Theorem: ProbabilityTheory.Kernel.densityProcess_mono_kernel_left
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
κ' : Kernel α (γ × β)
hκκ' : κ ≤ κ'
hκ'ν : κ'.fst ≤ ν
n : ℕ
a : α
x : γ
s : Set β
⊢ 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
κ' : Kernel α (γ × β)
hκκ' : κ ≤ κ'
hκ'ν : κ'.fst ≤ ν
n : ℕ
a : α
x : γ
s : Set β
⊢ ((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
κ' : Kernel α (γ × β)
hκκ' : κ ≤ κ'
hκ'ν : κ'.fst ≤ ν
n : ℕ
a : α
x : γ
s : Set β
⊢ ((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
κ' : Kernel α (γ × β)
hκκ' : κ ≤ κ'
hκ'ν : κ'.fst ≤ ν
n : ℕ
a : α
x : γ
s : Set β
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
κ' : Kernel α (γ × β)
hκκ' : κ ≤ κ'
hκ'ν : κ'.fst ≤ ν
n : ℕ
a : α
x : γ
s : Set β
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
κ' : Kernel α (γ × β)
hκκ' : κ ≤ κ'
hκ'ν : κ'.fst ≤ ν
n : ℕ
a : α
x : γ
s : Set β
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
κ' : Kernel α (γ × β)
hκκ' : κ ≤ κ'
hκ'ν : κ'.fst ≤ ν
n : ℕ
a : α
x : γ
s : Set β
h0 : ¬(ν a) (countablePartitionSet n x) = 0
h_le : (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_le : κ' a (countablePartitionSet n x ×ˢ s) ≤ ν a (countablePartitionSet n x) :=
    meas_countablePartitionSet_le_of_fst_le hκ'ν 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
κ' : Kernel α (γ × β)
hκκ' : κ ≤ κ'
hκ'ν : κ'.fst ≤ ν
n : ℕ
a : α
x : γ
s : Set β
h0 : ¬(ν a) (countablePartitionSet n x) = 0
h_le : (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
κ' : Kernel α (γ × β)
hκκ' : κ ≤ κ'
hκ'ν : κ'.fst ≤ ν
n : ℕ
a : α
x : γ
s : Set β
h0 : ¬(ν a) (countablePartitionSet n x) = 0
h_le : (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)

case neg.ha
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
κ' : Kernel α (γ × β)
hκκ' : κ ≤ κ'
hκ'ν : κ'.fst ≤ ν
n : ℕ
a : α
x : γ
s : Set β
h0 : ¬(ν a) (countablePartitionSet n x) = 0
h_le : (κ' a) (countablePartitionSet n x ×ˢ s) ≤ (ν a) (countablePartitionSet n x)
⊢ (κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x) ≠ ⊤

case neg.hb
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
κ' : Kernel α (γ × β)
hκκ' : κ ≤ κ'
hκ'ν : κ'.fst ≤ ν
n : ℕ
a : α
x : γ
s : Set β
h0 : ¬(ν a) (countablePartitionSet n x) = 0
h_le : (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]