awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Dirac (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : Countable α
μ : Measure α
measurableAtoms : Set (Set α) := measurableAtom '' univ
h_nonempty : ∀ (s : (↑measurableAtoms : Type u_1)), (↑s : Set α).Nonempty
points : (↑measurableAtoms : Type u_1) → α := fun (s : (↑measurableAtoms : Type u_1)) ↦ ⋯.some (s := (↑s : Set α))
h_points_mem : ∀ (s : (↑measurableAtoms : Type u_1)), points s ∈ (↑s : Set α)
x : α
⊢ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (measurableAtom x) : ℝ≥0∞)
    (((sum fun (x : (↑(range points) : Type u_1)) ↦
              HSMul.hSMul (α := ℝ≥0∞) (β := Measure α) ((μ : (a : Set α) → ℝ≥0∞) (measurableAtom (↑x : α)) : ℝ≥0∞)
                (dirac (↑x : α)) :
            Measure α) :
          (a : Set α) → ℝ≥0∞)
        (measurableAtom x) :
      ℝ≥0∞)
have rw : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (measurableAtom x) : ℝ≥0∞)
  (∑' (i : (↑(range points) : Type u_1)),
      (HSMul.hSMul (α := ℝ≥0∞) (β := Measure α) ((μ : (a : Set α) → ℝ≥0∞) (measurableAtom (↑i : α)) : ℝ≥0∞)
            (dirac (↑i : α)) :
          (a : Set α) → ℝ≥0∞)
        (measurableAtom x) :
    ℝ≥0∞)
Tactic: rw [sum_apply _ (MeasurableSet.measurableAtom_of_countable x)]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Dirac (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : Countable α
μ : Measure α
measurableAtoms : Set (Set α) := measurableAtom '' univ
h_nonempty : ∀ (s : (↑measurableAtoms : Type u_1)), (↑s : Set α).Nonempty
points : (↑measurableAtoms : Type u_1) → α := fun (s : (↑measurableAtoms : Type u_1)) ↦ ⋯.some (s := (↑s : Set α))
h_points_mem : ∀ (s : (↑measurableAtoms : Type u_1)), points s ∈ (↑s : Set α)
x : α
⊢ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (measurableAtom x) : ℝ≥0∞)
    (∑' (i : (↑(range points) : Type u_1)),
        (HSMul.hSMul (α := ℝ≥0∞) (β := Measure α) ((μ : (a : Set α) → ℝ≥0∞) (measurableAtom (↑i : α)) : ℝ≥0∞)
              (dirac (↑i : α)) :
            (a : Set α) → ℝ≥0∞)
          (measurableAtom x) :
      ℝ≥0∞)
have simp : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (measurableAtom x) : ℝ≥0∞)
  (∑' (i : (↑(range points) : Type u_1)),
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (measurableAtom (↑i : α)) : ℝ≥0∞)
        (((dirac (↑i : α) : Measure α) : (a : Set α) → ℝ≥0∞) (measurableAtom x) : ℝ≥0∞) :
    ℝ≥0∞)
Tactic: simp only [Measure.smul_apply, smul_eq_mul]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Dirac (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : Countable α
μ : Measure α
measurableAtoms : Set (Set α) := measurableAtom '' univ
h_nonempty : ∀ (s : (↑measurableAtoms : Type u_1)), (↑s : Set α).Nonempty
points : (↑measurableAtoms : Type u_1) → α := fun (s : (↑measurableAtoms : Type u_1)) ↦ ⋯.some (s := (↑s : Set α))
h_points_mem : ∀ (s : (↑measurableAtoms : Type u_1)), points s ∈ (↑s : Set α)
x : α
⊢ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (measurableAtom x) : ℝ≥0∞)
    (∑' (i : (↑(range points) : Type u_1)),
        HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (measurableAtom (↑i : α)) : ℝ≥0∞)
          (((dirac (↑i : α) : Measure α) : (a : Set α) → ℝ≥0∞) (measurableAtom x) : ℝ≥0∞) :
      ℝ≥0∞)
have simp_rw : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (measurableAtom x) : ℝ≥0∞)
  (∑' (i : (↑(range points) : Type u_1)),
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (measurableAtom (↑i : α)) : ℝ≥0∞)
        ((measurableAtom x).indicator 1 (↑i : α) : ℝ≥0∞) :
    ℝ≥0∞)
Tactic: simp_rw [dirac_apply' _ (MeasurableSet.measurableAtom_of_countable x)]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Dirac (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : Countable α
μ : Measure α
measurableAtoms : Set (Set α) := measurableAtom '' univ
