awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Homotopy.Lifting (goal)
Goal:
E : Type u_1
X : Type u_2
A : Type u_3
inst : TopologicalSpace E
inst_1 : TopologicalSpace X
inst_2 : TopologicalSpace A
p : E → X
inst_3 : SimplyConnectedSpace A
inst_4 : LocPathConnectedSpace A
s : Set X
cov : IsCoveringMapOn p s
a₀ : A
f : ContinuousMap.{u_3, u_2} A (↑s : Type u_2)
hs :
  ∀ (a : A),
    Membership.mem (α := X) s
      ((ContinuousMap.comp
              { toFun := Subtype.val (p := fun (x : X) ↦ x ∈ s), continuous_toFun := ⋯ : C({ x : X // x ∈ s }, X) } f :
            (a : A) → X)
          a :
        X)
F : ContinuousMap.{u_3, u_1} A (↑(p ⁻¹' s) : Type u_1)
hF :
  Eq (α := A → (↑s : Type u_2)) (s.restrictPreimage p ∘ (⇑F : (a : A) → (↑(p ⁻¹' s) : Type u_1)))
    (⇑f : (a : A) → (↑s : Type u_2))
he :
  p (↑((F : (a : A) → (↑(p ⁻¹' s) : Type u_1)) a₀) : E) =
    (ContinuousMap.comp
          { toFun := Subtype.val (p := fun (x : X) ↦ x ∈ s), continuous_toFun := ⋯ : C({ x : X // x ∈ s }, X) } f :
        (a : A) → X)
      a₀
hF_unique :
  ∀ (y : ContinuousMap.{u_3, u_1} A (↑(p ⁻¹' s) : Type u_1)),
    (fun (F_1 : ContinuousMap.{u_3, u_1} A (↑(p ⁻¹' s) : Type u_1)) ↦
          Eq.{u_1 + 1} (α := (↑(p ⁻¹' s) : Type u_1))
              ((F_1 : (a : A) → (↑(p ⁻¹' s) : Type u_1)) a₀ : (↑(p ⁻¹' s) : Type u_1))
              ((F : (a : A) → (↑(p ⁻¹' s) : Type u_1)) a₀ : (↑(p ⁻¹' s) : Type u_1)) ∧
            Eq (α := A → (↑s : Type u_2)) (s.restrictPreimage p ∘ (⇑F_1 : (a : A) → (↑(p ⁻¹' s) : Type u_1)))
              (⇑f : (a : A) → (↑s : Type u_2)))
        y →
      y = F
F' : C(A, E)
hF'₁ : Eq.{u_1 + 1} (α := E) ((F' : (a : A) → E) a₀ : E) ((↑((F : (a : A) → (↑(p ⁻¹' s) : Type u_1)) a₀) : E) : E)
hF'₂ : ∀ (x : A), p ((F' : (a : A) → E) x) = (↑((f : (a : A) → { x : X // x ∈ s }) x) : X)
⊢ Eq (α := A → (↑s : Type u_2))
    (s.restrictPreimage p ∘
      (⇑{ toFun := fun (a : A) ↦ Subtype.mk (p := fun (x : E) ↦ x ∈ p ⁻¹' s) ((F' : (a : A) → E) a) ⋯,
            continuous_toFun := ⋯ } :
        (a : A) → (↑(p ⁻¹' s) : Type u_1)))
    (⇑f : (a : A) → (↑s : Type u_2))
have refine_2 : ∀ (x : A),
  (↑((s.restrictPreimage p ∘
            (⇑{ toFun := fun (a : A) ↦ Subtype.mk (p := fun (x : E) ↦ x ∈ p ⁻¹' s) ((F' : (a : A) → E) a) sorry,
                  continuous_toFun := sorry : ContinuousMap.{u_3, u_1} A (↑(p ⁻¹' s) : Type u_1) } :
              (a : A) → (↑(p ⁻¹' s) : Type u_1)))
          x) :
      X) =
    (↑((f : (a : A) → (↑s : Type u_2)) x) : X)
Tactic: ext
-- Lean 4 proof step from Mathlib.Topology.Homotopy.Lifting (goal)
Goal:
E : Type u_1
X : Type u_2
A : Type u_3
inst : TopologicalSpace E
inst_1 : TopologicalSpace X
inst_2 : TopologicalSpace A
p : E → X
inst_3 : SimplyConnectedSpace A
inst_4 : LocPathConnectedSpace A
s : Set X
cov : IsCoveringMapOn p s
