-- Mathlib/Analysis/Convex/Integral.lean
theorem StrictConvexOn.ae_eq_const_or_map_average_lt [IsFiniteMeasure μ] (hg : StrictConvexOn ℝ s g)
    (hgc : ContinuousOn g s) (hsc : IsClosed s) (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ)
    (hgi : Integrable (g ∘ f) μ) :
    f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨ g (⨍ x, f x ∂μ) < ⨍ x, g (f x) ∂μ

Proof state:
case intro.intro.intro.intro
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
g : E → ℝ
inst✝ : IsFiniteMeasure μ
hg : StrictConvexOn ℝ s g
hgc : ContinuousOn g s
hsc : IsClosed s
hfs : ∀ᵐ (x : α) ∂μ, f x ∈ s
hfi : Integrable f μ
hgi : Integrable (g ∘ f) μ
this : ∀ {t : Set α}, μ t ≠ 0 → ⨍ (x : α) in t, f x ∂μ ∈ s ∧ g (⨍ (x : α) in t, f x ∂μ) ≤ ⨍ (x : α) in t, g (f x) ∂μ
t : Set α
hm : MeasurableSet t
h₀ : μ t ≠ 0
h₀' : μ tᶜ ≠ 0
hne : ⨍ (x : α) in t, f x ∂μ ≠ ⨍ (x : α) in tᶜ, f x ∂μ
⊢ g (⨍ (x : α), f x ∂μ) < ⨍ (x : α), g (f x) ∂μ

Tactic:
rcases average_mem_openSegment_compl_self hm.nullMeasurableSet h₀ h₀' (hfi.prod_mk hgi) with
  ⟨a, b, ha, hb, hab, h_avg⟩

Result:
case intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
g : E → ℝ
inst✝ : IsFiniteMeasure μ
hg : StrictConvexOn ℝ s g
hgc : ContinuousOn g s
hsc : IsClosed s
hfs : ∀ᵐ (x : α) ∂μ, f x ∈ s
hfi : Integrable f μ
hgi : Integrable (g ∘ f) μ
this : ∀ {t : Set α}, μ t ≠ 0 → ⨍ (x : α) in t, f x ∂μ ∈ s ∧ g (⨍ (x : α) in t, f x ∂μ) ≤ ⨍ (x : α) in t, g (f x) ∂μ
t : Set α
hm : MeasurableSet t
h₀ : μ t ≠ 0
h₀' : μ tᶜ ≠ 0
hne : ⨍ (x : α) in t, f x ∂μ ≠ ⨍ (x : α) in tᶜ, f x ∂μ
a b : ℝ
ha : 0 < a
hb : 0 < b
hab : a + b = 1
h_avg :
  a • ⨍ (x : α) in t, (f x, (g ∘ f) x) ∂μ + b • ⨍ (x : α) in tᶜ, (f x, (g ∘ f) x) ∂μ = ⨍ (x : α), (f x, (g ∘ f) x) ∂μ
⊢ g (⨍ (x : α), f x ∂μ) < ⨍ (x : α), g (f x) ∂μ

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
g : E → ℝ
inst✝ : IsFiniteMeasure μ
hg : StrictConvexOn ℝ s g
hgc : ContinuousOn g s
hsc : IsClosed s
hfs : ∀ᵐ (x : α) ∂μ, f x ∈ s
hfi : Integrable f μ
hgi : Integrable (g ∘ f) μ
this : ∀ {t : Set α}, μ t ≠ 0 → ⨍ (x : α) in t, f x ∂μ ∈ s ∧ g (⨍ (x : α) in t, f x ∂μ) ≤ ⨍ (x : α) in t, g (f x) ∂μ
t : Set α
hm : MeasurableSet t
h₀ : μ t ≠ 0
h₀' : μ tᶜ ≠ 0
hne : ⨍ (x : α) in t, f x ∂μ ≠ ⨍ (x : α) in tᶜ, f x ∂μ
a b : ℝ
ha : 0 < a
hb : 0 < b
hab : a + b = 1
h_avg :
  a • ⨍ (x : α) in t, (f x, (g ∘ f) x) ∂μ + b • ⨍ (x : α) in tᶜ, (f x, (g ∘ f) x) ∂μ = ⨍ (x : α), (f x, (g ∘ f) x) ∂μ
⊢ g (⨍ (x : α), f x ∂μ) < ⨍ (x : α), g (f x) ∂μ

Tactic:
rw [average_pair hfi hgi, average_pair hfi.integrableOn hgi.integrableOn,
  average_pair hfi.integrableOn hgi.integrableOn, Prod.smul_mk,
  Prod.smul_mk, Prod.mk_add_mk, Prod.mk.inj_iff] at h_avg

Result:
case intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
g : E → ℝ
inst✝ : IsFiniteMeasure μ
hg : StrictConvexOn ℝ s g
hgc : ContinuousOn g s
hsc : IsClosed s
hfs : ∀ᵐ (x : α) ∂μ, f x ∈ s
hfi : Integrable f μ
hgi : Integrable (g ∘ f) μ
this : ∀ {t : Set α}, μ t ≠ 0 → ⨍ (x : α) in t, f x ∂μ ∈ s ∧ g (⨍ (x : α) in t, f x ∂μ) ≤ ⨍ (x : α) in t, g (f x) ∂μ
t : Set α
hm : MeasurableSet t
h₀ : μ t ≠ 0
h₀' : μ tᶜ ≠ 0
hne : ⨍ (x : α) in t, f x ∂μ ≠ ⨍ (x : α) in tᶜ, f x ∂μ
a b : ℝ
ha : 0 < a
hb : 0 < b
hab : a + b = 1
h_avg :
  a • ⨍ (x : α) in t, f x ∂μ + b • ⨍ (x : α) in tᶜ, f x ∂μ = ⨍ (x : α), f x ∂μ ∧
    a • ⨍ (x : α) in t, (g ∘ f) x ∂μ + b • ⨍ (x : α) in tᶜ, (g ∘ f) x ∂μ = ⨍ (x : α), (g ∘ f) x ∂μ
⊢ g (⨍ (x : α), f x ∂μ) < ⨍ (x : α), g (f x) ∂μ

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
g : E → ℝ
inst✝ : IsFiniteMeasure μ
hg : StrictConvexOn ℝ s g
hgc : ContinuousOn g s
hsc : IsClosed s
hfs : ∀ᵐ (x : α) ∂μ, f x ∈ s
hfi : Integrable f μ
hgi : Integrable (g ∘ f) μ
this : ∀ {t : Set α}, μ t ≠ 0 → ⨍ (x : α) in t, f x ∂μ ∈ s ∧ g (⨍ (x : α) in t, f x ∂μ) ≤ ⨍ (x : α) in t, g (f x) ∂μ
t : Set α
hm : MeasurableSet t
h₀ : μ t ≠ 0
h₀' : μ tᶜ ≠ 0
hne : ⨍ (x : α) in t, f x ∂μ ≠ ⨍ (x : α) in tᶜ, f x ∂μ
a b : ℝ
ha : 0 < a
hb : 0 < b
hab : a + b = 1
h_avg :
  a • ⨍ (x : α) in t, f x ∂μ + b • ⨍ (x : α) in tᶜ, f x ∂μ = ⨍ (x : α), f x ∂μ ∧
    a • ⨍ (x : α) in t, g (f x) ∂μ + b • ⨍ (x : α) in tᶜ, g (f x) ∂μ = ⨍ (x : α), g (f x) ∂μ
⊢ g (⨍ (x : α), f x ∂μ) < ⨍ (x : α), g (f x) ∂μ

Tactic:
rw [← h_avg.1, ← h_avg.2]

Result:
case intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
g : E → ℝ
inst✝ : IsFiniteMeasure μ
hg : StrictConvexOn ℝ s g
hgc : ContinuousOn g s
hsc : IsClosed s
hfs : ∀ᵐ (x : α) ∂μ, f x ∈ s
hfi : Integrable f μ
hgi : Integrable (g ∘ f) μ
this : ∀ {t : Set α}, μ t ≠ 0 → ⨍ (x : α) in t, f x ∂μ ∈ s ∧ g (⨍ (x : α) in t, f x ∂μ) ≤ ⨍ (x : α) in t, g (f x) ∂μ
t : Set α
hm : MeasurableSet t
h₀ : μ t ≠ 0
h₀' : μ tᶜ ≠ 0
hne : ⨍ (x : α) in t, f x ∂μ ≠ ⨍ (x : α) in tᶜ, f x ∂μ
a b : ℝ
ha : 0 < a
hb : 0 < b
hab : a + b = 1
h_avg :
  a • ⨍ (x : α) in t, f x ∂μ + b • ⨍ (x : α) in tᶜ, f x ∂μ = ⨍ (x : α), f x ∂μ ∧
    a • ⨍ (x : α) in t, g (f x) ∂μ + b • ⨍ (x : α) in tᶜ, g (f x) ∂μ = ⨍ (x : α), g (f x) ∂μ
⊢ g (a • ⨍ (x : α) in t, f x ∂μ + b • ⨍ (x : α) in tᶜ, f x ∂μ) <
    a • ⨍ (x : α) in t, g (f x) ∂μ + b • ⨍ (x : α) in tᶜ, g (f x) ∂μ

Proof state:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
g : E → ℝ
inst✝ : IsFiniteMeasure μ
hg : StrictConvexOn ℝ s g
hgc : ContinuousOn g s
hsc : IsClosed s
hfs : ∀ᵐ (x : α) ∂μ, f x ∈ s
hfi : Integrable f μ
hgi : Integrable (g ∘ f) μ
⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ g (⨍ (x : α), f x ∂μ) < ⨍ (x : α), g (f x) ∂μ

Tactic:
have : ∀ {t}, μ t ≠ 0 → (⨍ x in t, f x ∂μ) ∈ s ∧ g (⨍ x in t, f x ∂μ) ≤ ⨍ x in t, g (f x) ∂μ :=
  fun ht =>
  hg.convexOn.set_average_mem_epigraph hgc hsc ht (measure_ne_top _ _) (ae_restrict_of_ae hfs)
    hfi.integrableOn hgi.integrableOn

Result:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
g : E → ℝ
inst✝ : IsFiniteMeasure μ
hg : StrictConvexOn ℝ s g
hgc : ContinuousOn g s
hsc : IsClosed s
hfs : ∀ᵐ (x : α) ∂μ, f x ∈ s
hfi : Integrable f μ
hgi : Integrable (g ∘ f) μ
this : ∀ {t : Set α}, μ t ≠ 0 → ⨍ (x : α) in t, f x ∂μ ∈ s ∧ g (⨍ (x : α) in t, f x ∂μ) ≤ ⨍ (x : α) in t, g (f x) ∂μ
⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ g (⨍ (x : α), f x ∂μ) < ⨍ (x : α), g (f x) ∂μ