mathlib-initiative/mathlib-tactics huggingface dataset:
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
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 : F → G
f : E → F
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
x : E
t : Set F
s : Set E
hs : Set.MapsTo f s t
rg : ℝ≥0∞
Hg : HasFPowerSeriesWithinOnBall g q t (f x) rg
rf : ℝ≥0∞
Hf : HasFPowerSeriesWithinOnBall f p s x rf
r : ℝ≥0
r_pos : 0 < r
hr : Summable fun i ↦ ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst
δ : ℝ≥0∞
δpos : 0 < δ
hδ : ∀ {z : E}, z ∈ insert x s ∩ Metric.eball x δ → f z ∈ insert (f x) t ∩ Metric.eball (f x) rg
rf' : ℝ≥0∞ := min rf δ
min_pos : 0 < min rf' ↑r
y : E
h'y : x + y ∈ insert x s
hy : y ∈ Metric.eball 0 (min rf' ↑r)
y_mem : y ∈ Metric.eball 0 rf
fy_mem : f (x + y) ∈ insert (f x) t ∩ Metric.eball (f x) rg
A : Tendsto (fun n ↦ (n, ∑ a ∈ Finset.Ico 1 n, (p a) fun x ↦ y)) atTop (atTop ×ˢ 𝓝 (f (x + y) - f x))
B : Tendsto (fun n ↦ q.partialSum n (∑ a ∈ Finset.Ico 1 n, (p a) fun _b ↦ y)) atTop (𝓝 (g (f (x + y))))
C :
  Tendsto (fun n ↦ ∑ i ∈ compPartialSumTarget 0 n n, (q.compAlongComposition p i.snd) fun _j ↦ y) atTop
    (𝓝 (g (f (x + y))))
D : HasSum (fun i ↦ (q.compAlongComposition p i.snd) fun _j ↦ y) (g (f (x + y)))
n : ℕ
⊢ HasSum (fun c ↦ (q.compAlongComposition p ⟨n, c⟩.snd) fun _j ↦ y) ((q.comp p n) fun _j ↦ y)

Tactic:
simp only [compAlongComposition_apply, FormalMultilinearSeries.comp, sum_apply]

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
g : F → G
f : E → F
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
x : E
t : Set F
s : Set E
hs : Set.MapsTo f s t
rg : ℝ≥0∞
Hg : HasFPowerSeriesWithinOnBall g q t (f x) rg
rf : ℝ≥0∞
Hf : HasFPowerSeriesWithinOnBall f p s x rf
r : ℝ≥0
r_pos : 0 < r
hr : Summable fun i ↦ ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst
δ : ℝ≥0∞
δpos : 0 < δ
hδ : ∀ {z : E}, z ∈ insert x s ∩ Metric.eball x δ → f z ∈ insert (f x) t ∩ Metric.eball (f x) rg
rf' : ℝ≥0∞ := min rf δ
min_pos : 0 < min rf' ↑r
y : E
h'y : x + y ∈ insert x s
hy : y ∈ Metric.eball 0 (min rf' ↑r)
y_mem : y ∈ Metric.eball 0 rf
fy_mem : f (x + y) ∈ insert (f x) t ∩ Metric.eball (f x) rg
A : Tendsto (fun n ↦ (n, ∑ a ∈ Finset.Ico 1 n, (p a) fun x ↦ y)) atTop (atTop ×ˢ 𝓝 (f (x + y) - f x))
B : Tendsto (fun n ↦ q.partialSum n (∑ a ∈ Finset.Ico 1 n, (p a) fun _b ↦ y)) atTop (𝓝 (g (f (x + y))))
C :
  Tendsto (fun n ↦ ∑ i ∈ compPartialSumTarget 0 n n, (q.compAlongComposition p i.snd) fun _j ↦ y) atTop
    (𝓝 (g (f (x + y))))
D : HasSum (fun i ↦ (q.compAlongComposition p i.snd) fun _j ↦ y) (g (f (x + y)))
n : ℕ
⊢ HasSum (fun c ↦ (q c.length) (p.applyComposition c fun _j ↦ y)) (∑ c, (q c.length) (p.applyComposition c fun _j ↦ y))
Module: Mathlib.Analysis.Analytic.Within