a₀ : A
f : ContinuousMap.{u_3, u_2} A (↑s : Type u_2)
hs :
  ∀ (a : A),
    Membership.mem (α := X) s
      ((ContinuousMap.comp
              { toFun := Subtype.val (p := fun (x : X) ↦ x ∈ s), continuous_toFun := ⋯ : C({ x : X // x ∈ s }, X) } f :
            (a : A) → X)
          a :
        X)
F : ContinuousMap.{u_3, u_1} A (↑(p ⁻¹' s) : Type u_1)
hF :
  Eq (α := A → (↑s : Type u_2)) (s.restrictPreimage p ∘ (⇑F : (a : A) → (↑(p ⁻¹' s) : Type u_1)))
    (⇑f : (a : A) → (↑s : Type u_2))
he :
  p (↑((F : (a : A) → (↑(p ⁻¹' s) : Type u_1)) a₀) : E) =
    (ContinuousMap.comp
          { toFun := Subtype.val (p := fun (x : X) ↦ x ∈ s), continuous_toFun := ⋯ : C({ x : X // x ∈ s }, X) } f :
        (a : A) → X)
      a₀
F' : C(A, E)
hF'₁ : Eq.{u_1 + 1} (α := E) ((F' : (a : A) → E) a₀ : E) ((↑((F : (a : A) → (↑(p ⁻¹' s) : Type u_1)) a₀) : E) : E)
hF'₂ : ∀ (x : A), p ((F' : (a : A) → E) x) = (↑((f : (a : A) → { x : X // x ∈ s }) x) : X)
hF_unique :
  { toFun := fun (a : A) ↦ Subtype.mk (p := fun (x : E) ↦ x ∈ p ⁻¹' s) ((F' : (a : A) → E) a) ⋯,
      continuous_toFun := ⋯ } =
    F
⊢ F' = ContinuousMap.comp { toFun := Subtype.val (p := fun (x : E) ↦ x ∈ p ⁻¹' s), continuous_toFun := ⋯ } F
have refine_2 : ∀ (a : A),
  Eq.{u_1 + 1} (α := E) ((F' : (a : A) → E) a : E)
    ((ContinuousMap.comp
            { toFun := Subtype.val (p := fun (x : E) ↦ x ∈ p ⁻¹' s), continuous_toFun := sorry :
              C({ x : E // x ∈ p ⁻¹' s }, E) }
            F :
          (a : A) → E)
        a :
      E)
Tactic: ext
-- Lean 4 proof step from Mathlib.Topology.Homotopy.LocallyContractible (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
ι : Type u_3
p : X → ι → Prop
s : X → ι → Set X
h : ∀ (x : X), (𝓝 x).HasBasis (p x) (s x)
h' : ∀ (x : X) (i : ι), p x i → ContractibleSpace.{u_1} (↑(s x i) : Type u_1)
x : X
⊢ (𝓝 x).HasBasis (fun (s : Set X) ↦ s ∈ 𝓝 x ∧ ContractibleSpace.{u_1} (↑s : Type u_1)) id
have rw : ∀ t ∈ 𝓝 x, ∃ r ∈ 𝓝 x, ContractibleSpace.{u_1} (↑r : Type u_1) ∧ r ⊆ t
Tactic: rw [hasBasis_self]
-- Lean 4 proof step from Mathlib.Topology.Homotopy.LocallyContractible (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
ι : Type u_3
p : X → ι → Prop
s : X → ι → Set X
h : ∀ (x : X), (𝓝 x).HasBasis (p x) (s x)
h' : ∀ (x : X) (i : ι), p x i → ContractibleSpace.{u_1} (↑(s x i) : Type u_1)
x : X
t : Set X
ht : t ∈ 𝓝 x
⊢ ∃ r ∈ 𝓝 x, ContractibleSpace.{u_1} (↑r : Type u_1) ∧ r ⊆ t
have obtain : ∀ (i : ι) (hpi : p x i) (hi : s x i ⊆ t), ∃ r ∈ 𝓝 x, ContractibleSpace.{u_1} (↑r : Type u_1) ∧ r ⊆ t
Tactic: obtain ⟨i, hpi, hi⟩ := (h x).mem_iff.mp ht
-- Lean 4 proof step from Mathlib.Topology.Homotopy.LocallyContractible (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
x_1 y : X
ι : Type u_3
inst_2 : StronglyLocallyContractibleSpace X
x : X
⊢ (𝓝 x).HasBasis (fun (s : Set X) ↦ s ∈ 𝓝 x ∧ IsPathConnected s) id
have refine : ∀ (s : Set X),
  s ∈ 𝓝 x ∧ ContractibleSpace.{u_1} (↑s : Type u_1) →
    ∀ (hs : s ∈ 𝓝 x) (hs' : ContractibleSpace.{u_1} (↑s : Type u_1)), IsPathConnected s
Tactic: refine contractible_basis x |>.to_hasBasis'
      (fun s ⟨hs, hs'⟩ ↦ ⟨s, ⟨hs, ?_⟩, le_rfl⟩) (fun s hs ↦ hs.1)
-- Lean 4 proof step from Mathlib.Topology.Homotopy.LocallyContractible (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
x_1 y : X
ι : Type u_3
inst_2 : StronglyLocallyContractibleSpace X
x : X
s : Set X
h : s ∈ 𝓝 x ∧ ContractibleSpace.{u_1} (↑s : Type u_1)
hs : s ∈ 𝓝 x
hs' : ContractibleSpace.{u_1} (↑s : Type u_1)
⊢ IsPathConnected s
have rw : PathConnectedSpace.{u_1} (↑s : Type u_1)
Tactic: rw [isPathConnected_iff_pathConnectedSpace]
-- Lean 4 proof step from Mathlib.Topology.Homotopy.LocallyContractible (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
x y : X
ι : Type u_3
inst_2 : StronglyLocallyContractibleSpace X
inst_3 : StronglyLocallyContractibleSpace Y
⊢ StronglyLocallyContractibleSpace (X × Y)
have refine_1 : ∀ (x : X × Y),
  (𝓝 x).HasBasis (ι := Prod.{u_1, u_2} (Set X) (Set Y))
    ((fun (x : X × Y) (x_1 : Prod.{u_1, u_2} (Set X) (Set Y)) ↦
        (match (motive := X × Y → Prop) x with
          | (x, y) =>
            match (motive := Prod.{u_1, u_2} (Set X) (Set Y) → Prop) x_1 with
            | (Ux, Uy) =>
              (Ux ∈ 𝓝 x ∧ ContractibleSpace.{u_1} (↑Ux : Type u_1)) ∧
                Uy ∈ 𝓝 y ∧ ContractibleSpace.{u_2} (↑Uy : Type u_2) :
          Prop))
      x)
    ((fun (x : X × Y) (x_1 : Prod.{u_1, u_2} (Set X) (Set Y)) ↦
        (match (motive := Prod.{u_1, u_2} (Set X) (Set Y) → Set (X × Y)) x_1 with
          | (Ux, Uy) => Ux ×ˢ Uy :
          Set (X × Y)))
      x)
Tactic: refine .of_bases (ι := Set X × Set Y)
    (p := fun (x, y) (Ux, Uy) =>
      (Ux ∈ 𝓝 x ∧ ContractibleSpace Ux) ∧ (Uy ∈ 𝓝 y ∧ ContractibleSpace Uy))
    (s := fun _ (Ux, Uy) => Ux ×ˢ Uy) ?_ ?_
-- Lean 4 proof step from Mathlib.Topology.Homotopy.LocallyContractible (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
x y : X
ι : Type u_3
inst_2 : StronglyLocallyContractibleSpace X
inst_3 : StronglyLocallyContractibleSpace Y
refine_1 :
  ∀ (x : X × Y),
    (𝓝 x).HasBasis (ι := Prod.{u_1, u_2} (Set X) (Set Y))
      ((fun (x : X × Y) (x_1 : Prod.{u_1, u_2} (Set X) (Set Y)) ↦
          (match (motive := X × Y → Prop) x with
            | (x, y) =>
              match (motive := Prod.{u_1, u_2} (Set X) (Set Y) → Prop) x_1 with
              | (Ux, Uy) =>
                (Ux ∈ 𝓝 x ∧ ContractibleSpace.{u_1} (↑Ux : Type u_1)) ∧
                  Uy ∈ 𝓝 y ∧ ContractibleSpace.{u_2} (↑Uy : Type u_2) :
            Prop))
        x)
      ((fun (x : X × Y) (x_1 : Prod.{u_1, u_2} (Set X) (Set Y)) ↦