h_nonempty : ∀ (s : (↑measurableAtoms : Type u_1)), (↑s : Set α).Nonempty
points : (↑measurableAtoms : Type u_1) → α := fun (s : (↑measurableAtoms : Type u_1)) ↦ ⋯.some (s := (↑s : Set α))
h_points_mem : ∀ (s : (↑measurableAtoms : Type u_1)), points s ∈ (↑s : Set α)
x : α
⊢ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (measurableAtom x) : ℝ≥0∞)
    (∑' (i : (↑(range points) : Type u_1)),
        HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (measurableAtom (↑i : α)) : ℝ≥0∞)
          ((measurableAtom x).indicator 1 (↑i : α) : ℝ≥0∞) :
      ℝ≥0∞)
have rw : (μ : (a : Set α) → ℝ≥0∞) (measurableAtom x) =
  HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞)
    ((μ : (a : Set α) → ℝ≥0∞)
        (measurableAtom
          (↑(Subtype.mk (p := fun (x : α) ↦ x ∈ range points)
                (points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) sorry)) sorry) :
            α)) :
      ℝ≥0∞)
    ((measurableAtom x).indicator 1
        (↑(Subtype.mk (p := fun (x : α) ↦ x ∈ range points)
              (points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) sorry)) sorry) :
          α) :
      ℝ≥0∞)
Tactic: rw [tsum_eq_single ⟨points ⟨measurableAtom x, by simp [measurableAtoms]⟩, by simp⟩]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Dirac (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : Countable α
μ : Measure α
measurableAtoms : Set (Set α) := measurableAtom '' univ
h_nonempty : ∀ (s : (↑measurableAtoms : Type u_1)), (↑s : Set α).Nonempty
points : (↑measurableAtoms : Type u_1) → α := fun (s : (↑measurableAtoms : Type u_1)) ↦ ⋯.some (s := (↑s : Set α))
h_points_mem : ∀ (s : (↑measurableAtoms : Type u_1)), points s ∈ (↑s : Set α)
x : α
rw :
  (μ : (a : Set α) → ℝ≥0∞) (measurableAtom x) =
    HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞)
      ((μ : (a : Set α) → ℝ≥0∞)
          (measurableAtom
            (↑(Subtype.mk (p := fun (x : α) ↦ x ∈ range points)
                  (points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯)) ⋯) :
              α)) :
        ℝ≥0∞)
      ((measurableAtom x).indicator 1
          (↑(Subtype.mk (p := fun (x : α) ↦ x ∈ range points)
                (points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯)) ⋯) :
            α) :
        ℝ≥0∞)
⊢ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (measurableAtom x) : ℝ≥0∞)
    (∑' (i : (↑(range points) : Type u_1)),
        HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (measurableAtom (↑i : α)) : ℝ≥0∞)
          ((measurableAtom x).indicator 1 (↑i : α) : ℝ≥0∞) :
      ℝ≥0∞)
have rw₁ : ∀ (b' : (↑(range points) : Type u_1)),
  b' ≠
      Subtype.mk (p := fun (x : α) ↦ x ∈ range points)
        (points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) sorry)) sorry →
    HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (measurableAtom (↑b' : α)) : ℝ≥0∞)
        ((measurableAtom x).indicator 1 (↑b' : α) : ℝ≥0∞) =
      0
Tactic: rw [tsum_eq_single ⟨points ⟨measurableAtom x, by simp [measurableAtoms]⟩, by simp⟩]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Dirac (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : Countable α
μ : Measure α
measurableAtoms : Set (Set α) := measurableAtom '' univ
h_nonempty : ∀ (s : (↑measurableAtoms : Type u_1)), (↑s : Set α).Nonempty
points : (↑measurableAtoms : Type u_1) → α := fun (s : (↑measurableAtoms : Type u_1)) ↦ ⋯.some (s := (↑s : Set α))
h_points_mem : ∀ (s : (↑measurableAtoms : Type u_1)), points s ∈ (↑s : Set α)
x : α
⊢ (μ : (a : Set α) → ℝ≥0∞) (measurableAtom x) =
    HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞)
      ((μ : (a : Set α) → ℝ≥0∞)
          (measurableAtom
            (↑(Subtype.mk (p := fun (x : α) ↦ x ∈ range points)
                  (points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯)) ⋯) :
              α)) :
        ℝ≥0∞)
      ((measurableAtom x).indicator 1
          (↑(Subtype.mk (p := fun (x : α) ↦ x ∈ range points)
                (points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯)) ⋯) :
            α) :
        ℝ≥0∞)
have rw : (μ : (a : Set α) → ℝ≥0∞) (measurableAtom x) =
  HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞)
    ((μ : (a : Set α) → ℝ≥0∞)
        (measurableAtom
          (↑(Subtype.mk (p := fun (x : α) ↦ x ∈ range points)
                (points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) sorry)) sorry) :
            α)) :
      ℝ≥0∞)
    (1
        (↑(Subtype.mk (p := fun (x : α) ↦ x ∈ range points)
              (points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) sorry)) sorry) :
          α) :
      ℝ≥0∞)
Tactic: rw [indicator_of_mem]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Dirac (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : Countable α
μ : Measure α
measurableAtoms : Set (Set α) := measurableAtom '' univ
h_nonempty : ∀ (s : (↑measurableAtoms : Type u_1)), (↑s : Set α).Nonempty
points : (↑measurableAtoms : Type u_1) → α := fun (s : (↑measurableAtoms : Type u_1)) ↦ ⋯.some (s := (↑s : Set α))
h_points_mem : ∀ (s : (↑measurableAtoms : Type u_1)), points s ∈ (↑s : Set α)
x : α
rw :
  (μ : (a : Set α) → ℝ≥0∞) (measurableAtom x) =
    HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞)
      ((μ : (a : Set α) → ℝ≥0∞)
          (measurableAtom
            (↑(Subtype.mk (p := fun (x : α) ↦ x ∈ range points)
                  (points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯)) ⋯) :
              α)) :
        ℝ≥0∞)
      (1
          (↑(Subtype.mk (p := fun (x : α) ↦ x ∈ range points)
                (points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯)) ⋯) :
            α) :
        ℝ≥0∞)
⊢ (μ : (a : Set α) → ℝ≥0∞) (measurableAtom x) =
    HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞)
      ((μ : (a : Set α) → ℝ≥0∞)
          (measurableAtom
            (↑(Subtype.mk (p := fun (x : α) ↦ x ∈ range points)
                  (points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯)) ⋯) :
              α)) :
        ℝ≥0∞)
      ((measurableAtom x).indicator 1
          (↑(Subtype.mk (p := fun (x : α) ↦ x ∈ range points)
                (points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯)) ⋯) :
            α) :
        ℝ≥0∞)
have h : (↑(Subtype.mk (p := fun (x : α) ↦ x ∈ range points)
        (points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) sorry)) sorry) :
    α) ∈
  measurableAtom x
Tactic: rw [indicator_of_mem]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Dirac (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : Countable α
μ : Measure α
measurableAtoms : Set (Set α) := measurableAtom '' univ
h_nonempty : ∀ (s : (↑measurableAtoms : Type u_1)), (↑s : Set α).Nonempty
points : (↑measurableAtoms : Type u_1) → α := fun (s : (↑measurableAtoms : Type u_1)) ↦ ⋯.some (s := (↑s : Set α))
h_points_mem : ∀ (s : (↑measurableAtoms : Type u_1)), points s ∈ (↑s : Set α)
x : α
⊢ (μ : (a : Set α) → ℝ≥0∞) (measurableAtom x) =
    HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞)
      ((μ : (a : Set α) → ℝ≥0∞)
          (measurableAtom
            (↑(Subtype.mk (p := fun (x : α) ↦ x ∈ range points)
                  (points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯)) ⋯) :
              α)) :
        ℝ≥0∞)
      (1
          (↑(Subtype.mk (p := fun (x : α) ↦ x ∈ range points)
                (points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯)) ⋯) :
            α) :
        ℝ≥0∞)
have simp : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (measurableAtom x) : ℝ≥0∞)
  ((μ : (a : Set α) → ℝ≥0∞)
      (measurableAtom (points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) sorry))) :
    ℝ≥0∞)
Tactic: simp only [Pi.one_apply, mul_one]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Dirac (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : Countable α
μ : Measure α
measurableAtoms : Set (Set α) := measurableAtom '' univ
h_nonempty : ∀ (s : (↑measurableAtoms : Type u_1)), (↑s : Set α).Nonempty
points : (↑measurableAtoms : Type u_1) → α := fun (s : (↑measurableAtoms : Type u_1)) ↦ ⋯.some (s := (↑s : Set α))
h_points_mem : ∀ (s : (↑measurableAtoms : Type u_1)), points s ∈ (↑s : Set α)
x : α
⊢ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (measurableAtom x) : ℝ≥0∞)
    ((μ : (a : Set α) → ℝ≥0∞)
        (measurableAtom (points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯))) :
      ℝ≥0∞)
have h : measurableAtom x =
  measurableAtom (points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) sorry))
Tactic: congr 1
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Dirac (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : Countable α
μ : Measure α
measurableAtoms : Set (Set α) := measurableAtom '' univ
h_nonempty : ∀ (s : (↑measurableAtoms : Type u_1)), (↑s : Set α).Nonempty
points : (↑measurableAtoms : Type u_1) → α := fun (s : (↑measurableAtoms : Type u_1)) ↦ ⋯.some (s := (↑s : Set α))
h_points_mem : ∀ (s : (↑measurableAtoms : Type u_1)), points s ∈ (↑s : Set α)
x : α
⊢ measurableAtom x =
    measurableAtom (points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯))
have h : points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) sorry) ∈ measurableAtom x
Tactic: refine (measurableAtom_eq_of_mem ?_).symm
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Dirac (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : Countable α
μ : Measure α
measurableAtoms : Set (Set α) := measurableAtom '' univ
h_nonempty : ∀ (s : (↑measurableAtoms : Type u_1)), (↑s : Set α).Nonempty
points : (↑measurableAtoms : Type u_1) → α := fun (s : (↑measurableAtoms : Type u_1)) ↦ ⋯.some (s := (↑s : Set α))
h_points_mem : ∀ (s : (↑measurableAtoms : Type u_1)), points s ∈ (↑s : Set α)
x : α
⊢ points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯) ∈ measurableAtom x
have h : measurableAtom x = (↑(Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) sorry) : Set α)
Tactic: convert! h_points_mem _
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Dirac (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : Countable α
μ : Measure α
measurableAtoms : Set (Set α) := measurableAtom '' univ
h_nonempty : ∀ (s : (↑measurableAtoms : Type u_1)), (↑s : Set α).Nonempty
points : (↑measurableAtoms : Type u_1) → α := fun (s : (↑measurableAtoms : Type u_1)) ↦ ⋯.some (s := (↑s : Set α))
h_points_mem : ∀ (s : (↑measurableAtoms : Type u_1)), points s ∈ (↑s : Set α)
x : α
⊢ (↑(Subtype.mk (p := fun (x : α) ↦ x ∈ range points)
          (points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯)) ⋯) :
      α) ∈
    measurableAtom x
have h : measurableAtom x = (↑(Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) sorry) : Set α)
Tactic: convert! h_points_mem _
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Dirac (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : Countable α
μ : Measure α
measurableAtoms : Set (Set α) := measurableAtom '' univ
h_nonempty : ∀ (s : (↑measurableAtoms : Type u_1)), (↑s : Set α).Nonempty
points : (↑measurableAtoms : Type u_1) → α := fun (s : (↑measurableAtoms : Type u_1)) ↦ ⋯.some (s := (↑s : Set α))
h_points_mem : ∀ (s : (↑measurableAtoms : Type u_1)), points s ∈ (↑s : Set α)
x : α
⊢ ∀ (b' : (↑(range points) : Type u_1)),
    b' ≠
        Subtype.mk (p := fun (x : α) ↦ x ∈ range points)
          (points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯)) ⋯ →
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (measurableAtom (↑b' : α)) : ℝ≥0∞)
          ((measurableAtom x).indicator 1 (↑b' : α) : ℝ≥0∞) =
        0
have simp : ∀ (a : α) (x_1 : Set α) (x_2 : x_1 ∈ measurableAtoms)
  (h : points (Subtype.mk (p := fun (a : Set α) ↦ a ∈ measurableAtoms) x_1 x_2) = a),
  ¬a = points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) sorry) →
    Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (measurableAtom a) : ℝ≥0∞) (0 : ℝ≥0∞) ∨ a ∉ measurableAtom x
Tactic: simp only [ne_eq, mul_eq_zero, indicator_apply_eq_zero, Pi.one_apply, one_ne_zero, imp_false,
      Subtype.forall, Set.mem_range, Subtype.exists, Subtype.mk.injEq, forall_exists_index]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Dirac (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : Countable α
μ : Measure α
measurableAtoms : Set (Set α) := measurableAtom '' univ
h_nonempty : ∀ (s : (↑measurableAtoms : Type u_1)), (↑s : Set α).Nonempty
points : (↑measurableAtoms : Type u_1) → α := fun (s : (↑measurableAtoms : Type u_1)) ↦ ⋯.some (s := (↑s : Set α))
h_points_mem : ∀ (s : (↑measurableAtoms : Type u_1)), points s ∈ (↑s : Set α)
x : α
⊢ ∀ (a : α) (x_1 : Set α) (x_2 : x_1 ∈ measurableAtoms)
    (h : points (Subtype.mk (p := fun (a : Set α) ↦ a ∈ measurableAtoms) x_1 x_2) = a),
    ¬a = points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯) →
      Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (measurableAtom a) : ℝ≥0∞) (0 : ℝ≥0∞) ∨ a ∉ measurableAtom x
have refine : ∀ (y : α) (s : Set α) (hs : s ∈ measurableAtoms)
  (hsy : points (Subtype.mk (p := fun (a : Set α) ↦ a ∈ measurableAtoms) s hs) = y)
  (hyx : ¬y = points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) sorry))
  (hyx' : y ∈ measurableAtom x),
  y = points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) sorry)
Tactic: refine fun y s hs hsy hyx ↦ .inr fun hyx' ↦ hyx ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Dirac (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : Countable α
μ : Measure α
measurableAtoms : Set (Set α) := measurableAtom '' univ
h_nonempty : ∀ (s : (↑measurableAtoms : Type u_1)), (↑s : Set α).Nonempty
points : (↑measurableAtoms : Type u_1) → α := fun (s : (↑measurableAtoms : Type u_1)) ↦ ⋯.some (s := (↑s : Set α))
h_points_mem : ∀ (s : (↑measurableAtoms : Type u_1)), points s ∈ (↑s : Set α)
x y : α
s : Set α
hs : s ∈ measurableAtoms
hsy : points (Subtype.mk (p := fun (a : Set α) ↦ a ∈ measurableAtoms) s hs) = y
hyx : ¬y = points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯)
hyx' : y ∈ measurableAtom x
⊢ y = points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯)
have rw : points (Subtype.mk (p := fun (a : Set α) ↦ a ∈ measurableAtoms) s hs) =
  points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) sorry)
Tactic: rw [← hsy]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Dirac (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : Countable α
μ : Measure α
measurableAtoms : Set (Set α) := measurableAtom '' univ
h_nonempty : ∀ (s : (↑measurableAtoms : Type u_1)), (↑s : Set α).Nonempty
points : (↑measurableAtoms : Type u_1) → α := fun (s : (↑measurableAtoms : Type u_1)) ↦ ⋯.some (s := (↑s : Set α))
h_points_mem : ∀ (s : (↑measurableAtoms : Type u_1)), points s ∈ (↑s : Set α)
x y : α
s : Set α
hs : s ∈ measurableAtoms
hsy : points (Subtype.mk (p := fun (a : Set α) ↦ a ∈ measurableAtoms) s hs) = y
hyx : ¬y = points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯)
hyx' : y ∈ measurableAtom x
⊢ points (Subtype.mk (p := fun (a : Set α) ↦ a ∈ measurableAtoms) s hs) =
    points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯)
have e_a : s = measurableAtom x
Tactic: congr
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Dirac (hypothesis)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : Countable α
μ : Measure α
measurableAtoms : Set (Set α) := measurableAtom '' univ
h_nonempty : ∀ (s : (↑measurableAtoms : Type u_1)), (↑s : Set α).Nonempty