Goal state before:
pp: case mpr.refine_2
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
g : E → F
r : ℝ≥0∞
hg : HasFPowerSeriesOnBall g p x r
e : ℝ
e0 : e > 0
hfg : ∀ ⦃y : E⦄, dist y x < e → y ∈ insert x s → f y = g y
⊢ HasFPowerSeriesOnBall g p x (min r (ENNReal.ofReal e))

Tactic:
exact hg.mono (lt_min hg.r_pos (by positivity)) (min_le_left _ _)

Goal state after:
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
n : ℕ
a : Composition n
b : Composition a.length
j : List ℕ
hj : j ∈ a.blocks.splitWrtComposition b
i : ℕ
hi : i ∈ j
⊢ i ∈ a.blocks

Tactic:
← a.blocks.flatten_splitWrtComposition b

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
n : ℕ
a : Composition n
b : Composition a.length
j : List ℕ
hj : j ∈ a.blocks.splitWrtComposition b
i : ℕ
hi : i ∈ j
⊢ i ∈ (a.blocks.splitWrtComposition b).flatten
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
n : ℕ
a : Composition n
b : Composition a.length
i✝ : Fin (a.gather b).length
i : ℕ
hi : i ∈ (a.blocks.splitWrtComposition b)[↑i✝]
⊢ i ∈ a.blocks

Tactic:
← a.blocks.flatten_splitWrtComposition b

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
n : ℕ
a : Composition n
b : Composition a.length
i✝ : Fin (a.gather b).length
i : ℕ
hi : i ∈ (a.blocks.splitWrtComposition b)[↑i✝]
⊢ i ∈ (a.blocks.splitWrtComposition b).flatten
Module: Mathlib.Analysis.Analytic.Inverse

Goal state before:
pp: case e_x
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
f : E → G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
x : E
hq : 0 < q.radius
hp : 0 < p.radius
r0 : ℝ≥0∞
h0 : HasFPowerSeriesOnBall f (q.comp p) x r0
r1 : NNReal
r1_pos : 0 < r1
hr1 : Summable fun i ↦ ‖q.compAlongComposition p i.snd‖₊ * r1 ^ i.fst
r : ℝ≥0∞ := min r0 ↑r1
this : Metric.eball 0 r ∈ 𝓝 0
y : E
hy : y ∈ Metric.eball 0 r
hy0 : y ∈ Metric.eball 0 r0
A : HasSum (fun i ↦ (q.compAlongComposition p i.snd) fun _j ↦ y) (f (x + y))
B :
  Tendsto (fun n ↦ ∑ i ∈ compPartialSumTarget 0 n.1 n.2, (q.compAlongComposition p i.snd) fun _j ↦ y) atTop
    (𝓝 (f (x + y)))
C : Tendsto (fun n ↦ q.partialSum n.1 (∑ a ∈ Ico 1 n.2, (p a) fun _b ↦ y)) atTop (𝓝 (f (x + y)))
fst✝ n : ℕ
hn : 1 ≤ n
⊢ (∑ a ∈ Ico 1 (fst✝, n).2, (p a) fun _b ↦ y) = (∑ k ∈ Finset.range (fst✝, n).2, (p k) fun x ↦ y) - (p 0) fun x ↦ 0

Tactic:
eq_sub_iff_add_eq',

Goal state after:
case e_x
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
f : E → G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
x : E
hq : 0 < q.radius
hp : 0 < p.radius
r0 : ℝ≥0∞
h0 : HasFPowerSeriesOnBall f (q.comp p) x r0
r1 : NNReal
r1_pos : 0 < r1
hr1 : Summable fun i ↦ ‖q.compAlongComposition p i.snd‖₊ * r1 ^ i.fst
r : ℝ≥0∞ := min r0 ↑r1
this : Metric.eball 0 r ∈ 𝓝 0
y : E
hy : y ∈ Metric.eball 0 r
hy0 : y ∈ Metric.eball 0 r0
A : HasSum (fun i ↦ (q.compAlongComposition p i.snd) fun _j ↦ y) (f (x + y))
B :
  Tendsto (fun n ↦ ∑ i ∈ compPartialSumTarget 0 n.1 n.2, (q.compAlongComposition p i.snd) fun _j ↦ y) atTop
    (𝓝 (f (x + y)))
C : Tendsto (fun n ↦ q.partialSum n.1 (∑ a ∈ Ico 1 n.2, (p a) fun _b ↦ y)) atTop (𝓝 (f (x + y)))
fst✝ n : ℕ
hn : 1 ≤ n
⊢ (((p 0) fun x ↦ 0) + ∑ a ∈ Ico 1 (fst✝, n).2, (p a) fun _b ↦ y) = ∑ k ∈ Finset.range (fst✝, n).2, (p k) fun x ↦ y
Module: Mathlib.Analysis.Analytic.Inverse

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
f : E → F
g : F → G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
x : E
hgf : HasFPowerSeriesAt (g ∘ f) (q.comp p) x
hf : HasFPowerSeriesAt f p x
hq : 0 < q.radius
this : ∀ᶠ (y : E) in 𝓝 0, f (x + y) - f x ∈ Metric.eball 0 q.radius
y : E
hy : Tendsto (fun a ↦ q.partialSum a.1 (p.partialSum a.2 y - (p 0) fun x ↦ 0)) atTop (𝓝 ((g ∘ f) (x + y)))
h'y : Tendsto (fun n ↦ p.partialSum n y) atTop (𝓝 (f (x + y)))
h''y : f (x + y) - f x ∈ Metric.eball 0 q.radius
u : Set G
hu : u ∈ 𝓝 (g (f (x + y)))
u_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
v : Set G
vu : v ⊆ u
v_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] v
hv : g (f (x + y)) ∈ v
a₀ b₀ : ℕ
hab : ∀ (a b : ℕ), a₀ ≤ a → b₀ ≤ b → q.partialSum a (p.partialSum b y - (p 0) fun x ↦ 0) ∈ v
⊢ ∃ a, ∀ (b : ℕ), a ≤ b → q.partialSum b (f (x + y) - f x) ∈ u

Tactic:
refine ⟨a₀, fun a ha ↦ ?_⟩

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
f : E → F
g : F → G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
x : E
hgf : HasFPowerSeriesAt (g ∘ f) (q.comp p) x
hf : HasFPowerSeriesAt f p x
hq : 0 < q.radius
this : ∀ᶠ (y : E) in 𝓝 0, f (x + y) - f x ∈ Metric.eball 0 q.radius
y : E
hy : Tendsto (fun a ↦ q.partialSum a.1 (p.partialSum a.2 y - (p 0) fun x ↦ 0)) atTop (𝓝 ((g ∘ f) (x + y)))
h'y : Tendsto (fun n ↦ p.partialSum n y) atTop (𝓝 (f (x + y)))
h''y : f (x + y) - f x ∈ Metric.eball 0 q.radius
u : Set G
hu : u ∈ 𝓝 (g (f (x + y)))
u_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
v : Set G
vu : v ⊆ u
v_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] v
hv : g (f (x + y)) ∈ v
a₀ b₀ : ℕ
hab : ∀ (a b : ℕ), a₀ ≤ a → b₀ ≤ b → q.partialSum a (p.partialSum b y - (p 0) fun x ↦ 0) ∈ v
a : ℕ
ha : a₀ ≤ a
⊢ q.partialSum a (f (x + y) - f x) ∈ u
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
s : Set G
x : G
f' : G →L[𝕜] F
⊢ HasFDerivWithinAt (⇑iso ∘ f) f' s x ↔ HasFDerivWithinAt f (↑iso.symm ∘SL f') s x

Tactic:
by
  rw [← iso.comp_hasFDerivWithinAt_iff, ← ContinuousLinearMap.comp_assoc, iso.coe_comp_coe_symm,
    ContinuousLinearMap.id_comp]

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

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'
x : E
s : Set E
L : G ≃L[𝕜] G'
f : E → F →L[𝕜] G
hs : UniqueDiffWithinAt 𝕜 s x
⊢ fderivWithin 𝕜 (fun x ↦ ↑L ∘SL f x) s x = ↑((ContinuousLinearEquiv.refl 𝕜 F).arrowCongr L) ∘SL fderivWithin 𝕜 f s x

Tactic:
change fderivWithin 𝕜 (((ContinuousLinearEquiv.refl 𝕜 F).arrowCongr L) ∘ f) s x = _
  rw [ContinuousLinearEquiv.comp_fderivWithin _ hs]

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

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
x : E
n : ℕ
F' : Fin n.succ → Type u_6
inst✝¹ : (i : Fin n.succ) → NormedAddCommGroup (F' i)
inst✝ : (i : Fin n.succ) → NormedSpace 𝕜 (F' i)
φ : E → F' 0
φs : E → (i : Fin n) → F' i.succ
⊢ DifferentiableAt 𝕜 (fun x ↦ Fin.cons (φ x) (φs x)) x ↔ DifferentiableAt 𝕜 φ x ∧ DifferentiableAt 𝕜 φs x

Tactic:
rw [differentiableAt_pi, Fin.forall_fin_succ, differentiableAt_pi]
  simp only [Fin.cons_zero, Fin.cons_succ]

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
x : E
f : E → F
h : UniqueDiffWithinAt 𝕜 s x
⊢ ‖iteratedFDerivWithin 𝕜 1 f s x‖ = ‖fderivWithin 𝕜 f s x‖

Tactic:
by
  simp only [← norm_fderivWithin_iteratedFDerivWithin,
    iteratedFDerivWithin_zero_eq_comp, LinearIsometryEquiv.comp_fderivWithin _ h]
  apply (continuousMultilinearCurryFin0 𝕜 E F).symm.toLinearIsometry.norm_toContinuousLinearMap_comp

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
n : ℕ
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
⊢ EqOn (iteratedFDerivWithin 𝕜 n f s) (iteratedFDeriv 𝕜 n f) s

Tactic:
by
  intro x hx
  rw [← iteratedFDerivWithin_univ]
  exact iteratedFDerivWithin_congr_set (Filter.eventuallyEq_univ.mpr <| hs.mem_nhds hx) n

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
f : E → F
x : E
m : Fin 1 → E
⊢ ((iteratedFDeriv 𝕜 0 (fun y ↦ fderiv 𝕜 f y) x) (init m)) (m (last 0)) = (fderiv 𝕜 f x) (m 0)

Tactic:
iteratedFDeriv_zero_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
m : Fin 1 → E
⊢ (fderiv 𝕜 f x) (m (last 0)) = (fderiv 𝕜 f x) (m 0)
Module: Mathlib.Analysis.Calculus.Deriv.Add

Goal state before:
pp: 𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
x c : 𝕜
⊢ deriv (fun x ↦ x) x = 1

Tactic:
deriv_id''

Goal state after:
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
x c : 𝕜
⊢ (fun x ↦ 1) x = 1
Module: Mathlib.Analysis.Calculus.Deriv.Add

Goal state before:
pp: 𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
x c : 𝕜
⊢ -deriv (fun x ↦ x) x = -1

Tactic:
deriv_id''

Goal state after:
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
x c : 𝕜
⊢ -(fun x ↦ 1) x = -1
Module: Mathlib.Analysis.Calculus.Deriv.Slope

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
s t : Set 𝕜
h : s ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (s ∩ t)
x : 𝕜
H : UniqueDiffWithinAt 𝕜 s x
H' : DifferentiableWithinAt 𝕜 f s x
I : (𝓝[(s ∩ t) \ {x}] x).NeBot
⊢ 𝓝[t ∩ (s \ {x})] x ≤ 𝓝[s \ {x}] x

Tactic:
exact nhdsWithin_mono _ inter_subset_right

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

Goal state before:
pp: 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
f' : E →L[𝕜'] F
s : Set E
x : E
g' : E →L[𝕜] F
h : HasFDerivWithinAt f g' s x
H : ContinuousLinearMap.restrictScalars 𝕜 f' = g'
⊢ HasFDerivWithinAt f f' s x

Tactic:
by
  rw [← H] at h
  exact .of_isLittleO h.isLittleO

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

Goal state before:
pp: 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
f' : E →L[𝕜'] F
x : E
g' : E →L[𝕜] F
h : HasFDerivAt f g' x
H : restrictScalars 𝕜 f' = g'
⊢ HasFDerivAt f f' x

Tactic:
by
  rw [← H] at h
  exact .of_isLittleO h.isLittleO

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]
    rfl

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
a b : 𝕜
s : Set 𝕜
h : ContinuousWithinAt (dslope f a) s b
⊢ ContinuousWithinAt f s b

Tactic:
have : ContinuousWithinAt (fun x => (x - a) • dslope f a x + f a) s b :=
    ((continuousWithinAt_id.sub continuousWithinAt_const).smul h).add continuousWithinAt_const

