mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

Goal state before:
pp: 𝕜 : Type u
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
f : E → F
x : E
n : ℕ
m : Fin (n + 1) → E
⊢ ((iteratedFDeriv 𝕜 n (fun y ↦ fderiv 𝕜 f y) x) (init m)) (m (last n)) =
    ((continuousMultilinearCurryRightEquiv' 𝕜 n E F).symm (iteratedFDeriv 𝕜 n (fun y ↦ fderiv 𝕜 f y) x)) m

Tactic:
continuousMultilinearCurryRightEquiv_symm_apply'

Goal state after:
𝕜 : Type u
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
f : E → F
x : E
n : ℕ
m : Fin (n + 1) → E
⊢ ((iteratedFDeriv 𝕜 n (fun y ↦ fderiv 𝕜 f y) x) (init m)) (m (last n)) =
    ((iteratedFDeriv 𝕜 n (fun y ↦ fderiv 𝕜 f y) x) (init m)) (m (last n))
Module: Mathlib.Analysis.Calculus.Deriv.Add

Goal state before:
pp: case e'_11
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
a : 𝕜
H : DifferentiableAt 𝕜 (fun x ↦ f (-x)) a
x✝ : 𝕜
⊢ f x✝ = ((fun x ↦ f (-x)) ∘ Neg.neg) x✝

Tactic:
simp only [Function.comp_apply, neg_neg]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
h : HasFiniteFPowerSeriesOnBall f p x 0 r
y : E
hy : y ∈ Metric.eball x r
z : E
hz : z ∈ Metric.eball x r
⊢ f z = (fun x ↦ 0) z

Tactic:
by rw [h.eq_zero_of_bound_zero z hz]

Goal state after:
Module: Mathlib.Analysis.Analytic.Uniqueness

Goal state before:
pp: 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
y : E
c : ℝ
c_pos : c > 0
t : Set E
t_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
δ : ℝ
δ_pos : δ > 0
δε : Metric.ball 0 δ ⊆ t
n : ℕ
p : E [×n + 1]→L[𝕜] F
ht : ∀ y ∈ t, ‖p fun x ↦ y‖ ≤ c * ‖‖y‖ ^ (n + 1 + 1)‖
hy : 0 < ‖y‖
ε : ℝ
ε_pos : 0 < ε
h₀ : 0 < c * ‖y‖ ^ (n.succ + 1)
k : 𝕜
k_pos : 0 < ‖k‖
k_norm : ‖k‖ < min (δ * ‖y‖⁻¹) (ε * (c * ‖y‖ ^ (n.succ + 1))⁻¹)
⊢ ‖k • y‖ < δ

Tactic:
norm_smul

Goal state after:
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
y : E
c : ℝ
c_pos : c > 0
t : Set E
t_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
δ : ℝ
δ_pos : δ > 0
δε : Metric.ball 0 δ ⊆ t
n : ℕ
p : E [×n + 1]→L[𝕜] F
ht : ∀ y ∈ t, ‖p fun x ↦ y‖ ≤ c * ‖‖y‖ ^ (n + 1 + 1)‖
hy : 0 < ‖y‖
ε : ℝ
ε_pos : 0 < ε
h₀ : 0 < c * ‖y‖ ^ (n.succ + 1)
k : 𝕜
k_pos : 0 < ‖k‖
k_norm : ‖k‖ < min (δ * ‖y‖⁻¹) (ε * (c * ‖y‖ ^ (n.succ + 1))⁻¹)
⊢ ‖k‖ * ‖y‖ < δ
Module: Mathlib.Analysis.Analytic.Uniqueness

Goal state before:
pp: 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 g : E → F
U : Set E
hf : AnalyticOnNhd 𝕜 f U
hg : AnalyticOnNhd 𝕜 g U
hU : IsPreconnected U
z₀ : E
h₀ : z₀ ∈ U
hfg : f =ᶠ[𝓝 z₀] g
⊢ EqOn f g U

Tactic:
have hfg' : f - g =ᶠ[𝓝 z₀] 0 := hfg.mono fun z h => by simp [h]
  simpa [sub_eq_zero] using! fun z hz =>
    (hf.sub hg).eqOn_zero_of_preconnected_of_eventuallyEq_zero hU h₀ hfg' hz

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

Goal state before:
pp: case e'_2.a
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
x : E
x✝ : { s // s.card = n }
a✝ : x✝ ∈ Finset.univ
⊢ ((p.changeOriginSeriesTerm 1 n ↑x✝ ⋯) fun x_1 ↦ x) (Fin.snoc ![] x) = (p (n + 1)) fun x_1 ↦ x

Tactic:
Fin.snoc_zero,

Goal state after:
case e'_2.a
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
x : E
x✝ : { s // s.card = n }
a✝ : x✝ ∈ Finset.univ
⊢ (((p.changeOriginSeriesTerm 1 n ↑x✝ ⋯) fun x_1 ↦ x) fun x_1 ↦ x) = (p (n + 1)) fun x_1 ↦ x
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
r : ℝ≥0∞
y : E
n : ℕ
⊢ ∀ {F : Type (max u v)} [inst : NormedAddCommGroup F] [inst_1 : NormedSpace 𝕜 F] [CompleteSpace F]
    {p : FormalMultilinearSeries 𝕜 E F} {f : E → F},
    HasFPowerSeriesOnBall f p x r → (n ! • (p n) fun x ↦ y) = (iteratedFDeriv 𝕜 n f x) fun x ↦ y

Tactic:
induction n with | zero => _ | succ n ih => _ <;> intro F _ _ _ p f h
  · rw [factorial_zero, one_smul, h.iteratedFDeriv_zero_apply_diag]
  · rw [factorial_succ, mul_comm, mul_smul, ← derivSeries_apply_diag,
      ← _root_.smul_apply, ih h.fderiv, iteratedFDeriv_succ_apply_right]
    rfl

Goal state after:
Module: Mathlib.Analysis.Calculus.LogDeriv

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
f g : 𝕜 → 𝕜'
x : 𝕜
hf : f x ≠ 0
hg : g x ≠ 0
hdf : DifferentiableAt 𝕜 f x
hdg : DifferentiableAt 𝕜 g x
⊢ logDeriv (fun z ↦ f z * g z) x = logDeriv f x + logDeriv g x

Tactic:
simp [field, logDeriv_apply, *]

Goal state after:
Module: Mathlib.Analysis.Calculus.LogDeriv

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
f g : 𝕜 → 𝕜'
x : 𝕜
hf : f x ≠ 0
hg : g x ≠ 0
hdf : DifferentiableAt 𝕜 f x
hdg : DifferentiableAt 𝕜 g x
⊢ logDeriv (fun z ↦ f z / g z) x = logDeriv f x - logDeriv g x

Tactic:
simp [field, logDeriv_apply, *]

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

Goal state before:
pp: case inl
f f' : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x
s : Set ℝ := {x | f x ≤ B x} ∩ Icc a b
A : ContinuousOn (fun x ↦ (f x, B x)) (Icc a b)
this✝ : IsClosed s
x : ℝ
hxB✝ : f x ≤ B x
xab : x ∈ Ico a b
y : ℝ
hy : y ∈ Ioi x
hxB : f x < B x
this : ∀ᶠ (x : ℝ) in 𝓝[Icc a b] x, f x < B x
⊢ {x | f x ≤ B x} ∈ 𝓝[>] x

Tactic:
have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsGT_of_mem xab) this

Goal state after:
case inl
f f' : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x
s : Set ℝ := {x | f x ≤ B x} ∩ Icc a b
A : ContinuousOn (fun x ↦ (f x, B x)) (Icc a b)
this✝¹ : IsClosed s
x : ℝ
hxB✝ : f x ≤ B x
xab : x ∈ Ico a b
y : ℝ
hy : y ∈ Ioi x
hxB : f x < B x
this✝ : ∀ᶠ (x : ℝ) in 𝓝[Icc a b] x, f x < B x
this : ∀ᶠ (x : ℝ) in 𝓝[>] x, f x < B x
⊢ {x | f x ≤ B x} ∈ 𝓝[>] x
Module: Mathlib.Analysis.Calculus.MeanValue

Goal state before:
pp: f : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x ∈ Ico a b, ∀ (r : ℝ), B' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
x : ℝ
hx : x ∈ Icc a b
r : ℝ
hr : r > 0
⊢ f x ≤ B x + r * (x - a)

Tactic:
by
    apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound
    · rwa [sub_self, mul_zero, add_zero]
    · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const))
    · intro x hx
      exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r)
    · intro x _ _
      rw [mul_one]
      exact (lt_add_iff_pos_right _).2 hr
    exact hx

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

Goal state before:
pp: case zero
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
c : F
s : Set E
hn : 0 ≠ 0
⊢ iteratedFDerivWithin 𝕜 0 (fun x ↦ c) s = 0

Tactic:
contradiction

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

Goal state before:
pp: n : ℕ
c : OrderedFinpartition n
j : Fin c.length
h : range ((c.extendMiddle j).emb j) = {0}
⊢ ∃ y, (c.extendMiddle j).emb j y = (c.emb j 0).succ

Tactic:
have A : (c.extendMiddle j).partSize j = c.partSize j + 1 := by simp [extendMiddle]

Goal state after:
n : ℕ
c : OrderedFinpartition n
j : Fin c.length
h : range ((c.extendMiddle j).emb j) = {0}
A : (c.extendMiddle j).partSize j = c.partSize j + 1
⊢ ∃ y, (c.extendMiddle j).emb j y = (c.emb j 0).succ
Module: Mathlib.Analysis.Normed.Algebra.Exponential

Goal state before:
pp: 𝔸 : Type u_2
inst✝³ : DivisionRing 𝔸
inst✝² : TopologicalSpace 𝔸
inst✝¹ : IsTopologicalRing 𝔸
inst✝ : CharZero 𝔸
⊢ exp = fun x ↦ ∑' (n : ℕ), x ^ n / ↑n !

Tactic:
exp_eq_expSeries_sum ℚ

Goal state after:
𝔸 : Type u_2
inst✝³ : DivisionRing 𝔸
inst✝² : TopologicalSpace 𝔸
inst✝¹ : IsTopologicalRing 𝔸
inst✝ : CharZero 𝔸
⊢ (expSeries ℚ 𝔸).sum = fun x ↦ ∑' (n : ℕ), x ^ n / ↑n !
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
n : ℕ
⊢ (↑n !)⁻¹ ≠ 0

Tactic:
· simp [this]

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.Exponential

Goal state before:
pp: case inr
𝔸 : Type u_1
inst✝² : NormedDivisionRing 𝔸
inst✝¹ : NormedAlgebra ℚ 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
n : ℕ
⊢ exp (-↑n • x) = exp x ^ (-↑n)

Tactic:
· rw [zpow_neg, zpow_natCast, neg_smul, exp_neg, natCast_zsmul, exp_nsmul]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ExpDeriv

Goal state before:
pp: ⊢ AnalyticOnNhd ℂ cexp univ

Tactic:
by
  rw [Complex.exp_eq_exp_ℂ]
  exact fun x _ ↦ NormedSpace.exp_analytic x

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

Goal state before:
pp: case neg.hB
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)
hab : a ≠ b
⊢ Icc a b ⊆ closure[inst✝⁵] (Ioo a b)

Tactic:
rw [closure_Ioo hab]

Goal state after:
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
m k : ℕ
⊢ iteratedDeriv k (fun x ↦ x ^ m) x = ↑(m.descFactorial k) * x ^ (m - k)

Tactic:
simpa using iteratedDerivWithin_pow (Set.mem_univ x) uniqueDiffOn_univ m k

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

Goal state before:
pp: case neg.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
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
x : α
⊢ (piecewise s hs (const α c) (const α 0)) x ≤ u.indicator (fun x ↦ c) x

Tactic:
simp only [SimpleFunc.coe_const, SimpleFunc.const_zero, SimpleFunc.coe_zero,
        Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, ← Function.const_def]
      grw [su]

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

Goal state before:
pp: case refine_2.hf
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
fint : Integrable (fun x ↦ ↑(f x)) μ
fmeas : AEMeasurable f μ
ε : ℝ≥0
εpos : 0 < ↑ε
δ : ℝ≥0
δpos : 0 < δ
hδε : δ < ε
int_f_ne_top : ∫⁻ (a : α), ↑(f a) ∂μ ≠ ∞
g : α → ℝ≥0∞
f_lt_g : ∀ (x : α), ↑(f x) < g x
gcont : LowerSemicontinuous g
gint : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ↑δ
gint_ne : ∫⁻ (x : α), g x ∂μ ≠ ∞
g_lt_top : ∀ᵐ (x : α) ∂μ, g x < ∞
Ig : ∫⁻ (a : α), ENNReal.ofReal (g a).toReal ∂μ = ∫⁻ (a : α), g a ∂μ
⊢ 0 ≤ᵐ[μ] fun x ↦ ↑(f x)

Tactic:
apply Filter.Eventually.of_forall fun x => _; simp

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

Goal state before:
pp: case refine_2.hf
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
fint : Integrable (fun x ↦ ↑(f x)) μ
fmeas : AEMeasurable f μ
ε : ℝ≥0
εpos : 0 < ↑ε
δ : ℝ≥0
δpos : 0 < δ
hδε : δ < ε
int_f_ne_top : ∫⁻ (a : α), ↑(f a) ∂μ ≠ ∞
g : α → ℝ≥0∞
f_lt_g : ∀ (x : α), ↑(f x) < g x
gcont : LowerSemicontinuous g
gint : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ↑δ
gint_ne : ∫⁻ (x : α), g x ∂μ ≠ ∞
g_lt_top : ∀ᵐ (x : α) ∂μ, g x < ∞
Ig : ∫⁻ (a : α), ENNReal.ofReal (g a).toReal ∂μ = ∫⁻ (a : α), g a ∂μ
⊢ 0 ≤ᵐ[μ] fun x ↦ (g x).toReal

Tactic:
apply Filter.Eventually.of_forall fun x => _; simp

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: x : ℝ
h : |x| < 1
⊢ Summable fun n ↦ x ^ (n + 1) / (↑n + 1)

Tactic:
show Summable fun n : ℕ => x ^ (n + 1) / (n + 1)

Goal state after:
x : ℝ
h : |x| < 1
⊢ Summable fun n ↦ x ^ (n + 1) / (↑n + 1)
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
x : E
⊢ sInf {r | r ∈ Ioi 0 ∧ r⁻¹ • x ∈ 0} = 0 x

Tactic:
obtain rfl | hx := eq_or_ne x 0

Goal state after:
case inl
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
⊢ sInf {r | r ∈ Ioi 0 ∧ r⁻¹ • 0 ∈ 0} = 0 0

case inr
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
x : E
hx : x ≠ 0
⊢ sInf {r | r ∈ Ioi 0 ∧ r⁻¹ • x ∈ 0} = 0 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
hs : Convex ℝ s
absorbs : Absorbent ℝ s
ε : ℝ
hε : 0 < ε
a : ℝ
ha : 0 < a
x : E
hx : x ∈ s
ha' : a < gauge s ((fun x ↦ a • x) x) + ε / 2
b : ℝ
hb : 0 < b
y : E
hy : y ∈ s
hb' : b < gauge s ((fun x ↦ b • x) y) + ε / 2
⊢ a • x + b • y ∈ (a + b) • s

Tactic:
by
      rw [hs.add_smul ha.le hb.le]
      exact add_mem_add (smul_mem_smul_set hx) (smul_mem_smul_set hy)

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
⊢ ∃ g, (∀ (x : ↥f.domain), g ↑x = ↑f x) ∧ ∀ (x : E), g x ≤ N x

Tactic:
have N_0 : N 0 = 0 := by grind [N_hom 2 (by norm_num) 0, smul_zero]

Goal state after:
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
⊢ ∃ g, (∀ (x : ↥f.domain), g ↑x = ↑f x) ∧ ∀ (x : E), g x ≤ N x
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: case inr.a
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
r : ℝ
hr : 0 ≤ r
x : E
hr' : 0 < r
⊢ gauge (closedBall 0 r) x ≤ ‖x‖ / r

Tactic:
← gauge_ball hr

Goal state after:
case inr.a
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
r : ℝ
hr : 0 ≤ r
x : E
hr' : 0 < r
⊢ gauge (closedBall 0 r) x ≤ gauge (ball 0 r) x
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: ε : Type u_3
inst✝² : TopologicalSpace ε
inst✝¹ : ENormedAddMonoid ε
inst✝ : PseudoMetrizableSpace ε
f : ℝ → ε
a b : ℝ
μ : Measure ℝ
g : ℝ → ε
h : f =ᵐ[μ.restrict (Ι a b)] g
⊢ IntervalIntegrable f μ a b ↔ IntervalIntegrable g μ a b

Tactic:
intervalIntegrable_iff,

Goal state after:
ε : Type u_3
inst✝² : TopologicalSpace ε
inst✝¹ : ENormedAddMonoid ε
inst✝ : PseudoMetrizableSpace ε
f : ℝ → ε
a b : ℝ
μ : Measure ℝ
g : ℝ → ε
h : f =ᵐ[μ.restrict (Ι a b)] g
⊢ IntegrableOn f (Ι a b) μ ↔ IntervalIntegrable g μ a b
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: ε : Type u_3
inst✝² : TopologicalSpace ε
inst✝¹ : ENormedAddMonoid ε
inst✝ : PseudoMetrizableSpace ε
f : ℝ → ε
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
⊢ IntervalIntegrable f μ a b ↔ IntegrableOn f (Ioc a b) μ

Tactic:
intervalIntegrable_iff,

Goal state after:
ε : Type u_3
inst✝² : TopologicalSpace ε
inst✝¹ : ENormedAddMonoid ε
inst✝ : PseudoMetrizableSpace ε
f : ℝ → ε
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
⊢ IntegrableOn f (Ι a b) μ ↔ IntegrableOn f (Ioc a b) μ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: ε : Type u_3
inst✝³ : TopologicalSpace ε
inst✝² : ENormedAddMonoid ε
inst✝¹ : PseudoMetrizableSpace ε
f : ℝ → ε
a b : ℝ
μ : Measure ℝ
inst✝ : NullSingletonClass μ
h : ‖f (min a b)‖ₑ ≠ ∞
⊢ IntervalIntegrable f μ a b ↔ IntegrableOn f [[a, b]] μ

Tactic:
intervalIntegrable_iff,

Goal state after:
ε : Type u_3
inst✝³ : TopologicalSpace ε
inst✝² : ENormedAddMonoid ε
inst✝¹ : PseudoMetrizableSpace ε
f : ℝ → ε
a b : ℝ
μ : Measure ℝ
inst✝ : NullSingletonClass μ
h : ‖f (min a b)‖ₑ ≠ ∞
⊢ IntegrableOn f (Ι a b) μ ↔ IntegrableOn f [[a, b]] μ
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)
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 := k - 1
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‖
⊢ ‖(y - x) • (L e (n e) m - f')‖ ≤ ‖y - x‖ * (12 * (1 / 2) ^ e)

Tactic:
norm_smul

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
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 := k - 1
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‖
⊢ ‖y - x‖ * ‖L e (n e) m - f'‖ ≤ ‖y - x‖ * (12 * (1 / 2) ^ e)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: ε : Type u_3
inst✝² : TopologicalSpace ε
inst✝¹ : ENormedAddMonoid ε
f : ℝ → ε
inst✝ : PseudoMetrizableSpace ε
μ : Measure ℝ
a b : ℝ
hf : AEStronglyMeasurable f (μ.restrict (Ι a b))
⊢ IntervalIntegrable (fun t ↦ ‖f t‖ₑ) μ a b ↔ IntervalIntegrable f μ a b

Tactic:
intervalIntegrable_iff,

Goal state after:
ε : Type u_3
inst✝² : TopologicalSpace ε
inst✝¹ : ENormedAddMonoid ε
f : ℝ → ε
inst✝ : PseudoMetrizableSpace ε
μ : Measure ℝ
a b : ℝ
hf : AEStronglyMeasurable f (μ.restrict (Ι a b))
⊢ IntegrableOn (fun t ↦ ‖f t‖ₑ) (Ι a b) μ ↔ IntegrableOn f (Ι a b) μ
Module: Mathlib.Analysis.Calculus.TangentCone.Prod

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁰ : Semiring 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : TopologicalSpace E
inst✝⁶ : ContinuousAdd E
inst✝⁵ : ContinuousConstSMul 𝕜 E
inst✝⁴ : AddCommGroup F
inst✝³ : Module 𝕜 F
inst✝² : TopologicalSpace F
inst✝¹ : ContinuousAdd F
inst✝ : ContinuousConstSMul 𝕜 F
x : E
s : Set E
y : F
t : Set F
hs : x ∈ closure[inst✝⁷] s
z : F
hz : z ∈ tangentConeAt 𝕜 t y
ι : Type (max u_1 u_3)
l : Filter ι
hl : l.NeBot
c : ι → 𝕜
d : ι → F
hd₀ : Tendsto d l (𝓝 0)
hds : ∀ᶠ (n : ι) in l, y + d n ∈ t
hcd : Tendsto (fun n ↦ c n • d n) l (𝓝 z)
⊢ (LinearMap.inr 𝕜 E F) z ∈ tangentConeAt 𝕜 (closure[inst✝⁷] s ×ˢ closure[inst✝²] t) (x, y)

Tactic:
refine mem_tangentConeAt_of_seq l c (fun n ↦ (0, d n)) (tendsto_const_nhds.prodMk_nhds hd₀)
    (hds.mono fun n hn ↦ by simp [hs, subset_closure hn]) ?_

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁰ : Semiring 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : TopologicalSpace E
inst✝⁶ : ContinuousAdd E
inst✝⁵ : ContinuousConstSMul 𝕜 E
inst✝⁴ : AddCommGroup F
inst✝³ : Module 𝕜 F
inst✝² : TopologicalSpace F
inst✝¹ : ContinuousAdd F
inst✝ : ContinuousConstSMul 𝕜 F
x : E
s : Set E
y : F
t : Set F
hs : x ∈ closure[inst✝⁷] s
z : F
hz : z ∈ tangentConeAt 𝕜 t y
ι : Type (max u_1 u_3)
l : Filter ι
hl : l.NeBot
c : ι → 𝕜
d : ι → F
hd₀ : Tendsto d l (𝓝 0)
hds : ∀ᶠ (n : ι) in l, y + d n ∈ t
hcd : Tendsto (fun n ↦ c n • d n) l (𝓝 z)
⊢ Tendsto (fun n ↦ c n • (0, d n)) l (𝓝 ((LinearMap.inr 𝕜 E F) z))
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝ : NormedAddCommGroup E
f : ℝ → E
μ : Measure ℝ
a b : ℝ
hf : AEStronglyMeasurable f (μ.restrict (Ι a b))
⊢ IntervalIntegrable (fun t ↦ ‖f t‖) μ a b ↔ IntervalIntegrable f μ a b

Tactic:
intervalIntegrable_iff,

Goal state after:
E : Type u_5
inst✝ : NormedAddCommGroup E
f : ℝ → E
μ : Measure ℝ
a b : ℝ
hf : AEStronglyMeasurable f (μ.restrict (Ι a b))
⊢ IntegrableOn (fun t ↦ ‖f t‖) (Ι a b) μ ↔ IntegrableOn f (Ι a b) μ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: ε : Type u_3
inst✝² : TopologicalSpace ε
inst✝¹ : ENormedAddMonoid ε
f : ℝ → ε
a b : ℝ
inst✝ : PseudoMetrizableSpace ε
hf : IntervalIntegrable f volume a b
c : ℝ
h : ‖f (min a b)‖ₑ ≠ ∞
h' : ‖f (min (a - c) (b - c) + c)‖ₑ ≠ ∞
this :
  ∀ {a b : ℝ},
    IntervalIntegrable f volume a b →
      ‖f (min a b)‖ₑ ≠ ∞ →
        ‖f (min (a - c) (b - c) + c)‖ₑ ≠ ∞ → a ≤ b → IntervalIntegrable (fun x ↦ f (x + c)) volume (a - c) (b - c)
hab : ¬a ≤ b
⊢ ‖f (min (a - c) (b - c) + c)‖ₑ ≠ ∞

Tactic:
exact h'

Goal state after: