Theorem: ProbabilityTheory.condCount_disjoint_union
File: Mathlib/Probability/CondCount.lean

Proof steps:
state_before: Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
⊢ (condCount s) u * (condCount (s ∪ t)) s + (condCount t) u * (condCount (s ∪ t)) t = (condCount (s ∪ t)) u
state_after: case inl.inl
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
u : Set Ω
hs ht : ∅.Finite
hst : Disjoint ∅ ∅
⊢ (condCount ∅) u * (condCount (∅ ∪ ∅)) ∅ + (condCount ∅) u * (condCount (∅ ∪ ∅)) ∅ = (condCount (∅ ∪ ∅)) u

case inl.inr
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
t u : Set Ω
ht : t.Finite
hs : ∅.Finite
hst : Disjoint ∅ t
ht' : t.Nonempty
⊢ (condCount ∅) u * (condCount (∅ ∪ t)) ∅ + (condCount t) u * (condCount (∅ ∪ t)) t = (condCount (∅ ∪ t)) u

case inr.inl
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s u : Set Ω
hs : s.Finite
hs' : s.Nonempty
ht : ∅.Finite
hst : Disjoint s ∅
⊢ (condCount s) u * (condCount (s ∪ ∅)) s + (condCount ∅) u * (condCount (s ∪ ∅)) ∅ = (condCount (s ∪ ∅)) u

case inr.inr
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ (condCount s) u * (condCount (s ∪ t)) s + (condCount t) u * (condCount (s ∪ t)) t = (condCount (s ∪ t)) u
tactic: rcases s.eq_empty_or_nonempty with (rfl | hs') <;> rcases t.eq_empty_or_nonempty with (rfl | ht')
state_before: case inr.inr
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ (condCount s) u * (condCount (s ∪ t)) s + (condCount t) u * (condCount (s ∪ t)) t = (condCount (s ∪ t)) u
state_after: case inr.inr
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ (Measure.count s)⁻¹ * Measure.count (s ∩ u) * ((Measure.count (s ∪ t))⁻¹ * Measure.count ((s ∪ t) ∩ s)) +
      (Measure.count t)⁻¹ * Measure.count (t ∩ u) * ((Measure.count (s ∪ t))⁻¹ * Measure.count ((s ∪ t) ∩ t)) =
    (Measure.count (s ∪ t))⁻¹ * Measure.count ((s ∪ t) ∩ u)
tactic: rw [condCount, condCount, condCount, cond_apply _ hs.measurableSet,
    cond_apply _ ht.measurableSet, cond_apply _ (hs.union ht).measurableSet,
    cond_apply _ (hs.union ht).measurableSet, cond_apply _ (hs.union ht).measurableSet]
state_before: case inr.inr
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ (Measure.count s)⁻¹ * Measure.count (s ∩ u) * ((Measure.count (s ∪ t))⁻¹ * Measure.count ((s ∪ t) ∩ s)) +
      (Measure.count t)⁻¹ * Measure.count (t ∩ u) * ((Measure.count (s ∪ t))⁻¹ * Measure.count ((s ∪ t) ∩ t)) =
    (Measure.count (s ∪ t))⁻¹ * Measure.count ((s ∪ t) ∩ u)
state_after: case inr.inr
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ Measure.count s * (Measure.count s)⁻¹ * Measure.count (s ∩ u) * (Measure.count (s ∪ t))⁻¹ +
      Measure.count t * (Measure.count t)⁻¹ * Measure.count (t ∩ u) * (Measure.count (s ∪ t))⁻¹ =
    (Measure.count (s ∪ t))⁻¹ * Measure.count ((s ∪ t) ∩ u)
tactic: conv_lhs =>
    rw [Set.union_inter_cancel_left, Set.union_inter_cancel_right,
      mul_comm (Measure.count (s ∪ t))⁻¹, mul_comm (Measure.count (s ∪ t))⁻¹, ← mul_assoc,
      ← mul_assoc, mul_comm _ (Measure.count s), mul_comm _ (Measure.count t), ← mul_assoc,
      ← mul_assoc]
state_before: case inr.inr
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ Measure.count s * (Measure.count s)⁻¹ * Measure.count (s ∩ u) * (Measure.count (s ∪ t))⁻¹ +
      Measure.count t * (Measure.count t)⁻¹ * Measure.count (t ∩ u) * (Measure.count (s ∪ t))⁻¹ =
    (Measure.count (s ∪ t))⁻¹ * Measure.count ((s ∪ t) ∩ u)
state_after: case inr.inr.hd
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ Disjoint (s ∩ u) (t ∩ u)

case inr.inr.h
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ MeasurableSet (t ∩ u)

case inr.inr.h0
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ Measure.count t ≠ 0

case inr.inr.ht
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ Measure.count t ≠ ⊤

case inr.inr.h0
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ Measure.count s ≠ 0

case inr.inr.ht
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ Measure.count s ≠ ⊤
tactic: rw [ENNReal.mul_inv_cancel, ENNReal.mul_inv_cancel, one_mul, one_mul, ← add_mul, ← measure_union,
    Set.union_inter_distrib_right, mul_comm]
state_before: case inr.inr.hd
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ Disjoint (s ∩ u) (t ∩ u)

case inr.inr.h
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ MeasurableSet (t ∩ u)

case inr.inr.h0
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ Measure.count t ≠ 0

case inr.inr.ht
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ Measure.count t ≠ ⊤

case inr.inr.h0
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ Measure.count s ≠ 0

case inr.inr.ht
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ Measure.count s ≠ ⊤
state_after: no goals
tactic: exacts [hst.mono inf_le_left inf_le_left, (ht.inter_of_left _).measurableSet,
    Measure.count_ne_zero ht', (Measure.count_apply_lt_top.2 ht).ne, Measure.count_ne_zero hs',
    (Measure.count_apply_lt_top.2 hs).ne]