mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Analytic.Composition

Goal state before:
pp: case h₀
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
p : FormalMultilinearSeries 𝕜 E F
x : E
n : ℕ
b : Composition n
a✝ : b ∈ Finset.univ
hb : b ≠ Composition.ones n
k : ℕ
hk : k ∈ b.blocks
lt_k : 1 < k
i : Fin b.blocks.length
hi : b.blocks[i] = k
⊢ p.compAlongComposition (id 𝕜 E x) b = 0

Tactic:
let j : Fin b.length := ⟨i.val, b.blocks_length ▸ i.prop⟩

Goal state after:
case h₀
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
p : FormalMultilinearSeries 𝕜 E F
x : E
n : ℕ
b : Composition n
a✝ : b ∈ Finset.univ
hb : b ≠ Composition.ones n
k : ℕ
hk : k ∈ b.blocks
lt_k : 1 < k
i : Fin b.blocks.length
hi : b.blocks[i] = k
j : Fin b.length := ⟨↑i, ⋯⟩
⊢ p.compAlongComposition (id 𝕜 E x) b = 0
Module: Mathlib.Analysis.Analytic.Inverse

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
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
n : ℕ
v : Fin n → F
⊢ (p.comp (p.rightInv i x) n) v = (id 𝕜 F ((p 0) 0) n) v

Tactic:
match n with
  | 0 =>
    simp only [comp_coeff_zero', Matrix.zero_empty, id_apply_zero]
    congr
    ext i
    exact i.elim0
  | 1 =>
    simp only [comp_coeff_one, h, rightInv_coeff_one, ContinuousLinearEquiv.apply_symm_apply,
      id_apply_one, ContinuousLinearEquiv.coe_apply, continuousMultilinearCurryFin1_symm_apply]
  | n + 2 =>
    have N : 0 < n + 2 := by simp
    simp [comp_rightInv_aux1 N, h, rightInv, comp_rightInv_aux2, -Set.toFinset_ofPred]

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

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
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
⊢ ((p.leftInv i x).comp p).comp (p.rightInv i x) = (id 𝕜 E x).comp (p.rightInv i x)

Tactic:
leftInv_comp p i _ h

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
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
⊢ (id 𝕜 E x).comp (p.rightInv i x) = (id 𝕜 E x).comp (p.rightInv i x)
Module: Mathlib.Analysis.Analytic.Inverse

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
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
hp : 0 < p.radius
C r : ℝ
Cpos : 0 < C
rpos : 0 < r
ple : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖↑i.symm‖
a : ℝ
apos : 0 < a
ha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a ≤ 1
ha2 : r * (I + 1) * a ≤ 1 / 2
S : ℕ → ℝ := fun n ↦ ∑ k ∈ Ico 1 n, a ^ k * ‖p.rightInv i x k‖
IRec : ∀ (n : ℕ), 1 ≤ n → S n ≤ (I + 1) * a
a' : NNReal := ⟨a, ⋯⟩
H : ↑a' ≤ (p.rightInv i x).radius
⊢ 0 < (p.rightInv i x).radius

Tactic:
by
    apply lt_of_lt_of_le _ H
    -- Prior to https://github.com/leanprover/lean4/pull/2734, this was `exact_mod_cast apos`.
    simpa only [ENNReal.coe_pos]

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

Goal state before:
pp: n : ℕ
v : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i))
a : Composition n
b : (i : Fin a.length) → Composition (a.blocksFun i)
H : (ofFn fun i ↦ (⟨a, b⟩.snd i).blocks) = ofFn fun i ↦ (v.snd i).blocks
⊢ ⟨a, b⟩ = v

Tactic:
rcases v with ⟨a', b'⟩

Goal state after:
n : ℕ
a : Composition n
b : (i : Fin a.length) → Composition (a.blocksFun i)
a' : Composition n
b' : (i : Fin a'.length) → Composition (a'.blocksFun i)
H : (ofFn fun i ↦ (⟨a, b⟩.snd i).blocks) = ofFn fun i ↦ (⟨a', b'⟩.snd i).blocks
⊢ ⟨a, b⟩ = ⟨a', b'⟩
Module: Mathlib.Analysis.Analytic.Composition

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
r : FormalMultilinearSeries 𝕜 G H
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
v : Fin n → E
f : (a : Composition n) × Composition a.length → H :=
  fun c ↦ (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))
g : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=
  fun c ↦ (r c.fst.length) fun i ↦ (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))
⊢ ((r.comp q).comp p n) v = (r.comp (q.comp p) n) v

Tactic:
suffices ∑ c, f c = ∑ c, g c by
    simpa +unfoldPartialApp only [FormalMultilinearSeries.comp, sum_apply,
      compAlongComposition_apply, Finset.sum_sigma', applyComposition,
      ContinuousMultilinearMap.map_sum]

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
r : FormalMultilinearSeries 𝕜 G H
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
v : Fin n → E
f : (a : Composition n) × Composition a.length → H :=
  fun c ↦ (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))
g : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=
  fun c ↦ (r c.fst.length) fun i ↦ (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))
⊢ ∑ c, f c = ∑ c, g c
Module: Mathlib.Analysis.Calculus.FDeriv.Equiv

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
f' : E →L[𝕜] F
x : E
s : Set E
c : F
h : HasFDerivWithinAt f f' s x
hf' : ∃ C, AntilipschitzWith C ⇑f'
⊢ ∀ᶠ (b : F) in map f (𝓝[s \ {x}] x), b ≠ c

Tactic:
apply Eventually.filter_mono (h.tendsto_nhdsWithin_nhdsNE hf')

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
f' : E →L[𝕜] F
x : E
s : Set E
c : F
h : HasFDerivWithinAt f f' s x
hf' : ∃ C, AntilipschitzWith C ⇑f'
⊢ ∀ᶠ (x : F) in 𝓝[≠] f x, x ≠ c
Module: Mathlib.Analysis.Calculus.FDeriv.Prod

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
ι : Type u_6
F' : ι → Type u_7
inst✝¹ : (i : ι) → NormedAddCommGroup (F' i)
inst✝ : (i : ι) → NormedSpace 𝕜 (F' i)
i : ι
⊢ Differentiable 𝕜 fun f ↦ f i

Tactic:
intro x; apply differentiableAt_apply

Goal state after:
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
f₁ f₂ : E → F
x : E
h : f₁ =ᶠ[𝓝 x] f₂
n : ℕ
⊢ iteratedFDeriv 𝕜 n f₁ =ᶠ[𝓝 x] iteratedFDeriv 𝕜 n f₂

Tactic:
simp_all [← nhdsWithin_univ, ← iteratedFDerivWithin_univ, EventuallyEq.iteratedFDerivWithin]

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

Goal state before:
pp: 𝕜 : Type u_1
𝔸 : Type u_2
E : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedRing 𝔸
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAlgebra 𝕜 𝔸
inst✝ : NormedSpace 𝕜 E
f : E → 𝔸
f' : E →L[𝕜] 𝔸
x : E
n : ℕ
⊢ ∑ x_1 ∈ Finset.range (n + 1), f x •> (f x ^ ((n + 1).pred - x_1) •> f' <• f x ^ x_1) + f' <• f x ^ (n + 1) =
    ∑ x_1 ∈ Finset.range (n + 1), f x ^ ((n + 1 + 1).pred - x_1) •> f' <• f x ^ x_1 +
      f x ^ ((n + 1 + 1).pred - (n + 1)) •> f' <• f x ^ (n + 1)

Tactic:
simp only [Nat.pred_eq_sub_one, add_tsub_cancel_right, tsub_self, pow_zero, one_smul]

Goal state after:
𝕜 : Type u_1
𝔸 : Type u_2
E : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedRing 𝔸
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAlgebra 𝕜 𝔸
inst✝ : NormedSpace 𝕜 E
f : E → 𝔸
f' : E →L[𝕜] 𝔸
x : E
n : ℕ
⊢ ∑ x_1 ∈ Finset.range (n + 1), f x •> (f x ^ (n - x_1) •> f' <• f x ^ x_1) + f' <• f x ^ (n + 1) =
    ∑ x_1 ∈ Finset.range (n + 1), f x ^ (n + 1 - x_1) •> f' <• f x ^ x_1 + f' <• f x ^ (n + 1)
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
f : 𝕜 → F
f' : F
x : 𝕜
𝕜' : Type u_2
inst✝⁴ : NormedRing 𝕜'
inst✝³ : NormedAlgebra 𝕜 𝕜'
inst✝² : Module 𝕜' F
inst✝¹ : IsBoundedSMul 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
c : 𝕜 → 𝕜'
c' : 𝕜'
hc : HasStrictDerivAt c c' x
hf : HasStrictDerivAt f f' x
⊢ HasStrictDerivAt (c • f) (c x • f' + c' • f x) x

Tactic:
by
  simpa using (HasStrictFDerivAt.smul hc hf).hasStrictDerivAt

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

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
L : Filter (𝕜 × 𝕜)
ι : Type u_1
u : Finset ι
A : ι → 𝕜 → F
A' : ι → F
h : ∀ i ∈ u, HasDerivAtFilter (A i) (A' i) L
⊢ HasDerivAtFilter (∑ i ∈ u, A i) (∑ i ∈ u, A' i) L

Tactic:
convert! HasDerivAtFilter.fun_sum h

Goal state after:
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
L : Filter (𝕜 × 𝕜)
ι : Type u_1
u : Finset ι
A : ι → 𝕜 → F
A' : ι → F
h : ∀ i ∈ u, HasDerivAtFilter (A i) (A' i) L
x✝ : 𝕜
⊢ (∑ i ∈ u, A i) x✝ = ∑ i ∈ u, A i x✝
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
x : 𝕜
G : Type u_2
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
c : 𝕜 → F →L[𝕜] G
c' : F →L[𝕜] G
u : 𝕜 → F
u' : F
hc : HasDerivAt c c' x
hu : HasDerivAt u u' x
⊢ HasDerivAt (fun y ↦ (c y) (u y)) (c' (u x) + (c x) u') x

Tactic:
simpa [add_comm] using (hc.hasFDerivAt.clm_apply hu.hasFDerivAt).hasDerivAt

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Slope

Goal state before:
pp: case inl
k : Type u_1
E : Type u_2
inst✝⁷ : Field k
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module k E
inst✝⁴ : LinearOrder k
inst✝³ : IsStrictOrderedRing k
inst✝² : PartialOrder E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : PosSMulMono k E
f : k → E
x y : k
s : Set k
hf : StrictMonoOn f s
hx : x ∈ s
hy : y ∈ s
hxy✝ : x ≠ y
hxy : x < y
⊢ 0 < slope f x y

Tactic:
exact (slope_pos_iff_of_le hxy.le).mpr (hf hx hy hxy)

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

Goal state before:
pp: case mp
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
𝕜' : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜'
inst✝⁸ : NormedAlgebra 𝕜 𝕜'
E : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedSpace 𝕜' E
inst✝⁴ : IsScalarTower 𝕜 𝕜' E
F : Type u_4
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
f : E → F
s : Set E
x : E
hf : DifferentiableWithinAt 𝕜 f s x
hs : UniqueDiffWithinAt 𝕜 s x
g' : E →L[𝕜'] F
hg' : HasFDerivWithinAt f g' s x
⊢ ∃ g', restrictScalars 𝕜 g' = fderivWithin 𝕜 f s x

Tactic:
exact ⟨g', hs.eq (hg'.restrictScalars 𝕜) hf.hasFDerivWithinAt⟩

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

Goal state before:
pp: case succ
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
f : E → F
x : E
r : ℝ≥0∞
h : HasFPowerSeriesOnBall f p x r
y : E
inst✝ : CompleteSpace F
n✝ : ℕ
⊢ ((n✝ + 1)! • (p (n✝ + 1)) fun x ↦ y) = (iteratedFDeriv 𝕜 (n✝ + 1) f x) fun x ↦ y

Tactic:
rw [factorial_succ, mul_comm, mul_smul, ← derivSeries_apply_diag,
      ← _root_.smul_apply, factorial_smul' _ h.fderiv, iteratedFDeriv_succ_apply_right]

Goal state after:
case succ
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
f : E → F
x : E
r : ℝ≥0∞
h : HasFPowerSeriesOnBall f p x r
y : E
inst✝ : CompleteSpace F
n✝ : ℕ
⊢ ((iteratedFDeriv 𝕜 n✝ (fderiv 𝕜 f) x) fun x ↦ y) y =
    ((iteratedFDeriv 𝕜 n✝ (fun y ↦ fderiv 𝕜 f y) x) (Fin.init fun x ↦ y)) y
Module: Mathlib.Analysis.Calculus.Deriv.Slope

Goal state before:
pp: 𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
x : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
inst✝ : OrderTopology 𝕜
g : 𝕜 → 𝕜
g' : 𝕜
hd : HasDerivAt g g' x
hg : Monotone g
⊢ 0 ≤ g'

Tactic:
by
  rw [← hasDerivWithinAt_univ] at hd
  apply hd.nonneg_of_monotoneOn _ (hg.monotoneOn _)
  exact PerfectSpace.univ_preperfect _ (mem_univ _)

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

Goal state before:
pp: 𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
inst✝ : OrderTopology 𝕜
g : 𝕜 → 𝕜
g' : 𝕜
hx : AccPt x (𝓟 s)
hd : HasDerivWithinAt g g' s x
hg : AntitoneOn g s
this : MonotoneOn (-g) s
⊢ g' ≤ 0

Tactic:
simpa using hd.neg.nonneg_of_monotoneOn hx this

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

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
f' : E
s : Set ℝ
x r : ℝ
hf : HasDerivWithinAt f f' s x
hr : ‖f'‖ < r
hr₀ : 0 < r
A : ∀ᶠ (z : ℝ) in 𝓝[s \ {x}] x, ‖(z - x)⁻¹ • (f z - f x)‖ ∈ Iio r
B : ∀ᶠ (z : ℝ) in 𝓝[{x}] x, ‖(z - x)⁻¹ • (f z - f x)‖ ∈ Iio r
C : {x_1 | (fun z ↦ ‖(z - x)⁻¹ • (f z - f x)‖ ∈ Iio r) x_1} ∈ 𝓝[s \ {x}] x ⊔ 𝓝[{x}] x
⊢ ∀ᶠ (z : ℝ) in 𝓝[s] x, ‖z - x‖⁻¹ * ‖f z - f x‖ < r

Tactic:
rw [← nhdsWithin_union, sdiff_union_self, nhdsWithin_union, mem_sup] at C

Goal state after:
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
f' : E
s : Set ℝ
x r : ℝ
hf : HasDerivWithinAt f f' s x
hr : ‖f'‖ < r
hr₀ : 0 < r
A : ∀ᶠ (z : ℝ) in 𝓝[s \ {x}] x, ‖(z - x)⁻¹ • (f z - f x)‖ ∈ Iio r
B : ∀ᶠ (z : ℝ) in 𝓝[{x}] x, ‖(z - x)⁻¹ • (f z - f x)‖ ∈ Iio r
C :
  {x_1 | (fun z ↦ ‖(z - x)⁻¹ • (f z - f x)‖ ∈ Iio r) x_1} ∈ 𝓝[s] x ∧
    {x_1 | (fun z ↦ ‖(z - x)⁻¹ • (f z - f x)‖ ∈ Iio r) x_1} ∈ 𝓝[{x}] x
⊢ ∀ᶠ (z : ℝ) in 𝓝[s] x, ‖z - x‖⁻¹ * ‖f z - f x‖ < r
Module: Mathlib.Analysis.Calculus.Deriv.Comp

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
h : 𝕜 → 𝕜'
h₂ : 𝕜' → 𝕜'
h' h₂' y : 𝕜'
hh₂ : HasDerivAt h₂ h₂' y
hh : HasDerivAt h h' x
hy : y = h x
⊢ HasDerivAt (h₂ ∘ h) (h₂' * h') x

Tactic:
by
  rw [hy] at hh₂; exact hh₂.comp x hh

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

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
h : 𝕜 → 𝕜'
h₂ : 𝕜' → 𝕜'
h' h₂' y : 𝕜'
hh₂ : HasStrictDerivAt h₂ h₂' y
hh : HasStrictDerivAt h h' x
hy : y = h x
⊢ HasStrictDerivAt (h₂ ∘ h) (h₂' * h') x

Tactic:
by
  rw [hy] at hh₂; exact hh₂.comp x hh

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

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
U : Set 𝕜
A : Type u_3
inst✝⁶ : NormedRing A
inst✝⁵ : IsDomain A
inst✝⁴ : NormedAlgebra 𝕜 A
B : Type u_4
inst✝³ : NormedAddCommGroup B
inst✝² : NormedSpace 𝕜 B
inst✝¹ : Module A B
inst✝ : IsTorsionFree A B
f : 𝕜 → A
g : 𝕜 → B
hf : AnalyticOnNhd 𝕜 f U
hg : AnalyticOnNhd 𝕜 g U
hfg : ∀ z ∈ U, f z • g z = 0
hU : IsPreconnected U
hU' : ¬U = ∅
z : 𝕜
hz : z ∈ U
hU'' : U.Nontrivial
this : ∃ᶠ (w : 𝕜) in 𝓝[≠] z, w ∈ U
⊢ (∀ z ∈ U, f z = 0) ∨ ∀ z ∈ U, g z = 0

Tactic:
have : ∃ᶠ w in 𝓝[≠] z, f w = 0 ∨ g w = 0 :=
    this.mp <| by filter_upwards with w hw using smul_eq_zero.mp (hfg w hw)

Goal state after:
case neg
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
U : Set 𝕜
A : Type u_3
inst✝⁶ : NormedRing A
inst✝⁵ : IsDomain A
inst✝⁴ : NormedAlgebra 𝕜 A
B : Type u_4
inst✝³ : NormedAddCommGroup B
inst✝² : NormedSpace 𝕜 B
inst✝¹ : Module A B
inst✝ : IsTorsionFree A B
f : 𝕜 → A
g : 𝕜 → B
hf : AnalyticOnNhd 𝕜 f U
hg : AnalyticOnNhd 𝕜 g U
hfg : ∀ z ∈ U, f z • g z = 0
hU : IsPreconnected U
hU' : ¬U = ∅
z : 𝕜
hz : z ∈ U
hU'' : U.Nontrivial
this✝ : ∃ᶠ (w : 𝕜) in 𝓝[≠] z, w ∈ U
this : ∃ᶠ (w : 𝕜) in 𝓝[≠] z, f w = 0 ∨ g w = 0
⊢ (∀ z ∈ U, f z = 0) ∨ ∀ z ∈ U, g z = 0
Module: Mathlib.Analysis.Calculus.MeanValue

Goal state before:
pp: case inr
f 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, 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
r : ℝ
hf' : ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
hfr : f' x < r
hrB : r < B' x
HB : ∀ᶠ (z : ℝ) in 𝓝[>] x, r < slope B x z
z : ℝ
hfz : slope f x z < r
hzB : r < slope B x z
hz : z ∈ Ioc x y
this : f z - B x ≤ B z - B x
⊢ z ∈ {x | f x ≤ B x}

Tactic:
sub_le_sub_iff_right

Goal state after:
case inr
f 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, 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
r : ℝ
hf' : ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
hfr : f' x < r
hrB : r < B' x
HB : ∀ᶠ (z : ℝ) in 𝓝[>] x, r < slope B x z
z : ℝ
hfz : slope f x z < r
hzB : r < slope B x z
hz : z ∈ Ioc x y
this : f z ≤ B z
⊢ z ∈ {x | f x ≤ B x}
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

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
s : Set E
f : E → F
x : E
n : ℕ∞ω
h : ContDiffWithinAt 𝕜 n f s x
u : Set E
hu : u ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
H : HasFTaylorSeriesUpToOn 0 f p u
⊢ ContinuousWithinAt f s x

Tactic:
rw [mem_nhdsWithin_insert] at hu

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
s : Set E
f : E → F
x : E
n : ℕ∞ω
h : ContDiffWithinAt 𝕜 n f s x
u : Set E
hu : x ∈ u ∧ u ∈ 𝓝[s] x
p : E → FormalMultilinearSeries 𝕜 E F
H : HasFTaylorSeriesUpToOn 0 f p u
⊢ ContinuousWithinAt f s x
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: case inl
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
⊢ ContDiffWithinAt 𝕜 n f (insert x s) x ↔ ContDiffWithinAt 𝕜 n f s x

Tactic:
exact contDiffWithinAt_insert_self

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
f : 𝕜 → F
s : Set 𝕜
n : ℕ∞ω
m : ℕ
h : ContDiffOn 𝕜 n f s
hmn : ↑m ≤ n
hs : UniqueDiffOn 𝕜 s
⊢ ContinuousOn (iteratedDerivWithin m f s) s

Tactic:
by
  simpa only [iteratedDerivWithin_eq_equiv_comp, LinearIsometryEquiv.comp_continuousOn_iff] using
    h.continuousOn_iteratedFDerivWithin hmn hs

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

Goal state before:
pp: case refine_1.cont
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
h : ∀ (m : ℕ), AnalyticOn 𝕜 (iteratedFDerivWithin 𝕜 m f s) s
x : E
hx : x ∈ s
k : ℕ
a✝ : ↑k ≤ ω
⊢ ContinuousOn (fun x ↦ ftaylorSeriesWithin 𝕜 f s x k) s

Tactic:
exact (h k).continuousOn

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

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
s : Set E
f : E → F
n : ℕ
hs : UniqueDiffOn 𝕜 s
⊢ ContDiffOn 𝕜 (↑n) f s ↔
    (∀ m ≤ n, ContinuousOn (fun x ↦ iteratedFDerivWithin 𝕜 m f s x) s) ∧
      ∀ m < n, DifferentiableOn 𝕜 (fun x ↦ iteratedFDerivWithin 𝕜 m f s x) s

Tactic:
rw [← WithTop.coe_natCast, contDiffOn_iff_continuousOn_differentiableOn hs]

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
s : Set E
f : E → F
n : ℕ
hs : UniqueDiffOn 𝕜 s
⊢ ((∀ (m : ℕ), ↑m ≤ ↑n → ContinuousOn (fun x ↦ iteratedFDerivWithin 𝕜 m f s x) s) ∧
      ∀ (m : ℕ), ↑m < ↑n → DifferentiableOn 𝕜 (fun x ↦ iteratedFDerivWithin 𝕜 m f s x) s) ↔
    (∀ m ≤ n, ContinuousOn (fun x ↦ iteratedFDerivWithin 𝕜 m f s x) s) ∧
      ∀ m < n, DifferentiableOn 𝕜 (fun x ↦ iteratedFDerivWithin 𝕜 m f s x) s
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: case h
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
hs : UniqueDiffOn 𝕜 s
h : ContDiffAt 𝕜 (↑n) f x
hx : x ∈ s
u : Set E
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
xu : x ∈ u
hu : ContDiffOn 𝕜 (↑n) f (insert x univ ∩ u)
⊢ ContDiffOn 𝕜 (↑n) f (univ ∩ u)

pp: case hx
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
hs : UniqueDiffOn 𝕜 s
h : ContDiffAt 𝕜 (↑n) f x
hx : x ∈ s
u : Set E
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
xu : x ∈ u
hu : ContDiffOn 𝕜 (↑n) f (insert x univ ∩ u)
⊢ x ∈ s ∩ u

Tactic:
· simpa using hu

Goal state after:
case hx
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
hs : UniqueDiffOn 𝕜 s
h : ContDiffAt 𝕜 (↑n) f x
hx : x ∈ s
u : Set E
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
xu : x ∈ u
hu : ContDiffOn 𝕜 (↑n) f (insert x univ ∩ u)
⊢ x ∈ s ∩ u
Module: Mathlib.Analysis.Normed.Algebra.Exponential

Goal state before:
pp: 𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : Field 𝕂
inst✝³ : DivisionRing 𝔸
inst✝² : Algebra 𝕂 𝔸
inst✝¹ : TopologicalSpace 𝔸
inst✝ : IsTopologicalRing 𝔸
x : 𝔸
n : ℕ
⊢ ((expSeries 𝕂 𝔸 n) fun x_1 ↦ x) = (↑n !)⁻¹ * x ^ n

Tactic:
← smul_eq_mul,

Goal state after:
𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : Field 𝕂
inst✝³ : DivisionRing 𝔸
inst✝² : Algebra 𝕂 𝔸
inst✝¹ : TopologicalSpace 𝔸
inst✝ : IsTopologicalRing 𝔸
x : 𝔸
n : ℕ
⊢ ((expSeries 𝕂 𝔸 n) fun x_1 ↦ x) = (↑n !)⁻¹ • x ^ n
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: case e_f
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : 𝕜
s : Set 𝕜
f : 𝕜 → F
c : F
n : ℕ
hn : 0 < n.succ
y : 𝕜
⊢ derivWithin (fun z ↦ c + f z) s y = derivWithin f s y

Tactic:
exact derivWithin_const_add _

Goal state after: