mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Algebra.Exponential

Goal state before:
pp: 𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕂
inst✝³ : CharZero 𝕂
inst✝² : ContinuousSMul ℚ 𝕂
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕂 𝔸
this : ∀ {n : ℕ}, ↑n ! ≠ 0
⊢ Tendsto (fun n ↦ ‖(↑(n + 1))⁻¹‖) atTop (𝓝 0)

Tactic:
apply norm_zero (E := 𝕂) ▸ Filter.Tendsto.norm

Goal state after:
𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕂
inst✝³ : CharZero 𝕂
inst✝² : ContinuousSMul ℚ 𝕂
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕂 𝔸
this : ∀ {n : ℕ}, ↑n ! ≠ 0
⊢ Tendsto (fun x ↦ (↑(x + 1))⁻¹) atTop (𝓝 0)
Module: Mathlib.Analysis.Normed.Algebra.Exponential

Goal state before:
pp: ι : Type u_3
𝔸 : ι → Type u_4
inst✝⁴ : Finite ι
inst✝³ : DecidableEq ι
inst✝² : (i : ι) → NormedRing (𝔸 i)
inst✝¹ : (i : ι) → NormedAlgebra ℚ (𝔸 i)
inst✝ : ∀ (i : ι), CompleteSpace (𝔸 i)
x : (i : ι) → 𝔸 i
j : ι
xj : 𝔸 j
i : ι
⊢ Function.update (exp x) j (exp xj) i = exp (Function.update x j xj) i

Tactic:
[

Goal state after:
ι : Type u_3
𝔸 : ι → Type u_4
inst✝⁴ : Finite ι
inst✝³ : DecidableEq ι
inst✝² : (i : ι) → NormedRing (𝔸 i)
inst✝¹ : (i : ι) → NormedAlgebra ℚ (𝔸 i)
inst✝ : ∀ (i : ι), CompleteSpace (𝔸 i)
x : (i : ι) → 𝔸 i
j : ι
xj : 𝔸 j
i : ι
⊢ Function.update (exp x) j (exp xj) i = exp (Function.update x j xj) i
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
f g : 𝕜 → F
hf : ContDiffWithinAt 𝕜 (↑n) f s x
hg : ContDiffWithinAt 𝕜 (↑n) g s x
⊢ iteratedDerivWithin n (f + g) s x = iteratedDerivWithin n f s x + iteratedDerivWithin n g s x

Tactic:
[

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
f g : 𝕜 → F
hf : ContDiffWithinAt 𝕜 (↑n) f s x
hg : ContDiffWithinAt 𝕜 (↑n) g s x
⊢ iteratedDerivWithin n (f + g) s x = iteratedDerivWithin n f s x + iteratedDerivWithin n g s x
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: case succ
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
s : Set 𝕜
𝕝 : Type u_4
inst✝³ : DivisionSemiring 𝕝
inst✝² : Module 𝕝 F
inst✝¹ : SMulCommClass 𝕜 𝕝 F
inst✝ : ContinuousConstSMul 𝕝 F
c : 𝕝
n : ℕ
IH : ∀ {x : 𝕜} (f : 𝕜 → F), iteratedDerivWithin n (c • f) s x = c • iteratedDerivWithin n f s x
x : 𝕜
f : 𝕜 → F
⊢ iteratedDerivWithin (n + 1) (c • f) s x = c • iteratedDerivWithin (n + 1) f s x

Tactic:
[

Goal state after:
case succ
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
s : Set 𝕜
𝕝 : Type u_4
inst✝³ : DivisionSemiring 𝕝
inst✝² : Module 𝕝 F
inst✝¹ : SMulCommClass 𝕜 𝕝 F
inst✝ : ContinuousConstSMul 𝕝 F
c : 𝕝
n : ℕ
IH : ∀ {x : 𝕜} (f : 𝕜 → F), iteratedDerivWithin n (c • f) s x = c • iteratedDerivWithin n f s x
x : 𝕜
f : 𝕜 → F
⊢ iteratedDerivWithin (n + 1) (c • f) s x = c • iteratedDerivWithin (n + 1) f s x
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: case succ
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
s : Set 𝕜
𝕜' : Type u_6
inst✝¹ : NormedDivisionRing 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
d : 𝕜'
n : ℕ
IH : ∀ {x : 𝕜} (f : 𝕜 → 𝕜'), iteratedDerivWithin n (fun z ↦ f z * d) s x = iteratedDerivWithin n f s x * d
x : 𝕜
f : 𝕜 → 𝕜'
⊢ iteratedDerivWithin (n + 1) (fun z ↦ f z * d) s x = iteratedDerivWithin (n + 1) f s x * d

Tactic:
[

Goal state after:
case succ
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
s : Set 𝕜
𝕜' : Type u_6
inst✝¹ : NormedDivisionRing 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
d : 𝕜'
n : ℕ
IH : ∀ {x : 𝕜} (f : 𝕜 → 𝕜'), iteratedDerivWithin n (fun z ↦ f z * d) s x = iteratedDerivWithin n f s x * d
x : 𝕜
f : 𝕜 → 𝕜'
⊢ iteratedDerivWithin (n + 1) (fun z ↦ f z * d) s x = iteratedDerivWithin (n + 1) f s x * d
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
s : Set 𝕜
f : 𝕜 → F
a : 𝕜
⊢ iteratedDerivWithin n (fun x ↦ f (-x)) s a = (-1) ^ n • iteratedDerivWithin n f (-s) (-a)

Tactic:
simp [iteratedDerivWithin, iteratedFDerivWithin_comp_neg n a]

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
m : ℕ
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
c : OrderedFinpartition m
⊢ (q.compAlongOrderedFinpartition p (c.extend none)).curryLeft =
    ((compAlongOrderedFinpartitionL 𝕜 E F G c).flipMultilinear fun i ↦ p (c.partSize i)) ∘SL
      (q (c.length + 1)).curryLeft ∘SL (continuousMultilinearCurryFin1 𝕜 E F) (p 1)

Tactic:
ext e v
  simp only [Nat.succ_eq_add_one, OrderedFinpartition.extend, extendLeft,
    ContinuousMultilinearMap.curryLeft_apply,
    FormalMultilinearSeries.compAlongOrderedFinpartition_apply, applyOrderedFinpartition_apply,
    ContinuousLinearMap.comp_apply, continuousMultilinearCurryFin1_apply,
    Matrix.zero_empty, ContinuousLinearMap.flipMultilinear_apply_apply,
    compAlongOrderedFinpartitionL_apply, compAlongOrderFinpartition_apply]
  congr
  ext j
  exact Fin.cases rfl (fun i ↦ rfl) j

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : E
i : ℕ
f g : E → F
hf : ContDiffAt 𝕜 (↑i) f x
hg : ContDiffAt 𝕜 (↑i) g x
⊢ iteratedFDeriv 𝕜 i (f + g) x = iteratedFDeriv 𝕜 i f x + iteratedFDeriv 𝕜 i g x

Tactic:
[

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : E
i : ℕ
f g : E → F
hf : ContDiffAt 𝕜 (↑i) f x
hg : ContDiffAt 𝕜 (↑i) g x
⊢ iteratedFDeriv 𝕜 i (f + g) x = iteratedFDeriv 𝕜 i f x + iteratedFDeriv 𝕜 i g x
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : E
i : ℕ
f : E → F
⊢ iteratedFDeriv 𝕜 i (-f) x = -iteratedFDeriv 𝕜 i f x

Tactic:
[

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : E
i : ℕ
f : E → F
⊢ iteratedFDeriv 𝕜 i (-f) x = -iteratedFDeriv 𝕜 i f x
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: s : Set ℝ
a : ℝ
⊢ (∃ᶠ (x : ℝ) in 𝓝 a, x ∈ s ∩ Ioi a) ↔ ∃ᶠ (x : ℝ) in 𝓝 a, x ∈ s ∧ x ∈ Ioi a

Tactic:
[

Goal state after:
s : Set ℝ
a : ℝ
⊢ (∃ᶠ (x : ℝ) in 𝓝 a, x ∈ s ∩ Ioi a) ↔ ∃ᶠ (x : ℝ) in 𝓝 a, x ∈ s ∧ x ∈ Ioi a
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : E
i : ℕ
f g : E → F
hf : ContDiffAt 𝕜 (↑i) f x
hg : ContDiffAt 𝕜 (↑i) g x
⊢ iteratedFDeriv 𝕜 i (f - g) x = iteratedFDeriv 𝕜 i f x - iteratedFDeriv 𝕜 i g x

Tactic:
[

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : E
i : ℕ
f g : E → F
hf : ContDiffAt 𝕜 (↑i) f x
hg : ContDiffAt 𝕜 (↑i) g x
⊢ iteratedFDeriv 𝕜 i (f - g) x = iteratedFDeriv 𝕜 i f x - iteratedFDeriv 𝕜 i g x
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type uF
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
n : ℕ∞ω
inst✝ : CompleteSpace E
e : E ≃L[𝕜] F
a✝ : Nontrivial E
O₁ : (E →L[𝕜] E) → F →L[𝕜] E := fun f ↦ f ∘SL ↑e.symm
O₂ : (E →L[𝕜] F) → E →L[𝕜] E := fun f ↦ ↑e.symm ∘SL f
this : inverse = O₁ ∘ Ring.inverse ∘ O₂
⊢ ContDiffAt 𝕜 n inverse ↑e

Tactic:
this

Goal state after:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type uF
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
n : ℕ∞ω
inst✝ : CompleteSpace E
e : E ≃L[𝕜] F
a✝ : Nontrivial E
O₁ : (E →L[𝕜] E) → F →L[𝕜] E := fun f ↦ f ∘SL ↑e.symm
O₂ : (E →L[𝕜] F) → E →L[𝕜] E := fun f ↦ ↑e.symm ∘SL f
this : inverse = O₁ ∘ Ring.inverse ∘ O₂
⊢ ContDiffAt 𝕜 n (O₁ ∘ Ring.inverse ∘ O₂) ↑e
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
c : ℝ≥0
s : Set α
hs : MeasurableSet s
ε : ℝ≥0∞
ε0 : ε ≠ 0
f : α →ₛ ℝ≥0 := piecewise s hs (const α c) (const α 0)
h : ¬∫⁻ (x : α), ↑(f x) ∂μ = ∞
hc : ¬c = 0
ne_top : μ s ≠ ∞
this : μ s < μ s + ε / ↑c
u : Set α
su : u ⊇ s
u_open : IsOpen u
μu : μ u < μ s + ε / ↑c
⊢ ↑c * (μ s + ε / ↑c) = ↑c * μ s + ε

Tactic:
[

Goal state after:
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
c : ℝ≥0
s : Set α
hs : MeasurableSet s
ε : ℝ≥0∞
ε0 : ε ≠ 0
f : α →ₛ ℝ≥0 := piecewise s hs (const α c) (const α 0)
h : ¬∫⁻ (x : α), ↑(f x) ∂μ = ∞
hc : ¬c = 0
ne_top : μ s ≠ ∞
this : μ s < μ s + ε / ↑c
u : Set α
su : u ⊇ s
u_open : IsOpen u
μu : μ u < μ s + ε / ↑c
⊢ ↑c * (μ s + ε / ↑c) = ↑c * μ s + ε
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f : α → ℝ≥0∞
hf : Measurable f
ε : ℝ≥0∞
εpos : ε ≠ 0
δ : ℕ → ℝ≥0∞
δpos : ∀ (i : ℕ), 0 < δ i
hδ : ∑' (i : ℕ), δ i < ε
g : ℕ → α → ℝ≥0
f_le_g : ∀ (n : ℕ) (x : α), (eapproxDiff f n) x ≤ g n x
gcont : ∀ (n : ℕ), LowerSemicontinuous (g n)
hg : ∀ (n : ℕ), ∫⁻ (x : α), ↑(g n x) ∂μ ≤ ∫⁻ (x : α), ↑((eapproxDiff f n) x) ∂μ + δ n
⊢ ∫⁻ (a : α), ∑' (i : ℕ), ↑((eapproxDiff f i) a) ∂μ ≤ ∫⁻ (x : α), f x ∂μ

Tactic:
[

Goal state after:
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f : α → ℝ≥0∞
hf : Measurable f
ε : ℝ≥0∞
εpos : ε ≠ 0
δ : ℕ → ℝ≥0∞
δpos : ∀ (i : ℕ), 0 < δ i
hδ : ∑' (i : ℕ), δ i < ε
g : ℕ → α → ℝ≥0
f_le_g : ∀ (n : ℕ) (x : α), (eapproxDiff f n) x ≤ g n x
gcont : ∀ (n : ℕ), LowerSemicontinuous (g n)
hg : ∀ (n : ℕ), ∫⁻ (x : α), ↑(g n x) ∂μ ≤ ∫⁻ (x : α), ↑((eapproxDiff f n) x) ∂μ + δ n
⊢ ∫⁻ (a : α), ∑' (i : ℕ), ↑((eapproxDiff f i) a) ∂μ ≤ ∫⁻ (x : α), f x ∂μ
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : DifferentiableOn ℝ f (interior D)
C : ℝ
hf'_ge : ∀ x ∈ interior D, C ≤ deriv f x
x : ℝ
hx : x ∈ D
y : ℝ
hy : y ∈ D
hxy : x ≤ y
⊢ C * (y - x) ≤ f y - f x

Tactic:
rcases eq_or_lt_of_le hxy with hxy' | hxy'

Goal state after:
case inl
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : DifferentiableOn ℝ f (interior D)
C : ℝ
hf'_ge : ∀ x ∈ interior D, C ≤ deriv f x
x : ℝ
hx : x ∈ D
y : ℝ
hy : y ∈ D
hxy : x ≤ y
hxy' : x = y
⊢ C * (y - x) ≤ f y - f x

case inr
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : DifferentiableOn ℝ f (interior D)
C : ℝ
hf'_ge : ∀ x ∈ interior D, C ≤ deriv f x
x : ℝ
hx : x ∈ D
y : ℝ
hy : y ∈ D
hxy : x ≤ y
hxy' : x < y
⊢ C * (y - x) ≤ f y - f x
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : DifferentiableOn ℝ f (interior D)
C : ℝ
hf'_ge : ∀ x ∈ interior D, C ≤ deriv f x
⊢ ∀ x ∈ D, ∀ y ∈ D, x ≤ y → C * (y - x) ≤ f y - f x

Tactic:
by
  intro x hx y hy hxy
  rcases eq_or_lt_of_le hxy with hxy' | hxy'
  · rw [hxy', sub_self, sub_self, mul_zero]
  have hxyD : Icc x y ⊆ D := hD.ordConnected.out hx hy
  have hxyD' : Ioo x y ⊆ interior D :=
    subset_sUnion_of_mem ⟨isOpen_Ioo, Ioo_subset_Icc_self.trans hxyD⟩
  obtain ⟨a, a_mem, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x) :=
    exists_deriv_eq_slope f hxy' (hf.mono hxyD) (hf'.mono hxyD')
  have : C ≤ (f y - f x) / (y - x) := ha ▸ hf'_ge _ (hxyD' a_mem)
  exact (le_div_iff₀ (sub_pos.2 hxy')).1 this

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: case neg
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this✝ : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x ↦ f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x ↦ ∞) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x ↦ ∞) x ∂μ + ε / 2
x : α
h : x ∉ s
this : f x = AEMeasurable.mk f fmeas x
⊢ ↑(f x) < (fun x ↦ g0 x + g1 x) x

Tactic:
this

Goal state after:
case neg
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this✝ : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x ↦ f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x ↦ ∞) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x ↦ ∞) x ∂μ + ε / 2
x : α
h : x ∉ s
this : f x = AEMeasurable.mk f fmeas x
⊢ ↑(AEMeasurable.mk f fmeas x) < (fun x ↦ g0 x + g1 x) x
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: 𝕜 : Type u_1
M : Type u_4
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup M
inst✝ : Module 𝕜 M
s : Set M
x : M
hs : Convex 𝕜 s
hx : x ∈ hull 𝕜 s
r₁ : 𝕜
hr₁ : 0 < r₁
y : M
r₂ : 𝕜
hr₂ : 0 < r₂
hy : y ∈ r₂ • s
⊢ r₁ • y ∈ {y | ∃ r, 0 < r ∧ y ∈ r • s}

Tactic:
refine ⟨r₁ * r₂, mul_pos hr₁ hr₂, ?_⟩

Goal state after:
𝕜 : Type u_1
M : Type u_4
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup M
inst✝ : Module 𝕜 M
s : Set M
x : M
hs : Convex 𝕜 s
hx : x ∈ hull 𝕜 s
r₁ : 𝕜
hr₁ : 0 < r₁
y : M
r₂ : 𝕜
hr₂ : 0 < r₂
hy : y ∈ r₂ • s
⊢ r₁ • y ∈ (r₁ * r₂) • s
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: 𝕜 : Type u_1
M : Type u_4
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup M
inst✝ : Module 𝕜 M
C : ConvexCone 𝕜 M
s : Set M
hs : Convex 𝕜 s
⊢ Disjoint s ↑C → Disjoint (hull 𝕜 s) C

Tactic:
[

Goal state after:
𝕜 : Type u_1
M : Type u_4
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup M
inst✝ : Module 𝕜 M
C : ConvexCone 𝕜 M
s : Set M
hs : Convex 𝕜 s
⊢ Disjoint s ↑C → Disjoint (hull 𝕜 s) C
Module: Mathlib.Geometry.Convex.Cone.Pointed

Goal state before:
pp: R : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsOrderedRing R
inst✝¹ : AddCommMonoid E
inst✝ : Module R E
C✝ C₁ C₂ : PointedCone R E
x✝¹ : E
r : R
C : ConvexCone R E
hC : C.Pointed
x✝ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
x : E
hx : x ∈ ↑C
c : R
hc : 0 ≤ c
⊢ ⟨c, hc⟩ • x ∈ ↑C

Tactic:
[

Goal state after:
R : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsOrderedRing R
inst✝¹ : AddCommMonoid E
inst✝ : Module R E
C✝ C₁ C₂ : PointedCone R E
x✝¹ : E
r : R
C : ConvexCone R E
hC : C.Pointed
x✝ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
x : E
hx : x ∈ ↑C
c : R
hc : 0 ≤ c
⊢ ⟨c, hc⟩ • x ∈ ↑C
Module: Mathlib.Geometry.Convex.Cone.Pointed

Goal state before:
pp: R : Type u_1
E : Type u_2
inst✝⁴ : Ring R
inst✝³ : LinearOrder R
inst✝² : IsOrderedRing R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
C : PointedCone R E
⊢ C.lineal = sSup {S | ↑S ≤ C}

Tactic:
[

Goal state after:
R : Type u_1
E : Type u_2
inst✝⁴ : Ring R
inst✝³ : LinearOrder R
inst✝² : IsOrderedRing R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
C : PointedCone R E
⊢ C.lineal = sSup {S | ↑S ≤ C}
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
c : ℝ≥0
s : Set α
hs : MeasurableSet s
int_f : ∫⁻ (x : α), ↑((piecewise s hs (const α c) (const α 0)) x) ∂μ ≠ ∞
ε : ℝ≥0∞
ε0 : ε ≠ 0
hc : ¬c = 0
μs_lt_top : μ s < ∞
this : 0 < ε / ↑c
F : Set α
Fs : F ⊆ s
F_closed : IsClosed F
μF : μ s < μ F + ε / ↑c
⊢ ↑c * (μ F + ε / ↑c) = ↑c * μ F + ε

Tactic:
[

Goal state after:
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
c : ℝ≥0
s : Set α
hs : MeasurableSet s
int_f : ∫⁻ (x : α), ↑((piecewise s hs (const α c) (const α 0)) x) ∂μ ≠ ∞
ε : ℝ≥0∞
ε0 : ε ≠ 0
hc : ¬c = 0
μs_lt_top : μ s < ∞
this : 0 < ε / ↑c
F : Set α
Fs : F ⊆ s
F_closed : IsClosed F
μF : μ s < μ F + ε / ↑c
⊢ ↑c * (μ F + ε / ↑c) = ↑c * μ F + ε
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
symmetric : ∀ x ∈ s, -x ∈ s
x : E
this : ∀ (x : E), -x ∈ s ↔ x ∈ s
⊢ gauge s (-x) = gauge s x

Tactic:
[

Goal state after:
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
symmetric : ∀ x ∈ s, -x ∈ s
x : E
this : ∀ (x : E), -x ∈ s ↔ x ∈ s
⊢ gauge s (-x) = gauge s x
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
symmetric : ∀ x ∈ s, -x ∈ s
x : E
this : ∀ (x : E), -x ∈ s ↔ x ∈ s
⊢ sInf {r | r ∈ Ioi 0 ∧ -(r⁻¹ • x) ∈ s} = sInf {r | r ∈ Ioi 0 ∧ r⁻¹ • x ∈ s}

Tactic:
this

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

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
x : E
⊢ gauge (-s) (-x) = gauge s x

Tactic:
[

Goal state after:
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
x : E
⊢ gauge (-s) (-x) = gauge s x
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
a : ℝ
hs₁ : Convex ℝ s
hs₀ : 0 ∈ s
hs₂ : Absorbent ℝ s
ha : 0 ≤ a
x : E
⊢ x ∈ {x | gauge s x ≤ a} ↔ x ∈ ⋂ r, ⋂ (_ : a < r), r • s

Tactic:
[

Goal state after:
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
a : ℝ
hs₁ : Convex ℝ s
hs₀ : 0 ∈ s
hs₂ : Absorbent ℝ s
ha : 0 ≤ a
x : E
⊢ x ∈ {x | gauge s x ≤ a} ↔ x ∈ ⋂ r, ⋂ (_ : a < r), r • s
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
absorbs : Absorbent ℝ s
a : ℝ
x✝ : E
⊢ x✝ ∈ {x | gauge s x < a} ↔ x✝ ∈ ⋃ r, ⋃ (_ : 0 < r), ⋃ (_ : r < a), r • s

Tactic:
[

Goal state after:
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
absorbs : Absorbent ℝ s
a : ℝ
x✝ : E
⊢ x✝ ∈ {x | gauge s x < a} ↔ x✝ ∈ ⋃ r, ⋃ (_ : 0 < r), ⋃ (_ : r < a), r • s
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
absorbs : Absorbent ℝ s
a : ℝ
x✝ : E
⊢ x✝ ∈ {x | gauge s x < a} ↔ x✝ ∈ ⋃ r ∈ Ioo 0 a, r • s

Tactic:
[

Goal state after:
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
absorbs : Absorbent ℝ s
a : ℝ
x✝ : E
⊢ x✝ ∈ {x | gauge s x < a} ↔ x✝ ∈ ⋃ r ∈ Ioo 0 a, r • s
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: case inr
E : Type u_2
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module ℝ E
α : Type u_3
inst✝⁶ : Field α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsStrictOrderedRing α
inst✝³ : MulActionWithZero α ℝ
inst✝² : IsStrictOrderedModule α ℝ
inst✝¹ : MulActionWithZero α E
inst✝ : IsScalarTower α ℝ (Set E)
s : Set E
a : α
ha : 0 ≤ a
x : E
ha' : 0 < a
r : ℝ
⊢ r ∈ {r | r ∈ Ioi 0 ∧ r⁻¹ • a • x ∈ s} ↔ r ∈ a • {r | r ∈ Ioi 0 ∧ r⁻¹ • x ∈ s}

Tactic:
[

Goal state after:
case inr
E : Type u_2
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module ℝ E
α : Type u_3
inst✝⁶ : Field α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsStrictOrderedRing α
inst✝³ : MulActionWithZero α ℝ
inst✝² : IsStrictOrderedModule α ℝ
inst✝¹ : MulActionWithZero α E
inst✝ : IsScalarTower α ℝ (Set E)
s : Set E
a : α
ha : 0 ≤ a
x : E
ha' : 0 < a
r : ℝ
⊢ r ∈ {r | r ∈ Ioi 0 ∧ r⁻¹ • a • x ∈ s} ↔ r ∈ a • {r | r ∈ Ioi 0 ∧ r⁻¹ • x ∈ s}
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: case inr.mp
E : Type u_2
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module ℝ E
α : Type u_3
inst✝⁶ : Field α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsStrictOrderedRing α
inst✝³ : MulActionWithZero α ℝ
inst✝² : IsStrictOrderedModule α ℝ
inst✝¹ : MulActionWithZero α E
inst✝ : IsScalarTower α ℝ (Set E)
s : Set E
a : α
ha : 0 ≤ a
x : E
ha' : 0 < a
r : ℝ
hr : r ∈ Ioi 0
hx : r⁻¹ • a • x ∈ s
⊢ ∃ y, (y ∈ Ioi 0 ∧ y⁻¹ • x ∈ s) ∧ a • y = r

Tactic:
[

Goal state after:
case inr.mp
E : Type u_2
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module ℝ E
α : Type u_3
inst✝⁶ : Field α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsStrictOrderedRing α
inst✝³ : MulActionWithZero α ℝ
inst✝² : IsStrictOrderedModule α ℝ
inst✝¹ : MulActionWithZero α E
inst✝ : IsScalarTower α ℝ (Set E)
s : Set E
a : α
ha : 0 ≤ a
x : E
ha' : 0 < a
r : ℝ
hr : r ∈ Ioi 0
hx : r⁻¹ • a • x ∈ s
⊢ ∃ y, (y ∈ Ioi 0 ∧ y⁻¹ • x ∈ s) ∧ a • y = r
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: case inr
E : Type u_2
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module ℝ E
α : Type u_3
inst✝⁸ : Field α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : MulActionWithZero α ℝ
inst✝⁴ : IsStrictOrderedModule α ℝ
inst✝³ : MulActionWithZero α E
inst✝² : SMulCommClass α ℝ ℝ
inst✝¹ : IsScalarTower α ℝ ℝ
inst✝ : IsScalarTower α ℝ E
s : Set E
a : α
ha : 0 ≤ a
ha' : 0 < a
x : E
r : ℝ
⊢ r ∈ {r | r ∈ Ioi 0 ∧ r⁻¹ • x ∈ a • s} ↔ r ∈ a⁻¹ • {r | r ∈ Ioi 0 ∧ r⁻¹ • x ∈ s}

Tactic:
[

Goal state after:
case inr
E : Type u_2
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module ℝ E
α : Type u_3
inst✝⁸ : Field α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : MulActionWithZero α ℝ
inst✝⁴ : IsStrictOrderedModule α ℝ
inst✝³ : MulActionWithZero α E
inst✝² : SMulCommClass α ℝ ℝ
inst✝¹ : IsScalarTower α ℝ ℝ
inst✝ : IsScalarTower α ℝ E
s : Set E
a : α
ha : 0 ≤ a
ha' : 0 < a
x : E
r : ℝ
⊢ r ∈ {r | r ∈ Ioi 0 ∧ r⁻¹ • x ∈ a • s} ↔ r ∈ a⁻¹ • {r | r ∈ Ioi 0 ∧ r⁻¹ • x ∈ s}
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: case inr.mp
E : Type u_2
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module ℝ E
α : Type u_3
inst✝⁸ : Field α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : MulActionWithZero α ℝ
inst✝⁴ : IsStrictOrderedModule α ℝ
inst✝³ : MulActionWithZero α E
inst✝² : SMulCommClass α ℝ ℝ
inst✝¹ : IsScalarTower α ℝ ℝ
inst✝ : IsScalarTower α ℝ E
s : Set E
a : α
ha : 0 ≤ a
ha' : 0 < a
x : E
r : ℝ
hr : r ∈ Ioi 0
y : E
hy : y ∈ s
h : a • y = r⁻¹ • x
⊢ ∃ y, (y ∈ Ioi 0 ∧ y⁻¹ • x ∈ s) ∧ a⁻¹ • y = r

Tactic:
[

Goal state after:
case inr.mp
E : Type u_2
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module ℝ E
α : Type u_3
inst✝⁸ : Field α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : MulActionWithZero α ℝ
inst✝⁴ : IsStrictOrderedModule α ℝ
inst✝³ : MulActionWithZero α E
inst✝² : SMulCommClass α ℝ ℝ
inst✝¹ : IsScalarTower α ℝ ℝ
inst✝ : IsScalarTower α ℝ E
s : Set E
a : α
ha : 0 ≤ a
ha' : 0 < a
x : E
r : ℝ
hr : r ∈ Ioi 0
y : E
hy : y ∈ s
h : a • y = r⁻¹ • x
⊢ ∃ y, (y ∈ Ioi 0 ∧ y⁻¹ • x ∈ s) ∧ a⁻¹ • y = r