Theorem: MeasureTheory.Measure.iInf_rat_gt_prod_Iic
File: Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean

tactic: rw [← measure_iInter_eq_iInf]
state_before: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
t : ℚ
⊢ ⨅ r, ρ (s ×ˢ Iic ↑↑r) = ρ (s ×ˢ Iic ↑t)
state_after: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
t : ℚ
⊢ ρ (⋂ i, s ×ˢ Iic ↑↑i) = ρ (s ×ˢ Iic ↑t)

case h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
t : ℚ
⊢ ∀ (i : { r' // t < r' }), MeasurableSet (s ×ˢ Iic ↑↑i)

case hd
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
t : ℚ
⊢ Directed (fun x x_1 => x ⊇ x_1) fun r => s ×ˢ Iic ↑↑r

case hfin
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
t : ℚ
⊢ ∃ i, ρ (s ×ˢ Iic ↑↑i) ≠ ⊤

tactic: rw [← prod_iInter]
state_before: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
t : ℚ
⊢ ρ (⋂ i, s ×ˢ Iic ↑↑i) = ρ (s ×ˢ Iic ↑t)
state_after: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
t : ℚ
⊢ ρ (s ×ˢ ⋂ i, Iic ↑↑i) = ρ (s ×ˢ Iic ↑t)

tactic: congr with x : 1
state_before: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
t : ℚ
⊢ ρ (s ×ˢ ⋂ i, Iic ↑↑i) = ρ (s ×ˢ Iic ↑t)
state_after: case h.e_6.h.e_a.h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
t : ℚ
x : ℝ
⊢ x ∈ ⋂ i, Iic ↑↑i ↔ x ∈ Iic ↑t

tactic: simp only [mem_iInter, mem_Iic, Subtype.forall, Subtype.coe_mk]
state_before: case h.e_6.h.e_a.h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
t : ℚ
x : ℝ
⊢ x ∈ ⋂ i, Iic ↑↑i ↔ x ∈ Iic ↑t
state_after: case h.e_6.h.e_a.h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
t : ℚ
x : ℝ
⊢ (∀ (a : ℚ), t < a → x ≤ ↑a) ↔ x ≤ ↑t

tactic: refine ⟨fun h ↦ ?_, fun h a hta ↦ h.trans ?_⟩
state_before: case h.e_6.h.e_a.h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
t : ℚ
x : ℝ
⊢ (∀ (a : ℚ), t < a → x ≤ ↑a) ↔ x ≤ ↑t
state_after: case h.e_6.h.e_a.h.refine_1
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
t : ℚ
x : ℝ
h : ∀ (a : ℚ), t < a → x ≤ ↑a
⊢ x ≤ ↑t

case h.e_6.h.e_a.h.refine_2
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
t : ℚ
x : ℝ
h : x ≤ ↑t
a : ℚ
hta : t < a
⊢ ↑t ≤ ↑a

tactic: refine le_of_forall_lt_rat_imp_le fun q htq ↦ h q ?_
state_before: case h.e_6.h.e_a.h.refine_1
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
t : ℚ
x : ℝ
h : ∀ (a : ℚ), t < a → x ≤ ↑a
⊢ x ≤ ↑t
state_after: case h.e_6.h.e_a.h.refine_1
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
t : ℚ
x : ℝ
h : ∀ (a : ℚ), t < a → x ≤ ↑a
q : ℚ
htq : ↑t < ↑q
⊢ t < q

tactic: exact mod_cast htq
state_before: case h.e_6.h.e_a.h.refine_1
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
t : ℚ
x : ℝ
h : ∀ (a : ℚ), t < a → x ≤ ↑a
q : ℚ
htq : ↑t < ↑q
⊢ t < q
state_after: no goals

tactic: exact mod_cast hta.le
state_before: case h.e_6.h.e_a.h.refine_2
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
t : ℚ
x : ℝ
h : x ≤ ↑t
a : ℚ
hta : t < a
⊢ ↑t ≤ ↑a
state_after: no goals

tactic: exact fun _ => hs.prod measurableSet_Iic
state_before: case h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
t : ℚ
⊢ ∀ (i : { r' // t < r' }), MeasurableSet (s ×ˢ Iic ↑↑i)
state_after: no goals

tactic: refine Monotone.directed_ge fun r r' hrr' ↦ prod_subset_prod_iff.mpr (Or.inl ⟨subset_rfl, ?_⟩)
state_before: case hd
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
t : ℚ
⊢ Directed (fun x x_1 => x ⊇ x_1) fun r => s ×ˢ Iic ↑↑r
state_after: case hd
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
t : ℚ
r r' : { r' // t < r' }
hrr' : r ≤ r'
⊢ Iic ↑↑r ⊆ Iic ↑↑r'

tactic: refine Iic_subset_Iic.mpr ?_
state_before: case hd
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
t : ℚ
r r' : { r' // t < r' }
hrr' : r ≤ r'
⊢ Iic ↑↑r ⊆ Iic ↑↑r'
state_after: case hd
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
t : ℚ
r r' : { r' // t < r' }
hrr' : r ≤ r'
⊢ ↑↑r ≤ ↑↑r'

tactic: exact mod_cast hrr'
state_before: case hd
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
t : ℚ
r r' : { r' // t < r' }
hrr' : r ≤ r'
⊢ ↑↑r ≤ ↑↑r'
state_after: no goals

tactic: exact ⟨⟨t + 1, lt_add_one _⟩, measure_ne_top ρ _⟩
state_before: case hfin
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
t : ℚ
⊢ ∃ i, ρ (s ×ˢ Iic ↑↑i) ≠ ⊤
state_after: no goals