points : (↑measurableAtoms : Type u_1) → α := fun (s : (↑measurableAtoms : Type u_1)) ↦ ⋯.some (s := (↑s : Set α))
h_points_mem : ∀ (s : (↑measurableAtoms : Type u_1)), points s ∈ (↑s : Set α)
x y : α
s : Set α
hs : s ∈ measurableAtoms
hsy : points (Subtype.mk (p := fun (a : Set α) ↦ a ∈ measurableAtoms) s hs) = y
hyx : ¬y = points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯)
hyx' : y ∈ measurableAtom x
⊢ s = measurableAtom x
have h1 : measurableAtom y = measurableAtom x
Tactic: have h1 : measurableAtom y = measurableAtom x := measurableAtom_eq_of_mem hyx'
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Dirac (hypothesis)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : Countable α
μ : Measure α
measurableAtoms : Set (Set α) := measurableAtom '' univ
h_nonempty : ∀ (s : (↑measurableAtoms : Type u_1)), (↑s : Set α).Nonempty
points : (↑measurableAtoms : Type u_1) → α := fun (s : (↑measurableAtoms : Type u_1)) ↦ ⋯.some (s := (↑s : Set α))
h_points_mem : ∀ (s : (↑measurableAtoms : Type u_1)), points s ∈ (↑s : Set α)
x y : α
s : Set α
hs : s ∈ measurableAtoms
hsy : points (Subtype.mk (p := fun (a : Set α) ↦ a ∈ measurableAtoms) s hs) = y
hyx : ¬y = points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯)
hyx' : y ∈ measurableAtom x
h1 : measurableAtom y = measurableAtom x
⊢ measurableAtom y = s
have h_points_mem₁ : points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) s hs) ∈
  (↑(Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) s hs) : Set α)
Tactic: specialize h_points_mem ⟨s, hs⟩
Cleared hypotheses: h_points_mem
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Dirac (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : Countable α
μ : Measure α
measurableAtoms : Set (Set α) := measurableAtom '' univ
h_nonempty : ∀ (s : (↑measurableAtoms : Type u_1)), (↑s : Set α).Nonempty
points : (↑measurableAtoms : Type u_1) → α := fun (s : (↑measurableAtoms : Type u_1)) ↦ ⋯.some (s := (↑s : Set α))
x y : α
s : Set α
hs : s ∈ measurableAtoms
hsy : points (Subtype.mk (p := fun (a : Set α) ↦ a ∈ measurableAtoms) s hs) = y
hyx : ¬y = points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯)
hyx' : y ∈ measurableAtom x
h1 : measurableAtom y = measurableAtom x
h_points_mem :
  points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) s hs) ∈
    (↑(Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) s hs) : Set α)
⊢ measurableAtom y = s
have obtain : ∀ (z : α) (left : Membership.mem (γ := Set α) univ z) (hz : measurableAtom z = s)
  (hsy : points (Subtype.mk (p := fun (a : Set α) ↦ a ∈ measurableAtoms) s sorry) = y)
  (h_points_mem :
    points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) s sorry) ∈
      (↑(Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) s sorry) : Set α)),
  measurableAtom y = s
Tactic: obtain ⟨z, _, hz⟩ := hs
Cleared hypotheses: hs, hsy, h_points_mem
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Dirac (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : Countable α
μ : Measure α
measurableAtoms : Set (Set α) := measurableAtom '' univ
h_nonempty : ∀ (s : (↑measurableAtoms : Type u_1)), (↑s : Set α).Nonempty
points : (↑measurableAtoms : Type u_1) → α := fun (s : (↑measurableAtoms : Type u_1)) ↦ ⋯.some (s := (↑s : Set α))
x y : α
s : Set α
hyx : ¬y = points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯)
hyx' : y ∈ measurableAtom x
h1 : measurableAtom y = measurableAtom x
z : α
left : Membership.mem (γ := Set α) univ z
hz : measurableAtom z = s
hsy : points (Subtype.mk (p := fun (a : Set α) ↦ a ∈ measurableAtoms) s ⋯) = y
h_points_mem : points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) s ⋯) ∈ s
⊢ measurableAtom y = s
have rw : measurableAtom (points (Subtype.mk (p := fun (a : Set α) ↦ a ∈ measurableAtoms) s sorry)) = measurableAtom z
Tactic: rw [← hz, ← hsy]