awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.LocallyCompact (goal)
Goal:
K : Type u_1
Γ₀ : Type u_2
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ₀
inst_2 : Valued K Γ₀
inst_3 : Valuation.RankOne.{u_1, u_2} (R := K) v
h_1 : CompleteSpace.{u_1} ↥𝒪[K] ∧ IsDiscreteValuationRing.{u_1} ↥𝒪[K] ∧ Finite.{u_1 + 1} 𝓀[K]
left : CompleteSpace.{u_1} ↥𝒪[K]
left_1 : IsDiscreteValuationRing.{u_1} ↥𝒪[K]
h : TotallyBounded.{u_1} (α := ↥𝒪[K]) Set.univ
⊢ IsCompact.{u_1} (X := ↥𝒪[K]) Set.univ
have refine_2 : TotallyBounded.{u_1} (α := ↥𝒪[K]) Set.univ ∧ IsComplete.{u_1} (α := ↥𝒪[K]) Set.univ
Tactic: rw [isCompact_iff_totallyBounded_isComplete]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.LocallyCompact (goal)
Goal:
K : Type u_1
Γ₀ : Type u_2
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ₀
inst_2 : Valued K Γ₀
inst_3 : Valuation.RankOne.{u_1, u_2} (R := K) v
⊢ ProperSpace K ↔ CompactSpace.{u_1} ↥𝒪[K]
have simp : ProperSpace K ↔ IsCompact.{u_1} (X := K) (Metric.closedBall 0 1)
Tactic: simp only [← isCompact_univ_iff, Subtype.isCompact_iff, Set.image_univ, Subtype.range_coe_subtype,
             toNormedField.setOf_mem_integer_eq_closedBall]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.LocallyCompact (goal)
Goal:
K : Type u_1
Γ₀ : Type u_2
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ₀
inst_2 : Valued K Γ₀
inst_3 : Valuation.RankOne.{u_1, u_2} (R := K) v
⊢ ProperSpace K ↔ IsCompact.{u_1} (X := K) (Metric.closedBall 0 1)
have mp : ProperSpace K → IsCompact.{u_1} (X := K) (Metric.closedBall 0 1)
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.LocallyCompact (goal)
Goal:
K : Type u_1
Γ₀ : Type u_2
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ₀
inst_2 : Valued K Γ₀
inst_3 : Valuation.RankOne.{u_1, u_2} (R := K) v
mp : ProperSpace K → IsCompact.{u_1} (X := K) (Metric.closedBall 0 1)
⊢ ProperSpace K ↔ IsCompact.{u_1} (X := K) (Metric.closedBall 0 1)
have mpr : IsCompact.{u_1} (X := K) (Metric.closedBall 0 1) → ProperSpace K
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.LocallyCompact (goal)
Goal:
K : Type u_1
Γ₀ : Type u_2
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ₀
inst_2 : Valued K Γ₀
inst_3 : Valuation.RankOne.{u_1, u_2} (R := K) v
h : IsCompact.{u_1} (X := K) (Metric.closedBall 0 1)
⊢ ProperSpace K
have mpr : LocallyCompactSpace K
Tactic: suffices LocallyCompactSpace K from .of_nontriviallyNormedField_of_weaklyLocallyCompactSpace K
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuativeRel (goal)
Goal:
R_1 : Type u_1
inst : CommRing R_1
inst_1 : ValuativeRel R_1
inst_2 : TopologicalSpace R_1
inst_3 : IsValuativeTopology R_1
R : Type u_2
inst_4 : CommRing R
inst_5 : ValuativeRel R
inst_6 : UniformSpace R
inst_7 : IsUniformAddGroup R
inst_8 : IsValuativeTopology R
⊢ ∀ (s : Set R),
    Membership.mem (γ := Filter R) (𝓝 0) s ↔
      ∃ (γ : (MonoidWithZeroHom.ValueGroup₀ (F := Valuation R (ValueGroupWithZero R)) v)ˣ),
        {x : R |
            (Valuation.restrict v : (a : R) → MonoidWithZeroHom.ValueGroup₀ (↑v : R →*₀ ValueGroupWithZero R)) x <
              (↑γ : MonoidWithZeroHom.ValueGroup₀ v)} ⊆
          s
have simp_rw : ∀ (s : Set R),
  Membership.mem (γ := Filter R) (𝓝 0) s ↔
    ∃ (γ : (MonoidWithZeroHom.ValueGroup₀ (F := Valuation R (ValueGroupWithZero R)) v)ˣ),
      {x : R |
          LT.lt (α := ValueGroupWithZero R) ((v : (a : R) → ValueGroupWithZero R) x : ValueGroupWithZero R)
            ((MonoidWithZeroHom.ValueGroup₀.embedding : (a : MonoidWithZeroHom.ValueGroup₀ v) → ValueGroupWithZero R)
                (↑γ : MonoidWithZeroHom.ValueGroup₀ v) :
              ValueGroupWithZero R)} ⊆
        s
Tactic: simp_rw [Valuation.restrict_lt_iff_lt_embedding]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuativeRel (goal)
Goal:
R_1 : Type u_1
inst : CommRing R_1
inst_1 : ValuativeRel R_1
inst_2 : TopologicalSpace R_1
inst_3 : IsValuativeTopology R_1
R : Type u_2
inst_4 : CommRing R
inst_5 : ValuativeRel R
inst_6 : UniformSpace R
inst_7 : IsUniformAddGroup R
inst_8 : IsValuativeTopology R
⊢ ∀ (s : Set R),
    Membership.mem (γ := Filter R) (𝓝 0) s ↔
      ∃ (γ : (MonoidWithZeroHom.ValueGroup₀ (F := Valuation R (ValueGroupWithZero R)) v)ˣ),
        {x : R |
            LT.lt (α := ValueGroupWithZero R) ((v : (a : R) → ValueGroupWithZero R) x : ValueGroupWithZero R)
              ((MonoidWithZeroHom.ValueGroup₀.embedding : (a : MonoidWithZeroHom.ValueGroup₀ v) → ValueGroupWithZero R)
                  (↑γ : MonoidWithZeroHom.ValueGroup₀ v) :
                ValueGroupWithZero R)} ⊆
          s
have h : ∀ (a : Set R),
  (∃ (γ : (MonoidWithZeroHom.ValueGroup₀ (F := Valuation R (ValueGroupWithZero R)) v)ˣ),
      {x : R |
          LT.lt (α := ValueGroupWithZero R) ((v : (a : R) → ValueGroupWithZero R) x : ValueGroupWithZero R)
            ((MonoidWithZeroHom.ValueGroup₀.embedding : (a : MonoidWithZeroHom.ValueGroup₀ v) → ValueGroupWithZero R)
                (↑γ : MonoidWithZeroHom.ValueGroup₀ v) :
              ValueGroupWithZero R)} ⊆
        a) ↔
    ∃ (γ : (ValueGroupWithZero R)ˣ), {x : R | (v : (a : R) → ValueGroupWithZero R) x < (↑γ : ValueGroupWithZero R)} ⊆ a
Tactic: convert! mem_nhds_zero_iff (R := R)
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuativeRel (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : ValuativeRel R
inst_2 : TopologicalSpace R
inst_3 : IsValuativeTopology R
⊢ Continuous[inst_2, _] (⇑v : (a : R) → ValueGroupWithZero R)
have simp : ∀ (x : R),
  Tendsto (β := ValueGroupWithZero R) ((⇑v : (a : R) → ValueGroupWithZero R) : (a : R) → ValueGroupWithZero R) (𝓝 x)
    (𝓝 ((v : (a : R) → ValueGroupWithZero R) x) : Filter (ValueGroupWithZero R))
Tactic: simp only [continuous_iff_continuousAt, ContinuousAt]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuativeRel (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : ValuativeRel R
inst_2 : TopologicalSpace R
inst_3 : IsValuativeTopology R
x : R
⊢ Tendsto (β := ValueGroupWithZero R) ((⇑v : (a : R) → ValueGroupWithZero R) : (a : R) → ValueGroupWithZero R) (𝓝 x)
    (𝓝 ((v : (a : R) → ValueGroupWithZero R) x) : Filter (ValueGroupWithZero R))
have pos : ∀
  (hx :
    Eq (α := ValueGroupWithZero R) ((v : (a : R) → ValueGroupWithZero R) x : ValueGroupWithZero R)
      (0 : ValueGroupWithZero R)),
  Tendsto (β := ValueGroupWithZero R) ((⇑v : (a : R) → ValueGroupWithZero R) : (a : R) → ValueGroupWithZero R) (𝓝 x)
    (𝓝 ((v : (a : R) → ValueGroupWithZero R) x) : Filter (ValueGroupWithZero R))
Tactic: by_cases hx : v x = 0
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuativeRel (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : ValuativeRel R
inst_2 : TopologicalSpace R
inst_3 : IsValuativeTopology R
x : R
pos :
  ∀
    (hx :
      Eq (α := ValueGroupWithZero R) ((v : (a : R) → ValueGroupWithZero R) x : ValueGroupWithZero R)
        (0 : ValueGroupWithZero R)),
    Tendsto (β := ValueGroupWithZero R) ((⇑v : (a : R) → ValueGroupWithZero R) : (a : R) → ValueGroupWithZero R) (𝓝 x)
      (𝓝 ((v : (a : R) → ValueGroupWithZero R) x) : Filter (ValueGroupWithZero R))
⊢ Tendsto (β := ValueGroupWithZero R) ((⇑v : (a : R) → ValueGroupWithZero R) : (a : R) → ValueGroupWithZero R) (𝓝 x)
    (𝓝 ((v : (a : R) → ValueGroupWithZero R) x) : Filter (ValueGroupWithZero R))
have neg : ∀
  (hx :
    ¬Eq (α := ValueGroupWithZero R) ((v : (a : R) → ValueGroupWithZero R) x : ValueGroupWithZero R)
        (0 : ValueGroupWithZero R)),
  Tendsto (β := ValueGroupWithZero R) ((⇑v : (a : R) → ValueGroupWithZero R) : (a : R) → ValueGroupWithZero R) (𝓝 x)
    (𝓝 ((v : (a : R) → ValueGroupWithZero R) x) : Filter (ValueGroupWithZero R))
Tactic: by_cases hx : v x = 0
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Opens (goal)
Goal:
X Y : TopCat
f : X ⟶ Y
ι : Type u_1
U : ι → Opens (↑Y : Type u)
⊢ (map f).obj (iSup U) = iSup ((map f).obj ∘ U)
have h : ∀ (x : (↑X : Type u)),
  x ∈ (↑((map f).obj (iSup U)) : Set (↑X : Type u)) ↔ x ∈ (↑(iSup ((map f).obj ∘ U)) : Set (↑X : Type u))
Tactic: ext
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Opens (goal)
Goal:
X Y Z : TopCat
U_1 V W : Opens (↑X : Type u)
f g : X ⟶ Y
h : f = g
U : Opens (↑Y : Type u)
⊢ (map f).obj U = (map g).obj U
have rw : (map g).obj U = (map g).obj U
Tactic: rw [congr_arg map h]
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Opens (goal)
Goal:
X Y : TopCat
f g : X ⟶ Y
h : f = g
⊢ map f = map g
have subst : map f = map f
Tactic: subst h
Cleared hypotheses: g, h
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Opens (goal)
Goal:
X Y Z : TopCat
U_1 V W : Opens (↑X : Type u)
f g : X ⟶ Y
h : f = g
U : Opens (↑Y : Type u)
⊢ (map f).obj U = (map g).obj U
have rw : (map g).obj U = (map g).obj U
Tactic: rw [h]
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Opens (goal)
Goal:
X Y Z : TopCat
U_1 V W : Opens (↑X : Type u)
f g : X ⟶ Y
h : f = g
U : Opens (↑Y : Type u)
⊢ (map g).obj U = (map f).obj U
have rw : (map g).obj U = (map g).obj U
Tactic: rw [h]
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Opens (goal)
Goal:
X Y : TopCat
f : X ⟶ Y
hf :
  IsOpenMap (X := (↑X : Type u)) (Y := (↑Y : Type u))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier) f) :
      (a : (↑X : Type u)) → (↑Y : Type u))
H : Mono f
X_1 Y_1 : Opens (↑X : Type u)
i : hf.functor.obj X_1 ⟶ hf.functor.obj Y_1
x : (↑X : Type u)
hx : x ∈ X_1
⊢ x ∈ Y_1
have obtain : ∀ (y : (↑X : Type u)) (hy : y ∈ (↑Y_1 : Set (↑X : Type u)))
  (eq :
    Eq (α := (↑Y : Type u))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier) f :
            (a : (↑X : Type u)) → (↑Y : Type u))
          y :
        (↑Y : Type u))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier) f :
            (a : (↑X : Type u)) → (↑Y : Type u))
          x :
        (↑Y : Type u))),
  x ∈ Y_1
Tactic: obtain ⟨y, hy, eq⟩ := i.le ⟨x, hx, rfl⟩
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Opens (goal)
Goal:
X Y : TopCat
f : X ⟶ Y
hf :
  IsInducing (X := (↑X : Type u)) (Y := (↑Y : Type u))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier) f) :
      (a : (↑X : Type u)) → (↑Y : Type u))
U : Opens (↑X : Type u)
⊢ (Opens.map f).obj (hf.functorObj U) = U
have a : (Opens.map f).obj (hf.functorObj U) ≤ U
Tactic: apply le_antisymm
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Opens (goal)
Goal:
X Y : TopCat
f : X ⟶ Y
hf :
  IsInducing (X := (↑X : Type u)) (Y := (↑Y : Type u))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier) f) :
      (a : (↑X : Type u)) → (↑Y : Type u))
U : Opens (↑X : Type u)
a : (Opens.map f).obj (hf.functorObj U) ≤ U
⊢ (Opens.map f).obj (hf.functorObj U) = U
have a₁ : U ≤ (Opens.map f).obj (hf.functorObj U)
Tactic: apply le_antisymm
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Opens (goal)
Goal:
X Y : TopCat
f : X ⟶ Y
hf :
  IsInducing (X := (↑X : Type u)) (Y := (↑Y : Type u))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier) f) :
      (a : (↑X : Type u)) → (↑Y : Type u))
U : Opens (↑X : Type u)
x : (↑X : Type u)
hx : x ∈ U
⊢ x ∈ (Opens.map f).obj (hf.functorObj U)
have a : ∀ (U : Set (↑X : Type u)) (hU : IsOpen U) (hx : x ∈ { carrier := U, is_open' := hU : Opens (↑X : Type u) }),
  x ∈ (Opens.map f).obj (hf.functorObj { carrier := U, is_open' := hU : Opens (↑X : Type u) })
Tactic: obtain ⟨U, hU⟩ := U
Cleared hypotheses: U, hx
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Opens (goal)
Goal:
X Y : TopCat
f : X ⟶ Y
hf :
  IsInducing (X := (↑X : Type u)) (Y := (↑Y : Type u))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier) f) :
      (a : (↑X : Type u)) → (↑Y : Type u))
x : (↑X : Type u)
U : Set (↑X : Type u)
hU : IsOpen U
hx : x ∈ { carrier := U, is_open' := hU : Opens (↑X : Type u) }
⊢ x ∈ (Opens.map f).obj (hf.functorObj { carrier := U, is_open' := hU : Opens (↑X : Type u) })
have a : ∀ (t : Set (↑Y : Type u)) (ht : IsOpen t)
  (hU :
    IsOpen (X := (↑X : Type u))
      ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
              f) :
          (a : (↑X : Type u)) → (↑Y : Type u)) ⁻¹'
        t))
  (hx :
    x ∈
      {
        carrier :=
          (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                  TopCat.carrier) f) :
              (a : (↑X : Type u)) → (↑Y : Type u)) ⁻¹'
            t,
        is_open' := hU : Opens (↑X : Type u) }),
  x ∈
    (Opens.map f).obj
      (hf.functorObj
        {
          carrier :=
            (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                    TopCat.carrier) f) :
                (a : (↑X : Type u)) → (↑Y : Type u)) ⁻¹'
              t,
          is_open' := hU : Opens (↑X : Type u) })
Tactic: obtain ⟨t, ht, rfl⟩ := hf.isOpen_iff.mp hU
Cleared hypotheses: U, hU, hx
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Opens (goal)
Goal:
X Y : TopCat
f : X ⟶ Y
hf :
  IsInducing (X := (↑X : Type u)) (Y := (↑Y : Type u))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier) f) :
      (a : (↑X : Type u)) → (↑Y : Type u))
U : Opens (↑X : Type u)
x : (↑X : Type u)
| x ∈ U
have rw : x ∈ (Opens.map f).obj (hf.functorObj U)
Tactic: rw [← hf.map_functorObj U]
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Opens (goal)
Goal:
X Y : TopCat
f : X ⟶ Y
hf :
  IsInducing (X := (↑X : Type u)) (Y := (↑Y : Type u))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier) f) :
      (a : (↑X : Type u)) → (↑Y : Type u))
U : Opens (↑X : Type u)
x : (↑X : Type u)
⊢ Membership.mem (α := (↑Y : Type u)) (hf.functorObj U)
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier) f :
            (a : (↑X : Type u)) → (↑Y : Type u))
          x :
        (↑Y : Type u)) ↔
    x ∈ U
have conv_rhs : Membership.mem (α := (↑Y : Type u)) (hf.functorObj U)
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier) f :
          (a : (↑X : Type u)) → (↑Y : Type u))
        x :
      (↑Y : Type u)) ↔
  x ∈ (Opens.map f).obj (hf.functorObj U)
Tactic: conv_rhs => rw [← hf.map_functorObj U]
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Opens (goal)
Goal:
X Y : TopCat
f : X ⟶ Y
hf :
  IsInducing (X := (↑X : Type u)) (Y := (↑Y : Type u))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier) f) :
      (a : (↑X : Type u)) → (↑Y : Type u))
U : Opens (↑X : Type u)
V : Opens (↑Y : Type u)
⊢ V ≤ hf.functorObj U ↔ (Opens.map f).obj V ≤ U
have obtain : ∀ (U : Set (↑X : Type u)) (hU : IsOpen U),
  V ≤ hf.functorObj { carrier := U, is_open' := hU } ↔ (Opens.map f).obj V ≤ { carrier := U, is_open' := hU }
Tactic: obtain ⟨U, hU⟩ := U
Cleared hypotheses: U
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Opens (goal)
Goal:
X Y : TopCat
f : X ⟶ Y
hf :
  IsInducing (X := (↑X : Type u)) (Y := (↑Y : Type u))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier) f) :
      (a : (↑X : Type u)) → (↑Y : Type u))
V : Opens (↑Y : Type u)
U : Set (↑X : Type u)
hU : IsOpen U
⊢ V ≤ hf.functorObj { carrier := U, is_open' := hU } ↔ (Opens.map f).obj V ≤ { carrier := U, is_open' := hU }
have obtain : ∀ (t : Set (↑Y : Type u)) (ht : IsOpen t)
  (hU :
    IsOpen (X := (↑X : Type u))
      ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
              f) :
          (a : (↑X : Type u)) → (↑Y : Type u)) ⁻¹'
        t)),
  V ≤
      hf.functorObj
        {
          carrier :=
            (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                    TopCat.carrier) f) :
                (a : (↑X : Type u)) → (↑Y : Type u)) ⁻¹'
              t,
          is_open' := hU } ↔
    (Opens.map f).obj V ≤
      {
        carrier :=
          (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                  TopCat.carrier) f) :
              (a : (↑X : Type u)) → (↑Y : Type u)) ⁻¹'
            t,
        is_open' := hU }
Tactic: obtain ⟨t, ht, rfl⟩ := hf.isOpen_iff.mp hU
Cleared hypotheses: U, hU
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Opens (goal)
Goal:
X Y : TopCat
f : X ⟶ Y
hf :
  IsInducing (X := (↑X : Type u)) (Y := (↑Y : Type u))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier) f) :
      (a : (↑X : Type u)) → (↑Y : Type u))
V : Opens (↑Y : Type u)
t : Set (↑Y : Type u)
ht : IsOpen t
hU :
  IsOpen (X := (↑X : Type u))
    ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
            f) :
        (a : (↑X : Type u)) → (↑Y : Type u)) ⁻¹'
      t)
⊢ V ≤
      hf.functorObj
        {
          carrier :=
            (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                    TopCat.carrier) f) :
                (a : (↑X : Type u)) → (↑Y : Type u)) ⁻¹'
              t,
          is_open' := hU } ↔
    (Opens.map f).obj V ≤
      {
        carrier :=
          (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                  TopCat.carrier) f) :
              (a : (↑X : Type u)) → (↑Y : Type u)) ⁻¹'
            t,
        is_open' := hU }
have mp : V ≤
    hf.functorObj
      {
        carrier :=
          (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                  TopCat.carrier) f) :
              (a : (↑X : Type u)) → (↑Y : Type u)) ⁻¹'
            t,
        is_open' := hU } →
  (Opens.map f).obj V ≤
    {
      carrier :=
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
                f) :
            (a : (↑X : Type u)) → (↑Y : Type u)) ⁻¹'
          t,
      is_open' := hU }
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Opens (goal)
Goal:
X Y : TopCat
f : X ⟶ Y
hf :
  IsInducing (X := (↑X : Type u)) (Y := (↑Y : Type u))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier) f) :
      (a : (↑X : Type u)) → (↑Y : Type u))
V : Opens (↑Y : Type u)
t : Set (↑Y : Type u)
ht : IsOpen t
hU :
  IsOpen (X := (↑X : Type u))
    ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
            f) :
        (a : (↑X : Type u)) → (↑Y : Type u)) ⁻¹'
      t)
mp :
  V ≤
      hf.functorObj
        {
          carrier :=
            (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                    TopCat.carrier) f) :
                (a : (↑X : Type u)) → (↑Y : Type u)) ⁻¹'
              t,
          is_open' := hU } →
    (Opens.map f).obj V ≤
      {
        carrier :=
          (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                  TopCat.carrier) f) :
              (a : (↑X : Type u)) → (↑Y : Type u)) ⁻¹'
            t,
        is_open' := hU }
⊢ V ≤
      hf.functorObj
        {
          carrier :=
            (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                    TopCat.carrier) f) :
                (a : (↑X : Type u)) → (↑Y : Type u)) ⁻¹'
              t,
          is_open' := hU } ↔
    (Opens.map f).obj V ≤
      {
        carrier :=
          (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                  TopCat.carrier) f) :
              (a : (↑X : Type u)) → (↑Y : Type u)) ⁻¹'
            t,
        is_open' := hU }
have mpr : (Opens.map f).obj V ≤
    {
      carrier :=
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
                f) :
            (a : (↑X : Type u)) → (↑Y : Type u)) ⁻¹'
          t,
      is_open' := hU } →
  V ≤
    hf.functorObj
      {
        carrier :=
          (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                  TopCat.carrier) f) :
              (a : (↑X : Type u)) → (↑Y : Type u)) ⁻¹'
            t,
        is_open' := hU }
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Opens (goal)
Goal:
X Y : TopCat
f : X ⟶ Y
hf :
  IsInducing (X := (↑X : Type u)) (Y := (↑Y : Type u))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier) f) :
      (a : (↑X : Type u)) → (↑Y : Type u))
V : Opens (↑Y : Type u)
t : Set (↑Y : Type u)
ht : IsOpen t
hU :
  IsOpen (X := (↑X : Type u))
    ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
            f) :
        (a : (↑X : Type u)) → (↑Y : Type u)) ⁻¹'
      t)
h :
  (Opens.map f).obj V ≤
    {
      carrier :=
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
                f) :
            (a : (↑X : Type u)) → (↑Y : Type u)) ⁻¹'
          t,
      is_open' := hU }
x : (↑Y : Type u)
hx : x ∈ V
⊢ x ∈
    hf.functorObj
      {
        carrier :=
          (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                  TopCat.carrier) f) :
              (a : (↑X : Type u)) → (↑Y : Type u)) ⁻¹'
            t,
        is_open' := hU : Opens (↑X : Type u) }
have mpr : (↑((Opens.map f).obj { carrier := V.carrier ∪ t, is_open' := sorry : Opens (↑Y : Type u) }) : Set (↑X : Type u)) =
  (↑{
        carrier :=
          (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                  TopCat.carrier) f) :
              (a : (↑X : Type u)) → (↑Y : Type u)) ⁻¹'
            t,
        is_open' := hU : Opens (↑X : Type u) } :
    Set (↑X : Type u))
Tactic: refine Opens.mem_sSup.mpr ⟨⟨_, V.2.union ht⟩, Opens.ext ?_, Set.mem_union_left t hx⟩