mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.Deriv.ZPow

Goal state before:
pp: 𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
x : 𝕜
h : x ≠ 0 ∨ 0 ≤ m
⊢ HasStrictDerivAt (fun x ↦ x ^ m) (↑m * x ^ (m - 1)) x

Tactic:
have : ∀ m : ℤ, 0 < m → HasStrictDerivAt (· ^ m) ((m : 𝕜) * x ^ (m - 1)) x := fun m hm ↦ by
    lift m to ℕ using hm.le
    simp only [zpow_natCast, Int.cast_natCast]
    convert hasStrictDerivAt_pow m x
    rw [← Int.ofNat_one, ← Int.ofNat_sub, zpow_natCast]
    norm_cast at hm
  rcases lt_trichotomy m 0 with (hm | hm | hm)
  · have hx : x ≠ 0 := h.resolve_right hm.not_ge
    have := (hasStrictDerivAt_inv ?_).scomp _ (this (-m) (neg_pos.2 hm)) <;>
      [skip; exact zpow_ne_zero _ hx]
    simp only [Function.comp_def, zpow_neg, inv_inv, smul_eq_mul] at this
    convert! this using 1
    rw [sq, mul_inv, inv_inv, Int.cast_neg, neg_mul, neg_mul_neg, ← zpow_add₀ hx, mul_assoc, ←
      zpow_add₀ hx]
    congr
    abel
  · simp only [hm, zpow_zero, Int.cast_zero, zero_mul, hasStrictDerivAt_const]
  · exact this m hm

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

Goal state before:
pp: 𝕜 : Type u_3
G : Type u_4
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 𝕜
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : E → G
C : ℝ≥0
hf : Differentiable 𝕜 f
bound : ∀ (x : E), ‖fderiv 𝕜 f x‖₊ ≤ C
this : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
A : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
⊢ LipschitzOnWith C f univ

Tactic:
exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n : ℕ∞ω
inst✝ : Subsingleton F
⊢ ContDiffWithinAt 𝕜 n f s x

Tactic:
by
  rw [Subsingleton.elim f fun _ => 0]; exact contDiffWithinAt_const

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

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
𝕜 : Type u_3
G : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E → G
s : Set E
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hs' : IsPreconnected s
hf : DifferentiableOn 𝕜 f s
hf' : EqOn (fderiv 𝕜 f) 0 s
y : E
hy : y ∈ s
x : E
hx : x ∈ s
h₁ : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (s ∩ f ⁻¹' {f y})
h₂ : Continuous[instTopologicalSpaceSubtype, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f ∘ Subtype.val)
h₃ : ¬f x = f y
t : Set E
ht : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
ht' : Subtype.val ⁻¹' t = (f ∘ Subtype.val ⁻¹' {f y})ᶜ
ht'' : ∀ a ∈ s, a ∈ t ↔ f a ≠ f y
⊢ y ∈ s ∩ (s ∩ f ⁻¹' {f y})

Tactic:
by simpa

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

Goal state before:
pp: case e'_12
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
f : E → F
n : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
hf : HasFTaylorSeriesUpToOn n f p s
g : G →ᴬ[𝕜] E
A : (m : ℕ) → (E [×m]→L[𝕜] F) → G [×m]→L[𝕜] F := fun m h ↦ h.compContinuousLinearMap fun x ↦ g.contLinear
hA : ∀ (m : ℕ), IsBoundedLinearMap 𝕜 (A m)
m : ℕ
hm : ↑m < n
x : G
hx : x ∈ ⇑g ⁻¹' s
e_8✝ : ContinuousMultilinearMap.instAddCommGroup = ContinuousMultilinearMap.normedAddCommGroup'.toAddCommGroup
e_9✝ : ContinuousMultilinearMap.instModule ≍ ContinuousMultilinearMap.normedSpace'.toModule
e_10✝ : ContinuousMultilinearMap.instTopologicalSpace = PseudoMetricSpace.toUniformSpace.toTopologicalSpace
y : G
v : Fin m → G
⊢ (p (g x) m.succ) (⇑g.contLinear ∘ cons y v) = (p (g x) m.succ) (cons (g.contLinear y) (⇑g.contLinear ∘ v))

Tactic:
rw [comp_cons]

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

Goal state before:
pp: 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
s : Set E
f : E → F
x : E
n : ℕ∞ω
k : ℕ
hf : ContDiffWithinAt 𝕜 n f s x
hs : UniqueDiffOn 𝕜 s
hk : ↑k ≤ n
hx : x ∈ s
⊢ 0 + ↑k ≤ n

Tactic:
by simpa

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

Goal state before:
pp: 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
f : E → F
x : E
n : ℕ∞ω
k : ℕ
hf : ContDiffAt 𝕜 n f x
hk : ↑k ≤ n
⊢ 0 + ↑k ≤ n

Tactic:
by simpa

Goal state after:
Module: Mathlib.Topology.OpenPartialHomeomorph.IsImage

Goal state before:
pp: X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
s : Set X
t : Set Y
⊢ e.IsImage s t ↔ e.target ∩ ↑e.symm ⁻¹' (e.source ∩ s) = e.target ∩ t

Tactic:
iff_symm_preimage_eq,

Goal state after:
X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
s : Set X
t : Set Y
⊢ e.target ∩ ↑e.symm ⁻¹' s = e.target ∩ t ↔ e.target ∩ ↑e.symm ⁻¹' (e.source ∩ s) = e.target ∩ t
Module: Mathlib.Analysis.Calculus.FDeriv.OfCompLeft

Goal state before:
pp: case refine_2
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
g : E → F
f : F → G
h : E → G
g' : E →L[𝕜] F
f' : F →L[𝕜] G
a : E
s : Set E
t : Set F
hg : Tendsto g (𝓝[s] a) (𝓝[t] g a)
hf : HasFDerivWithinAt f f' t (g a)
hf' : Topology.IsEmbedding ⇑f'
hh : HasFDerivWithinAt h (f' ∘SL g') s a
hcomp : f ∘ g =ᶠ[𝓝[s] a] h
ha : a ∈ s
⊢ Prod.map (f ∘ g) (f ∘ g) =ᶠ[𝓝[s] a ×ˢ pure a] Prod.map h h

Tactic:
· exact hcomp.prodMap <| hcomp.self_of_nhdsWithin ha

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

Goal state before:
pp: n : ℕ
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) ≠ {0}
⊢ c.partSize (c.index 0) = Nat.card ↑(range (c.emb (c.index 0)))

Tactic:
Nat.card_range_of_injective (c.emb_strictMono _).injective

Goal state after:
n : ℕ
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) ≠ {0}
⊢ c.partSize (c.index 0) = Nat.card (Fin (c.partSize (c.index 0)))
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: case mp
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
s : Set E
f : E → F
n : ℕ∞ω
hn : n ≠ ∞
⊢ ContDiffOn 𝕜 (n + 1) f s →
    ∀ x ∈ s,
      ∃ u ∈ 𝓝[insert x s] x,
        (n = ω → AnalyticOn 𝕜 f u) ∧ ∃ f', (∀ x ∈ u, HasFDerivWithinAt f (f' x) u x) ∧ ContDiffOn 𝕜 n f' u

pp: case mpr
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
s : Set E
f : E → F
n : ℕ∞ω
hn : n ≠ ∞
⊢ (∀ x ∈ s,
      ∃ u ∈ 𝓝[insert x s] x,
        (n = ω → AnalyticOn 𝕜 f u) ∧ ∃ f', (∀ x ∈ u, HasFDerivWithinAt f (f' x) u x) ∧ ContDiffOn 𝕜 n f' u) →
    ContDiffOn 𝕜 (n + 1) f s

