Theorem: ProbabilityTheory.kernel.meas_countablePartitionSet_le_of_fst_le
File: Mathlib/Probability/Kernel/Disintegration/Density.lean

tactic: calc κ a (countablePartitionSet n x ×ˢ s)
  ≤ fst κ a (countablePartitionSet n x) := by
      rw [fst_apply' _ _ (measurableSet_countablePartitionSet _ _)]
      refine measure_mono (fun x ↦ ?_)
      simp only [mem_prod, mem_setOf_eq, and_imp]
      exact fun h _ ↦ h
_ ≤ ν a (countablePartitionSet n x) := hκν a _
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
n : ℕ
a : α
x : γ
s : Set β
⊢ (κ a) (countablePartitionSet n x ×ˢ s) ≤ (ν a) (countablePartitionSet n x)
state_after: no goals

tactic: rw [fst_apply' _ _ (measurableSet_countablePartitionSet _ _)]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
n : ℕ
a : α
x : γ
s : Set β
⊢ (κ a) (countablePartitionSet n x ×ˢ s) ≤ ((fst κ) a) (countablePartitionSet n x)
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
n : ℕ
a : α
x : γ
s : Set β
⊢ (κ a) (countablePartitionSet n x ×ˢ s) ≤ (κ a) {p | p.1 ∈ countablePartitionSet n x}

tactic: refine measure_mono (fun x ↦ ?_)
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
n : ℕ
a : α
x : γ
s : Set β
⊢ (κ a) (countablePartitionSet n x ×ˢ s) ≤ (κ a) {p | p.1 ∈ countablePartitionSet n x}
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
n : ℕ
a : α
x✝ : γ
s : Set β
x : γ × β
⊢ x ∈ countablePartitionSet n x✝ ×ˢ s → x ∈ {p | p.1 ∈ countablePartitionSet n x✝}

tactic: simp only [mem_prod, mem_setOf_eq, and_imp]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
n : ℕ
a : α
x✝ : γ
s : Set β
x : γ × β
⊢ x ∈ countablePartitionSet n x✝ ×ˢ s → x ∈ {p | p.1 ∈ countablePartitionSet n x✝}
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
n : ℕ
a : α
x✝ : γ
s : Set β
x : γ × β
⊢ x.1 ∈ countablePartitionSet n x✝ → x.2 ∈ s → x.1 ∈ countablePartitionSet n x✝

tactic: exact fun h _ ↦ h
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
n : ℕ
a : α
x✝ : γ
s : Set β
x : γ × β
⊢ x.1 ∈ countablePartitionSet n x✝ → x.2 ∈ s → x.1 ∈ countablePartitionSet n x✝
state_after: no goals