          (match (motive := Prod.{u_1, u_2} (Set X) (Set Y) → Set (X × Y)) x_1 with
            | (Ux, Uy) => Ux ×ˢ Uy :
            Set (X × Y)))
        x)
⊢ StronglyLocallyContractibleSpace (X × Y)
have refine_2 : ∀ (x : X × Y) (i : Prod.{u_1, u_2} (Set X) (Set Y)),
  (fun (x : X × Y) (x_1 : Prod.{u_1, u_2} (Set X) (Set Y)) ↦
        (match (motive := X × Y → Prop) x with
          | (x, y) =>
            match (motive := Prod.{u_1, u_2} (Set X) (Set Y) → Prop) x_1 with
            | (Ux, Uy) =>
              (Ux ∈ 𝓝 x ∧ ContractibleSpace.{u_1} (↑Ux : Type u_1)) ∧
                Uy ∈ 𝓝 y ∧ ContractibleSpace.{u_2} (↑Uy : Type u_2) :
          Prop))
      x i →
    ContractibleSpace
      (↑((fun (x : X × Y) (x_1 : Prod.{u_1, u_2} (Set X) (Set Y)) ↦
              (match (motive := Prod.{u_1, u_2} (Set X) (Set Y) → Set (X × Y)) x_1 with
                | (Ux, Uy) => Ux ×ˢ Uy :
                Set (X × Y)))
            x i) :
        Type (max u_1 u_2))
Tactic: refine .of_bases (ι := Set X × Set Y)
    (p := fun (x, y) (Ux, Uy) =>
      (Ux ∈ 𝓝 x ∧ ContractibleSpace Ux) ∧ (Uy ∈ 𝓝 y ∧ ContractibleSpace Uy))
    (s := fun _ (Ux, Uy) => Ux ×ˢ Uy) ?_ ?_
-- Lean 4 proof step from Mathlib.Topology.Homotopy.LocallyContractible (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
x_1 y_1 : X
ι : Type u_3
inst_2 : StronglyLocallyContractibleSpace X
inst_3 : StronglyLocallyContractibleSpace Y
x : X
y : Y
⊢ (𝓝 (x, y)).HasBasis (ι := Prod.{u_1, u_2} (Set X) (Set Y))
    ((fun (x : X × Y) (x_2 : Prod.{u_1, u_2} (Set X) (Set Y)) ↦
        (match (motive := X × Y → Prop) x with
          | (x, y) =>
            match (motive := Prod.{u_1, u_2} (Set X) (Set Y) → Prop) x_2 with
            | (Ux, Uy) =>
              (Ux ∈ 𝓝 x ∧ ContractibleSpace.{u_1} (↑Ux : Type u_1)) ∧
                Uy ∈ 𝓝 y ∧ ContractibleSpace.{u_2} (↑Uy : Type u_2) :
          Prop))
      (x, y))
    ((fun (x : X × Y) (x_2 : Prod.{u_1, u_2} (Set X) (Set Y)) ↦
        (match (motive := Prod.{u_1, u_2} (Set X) (Set Y) → Set (X × Y)) x_2 with
          | (Ux, Uy) => Ux ×ˢ Uy :
          Set (X × Y)))
      (x, y))
have refine_1 : (𝓝 x ×ˢ 𝓝 y).HasBasis (ι := Prod.{u_1, u_2} (Set X) (Set Y))
  ((fun (x : X × Y) (x_2 : Prod.{u_1, u_2} (Set X) (Set Y)) ↦
      (match (motive := X × Y → Prop) x with
        | (x, y) =>
          match (motive := Prod.{u_1, u_2} (Set X) (Set Y) → Prop) x_2 with
          | (Ux, Uy) =>
            (Ux ∈ 𝓝 x ∧ ContractibleSpace.{u_1} (↑Ux : Type u_1)) ∧
              Uy ∈ 𝓝 y ∧ ContractibleSpace.{u_2} (↑Uy : Type u_2) :
        Prop))
    (x, y))
  ((fun (x : X × Y) (x_2 : Prod.{u_1, u_2} (Set X) (Set Y)) ↦
      (match (motive := Prod.{u_1, u_2} (Set X) (Set Y) → Set (X × Y)) x_2 with
        | (Ux, Uy) => Ux ×ˢ Uy :
        Set (X × Y)))
    (x, y))
Tactic: rw [nhds_prod_eq]
-- Lean 4 proof step from Mathlib.Topology.Homotopy.LocallyContractible (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
x_1 y_1 : X
ι : Type u_3
inst_2 : StronglyLocallyContractibleSpace X
inst_3 : StronglyLocallyContractibleSpace Y
x : X
y : Y
⊢ ∀ (i : Prod.{u_1, u_2} (Set X) (Set Y)),
    (fun (x : X × Y) (x_2 : Prod.{u_1, u_2} (Set X) (Set Y)) ↦
          (match (motive := X × Y → Prop) x with
            | (x, y) =>
              match (motive := Prod.{u_1, u_2} (Set X) (Set Y) → Prop) x_2 with
              | (Ux, Uy) =>
                (Ux ∈ 𝓝 x ∧ ContractibleSpace.{u_1} (↑Ux : Type u_1)) ∧
                  Uy ∈ 𝓝 y ∧ ContractibleSpace.{u_2} (↑Uy : Type u_2) :
            Prop))
        (x, y) i →
      ContractibleSpace
        (↑((fun (x : X × Y) (x_2 : Prod.{u_1, u_2} (Set X) (Set Y)) ↦
                (match (motive := Prod.{u_1, u_2} (Set X) (Set Y) → Set (X × Y)) x_2 with
                  | (Ux, Uy) => Ux ×ˢ Uy :
                  Set (X × Y)))
              (x, y) i) :
          Type (max u_1 u_2))
have refine_2 : ∀ (Ux : Set X) (Uy : Set Y),
  (fun (x : X × Y) (x_2 : Prod.{u_1, u_2} (Set X) (Set Y)) ↦
        (match (motive := X × Y → Prop) x with
          | (x, y) =>
            match (motive := Prod.{u_1, u_2} (Set X) (Set Y) → Prop) x_2 with
            | (Ux, Uy) =>
              (Ux ∈ 𝓝 x ∧ ContractibleSpace.{u_1} (↑Ux : Type u_1)) ∧
                Uy ∈ 𝓝 y ∧ ContractibleSpace.{u_2} (↑Uy : Type u_2) :
          Prop))
      (x, y) (Ux, Uy) →
    ContractibleSpace
      (↑((fun (x : X × Y) (x_2 : Prod.{u_1, u_2} (Set X) (Set Y)) ↦
              (match (motive := Prod.{u_1, u_2} (Set X) (Set Y) → Set (X × Y)) x_2 with
                | (Ux, Uy) => Ux ×ˢ Uy :
                Set (X × Y)))
            (x, y) (Ux, Uy)) :
        Type (max u_1 u_2))
Tactic: (Ux, Uy)
-- Lean 4 proof step from Mathlib.Topology.Homotopy.LocallyContractible (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
x_1 y_1 : X
ι : Type u_3
inst_2 : StronglyLocallyContractibleSpace X
inst_3 : StronglyLocallyContractibleSpace Y
x : X
y : Y
Ux : Set X
Uy : Set Y
⊢ (fun (x : X × Y) (x_2 : Prod.{u_1, u_2} (Set X) (Set Y)) ↦
        (match (motive := X × Y → Prop) x with
          | (x, y) =>
            match (motive := Prod.{u_1, u_2} (Set X) (Set Y) → Prop) x_2 with
            | (Ux, Uy) =>
              (Ux ∈ 𝓝 x ∧ ContractibleSpace.{u_1} (↑Ux : Type u_1)) ∧
                Uy ∈ 𝓝 y ∧ ContractibleSpace.{u_2} (↑Uy : Type u_2) :
          Prop))
      (x, y) (Ux, Uy) →
    ContractibleSpace
      (↑((fun (x : X × Y) (x_2 : Prod.{u_1, u_2} (Set X) (Set Y)) ↦
              (match (motive := Prod.{u_1, u_2} (Set X) (Set Y) → Set (X × Y)) x_2 with
                | (Ux, Uy) => Ux ×ˢ Uy :
                Set (X × Y)))
            (x, y) (Ux, Uy)) :
        Type (max u_1 u_2))
have refine_2 : ∀ (hUx : Ux ∈ 𝓝 x ∧ ContractibleSpace.{u_1} (↑Ux : Type u_1))
  (hUy : Uy ∈ 𝓝 y ∧ ContractibleSpace.{u_2} (↑Uy : Type u_2)),
  ContractibleSpace
    (↑((fun (x : X × Y) (x_2 : Prod.{u_1, u_2} (Set X) (Set Y)) ↦
            (match (motive := Prod.{u_1, u_2} (Set X) (Set Y) → Set (X × Y)) x_2 with
              | (Ux, Uy) => Ux ×ˢ Uy :
              Set (X × Y)))
          (x, y) (Ux, Uy)) :
      Type (max u_1 u_2))
Tactic: ⟨hUx, hUy⟩
-- Lean 4 proof step from Mathlib.Topology.Homotopy.LocallyContractible (hypothesis)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
x_1 y_1 : X
ι : Type u_3
inst_2 : StronglyLocallyContractibleSpace X
inst_3 : StronglyLocallyContractibleSpace Y
x : X
y : Y
Ux : Set X
Uy : Set Y
hUx : Ux ∈ 𝓝 x ∧ ContractibleSpace.{u_1} (↑Ux : Type u_1)
hUy : Uy ∈ 𝓝 y ∧ ContractibleSpace.{u_2} (↑Uy : Type u_2)
⊢ ContractibleSpace
    (↑((fun (x : X × Y) (x_2 : Prod.{u_1, u_2} (Set X) (Set Y)) ↦
            (match (motive := Prod.{u_1, u_2} (Set X) (Set Y) → Set (X × Y)) x_2 with
              | (Ux, Uy) => Ux ×ˢ Uy :
              Set (X × Y)))
          (x, y) (Ux, Uy)) :
      Type (max u_1 u_2))
have this : ContractibleSpace.{u_1} (↑Ux : Type u_1)
Tactic: haveI : ContractibleSpace Ux := hUx.2
-- Lean 4 proof step from Mathlib.Topology.Homotopy.LocallyContractible (hypothesis)
Goal:
X : Type u_1
Y : Type u_2
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
x_1 y_1 : X
ι : Type u_3
inst_2 : StronglyLocallyContractibleSpace X
inst_3 : StronglyLocallyContractibleSpace Y
x : X
y : Y
Ux : Set X
Uy : Set Y
hUx : Ux ∈ 𝓝 x ∧ ContractibleSpace.{u_1} (↑Ux : Type u_1)
hUy : Uy ∈ 𝓝 y ∧ ContractibleSpace.{u_2} (↑Uy : Type u_2)
this : ContractibleSpace.{u_1} (↑Ux : Type u_1)
⊢ ContractibleSpace
    (↑((fun (x : X × Y) (x_2 : Prod.{u_1, u_2} (Set X) (Set Y)) ↦
            (match (motive := Prod.{u_1, u_2} (Set X) (Set Y) → Set (X × Y)) x_2 with
              | (Ux, Uy) => Ux ×ˢ Uy :
              Set (X × Y)))
          (x, y) (Ux, Uy)) :
      Type (max u_1 u_2))
have this₁ : ContractibleSpace.{u_2} (↑Uy : Type u_2)
Tactic: haveI : ContractibleSpace Uy := hUy.2
-- Lean 4 proof step from Mathlib.Topology.Homotopy.LocallyContractible (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
inst_1 : StronglyLocallyContractibleSpace X
x : X
U : Set X
hU : U ∈ 𝓝 x
⊢ ∃ (V : Set X) (hVU : V ⊆ U), V ∈ 𝓝 x ∧ Nullhomotopic (ContinuousMap.inclusion (s := V) (t := U) hVU)
have obtain : ∀ (V : Set X) (hVU : id V ⊆ U) (hVmem : V ∈ 𝓝 x) (hVcontractible : ContractibleSpace.{u_1} (↑V : Type u_1)),
  ∃ (V : Set X) (hVU : V ⊆ U), V ∈ 𝓝 x ∧ Nullhomotopic (ContinuousMap.inclusion (s := V) (t := U) hVU)
Tactic: obtain ⟨V, ⟨hVmem, hVcontractible⟩, hVU⟩ := (contractible_basis x).mem_iff.mp hU
