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

Goal state before:
pp: 𝕂 : Type u_1
𝕊 : Type u_2
𝔸 : Type u_3
inst✝⁹ : NontriviallyNormedField 𝕂
inst✝⁸ : CharZero 𝕂
inst✝⁷ : NormedCommRing 𝕊
inst✝⁶ : NormedRing 𝔸
inst✝⁵ : NormedSpace 𝕂 𝕊
inst✝⁴ : NormedAlgebra 𝕂 𝔸
inst✝³ : Algebra 𝕊 𝔸
inst✝² : ContinuousSMul 𝕊 𝔸
inst✝¹ : IsScalarTower 𝕂 𝕊 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
t : 𝕊
htx : t • x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
this✝¹ : Tendsto (fun h ↦ h • x) (𝓝 0) (𝓝 0)
this✝ : ∀ᶠ (h : 𝕊) in 𝓝 0, h • x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
h : 𝕊
hh : h • x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
this : Commute (t • x) (h • x)
⊢ exp (t • x) * (exp (h • x) - exp 0 - (ContinuousLinearMap.smulRight 1 x) h) =
    exp (t • x) * exp (h • x) - exp (t • x) - (exp (t • x) • ContinuousLinearMap.smulRight 1 x) h

Tactic:
exp_zero,

Goal state after:
𝕂 : Type u_1
𝕊 : Type u_2
𝔸 : Type u_3
inst✝⁹ : NontriviallyNormedField 𝕂
inst✝⁸ : CharZero 𝕂
inst✝⁷ : NormedCommRing 𝕊
inst✝⁶ : NormedRing 𝔸
inst✝⁵ : NormedSpace 𝕂 𝕊
inst✝⁴ : NormedAlgebra 𝕂 𝔸
inst✝³ : Algebra 𝕊 𝔸
inst✝² : ContinuousSMul 𝕊 𝔸
inst✝¹ : IsScalarTower 𝕂 𝕊 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
t : 𝕊
htx : t • x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
this✝¹ : Tendsto (fun h ↦ h • x) (𝓝 0) (𝓝 0)
this✝ : ∀ᶠ (h : 𝕊) in 𝓝 0, h • x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
h : 𝕊
hh : h • x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
this : Commute (t • x) (h • x)
⊢ exp (t • x) * (exp (h • x) - 1 - (ContinuousLinearMap.smulRight 1 x) h) =
    exp (t • x) * exp (h • x) - exp (t • x) - (exp (t • x) • ContinuousLinearMap.smulRight 1 x) h
Module: Mathlib.Analysis.SpecialFunctions.ExpDeriv

Goal state before:
pp: x : ℂ
⊢ HasDerivAt cexp (cexp x) x

Tactic:
rw [hasDerivAt_iff_isLittleO_nhds_zero]

Goal state after:
x : ℂ
⊢ (fun h ↦ cexp (x + h) - cexp x - h • cexp x) =o[𝓝 0] fun h ↦ h
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
s : Set E
x : E
i : ℕ
f : E → F
hu : UniqueDiffOn 𝕜 s
hx : x ∈ s
⊢ iteratedFDerivWithin 𝕜 i (-f) s x = -iteratedFDerivWithin 𝕜 i f s x

Tactic:
by
  induction i generalizing x with ext h
  | zero => simp
  | succ i hi =>
    calc
      iteratedFDerivWithin 𝕜 (i + 1) (-f) s x h =
          fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 i (-f) s) s x (h 0) (Fin.tail h) :=
        iteratedFDerivWithin_succ_apply_left _
      _ = fderivWithin 𝕜 (-iteratedFDerivWithin 𝕜 i f s) s x (h 0) (Fin.tail h) := by
        rw [fderivWithin_congr' (@hi) hx, Pi.neg_def]
      _ = -(fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 i f s) s) x (h 0) (Fin.tail h) := by
        rw [fderivWithin_neg (hu x hx), neg_apply, neg_apply]
      _ = -(iteratedFDerivWithin 𝕜 (i + 1) f s) x h := by
        rw [iteratedFDerivWithin_succ_apply_left]

Goal state after:
Module: Mathlib.Topology.Order.ExtendFrom

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : DenselyOrdered α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
f : α → β
a b : α
la lb : β
inst✝ : RegularSpace β
hf : ContinuousOn f (Ioo a b)
ha : Tendsto f (𝓝[>] a) (𝓝 la)
hb : Tendsto f (𝓝[<] b) (𝓝 lb)
⊢ ContinuousOn (extendFrom (Ioo a b) f) (Icc a b)

