Theorem: MeasureTheory.lintegral_sub_le'
File: Mathlib/MeasureTheory/Integral/Lebesgue.lean

tactic: rw [tsub_le_iff_right]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
⊢ ∫⁻ (x : α), g x ∂μ - ∫⁻ (x : α), f x ∂μ ≤ ∫⁻ (x : α), g x - f x ∂μ
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
⊢ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), g x - f x ∂μ + ∫⁻ (x : α), f x ∂μ

tactic: by_cases hfi : ∫⁻ x, f x ∂μ = ∞
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
⊢ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), g x - f x ∂μ + ∫⁻ (x : α), f x ∂μ
state_after: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hfi : ∫⁻ (x : α), f x ∂μ = ⊤
⊢ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), g x - f x ∂μ + ∫⁻ (x : α), f x ∂μ

case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hfi : ¬∫⁻ (x : α), f x ∂μ = ⊤
⊢ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), g x - f x ∂μ + ∫⁻ (x : α), f x ∂μ

tactic: rw [hfi, add_top]
state_before: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hfi : ∫⁻ (x : α), f x ∂μ = ⊤
⊢ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), g x - f x ∂μ + ∫⁻ (x : α), f x ∂μ
state_after: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hfi : ∫⁻ (x : α), f x ∂μ = ⊤
⊢ ∫⁻ (x : α), g x ∂μ ≤ ⊤

tactic: exact le_top
state_before: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hfi : ∫⁻ (x : α), f x ∂μ = ⊤
⊢ ∫⁻ (x : α), g x ∂μ ≤ ⊤
state_after: no goals

tactic: rw [← lintegral_add_right' _ hf]
state_before: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hfi : ¬∫⁻ (x : α), f x ∂μ = ⊤
⊢ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), g x - f x ∂μ + ∫⁻ (x : α), f x ∂μ
state_after: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hfi : ¬∫⁻ (x : α), f x ∂μ = ⊤
⊢ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (a : α), g a - f a + f a ∂μ

tactic: gcongr
state_before: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hfi : ¬∫⁻ (x : α), f x ∂μ = ⊤
⊢ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (a : α), g a - f a + f a ∂μ
state_after: case neg.hfg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hfi : ¬∫⁻ (x : α), f x ∂μ = ⊤
x✝ : α
⊢ g x✝ ≤ g x✝ - f x✝ + f x✝

tactic: exact le_tsub_add
state_before: case neg.hfg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hfi : ¬∫⁻ (x : α), f x ∂μ = ⊤
x✝ : α
⊢ g x✝ ≤ g x✝ - f x✝ + f x✝
state_after: no goals