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

Goal state before:
pp: case convert_2.inl
𝕜 : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
G : Type u_5
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
u : E →L[𝕜] F
f : F → G
pf : FormalMultilinearSeries 𝕜 F G
s : Set F
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f pf s (u x) r
y✝ : E
hy2 : y✝ ∈ Metric.eball 0 (r / ‖u‖ₑ)
hy1 : y✝ = 0
⊢ ↑u y✝ = 0 ∨ u x + ↑u y✝ ∈ s

Tactic:
simp [hy1]

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

Goal state before:
pp: case convert_2.inr
𝕜 : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
G : Type u_5
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
u : E →L[𝕜] F
f : F → G
pf : FormalMultilinearSeries 𝕜 F G
s : Set F
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f pf s (u x) r
y✝ : E
hy2 : y✝ ∈ Metric.eball 0 (r / ‖u‖ₑ)
hy1 : u x + u y✝ ∈ s
⊢ ↑u y✝ = 0 ∨ u x + ↑u y✝ ∈ s

Tactic:
simp [hy1]

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 : F → G
s : Set F
x : E
H : DifferentiableWithinAt 𝕜 (f ∘ ⇑iso) (⇑iso ⁻¹' s) x
this : DifferentiableWithinAt 𝕜 ((f ∘ ⇑iso) ∘ ⇑iso.symm) s (iso x)
⊢ DifferentiableWithinAt 𝕜 f s (iso x)

Tactic:
Function.comp_assoc,

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 : F → G
s : Set F
x : E
H : DifferentiableWithinAt 𝕜 (f ∘ ⇑iso) (⇑iso ⁻¹' s) x
this : DifferentiableWithinAt 𝕜 (f ∘ ⇑iso ∘ ⇑iso.symm) s (iso x)
⊢ DifferentiableWithinAt 𝕜 f s (iso 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
s : Set F
x : E
f' : F →L[𝕜] G
H : HasFDerivWithinAt (f ∘ ⇑iso) (f' ∘SL ↑iso) (⇑iso ⁻¹' s) (iso.symm (iso x))
⊢ f = (f ∘ ⇑iso) ∘ ⇑iso.symm

Tactic:
Function.comp_assoc,

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 : F → G
s : Set F
x : E
f' : F →L[𝕜] G
H : HasFDerivWithinAt (f ∘ ⇑iso) (f' ∘SL ↑iso) (⇑iso ⁻¹' s) (iso.symm (iso x))
⊢ f = f ∘ ⇑iso ∘ ⇑iso.symm
Module: Mathlib.Analysis.Calculus.FDeriv.Equiv

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
f' : E →L[ℝ] F
x : E
L : Filter E
⊢ Tendsto (fun x' ↦ ‖x' - x‖⁻¹ • (f x' - f x - f' (x' - x))) L (𝓝 0) ↔
    Tendsto (fun x' ↦ ‖x' - x‖⁻¹ * ‖f x' - f x - f' (x' - x)‖) L (𝓝 0)

Tactic:
tendsto_iff_norm_sub_tendsto_zero

Goal state after:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
f' : E →L[ℝ] F
x : E
L : Filter E
⊢ Tendsto (fun e ↦ ‖‖e - x‖⁻¹ • (f e - f x - f' (e - x)) - 0‖) L (𝓝 0) ↔
    Tendsto (fun x' ↦ ‖x' - x‖⁻¹ * ‖f x' - f x - f' (x' - x)‖) L (𝓝 0)
Module: Mathlib.Analysis.Calculus.FDeriv.Equiv

Goal state before:
pp: case inr
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
s : Set E
x : E
c : 𝕜
hc : c ≠ 0
⊢ fderivWithin 𝕜 (fun x ↦ f (c • x)) s x = fderivWithin 𝕜 (c • f) (c • s) (c • x)

Tactic:
classical
    simp only [fderivWithin, DifferentiableWithinAt, hasFDerivWithinAt_comp_smul_iff_smul hc]

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

Goal state before:
pp: case e'_6.e'_1.inr
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
inst✝ : DecidableEq ι
f : F [⋀^ι]→L[𝕜] G
g dg : E →L[𝕜] F
v : ι → E
a b : ι
heq : v a = v b
hne : a ≠ b
i : ι
hia : i ≠ a
⊢ (Function.update (fun x ↦ g) b dg i) (v i) =
    ((fun j ↦ (Function.update (fun x ↦ g) a dg j) (v j)) ∘ ⇑(Equiv.swap a b)) i

Tactic:
rcases eq_or_ne i b with rfl | hib
          · simp [Function.update_apply, heq]
          · simp [Function.update_apply, Equiv.swap_apply_of_ne_of_ne, *]

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
inst✝ : DecidableEq ι
f : F [⋀^ι]→L[𝕜] G
g dg : E →L[𝕜] F
v : ι → E
a b : ι
heq : v a = v b
hne : a ≠ b
i : ι
hia : ¬i = a
hib : ¬i = b
⊢ (f fun j ↦ (Function.update (fun x ↦ g) i dg j) (v j)) = 0

Tactic:
apply f.map_eq_zero_of_eq _ _ hne

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
inst✝ : DecidableEq ι
f : F [⋀^ι]→L[𝕜] G
g dg : E →L[𝕜] F
v : ι → E
a b : ι
heq : v a = v b
hne : a ≠ b
i : ι
hia : ¬i = a
hib : ¬i = b
⊢ (Function.update (fun x ↦ g) i dg a) (v a) = (Function.update (fun x ↦ g) i dg b) (v b)
Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

Goal state before:
pp: case succ
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 t : Set E
f : E → F
y✝ : E
n : ℕ
ihn : ∀ {x : E}, s =ᶠ[𝓝[{y✝}ᶜ] x] t → iteratedFDerivWithin 𝕜 n f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 n f t
x : E
h : s =ᶠ[𝓝[{y✝}ᶜ] x] t
y : E
hy : ∀ᶠ (x : E) in 𝓝[{y✝}ᶜ] y, s x = t x
⊢ iteratedFDerivWithin 𝕜 (n + 1) f s y = iteratedFDerivWithin 𝕜 (n + 1) f t y

Tactic:
simp only [iteratedFDerivWithin_succ_eq_comp_left, (· ∘ ·)]

Goal state after:
case succ
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 t : Set E
f : E → F
y✝ : E
n : ℕ
ihn : ∀ {x : E}, s =ᶠ[𝓝[{y✝}ᶜ] x] t → iteratedFDerivWithin 𝕜 n f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 n f t
x : E
h : s =ᶠ[𝓝[{y✝}ᶜ] x] t
y : E
hy : ∀ᶠ (x : E) in 𝓝[{y✝}ᶜ] y, s x = t x
⊢ (continuousMultilinearCurryLeftEquiv 𝕜 (fun x ↦ E) F).symm (fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s y) =
    (continuousMultilinearCurryLeftEquiv 𝕜 (fun x ↦ E) F).symm (fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f t) t y)
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
⊢ ‖iteratedFDeriv 𝕜 0 f x‖ = ‖f x‖

Tactic:
rw [iteratedFDeriv_zero_eq_comp, comp_apply, LinearIsometryEquiv.norm_map]

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

Goal state before:
pp: case pos
𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
E : Type w
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
G : Type u_1
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
x : 𝕜
s : Set 𝕜
B : E →L[𝕜] F →L[𝕜] G
u : 𝕜 → E
v : 𝕜 → F
hu : DifferentiableWithinAt 𝕜 u s x
hv : DifferentiableWithinAt 𝕜 v s x
hsx : UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (fun y ↦ (B (u y)) (v y)) s x = (B (u x)) (derivWithin v s x) + (B (derivWithin u s x)) (v x)

pp: case neg
𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
E : Type w
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
G : Type u_1
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
x : 𝕜
s : Set 𝕜
B : E →L[𝕜] F →L[𝕜] G
u : 𝕜 → E
v : 𝕜 → F
hu : DifferentiableWithinAt 𝕜 u s x
hv : DifferentiableWithinAt 𝕜 v s x
hsx : ¬UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (fun y ↦ (B (u y)) (v y)) s x = (B (u x)) (derivWithin v s x) + (B (derivWithin u s x)) (v x)

Tactic:
· exact (B.hasDerivWithinAt_of_bilinear hu.hasDerivWithinAt hv.hasDerivWithinAt).derivWithin hsx

Goal state after:
case neg
𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
E : Type w
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
G : Type u_1
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
x : 𝕜
s : Set 𝕜
B : E →L[𝕜] F →L[𝕜] G
u : 𝕜 → E
v : 𝕜 → F
hu : DifferentiableWithinAt 𝕜 u s x
hv : DifferentiableWithinAt 𝕜 v s x
hsx : ¬UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (fun y ↦ (B (u y)) (v y)) s x = (B (u x)) (derivWithin v s x) + (B (derivWithin u s x)) (v x)
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
ι : Type u_2
𝔸' : Type u_3
inst✝¹ : NormedCommRing 𝔸'
inst✝ : NormedAlgebra 𝕜 𝔸'
u : Finset ι
f : ι → 𝕜 → 𝔸'
hd : ∀ i ∈ u, DifferentiableAt 𝕜 (f i) x
⊢ DifferentiableAt 𝕜 (fun x ↦ ∏ i ∈ u, f i x) x

Tactic:
by
  classical
  exact
    (HasDerivAt.fun_finsetProd (fun i hi ↦ DifferentiableAt.hasDerivAt (hd i hi))).differentiableAt

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

Goal state before:
pp: case inr.refine_3
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
x : (i : ι) → E i
inst✝ : DecidableEq ι
y : (i : ι) → E i
h✝ : Nonempty ι
i : ι
⊢ f (Function.update x i (y i)) =
    (f.compContinuousLinearMap ContinuousLinearMap.proj) fun i_1 ↦
      Function.update (fun x_1 ↦ x) ((Fintype.equivFinOfCardEq ⋯) i) y ((Fintype.equivFinOfCardEq ⋯) i_1)

Tactic:
compContinuousLinearMap_apply,

Goal state after:
case inr.refine_3
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
x : (i : ι) → E i
inst✝ : DecidableEq ι
y : (i : ι) → E i
h✝ : Nonempty ι
i : ι
⊢ f (Function.update x i (y i)) =
    f fun i_1 ↦
      (ContinuousLinearMap.proj i_1)
        (Function.update (fun x_1 ↦ x) ((Fintype.equivFinOfCardEq ⋯) i) y ((Fintype.equivFinOfCardEq ⋯) i_1))
Module: Mathlib.Analysis.Calculus.Deriv.Slope

Goal state before:
pp: 𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SeparableSpace 𝕜
f : 𝕜 → F
s t : Set 𝕜
ts : t ⊆ s
t_count : t.Countable
ht : s ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
⊢ s ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (s ∩ t)

Tactic:
inter_eq_self_of_subset_right ts

Goal state after:
𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SeparableSpace 𝕜
f : 𝕜 → F
s t : Set 𝕜
ts : t ⊆ s
t_count : t.Countable
ht : s ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
⊢ s ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: s t : Set ℝ
⊢ interior (re ⁻¹' s ∩ im ⁻¹' t) = re ⁻¹' interior s ∩ im ⁻¹' interior t

Tactic:
interior_inter,

Goal state after:
s t : Set ℝ
⊢ interior (re ⁻¹' s) ∩ interior (im ⁻¹' t) = re ⁻¹' interior s ∩ im ⁻¹' interior t
Module: Mathlib.Analysis.Calculus.FDeriv.Mul

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
x : E
𝕜' : Type u_5
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 : E → 𝕜'
c' : E →L[𝕜] 𝕜'
hc : HasFDerivAt c c' x
f : F
⊢ HasFDerivAt (fun y ↦ c y • f) (c'.smulRight f) x

Tactic:
simpa only [smul_zero, zero_add] using! hc.smul (hasFDerivAt_const f x)

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' : 𝕜'
t : Set 𝕜'
hh : HasDerivAt h h' x
ht : ∀ᶠ (x' : 𝕜) in 𝓝 x, h x' ∈ t
hh₂ : HasDerivWithinAt h₂ h₂' t (h x)
⊢ HasDerivAt (h₂ ∘ h) (h₂' * h') x

Tactic:
exact hh₂.comp_hasDerivAt x hh ht

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
n : ℕ
g : 𝕜 → E
hg_an : AnalyticAt 𝕜 g z₀
hg_eq : ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • g z
hg_ne : ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0
z : 𝕜
⊢ f z = (z - z₀) ^ n • g z → f z = 0 → z ∈ {z₀}ᶜ → g z = 0

Tactic:
hf_eq

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
n : ℕ
g : 𝕜 → E
hg_an : AnalyticAt 𝕜 g z₀
hg_eq : ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • g z
hg_ne : ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0
z : 𝕜
hf_eq : f z = (z - z₀) ^ n • g z
⊢ f z = 0 → z ∈ {z₀}ᶜ → g z = 0
Module: Mathlib.Analysis.Analytic.IsolatedZeros

Goal state before:
pp: case h
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : 𝕜
f : 𝕜 → E
s : Set E
hfx : AnalyticAt 𝕜 f x
hs : s ∈ 𝓝[≠] f x
this : ∀ᶠ (z : 𝕜) in 𝓝 x, f z ∈ insert (f x) s
h : ¬f ⁻¹' s ∈ 𝓝[≠] x
⊢ ∃ᶠ (z : 𝕜) in 𝓝[≠] x, f z = f x

Tactic:
apply ((frequently_imp_distrib_right.2 h).and_eventually
    (eventually_nhdsWithin_of_eventually_nhds this)).mono

Goal state after:
case h
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : 𝕜
f : 𝕜 → E
s : Set E
hfx : AnalyticAt 𝕜 f x
hs : s ∈ 𝓝[≠] f x
this : ∀ᶠ (z : 𝕜) in 𝓝 x, f z ∈ insert (f x) s
h : ¬f ⁻¹' s ∈ 𝓝[≠] x
⊢ ∀ (x_1 : 𝕜), (f x_1 ∈ s → False) ∧ f x_1 ∈ insert (f x) s → f x_1 = f x
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
f : E → F
g : F →L[𝕜] G
hf : ContDiffAt 𝕜 n f x
i : ℕ
hi : ↑i ≤ n
⊢ iteratedFDerivWithin 𝕜 i (⇑g ∘ f) univ x = g.compContinuousMultilinearMap (iteratedFDerivWithin 𝕜 i f univ x)

Tactic:
exact g.iteratedFDerivWithin_comp_left hf.contDiffWithinAt uniqueDiffOn_univ (mem_univ x) hi

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

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
𝕜 : Type u_3
G : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E → G
s : Set E
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hs' : IsPreconnected s
hf : DifferentiableOn 𝕜 f s
hf' : EqOn (fderiv 𝕜 f) 0 s
⊢ ∃ a, ∀ x ∈ s, f x = a

Tactic:
obtain (rfl | ⟨y, hy⟩) := s.eq_empty_or_nonempty
  · exact ⟨0, by simp⟩
  · refine ⟨f y, fun x hx ↦ ?_⟩
    have h₁ := hs.isOpen_inter_preimage_of_fderiv_eq_zero hf hf' {f y}
    have h₂ := hf.continuousOn.comp_continuous continuous_subtype_val (fun x ↦ x.2)
    by_contra h₃
    obtain ⟨t, ht, ht'⟩ := (isClosed_singleton (x := f y)).preimage h₂
    have ht'' : ∀ a ∈ s, a ∈ t ↔ f a ≠ f y := by simpa [Set.ext_iff] using ht'
    obtain ⟨z, H₁, H₂, H₃⟩ := hs' _ _ h₁ ht (fun x h ↦ by simp [h, ht'', eq_or_ne]) ⟨y, by simpa⟩
      ⟨x, by simp [ht'' _ hx, hx, h₃]⟩
    exact (ht'' _ H₁).mp H₃ H₂.2

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

Goal state before:
pp: case refine_1
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → E
x₀ : ℝ
n : ℕ
s : Set ℝ
hs : Convex ℝ s
hx₀s : x₀ ∈ s
hff' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf' : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℝ) in 𝓝[s] x₀, ‖f' x‖ ≤ c * ‖(x - x₀) ^ n‖
⊢ ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℝ) in 𝓝[s] x₀, ‖toSpanSingleton ℝ (f' x)‖ ≤ c * ‖‖x - x₀‖ ^ n‖

Tactic:
convert! hf' using 4 with c hc x

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → E
x₀ : ℝ
n : ℕ
s : Set ℝ
hs : Convex ℝ s
hx₀s : x₀ ∈ s
hff' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf' : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℝ) in 𝓝[s] x₀, ‖f' x‖ ≤ c * ‖(x - x₀) ^ n‖
c : ℝ
hc : 0 < c
x : ℝ
⊢ ‖toSpanSingleton ℝ (f' x)‖ ≤ c * ‖‖x - x₀‖ ^ n‖ ↔ ‖f' x‖ ≤ c * ‖(x - x₀) ^ n‖
Module: Mathlib.Analysis.Calculus.ContDiff.Deriv

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
h : ContDiff 𝕜 2 f
⊢ Differentiable 𝕜 (deriv f)

Tactic:
by
  unfold deriv; fun_prop

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

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ∞ω
f : 𝕜 → F
h : ContDiff 𝕜 (n + 1) f
⊢ ContDiff 𝕜 n (deriv f)

Tactic:
by
  unfold deriv; fun_prop

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

Goal state before:
pp: case e'_10
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 ≠ ∞
u : Set E
hu : u ∈ 𝓝[insert x✝ s] x✝
hf : n = ω → AnalyticOn 𝕜 f u
f' : E → E →L[𝕜] F
f'_eq_deriv : ∀ x ∈ u, HasFDerivWithinAt f (f' x) u x
Hf' : ContDiffWithinAt 𝕜 n f' u x✝
v : Set E
hv : v ∈ 𝓝[insert x✝ u] x✝
p' : E → FormalMultilinearSeries 𝕜 E (E →L[𝕜] F)
Hp' : HasFTaylorSeriesUpToOn n f' p' v
p'_an : n = ω → ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ p' x i) v
x y : E
⊢ ((continuousMultilinearCurryFin1 𝕜 E F) ((p' x).unshift (f x) 1)) y = (p' x 0).curry0 y

Tactic:
change p' x 0 (init (@snoc 0 (fun _ : Fin 1 => E) 0 y)) y = p' x 0 0 y

Goal state after:
case e'_10
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 ≠ ∞
u : Set E
hu : u ∈ 𝓝[insert x✝ s] x✝
hf : n = ω → AnalyticOn 𝕜 f u
f' : E → E →L[𝕜] F
f'_eq_deriv : ∀ x ∈ u, HasFDerivWithinAt f (f' x) u x
Hf' : ContDiffWithinAt 𝕜 n f' u x✝
v : Set E
hv : v ∈ 𝓝[insert x✝ u] x✝
p' : E → FormalMultilinearSeries 𝕜 E (E →L[𝕜] F)
Hp' : HasFTaylorSeriesUpToOn n f' p' v
p'_an : n = ω → ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ p' x i) v
x y : E
⊢ ((p' x 0) (init (snoc 0 y))) y = ((p' x 0) 0) y
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
m✝ n : ℕ∞ω
h : ContDiffWithinAt 𝕜 n f s x
hn : n ≠ ω
m : ℕ
hm : ↑m ≤ n
h' : ↑m = ∞ → n = ω
⊢ ∃ u, IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u ∧ x ∈ u ∧ ContDiffOn 𝕜 (↑m) f (insert x s ∩ u)

Tactic:
rcases contDiffWithinAt_nat.1 (h.of_le hm) with ⟨t, ht, p, hp⟩
      rcases mem_nhdsWithin.1 ht with ⟨u, huo, hxu, hut⟩
      rw [inter_comm] at hut
      exact ⟨u, huo, hxu, (hp.mono hut).contDiffOn⟩

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

Goal state before:
pp: case 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 : ∃ u ∈ 𝓝[insert x s] x, ContDiffOn 𝕜 n f u
⊢ ContDiffWithinAt 𝕜 n f s x

Tactic:
rcases h with ⟨u, u_mem, hu⟩

Goal state after:
case 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 ≠ ∞
u : Set E
u_mem : u ∈ 𝓝[insert x s] x
hu : ContDiffOn 𝕜 n f u
⊢ ContDiffWithinAt 𝕜 n f s x
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
a' : Fin (c.partSize m + 1)
⊢ ∀ (b' : Fin (c.partSize m + 1)), a' < b' → cases 0 (succ ∘ c.emb m) a' < cases 0 (succ ∘ c.emb m) b'

Tactic:
b'

Goal state after:
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
a' b' : Fin (c.partSize m + 1)
⊢ a' < b' → cases 0 (succ ∘ c.emb m) a' < cases 0 (succ ∘ c.emb m) b'
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
i : Fin c.length
hi : i ∈ univ
j : Fin c.length
hj : j ∈ univ
hij h : i ≠ j
a' : Fin (c.partSize i)
⊢ ∀ (b' : Fin (c.partSize j + 1)), (c.emb i a').succ ≠ cases 0 (succ ∘ c.emb j) b'

Tactic:
b'

Goal state after:
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
i : Fin c.length
hi : i ∈ univ
j : Fin c.length
hj : j ∈ univ
hij h : i ≠ j
a' : Fin (c.partSize i)
b' : Fin (c.partSize j + 1)
⊢ (c.emb i a').succ ≠ cases 0 (succ ∘ c.emb j) b'