Theorem: MeasureTheory.FiniteMeasure.testAgainstNN_lipschitz_estimate
File: Mathlib/MeasureTheory/Measure/FiniteMeasure.lean

tactic: simp only [← μ.testAgainstNN_const (nndist f g), ← testAgainstNN_add, ← ENNReal.coe_le_coe,
  BoundedContinuousFunction.coe_add, const_apply, ENNReal.coe_add, Pi.add_apply,
  coe_nnreal_ennreal_nndist, testAgainstNN_coe_eq]
state_before: Ω : Type u_1
inst✝⁶ : MeasurableSpace Ω
R : Type u_2
inst✝⁵ : SMul R ℝ≥0
inst✝⁴ : SMul R ℝ≥0∞
inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
f g : Ω →ᵇ ℝ≥0
⊢ μ.testAgainstNN f ≤ μ.testAgainstNN g + nndist f g * μ.mass
state_after: Ω : Type u_1
inst✝⁶ : MeasurableSpace Ω
R : Type u_2
inst✝⁵ : SMul R ℝ≥0
inst✝⁴ : SMul R ℝ≥0∞
inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
f g : Ω →ᵇ ℝ≥0
⊢ ∫⁻ (ω : Ω), ↑(f ω) ∂↑μ ≤ ∫⁻ (ω : Ω), ↑(g ω) + edist f g ∂↑μ

tactic: apply lintegral_mono
state_before: Ω : Type u_1
inst✝⁶ : MeasurableSpace Ω
R : Type u_2
inst✝⁵ : SMul R ℝ≥0
inst✝⁴ : SMul R ℝ≥0∞
inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
f g : Ω →ᵇ ℝ≥0
⊢ ∫⁻ (ω : Ω), ↑(f ω) ∂↑μ ≤ ∫⁻ (ω : Ω), ↑(g ω) + edist f g ∂↑μ
state_after: case hfg
Ω : Type u_1
inst✝⁶ : MeasurableSpace Ω
R : Type u_2
inst✝⁵ : SMul R ℝ≥0
inst✝⁴ : SMul R ℝ≥0∞
inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
f g : Ω →ᵇ ℝ≥0
⊢ (fun a => ↑(f a)) ≤ fun a => ↑(g a) + edist f g

tactic: have le_dist : ∀ ω, dist (f ω) (g ω) ≤ nndist f g := BoundedContinuousFunction.dist_coe_le_dist
state_before: case hfg
Ω : Type u_1
inst✝⁶ : MeasurableSpace Ω
R : Type u_2
inst✝⁵ : SMul R ℝ≥0
inst✝⁴ : SMul R ℝ≥0∞
inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
f g : Ω →ᵇ ℝ≥0
⊢ (fun a => ↑(f a)) ≤ fun a => ↑(g a) + edist f g
state_after: case hfg
Ω : Type u_1
inst✝⁶ : MeasurableSpace Ω
R : Type u_2
inst✝⁵ : SMul R ℝ≥0
inst✝⁴ : SMul R ℝ≥0∞
inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
f g : Ω →ᵇ ℝ≥0
le_dist : ∀ (ω : Ω), dist (f ω) (g ω) ≤ ↑(nndist f g)
⊢ (fun a => ↑(f a)) ≤ fun a => ↑(g a) + edist f g

tactic: intro ω
state_before: case hfg
Ω : Type u_1
inst✝⁶ : MeasurableSpace Ω
R : Type u_2
inst✝⁵ : SMul R ℝ≥0
inst✝⁴ : SMul R ℝ≥0∞
inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
f g : Ω →ᵇ ℝ≥0
le_dist : ∀ (ω : Ω), dist (f ω) (g ω) ≤ ↑(nndist f g)
⊢ (fun a => ↑(f a)) ≤ fun a => ↑(g a) + edist f g
state_after: case hfg
Ω : Type u_1
inst✝⁶ : MeasurableSpace Ω
R : Type u_2
inst✝⁵ : SMul R ℝ≥0
inst✝⁴ : SMul R ℝ≥0∞
inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
f g : Ω →ᵇ ℝ≥0
le_dist : ∀ (ω : Ω), dist (f ω) (g ω) ≤ ↑(nndist f g)
ω : Ω
⊢ (fun a => ↑(f a)) ω ≤ (fun a => ↑(g a) + edist f g) ω

