Theorem: MeasureTheory.toReal_rnDeriv_tilted_right
File: Mathlib/MeasureTheory/Measure/Tilted.lean

tactic: filter_upwards [rnDeriv_tilted_right μ ν hf] with x hx
state_before: α : Type u_1
mα : MeasurableSpace α
μ✝ : Measure α
f : α → ℝ
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hf : Integrable (fun x => rexp (f x)) ν
⊢ (fun x => (μ.rnDeriv (ν.tilted f) x).toReal) =ᶠ[ae ν] fun x =>
    (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * (μ.rnDeriv ν x).toReal
state_after: case h
α : Type u_1
mα : MeasurableSpace α
μ✝ : Measure α
f : α → ℝ
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hf : Integrable (fun x => rexp (f x)) ν
x : α
hx : μ.rnDeriv (ν.tilted f) x = ENNReal.ofReal (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * μ.rnDeriv ν x
⊢ (μ.rnDeriv (ν.tilted f) x).toReal = (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * (μ.rnDeriv ν x).toReal

tactic: rw [hx]
state_before: case h
α : Type u_1
mα : MeasurableSpace α
μ✝ : Measure α
f : α → ℝ
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hf : Integrable (fun x => rexp (f x)) ν
x : α
hx : μ.rnDeriv (ν.tilted f) x = ENNReal.ofReal (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * μ.rnDeriv ν x
⊢ (μ.rnDeriv (ν.tilted f) x).toReal = (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * (μ.rnDeriv ν x).toReal
state_after: case h
α : Type u_1
mα : MeasurableSpace α
μ✝ : Measure α
f : α → ℝ
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hf : Integrable (fun x => rexp (f x)) ν
x : α
hx : μ.rnDeriv (ν.tilted f) x = ENNReal.ofReal (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * μ.rnDeriv ν x
⊢ (ENNReal.ofReal (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * μ.rnDeriv ν x).toReal =
    (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * (μ.rnDeriv ν x).toReal

tactic: simp only [ENNReal.toReal_mul, gt_iff_lt, mul_eq_mul_right_iff, ENNReal.toReal_ofReal_eq_iff]
state_before: case h
α : Type u_1
mα : MeasurableSpace α
μ✝ : Measure α
f : α → ℝ
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hf : Integrable (fun x => rexp (f x)) ν
x : α
hx : μ.rnDeriv (ν.tilted f) x = ENNReal.ofReal (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * μ.rnDeriv ν x
⊢ (ENNReal.ofReal (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * μ.rnDeriv ν x).toReal =
    (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * (μ.rnDeriv ν x).toReal
state_after: case h
α : Type u_1
mα : MeasurableSpace α
μ✝ : Measure α
f : α → ℝ
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hf : Integrable (fun x => rexp (f x)) ν
x : α
hx : μ.rnDeriv (ν.tilted f) x = ENNReal.ofReal (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * μ.rnDeriv ν x
⊢ 0 ≤ rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν ∨ (μ.rnDeriv ν x).toReal = 0

tactic: exact Or.inl (by positivity)
state_before: case h
α : Type u_1
mα : MeasurableSpace α
μ✝ : Measure α
f : α → ℝ
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hf : Integrable (fun x => rexp (f x)) ν
x : α
hx : μ.rnDeriv (ν.tilted f) x = ENNReal.ofReal (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * μ.rnDeriv ν x
⊢ 0 ≤ rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν ∨ (μ.rnDeriv ν x).toReal = 0
state_after: no goals

tactic: positivity
state_before: α : Type u_1
mα : MeasurableSpace α
μ✝ : Measure α
f : α → ℝ
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hf : Integrable (fun x => rexp (f x)) ν
x : α
hx : μ.rnDeriv (ν.tilted f) x = ENNReal.ofReal (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * μ.rnDeriv ν x
⊢ 0 ≤ rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν
state_after: no goals