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

Proof steps:
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 : κ.fst ≤ ν
n : ℕ
a : α
x : γ
s s' : Set β
h : s ⊆ s'
⊢ κ.densityProcess ν n a x s ≤ κ.densityProcess ν n a x s'
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 : κ.fst ≤ ν
n : ℕ
a : α
x : γ
s s' : Set β
h : s ⊆ s'
⊢ ((κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x)).toReal ≤
    ((κ a) (countablePartitionSet n x ×ˢ s') / (ν a) (countablePartitionSet n x)).toReal
tactic: unfold densityProcess
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 : κ.fst ≤ ν
n : ℕ
a : α
x : γ
s s' : Set β
h : s ⊆ s'
⊢ ((κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x)).toReal ≤
    ((κ a) (countablePartitionSet n x ×ˢ s') / (ν a) (countablePartitionSet n x)).toReal
state_after: case pos
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 : κ.fst ≤ ν
n : ℕ
a : α
x : γ
s s' : Set β
h : s ⊆ s'
h0 : (ν a) (countablePartitionSet n x) = 0
⊢ ((κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x)).toReal ≤
    ((κ a) (countablePartitionSet n x ×ˢ s') / (ν a) (countablePartitionSet n x)).toReal

case neg
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 : κ.fst ≤ ν
n : ℕ
a : α
x : γ
s s' : Set β
h : s ⊆ s'
h0 : ¬(ν a) (countablePartitionSet n x) = 0
⊢ ((κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x)).toReal ≤
    ((κ a) (countablePartitionSet n x ×ˢ s') / (ν a) (countablePartitionSet n x)).toReal
tactic: by_cases h0 : ν a (countablePartitionSet n x) = 0
state_before: case neg
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 : κ.fst ≤ ν
n : ℕ
a : α
x : γ
s s' : Set β
h : s ⊆ s'
h0 : ¬(ν a) (countablePartitionSet n x) = 0
⊢ ((κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x)).toReal ≤
    ((κ a) (countablePartitionSet n x ×ˢ s') / (ν a) (countablePartitionSet n x)).toReal
state_after: case neg
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 : κ.fst ≤ ν
n : ℕ
a : α
x : γ
s s' : Set β
h : s ⊆ s'
h0 : ¬(ν a) (countablePartitionSet n x) = 0
h_ne_top : ∀ (s : Set β), (κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x) ≠ ⊤
⊢ ((κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x)).toReal ≤
    ((κ a) (countablePartitionSet n x ×ˢ s') / (ν a) (countablePartitionSet n x)).toReal
tactic: have h_ne_top : ∀ s,
      κ a (countablePartitionSet n x ×ˢ s) / ν a (countablePartitionSet n x) ≠ ⊤ := by
    intro s
    rw [ne_eq, ENNReal.div_eq_top]
    simp only [ne_eq, h0, and_false, false_or, not_and, not_not]
    refine fun h_top ↦ eq_top_mono ?_ h_top
    exact meas_countablePartitionSet_le_of_fst_le hκν n a x s
state_before: case neg
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 : κ.fst ≤ ν
n : ℕ
a : α
x : γ
s s' : Set β
h : s ⊆ s'
h0 : ¬(ν a) (countablePartitionSet n x) = 0
h_ne_top : ∀ (s : Set β), (κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x) ≠ ⊤
⊢ ((κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x)).toReal ≤
    ((κ a) (countablePartitionSet n x ×ˢ s') / (ν a) (countablePartitionSet n x)).toReal
state_after: case neg
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 : κ.fst ≤ ν
n : ℕ
a : α
x : γ
s s' : Set β
h : s ⊆ s'
h0 : ¬(ν a) (countablePartitionSet n x) = 0
h_ne_top : ∀ (s : Set β), (κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x) ≠ ⊤
⊢ (κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x) ≤
    (κ a) (countablePartitionSet n x ×ˢ s') / (ν a) (countablePartitionSet n x)
tactic: rw [ENNReal.toReal_le_toReal (h_ne_top s) (h_ne_top s')]
state_before: case neg
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 : κ.fst ≤ ν
n : ℕ
a : α
x : γ
s s' : Set β
h : s ⊆ s'
h0 : ¬(ν a) (countablePartitionSet n x) = 0
h_ne_top : ∀ (s : Set β), (κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x) ≠ ⊤
⊢ (κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x) ≤
    (κ a) (countablePartitionSet n x ×ˢ s') / (ν a) (countablePartitionSet n x)
state_after: no goals
tactic: gcongr