-- Mathlib/Probability/Kernel/WithDensity.lean
theorem isSFiniteKernel_withDensity_of_isFiniteKernel (κ : Kernel α β) [IsFiniteKernel κ]
    (hf_ne_top : ∀ a b, f a b ≠ ∞) : IsSFiniteKernel (withDensity κ f)

Proof state:
case h.h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β → ℝ≥0∞
κ : Kernel α β
inst✝ : IsFiniteKernel κ
hf_ne_top : ∀ (a : α) (b : β), f a b ≠ ⊤
hf : Measurable (Function.uncurry f)
fs : ℕ → α → β → ℝ≥0∞ := fun n a b => f a b ⊓ (↑n + 1) - f a b ⊓ ↑n
h_le : ∀ (a : α) (b : β) (n : ℕ), ⌈(f a b).toReal⌉₊ ≤ n → f a b ≤ ↑n
h_zero : ∀ (a : α) (b : β) (n : ℕ), ⌈(f a b).toReal⌉₊ ≤ n → fs n a b = 0
h_sum_a : ∀ (a : α), Summable fun n => fs n a
a : α
b : β
h_finset_sum : ∀ (n : ℕ), ∑ i ∈ Finset.range n, fs i a b = f a b ⊓ ↑n
⊢ (fun x => f a b) =ᶠ[Filter.atTop] fun n => f a b ⊓ ↑n

Tactic:
rw [Filter.EventuallyEq, Filter.eventually_atTop]

Result:
case h.h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β → ℝ≥0∞
κ : Kernel α β
inst✝ : IsFiniteKernel κ
hf_ne_top : ∀ (a : α) (b : β), f a b ≠ ⊤
hf : Measurable (Function.uncurry f)
fs : ℕ → α → β → ℝ≥0∞ := fun n a b => f a b ⊓ (↑n + 1) - f a b ⊓ ↑n
h_le : ∀ (a : α) (b : β) (n : ℕ), ⌈(f a b).toReal⌉₊ ≤ n → f a b ≤ ↑n
h_zero : ∀ (a : α) (b : β) (n : ℕ), ⌈(f a b).toReal⌉₊ ≤ n → fs n a b = 0
h_sum_a : ∀ (a : α), Summable fun n => fs n a
a : α
b : β
h_finset_sum : ∀ (n : ℕ), ∑ i ∈ Finset.range n, fs i a b = f a b ⊓ ↑n
⊢ ∃ a_1, ∀ b_1 ≥ a_1, f a b = f a b ⊓ ↑b_1

Proof state:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β → ℝ≥0∞
κ : Kernel α β
inst✝ : IsFiniteKernel κ
hf_ne_top : ∀ (a : α) (b : β), f a b ≠ ⊤
hf : Measurable (Function.uncurry f)
fs : ℕ → α → β → ℝ≥0∞ := fun n a b => f a b ⊓ (↑n + 1) - f a b ⊓ ↑n
h_le : ∀ (a : α) (b : β) (n : ℕ), ⌈(f a b).toReal⌉₊ ≤ n → f a b ≤ ↑n
h_zero : ∀ (a : α) (b : β) (n : ℕ), ⌈(f a b).toReal⌉₊ ≤ n → fs n a b = 0
hf_eq_tsum : f = ∑' (n : ℕ), fs n
n : ℕ
this✝ this : IsFiniteKernel (κ.withDensity (fs n))
⊢ IsSFiniteKernel (κ.withDensity (fs n))

Tactic:
infer_instance

Result:
no goals

Proof state:
case succ
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β → ℝ≥0∞
κ : Kernel α β
inst✝ : IsFiniteKernel κ
hf_ne_top : ∀ (a : α) (b : β), f a b ≠ ⊤
hf : Measurable (Function.uncurry f)
fs : ℕ → α → β → ℝ≥0∞ := fun n a b => f a b ⊓ (↑n + 1) - f a b ⊓ ↑n
h_le : ∀ (a : α) (b : β) (n : ℕ), ⌈(f a b).toReal⌉₊ ≤ n → f a b ≤ ↑n
h_zero : ∀ (a : α) (b : β) (n : ℕ), ⌈(f a b).toReal⌉₊ ≤ n → fs n a b = 0
h_sum_a : ∀ (a : α), Summable fun n => fs n a
a : α
b : β
n : ℕ
hn : ∑ i ∈ Finset.range n, fs i a b = f a b ⊓ ↑n
⊢ f a b ⊓ ↑n + fs n a b = f a b ⊓ ↑(n + 1)

Tactic:
simp [fs]

Result:
no goals

Proof state:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β → ℝ≥0∞
κ : Kernel α β
inst✝ : IsFiniteKernel κ
hf_ne_top : ∀ (a : α) (b : β), f a b ≠ ⊤
hf : Measurable (Function.uncurry f)
fs : ℕ → α → β → ℝ≥0∞ := fun n a b => f a b ⊓ (↑n + 1) - f a b ⊓ ↑n
h_le : ∀ (a : α) (b : β) (n : ℕ), ⌈(f a b).toReal⌉₊ ≤ n → f a b ≤ ↑n
h_zero : ∀ (a : α) (b : β) (n : ℕ), ⌈(f a b).toReal⌉₊ ≤ n → fs n a b = 0
hf_eq_tsum : f = ∑' (n : ℕ), fs n
⊢ ∀ (n : ℕ), Measurable (Function.uncurry (fs n))

Tactic:
fun_prop

Result:
no goals

Proof state:
case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β → ℝ≥0∞
κ : Kernel α β
inst✝ : IsFiniteKernel κ
hf_ne_top : ∀ (a : α) (b : β), f a b ≠ ⊤
hf : Measurable (Function.uncurry f)
fs : ℕ → α → β → ℝ≥0∞ := fun n a b => f a b ⊓ (↑n + 1) - f a b ⊓ ↑n
h_le : ∀ (a : α) (b : β) (n : ℕ), ⌈(f a b).toReal⌉₊ ≤ n → f a b ≤ ↑n
h_zero : ∀ (a : α) (b : β) (n : ℕ), ⌈(f a b).toReal⌉₊ ≤ n → fs n a b = 0
hf_eq_tsum : f = ∑' (n : ℕ), fs n
n : ℕ
⊢ IsFiniteKernel (κ.withDensity (fs n))

Tactic:
refine isFiniteKernel_withDensity_of_bounded _ (ENNReal.coe_ne_top : ↑n + 1 ≠ ∞) fun a b => ?_

Result:
case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β → ℝ≥0∞
κ : Kernel α β
inst✝ : IsFiniteKernel κ
hf_ne_top : ∀ (a : α) (b : β), f a b ≠ ⊤
hf : Measurable (Function.uncurry f)
fs : ℕ → α → β → ℝ≥0∞ := fun n a b => f a b ⊓ (↑n + 1) - f a b ⊓ ↑n
h_le : ∀ (a : α) (b : β) (n : ℕ), ⌈(f a b).toReal⌉₊ ≤ n → f a b ≤ ↑n
h_zero : ∀ (a : α) (b : β) (n : ℕ), ⌈(f a b).toReal⌉₊ ≤ n → fs n a b = 0
hf_eq_tsum : f = ∑' (n : ℕ), fs n
n : ℕ
a : α
b : β
⊢ fs n a b ≤ ↑((fun x1 x2 => x1 + x2) (↑n) 1)

Proof state:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β → ℝ≥0∞
κ : Kernel α β
inst✝ : IsFiniteKernel κ
hf_ne_top : ∀ (a : α) (b : β), f a b ≠ ⊤
hf : Measurable (Function.uncurry f)
fs : ℕ → α → β → ℝ≥0∞ := fun n a b => f a b ⊓ (↑n + 1) - f a b ⊓ ↑n
h_le : ∀ (a : α) (b : β) (n : ℕ), ⌈(f a b).toReal⌉₊ ≤ n → f a b ≤ ↑n
h_zero : ∀ (a : α) (b : β) (n : ℕ), ⌈(f a b).toReal⌉₊ ≤ n → fs n a b = 0
hf_eq_tsum : f = ∑' (n : ℕ), fs n
n : ℕ
a : α
b : β
⊢ ↑n + 1 = ↑(n + 1)

Tactic:
norm_cast

Result:
no goals

Proof state:
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β → ℝ≥0∞
κ : Kernel α β
inst✝ : IsFiniteKernel κ
hf_ne_top : ∀ (a : α) (b : β), f a b ≠ ⊤
hf : ¬Measurable (Function.uncurry f)
⊢ IsSFiniteKernel 0

Tactic:
infer_instance

Result:
no goals

Proof state:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β → ℝ≥0∞
κ : Kernel α β
inst✝ : IsFiniteKernel κ
hf_ne_top : ∀ (a : α) (b : β), f a b ≠ ⊤
hf : Measurable (Function.uncurry f)
fs : ℕ → α → β → ℝ≥0∞ := fun n a b => f a b ⊓ (↑n + 1) - f a b ⊓ ↑n
h_le : ∀ (a : α) (b : β) (n : ℕ), ⌈(f a b).toReal⌉₊ ≤ n → f a b ≤ ↑n
h_zero : ∀ (a : α) (b : β) (n : ℕ), ⌈(f a b).toReal⌉₊ ≤ n → fs n a b = 0
⊢ f = ∑' (n : ℕ), fs n

Tactic:
have h_sum_a : ∀ a, Summable fun n => fs n a := by
  refine fun a => Pi.summable.mpr fun b => ?_
  suffices ∀ n, n ∉ Finset.range ⌈(f a b).toReal⌉₊ → fs n a b = 0 from
    summable_of_ne_finset_zero this
  intro n hn_not_mem
  rw [Finset.mem_range, not_lt] at hn_not_mem
  exact h_zero a b n hn_not_mem

Result:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β → ℝ≥0∞
κ : Kernel α β
inst✝ : IsFiniteKernel κ
hf_ne_top : ∀ (a : α) (b : β), f a b ≠ ⊤
hf : Measurable (Function.uncurry f)
fs : ℕ → α → β → ℝ≥0∞ := fun n a b => f a b ⊓ (↑n + 1) - f a b ⊓ ↑n
h_le : ∀ (a : α) (b : β) (n : ℕ), ⌈(f a b).toReal⌉₊ ≤ n → f a b ≤ ↑n
h_zero : ∀ (a : α) (b : β) (n : ℕ), ⌈(f a b).toReal⌉₊ ≤ n → fs n a b = 0
h_sum_a : ∀ (a : α), Summable fun n => fs n a
⊢ f = ∑' (n : ℕ), fs n

Proof state:
case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β → ℝ≥0∞
κ : Kernel α β
inst✝ : IsFiniteKernel κ
hf_ne_top : ∀ (a : α) (b : β), f a b ≠ ⊤
hf : Measurable (Function.uncurry f)
⊢ IsSFiniteKernel (κ.withDensity f)

Tactic:
let fs : ℕ → α → β → ℝ≥0∞ := fun n a b => min (f a b) (n + 1) - min (f a b) n

Result:
case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β → ℝ≥0∞
κ : Kernel α β
inst✝ : IsFiniteKernel κ
hf_ne_top : ∀ (a : α) (b : β), f a b ≠ ⊤
hf : Measurable (Function.uncurry f)
fs : ℕ → α → β → ℝ≥0∞ := fun n a b => f a b ⊓ (↑n + 1) - f a b ⊓ ↑n
⊢ IsSFiniteKernel (κ.withDensity f)