awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic
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
h𝓤GN :
  HasBasis.{u, 1} (α := (G ⧸ N) × G ⧸ N) (𝓤 (G ⧸ N)) (fun (x : ℕ) ↦ True) fun (i : ℕ) ↦
    {x : (G ⧸ N) × G ⧸ N | Membership.mem (γ := Set (G ⧸ N)) (mk '' u i) (x.2 / x.1)}
hx : ∀ (i : ℕ), ∃ (N_1 : ℕ), ∀ (m : ℕ), N_1 ≤ m → ∀ (n : ℕ), N_1 ≤ n → ∃ x_1 ∈ u i, (↑x_1 : G ⧸ N) = x n / x m
i j M : ℕ
hM : ∀ (m : ℕ), M ≤ m → ∀ (n : ℕ), M ≤ n → ∃ x_1 ∈ u i, (↑x_1 : G ⧸ N) = x n / x m
a b : ℕ
ha : max j M + 1 ≤ a
hb : max j M + 1 ≤ b
g : G
hg : x b = (↑g : G ⧸ N)
y : G
y_mem : y ∈ u i
hy : (↑y : G ⧸ N) = x b / x a
simpa : x a = (↑(y⁻¹ * g) : G ⧸ N)
⊢ ∃ (g' : G), g / g' ∈ u i ∧ x a = (↑g' : G ⧸ N)

Tactic:
refine
      ⟨y⁻¹ * g, by
        simpa only [div_eq_mul_inv, mul_inv_rev, inv_inv, mul_inv_cancel_left] using y_mem, ?_⟩
-- Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic
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
h𝓤GN :
  HasBasis.{u, 1} (α := (G ⧸ N) × G ⧸ N) (𝓤 (G ⧸ N)) (fun (x : ℕ) ↦ True) fun (i : ℕ) ↦
    {x : (G ⧸ N) × G ⧸ N | Membership.mem (γ := Set (G ⧸ N)) (mk '' u i) (x.2 / x.1)}
hx : ∀ (i : ℕ), ∃ (N_1 : ℕ), ∀ (m : ℕ), N_1 ≤ m → ∀ (n : ℕ), N_1 ≤ n → ∃ x_1 ∈ u i, (↑x_1 : G ⧸ N) = x n / x m
i j M : ℕ
hM : ∀ (m : ℕ), M ≤ m → ∀ (n : ℕ), M ≤ n → ∃ x_1 ∈ u i, (↑x_1 : G ⧸ N) = x n / x m
a b : ℕ
ha : max j M + 1 ≤ a
hb : max j M + 1 ≤ b
g : G
hg : x b = (↑g : G ⧸ N)
y : G
y_mem : y ∈ u i
hy : (↑y : G ⧸ N) = x b / x a
rw : x a = HMul.hMul (α := G ⧸ N) (↑y⁻¹ : G ⧸ N) (↑g : G ⧸ N)
rw₁ : x a = HMul.hMul (α := G ⧸ N) (↑y : G ⧸ N)⁻¹ (↑g : G ⧸ N)
rw₂ : x a = (x b / x a)⁻¹ * (↑g : G ⧸ N)
rw₃ : x a = ((↑g : G ⧸ N) / x a)⁻¹ * (↑g : G ⧸ N)
rw₄ : x a = x a / (↑g : G ⧸ N) * (↑g : G ⧸ N)
rw₅ rw₆ : x a = x a
⊢ x a = (↑(y⁻¹ * g) : G ⧸ N)

Tactic:
rw [QuotientGroup.mk_mul, QuotientGroup.mk_inv, hy, hg, inv_div, div_mul_cancel]

Premises used: QuotientGroup.mk_inv, inv_div, div_mul_cancel, QuotientGroup.mk_mul
-- Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic
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
⊢ ∀ (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)

Tactic:
have key₀ : ∀ i j : ℕ, ∃ M : ℕ, j < M ∧ ∀ a b : ℕ, M ≤ a → M ≤ b →
      ∀ g : G, x b = g → ∃ g' : G, g / g' ∈ u i ∧ x a = g' := by
    have h𝓤GN : (𝓤 (G ⧸ N)).HasBasis (fun _ ↦ True) fun i ↦ { x | x.snd / x.fst ∈ (↑) '' u i } := by
      simpa [uniformity_eq_comap_nhds_one', div_eq_mul_inv] using hv.comap _
    rw [h𝓤GN.cauchySeq_iff] at hx
    simp only [mem_setOf_eq, forall_true_left, mem_image] at hx
    intro i j
    rcases hx i with ⟨M, hM⟩
    refine ⟨max j M + 1, (le_max_left _ _).trans_lt (lt_add_one _), fun a b ha hb g hg => ?_⟩
    obtain ⟨y, y_mem, hy⟩ :=
      hM a (((le_max_right j _).trans (lt_add_one _).le).trans ha) b
        (((le_max_right j _).trans (lt_add_one _).le).trans hb)
    refine
      ⟨y⁻¹ * g, by
        simpa only [div_eq_mul_inv, mul_inv_rev, inv_inv, mul_inv_cancel_left] using y_mem, ?_⟩
    rw [QuotientGroup.mk_mul, QuotientGroup.mk_inv, hy, hg, inv_div, div_mul_cancel]
-- Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic
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))
        ⋯
⊢ ∀ (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))
-- Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic
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))
          ⋯)
        ⋯
⊢ ∀ (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
-- Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic
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)
⊢ (𝓤 G).HasBasis (fun (x : ℕ) ↦ True) fun (i : ℕ) ↦ {x : G × G | x.2 / x.1 ∈ u i}

Tactic:
simpa [uniformity_eq_comap_nhds_one', div_eq_mul_inv] using hu.toHasBasis.comap _

Premises used: Filter.HasAntitoneBasis.toHasBasis, div_eq_mul_inv, Filter.HasBasis.comap, uniformity_eq_comap_nhds_one'
-- Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic
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)
⊢ (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 _
-- Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic
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}
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)))
⊢ CauchySeq.{u, 0} (α := G) fun (n : ℕ) ↦ (x' n).fst (β := fun (g : G) ↦ x (φ (n + 1)) = (↑g : G ⧸ N))

Tactic:
rw [h𝓤G.cauchySeq_iff']

Premises used: Filter.HasBasis.cauchySeq_iff'
-- Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic
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}
m n : ℕ
hmn : n ≥ m
⊢ (x' n).fst (β := fun (g : G) ↦ x (φ (n + 1)) = (↑g : G ⧸ N)) /
      (x' n).fst (β := fun (g : G) ↦ x (φ (n + 1)) = (↑g : G ⧸ N)) ∈
    u n

Tactic:
simpa only [div_self'] using mem_of_mem_nhds (hu.mem _)

Premises used: mem_of_mem_nhds, Filter.HasAntitoneBasis.mem, div_self'
-- Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic
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),
    ∃ (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:
exact fun m =>
      ⟨m, fun n hmn =>
        Nat.decreasingInduction'
          (fun k _ _ hk => u_mul k ⟨_, hx' k, _, hk, div_mul_div_cancel _ _ _⟩) hmn
          (by simpa only [div_self'] using mem_of_mem_nhds (hu.mem _))⟩