → G ⧸ N
hx : CauchySeq x
key₀ :
  ∀ (i j : ℕ),
    ∃ (M : ℕ),
      j < M ∧
        ∀ (a b : ℕ), M ≤ a → M ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u i ∧ x a = (↑g' : G ⧸ N)
φ : ℕ → ℕ :=
  fun (n : ℕ) ↦
    Nat.recOn (motive := fun (x : ℕ) ↦ ℕ) n
      (Classical.choose (p := fun (x_1 : ℕ) ↦
        0 < x_1 ∧
          ∀ (a b : ℕ),
            x_1 ≤ a → x_1 ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u 0 ∧ x a = (↑g' : G ⧸ N))
        ⋯)
      fun (k yk : ℕ) ↦
      Classical.choose (p := fun (x_1 : ℕ) ↦
        yk < x_1 ∧
          ∀ (a b : ℕ),
            x_1 ≤ a → x_1 ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u (k + 1) ∧ x a = (↑g' : G ⧸ N))
        ⋯
hφ :
  ∀ (n : ℕ),
    φ n < φ (n + 1) ∧
      ∀ (a b : ℕ),
        φ (n + 1) ≤ a →
          φ (n + 1) ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u (n + 1) ∧ x a = (↑g' : G ⧸ N)
x' : (n : ℕ) → (g : G) ×' x (φ (n + 1)) = (↑g : G ⧸ N) :=
  fun (n : ℕ) ↦
    Nat.recOn (motive := fun (x_1 : ℕ) ↦ (g : G) ×' x (φ (x_1 + 1)) = (↑g : G ⧸ N)) n
      (PSigma.mk (β := fun (g : G) ↦ x (φ (Nat.zero + 1)) = (↑g : G ⧸ N))
        (Classical.choose (p := fun (x_1 : G) ↦ (↑x_1 : G ⧸ N) = x (φ 1)) ⋯) ⋯)
      fun (k : ℕ) (hk : (g : G) ×' x (φ (k + 1)) = (↑g : G ⧸ N)) ↦
      PSigma.mk (β := fun (g : G) ↦ x (φ (k.succ + 1)) = (↑g : G ⧸ N))
        (Classical.choose (p := fun (x_1 : G) ↦
          hk.fst (β := fun (g : G) ↦ x (φ (k + 1)) = (↑g : G ⧸ N)) / x_1 ∈ u (k + 1) ∧
            x (φ (k + 1 + 1)) = (↑x_1 : G ⧸ N))
          ⋯)
        ⋯
hx' :
  ∀ (n : ℕ),
    (x' n).fst (β := fun (g : G) ↦ x (φ (n + 1)) = (↑g : G ⧸ N)) /
        PSigma.fst (β := fun (g : G) ↦ x (φ (n + 1 + 1)) = (↑g : G ⧸ N))
          (x' (n + 1) : (g : G) ×' x (φ (n + 1 + 1)) = (↑g : G ⧸ N)) ∈
      u (n + 1)
x'_cauchy : CauchySeq.{u, 0} (α := G) fun (n : ℕ) ↦ (x' n).fst (β := fun (g : G) ↦ x (φ (n + 1)) = (↑g : G ⧸ N))
x₀ : G
hx₀ : Tendsto (fun (n : ℕ) ↦ (x' n).fst (β := fun (g : G) ↦ x (φ (n + 1)) = (↑g : G ⧸ N))) atTop (𝓝 x₀)
⊢ ∃ (a : G ⧸ N), Tendsto x atTop (𝓝 a)
have tendsto_nhds_of_cauchySeq_of_subseq : Tendsto (α := ℕ) (x ∘ fun (n : ℕ) ↦ φ (n + 1)) atTop (𝓝 (↑x₀ : G ⧸ N))
Tactic: refine
    ⟨↑x₀,
      tendsto_nhds_of_cauchySeq_of_subseq hx
        (strictMono_nat_of_lt_succ fun n => (hφ (n + 1)).1).tendsto_atTop ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.IsUniformGroup.Basic (goal)
Goal:
G : Type u
inst : Group G
inst_1 : TopologicalSpace G
inst_2 : IsTopologicalGroup G
inst_3 : FirstCountableTopology G
N : Subgroup G
inst_4 : N.Normal
inst_5 : CompleteSpace G
this_1 : UniformSpace (G ⧸ N) := IsTopologicalGroup.rightUniformSpace (G ⧸ N)
this_2 : UniformSpace G := IsTopologicalGroup.rightUniformSpace G
this : IsCountablyGenerated.{u} (α := (G ⧸ N) × G ⧸ N) (𝓤 (G ⧸ N))
u : ℕ → Set G
hu : HasAntitoneBasis (𝓝 1) u
u_mul : ∀ (n : ℕ), u (n + 1) * u (n + 1) ⊆ u n
hv : HasBasis.{u, 1} (α := G ⧸ N) (𝓝 1) (fun (x : ℕ) ↦ True) fun (x : ℕ) ↦ mk '' u x
v_anti : Antitone.{0, u} (β := Set (G ⧸ N)) fun (x : ℕ) ↦ mk '' u x
x : ℕ → G ⧸ N
hx : CauchySeq x
key₀ :
  ∀ (i j : ℕ),
    ∃ (M : ℕ),
      j < M ∧
        ∀ (a b : ℕ), M ≤ a → M ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u i ∧ x a = (↑g' : G ⧸ N)
φ : ℕ → ℕ :=
  fun (n : ℕ) ↦
    Nat.recOn (motive := fun (x : ℕ) ↦ ℕ) n
      (Classical.choose (p := fun (x_1 : ℕ) ↦
        0 < x_1 ∧
          ∀ (a b : ℕ),
            x_1 ≤ a → x_1 ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u 0 ∧ x a = (↑g' : G ⧸ N))
        ⋯)
      fun (k yk : ℕ) ↦
      Classical.choose (p := fun (x_1 : ℕ) ↦
        yk < x_1 ∧
          ∀ (a b : ℕ),
            x_1 ≤ a → x_1 ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u (k + 1) ∧ x a = (↑g' : G ⧸ N))
        ⋯
hφ :
  ∀ (n : ℕ),
    φ n < φ (n + 1) ∧
      ∀ (a b : ℕ),
        φ (n + 1) ≤ a →
          φ (n + 1) ≤ b → ∀ (g : G), x b = (↑g : G ⧸ N) → ∃ (g' : G), g / g' ∈ u (n + 1) ∧ x a = (↑g' : G ⧸ N)
x' : (n : ℕ) → (g : G) ×' x (φ (n + 1)) = (↑g : G ⧸ N) :=
  fun (n : ℕ) ↦
    Nat.recOn (motive := fun (x_1 : ℕ) ↦ (g : G) ×' x (φ (x_1 + 1)) = (↑g : G ⧸ N)) n
      (PSigma.mk (β := fun (g : G) ↦ x (φ (Nat.zero + 1)) = (↑g : G ⧸ N))
        (Classical.choose (p := fun (x_1 : G) ↦ (↑x_1 : G ⧸ N) = x (φ 1)) ⋯) ⋯)
      fun (k : ℕ) (hk : (g : G) ×' x (φ (k + 1)) = (↑g : G ⧸ N)) ↦
      PSigma.mk (β := fun (g : G) ↦ x (φ (k.succ + 1)) = (↑g : G ⧸ N))
        (Classical.choose (p := fun (x_1 : G) ↦
          hk.fst (β := fun (g : G) ↦ x (φ (k + 1)) = (↑g : G ⧸ N)) / x_1 ∈ u (k + 1) ∧
            x (φ (k + 1 + 1)) = (↑x_1 : G ⧸ N))
          ⋯)
        ⋯
hx' :
  ∀ (n : ℕ),
    (x' n).fst (β := fun (g : G) ↦ x (φ (n + 1)) = (↑g : G ⧸ N)) /
        PSigma.fst (β := fun (g : G) ↦ x (φ (n + 1 + 1)) = (↑g : G ⧸ N))
          (x' (n + 1) : (g : G) ×' x (φ (n + 1 + 1)) = (↑g : G ⧸ N)) ∈
      u (n + 1)
x'_cauchy : CauchySeq.{u, 0} (α := G) fun (n : ℕ) ↦ (x' n).fst (β := fun (g : G) ↦ x (φ (n + 1)) = (↑g : G ⧸ N))
x₀ : G
hx₀ : Tendsto (fun (n : ℕ) ↦ (x' n).fst (β := fun (g : G) ↦ x (φ (n + 1)) = (↑g : G ⧸ N))) atTop (𝓝 x₀)
⊢ Tendsto (α := ℕ) (x ∘ fun (n : ℕ) ↦ φ (n + 1)) atTop (𝓝 (↑x₀ : G ⧸ N))
have h : Eq (α := ℕ → G ⧸ N) (x ∘ fun (n : ℕ) ↦ φ (n + 1))
  (mk ∘ fun (n : ℕ) ↦ (x' n).fst (β := fun (g : G) ↦ x (φ (n + 1)) = (↑g : G ⧸ N)))
Tactic: convert! ((continuous_coinduced_rng : Continuous ((↑) : G → G ⧸ N)).tendsto x₀).comp hx₀
-- Lean 4 proof step from Mathlib.Topology.Algebra.IsUniformGroup.Basic (goal)
Goal:
G : Type u_1
inst : Group G
us : UniformSpace G
inst_1 : IsRightUniformGroup G
inst_2 : FirstCountableTopology G
N : Subgroup G
inst_3 : N.Normal
hG : CompleteSpace G
⊢ IsTopologicalGroup.rightUniformSpace G = us
have h : 𝓤 G = 𝓤 G
Tactic: ext : 1
-- Lean 4 proof step from Mathlib.Topology.Algebra.IsUniformGroup.Basic (goal)
Goal:
G : Type u_1
inst : Group G
us : UniformSpace G
inst_1 : IsRightUniformGroup G
inst_2 : FirstCountableTopology G
N : Subgroup G
inst_3 : N.Normal
hG : CompleteSpace G
⊢ 𝓤 G = 𝓤 G
have h : 𝓤 G = comap (β := G) (fun (x : G × G) ↦ x.2 * x.1⁻¹) (𝓝 1)
Tactic: rw [@IsRightUniformGroup.uniformity_eq (G := G) us _ _]
-- Lean 4 proof step from Mathlib.Topology.Algebra.IsUniformGroup.Basic (hypothesis)
Goal:
G : Type u_1
inst : Group G
us : UniformSpace G
inst_1 : IsRightUniformGroup G
inst_2 : FirstCountableTopology G
N : Subgroup G
inst_3 : N.Normal
hG : CompleteSpace G
⊢ CompleteSpace.{u_1} (G ⧸ N)
have this : IsTopologicalGroup.rightUniformSpace G = us
Tactic: have : IsTopologicalGroup.rightUniformSpace G = us := by
    ext : 1
    rw [@IsRightUniformGroup.uniformity_eq (G := G) us _ _]
    rfl