-- Lean 4 proof step from Mathlib.Topology.Homotopy.LocallyContractible (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
inst_1 : StronglyLocallyContractibleSpace X
x : X
U : Set X
hU : U ∈ 𝓝 x
V : Set X
hVU : id V ⊆ U
hVmem : V ∈ 𝓝 x
hVcontractible : ContractibleSpace.{u_1} (↑V : Type u_1)
⊢ ∃ (V : Set X) (hVU : V ⊆ U), V ∈ 𝓝 x ∧ Nullhomotopic (ContinuousMap.inclusion (s := V) (t := U) hVU)
have refine : Nullhomotopic (ContinuousMap.inclusion (s := V) (t := U) hVU)
Tactic: refine ⟨V, hVU, hVmem, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Homotopy.LocallyContractible (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
inst_1 : StronglyLocallyContractibleSpace X
x : X
U : Set X
hU : U ∈ 𝓝 x
V : Set X
hVU : id V ⊆ U
hVmem : V ∈ 𝓝 x
hVcontractible : ContractibleSpace.{u_1} (↑V : Type u_1)
⊢ Nullhomotopic (ContinuousMap.inclusion (s := V) (t := U) hVU)
have obtain : ∀ (v₀ : (↑V : Type u_1)) (hid : Homotopic (ContinuousMap.id (↑V : Type u_1)) (ContinuousMap.const (↑V : Type u_1) v₀)),
  Nullhomotopic (ContinuousMap.inclusion (s := V) (t := U) hVU)
Tactic: obtain ⟨v₀, hid⟩ := id_nullhomotopic V
-- Lean 4 proof step from Mathlib.Topology.Homotopy.LocallyContractible (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
inst_1 : StronglyLocallyContractibleSpace X
x : X
U : Set X
hU : U ∈ 𝓝 x
V : Set X
hVU : id V ⊆ U
hVmem : V ∈ 𝓝 x
hVcontractible : ContractibleSpace.{u_1} (↑V : Type u_1)
v₀ : (↑V : Type u_1)
hid : Homotopic (ContinuousMap.id (↑V : Type u_1)) (ContinuousMap.const (↑V : Type u_1) v₀)
⊢ Nullhomotopic (ContinuousMap.inclusion (s := V) (t := U) hVU)
have refine : Homotopic (ContinuousMap.inclusion (s := V) (t := U) hVU)
  (ContinuousMap.const (↑V : Type u_1)
    ((ContinuousMap.inclusion (s := id V) (t := U) hVU : (a : (↑(id V) : Type u_1)) → (↑U : Type u_1)) v₀))
Tactic: refine ⟨ContinuousMap.inclusion hVU v₀, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Homotopy.LocallyContractible (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
inst_1 : StronglyLocallyContractibleSpace X
x : X
U : Set X
hU : U ∈ 𝓝 x
V : Set X
hVU : id V ⊆ U
hVmem : V ∈ 𝓝 x
hVcontractible : ContractibleSpace.{u_1} (↑V : Type u_1)
v₀ : (↑V : Type u_1)
hid : Homotopic (ContinuousMap.id (↑V : Type u_1)) (ContinuousMap.const (↑V : Type u_1) v₀)
⊢ Homotopic (ContinuousMap.inclusion (s := V) (t := U) hVU)
    (ContinuousMap.const (↑V : Type u_1)
      ((ContinuousMap.inclusion (s := id V) (t := U) hVU : (a : (↑(id V) : Type u_1)) → (↑U : Type u_1)) v₀))
have h : Eq (α := C((↑V : Type u_1), (↑U : Type u_1))) (ContinuousMap.inclusion (s := V) (t := U) hVU)
  (ContinuousMap.comp (ContinuousMap.inclusion (s := id V) (t := U) hVU) (ContinuousMap.id (↑V : Type u_1)))
Tactic: convert! Homotopic.comp (.refl _) hid
-- Lean 4 proof step from Mathlib.Topology.Homotopy.LocallyContractible (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
