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

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
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f p s x r
r' : ℝ≥0
hr : ↑r' < r
hr'0 : 0 < r'
a : ℝ
ha : a ∈ Ioo 0 1
C : ℝ
hC : 0 < C
hp : ∀ (n : ℕ), ‖p n‖ ≤ C * a ^ n / ↑r' ^ n
L : E × E → ℝ := fun y ↦ C * (a / ↑r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
y : E × E
hy' : y ∈ Metric.eball (x, x) ↑r'
ys : y ∈ insert x s ×ˢ insert x s
hy : y ∈ Metric.eball x r ×ˢ Metric.eball x r
A : ℕ → F := fun n ↦ ((p n) fun x_1 ↦ y.1 - x) - (p n) fun x_1 ↦ y.2 - x
⊢ ((p 1) fun x ↦ y.1 - y.2) = ∑ i ∈ Finset.range 2, (((p i) fun x_1 ↦ y.1 - x) - (p i) fun x_1 ↦ y.2 - x)

Tactic:
Finset.sum_range_succ,

Goal state after:
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
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f p s x r
r' : ℝ≥0
hr : ↑r' < r
hr'0 : 0 < r'
a : ℝ
ha : a ∈ Ioo 0 1
C : ℝ
hC : 0 < C
hp : ∀ (n : ℕ), ‖p n‖ ≤ C * a ^ n / ↑r' ^ n
L : E × E → ℝ := fun y ↦ C * (a / ↑r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
y : E × E
hy' : y ∈ Metric.eball (x, x) ↑r'
ys : y ∈ insert x s ×ˢ insert x s
hy : y ∈ Metric.eball x r ×ˢ Metric.eball x r
A : ℕ → F := fun n ↦ ((p n) fun x_1 ↦ y.1 - x) - (p n) fun x_1 ↦ y.2 - x
⊢ ((p 1) fun x ↦ y.1 - y.2) =
    ∑ x_1 ∈ Finset.range 1, (((p x_1) fun x_2 ↦ y.1 - x) - (p x_1) fun x_2 ↦ y.2 - x) +
      (((p 1) fun x_1 ↦ y.1 - x) - (p 1) fun x_1 ↦ y.2 - x)
Module: Mathlib.Analysis.Analytic.Basic

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
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f p s x r
r' : ℝ≥0
hr : ↑r' < r
hr'0 : 0 < r'
a : ℝ
ha : a ∈ Ioo 0 1
C : ℝ
hC : 0 < C
hp : ∀ (n : ℕ), ‖p n‖ ≤ C * a ^ n / ↑r' ^ n
L : E × E → ℝ := fun y ↦ C * (a / ↑r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
y : E × E
hy' : ‖y - (x, x)‖₊ < r'
ys : y ∈ insert x s ×ˢ insert x s
hy : y ∈ Metric.eball x r ×ˢ Metric.eball x r
A : ℕ → F := fun n ↦ ((p n) fun x_1 ↦ y.1 - x) - (p n) fun x_1 ↦ y.2 - x
hA : HasSum (fun n ↦ A (n + 2)) (f y.1 - f y.2 - (p 1) fun x ↦ y.1 - y.2)
B : ℕ → ℝ := fun n ↦ C * (a / ↑r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * ((↑n + 2) * a ^ n)
hAB : ∀ (n : ℕ), ‖A (n + 2)‖ ≤ B n
⊢ HasSum B (L y)

Tactic:
apply HasSum.mul_left

Goal state after:
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
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f p s x r
r' : ℝ≥0
hr : ↑r' < r
hr'0 : 0 < r'
a : ℝ
ha : a ∈ Ioo 0 1
C : ℝ
hC : 0 < C
hp : ∀ (n : ℕ), ‖p n‖ ≤ C * a ^ n / ↑r' ^ n
L : E × E → ℝ := ⋯
y : E × E
hy' : ‖y - (x, x)‖₊ < r'
ys : y ∈ insert x s ×ˢ insert x s
hy : y ∈ Metric.eball x r ×ˢ Metric.eball x r
A : ℕ → F := ⋯
hA : HasSum (fun n ↦ A (n + 2)) (f y.1 - f y.2 - (p 1) fun x ↦ y.1 - y.2)
B : ℕ → ℝ := ⋯
hAB : ∀ (n : ℕ), ‖A (n + 2)‖ ≤ B n
⊢ HasSum (fun i ↦ (↑i + 2) * a ^ i) (a / (1 - a) ^ 2 + 2 / (1 - a))
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
m M N n : ℕ
f : Fin n → ℕ
hi : (m ≤ ⟨n, f⟩.fst ∧ ⟨n, f⟩.fst < M) ∧ ∀ (a : Fin ⟨n, f⟩.fst), 1 ≤ ⟨n, f⟩.snd a ∧ ⟨n, f⟩.snd a < N
i : ℕ
hi' : i ∈ Set.range fun a ↦ f a
⊢ ∃ j, f j = i

Tactic:
Set.mem_range

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
m M N n : ℕ
f : Fin n → ℕ
hi : (m ≤ ⟨n, f⟩.fst ∧ ⟨n, f⟩.fst < M) ∧ ∀ (a : Fin ⟨n, f⟩.fst), 1 ≤ ⟨n, f⟩.snd a ∧ ⟨n, f⟩.snd a < N
i : ℕ
hi' : ∃ y, f y = i
⊢ ∃ j, f j = i
Module: Mathlib.Analysis.Analytic.Inverse

Goal state before:
pp: case a.a
n : ℕ
p : ℕ → ℝ
hp : ∀ (k : ℕ), 0 ≤ p k
r a : ℝ
hr : 0 ≤ r
ha : 0 ≤ a
k : ℕ
a✝¹ : k ∈ Ico 2 (n + 1)
c : Composition k
a✝ : c ∈ {c | 1 < c.length}.toFinset
⊢ a ^ k * (r ^ c.length * ∏ j, p (c.blocksFun j)) = (∏ x, r) * ((a ^ ∑ i, c.blocksFun i) * ∏ x, p (c.blocksFun x))

Tactic:
Composition.sum_blocksFun,

Goal state after:
case a.a
n : ℕ
p : ℕ → ℝ
hp : ∀ (k : ℕ), 0 ≤ p k
r a : ℝ
hr : 0 ≤ r
ha : 0 ≤ a
k : ℕ
a✝¹ : k ∈ Ico 2 (n + 1)
c : Composition k
a✝ : c ∈ {c | 1 < c.length}.toFinset
⊢ a ^ k * (r ^ c.length * ∏ j, p (c.blocksFun j)) = (∏ x, r) * (a ^ k * ∏ x, p (c.blocksFun x))
Module: Mathlib.Analysis.Analytic.Composition

Goal state before:
pp: case h
α : Type u_6
inst✝ : AddCommMonoid α
m M N : ℕ
f : (n : ℕ) × (Fin n → ℕ) → α
g : (n : ℕ) × Composition n → α
h : ∀ (e : (n : ℕ) × (Fin n → ℕ)) (he : e ∈ compPartialSumSource m M N), f e = g (compChangeOfVariables m M N e he)
⊢ ∀ (a : (n : ℕ) × (Fin n → ℕ)) (ha : a ∈ compPartialSumSource m M N), f a = g (compChangeOfVariables m M N a ha)

Tactic:
· assumption

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
x : E
h : {x} ∈ 𝓝[s] x
t : Set E
ot : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
xt : x ∈ t
r : ℝ
r0 : r > 0
rt : ball x r ⊆ t
y : E
yr : y ∈ eball 0 (ENNReal.ofReal r)
st : ∀ y ∈ t, y ∈ s → y = x
ys : x + y ∈ s
⊢ x + y ∈ ball x r

Tactic:
simpa using yr

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

Goal state before:

Tactic:
_ = fun x ↦ h.deriv (x.1 - x.2) (x.2.1, x.1.2) := by
      ext ⟨⟨x₁, y₁⟩, ⟨x₂, y₂⟩⟩
      rcases p with ⟨x, y⟩
      simp only [map_sub, deriv_apply, Function.comp_apply, Prod.mk_add_mk, h.add_right, h.add_left,
        Prod.mk_sub_mk, h.map_sub_left, h.map_sub_right, sub_add_sub_cancel]
      abel
    -- _ =O[𝓝 (0 : T)] fun x ↦ ‖x.1 - x.2‖ * ‖(x.2.1, x.1.2)‖ :=
    --     h.toContinuousLinearMap.deriv₂.isBoundedBilinearMap.isBigO_comp
    -- _ = o[𝓝 0] fun x ↦ ‖x.1 - x.2‖ * 1 := _
    _ =o[𝓝 (0 : T)] fun x ↦ x.1 - x.2 := by
      -- TODO : add 2 `calc` steps instead of the next 3 lines
      refine h.toContinuousLinearMap.deriv₂.isBoundedBilinearMap.isBigO_comp.trans_isLittleO ?_
      suffices (fun x : T ↦ ‖x.1 - x.2‖ * ‖(x.2.1, x.1.2)‖) =o[𝓝 0] fun x ↦ ‖x.1 - x.2‖ * 1 by
        simpa only [mul_one, isLittleO_norm_right] using this
      refine (isBigO_refl _ _).mul_isLittleO ((isLittleO_one_iff _).2 ?_)
      -- TODO: `continuity` fails
      exact (continuous_snd.fst.prodMk continuous_fst.snd).norm.tendsto' _ _ (by simp)
    _ = _ := by simp [T, Function.comp_def]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
B : E →L[𝕜] F →L[𝕜] G
f : G' → E
g : G' → F
f' : G' →L[𝕜] E
g' : G' →L[𝕜] F
x : G'
hf : HasFDerivAt f f' x
hg : HasFDerivAt g g' x
⊢ HasFDerivAt (fun y ↦ (B (f y)) (g y)) (((precompR G' B) (f x)) g' + ((precompL G' B) f') (g x)) x

Tactic:
exact (B.isBoundedBilinearMap.hasFDerivAt (f x, g x)).comp x (hf.prodMk hg)

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

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
iso : E ≃L[𝕜] F
f : G → E
x : G
f' : G →L[𝕜] E
⊢ HasStrictFDerivAt (⇑iso ∘ f) (↑iso ∘SL f') x ↔ HasStrictFDerivAt f f' x

Tactic:
refine ⟨fun H => ?_, fun H => iso.hasStrictFDerivAt.comp x H⟩

Goal state after:
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
iso : E ≃L[𝕜] F
f : G → E
x : G
f' : G →L[𝕜] E
H : HasStrictFDerivAt (⇑iso ∘ f) (↑iso ∘SL f') x
⊢ HasStrictFDerivAt f f' x
Module: Mathlib.Analysis.Calculus.FDeriv.Add

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
x : E
s : Set E
c : F
⊢ fderivWithin 𝕜 (fun y ↦ f y + c) s x = fderivWithin 𝕜 f s x

Tactic:
by
  classical simp [fderivWithin]

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

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
iso : E ≃L[𝕜] F
f : G → E
s : Set G
x : G
hxs : UniqueDiffWithinAt 𝕜 s x
h : DifferentiableWithinAt 𝕜 f s x
⊢ fderivWithin 𝕜 (⇑iso ∘ f) s x = ↑iso ∘SL fderivWithin 𝕜 f s x

pp: case neg
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
iso : E ≃L[𝕜] F
f : G → E
s : Set G
x : G
hxs : UniqueDiffWithinAt 𝕜 s x
h : ¬DifferentiableWithinAt 𝕜 f s x
⊢ fderivWithin 𝕜 (⇑iso ∘ f) s x = ↑iso ∘SL fderivWithin 𝕜 f s x

Tactic:
· rw [fderiv_comp_fderivWithin x iso.differentiableAt h hxs, iso.fderiv]

Goal state after:
case neg
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
iso : E ≃L[𝕜] F
f : G → E
s : Set G
x : G
hxs : UniqueDiffWithinAt 𝕜 s x
h : ¬DifferentiableWithinAt 𝕜 f s x
⊢ fderivWithin 𝕜 (⇑iso ∘ f) s x = ↑iso ∘SL fderivWithin 𝕜 f s x
Module: Mathlib.Analysis.Calculus.FDeriv.Equiv

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
iso : E ≃L[𝕜] F
f : F → G
⊢ Differentiable 𝕜 (f ∘ ⇑iso) ↔ Differentiable 𝕜 f

Tactic:
by
  simp only [← differentiableOn_univ, ← iso.comp_right_differentiableOn_iff, preimage_univ]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Alternating.Basic

Goal state before:
pp: 𝕜 : Type u
n : ℕ
E : Type wE
F : Type wF
G : Type wG
ι : Type v
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : SeminormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : Fintype ι
ι' : Type u_1
inst✝ : Fintype ι'
f : E [⋀^ι]→ₗ[𝕜] F [⋀^ι']→ₗ[𝕜] G
C : ℝ
H : ∀ (m₁ : ι → E) (m₂ : ι' → F), ‖(f m₁) m₂‖ ≤ (C * ∏ i, ‖m₁ i‖) * ∏ i, ‖m₂ i‖
v : ι → E
i j : ι
hv : v i = v j
hij : i ≠ j
v' : ι' → F
⊢ ((f v).mkContinuous (C * ∏ i, ‖v i‖) ⋯) v' = 0 v'

Tactic:
have : f v = 0 := by simpa using f.map_eq_zero_of_eq' v i j hv hij

Goal state after:
𝕜 : Type u
n : ℕ
E : Type wE
F : Type wF
G : Type wG
ι : Type v
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : SeminormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : Fintype ι
ι' : Type u_1
inst✝ : Fintype ι'
f : E [⋀^ι]→ₗ[𝕜] F [⋀^ι']→ₗ[𝕜] G
C : ℝ
H : ∀ (m₁ : ι → E) (m₂ : ι' → F), ‖(f m₁) m₂‖ ≤ (C * ∏ i, ‖m₁ i‖) * ∏ i, ‖m₂ i‖
v : ι → E
i j : ι
hv : v i = v j
hij : i ≠ j
v' : ι' → F
this : f v = 0
⊢ ((f v).mkContinuous (C * ∏ i, ‖v i‖) ⋯) v' = 0 v'
Module: Mathlib.Topology.MetricSpace.Completion

Goal state before:
pp: case refine_2
α : Type u
inst✝ : PseudoMetricSpace α
x y z : Completion α
a b c : α
⊢ dist a c ≤ dist a b + dist b c

Tactic:
exact dist_triangle a b c

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
s : Set E
f : E → F
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
⊢ HasFTaylorSeriesUpToOn (↑(n + 1)) f p s ↔
    (∀ x ∈ s, (p x 0).curry0 = f x) ∧
      (∀ x ∈ s, HasFDerivWithinAt (fun y ↦ p y 0) (p x 1).curryLeft s x) ∧
        HasFTaylorSeriesUpToOn (↑n) (fun x ↦ (continuousMultilinearCurryFin1 𝕜 E F) (p x 1)) (fun x ↦ (p x).shift) s

Tactic:
by
  constructor
  · intro H
    refine ⟨H.zero_eq, H.fderivWithin 0 (Nat.cast_lt.2 (Nat.succ_pos n)), ?_⟩
    exact H.shift_of_succ
  · rintro ⟨Hzero_eq, Hfderiv_zero, Htaylor⟩
    constructor
    · exact Hzero_eq
    · intro m (hm : (m : ℕ∞ω) < n.succ) x (hx : x ∈ s)
      rcases m with - | m
      · exact Hfderiv_zero x hx
      · have A : (m : ℕ∞ω) < n := by
          rw [Nat.cast_lt] at hm ⊢
          exact Nat.lt_of_succ_lt_succ hm
        have :
          HasFDerivWithinAt (𝕜 := 𝕜) (continuousMultilinearCurryRightEquiv' 𝕜 m E F ∘ (p · m.succ))
            ((p x).shift m.succ).curryLeft s x := Htaylor.fderivWithin _ A x hx
        rw [LinearIsometryEquiv.comp_hasFDerivWithinAt_iff'
            (f' := ((p x).shift m.succ).curryLeft)] at this
        convert! this
        ext y v
        change
          (p x (Nat.succ (Nat.succ m))) (cons y v) =
            (p x m.succ.succ) (snoc (cons y (init v)) (v (last _)))
        rw [← cons_snoc_eq_snoc_cons, snoc_init_self]
    · intro m (hm : (m : ℕ∞ω) ≤ n.succ)
      rcases m with - | m
      · have : DifferentiableOn 𝕜 (fun x => p x 0) s := fun x hx =>
          (Hfderiv_zero x hx).differentiableWithinAt
        exact this.continuousOn
      · refine (continuousMultilinearCurryRightEquiv' 𝕜 m E F).comp_continuousOn_iff.mp ?_
        refine Htaylor.cont _ ?_
        rw [Nat.cast_le] at hm ⊢
        exact Nat.lt_succ_iff.mp hm

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
s : Set E
f : E → F
x : E
⊢ ‖(⇑(continuousMultilinearCurryFin0 𝕜 E F).symm ∘ f) x‖ = ‖f x‖

Tactic:
comp_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
s : Set E
f : E → F
x : E
⊢ ‖(continuousMultilinearCurryFin0 𝕜 E F).symm (f x)‖ = ‖f x‖
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
s : Set E
f : E → F
x : E
n : ℕ
⊢ ‖fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s x‖ =
    ‖(⇑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x ↦ E) F).symm ∘ fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s) x‖

Tactic:
comp_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
s : Set E
f : E → F
x : E
n : ℕ
⊢ ‖fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s x‖ =
    ‖(continuousMultilinearCurryLeftEquiv 𝕜 (fun x ↦ E) F).symm (fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s x)‖
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
s : Set E
f : E → F
x : E
n : ℕ
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
⊢ ‖iteratedFDerivWithin 𝕜 n (fderivWithin 𝕜 f s) s x‖ =
    ‖(⇑(continuousMultilinearCurryRightEquiv' 𝕜 n E F).symm ∘ iteratedFDerivWithin 𝕜 n (fun y ↦ fderivWithin 𝕜 f s y) s)
        x‖

Tactic:
comp_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
s : Set E
f : E → F
x : E
n : ℕ
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
⊢ ‖iteratedFDerivWithin 𝕜 n (fderivWithin 𝕜 f s) s x‖ =
    ‖(continuousMultilinearCurryRightEquiv' 𝕜 n E F).symm (iteratedFDerivWithin 𝕜 n (fun y ↦ fderivWithin 𝕜 f s y) s x)‖
Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

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
f : E → F
n : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
⊢ HasFTaylorSeriesUpToOn n f p univ → HasFTaylorSeriesUpTo n f p

Tactic:
refine fun H ↦ ⟨by simpa using H.zero_eq, ?_, by simpa using H.cont⟩

Goal state after:
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
f : E → F
n : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
H : HasFTaylorSeriesUpToOn n f p univ
⊢ ∀ (m : ℕ), ↑m < n → ∀ (x : E), HasFDerivAt (fun y ↦ p y m) (p x m.succ).curryLeft x
Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

Goal state before:
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
f : E → F
n : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
⊢ HasFTaylorSeriesUpTo n f p → HasFTaylorSeriesUpToOn n f p univ

Tactic:
refine fun H ↦ ⟨by simpa using H.zero_eq, ?_, by simpa using H.cont⟩

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
f : E → F
n : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
H : HasFTaylorSeriesUpTo n f p
⊢ ∀ (m : ℕ), ↑m < n → ∀ x ∈ univ, HasFDerivWithinAt (fun x ↦ p x m) (p x m.succ).curryLeft univ x
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
⊢ ‖(⇑(continuousMultilinearCurryFin0 𝕜 E F).symm ∘ f) x‖ = ‖f x‖

Tactic:
comp_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
⊢ ‖(continuousMultilinearCurryFin0 𝕜 E F).symm (f x)‖ = ‖f x‖
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
⊢ ‖fderiv 𝕜 (iteratedFDeriv 𝕜 n f) x‖ =
    ‖(⇑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x ↦ E) F).symm ∘ fderiv 𝕜 (iteratedFDeriv 𝕜 n f)) x‖

Tactic:
comp_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
⊢ ‖fderiv 𝕜 (iteratedFDeriv 𝕜 n f) x‖ =
    ‖(continuousMultilinearCurryLeftEquiv 𝕜 (fun x ↦ E) F).symm (fderiv 𝕜 (iteratedFDeriv 𝕜 n f) x)‖
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:
comp_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)) =
    ((continuousMultilinearCurryRightEquiv' 𝕜 n E F).symm (iteratedFDeriv 𝕜 n (fun y ↦ fderiv 𝕜 f y) x)) m
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
⊢ ‖iteratedFDeriv 𝕜 n (fderiv 𝕜 f) x‖ =
    ‖(⇑(continuousMultilinearCurryRightEquiv' 𝕜 n E F).symm ∘ iteratedFDeriv 𝕜 n fun y ↦ fderiv 𝕜 f y) x‖

Tactic:
comp_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
⊢ ‖iteratedFDeriv 𝕜 n (fderiv 𝕜 f) x‖ =
    ‖(continuousMultilinearCurryRightEquiv' 𝕜 n E F).symm (iteratedFDeriv 𝕜 n (fun y ↦ fderiv 𝕜 f y) x)‖
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
z : E
m : Fin 2 → E
⊢ (iteratedFDeriv 𝕜 2 f z) m = ((fderiv 𝕜 (fderiv 𝕜 f) z) (m 0)) (m 1)

Tactic:
simp [iteratedFDeriv_succ_apply_right, init]

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

Goal state before:
pp: 𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
x : 𝕜
𝕝 : Type u_3
inst✝³ : DivisionSemiring 𝕝
inst✝² : Module 𝕝 F
inst✝¹ : SMulCommClass 𝕜 𝕝 F
inst✝ : ContinuousConstSMul 𝕝 F
c : 𝕝
f : 𝕜 → F
⊢ deriv (fun y ↦ c • f y) x = c • deriv f x

Tactic:
by
  simp only [← derivWithin_univ, derivWithin_fun_const_smul_field]

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
  simp

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

Goal state before:
pp: k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a b : k
⊢ (b - a) • slope f a b = f b -ᵥ f a

Tactic:
rcases eq_or_ne a b with (rfl | hne)
  · rw [sub_self, zero_smul, vsub_self]
  · rw [slope, smul_inv_smul₀ (sub_ne_zero.2 hne.symm)]

Goal state after: