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

Goal state before:
pp: 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
g : F → G
x : E
n : ℕ∞ω
t : Set F
hg : ContDiffWithinAt 𝕜 n g t (f x)
hf : ContDiffWithinAt 𝕜 n f s x
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hxs : x ∈ s
hst : ∀ᶠ (y : E) in 𝓝[s] x, f y ∈ t
i : ℕ
hi : ↑i ≤ n
hxt : f x ∈ t
hf_tendsto : Tendsto f (𝓝[s] x) (𝓝[t] f x)
H₁ : ∀ᶠ (u : Set E) in (𝓝[s] x).smallSets, HasFTaylorSeriesUpToOn (↑i) f (ftaylorSeriesWithin 𝕜 f s) u
H₂ : ∀ᶠ (u : Set E) in (𝓝[s] x).smallSets, HasFTaylorSeriesUpToOn (↑i) g (ftaylorSeriesWithin 𝕜 g t) (f '' u)
u : Set E
hu :
  ∀ ⦃x : Set E⦄,
    x ∈ 𝒫 (u ∩ s) →
      HasFTaylorSeriesUpToOn (↑i) f (ftaylorSeriesWithin 𝕜 f s) x ∧
        HasFTaylorSeriesUpToOn (↑i) g (ftaylorSeriesWithin 𝕜 g t) (f '' x)
hxu : x ∈ u
huo : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
⊢ ∃ u,
    x ∈ u ∧
      IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u ∧
        HasFTaylorSeriesUpToOn (↑i) f (ftaylorSeriesWithin 𝕜 f s) (s ∩ u) ∧
          HasFTaylorSeriesUpToOn (↑i) g (ftaylorSeriesWithin 𝕜 g t) (f '' (s ∩ u))

Tactic:
exact ⟨u, hxu, huo, hu (by simp [inter_comm])⟩

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
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
x : E
n : ℕ∞ω
s : Set E
f : E → F
g : E → G
u : Set E
hu : u ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
hp : HasFTaylorSeriesUpToOn ω f p u
h'p : ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ p x i) u
v : Set E
hv : v ∈ 𝓝[insert x s] x
q : E → FormalMultilinearSeries 𝕜 E G
hq : HasFTaylorSeriesUpToOn ω g q v
h'q : ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ q x i) v
i : ℕ
⊢ AnalyticOn 𝕜 (fun x ↦ (p x i).prod (q x i)) (u ∩ v)

Tactic:
change AnalyticOn 𝕜 (fun x ↦ ContinuousMultilinearMap.prodL _ _ _ _ (p x i, q x i)) (u ∩ v)

Goal state after:
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
x : E
n : ℕ∞ω
s : Set E
f : E → F
g : E → G
u : Set E
hu : u ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
hp : HasFTaylorSeriesUpToOn ω f p u
h'p : ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ p x i) u
v : Set E
hv : v ∈ 𝓝[insert x s] x
q : E → FormalMultilinearSeries 𝕜 E G
hq : HasFTaylorSeriesUpToOn ω g q v
h'q : ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ q x i) v
i : ℕ
⊢ AnalyticOn 𝕜 (fun x ↦ (ContinuousMultilinearMap.prodL 𝕜 (fun _i ↦ E) F G) (p x i, q x i)) (u ∩ v)
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: case mp.refine_1
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
x : E
n : ℕ∞ω
hn : n ≠ ∞
h'n : n + 1 ≠ ∞
h : ContDiffWithinAt 𝕜 (n + 1) f s x
u : Set E
hu : u ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
Hp : HasFTaylorSeriesUpToOn (n + 1) f p u
H'p : n + 1 = ω → ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ p x i) u
⊢ n = ω → AnalyticOn 𝕜 f u

pp: case mp.refine_2
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
x : E
n : ℕ∞ω
hn : n ≠ ∞
h'n : n + 1 ≠ ∞
h : ContDiffWithinAt 𝕜 (n + 1) f s x
u : Set E
hu : u ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
Hp : HasFTaylorSeriesUpToOn (n + 1) f p u
H'p : n + 1 = ω → ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ p x i) u
⊢ ContDiffWithinAt 𝕜 n (fun y ↦ (continuousMultilinearCurryFin1 𝕜 E F) (p y 1)) u x

Tactic:
· rintro rfl
      exact Hp.analyticOn (H'p rfl 0)

Goal state after:
case mp.refine_2
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
x : E
n : ℕ∞ω
hn : n ≠ ∞
h'n : n + 1 ≠ ∞
h : ContDiffWithinAt 𝕜 (n + 1) f s x
u : Set E
hu : u ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
Hp : HasFTaylorSeriesUpToOn (n + 1) f p u
H'p : n + 1 = ω → ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ p x i) u
⊢ ContDiffWithinAt 𝕜 n (fun y ↦ (continuousMultilinearCurryFin1 𝕜 E F) (p y 1)) u x
Module: Mathlib.Analysis.Calculus.ContDiff.Comp

Goal state before:
pp: 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
x₀ : E
m n : ℕ∞ω
f : E → F → G
g : E → F
hf : ContDiffAt 𝕜 n (uncurry f) (x₀, g x₀)
hg : ContDiffAt 𝕜 m g x₀
hmn : m + 1 ≤ n
⊢ ContDiffAt 𝕜 m (fun x ↦ fderiv 𝕜 (f x) (g x)) x₀

Tactic:
by
  simp_rw [← fderivWithin_univ]
  refine (ContDiffWithinAt.fderivWithin hf.contDiffWithinAt hg.contDiffWithinAt uniqueDiffOn_univ
    hmn (mem_univ x₀) ?_).contDiffAt univ_mem
  rw [preimage_univ]

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

Goal state before:
pp: case pos
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
s : Set 𝕜
x : 𝕜
t : Set F
ht : ¬AccPt (f x) (𝓟 t)
hf : DifferentiableWithinAt 𝕜 f s x
h : derivWithin f s x ≠ 0
⊢ ∀ᶠ (x : 𝕜) in 𝓝[s \ {x}] x, f x ∉ t

Tactic:
exact hf.hasDerivWithinAt.eventually_notMem h t ht

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
x : X
h : e.IsImage s t
hx : x ∈ e.source
⊢ map (↑e) (𝓝[s] x) = 𝓝[t] ↑e x

Tactic:
rw [e.map_nhdsWithin_eq hx, h.image_eq, e.nhdsWithin_target_inter (e.map_source hx)]

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

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
f : 𝕜 → F
s : Set 𝕜
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
⊢ EqOn (iteratedDeriv n f) (deriv^[n] f) s

Tactic:
rw [iteratedDeriv_eq_iterate]

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
f : 𝕜 → F
s : Set 𝕜
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
⊢ EqOn (deriv^[n] f) (deriv^[n] f) s
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: case pos
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✝ c : OrderedFinpartition n
k m : Fin c.length
h : m = k
⊢ Fin (update c.partSize k (c.partSize k + 1) m) → Fin (n + 1)

Tactic:
have : update c.partSize k (c.partSize k + 1) m = c.partSize k + 1 := by rw [h]; simp

Goal state after:
case pos
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✝ c : OrderedFinpartition n
k m : Fin c.length
h : m = k
this : update c.partSize k (c.partSize k + 1) m = c.partSize k + 1
⊢ Fin (update c.partSize k (c.partSize k + 1) m) → Fin (n + 1)
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Defs

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
⊢ iteratedDeriv n 0 x = 0

Tactic:
by
  simp [Pi.zero_def]

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

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
⊢ iteratedDerivWithin n 0 s x = 0

Tactic:
by
  simp [Pi.zero_def]

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

Goal state before:
pp: case inl
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✝ c : OrderedFinpartition n
m : Fin c.length
⊢ StrictMono
    (if h : m = m then
      have this := ⋯;
      (fun i ↦ cases 0 (succ ∘ c.emb m) i) ∘ Fin.cast this
    else
      have this := ⋯;
      succ ∘ c.emb m ∘ Fin.cast this)

pp: case inr
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✝ c : OrderedFinpartition n
k m : Fin c.length
hm : m ≠ k
⊢ StrictMono
    (if h : m = k then
      have this := ⋯;
      (fun i ↦ cases 0 (succ ∘ c.emb k) i) ∘ Fin.cast this
    else
      have this := ⋯;
      succ ∘ c.emb m ∘ Fin.cast this)

Tactic:
· suffices ∀ (a' b' : Fin (c.partSize m + 1)), a' < b' →
          (cases (motive := fun _ ↦ Fin (n + 1)) 0 (succ ∘ c.emb m)) a' <
          (cases (motive := fun _ ↦ Fin (n + 1)) 0 (succ ∘ c.emb m)) b' by
        simp only [↓reduceDIte]
        intro a b hab
        exact this _ _ hab
      intro a' b' h'
      induction b' using Fin.induction with
      | zero => simp at h'
      | succ b =>
        induction a' using Fin.induction with
        | zero => simp
        | succ a' =>
          simp only [cases_succ, comp_apply, succ_lt_succ_iff]
          exact c.emb_strictMono m (by simpa using h')

Goal state after:
case inr
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✝ c : OrderedFinpartition n
k m : Fin c.length
hm : m ≠ k
⊢ StrictMono
    (if h : m = k then
      have this := ⋯;
      (fun i ↦ cases 0 (succ ∘ c.emb k) i) ∘ Fin.cast this
    else
      have this := ⋯;
      succ ∘ c.emb m ∘ Fin.cast this)
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: case inl.refine_2
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
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) ≠ {0}
x : Fin n
j : Fin (c.partSize (c.index 0))
hij : c.emb (c.index 0) j = x.succ
j_ne : j ≠ 0
je_ne' : ↑j ≠ 0
A : c.partSize (c.index 0) - 1 + 1 = c.partSize (c.index 0)
B : update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) (c.index 0) = c.partSize (c.index 0) - 1
this : x = x.succ.pred ⋯
⊢ (c.emb (c.index 0) (Fin.cast ⋯ (Fin.cast ⋯ ((Fin.cast ⋯ j).pred ⋯)).succ)).pred ⋯ = x.succ.pred ⋯

Tactic:
simp only [pred_inj, ← hij]

Goal state after:
case inl.refine_2
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
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) ≠ {0}
x : Fin n
j : Fin (c.partSize (c.index 0))
hij : c.emb (c.index 0) j = x.succ
j_ne : j ≠ 0
je_ne' : ↑j ≠ 0
A : c.partSize (c.index 0) - 1 + 1 = c.partSize (c.index 0)
B : update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) (c.index 0) = c.partSize (c.index 0) - 1
this : x = x.succ.pred ⋯
⊢ c.emb (c.index 0) (Fin.cast ⋯ (Fin.cast ⋯ ((Fin.cast ⋯ j).pred ⋯)).succ) = c.emb (c.index 0) j
Module: Mathlib.Analysis.Complex.RealDeriv

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
s : Set ℂ
x : ℂ
f' : E
h : HasDerivWithinAt f f' s x
⊢ HasFDerivWithinAt f (reCLM.smulRight f' + I • imCLM.smulRight f') s x

Tactic:
by
  simpa only [Complex.restrictScalars_toSpanSingleton'] using h.hasFDerivWithinAt.restrictScalars ℝ

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

Goal state before:
pp: case inr
𝔸 : Type u_2
inst✝³ : Ring 𝔸
inst✝² : TopologicalSpace 𝔸
inst✝¹ : IsTopologicalRing 𝔸
inst✝ : T2Space 𝔸
x : 𝔸
val✝ : Algebra ℚ 𝔸
⊢ ∑' (n : ℕ), MulOpposite.op ((↑n !)⁻¹ • x ^ n) = MulOpposite.op (∑' (n : ℕ), (↑n !)⁻¹ • x ^ n)

Tactic:
tsum_op

Goal state after:
Module: Mathlib.Topology.MetricSpace.CauSeqFilter

Goal state before:
pp: β : Type v
inst✝¹ : NormedField β
inst✝ : CauSeq.IsComplete β norm
u : ℕ → β
hu : CauchySeq u
C : IsCauSeq norm u
ε : ℝ
⊢ ε > 0 → ∃ N, ∀ n ≥ N, dist (u n) (CauSeq.lim ⟨u, C⟩) < ε

Tactic:
εpos

Goal state after:
β : Type v
inst✝¹ : NormedField β
inst✝ : CauSeq.IsComplete β norm
u : ℕ → β
hu : CauchySeq u
C : IsCauSeq norm u
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, dist (u n) (CauSeq.lim ⟨u, C⟩) < ε
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
⊢ HasFDerivAt exp 1 (0 • x)

Tactic:
zero_smul

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 :
  (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
⊢ HasFDerivAt exp 1 0
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
s : Set E
a y : E
h : IsLocalMaxOn f s a
hy : y ∈ posTangentConeAt s a
hf : ¬DifferentiableWithinAt ℝ f s a
⊢ (fderivWithin ℝ f s a) y ≤ 0

Tactic:
rw [fderivWithin_zero_of_not_differentiableWithinAt hf]; rfl

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

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
s : Set E
a y : E
h : IsLocalMaxOn f s a
hy : y ∈ posTangentConeAt s a
hy' : -y ∈ posTangentConeAt s a
hf : ¬DifferentiableWithinAt ℝ f s a
⊢ (fderivWithin ℝ f s a) y = 0

Tactic:
rw [fderivWithin_zero_of_not_differentiableWithinAt hf]; rfl

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

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
s : Set E
a y : E
h : IsLocalMinOn f s a
hy : y ∈ posTangentConeAt s a
hf : ¬DifferentiableWithinAt ℝ f s a
⊢ 0 ≤ (fderivWithin ℝ f s a) y

Tactic:
rw [fderivWithin_zero_of_not_differentiableWithinAt hf]; rfl

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

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
s : Set E
a y : E
h : IsLocalMinOn f s a
hy : y ∈ posTangentConeAt s a
hy' : -y ∈ posTangentConeAt s a
hf : ¬DifferentiableWithinAt ℝ f s a
⊢ (fderivWithin ℝ f s a) y = 0

Tactic:
rw [fderivWithin_zero_of_not_differentiableWithinAt hf]; rfl

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝⁶ : ConditionallyCompleteLinearOrder X
inst✝⁵ : DenselyOrdered X
inst✝⁴ : TopologicalSpace X
inst✝³ : OrderTopology X
inst✝² : LinearOrder Y
inst✝¹ : TopologicalSpace Y
inst✝ : OrderTopology Y
f : X → Y
a b : X
l : Y
hab : a < b
hfc : ContinuousOn f (Ioo a b)
ha : Tendsto f (𝓝[>] a) (𝓝 l)
hb : Tendsto f (𝓝[<] b) (𝓝 l)
h : EqOn (extendFrom (Ioo a b) f) f (Ioo a b)
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Ioo a b) c

Tactic:
obtain ⟨c, hc, hfc⟩ : ∃ c ∈ Ioo a b, IsExtrOn (extendFrom (Ioo a b) f) (Icc a b) c :=
    exists_Ioo_extr_on_Icc hab (continuousOn_Icc_extendFrom_Ioo hfc ha hb)
      ((eq_lim_at_left_extendFrom_Ioo hab ha).trans (eq_lim_at_right_extendFrom_Ioo hab hb).symm)

Goal state after:
X : Type u_1
Y : Type u_2
inst✝⁶ : ConditionallyCompleteLinearOrder X
inst✝⁵ : DenselyOrdered X
inst✝⁴ : TopologicalSpace X
inst✝³ : OrderTopology X
inst✝² : LinearOrder Y
inst✝¹ : TopologicalSpace Y
inst✝ : OrderTopology Y
f : X → Y
a b : X
l : Y
hab : a < b
hfc✝ : ContinuousOn f (Ioo a b)
ha : Tendsto f (𝓝[>] a) (𝓝 l)
hb : Tendsto f (𝓝[<] b) (𝓝 l)
h : EqOn (extendFrom (Ioo a b) f) f (Ioo a b)
c : X
hc : c ∈ Ioo a b
hfc : IsExtrOn (extendFrom (Ioo a b) f) (Icc a b) c
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Ioo a b) c
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
hcont_at_a : ¬ContinuousWithinAt f (Ici a) a
hcontra : DifferentiableWithinAt ℝ f (Ioi a) a
this : ContinuousWithinAt f (Ioi a) a
⊢ False

Tactic:
← ContinuousWithinAt.sdiff_iff this

Goal state after:
f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
hcont_at_a : ¬ContinuousWithinAt f (Ici a \ Ioi a) a
hcontra : DifferentiableWithinAt ℝ f (Ioi a) a
this : ContinuousWithinAt f (Ioi a) a
⊢ False
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: x : ℝ
h : |x| < 1
n : ℕ
F : ℝ → ℝ := fun x ↦ 1 / 2 * log ((1 + x) / (1 - x)) - ∑ i ∈ Finset.range n, x ^ (2 * i + 1) / (2 * ↑i + 1)
F' : ℝ → ℝ := fun y ↦ (y ^ 2) ^ n / (1 - y ^ 2)
hI : Set.Icc (-|x|) |x| ⊆ Set.Ioo (-1) 1
A : ∀ y ∈ Set.Ioo (-1) 1, HasDerivAt F (F' y) y
y : ℝ
hy : y ∈ Set.Icc (-|x|) |x|
this : y ^ 2 ≤ x ^ 2
⊢ ‖F' y‖ = (y ^ 2) ^ n / |1 - y ^ 2|

Tactic:
by simp [F']

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
s : PointedCone ℝ E
f : E →ₗ.[ℝ] ℝ
nonneg : ∀ (x : ↥f.domain), ↑x ∈ s → 0 ≤ ↑f x
dense : ∀ (y : E), ∃ x, ↑x + y ∈ s
hdom : f.domain ≠ ⊤
y : E
hy : y ∉ f.domain
c : ℝ
le_c : ∀ (x : ↥f.domain), -↑x - y ∈ s → ↑f x ≤ c
c_le : ∀ (x : ↥f.domain), ↑x + y ∈ s → c ≤ ↑f x
x : E
hx : x ∈ f.domain
r : ℝ
hy' : r • y ∈ (LinearPMap.mkSpanSingleton y (-c) ⋯).domain
hz : x + r • y ∈ (f.supSpanSingleton y (-c) hy).domain
hzs : x + r • y ∈ s
hr : r < 0
⊢ -(r⁻¹ • x) - y ∈ s

Tactic:
rwa [← s.smul_mem_iff (neg_pos.2 hr), smul_sub, smul_neg, neg_smul, neg_neg, smul_smul,
          mul_inv_cancel₀ hr.ne, one_smul, sub_eq_add_neg, neg_smul, neg_neg]

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
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 := fun e ↦ L e (n e) (n e)
ε : ℝ
⊢ ε > 0 → ∃ N, ∀ n ≥ N, dist (L0 n) (L0 N) < ε

Tactic:
εpos

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
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 := fun e ↦ L e (n e) (n e)
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, dist (L0 n) (L0 N) < ε
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: a : ℝ
h : 0 ≤ a
⊢ HasSum (fun k ↦ 2 * (1 / (2 * ↑k + 1)) * (a / (a + 2)) ^ (2 * k + 1)) (log (1 + a))

Tactic:
obtain (rfl | ha0) := eq_or_ne a 0

Goal state after:
case inl
h : 0 ≤ 0
⊢ HasSum (fun k ↦ 2 * (1 / (2 * ↑k + 1)) * (0 / (0 + 2)) ^ (2 * k + 1)) (log (1 + 0))

case inr
a : ℝ
h : 0 ≤ a
ha0 : a ≠ 0
⊢ HasSum (fun k ↦ 2 * (1 / (2 * ↑k + 1)) * (a / (a + 2)) ^ (2 * k + 1)) (log (1 + a))
Module: Mathlib.Analysis.Convex.Cone.Extension

Goal state before:
pp: case refine_2.inr.inl
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : PointedCone ℝ E
f : E →ₗ.[ℝ] ℝ
nonneg : ∀ (x : ↥f.domain), ↑x ∈ s → 0 ≤ ↑f x
dense : ∀ (y : E), ∃ x, ↑x + y ∈ s
hdom : f.domain ≠ ⊤
y : E
hy : y ∉ f.domain
c : ℝ
le_c : ∀ (x : ↥f.domain), -↑x - y ∈ s → ↑f x ≤ c
c_le : ∀ (x : ↥f.domain), ↑x + y ∈ s → c ≤ ↑f x
x : E
hx : x ∈ f.domain
hy' : 0 • y ∈ (LinearPMap.mkSpanSingleton y (-c) ⋯).domain
hz : x + 0 • y ∈ (f.supSpanSingleton y (-c) hy).domain
hzs : x ∈ s
⊢ 0 • c ≤ ↑f ⟨x, hx⟩

Tactic:
zero_smul

Goal state after:
case refine_2.inr.inl
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : PointedCone ℝ E
f : E →ₗ.[ℝ] ℝ
nonneg : ∀ (x : ↥f.domain), ↑x ∈ s → 0 ≤ ↑f x
dense : ∀ (y : E), ∃ x, ↑x + y ∈ s
hdom : f.domain ≠ ⊤
y : E
hy : y ∉ f.domain
c : ℝ
le_c : ∀ (x : ↥f.domain), -↑x - y ∈ s → ↑f x ≤ c
c_le : ∀ (x : ↥f.domain), ↑x + y ∈ s → c ≤ ↑f x
x : E
hx : x ∈ f.domain
hy' : 0 • y ∈ (LinearPMap.mkSpanSingleton y (-c) ⋯).domain
hz : x + 0 • y ∈ (f.supSpanSingleton y (-c) hy).domain
hzs : x ∈ s
⊢ 0 ≤ ↑f ⟨x, hx⟩