awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Measure.Dirac
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∞)
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) ⋯)) ⋯ →
      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
⊢ 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:
rw [tsum_eq_single ⟨points ⟨measurableAtom x, by simp [measurableAtoms]⟩, by simp⟩]

Premises used: tsum_eq_single
-- Module: Mathlib.MeasureTheory.Measure.Dirac
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∞)
h :
  (↑(Subtype.mk (p := fun (x : α) ↦ x ∈ range points)
          (points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯)) ⋯) :
      α) ∈
    measurableAtom x
⊢ (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∞)

Tactic:
rw [indicator_of_mem]

Premises used: Set.indicator_of_mem
-- Module: Mathlib.MeasureTheory.Measure.Dirac
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
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) ⋯))) :
      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∞)
      (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∞)

Tactic:
simp only [Pi.one_apply, mul_one]

Premises used: Pi.one_apply, mul_one
-- Module: Mathlib.MeasureTheory.Measure.Dirac
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
h :
  measurableAtom x =
    measurableAtom (points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯))
⊢ 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∞)

Tactic:
congr 1
-- Module: Mathlib.MeasureTheory.Measure.Dirac
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) ⋯))

Tactic:
(measurableAtom_eq_of_mem ?_).symm

Premises used: measurableAtom_eq_of_mem, Eq.symm
-- Module: Mathlib.MeasureTheory.Measure.Dirac
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
h : points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯) ∈ measurableAtom x
⊢ measurableAtom x =
    measurableAtom (points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯))

Tactic:
refine (measurableAtom_eq_of_mem ?_).symm

Premises used: measurableAtom_eq_of_mem, Eq.symm
-- Module: Mathlib.MeasureTheory.Measure.Dirac
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) ⋯) ∈
    (↑(Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯) : Set α)

Tactic:
h_points_mem _
-- Module: Mathlib.MeasureTheory.Measure.Dirac
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
h : measurableAtom x = (↑(Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯) : Set α)
⊢ points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯) ∈ measurableAtom x

Tactic:
convert! h_points_mem _
-- Module: Mathlib.MeasureTheory.Measure.Dirac
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 = (↑(Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯) : Set α)

Tactic:
simp
-- Module: Mathlib.MeasureTheory.Measure.Dirac
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
h : measurableAtom x = (↑(Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯) : Set 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

Tactic:
convert! h_points_mem _
-- Module: Mathlib.MeasureTheory.Measure.Dirac
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
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) ⋯) →
      Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (measurableAtom a) : ℝ≥0∞) (0 : ℝ≥0∞) ∨ a ∉ measurableAtom x
⊢ ∀ (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

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]

Premises used: Set.mem_range, imp_false, Pi.one_apply, Subtype.exists, forall_exists_index, Subtype.forall, mul_eq_zero, ne_eq, Subtype.mk.injEq, one_ne_zero, Set.indicator_apply_eq_zero
-- Module: Mathlib.MeasureTheory.Measure.Dirac
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

Tactic:
fun y s hs hsy hyx ↦ .inr fun hyx' ↦ hyx ?_

Premises used: Or.inr
-- Module: Mathlib.MeasureTheory.Measure.Dirac
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
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) ⋯))
    (hyx' : y ∈ measurableAtom x),
    y = points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯)
⊢ ∀ (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

Tactic:
refine fun y s hs hsy hyx ↦ .inr fun hyx' ↦ hyx ?_
-- Module: Mathlib.MeasureTheory.Measure.Dirac
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
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) ⋯)
⊢ y = points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) (measurableAtom x) ⋯)

Tactic:
rw [← hsy]
-- Module: Mathlib.MeasureTheory.Measure.Dirac
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
e_a : s = 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) ⋯)

Tactic:
congr
-- Module: Mathlib.MeasureTheory.Measure.Dirac
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
⊢ measurableAtom y = measurableAtom x

Tactic:
have h1 : measurableAtom y = measurableAtom x := measurableAtom_eq_of_mem hyx'
-- Module: Mathlib.MeasureTheory.Measure.Dirac
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
⊢ 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⟩

Premises used: Subtype.mk
-- Module: Mathlib.MeasureTheory.Measure.Dirac
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)
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 ⋯) = y)
    (h_points_mem :
      points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) s ⋯) ∈
        (↑(Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) s ⋯) : Set α)),
    measurableAtom y = s
⊢ measurableAtom y = s

Tactic:
obtain ⟨z, _, hz⟩ := hs
-- Module: Mathlib.MeasureTheory.Measure.Dirac
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
rw : measurableAtom (points (Subtype.mk (p := fun (a : Set α) ↦ a ∈ measurableAtoms) s ⋯)) = measurableAtom z
⊢ measurableAtom y = s

Tactic:
rw [← hz, ← hsy]
-- Module: Mathlib.MeasureTheory.Measure.Dirac
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 (points (Subtype.mk (p := fun (a : Set α) ↦ a ∈ measurableAtoms) s ⋯)) = measurableAtom z

Tactic:
measurableAtom_eq_of_mem ?_

Premises used: measurableAtom_eq_of_mem
-- Module: Mathlib.MeasureTheory.Measure.Dirac
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
refine : points (Subtype.mk (p := fun (a : Set α) ↦ a ∈ measurableAtoms) s ⋯) ∈ measurableAtom z
⊢ measurableAtom (points (Subtype.mk (p := fun (a : Set α) ↦ a ∈ measurableAtoms) s ⋯)) = measurableAtom z

Tactic:
refine measurableAtom_eq_of_mem ?_

Premises used: measurableAtom_eq_of_mem
-- Module: Mathlib.MeasureTheory.Measure.Dirac
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
⊢ points (Subtype.mk (p := fun (x : Set α) ↦ x ∈ measurableAtoms) s ⋯) ∈ s

Tactic:
h_points_mem
-- Module: Mathlib.MeasureTheory.Measure.Dirac
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
⊢ points (Subtype.mk (p := fun (a : Set α) ↦ a ∈ measurableAtoms) s ⋯) ∈ measurableAtom z

Tactic:
convert! h_points_mem