Tactic:
· intro h x hx
    rcases (contDiffWithinAt_succ_iff_hasFDerivWithinAt hn).1 (h x hx) with
      ⟨u, hu, f_an, f', hf', Hf'⟩
    rcases Hf'.contDiffOn le_rfl (by simp [hn]) with ⟨v, vu, v'u, hv⟩
    rw [insert_eq_of_mem hx] at hu ⊢
    have xu : x ∈ u := mem_of_mem_nhdsWithin hx hu
    rw [insert_eq_of_mem xu] at vu v'u
    exact ⟨v, nhdsWithin_le_of_mem hu vu, fun h ↦ (f_an h).mono v'u, f',
      fun y hy ↦ (hf' y (v'u hy)).mono v'u, hv⟩

Goal state after:
case mpr
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
s : Set E
f : E → F
n : ℕ∞ω
hn : n ≠ ∞
⊢ (∀ x ∈ s,
      ∃ u ∈ 𝓝[insert x s] x,
        (n = ω → AnalyticOn 𝕜 f u) ∧ ∃ f', (∀ x ∈ u, HasFDerivWithinAt f (f' x) u x) ∧ ContDiffOn 𝕜 n f' u) →
    ContDiffOn 𝕜 (n + 1) f s
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) * ↑n !)⁻¹ / (↑n !)⁻¹‖) atTop (𝓝 0)

Tactic:
mul_inv_rev,

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 n ↦ ‖(↑n !)⁻¹ * (↑(n + 1))⁻¹ / (↑n !)⁻¹‖) atTop (𝓝 0)
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
s : Set E
t : Set F
q✝ : F → FormalMultilinearSeries 𝕜 F G
p✝ : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c : OrderedFinpartition n
f : F [×c.length]→L[𝕜] G
p : (i : Fin c.length) → E [×c.partSize i]→L[𝕜] F
m : Fin c.length
q q' : E [×c.partSize m]→L[𝕜] F
⊢ c.compAlongOrderedFinpartition f (update p m (q + q')) =
    c.compAlongOrderedFinpartition f (update p m q) + c.compAlongOrderedFinpartition f (update p m q')

Tactic:
ext v
        simp [applyOrderedFinpartition_update_left]

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
s : Set E
t : Set F
q✝ : F → FormalMultilinearSeries 𝕜 F G
p✝ : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c : OrderedFinpartition n
f : F [×c.length]→L[𝕜] G
p : (i : Fin c.length) → E [×c.partSize i]→L[𝕜] F
m : Fin c.length
a : 𝕜
q : E [×c.partSize m]→L[𝕜] F
⊢ c.compAlongOrderedFinpartition f (update p m (a • q)) = a • c.compAlongOrderedFinpartition f (update p m q)

Tactic:
ext v
        simp [applyOrderedFinpartition_update_left]

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

Goal state before:
pp: ⊢ cexp = NormedSpace.exp

Tactic:
by
  refine funext fun x => ?_
  rw [Complex.exp, exp_eq_tsum_div]
  exact tendsto_nhds_unique x.exp'.tendsto_limit (expSeries_div_summable x).hasSum.tendsto_sum_nat

Goal state after:
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 :
  (fun h ↦ exp (t • x) * (exp ((0 + h) • x) - exp (0 • x) - (ContinuousLinearMap.smulRight 1 x) h)) =ᶠ[𝓝 0] fun h ↦
    exp ((t + h) • x) - exp (t • x) - (exp (t • x) • ContinuousLinearMap.smulRight 1 x) h
⊢ (fun h ↦ exp ((t + h) • x) - exp (t • x) - (exp (t • x) • ContinuousLinearMap.smulRight 1 x) h) =o[𝓝 0] fun h ↦ h

Tactic:
apply (IsLittleO.const_mul_left _ _).congr' this (EventuallyEq.refl _ _)
    rw [← hasFDerivAt_iff_isLittleO_nhds_zero (f := fun u => exp (u • x))
      (f' := (1 : 𝕊 →L[𝕂] 𝕊).smulRight x) (x := 0)]
    have : HasFDerivAt exp (1 : 𝔸 →L[𝕂] 𝔸) ((1 : 𝕊 →L[𝕂] 𝕊).smulRight x 0) := by
      rw [ContinuousLinearMap.smulRight_apply, one_apply_eq_self, zero_smul]
      exact hasFDerivAt_exp_zero_of_radius_pos htx.pos
    exact this.comp 0 ((1 : 𝕊 →L[𝕂] 𝕊).smulRight x).hasFDerivAt

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

Goal state before:
pp: case neg.hf.inr.inr
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
x : α
x_in : x ∈ Icc a b
h : x ∈ Ioo a b
⊢ ∃ y, Tendsto f (𝓝[Ioo a b] x) (𝓝 y)

Tactic:
· exact ⟨f x, hf x h⟩

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

Goal state before:
pp: case neg.hf.inr
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 : β
inst✝ : RegularSpace β
hf : ContinuousOn f (Ioo a b)
ha : Tendsto f (𝓝[>] a) (𝓝 la)
hab : a < b
x : α
x_in : x ∈ Ico a b
h : x ∈ Ioo a b
⊢ ∃ y, Tendsto f (𝓝[Ioo a b] x) (𝓝 y)

Tactic:
· exact ⟨f x, hf x h⟩

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

Goal state before:
pp: f : ℝ → ℝ
a : ℝ
hf : Tendsto (deriv f) (𝓝[>] a) atTop
x : ℝ
hx : x ∈ Ioi a
⊢ deriv f x = derivWithin f (Ioi a) x

Tactic:
have : Ioi a ∈ 𝓝 x := by simp [← mem_interior_iff_mem_nhds, hx]

Goal state after:
f : ℝ → ℝ
a : ℝ
hf : Tendsto (deriv f) (𝓝[>] a) atTop
x : ℝ
hx : x ∈ Ioi a
this : Ioi a ∈ 𝓝 x
⊢ deriv f x = derivWithin f (Ioi a) x
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: f : ℝ → ℝ
a : ℝ
hf : Tendsto (deriv f) (𝓝[<] a) atBot
⊢ ¬DifferentiableWithinAt ℝ f (Iio a) a

Tactic:
let f' := f ∘ Neg.neg

Goal state after:
f : ℝ → ℝ
a : ℝ
hf : Tendsto (deriv f) (𝓝[<] a) atBot
f' : ℝ → ℝ := f ∘ Neg.neg
⊢ ¬DifferentiableWithinAt ℝ f (Iio a) a
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

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 : E → F
L : E →L[𝕜] F
r ε : ℝ
x : E
r' : ℝ
r'_mem : r' ∈ Ioc (r / 2) r
hr' : ∀ y ∈ ball x r', ∀ z ∈ ball x r', ‖f z - f y - L (z - y)‖ < ε * r
s : ℝ
s_gt : r / 2 < s
s_lt : s < r'
⊢ ∃ ε_1 > 0, ball x ε_1 ⊆ A f L r ε

Tactic:
have : s ∈ Ioc (r / 2) r := ⟨s_gt, le_of_lt (s_lt.trans_le r'_mem.2)⟩

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
f : E → F
L : E →L[𝕜] F
r ε : ℝ
x : E
r' : ℝ
r'_mem : r' ∈ Ioc (r / 2) r
hr' : ∀ y ∈ ball x r', ∀ z ∈ ball x r', ‖f z - f y - L (z - y)‖ < ε * r
s : ℝ
s_gt : r / 2 < s
s_lt : s < r'
this : s ∈ Ioc (r / 2) r
⊢ ∃ ε_1 > 0, ball x ε_1 ⊆ A f L r ε
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: n : ℕ∞ω
x : ℝ
hx : x ≠ 0
y : ℝ
hy : 0 < y
⊢ y ∈ expPartialHomeomorph.target

Tactic:
by simpa

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≥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
⊢ ∫⁻ (x : α), ↑(f x) ∂μ + ↑δ ≠ ∞

Tactic:
by simpa

Goal state after:
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
y : M
hy : y ∈ s
⊢ 0 < 1 ∧ y ∈ 1 • s

Tactic:
by simpa

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Pointed

Goal state before:
pp: R : Type u_1
E : Type u_2
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsOrderedRing R
inst✝¹ : AddCommMonoid E
inst✝ : Module R E
s : Set (Submodule R E)
⊢ ↑(⨅ S ∈ s, S) = ⨅ S ∈ s, ↑S

Tactic:
rw [← sInf_eq_iInf, ofSubmodule_sInf, sInf_eq_iInf, iInf_image]

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

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 : 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
e : ℕ
this : x ∈ ⋃ n, ⋂ p, ⋂ (_ : p ≥ n), ⋂ q, ⋂ (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)
n p q : ℕ
hp : n ≤ p
hq : n ≤ q
hn : ∀ i ≥ n, ∀ i_2 ≥ n, x ∈ B f K ((1 / 2) ^ i) ((1 / 2) ^ i_2) ((1 / 2) ^ e)
L : E →L[𝕜] F
hL : x ∈ ⋃ (_ : L ∈ K), A f L ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f L ((1 / 2) ^ q) ((1 / 2) ^ e)
⊢ ∃ L ∈ K, x ∈ A f L ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f L ((1 / 2) ^ q) ((1 / 2) ^ e)

Tactic:
exact ⟨L, exists_prop.mp <| mem_iUnion.1 hL⟩

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
⊢ g (0, N x) - g (x, N x) = N x - g (x, N x)

Tactic:
simpa using g_eq 0 (N x)

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

Goal state before:
pp: E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s : Set E
x : E
inst✝¹ : TopologicalSpace E
inst✝ : T1Space E
hs : Absorbent ℝ s
hb : Bornology.IsVonNBounded ℝ s
⊢ 0 < gauge s x ↔ x ≠ 0

Tactic:
simp only [(gauge_nonneg _).lt_iff_ne', Ne, gauge_eq_zero hs hb]

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.Separation

Goal state before:
pp: E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
t : Set E
x : E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul ℝ E
ht₁ : Convex ℝ t
ht₂ : IsOpen t
disj : x ∉ t
f : StrongDual ℝ E
hf : ∀ a ∈ t, f a < f x
⊢ ∀ b ∈ t, (-f) x < (-f) b

Tactic:
by simpa

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

Goal state before:
pp: E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
r : ℝ
hr : 0 ≤ r
x : E
hr' : 0 < r
this : ∀ᶠ (R : ℝ) in 𝓝[>] r, ‖x‖ / R ≤ gauge (closedBall 0 r) x
⊢ ‖x‖ / r ≤ gauge (closedBall 0 r) x

Tactic:
refine le_of_tendsto ?_ this
        exact tendsto_const_nhds.div inf_le_left hr'.ne'

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.Separation

Goal state before:
pp: E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s t : Set E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul ℝ E
hs : Convex ℝ s
ht : Convex ℝ t
hst : Disjoint (interior s) t
hsint : (interior s).Nonempty
htne : t.Nonempty
f : StrongDual ℝ E
u : ℝ
hfA : ∀ a ∈ interior s, f a < u
hfB : ∀ b ∈ t, u ≤ f b
⊢ ∃ f u, f ≠ 0 ∧ (∀ a ∈ s, f a ≤ u) ∧ ∀ b ∈ t, u ≤ f b

Tactic:
refine ⟨f, u, ?_, fun a ha ↦ ?_, hfB⟩

Goal state after:
case refine_1
E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s t : Set E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul ℝ E
hs : Convex ℝ s
ht : Convex ℝ t
hst : Disjoint (interior s) t
hsint : (interior s).Nonempty
htne : t.Nonempty
f : StrongDual ℝ E
u : ℝ
hfA : ∀ a ∈ interior s, f a < u
hfB : ∀ b ∈ t, u ≤ f b
⊢ f ≠ 0

case refine_2
E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s t : Set E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul ℝ E
hs : Convex ℝ s
ht : Convex ℝ t
hst : Disjoint (interior s) t
hsint : (interior s).Nonempty
htne : t.Nonempty
f : StrongDual ℝ E
u : ℝ
hfA : ∀ a ∈ interior s, f a < u
hfB : ∀ b ∈ t, u ≤ f b
a : E
ha : a ∈ s
⊢ f a ≤ u
Module: Mathlib.Analysis.LocallyConvex.Separation

Goal state before:
pp: case pos
E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s t : Set E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul ℝ E
hs : Convex ℝ s
ht : Convex ℝ t
hst : Disjoint (interior s) t
hsint : (interior s).Nonempty
htne : t.Nonempty
⊢ ∃ f u, (∀ a ∈ s, f a ≤ u) ∧ ∀ b ∈ t, u ≤ f b

Tactic:
obtain ⟨f, u, -, hs', ht'⟩ :=
      geometric_hahn_banach_of_nonempty_interior hs ht hst hsint htne

Goal state after:
case pos
E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s t : Set E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul ℝ E
hs : Convex ℝ s
ht : Convex ℝ t
hst : Disjoint (interior s) t
hsint : (interior s).Nonempty
htne : t.Nonempty
f : StrongDual ℝ E
u : ℝ
hs' : ∀ a ∈ s, f a ≤ u
ht' : ∀ b ∈ t, u ≤ f b
⊢ ∃ f u, (∀ a ∈ s, f a ≤ u) ∧ ∀ b ∈ t, u ≤ f b
Module: Mathlib.Analysis.LocallyConvex.Separation

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
t : Set E
x : E
inst✝⁴ : RCLike 𝕜
inst✝³ : Module 𝕜 E
inst✝² : IsScalarTower ℝ 𝕜 E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul 𝕜 E
ht₁ : Convex ℝ t
ht₂ : IsOpen t
disj : x ∉ t
f : StrongDual 𝕜 E
hf : ∀ a ∈ t, re (f a) < re (f x)
⊢ ∀ b ∈ t, re ((-f) x) < re ((-f) b)

Tactic:
by simpa

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.Separation

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁹ : TopologicalSpace E
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module ℝ E
s : Set E
inst✝⁶ : RCLike 𝕜
inst✝⁵ : Module 𝕜 E
inst✝⁴ : IsScalarTower ℝ 𝕜 E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : LocallyConvexSpace ℝ E
inst✝ : HereditarilyLindelofSpace E
hs₁ : Convex ℝ s
hs₂ : IsClosed s
ι : Type (max u_1 u_2) := (l : StrongDual 𝕜 E) × { c // ∀ y ∈ s, re (l y) ≤ c }
l : ι → StrongDual 𝕜 E := fun lc ↦ lc.fst
⊢ ∃ l c, ⋂ n, {x | re ((l n) x) ≤ c n} = s

Tactic:
set c : ι → ℝ := fun lc ↦ lc.2.val

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : TopologicalSpace E
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module ℝ E
s : Set E
inst✝⁶ : RCLike 𝕜
inst✝⁵ : Module 𝕜 E
inst✝⁴ : IsScalarTower ℝ 𝕜 E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : LocallyConvexSpace ℝ E
inst✝ : HereditarilyLindelofSpace E
hs₁ : Convex ℝ s
hs₂ : IsClosed s
ι : Type (max u_1 u_2) := (l : StrongDual 𝕜 E) × { c // ∀ y ∈ s, re (l y) ≤ c }
l : ι → StrongDual 𝕜 E := fun lc ↦ lc.fst
c : ι → ℝ := fun lc ↦ ↑lc.snd
⊢ ∃ l c, ⋂ n, {x | re ((l n) x) ≤ c n} = s
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
e : ℕ
this : x ∈ ⋃ n, ⋂ p, ⋂ (_ : p ≥ n), ⋂ q, ⋂ (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)
n p q : ℕ
hp : n ≤ p
hq : n ≤ q
hn : ∀ i ≥ n, ∀ i_2 ≥ n, x ∈ B f K ((1 / 2) ^ i) ((1 / 2) ^ i_2) ((1 / 2) ^ e)
L : F
hL : x ∈ ⋃ (_ : L ∈ K), A f L ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f L ((1 / 2) ^ q) ((1 / 2) ^ e)
⊢ ∃ L ∈ K, x ∈ A f L ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f L ((1 / 2) ^ q) ((1 / 2) ^ e)

Tactic:
exact ⟨L, exists_prop.mp <| mem_iUnion.1 hL⟩

Goal state after: