mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
x₀ : E
ε r M : ℝ
hf : ConvexOn ℝ (ball x₀ r) f
hε : 0 < ε
hM : ∀ (a : E), dist a x₀ < r → |f a| ≤ M
K : ℝ := 2 * M / ε
hK : K = 2 * M / ε
⊢ LipschitzOnWith K.toNNReal f (ball x₀ (r - ε))

Tactic:
have oneside {x y : E} (hx : x ∈ ball x₀ (r - ε)) (hy : y ∈ ball x₀ (r - ε)) :
      f x - f y ≤ K * ‖x - y‖ := by
    obtain rfl | hxy := eq_or_ne x y
    · simp
    have hx₀r : ball x₀ (r - ε) ⊆ ball x₀ r := ball_subset_ball <| by linarith
    have hx' : x ∈ ball x₀ r := hx₀r hx
    have hy' : y ∈ ball x₀ r := hx₀r hy
    let z := x + (ε / ‖x - y‖) • (x - y)
    replace hxy : 0 < ‖x - y‖ := by rwa [norm_sub_pos_iff]
    have hz : z ∈ ball x₀ r := mem_ball_iff_norm.2 <| by
      calc
        _ = ‖(x - x₀) + (ε / ‖x - y‖) • (x - y)‖ := by simp only [z, add_sub_right_comm]
        _ ≤ ‖x - x₀‖ + ‖(ε / ‖x - y‖) • (x - y)‖ := norm_add_le ..
        _ < r - ε + ε :=
          add_lt_add_of_lt_of_le (mem_ball_iff_norm.1 hx) <| by
            simp [norm_smul, abs_of_nonneg, hε.le, hxy.ne']
        _ = r := by simp
    let a := ε / (ε + ‖x - y‖)
    let b := ‖x - y‖ / (ε + ‖x - y‖)
    have hab : a + b = 1 := by simp [field, a, b]
    have hxyz : x = a • y + b • z := by
      calc
        x = a • x + b • x := by rw [Convex.combo_self hab]
        _ = a • y + b • z := by simp [z, a, b, smul_smul, hxy.ne', smul_sub]; abel
    rw [hK, mul_comm, ← mul_div_assoc, le_div_iff₀' hε]
    calc
      ε * (f x - f y) ≤ ‖x - y‖ * (f z - f x) := by
        have h := hf.2 hy' hz (by positivity) (by positivity) hab
        simp only [← hxyz, smul_eq_mul, a, b] at h
        field_simp at h
        linear_combination h
      _ ≤ _ := by
        rw [sub_eq_add_neg (f _), two_mul]
        gcongr
        · exact (le_abs_self _).trans <| hM _ hz
        · exact (neg_le_abs _).trans <| hM _ hx'

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
x₀ : E
ε r M : ℝ
hf : ConvexOn ℝ (ball x₀ r) f
hε : 0 < ε
hM : ∀ (a : E), dist a x₀ < r → |f a| ≤ M
K : ℝ := 2 * M / ε
hK : K = 2 * M / ε
oneside : ∀ {x y : E}, x ∈ ball x₀ (r - ε) → y ∈ ball x₀ (r - ε) → f x - f y ≤ K * ‖x - y‖
⊢ LipschitzOnWith K.toNNReal f (ball x₀ (r - ε))
Module: Mathlib.Analysis.Convex.GaugeRescale

Goal state before:
pp: E : Type u_1
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℝ E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : T1Space E
s t : Set E
hsc : Convex ℝ s
hsb : IsVonNBounded ℝ s
hst : Convex ℝ t
htb : IsVonNBounded ℝ t
x : E
hx : x ∈ interior s
y : E
hy : y ∈ interior t
⊢ E ≃ₜ E

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.GaugeRescale

Goal state before:
pp: E : Type u_1
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℝ E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : T1Space E
s t : Set E
hsc : Convex ℝ s
hsb : IsVonNBounded ℝ s
hst : Convex ℝ t
htb : IsVonNBounded ℝ t
x : E
hx : x ∈ interior s
y : E
hy : y ∈ interior t
h : E ≃ₜ E := gaugeRescaleHomeomorph (-x +ᵥ s) (-y +ᵥ t) ⋯ ⋯ ⋯ ⋯ ⋯ ⋯
⊢ (fun a ↦ y + h (-x + a)) '' interior s = y +ᵥ ⇑h '' interior (-x +ᵥ s)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.GaugeRescale

Goal state before:
pp: E : Type u_1
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℝ E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : T1Space E
s t : Set E
hsc : Convex ℝ s
hsb : IsVonNBounded ℝ s
hst : Convex ℝ t
htb : IsVonNBounded ℝ t
x : E
hx : x ∈ interior s
y : E
hy : y ∈ interior t
h : E ≃ₜ E := gaugeRescaleHomeomorph (-x +ᵥ s) (-y +ᵥ t) ⋯ ⋯ ⋯ ⋯ ⋯ ⋯
⊢ y +ᵥ ⇑h '' interior (-x +ᵥ s) = interior t

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
x₀ : E
ε r M : ℝ
hf : ConvexOn ℝ (ball x₀ r) f
hε : 0 < ε
hM : ∀ (a : E), dist a x₀ < r → |f a| ≤ M
⊢ LipschitzOnWith (2 * M / ε).toNNReal f (ball x₀ (r - ε))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.GaugeRescale

Goal state before:
pp: E : Type u_1
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℝ E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : T1Space E
s t : Set E
hsc : Convex ℝ s
hsb : IsVonNBounded ℝ s
hst : Convex ℝ t
htb : IsVonNBounded ℝ t
x : E
hx : x ∈ interior s
y : E
hy : y ∈ interior t
h : E ≃ₜ E := gaugeRescaleHomeomorph (-x +ᵥ s) (-y +ᵥ t) ⋯ ⋯ ⋯ ⋯ ⋯ ⋯
⊢ (fun a ↦ y + h (-x + a)) '' closure[inst✝³] s = y +ᵥ ⇑h '' closure[inst✝³] (-x +ᵥ s)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
x₀ : E
ε r M : ℝ
hf : ConcaveOn ℝ (ball x₀ r) f
hε : 0 < ε
hM : ∀ (a : E), dist a x₀ < r → |f a| ≤ M
⊢ ∀ (a : E), dist a x₀ < r → |(-f) a| ≤ M

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.GaugeRescale

Goal state before:
pp: E : Type u_1
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℝ E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : T1Space E
s t : Set E
hsc : Convex ℝ s
hsb : IsVonNBounded ℝ s
hst : Convex ℝ t
htb : IsVonNBounded ℝ t
x : E
hx : x ∈ interior s
y : E
hy : y ∈ interior t
h : E ≃ₜ E := gaugeRescaleHomeomorph (-x +ᵥ s) (-y +ᵥ t) ⋯ ⋯ ⋯ ⋯ ⋯ ⋯
⊢ y +ᵥ ⇑h '' closure[inst✝³] (-x +ᵥ s) = closure[inst✝³] t

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
x₀ : E
ε r M : ℝ
hf : ConcaveOn ℝ (ball x₀ r) f
hε : 0 < ε
hM : ∀ (a : E), dist a x₀ < r → |f a| ≤ M
⊢ LipschitzOnWith (2 * M / ε).toNNReal f (ball x₀ (r - ε))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.GaugeRescale

Goal state before:
pp: E : Type u_1
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℝ E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : T1Space E
s t : Set E
hsc : Convex ℝ s
hsne : (interior s).Nonempty
hsb : IsVonNBounded ℝ s
hst : Convex ℝ t
htne : (interior t).Nonempty
htb : IsVonNBounded ℝ t
⊢ ∃ e, ⇑e '' interior s = interior t ∧ ⇑e '' closure[inst✝³] s = closure[inst✝³] t ∧ ⇑e '' frontier s = frontier t

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.GaugeRescale

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set E
hc : Convex ℝ s
hne : (interior s).Nonempty
hb : Bornology.IsBounded s
⊢ (interior (ball 0 1)).Nonempty

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.GaugeRescale

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set E
hc : Convex ℝ s
hne : (interior s).Nonempty
hb : Bornology.IsBounded s
⊢ ∃ h,
    ⇑h '' interior s = ball 0 1 ∧
      ⇑h '' closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s = closedBall 0 1 ∧
        ⇑h '' frontier s = sphere 0 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.Convex

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s t : Set E
hs₁ : Convex ℝ s
hs₂ : IsCompact s
ht : t ∈ 𝓝ˢ s
⊢ ∃ u, s ⊆ interior ((convexHull ℝ) ↑u) ∧ (convexHull ℝ) ↑u ⊆ t

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
x₀ : E
r r' : ℝ
hf : ConvexOn ℝ (ball x₀ r) f
hr : r' < r
hf' : IsBounded (f '' ball x₀ r)
⊢ ∃ K, LipschitzOnWith K f (ball x₀ r')

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
x₀ : E
r r' : ℝ
hf : ConcaveOn ℝ (ball x₀ r) f
hr : r' < r
hf' : IsBounded (f '' ball x₀ r)
⊢ IsBounded ((-f) '' ball x₀ r)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
x₀ : E
r r' : ℝ
hf : ConcaveOn ℝ (ball x₀ r) f
hr : r' < r
hf' : IsBounded (f '' ball x₀ r)
⊢ ∃ K, LipschitzOnWith K f (ball x₀ r')

Tactic:
replace hf' : IsBounded ((-f) '' ball x₀ r) := by convert! hf'.neg; ext; simp [neg_eq_iff_eq_neg]
  simpa using hf.neg.exists_lipschitzOnWith_of_isBounded hr hf'

Goal state after:
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
x₀ : E
r r' : ℝ
hf : ConcaveOn ℝ (ball x₀ r) f
hr : r' < r
hf' : IsBounded (f '' ball x₀ r)
⊢ ∃ K, LipschitzOnWith K f (ball x₀ r')

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
C : Set E
f : E → ℝ
hf : ConvexOn ℝ C f
x₀ : E
hC : C ∈ 𝓝 x₀
r : ℝ
hr : ∀ᶠ (x : ℝ) in Filter.map f (𝓝 x₀), (fun x1 x2 ↦ x1 ≤ x2) x r
x1✝ : ℝ
x2✝ : x1✝ > 0
⊢ ∀ ⦃x : E⦄, dist x x₀ < x1✝ → dist (2 • x₀ - x) x₀ < x1✝

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
C : Set E
f : E → ℝ
hf : ConvexOn ℝ C f
x₀ : E
hC : C ∈ 𝓝 x₀
r : ℝ
this : Filter.Tendsto (fun y ↦ 2 • x₀ - y) (𝓝 x₀) (𝓝 x₀)
hr : ∀ᶠ (a : E) in 𝓝 x₀, f a ≤ r
y : E
hx : 2 • x₀ - y ∈ C
hx' : y ∈ C
hfr : f y ≤ r
hfr' : f (2 • x₀ - y) ≤ r
⊢ |r| ≤ |r| + 2 * |f x₀|

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
C : Set E
f : E → ℝ
hf : ConvexOn ℝ C f
x₀ : E
hC : C ∈ 𝓝 x₀
r : ℝ
this : Filter.Tendsto (fun y ↦ 2 • x₀ - y) (𝓝 x₀) (𝓝 x₀)
hr : ∀ᶠ (a : E) in 𝓝 x₀, f a ≤ r
y : E
hx : 2 • x₀ - y ∈ C
hx' : y ∈ C
hfr : f y ≤ r
hfr' : f (2 • x₀ - y) ≤ r
⊢ 0 ≤ 1 / 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Integral

Goal state before:
pp: 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
inst✝ : IsProbabilityMeasure μ
hs : Convex ℝ s
hsc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hf : ∀ᵐ (x : α) ∂μ, f x ∈ s
hfi : Integrable f μ
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : α → E
hgm : StronglyMeasurable g
hfg : f =ᵐ[μ] g
this : SeparableSpace ↑(range g ∩ s)
x₀ : α
h₀ : f x₀ ∈ s ∧ f x₀ = g x₀
⊢ f x₀ ∈ range g

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Integral

Goal state before:
pp: 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
inst✝ : IsProbabilityMeasure μ
hs : Convex ℝ s
hsc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hf : ∀ᵐ (x : α) ∂μ, f x ∈ s
hfi : Integrable f μ
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : α → E
hgm : StronglyMeasurable g
hfg : f =ᵐ[μ] g
this : SeparableSpace ↑(range g ∩ s)
⊢ (range g ∩ s).Nonempty

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
C : Set E
f : E → ℝ
hf : ConvexOn ℝ C f
x₀ : E
hC : C ∈ 𝓝 x₀
r : ℝ
this : Filter.Tendsto (fun y ↦ 2 • x₀ - y) (𝓝 x₀) (𝓝 x₀)
hr : ∀ᶠ (a : E) in 𝓝 x₀, f a ≤ r
y : E
hx : 2 • x₀ - y ∈ C
hx' : y ∈ C
hfr : f y ≤ r
hfr' : f (2 • x₀ - y) ≤ r
⊢ 2 * f x₀ ≤ f y + f (2 • x₀ - y)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
C : Set E
f : E → ℝ
hf : ConvexOn ℝ C f
x₀ : E
hC : C ∈ 𝓝 x₀
r : ℝ
this : Filter.Tendsto (fun y ↦ 2 • x₀ - y) (𝓝 x₀) (𝓝 x₀)
hr : ∀ᶠ (a : E) in 𝓝 x₀, f a ≤ r
y : E
hx : 2 • x₀ - y ∈ C
hx' : y ∈ C
hfr : f y ≤ r
hfr' : f (2 • x₀ - y) ≤ r
⊢ f y + f (2 • x₀ - y) ≤ f y + |r|

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
C : Set E
f : E → ℝ
hf : ConvexOn ℝ C f
x₀ : E
hC : C ∈ 𝓝 x₀
⊢ Filter.IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x₀) |f| ↔ Filter.IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x₀) f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
C : Set E
f : E → ℝ
hf : ConcaveOn ℝ C f
x₀ : E
hC : C ∈ 𝓝 x₀
⊢ Filter.IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x₀) |f| ↔ Filter.IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) (𝓝 x₀) f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
C : Set E
f : E → ℝ
hC : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] C
hC' : C.Nonempty
hf : ConvexOn ℝ C f
tfae_1_to_2 : LocallyLipschitzOn C f → ContinuousOn f C
⊢ ContinuousOn f C → ∃ x₀ ∈ C, ContinuousAt f x₀

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
C : Set E
f : E → ℝ
hC : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] C
hC' : C.Nonempty
hf : ConvexOn ℝ C f
tfae_1_to_2 : LocallyLipschitzOn C f → ContinuousOn f C
tfae_2_to_3 : ContinuousOn f C → ∃ x₀ ∈ C, ContinuousAt f x₀
x₀ : E
hx₀ : x₀ ∈ C
h : ContinuousAt f x₀
⊢ f x₀ < f x₀ + 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Join

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
s t : Set E
x : E
⊢ x ∈ convexJoin 𝕜 s t ↔ ∃ a ∈ s, ∃ b ∈ t, x ∈ segment 𝕜 a b

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Join

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
s t : Set E
⊢ ⋃ i' ∈ t, ⋃ i ∈ s, segment 𝕜 i i' = convexJoin 𝕜 t s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
C : Set E
f : E → ℝ
hC : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] C
hC' : C.Nonempty
hf : ConvexOn ℝ C f
tfae_1_to_2 : LocallyLipschitzOn C f → ContinuousOn f C
tfae_2_to_3 : ContinuousOn f C → ∃ x₀ ∈ C, ContinuousAt f x₀
x₀ : E
hx₀ : x₀ ∈ C
h : ContinuousAt f x₀
⊢ ∀ᶠ (x : ℝ) in Filter.map f (𝓝 x₀), (fun x1 x2 ↦ x1 ≤ x2) x (f x₀ + 1)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Join

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
t : Set E
⊢ convexJoin 𝕜 ∅ t = ∅

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Join

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
s : Set E
⊢ convexJoin 𝕜 s ∅ = ∅

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Join

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
t : Set E
x : E
⊢ convexJoin 𝕜 {x} t = ⋃ y ∈ t, segment 𝕜 x y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Join

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
s : Set E
y : E
⊢ convexJoin 𝕜 s {y} = ⋃ x ∈ s, segment 𝕜 x y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Join

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
y x : E
⊢ convexJoin 𝕜 {x} {y} = segment 𝕜 x y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Join

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
s₁ s₂ t : Set E
⊢ convexJoin 𝕜 (s₁ ∪ s₂) t = convexJoin 𝕜 s₁ t ∪ convexJoin 𝕜 s₂ t

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Join

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
s t₁ t₂ : Set E
⊢ convexJoin 𝕜 s (t₁ ∪ t₂) = convexJoin 𝕜 s t₁ ∪ convexJoin 𝕜 s t₂

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
C : Set E
f : E → ℝ
hC : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] C
hC' : C.Nonempty
hf : ConvexOn ℝ C f
tfae_1_to_2 : LocallyLipschitzOn C f → ContinuousOn f C
tfae_2_to_3 : ContinuousOn f C → ∃ x₀ ∈ C, ContinuousAt f x₀
tfae_3_to_4 : (∃ x₀ ∈ C, ContinuousAt f x₀) → ∃ x₀ ∈ C, Filter.IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x₀) f
x₀ : E
hx₀ : x₀ ∈ C
r : ℝ
hr : ∀ᶠ (x : ℝ) in Filter.map f (𝓝 x₀), (fun x1 x2 ↦ x1 ≤ x2) x r
x : E
hx : x ∈ C
⊢ ContinuousAt (fun δ ↦ (1 - δ)⁻¹ • x - (δ / (1 - δ)) • x₀) 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Join

Goal state before:
pp: ι : Sort u_1
𝕜 : Type u_2
E : Type u_3
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
s : ι → Set E
t : Set E
⊢ convexJoin 𝕜 (⋃ i, s i) t = ⋃ i, convexJoin 𝕜 (s i) t

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Join

Goal state before:
pp: ι : Sort u_1
𝕜 : Type u_2
E : Type u_3
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
s : Set E
t : ι → Set E
⊢ convexJoin 𝕜 s (⋃ i, t i) = ⋃ i, convexJoin 𝕜 s (t i)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
C : Set E
f : E → ℝ
hC : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] C
hC' : C.Nonempty
hf : ConvexOn ℝ C f
tfae_1_to_2 : LocallyLipschitzOn C f → ContinuousOn f C
tfae_2_to_3 : ContinuousOn f C → ∃ x₀ ∈ C, ContinuousAt f x₀
tfae_3_to_4 : (∃ x₀ ∈ C, ContinuousAt f x₀) → ∃ x₀ ∈ C, Filter.IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x₀) f
x₀ : E
hx₀ : x₀ ∈ C
r : ℝ
hr : ∀ᶠ (x : ℝ) in Filter.map f (𝓝 x₀), (fun x1 x2 ↦ x1 ≤ x2) x r
x : E
hx : x ∈ C
h : ContinuousAt (fun δ ↦ (1 - δ)⁻¹ • x - (δ / (1 - δ)) • x₀) 0
⊢ C ∈ 𝓝 ((fun δ ↦ (1 - δ)⁻¹ • x - (δ / (1 - δ)) • x₀) 0)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
C : Set E
f : E → ℝ
hC : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] C
hC' : C.Nonempty
hf : ConvexOn ℝ C f
tfae_1_to_2 : LocallyLipschitzOn C f → ContinuousOn f C
tfae_2_to_3 : ContinuousOn f C → ∃ x₀ ∈ C, ContinuousAt f x₀
tfae_3_to_4 : (∃ x₀ ∈ C, ContinuousAt f x₀) → ∃ x₀ ∈ C, Filter.IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x₀) f
x₀ : E
hx₀ : x₀ ∈ C
r : ℝ
hr : ∀ᶠ (x : ℝ) in Filter.map f (𝓝 x₀), (fun x1 x2 ↦ x1 ≤ x2) x r
x : E
hx : x ∈ C
⊢ ∀ᶠ (δ : ℝ) in 𝓝 0, (1 - δ)⁻¹ • x - (δ / (1 - δ)) • x₀ ∈ C

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Integral

Goal state before:
pp: 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
inst✝ : IsProbabilityMeasure μ
hs : Convex ℝ s
hsc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hf : ∀ᵐ (x : α) ∂μ, f x ∈ s
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : α → E
hfi : Integrable g μ
hgm : StronglyMeasurable g
hfg : f =ᵐ[μ] g
this : SeparableSpace ↑(range g ∩ s)
y₀ : E
h₀ : y₀ ∈ range g ∩ s
⊢ ∀ᵐ (x : α) ∂μ, g x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (range g ∩ s)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Join

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t u : Set E
x : E
hx : x ∈ s
y : E
hy : y ∈ t
a₁ b₁ : 𝕜
ha₁ : 0 ≤ a₁
hb₁ : 0 ≤ b₁
hab₁ : a₁ + b₁ = 1
z : E
hz : z ∈ u
a₂ b₂ : 𝕜
ha₂ : 0 ≤ a₂
hb₂✝ : 0 ≤ b₂
hab₂ : a₂ + b₂ = 1
hb₂ : 0 < b₂
⊢ 0 ≤ a₂ * b₁ / (a₂ * b₁ + b₂)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Join

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t u : Set E
x : E
hx : x ∈ s
y : E
hy : y ∈ t
a₁ b₁ : 𝕜
ha₁ : 0 ≤ a₁
hb₁ : 0 ≤ b₁
hab₁ : a₁ + b₁ = 1
z : E
hz : z ∈ u
a₂ b₂ : 𝕜
ha₂ : 0 ≤ a₂
hb₂✝ : 0 ≤ b₂
hab₂ : a₂ + b₂ = 1
hb₂ : 0 < b₂
⊢ 0 ≤ b₂ / (a₂ * b₁ + b₂)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Join

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t u : Set E
x : E
hx : x ∈ s
y : E
hy : y ∈ t
a₁ b₁ : 𝕜
ha₁ : 0 ≤ a₁
hb₁ : 0 ≤ b₁
hab₁ : a₁ + b₁ = 1
z : E
hz : z ∈ u
a₂ b₂ : 𝕜
ha₂ : 0 ≤ a₂
hb₂✝ : 0 ≤ b₂
hab₂ : a₂ + b₂ = 1
hb₂ : 0 < b₂
⊢ a₂ * b₁ / (a₂ * b₁ + b₂) + b₂ / (a₂ * b₁ + b₂) = 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Join

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t u : Set E
x : E
hx : x ∈ s
y : E
hy : y ∈ t
a₁ b₁ : 𝕜
ha₁ : 0 ≤ a₁
hb₁ : 0 ≤ b₁
hab₁ : a₁ + b₁ = 1
z : E
hz : z ∈ u
a₂ b₂ : 𝕜
ha₂ : 0 ≤ a₂
hb₂✝ : 0 ≤ b₂
hab₂ : a₂ + b₂ = 1
hb₂ : 0 < b₂
⊢ 0 ≤ a₂ * a₁

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Join

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t u : Set E
x : E
hx : x ∈ s
y : E
hy : y ∈ t
a₁ b₁ : 𝕜
ha₁ : 0 ≤ a₁
hb₁ : 0 ≤ b₁
hab₁ : a₁ + b₁ = 1
z : E
hz : z ∈ u
a₂ b₂ : 𝕜
ha₂ : 0 ≤ a₂
hb₂✝ : 0 ≤ b₂
hab₂ : a₂ + b₂ = 1
hb₂ : 0 < b₂
⊢ 0 ≤ a₂ * b₁ + b₂

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
C : Set E
f : E → ℝ
hC : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] C
hC' : C.Nonempty
hf : ConvexOn ℝ C f
tfae_1_to_2 : LocallyLipschitzOn C f → ContinuousOn f C
tfae_2_to_3 : ContinuousOn f C → ∃ x₀ ∈ C, ContinuousAt f x₀
tfae_3_to_4 : (∃ x₀ ∈ C, ContinuousAt f x₀) → ∃ x₀ ∈ C, Filter.IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x₀) f
x₀ : E
hx₀ : x₀ ∈ C
r : ℝ
x : E
hx : x ∈ C
this : ∀ᶠ (δ : ℝ) in 𝓝 0, (1 - δ)⁻¹ • x - (δ / (1 - δ)) • x₀ ∈ C
δ : ℝ
hδ₀ : 0 < δ
hδ₁ : δ < 1
y : E := (1 - δ)⁻¹ • x - (δ / (1 - δ)) • x₀
hy : y ∈ C
hr✝ : ∀ᶠ (a : E) in 𝓝 x₀, f a ≤ r
ε : ℝ
hε : ε > 0
hr : ∀ ⦃y : E⦄, dist y x₀ < ε → f y ≤ r ∧ y ∈ C
⊢ ε * δ > 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
C : Set E
f : E → ℝ
hC : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] C
hC' : C.Nonempty
hf : ConvexOn ℝ C f
tfae_1_to_2 : LocallyLipschitzOn C f → ContinuousOn f C
tfae_2_to_3 : ContinuousOn f C → ∃ x₀ ∈ C, ContinuousAt f x₀
tfae_3_to_4 : (∃ x₀ ∈ C, ContinuousAt f x₀) → ∃ x₀ ∈ C, Filter.IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x₀) f
x₀ : E
hx₀ : x₀ ∈ C
r : ℝ
x : E
hx : x ∈ C
this : ∀ᶠ (δ : ℝ) in 𝓝 0, (1 - δ)⁻¹ • x - (δ / (1 - δ)) • x₀ ∈ C
δ : ℝ
hδ₀ : 0 < δ
hδ₁ : δ < 1
y : E := (1 - δ)⁻¹ • x - (δ / (1 - δ)) • x₀
hy : y ∈ C
hr✝ : ∀ᶠ (a : E) in 𝓝 x₀, f a ≤ r
ε : ℝ
hε : ε > 0
hr : ∀ ⦃y : E⦄, dist y x₀ < ε → f y ≤ r ∧ y ∈ C
z : E
hz : dist z x < ε * δ
⊢ (fun x ↦ (1 - δ) • x) (δ⁻¹ • (x - y) + y) = (fun x ↦ (1 - δ) • x) x₀

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.Join

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t u : Set E
⊢ convexJoin 𝕜 (convexJoin 𝕜 s t) u ⊆ convexJoin 𝕜 s (convexJoin 𝕜 t u)

Tactic:
by

Goal state after: