awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (goal)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : (n : ι) → Set Ω := fun (n : ι) ↦ if n = a then t2 else if n ∈ s then ft1 n else univ
n : ι
hn_mem_insert : n ∈ insert a s
inl : ∀ (hn_mem : n = a), (if n = a then t2 else if n ∈ s then ft1 n else univ) ∈ π n
⊢ (if n = a then t2 else if n ∈ s then ft1 n else univ) ∈ π n
have inr : ∀ (hn_mem : n ∈ s), (if n = a then t2 else if n ∈ s then ft1 n else univ) ∈ π n
Tactic: rcases Finset.mem_insert.mp hn_mem_insert with hn_mem | hn_mem
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (hypothesis)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : (n : ι) → Set Ω := fun (n : ι) ↦ if n = a then t2 else if n ∈ s then ft1 n else univ
⊢ ∀ᵐ (a : α) ∂μ,
    (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ t2) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
        ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t2 : ℝ≥0∞)
have h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
Tactic: have h_f_mem : ∀ n ∈ insert a s, f n ∈ π n := by
    intro n hn_mem_insert
    dsimp only [f]
    rcases Finset.mem_insert.mp hn_mem_insert with hn_mem | hn_mem
    · simp [hn_mem, ht2_mem_pia]
    · grind
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (hypothesis)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : (n : ι) → Set Ω := fun (n : ι) ↦ if n = a then t2 else if n ∈ s then ft1 n else univ
h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
⊢ ∀ᵐ (a : α) ∂μ,
    (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ t2) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
        ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t2 : ℝ≥0∞)
have h_f_mem_pi : ∀ n ∈ s, f n ∈ π n
Tactic: have h_f_mem_pi : ∀ n ∈ s, f n ∈ π n := fun x hxS => h_f_mem x (by simp [hxS])
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (goal)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : (n : ι) → Set Ω := fun (n : ι) ↦ if n = a then t2 else if n ∈ s then ft1 n else univ
h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
h_f_mem_pi : ∀ n ∈ s, f n ∈ π n
⊢ t1 = ⋂ n ∈ s, f n
have suffices : ∀ n ∈ s, f n = ft1 n
Tactic: suffices h_forall : ∀ n ∈ s, f n = ft1 n by grind
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (hypothesis)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : (n : ι) → Set Ω := fun (n : ι) ↦ if n = a then t2 else if n ∈ s then ft1 n else univ
h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
h_f_mem_pi : ∀ n ∈ s, f n ∈ π n
n : ι
hnS : n ∈ s
⊢ f n = ft1 n
have hn_ne_a : n ≠ a
Tactic: have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hnS)
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (goal)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : (n : ι) → Set Ω := fun (n : ι) ↦ if n = a then t2 else if n ∈ s then ft1 n else univ
h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
h_f_mem_pi : ∀ n ∈ s, f n ∈ π n
n : ι
hnS : n ∈ s
hn_ne_a : n ≠ a
simp_rw : (if n = a then t2 else if n ∈ s then ft1 n else univ) = ft1 n
⊢ f n = ft1 n
have simp_rw₁ : (if n = a then t2 else ft1 n) = ft1 n
Tactic: simp_rw [f, if_pos hnS, if_neg hn_ne_a]
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (hypothesis)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : (n : ι) → Set Ω := fun (n : ι) ↦ if n = a then t2 else if n ∈ s then ft1 n else univ
h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
h_f_mem_pi : ∀ n ∈ s, f n ∈ π n
⊢ ∀ᵐ (a : α) ∂μ,
    (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ t2) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
        ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t2 : ℝ≥0∞)
have h_t1 : t1 = ⋂ n ∈ s, f n
Tactic: have h_t1 : t1 = ⋂ n ∈ s, f n := by
    suffices h_forall : ∀ n ∈ s, f n = ft1 n by grind
    intro n hnS
    have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hnS)
    simp_rw [f, if_pos hnS, if_neg hn_ne_a]
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (goal)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : (n : ι) → Set Ω := fun (n : ι) ↦ if n = a then t2 else if n ∈ s then ft1 n else univ
h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
h_f_mem_pi : ∀ n ∈ s, f n ∈ π n
h_t1 : t1 = ⋂ n ∈ s, f n
⊢ ∀ᵐ (a' : α) ∂μ,
    Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
      (∏ n ∈ s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : ℝ≥0∞)
have h : ∀ (a' : α)
  (ha' :
    Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋂ i ∈ s, f i) : ℝ≥0∞)
      (∏ i ∈ s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f i) : ℝ≥0∞)),
  Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
    (∏ n ∈ s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : ℝ≥0∞)
Tactic: filter_upwards [hp_ind s h_f_mem_pi] with a' ha'
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (goal)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : (n : ι) → Set Ω := fun (n : ι) ↦ if n = a then t2 else if n ∈ s then ft1 n else univ
h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
h_f_mem_pi : ∀ n ∈ s, f n ∈ π n
h_t1 : t1 = ⋂ n ∈ s, f n
a' : α
ha' h :
  Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋂ n ∈ s, f n) : ℝ≥0∞)
    (∏ n ∈ s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : ℝ≥0∞)
⊢ Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
    (∏ n ∈ s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : ℝ≥0∞)
have h₁ : Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋂ n ∈ s, f n) : ℝ≥0∞)
  ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋂ i ∈ s, f i) : ℝ≥0∞)
Tactic: rw [h_t1, ← ha']
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (hypothesis)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : (n : ι) → Set Ω := fun (n : ι) ↦ if n = a then t2 else if n ∈ s then ft1 n else univ
h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
h_f_mem_pi : ∀ n ∈ s, f n ∈ π n
h_t1 : t1 = ⋂ n ∈ s, f n
⊢ ∀ᵐ (a : α) ∂μ,
    (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ t2) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
        ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t2 : ℝ≥0∞)
have h_μ_t1 : ∀ᵐ (a' : α) ∂μ,
  Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
    (∏ n ∈ s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : ℝ≥0∞)
Tactic: have h_μ_t1 : ∀ᵐ a' ∂μ, κ a' t1 = ∏ n ∈ s, κ a' (f n) := by
    filter_upwards [hp_ind s h_f_mem_pi] with a' ha'
    rw [h_t1, ← ha']
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (hypothesis)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : (n : ι) → Set Ω := fun (n : ι) ↦ if n = a then t2 else if n ∈ s then ft1 n else univ
h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
h_f_mem_pi : ∀ n ∈ s, f n ∈ π n
h_t1 : t1 = ⋂ n ∈ s, f n
h_μ_t1 :
  ∀ᵐ (a' : α) ∂μ,
    Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
      (∏ n ∈ s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : ℝ≥0∞)
⊢ ∀ᵐ (a : α) ∂μ,
    (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ t2) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
        ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t2 : ℝ≥0∞)
have h_t2 : t2 = f a
Tactic: have h_t2 : t2 = f a := by simp [f]
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (goal)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : (n : ι) → Set Ω := fun (n : ι) ↦ if n = a then t2 else if n ∈ s then ft1 n else univ
h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
h_f_mem_pi : ∀ n ∈ s, f n ∈ π n
h_t1 : t1 = ⋂ n ∈ s, f n
h_μ_t1 :
  ∀ᵐ (a' : α) ∂μ,
    Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
      (∏ n ∈ s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : ℝ≥0∞)
h_t2 : t2 = f a
⊢ t1 ∩ t2 = ⋂ n ∈ insert a s, f n
have rw : (⋂ n ∈ s, f n) ∩ t2 = ⋂ n ∈ insert a s, f n
Tactic: rw [h_t1, h_t2, Finset.set_biInter_insert, Set.inter_comm]
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (goal)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : (n : ι) → Set Ω := fun (n : ι) ↦ if n = a then t2 else if n ∈ s then ft1 n else univ
h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
h_f_mem_pi : ∀ n ∈ s, f n ∈ π n
h_t1 : t1 = ⋂ n ∈ s, f n
h_μ_t1 :
  ∀ᵐ (a' : α) ∂μ,
    Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
      (∏ n ∈ s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : ℝ≥0∞)
h_t2 : t2 = f a
rw : (⋂ n ∈ s, f n) ∩ t2 = ⋂ n ∈ insert a s, f n
⊢ t1 ∩ t2 = ⋂ n ∈ insert a s, f n
have rw₁ : (⋂ n ∈ s, f n) ∩ f a = ⋂ n ∈ insert a s, f n
Tactic: rw [h_t1, h_t2, Finset.set_biInter_insert, Set.inter_comm]
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (goal)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : (n : ι) → Set Ω := fun (n : ι) ↦ if n = a then t2 else if n ∈ s then ft1 n else univ
h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
h_f_mem_pi : ∀ n ∈ s, f n ∈ π n
h_t1 : t1 = ⋂ n ∈ s, f n
h_μ_t1 :
  ∀ᵐ (a' : α) ∂μ,
    Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
      (∏ n ∈ s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : ℝ≥0∞)
h_t2 : t2 = f a
rw : (⋂ n ∈ s, f n) ∩ t2 = ⋂ n ∈ insert a s, f n
rw₁ : (⋂ n ∈ s, f n) ∩ f a = ⋂ n ∈ insert a s, f n
⊢ t1 ∩ t2 = ⋂ n ∈ insert a s, f n
have rw₂ : (⋂ n ∈ s, f n) ∩ f a = f a ∩ ⋂ x ∈ s, f x
Tactic: rw [h_t1, h_t2, Finset.set_biInter_insert, Set.inter_comm]
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (goal)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : (n : ι) → Set Ω := fun (n : ι) ↦ if n = a then t2 else if n ∈ s then ft1 n else univ
h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
h_f_mem_pi : ∀ n ∈ s, f n ∈ π n
h_t1 : t1 = ⋂ n ∈ s, f n
h_μ_t1 :
  ∀ᵐ (a' : α) ∂μ,
    Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
      (∏ n ∈ s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : ℝ≥0∞)
h_t2 : t2 = f a
rw : (⋂ n ∈ s, f n) ∩ t2 = ⋂ n ∈ insert a s, f n
rw₁ : (⋂ n ∈ s, f n) ∩ f a = ⋂ n ∈ insert a s, f n
rw₂ : (⋂ n ∈ s, f n) ∩ f a = f a ∩ ⋂ x ∈ s, f x
⊢ t1 ∩ t2 = ⋂ n ∈ insert a s, f n
have rw₃ : f a ∩ ⋂ n ∈ s, f n = f a ∩ ⋂ x ∈ s, f x
Tactic: rw [h_t1, h_t2, Finset.set_biInter_insert, Set.inter_comm]
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (hypothesis)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : (n : ι) → Set Ω := fun (n : ι) ↦ if n = a then t2 else if n ∈ s then ft1 n else univ
h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
h_f_mem_pi : ∀ n ∈ s, f n ∈ π n
h_t1 : t1 = ⋂ n ∈ s, f n
h_μ_t1 :
  ∀ᵐ (a' : α) ∂μ,
    Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
      (∏ n ∈ s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : ℝ≥0∞)
h_t2 : t2 = f a
⊢ ∀ᵐ (a' : α) ∂μ,
    Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ t2) : ℝ≥0∞)
      (∏ n ∈ insert a s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : ℝ≥0∞)
have h_t1_inter_t2 : t1 ∩ t2 = ⋂ n ∈ insert a s, f n
Tactic: have h_t1_inter_t2 : t1 ∩ t2 = ⋂ n ∈ insert a s, f n := by
      rw [h_t1, h_t2, Finset.set_biInter_insert, Set.inter_comm]
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (goal)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : (n : ι) → Set Ω := fun (n : ι) ↦ if n = a then t2 else if n ∈ s then ft1 n else univ
h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
h_f_mem_pi : ∀ n ∈ s, f n ∈ π n
h_t1 : t1 = ⋂ n ∈ s, f n
h_μ_t1 :
  ∀ᵐ (a' : α) ∂μ,
    Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
      (∏ n ∈ s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : ℝ≥0∞)
h_t2 : t2 = f a
h_t1_inter_t2 : t1 ∩ t2 = ⋂ n ∈ insert a s, f n
⊢ ∀ᵐ (a' : α) ∂μ,
    Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ t2) : ℝ≥0∞)
      (∏ n ∈ insert a s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : ℝ≥0∞)
have h : ∀ (a' : α)
  (ha' :
    Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋂ i ∈ insert a s, f i) : ℝ≥0∞)
      (∏ i ∈ insert a s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f i) : ℝ≥0∞)),
  Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ t2) : ℝ≥0∞)
    (∏ n ∈ insert a s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : ℝ≥0∞)
Tactic: filter_upwards [hp_ind (insert a s) h_f_mem] with a' ha'
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (goal)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : (n : ι) → Set Ω := fun (n : ι) ↦ if n = a then t2 else if n ∈ s then ft1 n else univ
h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
h_f_mem_pi : ∀ n ∈ s, f n ∈ π n
h_t1 : t1 = ⋂ n ∈ s, f n
h_μ_t1 :
  ∀ᵐ (a' : α) ∂μ,
    Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
      (∏ n ∈ s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : ℝ≥0∞)
h_t2 : t2 = f a
h_t1_inter_t2 : t1 ∩ t2 = ⋂ n ∈ insert a s, f n
a' : α
ha' h :
  Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋂ n ∈ insert a s, f n) : ℝ≥0∞)
    (∏ n ∈ insert a s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : ℝ≥0∞)
⊢ Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ t2) : ℝ≥0∞)
    (∏ n ∈ insert a s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : ℝ≥0∞)
have h₁ : Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋂ n ∈ insert a s, f n) : ℝ≥0∞)
  ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋂ i ∈ insert a s, f i) : ℝ≥0∞)
Tactic: rw [h_t1_inter_t2, ← ha']
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (hypothesis)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : (n : ι) → Set Ω := fun (n : ι) ↦ if n = a then t2 else if n ∈ s then ft1 n else univ
h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
h_f_mem_pi : ∀ n ∈ s, f n ∈ π n
h_t1 : t1 = ⋂ n ∈ s, f n
h_μ_t1 :
  ∀ᵐ (a' : α) ∂μ,
    Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
      (∏ n ∈ s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : ℝ≥0∞)
h_t2 : t2 = f a
⊢ ∀ᵐ (a : α) ∂μ,
    (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ t2) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
        ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t2 : ℝ≥0∞)
have h_μ_inter : ∀ᵐ (a' : α) ∂μ,
  Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ t2) : ℝ≥0∞)
    (∏ n ∈ insert a s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : ℝ≥0∞)
Tactic: have h_μ_inter : ∀ᵐ a' ∂μ, κ a' (t1 ∩ t2) = ∏ n ∈ insert a s, κ a' (f n) := by
    have h_t1_inter_t2 : t1 ∩ t2 = ⋂ n ∈ insert a s, f n := by
      rw [h_t1, h_t2, Finset.set_biInter_insert, Set.inter_comm]
    filter_upwards [hp_ind (insert a s) h_f_mem] with a' ha'
    rw [h_t1_inter_t2, ← ha']
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (hypothesis)
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : (n : ι) → Set Ω := fun (n : ι) ↦ if n = a then t2 else if n ∈ s then ft1 n else univ
h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
h_f_mem_pi : ∀ n ∈ s, f n ∈ π n
h_t1 : t1 = ⋂ n ∈ s, f n
h_μ_t1 :
  ∀ᵐ (a' : α) ∂μ,
    Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
      (∏ n ∈ s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : ℝ≥0∞)
h_t2 : t2 = f a
h_μ_inter :
  ∀ᵐ (a' : α) ∂μ,
    Eq (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ t2) : ℝ≥0∞)
      (∏ n ∈ insert a s, (((κ : (a : α) → Measure Ω) a' : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f n) : ℝ≥0∞)
⊢ ∀ᵐ (a : α) ∂μ,
    (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ t2) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
        ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t2 : ℝ≥0∞)
have has : a ∉ s
Tactic: have has : a ∉ s := fun has_mem => haS (hs_mem has_mem)