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
a b : 𝕜
s : Set 𝕜
h : ContinuousWithinAt (dslope f a) s b
this : ContinuousWithinAt (fun x ↦ (x - a) • dslope f a x + f a) s b
⊢ ContinuousWithinAt f s b
Module: Mathlib.Analysis.Calculus.FDeriv.Mul

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
ι : Type u_5
𝔸 : Type u_6
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸
l : List ι
x : { i // i ∈ l } → 𝔸
⊢ HasFDerivAt (fun x ↦ (List.map x l.attach).prod)
    (∑ i,
      (List.map x (List.take (↑i) l.attach)).prod •>
        (proj l.attach[Fin.cast ⋯ i] <• (List.map x (List.drop (↑i).succ l.attach)).prod))
    x

Tactic:
exact hasStrictFDerivAt_list_prod_attach'.hasFDerivAt

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.CircleMap

Goal state before:
pp: c : ℂ
R θ : ℝ
⊢ circleMap c R θ = c ↔ R = 0

Tactic:
by
  simp [circleMap, Complex.exp_ne_zero]

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

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type w
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
h₂ : 𝕜' → 𝕜'
h₂' y : 𝕜'
f : E → 𝕜'
f' : E →L[𝕜] 𝕜'
x : E
hh : HasStrictDerivAt h₂ h₂' y
hf : HasStrictFDerivAt f f' x
hy : y = f x
⊢ HasStrictFDerivAt (h₂ ∘ f) (h₂' • f') x

Tactic:
by
  rw [hy] at hh; exact hh.comp_hasStrictFDerivAt x hf

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

Goal state before:
pp: E : Type u_4
F : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
p : E → FormalMultilinearSeries ℝ E F
s : Set E
x : E
hf : HasFTaylorSeriesUpToOn 1 f p (insert x s)
hs : Convex ℝ s
K : ℝ≥0
hK : ‖p x 1‖₊ < K
f' : E → E →L[ℝ] F := fun y ↦ (continuousMultilinearCurryFin1 ℝ E F) (p y 1)
⊢ ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t

Tactic:
have hder : ∀ y ∈ s, HasFDerivWithinAt f (f' y) s y := fun y hy =>
    (hf.hasFDerivWithinAt one_ne_zero (subset_insert x s hy)).mono (subset_insert x s)

Goal state after:
E : Type u_4
F : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
p : E → FormalMultilinearSeries ℝ E F
s : Set E
x : E
hf : HasFTaylorSeriesUpToOn 1 f p (insert x s)
hs : Convex ℝ s
K : ℝ≥0
hK : ‖p x 1‖₊ < K
f' : E → E →L[ℝ] F := fun y ↦ (continuousMultilinearCurryFin1 ℝ E F) (p y 1)
hder : ∀ y ∈ s, HasFDerivWithinAt f (f' y) s y
⊢ ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t
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
h₁ : Differentiable 𝕜 f
h₂ : ∀ (x : E), fderiv 𝕜 f x = 0
⊢ IsLocallyConstant f

Tactic:
simpa using!
    isOpen_univ.isOpen_inter_preimage_of_fderiv_eq_zero h₁.differentiableOn fun _ _ ↦ h₂ _

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

Goal state before:
pp: case inr
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})
⊢ f x = f y

Tactic:
have h₂ := hf.continuousOn.comp_continuous continuous_subtype_val (fun x ↦ x.2)

Goal state after:
case inr
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)
⊢ f x = f y
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
s : Set E
f : E → F
x : E
n : ℕ∞ω
g : F →L[𝕜] 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
i : ℕ
⊢ AnalyticOn 𝕜 (fun x ↦ g.compContinuousMultilinearMap (p x i)) u

Tactic:
change AnalyticOn 𝕜
      (fun x ↦ (ContinuousLinearMap.compContinuousMultilinearMapL 𝕜
      (fun _ : Fin i ↦ E) F G g) (p x i)) u

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
s : Set E
f : E → F
x : E
n : ℕ∞ω
g : F →L[𝕜] 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
i : ℕ
⊢ AnalyticOn 𝕜 (fun x ↦ ((ContinuousLinearMap.compContinuousMultilinearMapL 𝕜 (fun x ↦ E) F G) g) (p x i)) u