inst_1 : StronglyLocallyContractibleSpace X
x : X
U : Set X
hU : U ∈ 𝓝 x
V : Set X
hVU : id V ⊆ U
hVmem : V ∈ 𝓝 x
hVcontractible : ContractibleSpace.{u_1} (↑V : Type u_1)
v₀ : (↑V : Type u_1)
hid : Homotopic (ContinuousMap.id (↑V : Type u_1)) (ContinuousMap.const (↑V : Type u_1) v₀)
⊢ Eq (α := C((↑V : Type u_1), (↑U : Type u_1))) (ContinuousMap.inclusion (s := V) (t := U) hVU)
    (ContinuousMap.comp (ContinuousMap.inclusion (s := id V) (t := U) hVU) (ContinuousMap.id (↑V : Type u_1)))
have h : ∀ (a : (↑V : Type u_1)),
  Eq.{u_1 + 1} (α := X)
    (↑((ContinuousMap.inclusion (s := V) (t := U) hVU : (a : (↑V : Type u_1)) → (↑U : Type u_1)) a) : X)
    (↑((ContinuousMap.comp (ContinuousMap.inclusion (s := id V) (t := U) hVU) (ContinuousMap.id (↑V : Type u_1)) :
            (a : (↑V : Type u_1)) → (↑U : Type u_1))
          a) :
      X)
Tactic: ext
-- Lean 4 proof step from Mathlib.Topology.Homotopy.Path (goal)
Goal:
X : Type u
inst : TopologicalSpace X
x₀ x₁ : X
p₀ p₁ : Path x₀ x₁
F : p₀.Homotopy p₁
⊢ F.eval 0 = p₀
have a : ∀ (t : (↑I : Type)), Eq (α := X) ((F.eval 0 : (a : (↑I : Type)) → X) t : X) ((p₀ : (a : (↑I : Type)) → X) t : X)
Tactic: ext t
-- Lean 4 proof step from Mathlib.Topology.Homotopy.Path (goal)
Goal:
X : Type u
inst : TopologicalSpace X
x₀ x₁ : X
p₀ p₁ : Path x₀ x₁
F : p₀.Homotopy p₁
⊢ F.eval 1 = p₁
have a : ∀ (t : (↑I : Type)), Eq (α := X) ((F.eval 1 : (a : (↑I : Type)) → X) t : X) ((p₁ : (a : (↑I : Type)) → X) t : X)
Tactic: ext t
-- Lean 4 proof step from Mathlib.Topology.Homotopy.Path (goal)
Goal:
X : Type u
Y : Type v
inst : TopologicalSpace X
inst_1 : TopologicalSpace Y
x₀ x₁ x₂ x₃ : X
p₀ q₀ : Path x₀ x₁
p₁ q₁ : Path x₁ x₂
F : p₀.Homotopy q₀
G : p₁.Homotopy q₁
x t : (↑I : Type)
ht : Membership.mem (γ := Set (↑I : Type)) {0, 1} t
⊢ Eq (α := X)
    (({
            toFun := fun (x_1 : (↑I : Type)) ↦
              if (↑(x, x_1).2 : ℝ) ≤ 1 / 2 then ((F.eval (x, x_1).1).extend : (a : ℝ) → X) (2 * (↑(x, x_1).2 : ℝ))
              else ((G.eval (x, x_1).1).extend : (a : ℝ) → X) (2 * (↑(x, x_1).2 : ℝ) - 1),
            continuous_toFun := ⋯ : C((↑I : Type), X) } :
          (a : (↑I : Type)) → X)
        t :
      X)
    (((p₀.trans p₁).toContinuousMap : (a : (↑I : Type)) → X) t : X)
have inl : ∀ (ht : t = 0),
  Eq (α := X)
    (({
            toFun := fun (x_1 : (↑I : Type)) ↦
              if (↑(x, x_1).2 : ℝ) ≤ 1 / 2 then ((F.eval (x, x_1).1).extend : (a : ℝ) → X) (2 * (↑(x, x_1).2 : ℝ))
              else ((G.eval (x, x_1).1).extend : (a : ℝ) → X) (2 * (↑(x, x_1).2 : ℝ) - 1),
            continuous_toFun := sorry : C((↑I : Type), X) } :
          (a : (↑I : Type)) → X)
        t :
      X)
    (((p₀.trans p₁).toContinuousMap : (a : (↑I : Type)) → X) t : X)
Tactic: rcases ht with ht | ht
Cleared hypotheses: ht