Tactic:
by_cases! hab : a = b
  · simp [hab]
  apply continuousOn_extendFrom
  · rw [closure_Ioo hab]
  · intro x x_in
    rcases eq_endpoints_or_mem_Ioo_of_mem_Icc x_in with (rfl | rfl | h)
    · exact ⟨la, ha.mono_left <| nhdsWithin_mono _ Ioo_subset_Ioi_self⟩
    · exact ⟨lb, hb.mono_left <| nhdsWithin_mono _ Ioo_subset_Iio_self⟩
    · exact ⟨f x, hf x h⟩

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: f : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfd : DifferentiableOn ℝ f (Ioo a b)
⊢ ∃ c ∈ Ioo a b, deriv f c = slope f a b

Tactic:
slope_def_field

Goal state after:
f : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfd : DifferentiableOn ℝ f (Ioo a b)
⊢ ∃ c ∈ Ioo a b, deriv f c = (f b - f a) / (b - a)
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: case pos.refine_1
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
⊢ ∀ (x : α), (piecewise s hs (const α c) (const α 0)) x ≤ (fun x ↦ 0) x

Tactic:
simp only [hc, Set.indicator_zero', Pi.zero_apply, SimpleFunc.const_zero, imp_true_iff,
          SimpleFunc.coe_zero, Set.piecewise_eq_indicator,
          SimpleFunc.coe_piecewise, le_zero_iff]

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

Goal state before:
pp: case refine_1
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
⊢ ↑(f x) < (fun x ↦ g0 x + g1 x) x

Tactic:
by_cases h : x ∈ s

Goal state after:
case pos
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
⊢ ↑(f x) < (fun x ↦ g0 x + g1 x) x

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
⊢ ↑(f x) < (fun x ↦ g0 x + g1 x) x
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: 𝕜 : Type u_1
R : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommMonoid M
inst✝ : SMul R M
C C₁✝ C₂✝ : ConvexCone R M
s : Set M
c : R
x : M
C₁ C₂ : ConvexCone R M
h : C₁.carrier = C₂.carrier
⊢ C₁ = C₂

Tactic:
cases C₁

Goal state after:
case mk
𝕜 : Type u_1
R : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommMonoid M
inst✝ : SMul R M
C C₁ C₂✝ : ConvexCone R M
s : Set M
c : R
x : M
C₂ : ConvexCone R M
carrier✝ : Set M
smul_mem'✝ : ∀ ⦃c : R⦄, 0 < c → ∀ ⦃x : M⦄, x ∈ carrier✝ → c • x ∈ carrier✝
add_mem'✝ : ∀ ⦃x : M⦄, x ∈ carrier✝ → ∀ ⦃y : M⦄, y ∈ carrier✝ → x + y ∈ carrier✝
h : { carrier := carrier✝, smul_mem' := smul_mem'✝, add_mem' := add_mem'✝ }.carrier = C₂.carrier
⊢ { carrier := carrier✝, smul_mem' := smul_mem'✝, add_mem' := add_mem'✝ } = C₂
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: case inr.inl
R : Type u_7
M : Type u_8
inst✝⁵ : Ring R
inst✝⁴ : LinearOrder R
inst✝³ : AddLeftStrictMono R
inst✝² : AddCommGroup M
inst✝¹ : Nontrivial M
inst✝ : Module R M
C : ConvexCone R M
h : Submodule.span R ↑C = ⊤
x : M
hne : (↑C).Nonempty
y : M
hy : y ∈ ↑C
z : M
hz : z ∈ ↑C
⊢ ∃ x ∈ C, ∃ y_1 ∈ C, x - y_1 = 0 • (y - z)

Tactic:
simp only [zero_smul]

Goal state after:
case inr.inl
R : Type u_7
M : Type u_8
inst✝⁵ : Ring R
inst✝⁴ : LinearOrder R
inst✝³ : AddLeftStrictMono R
inst✝² : AddCommGroup M
inst✝¹ : Nontrivial M
inst✝ : Module R M
C : ConvexCone R M
h : Submodule.span R ↑C = ⊤
x : M
hne : (↑C).Nonempty
y : M
hy : y ∈ ↑C
z : M
hz : z ∈ ↑C
⊢ ∃ x ∈ C, ∃ y ∈ C, x - y = 0
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
a : ℝ
ha : 0 ≤ a
hx : x ∈ a • s
⊢ gauge s x ≤ a

Tactic:
obtain rfl | ha' := ha.eq_or_lt
  · rw [mem_singleton_iff.1 (zero_smul_set_subset _ hx), gauge_zero]
  · exact csInf_le bddBelow_gauge_set ⟨ha', hx⟩

Goal state after:
Module: Mathlib.Analysis.Convex.Cone.Extension

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
f : E →ₗ.[ℝ] ℝ
N : E → ℝ
N_hom : ∀ (c : ℝ), 0 < c → ∀ (x : E), N (c • x) = c * N x
N_add : ∀ (x y : E), N (x + y) ≤ N x + N y
hf : ∀ (x : ↥f.domain), ↑f x ≤ N ↑x
N_0 : N 0 = 0
s : PointedCone ℝ (E × ℝ) := { carrier := {p | N p.1 ≤ p.2}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }
f' : E × ℝ →ₗ.[ℝ] ℝ := (-f).coprod (LinearMap.id.toPMap ⊤)
hf'_nonneg : ∀ (x : ↥f'.domain), ↑x ∈ s → 0 ≤ ↑f' x
hf'_dense : ∀ (y : E × ℝ), ∃ x, ↑x + y ∈ s
g : E × ℝ →ₗ[ℝ] ℝ
g_nonneg : ∀ x ∈ s, 0 ≤ g x
g_eq : ∀ (x : ↥f.domain) (y : ℝ), g (↑x, y) = y - ↑f x
x : E
⊢ N x - g (x, N x) ≤ N x

Tactic:
by simpa using g_nonneg ⟨x, N x⟩ (le_refl (N x))

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

Goal state before:
pp: 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
hf : Integrable f μ
ε : ℝ
εpos : 0 < ε
δ : ℝ≥0 := NNReal.mk (ε / 2) ⋯
δpos : 0 < δ
fp : α → ℝ≥0 := fun x ↦ (f x).toNNReal
int_fp : Integrable (fun x ↦ ↑(fp x)) μ
gp : α → ℝ≥0∞
fp_lt_gp : ∀ (x : α), ↑(fp x) < gp x
gpcont : LowerSemicontinuous gp
gp_lt_top : ∀ᵐ (x : α) ∂μ, gp x < ∞
gp_integrable : Integrable (fun x ↦ (gp x).toReal) μ
gpint : ∫ (x : α), (gp x).toReal ∂μ < ∫ (x : α), ↑(fp x) ∂μ + ↑δ
fm : α → ℝ≥0 := fun x ↦ (-f x).toNNReal
int_fm : Integrable (fun x ↦ ↑(fm x)) μ
gm : α → ℝ≥0
gm_le_fm : ∀ (x : α), gm x ≤ fm x
gmcont : UpperSemicontinuous gm
gm_integrable : Integrable (fun x ↦ ↑(gm x)) μ
gmint : ∫ (x : α), ↑(fm x) ∂μ - ↑δ ≤ ∫ (x : α), ↑(gm x) ∂μ
g : α → EReal := fun x ↦ ↑(gp x) - ↑↑(gm x)
ae_g : ∀ᵐ (x : α) ∂μ, (g x).toReal = (↑(gp x)).toReal - (↑↑(gm x)).toReal
⊢ ∫ (x : α), (g x).toReal ∂μ < ∫ (x : α), f x ∂μ + ε

Tactic:
show (∫ x : α, (g x).toReal ∂μ) < (∫ x : α, f x ∂μ) + ε

Goal state after:
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
hf : Integrable f μ
ε : ℝ
εpos : 0 < ε
δ : ℝ≥0 := NNReal.mk (ε / 2) ⋯
δpos : 0 < δ
fp : α → ℝ≥0 := fun x ↦ (f x).toNNReal
int_fp : Integrable (fun x ↦ ↑(fp x)) μ
gp : α → ℝ≥0∞
fp_lt_gp : ∀ (x : α), ↑(fp x) < gp x
gpcont : LowerSemicontinuous gp
gp_lt_top : ∀ᵐ (x : α) ∂μ, gp x < ∞
gp_integrable : Integrable (fun x ↦ (gp x).toReal) μ
gpint : ∫ (x : α), (gp x).toReal ∂μ < ∫ (x : α), ↑(fp x) ∂μ + ↑δ
fm : α → ℝ≥0 := fun x ↦ (-f x).toNNReal
int_fm : Integrable (fun x ↦ ↑(fm x)) μ
gm : α → ℝ≥0
gm_le_fm : ∀ (x : α), gm x ≤ fm x
gmcont : UpperSemicontinuous gm
gm_integrable : Integrable (fun x ↦ ↑(gm x)) μ
gmint : ∫ (x : α), ↑(fm x) ∂μ - ↑δ ≤ ∫ (x : α), ↑(gm x) ∂μ
g : α → EReal := fun x ↦ ↑(gp x) - ↑↑(gm x)
ae_g : ∀ᵐ (x : α) ∂μ, (g x).toReal = (↑(gp x)).toReal - (↑↑(gm x)).toReal
⊢ ∫ (x : α), (g x).toReal ∂μ < ∫ (x : α), f x ∂μ + ε
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: case hbc
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := ⋯
this : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
pos : 0 < 4 + 12 * ‖c‖
e : ℕ
he : (1 / 2) ^ e < ε / (4 + 12 * ‖c‖)
y : E
hy : y ∈ ball 0 ((1 / 2) ^ (n e + 1))
y_pos : ¬y = 0
yzero : 0 < ‖y‖
y_lt : ‖y‖ < (1 / 2) ^ (n e + 1)
yone : ‖y‖ ≤ 1
k : ℕ
k_gt : n e < k
m : ℕ := ⋯
h'k : ‖y‖ ≤ (1 / 2) ^ (m + 1)
hk : (1 / 2) ^ (m + 1 + 1) < ‖y‖
m_ge : n e ≤ m
km : k = m + 1
J1 : ‖f (x + y) - f x - (L e (n e) m) (x + y - x)‖ ≤ (1 / 2) ^ e * (1 / 2) ^ m
J2 : ‖f (x + y) - f x - (L e (n e) m) y‖ ≤ 4 * (1 / 2) ^ e * ‖y‖
⊢ ‖L e (n e) m - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e

Tactic:
exact Lf' _ _ m_ge

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

Goal state before:
pp: 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
hf : Integrable f μ
ε : ℝ
εpos : 0 < ε
δ : ℝ≥0 := ⋯
δpos : 0 < δ
fp : α → ℝ≥0 := ⋯
int_fp : Integrable (fun x ↦ ↑(fp x)) μ
gp : α → ℝ≥0∞
fp_lt_gp : ∀ (x : α), ↑(fp x) < gp x
gpcont : LowerSemicontinuous gp
gp_lt_top : ∀ᵐ (x : α) ∂μ, gp x < ∞
gp_integrable : Integrable (fun x ↦ (gp x).toReal) μ
gpint : ∫ (x : α), (gp x).toReal ∂μ < ∫ (x : α), ↑(fp x) ∂μ + ↑δ
fm : α → ℝ≥0 := ⋯
int_fm : Integrable (fun x ↦ ↑(fm x)) μ
gm : α → ℝ≥0
gm_le_fm : ∀ (x : α), gm x ≤ fm x
gmcont : UpperSemicontinuous gm
gm_integrable : Integrable (fun x ↦ ↑(gm x)) μ
gmint : ∫ (x : α), ↑(fm x) ∂μ - ↑δ ≤ ∫ (x : α), ↑(gm x) ∂μ
g : α → EReal := ⋯
ae_g : ∀ᵐ (x : α) ∂μ, (g x).toReal = (↑(gp x)).toReal - (↑↑(gm x)).toReal
⊢ ∫ (x : α), (↑(gp x)).toReal ∂μ < ∫ (x : α), ↑(fp x) ∂μ + ↑δ

Tactic:
convert! gpint

Goal state after:
case e'_3
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
hf : Integrable f μ
ε : ℝ
εpos : 0 < ε
δ : ℝ≥0 := NNReal.mk (ε / 2) ⋯
δpos : 0 < δ
fp : α → ℝ≥0 := fun x ↦ (f x).toNNReal
int_fp : Integrable (fun x ↦ ↑(fp x)) μ
gp : α → ℝ≥0∞
fp_lt_gp : ∀ (x : α), ↑(fp x) < gp x
gpcont : LowerSemicontinuous gp
gp_lt_top : ∀ᵐ (x : α) ∂μ, gp x < ∞
gp_integrable : Integrable (fun x ↦ (gp x).toReal) μ
gpint : ∫ (x : α), (gp x).toReal ∂μ < ∫ (x : α), ↑(fp x) ∂μ + ↑δ
fm : α → ℝ≥0 := fun x ↦ (-f x).toNNReal
int_fm : Integrable (fun x ↦ ↑(fm x)) μ
gm : α → ℝ≥0
gm_le_fm : ∀ (x : α), gm x ≤ fm x
gmcont : UpperSemicontinuous gm
gm_integrable : Integrable (fun x ↦ ↑(gm x)) μ
gmint : ∫ (x : α), ↑(fm x) ∂μ - ↑δ ≤ ∫ (x : α), ↑(gm x) ∂μ
g : α → EReal := fun x ↦ ↑(gp x) - ↑↑(gm x)
ae_g : ∀ᵐ (x : α) ∂μ, (g x).toReal = (↑(gp x)).toReal - (↑↑(gm x)).toReal
x✝ : α
⊢ (↑(gp x✝)).toReal = (gp x✝).toReal
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: case inr
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set E
r : ℝ
x : E
hs : ball 0 r ⊆ s
hr : 0 < r
⊢ gauge s x * r ≤ ‖x‖

Tactic:
← le_div_iff₀ hr,

Goal state after:
case inr
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set E
r : ℝ
x : E
hs : ball 0 r ⊆ s
hr : 0 < r
⊢ gauge s x ≤ ‖x‖ / r
Module: Mathlib.Analysis.LocallyConvex.SeparatingDual

Goal state before:
pp: R : Type u_1
V : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : AddCommGroup V
inst✝³ : TopologicalSpace V
inst✝² : TopologicalSpace R
inst✝¹ : Module R V
inst✝ : SeparatingDual R V
x y : V
⊢ x = y ↔ ∀ (g : StrongDual R V), g x = g y

Tactic:
rw [← sub_eq_zero, eq_zero_iff_forall_dual_eq_zero (R := R) (x - y)]

Goal state after:
R : Type u_1
V : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : AddCommGroup V
inst✝³ : TopologicalSpace V
inst✝² : TopologicalSpace R
inst✝¹ : Module R V
inst✝ : SeparatingDual R V
x y : V
⊢ (∀ (g : StrongDual R V), g (x - y) = 0) ↔ ∀ (g : StrongDual R V), g x = g y
Module: Mathlib.Analysis.LocallyConvex.SeparatingDual

Goal state before:
pp: R : Type u_1
V : Type u_2
inst✝⁶ : Field R
inst✝⁵ : AddCommGroup V
inst✝⁴ : TopologicalSpace R
inst✝³ : TopologicalSpace V
inst✝² : IsTopologicalRing R
inst✝¹ : Module R V
inst✝ : SeparatingDual R V
x : V
hx : x ≠ 0
⊢ ∃ f, f x = 1

Tactic:
by
  rcases exists_ne_zero (R := R) hx with ⟨f, hf⟩
  exact ⟨(f x)⁻¹ • f, inv_mul_cancel₀ hf⟩

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
x : ℝ
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ), n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * (1 / 2) ^ e
L0 : ℕ → F := fun e ↦ L e (n e) (n e)
⊢ x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}

Tactic:
have : CauchySeq L0 := by
    rw [Metric.cauchySeq_iff']
    intro ε εpos
    obtain ⟨e, he⟩ : ∃ e : ℕ, (1 / 2) ^ e < ε / 12 :=
      exists_pow_lt_of_lt_one (div_pos εpos (by norm_num)) (by norm_num)
    refine ⟨e, fun e' he' => ?_⟩
    rw [dist_comm, dist_eq_norm]
    calc
      ‖L0 e - L0 e'‖ ≤ 12 * (1 / 2) ^ e := M _ _ _ _ _ _ le_rfl le_rfl le_rfl le_rfl he'
      _ < 12 * (ε / 12) := mul_lt_mul' le_rfl he (le_of_lt P) (by norm_num)
      _ = ε := by ring

Goal state after:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
x : ℝ
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ), n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * (1 / 2) ^ e
L0 : ℕ → F := fun e ↦ L e (n e) (n e)
this : CauchySeq L0
⊢ x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: α : Type u_1
G : Type u_3
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
ι : Type u_4
l : Filter ι
inst✝ : l.IsCountablyGenerated
F : ι → α → G
f : α → G
bound : α → ℝ
hF_meas : ∀ᶠ (n : ι) in l, AEStronglyMeasurable (F n) μ
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) l (𝓝 (f a))
⊢ Tendsto (fun n ↦ ∫ (a : α), F n a ∂μ) l (𝓝 (∫ (a : α), f a ∂μ))

Tactic:
simp only [integral_eq_setToFun]
  exact tendsto_setToFun_filter_of_dominated_convergence (dominatedFinMeasAdditive_weightedSMul μ)
    bound hF_meas h_bound bound_integrable h_lim

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: case hbc
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
x : ℝ
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ), n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * (1 / 2) ^ e
L0 : ℕ → F := ⋯
this : CauchySeq L0
f' : F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
e : ℕ
he : (1 / 2) ^ e < ε / 16
y : ℝ
hy : y ∈ Icc x (x + (1 / 2) ^ (n e + 1))
xy : x < y
yzero : 0 < y - x
y_le : y - x ≤ (1 / 2) ^ (n e + 1)
yone : y - x ≤ 1
k : ℕ
k_gt : n e < k
m : ℕ := ⋯
h'k : y - x ≤ (1 / 2) ^ (m + 1)
hk : (1 / 2) ^ (m + 1 + 1) < y - x
m_ge : n e ≤ m
km : k = m + 1
J : ‖f y - f x - (y - x) • L e (n e) m‖ ≤ 4 * (1 / 2) ^ e * ‖y - x‖
⊢ ‖L e (n e) m - f'‖ ≤ 12 * (1 / 2) ^ e

Tactic:
exact Lf' _ _ m_ge

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c : ℝ
f : ℝ → E
hc : c ≠ 0
A : MeasurableEmbedding fun x ↦ x * c
⊢ ∫ (x : ℝ) in a..b, f (x * c) =
    c⁻¹ • ∫ (x : ℝ) in a * c..b * c, f x ∂ENNReal.ofReal |c| • Measure.map (fun x ↦ x * c) volume

Tactic:
integral_smul_measure,

Goal state after:
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c : ℝ
f : ℝ → E
hc : c ≠ 0
A : MeasurableEmbedding fun x ↦ x * c
⊢ ∫ (x : ℝ) in a..b, f (x * c) =
    c⁻¹ • (ENNReal.ofReal |c|).toReal • ∫ (x : ℝ) in a * c..b * c, f x ∂Measure.map (fun x ↦ x * c) volume
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
d : ℝ
A : MeasurableEmbedding fun x ↦ x + d
⊢ ∫ (x : ℝ) in a..b, f (x + d) = ∫ (x : ℝ) in a + d..b + d, f x ∂Measure.map (fun x ↦ x + d) volume

Tactic:
by
      simp [intervalIntegral, A.setIntegral_map]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: case refine_2
E : Type u_1
X : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : TopologicalSpace X
μ : Measure ℝ
inst✝ : FirstCountableTopology X
F : X → ℝ → E
bound : ℝ → ℝ
a b a₀ b₀ : ℝ
x₀ : X
hF_meas : ∀ (x : X), AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
h_bound : ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x t‖ ≤ bound t
bound_integrable : IntervalIntegrable bound μ a b
h_cont : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ContinuousAt (fun x ↦ F x t) x₀
ha₀ : a₀ ∈ Ioo a b
hb₀ : b₀ ∈ Ioo a b
hμb₀ : μ {b₀} = 0
hsub : ∀ {a₀ b₀ : ℝ}, a₀ ∈ Ioo a b → b₀ ∈ Ioo a b → Ι a₀ b₀ ⊆ Ι a b
Ioo_nhds : Ioo a b ∈ 𝓝 b₀
Icc_nhds : Icc a b ∈ 𝓝 b₀
hx₀ : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x₀ t‖ ≤ bound t
⊢ ContinuousAt (fun x ↦ ∫ (s : ℝ) in b₀..x.2, F x₀ s ∂μ) (x₀, b₀)

Tactic:
refine (?_ : ContinuousAt (fun t ↦ ∫ s in b₀..t, F x₀ s ∂μ) b₀).snd'
    apply ContinuousWithinAt.continuousAt _ (Icc_mem_nhds hb₀.1 hb₀.2)
    apply intervalIntegral.continuousWithinAt_primitive hμb₀
    rw [min_eq_right hb₀.1.le, max_eq_right hb₀.2.le]
    exact bound_integrable.mono_fun' (hF_meas x₀) hx₀

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: case pos
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
c d : ℝ
hc : c = 0
⊢ c⁻¹ • ∫ (x : ℝ) in a..b, f (x / c - d) = ∫ (x : ℝ) in a / c - d..b / c - d, f x

Tactic:
simp [hc, integral_comp_div_sub]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: case neg
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
c d : ℝ
hc : ¬c = 0
⊢ c⁻¹ • ∫ (x : ℝ) in a..b, f (x / c - d) = ∫ (x : ℝ) in a / c - d..b / c - d, f x

Tactic:
simp [hc, integral_comp_div_sub]

Goal state after: