-- Mathlib/MeasureTheory/PiSystem.lean
theorem induction_on_inter {m : MeasurableSpace α} {C : ∀ s : Set α, MeasurableSet s → Prop}
    {s : Set (Set α)} (h_eq : m = generateFrom s) (h_inter : IsPiSystem s)
    (empty : C ∅ .empty) (basic : ∀ t (ht : t ∈ s), C t <| h_eq ▸ .basic t ht)
    (compl : ∀ t (htm : MeasurableSet t), C t htm → C tᶜ htm.compl)
    (iUnion : ∀ (f : ℕ → Set α), Pairwise (Disjoint on f) → ∀ (hfm : ∀ i, MeasurableSet (f i)),
      (∀ i, C (f i) (hfm i)) → C (⋃ i, f i) (.iUnion hfm)) :
    ∀ t (ht : MeasurableSet t), C t ht

Proof state:
α : Type u_3
m : MeasurableSpace α
C : (s : Set α) → MeasurableSet s → Prop
s : Set (Set α)
h_eq : m = generateFrom s
h_inter : IsPiSystem s
empty : C ∅ ⋯
basic : ∀ (t : Set α) (ht : t ∈ s), C t ⋯
compl : ∀ (t : Set α) (htm : MeasurableSet t), C t htm → C tᶜ ⋯
iUnion :
  ∀ (f : ℕ → Set α),
    Pairwise (Disjoint on f) → ∀ (hfm : ∀ (i : ℕ), MeasurableSet (f i)), (∀ (i : ℕ), C (f i) ⋯) → C (⋃ i, f i) ⋯
eq : MeasurableSet = DynkinSystem.GenerateHas s
⊢ ∀ (t : Set α) (ht : DynkinSystem.GenerateHas s t), C t ⋯

Tactic:
intro t ht

Result:
α : Type u_3
m : MeasurableSpace α
C : (s : Set α) → MeasurableSet s → Prop
s : Set (Set α)
h_eq : m = generateFrom s
h_inter : IsPiSystem s
empty : C ∅ ⋯
basic : ∀ (t : Set α) (ht : t ∈ s), C t ⋯
compl : ∀ (t : Set α) (htm : MeasurableSet t), C t htm → C tᶜ ⋯
iUnion :
  ∀ (f : ℕ → Set α),
    Pairwise (Disjoint on f) → ∀ (hfm : ∀ (i : ℕ), MeasurableSet (f i)), (∀ (i : ℕ), C (f i) ⋯) → C (⋃ i, f i) ⋯
eq : MeasurableSet = DynkinSystem.GenerateHas s
t : Set α
ht : DynkinSystem.GenerateHas s t
⊢ C t ⋯

Proof state:
case iUnion
α : Type u_3
m : MeasurableSpace α
C : (s : Set α) → MeasurableSet s → Prop
s : Set (Set α)
h_eq : m = generateFrom s
h_inter : IsPiSystem s
empty : C ∅ ⋯
basic : ∀ (t : Set α) (ht : t ∈ s), C t ⋯
compl : ∀ (t : Set α) (htm : MeasurableSet t), C t htm → C tᶜ ⋯
iUnion :
  ∀ (f : ℕ → Set α),
    Pairwise (Disjoint on f) → ∀ (hfm : ∀ (i : ℕ), MeasurableSet (f i)), (∀ (i : ℕ), C (f i) ⋯) → C (⋃ i, f i) ⋯
eq : MeasurableSet = DynkinSystem.GenerateHas s
t : Set α
f : ℕ → Set α
hfd : Pairwise (Disjoint on f)
hf : ∀ (i : ℕ), DynkinSystem.GenerateHas s (f i)
ihf : ∀ (i : ℕ), C (f i) ⋯
⊢ C (⋃ i, f i) ⋯

Tactic:
exact iUnion f hfd (eq ▸ hf) ihf

Result:
no goals

Proof state:
α : Type u_3
m : MeasurableSpace α
C : (s : Set α) → MeasurableSet s → Prop
s : Set (Set α)
h_eq : m = generateFrom s
h_inter : IsPiSystem s
empty : C ∅ ⋯
basic : ∀ (t : Set α) (ht : t ∈ s), C t ⋯
compl : ∀ (t : Set α) (htm : MeasurableSet t), C t htm → C tᶜ ⋯
iUnion :
  ∀ (f : ℕ → Set α),
    Pairwise (Disjoint on f) → ∀ (hfm : ∀ (i : ℕ), MeasurableSet (f i)), (∀ (i : ℕ), C (f i) ⋯) → C (⋃ i, f i) ⋯
⊢ ∀ (t : Set α) (ht : MeasurableSet t), C t ht

Tactic:
have eq : MeasurableSet = DynkinSystem.GenerateHas s := by
  rw [h_eq, DynkinSystem.generateFrom_eq h_inter]
  rfl

Result:
α : Type u_3
m : MeasurableSpace α
C : (s : Set α) → MeasurableSet s → Prop
s : Set (Set α)
h_eq : m = generateFrom s
h_inter : IsPiSystem s
empty : C ∅ ⋯
basic : ∀ (t : Set α) (ht : t ∈ s), C t ⋯
compl : ∀ (t : Set α) (htm : MeasurableSet t), C t htm → C tᶜ ⋯
iUnion :
  ∀ (f : ℕ → Set α),
    Pairwise (Disjoint on f) → ∀ (hfm : ∀ (i : ℕ), MeasurableSet (f i)), (∀ (i : ℕ), C (f i) ⋯) → C (⋃ i, f i) ⋯
eq : MeasurableSet = DynkinSystem.GenerateHas s
⊢ ∀ (t : Set α) (ht : MeasurableSet t), C t ht

Proof state:
α : Type u_3
m : MeasurableSpace α
C : (s : Set α) → MeasurableSet s → Prop
s : Set (Set α)
h_eq : m = generateFrom s
h_inter : IsPiSystem s
empty : C ∅ ⋯
basic : ∀ (t : Set α) (ht : t ∈ s), C t ⋯
compl : ∀ (t : Set α) (htm : MeasurableSet t), C t htm → C tᶜ ⋯
iUnion :
  ∀ (f : ℕ → Set α),
    Pairwise (Disjoint on f) → ∀ (hfm : ∀ (i : ℕ), MeasurableSet (f i)), (∀ (i : ℕ), C (f i) ⋯) → C (⋃ i, f i) ⋯
eq : MeasurableSet = DynkinSystem.GenerateHas s
t : Set α
ht : DynkinSystem.GenerateHas s t
⊢ C t ⋯

Tactic:
induction ht with
| basic u hu => exact basic u hu
| empty => exact empty
| @compl u hu ihu => exact compl _ (eq ▸ hu) ihu
| @iUnion f hfd hf ihf => exact iUnion f hfd (eq ▸ hf) ihf

Result:
no goals

Proof state:
case empty
α : Type u_3
m : MeasurableSpace α
C : (s : Set α) → MeasurableSet s → Prop
s : Set (Set α)
h_eq : m = generateFrom s
h_inter : IsPiSystem s
empty : C ∅ ⋯
basic : ∀ (t : Set α) (ht : t ∈ s), C t ⋯
compl : ∀ (t : Set α) (htm : MeasurableSet t), C t htm → C tᶜ ⋯
iUnion :
  ∀ (f : ℕ → Set α),
    Pairwise (Disjoint on f) → ∀ (hfm : ∀ (i : ℕ), MeasurableSet (f i)), (∀ (i : ℕ), C (f i) ⋯) → C (⋃ i, f i) ⋯
eq : MeasurableSet = DynkinSystem.GenerateHas s
t : Set α
⊢ C ∅ ⋯

Tactic:
exact empty

Result:
no goals

Proof state:
α : Type u_3
m : MeasurableSpace α
C : (s : Set α) → MeasurableSet s → Prop
s : Set (Set α)
h_eq : m = generateFrom s
h_inter : IsPiSystem s
empty : C ∅ ⋯
basic : ∀ (t : Set α) (ht : t ∈ s), C t ⋯
compl : ∀ (t : Set α) (htm : MeasurableSet t), C t htm → C tᶜ ⋯
iUnion :
  ∀ (f : ℕ → Set α),
    Pairwise (Disjoint on f) → ∀ (hfm : ∀ (i : ℕ), MeasurableSet (f i)), (∀ (i : ℕ), C (f i) ⋯) → C (⋃ i, f i) ⋯
eq : MeasurableSet = DynkinSystem.GenerateHas s
⊢ ∀ (t : Set α) (ht : MeasurableSet t), C t ht

Tactic:
suffices ∀ t (ht : DynkinSystem.GenerateHas s t), C t (eq ▸ ht) from
  fun t ht ↦ this t (eq ▸ ht)

Result:
α : Type u_3
m : MeasurableSpace α
C : (s : Set α) → MeasurableSet s → Prop
s : Set (Set α)
h_eq : m = generateFrom s
h_inter : IsPiSystem s
empty : C ∅ ⋯
basic : ∀ (t : Set α) (ht : t ∈ s), C t ⋯
compl : ∀ (t : Set α) (htm : MeasurableSet t), C t htm → C tᶜ ⋯
iUnion :
  ∀ (f : ℕ → Set α),
    Pairwise (Disjoint on f) → ∀ (hfm : ∀ (i : ℕ), MeasurableSet (f i)), (∀ (i : ℕ), C (f i) ⋯) → C (⋃ i, f i) ⋯
eq : MeasurableSet = DynkinSystem.GenerateHas s
⊢ ∀ (t : Set α) (ht : DynkinSystem.GenerateHas s t), C t ⋯

Proof state:
case basic
α : Type u_3
m : MeasurableSpace α
C : (s : Set α) → MeasurableSet s → Prop
s : Set (Set α)
h_eq : m = generateFrom s
h_inter : IsPiSystem s
empty : C ∅ ⋯
basic : ∀ (t : Set α) (ht : t ∈ s), C t ⋯
compl : ∀ (t : Set α) (htm : MeasurableSet t), C t htm → C tᶜ ⋯
iUnion :
  ∀ (f : ℕ → Set α),
    Pairwise (Disjoint on f) → ∀ (hfm : ∀ (i : ℕ), MeasurableSet (f i)), (∀ (i : ℕ), C (f i) ⋯) → C (⋃ i, f i) ⋯
eq : MeasurableSet = DynkinSystem.GenerateHas s
t u : Set α
hu : u ∈ s
⊢ C u ⋯

Tactic:
exact basic u hu

Result:
no goals

Proof state:
case compl
α : Type u_3
m : MeasurableSpace α
C : (s : Set α) → MeasurableSet s → Prop
s : Set (Set α)
h_eq : m = generateFrom s
h_inter : IsPiSystem s
empty : C ∅ ⋯
basic : ∀ (t : Set α) (ht : t ∈ s), C t ⋯
compl : ∀ (t : Set α) (htm : MeasurableSet t), C t htm → C tᶜ ⋯
iUnion :
  ∀ (f : ℕ → Set α),
    Pairwise (Disjoint on f) → ∀ (hfm : ∀ (i : ℕ), MeasurableSet (f i)), (∀ (i : ℕ), C (f i) ⋯) → C (⋃ i, f i) ⋯
eq : MeasurableSet = DynkinSystem.GenerateHas s
t u : Set α
hu : DynkinSystem.GenerateHas s u
ihu : C u ⋯
⊢ C uᶜ ⋯

Tactic:
exact compl _ (eq ▸ hu) ihu

Result:
no goals

Proof state:
α : Type u_3
m : MeasurableSpace α
C : (s : Set α) → MeasurableSet s → Prop
s : Set (Set α)
h_eq : m = generateFrom s
h_inter : IsPiSystem s
empty : C ∅ ⋯
basic : ∀ (t : Set α) (ht : t ∈ s), C t ⋯
compl : ∀ (t : Set α) (htm : MeasurableSet t), C t htm → C tᶜ ⋯
iUnion :
  ∀ (f : ℕ → Set α),
    Pairwise (Disjoint on f) → ∀ (hfm : ∀ (i : ℕ), MeasurableSet (f i)), (∀ (i : ℕ), C (f i) ⋯) → C (⋃ i, f i) ⋯
⊢ MeasurableSet = DynkinSystem.GenerateHas s

Tactic:
rfl

Result:
no goals