tactic: have le' : f ω ≤ g ω + nndist f g := by
  apply (NNReal.le_add_nndist (f ω) (g ω)).trans
  rw [add_le_add_iff_left]
  exact dist_le_coe.mp (le_dist ω)
state_before: case hfg
Ω : Type u_1
inst✝⁶ : MeasurableSpace Ω
R : Type u_2
inst✝⁵ : SMul R ℝ≥0
inst✝⁴ : SMul R ℝ≥0∞
inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
f g : Ω →ᵇ ℝ≥0
le_dist : ∀ (ω : Ω), dist (f ω) (g ω) ≤ ↑(nndist f g)
ω : Ω
⊢ (fun a => ↑(f a)) ω ≤ (fun a => ↑(g a) + edist f g) ω
state_after: case hfg
Ω : Type u_1
inst✝⁶ : MeasurableSpace Ω
R : Type u_2
inst✝⁵ : SMul R ℝ≥0
inst✝⁴ : SMul R ℝ≥0∞
inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
f g : Ω →ᵇ ℝ≥0
le_dist : ∀ (ω : Ω), dist (f ω) (g ω) ≤ ↑(nndist f g)
ω : Ω
le' : f ω ≤ g ω + nndist f g
⊢ (fun a => ↑(f a)) ω ≤ (fun a => ↑(g a) + edist f g) ω

tactic: have le : (f ω : ℝ≥0∞) ≤ (g ω : ℝ≥0∞) + nndist f g := by
  rw [← ENNReal.coe_add];
  exact ENNReal.coe_mono le'
state_before: case hfg
Ω : Type u_1
inst✝⁶ : MeasurableSpace Ω
R : Type u_2
inst✝⁵ : SMul R ℝ≥0
inst✝⁴ : SMul R ℝ≥0∞
inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
f g : Ω →ᵇ ℝ≥0
le_dist : ∀ (ω : Ω), dist (f ω) (g ω) ≤ ↑(nndist f g)
ω : Ω
le' : f ω ≤ g ω + nndist f g
⊢ (fun a => ↑(f a)) ω ≤ (fun a => ↑(g a) + edist f g) ω
state_after: case hfg
Ω : Type u_1
inst✝⁶ : MeasurableSpace Ω
R : Type u_2
inst✝⁵ : SMul R ℝ≥0
inst✝⁴ : SMul R ℝ≥0∞
inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
f g : Ω →ᵇ ℝ≥0
le_dist : ∀ (ω : Ω), dist (f ω) (g ω) ≤ ↑(nndist f g)
ω : Ω
le' : f ω ≤ g ω + nndist f g
le : ↑(f ω) ≤ ↑(g ω) + ↑(nndist f g)
⊢ (fun a => ↑(f a)) ω ≤ (fun a => ↑(g a) + edist f g) ω

tactic: rwa [coe_nnreal_ennreal_nndist] at le
state_before: case hfg
Ω : Type u_1
inst✝⁶ : MeasurableSpace Ω
R : Type u_2
inst✝⁵ : SMul R ℝ≥0
inst✝⁴ : SMul R ℝ≥0∞
inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
f g : Ω →ᵇ ℝ≥0
le_dist : ∀ (ω : Ω), dist (f ω) (g ω) ≤ ↑(nndist f g)
ω : Ω
le' : f ω ≤ g ω + nndist f g
le : ↑(f ω) ≤ ↑(g ω) + ↑(nndist f g)
⊢ (fun a => ↑(f a)) ω ≤ (fun a => ↑(g a) + edist f g) ω
state_after: no goals

tactic: apply (NNReal.le_add_nndist (f ω) (g ω)).trans
state_before: Ω : Type u_1
inst✝⁶ : MeasurableSpace Ω
R : Type u_2
inst✝⁵ : SMul R ℝ≥0
inst✝⁴ : SMul R ℝ≥0∞
inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
f g : Ω →ᵇ ℝ≥0
le_dist : ∀ (ω : Ω), dist (f ω) (g ω) ≤ ↑(nndist f g)
ω : Ω
⊢ f ω ≤ g ω + nndist f g
state_after: Ω : Type u_1
inst✝⁶ : MeasurableSpace Ω
R : Type u_2
inst✝⁵ : SMul R ℝ≥0
inst✝⁴ : SMul R ℝ≥0∞
inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
f g : Ω →ᵇ ℝ≥0
le_dist : ∀ (ω : Ω), dist (f ω) (g ω) ≤ ↑(nndist f g)
ω : Ω
⊢ g ω + nndist (f ω) (g ω) ≤ g ω + nndist f g

tactic: rw [add_le_add_iff_left]
state_before: Ω : Type u_1
inst✝⁶ : MeasurableSpace Ω
R : Type u_2
inst✝⁵ : SMul R ℝ≥0
inst✝⁴ : SMul R ℝ≥0∞
inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
f g : Ω →ᵇ ℝ≥0
le_dist : ∀ (ω : Ω), dist (f ω) (g ω) ≤ ↑(nndist f g)
ω : Ω
⊢ g ω + nndist (f ω) (g ω) ≤ g ω + nndist f g
state_after: Ω : Type u_1
inst✝⁶ : MeasurableSpace Ω
R : Type u_2
inst✝⁵ : SMul R ℝ≥0
inst✝⁴ : SMul R ℝ≥0∞
inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
f g : Ω →ᵇ ℝ≥0
le_dist : ∀ (ω : Ω), dist (f ω) (g ω) ≤ ↑(nndist f g)
ω : Ω
⊢ nndist (f ω) (g ω) ≤ nndist f g

tactic: exact dist_le_coe.mp (le_dist ω)
state_before: Ω : Type u_1
inst✝⁶ : MeasurableSpace Ω
R : Type u_2
inst✝⁵ : SMul R ℝ≥0
inst✝⁴ : SMul R ℝ≥0∞
inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
f g : Ω →ᵇ ℝ≥0
le_dist : ∀ (ω : Ω), dist (f ω) (g ω) ≤ ↑(nndist f g)
ω : Ω
⊢ nndist (f ω) (g ω) ≤ nndist f g
state_after: no goals

tactic: rw [← ENNReal.coe_add]
state_before: Ω : Type u_1
inst✝⁶ : MeasurableSpace Ω
R : Type u_2
inst✝⁵ : SMul R ℝ≥0
inst✝⁴ : SMul R ℝ≥0∞
inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
f g : Ω →ᵇ ℝ≥0
le_dist : ∀ (ω : Ω), dist (f ω) (g ω) ≤ ↑(nndist f g)
ω : Ω
le' : f ω ≤ g ω + nndist f g
⊢ ↑(f ω) ≤ ↑(g ω) + ↑(nndist f g)
state_after: Ω : Type u_1
inst✝⁶ : MeasurableSpace Ω
R : Type u_2
inst✝⁵ : SMul R ℝ≥0
inst✝⁴ : SMul R ℝ≥0∞
inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
f g : Ω →ᵇ ℝ≥0
le_dist : ∀ (ω : Ω), dist (f ω) (g ω) ≤ ↑(nndist f g)
ω : Ω
le' : f ω ≤ g ω + nndist f g
⊢ ↑(f ω) ≤ ↑(g ω + nndist f g)

tactic: exact ENNReal.coe_mono le'
state_before: Ω : Type u_1
inst✝⁶ : MeasurableSpace Ω
R : Type u_2
inst✝⁵ : SMul R ℝ≥0
inst✝⁴ : SMul R ℝ≥0∞
inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
f g : Ω →ᵇ ℝ≥0
le_dist : ∀ (ω : Ω), dist (f ω) (g ω) ≤ ↑(nndist f g)
ω : Ω
le' : f ω ≤ g ω + nndist f g
⊢ ↑(f ω) ≤ ↑(g ω + nndist f g)
state_after: no goals