mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.FDeriv.Mul

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
ι : Type u_5
𝔸' : Type u_7
inst✝² : NormedCommRing 𝔸'
inst✝¹ : NormedAlgebra 𝕜 𝔸'
u : Finset ι
g : ι → E → 𝔸'
g' : ι → E →L[𝕜] 𝔸'
inst✝ : DecidableEq ι
x : E
hg : ∀ i ∈ u, HasStrictFDerivAt (g i) (g' i) x
⊢ HasStrictFDerivAt (fun x ↦ ∏ i ∈ u, g i x) (∑ i ∈ u, (∏ j ∈ u.erase i, g j x) •> g' i) x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
ι : Type u_5
𝔸' : Type u_7
inst✝² : NormedCommRing 𝔸'
inst✝¹ : NormedAlgebra 𝕜 𝔸'
u : Finset ι
g : ι → E → 𝔸'
g' : ι → E →L[𝕜] 𝔸'
inst✝ : DecidableEq ι
x : E
hg : ∀ i ∈ u, HasStrictFDerivAt (g i) (g' i) x
⊢ HasStrictFDerivAt (fun x ↦ ∏ x_1 ∈ u.attach, g (↑x_1) x) (∑ i ∈ u, (∏ j ∈ u.erase i, g j x) •> g' i) x
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
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
h2 : ContinuousAt ((swap dslope z₀)^[p.order] f) z₀
h3 : ∀ᶠ (z : 𝕜) in 𝓝 z₀, (swap dslope z₀)^[p.order] f z ≠ 0
z : 𝕜
e1 : (swap dslope z₀)^[p.order] f z ≠ 0
e2 : z ∈ {z₀}ᶜ
⊢ f z ≠ 0

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
h2 : ContinuousAt ((swap dslope z₀)^[p.order] f) z₀
h3 : ∀ᶠ (z : 𝕜) in 𝓝 z₀, (swap dslope z₀)^[p.order] f z ≠ 0
z : 𝕜
e1 : (swap dslope z₀)^[p.order] f z ≠ 0
e2 : z ∈ {z₀}ᶜ
⊢ z - z₀ = 0 → p.order = 0
Module: Mathlib.Analysis.Calculus.FDeriv.Mul

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
ι : Type u_5
𝔸' : Type u_7
inst✝² : NormedCommRing 𝔸'
inst✝¹ : NormedAlgebra 𝕜 𝔸'
u : Finset ι
g : ι → E → 𝔸'
g' : ι → E →L[𝕜] 𝔸'
inst✝ : DecidableEq ι
x : E
hg : ∀ i ∈ u, HasFDerivAt (g i) (g' i) x
⊢ HasFDerivAt (fun x ↦ ∏ i ∈ u, g i x) (∑ i ∈ u, (∏ j ∈ u.erase i, g j x) •> g' i) x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
ι : Type u_5
𝔸' : Type u_7
inst✝² : NormedCommRing 𝔸'
inst✝¹ : NormedAlgebra 𝕜 𝔸'
u : Finset ι
g : ι → E → 𝔸'
g' : ι → E →L[𝕜] 𝔸'
inst✝ : DecidableEq ι
x : E
hg : ∀ i ∈ u, HasFDerivAt (g i) (g' i) x
⊢ HasFDerivAt (fun x ↦ ∏ x_1 ∈ u.attach, g (↑x_1) x) (∑ i ∈ u, (∏ j ∈ u.erase i, g j x) •> g' i) x
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
⊢ (∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0) ∨ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z ≠ 0

Tactic:
rcases hf with ⟨p, hp⟩
  by_cases h : p = 0
  · exact Or.inl (HasFPowerSeriesAt.eventually_eq_zero (by rwa [h] at hp))
  · exact Or.inr (hp.locally_ne_zero h)

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 g : 𝕜 → E
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
hg : AnalyticAt 𝕜 g z₀
⊢ (∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = g z) ∨ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z ≠ g z

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f g : 𝕜 → E
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
hg : AnalyticAt 𝕜 g z₀
⊢ (∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = g z) ∨ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, ¬f z = g z
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 g : 𝕜 → E
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
hg : AnalyticAt 𝕜 g z₀
⊢ (∃ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = g z) ↔ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = g z

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f g : 𝕜 → E
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
hg : AnalyticAt 𝕜 g z₀
⊢ (∃ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = g z) ↔ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = g z
Module: Mathlib.Analysis.Calculus.FDeriv.Mul

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
s : Set E
ι : Type u_5
𝔸' : Type u_7
inst✝² : NormedCommRing 𝔸'
inst✝¹ : NormedAlgebra 𝕜 𝔸'
u : Finset ι
g : ι → E → 𝔸'
g' : ι → E →L[𝕜] 𝔸'
inst✝ : DecidableEq ι
x : E
hg : ∀ i ∈ u, HasFDerivWithinAt (g i) (g' i) s x
⊢ HasFDerivWithinAt (fun x ↦ ∏ i ∈ u, g i x) (∑ i ∈ u, (∏ j ∈ u.erase i, g j x) •> g' i) s x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
s : Set E
ι : Type u_5
𝔸' : Type u_7
inst✝² : NormedCommRing 𝔸'
inst✝¹ : NormedAlgebra 𝕜 𝔸'
u : Finset ι
g : ι → E → 𝔸'
g' : ι → E →L[𝕜] 𝔸'
inst✝ : DecidableEq ι
x : E
hg : ∀ i ∈ u, HasFDerivWithinAt (g i) (g' i) s x
⊢ HasFDerivWithinAt (fun x ↦ ∏ x_1 ∈ u.attach, g (↑x_1) x) (∑ i ∈ u, (∏ j ∈ u.erase i, g j x) •> g' i) s x
Module: Mathlib.Analysis.Calculus.FDeriv.Mul

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
R : Type u_5
inst✝¹ : NormedDivisionRing R
inst✝ : NormedAlgebra 𝕜 R
x : R
hx : x ≠ 0
⊢ HasStrictFDerivAt Inv.inv (-((mulLeftRight 𝕜 R) x⁻¹) x⁻¹) x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
R : Type u_5
inst✝¹ : NormedDivisionRing R
inst✝ : NormedAlgebra 𝕜 R
x : R
hx : x ≠ 0
⊢ HasStrictFDerivAt Inv.inv (-((mulLeftRight 𝕜 R) x⁻¹) x⁻¹) x
Module: Mathlib.Analysis.Calculus.FDeriv.Mul

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
R : Type u_5
inst✝¹ : NormedDivisionRing R
inst✝ : NormedAlgebra 𝕜 R
x : R
hx : x ≠ 0
⊢ HasFDerivAt Inv.inv (-((mulLeftRight 𝕜 R) x⁻¹) x⁻¹) x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
R : Type u_5
inst✝¹ : NormedDivisionRing R
inst✝ : NormedAlgebra 𝕜 R
x : R
hx : x ≠ 0
⊢ HasFDerivAt Inv.inv (-((mulLeftRight 𝕜 R) x⁻¹) x⁻¹) x
Module: Mathlib.Analysis.Calculus.Deriv.Shift

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
f' : F
a x : 𝕜
hf : HasDerivAt f f' (a + x)
⊢ HasDerivAt (fun x ↦ f (a + x)) f' x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
F : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
f' : F
a x : 𝕜
hf : HasDerivAt f f' (a + x)
⊢ HasDerivAt (fun x ↦ f (a + x)) f' x
Module: Mathlib.Analysis.Calculus.Deriv.Shift

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
f' : F
x a : 𝕜
hf : HasDerivAt f f' (x + a)
⊢ HasDerivAt (fun x ↦ f (x + a)) f' x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
F : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
f' : F
x a : 𝕜
hf : HasDerivAt f f' (x + a)
⊢ HasDerivAt (fun x ↦ f (x + a)) f' x
Module: Mathlib.Analysis.Calculus.Deriv.Shift

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
f' : F
a x : 𝕜
hf : HasDerivAt f f' (a - x)
⊢ HasDerivAt (fun x ↦ f (a - x)) (-f') x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
F : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
f' : F
a x : 𝕜
hf : HasDerivAt f f' (a - x)
⊢ HasDerivAt (fun x ↦ f (a - x)) (-f') x
Module: Mathlib.Analysis.Calculus.Deriv.Shift

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
f' : F
x a : 𝕜
hf : HasDerivAt f f' (x - a)
⊢ HasDerivAt (fun x ↦ f (x - a)) f' x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
F : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
f' : F
x a : 𝕜
hf : HasDerivAt f f' (x - a)
⊢ HasDerivAt (fun x ↦ f (x - a)) f' x
Module: Mathlib.Analysis.Calculus.Deriv.Shift

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
s : Set 𝕜
x : 𝕜
⊢ derivWithin (fun x ↦ f (-x)) s x = -derivWithin f (-s) (-x)

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
F : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
s : Set 𝕜
x : 𝕜
⊢ derivWithin (fun x ↦ f (-x)) s x = -derivWithin f (-s) (-x)
Module: Mathlib.Analysis.Calculus.Deriv.Shift

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
x : 𝕜
⊢ deriv (fun x ↦ f (-x)) x = -deriv f (-x)

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
F : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
x : 𝕜
⊢ deriv (fun x ↦ f (-x)) x = -deriv f (-x)
Module: Mathlib.Analysis.Calculus.Deriv.Shift

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
a x : 𝕜
⊢ deriv (fun x ↦ f (x + a)) x = deriv f (x + a)

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
F : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
a x : 𝕜
⊢ deriv (fun x ↦ f (a + x)) x = deriv f (a + x)
Module: Mathlib.Analysis.Calculus.Deriv.Comp

Goal state before:
pp: 𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
f : 𝕜 → 𝕜
f' : 𝕜
hf : HasDerivAt f f' x
hx : f x = x
n : ℕ
⊢ Tendsto (Prod.map f f) (𝓝 x ×ˢ pure x) (𝓝 x ×ˢ pure x)

Tactic:
simpa using

Goal state after:
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
f : 𝕜 → 𝕜
f' : 𝕜
hf : HasDerivAt f f' x
hx : f x = x
n : ℕ
⊢ Tendsto (Prod.map f f ∘ fun a ↦ (a, x)) (𝓝 x) (map (fun a ↦ (a, x)) (𝓝 x))
Module: Mathlib.Analysis.Calculus.Deriv.Comp

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → F
f' : F
x : 𝕜
s : Set 𝕜
l : F → E
l' : F →L[𝕜] E
t : Set F
hl : HasFDerivWithinAt l l' t (f x)
hf : HasDerivWithinAt f f' s x
hst : MapsTo f s t
⊢ HasDerivWithinAt (l ∘ f) (l' f') s x

Tactic:
simpa using

Goal state after:
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → F
f' : F
x : 𝕜
s : Set 𝕜
l : F → E
l' : F →L[𝕜] E
t : Set F
hl : HasFDerivWithinAt l l' t (f x)
hf : HasDerivWithinAt f f' s x
hst : MapsTo f s t
⊢ HasDerivWithinAt (l ∘ f) (l' f') s x
Module: Mathlib.Analysis.Calculus.Deriv.Comp

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → F
f' : F
x : 𝕜
l : F → E
l' : F →L[𝕜] E
t : Set F
hl : HasFDerivWithinAt l l' t (f x)
hf : HasDerivAt f f' x
ht : ∀ᶠ (x' : 𝕜) in 𝓝 x, f x' ∈ t
⊢ HasDerivAt (l ∘ f) (l' f') x

Tactic:
simpa using

Goal state after:
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → F
f' : F
x : 𝕜
l : F → E
l' : F →L[𝕜] E
t : Set F
hl : HasFDerivWithinAt l l' t (f x)
hf : HasDerivAt f f' x
ht : ∀ᶠ (x' : 𝕜) in 𝓝 x, f x' ∈ t
⊢ HasDerivAt (l ∘ f) (l' f') x
Module: Mathlib.Analysis.Calculus.Deriv.ZPow

Goal state before:
pp: 𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
k : ℕ
c d : 𝕜
⊢ (deriv^[k] fun x ↦ (c * x - d)⁻¹) = fun x ↦ (-1) ^ k * ↑k ! * c ^ k * (c * x - d) ^ (-1 - ↑k)

Tactic:
simpa using

Goal state after:
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
k : ℕ
c d : 𝕜
⊢ (deriv^[k] fun x ↦ (c * x + -d)⁻¹) = fun x ↦ (-1) ^ k * ↑k ! * c ^ k * (c * x + -d) ^ (-1 + -↑k)
Module: Mathlib.Analysis.Analytic.IsolatedZeros

Goal state before:
pp: case mp
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
⊢ ∀ᶠ (x : 𝕜) in 𝓝 z₀, g x = 0

Tactic:
← AnalyticAt.frequently_eq_iff_eventually_eq hg_an analyticAt_const

Goal state after:
case mp
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
⊢ ∃ᶠ (z : 𝕜) in 𝓝[≠] z₀, g z = 0
Module: Mathlib.Analysis.Analytic.IsolatedZeros

Goal state before:
pp: case mp
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 g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • g z) → ¬∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0

Tactic:
rintro ⟨n, g, hg_an, hg_ne, hg_eq⟩
    contrapose hg_ne
    apply EventuallyEq.eq_of_nhds
    rw [EventuallyEq, ← AnalyticAt.frequently_eq_iff_eventually_eq hg_an analyticAt_const]
    refine (eventually_nhdsWithin_iff.mpr ?_).frequently
    filter_upwards [hg_eq, hg_ne] with z hf_eq hf0 hz
    rwa [hf0, eq_comm, smul_eq_zero_iff_right] at hf_eq
    exact pow_ne_zero _ (sub_ne_zero.mpr hz)

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
a b : E
x : 𝕜
⊢ HasStrictDerivAt (⇑(lineMap a b)) (b - a) x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
a b : E
x : 𝕜
⊢ HasStrictDerivAt (⇑(lineMap a b)) (b - a) x
Module: Mathlib.Analysis.Analytic.IsolatedZeros

Goal state before:
pp: case pos
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 = {z}
⊢ (∀ z ∈ U, f z = 0) ∨ ∀ z ∈ U, g z = 0

Tactic:
simpa using

Goal state after:
case pos
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 = {z}
⊢ f z = 0 ∨ g z = 0
Module: Mathlib.Analysis.Calculus.LogDeriv

Goal state before:
pp: case cons.hg
𝕜 : Type u_1
𝕜' : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
ι : Type u_3
f : ι → 𝕜 → 𝕜'
x : 𝕜
a : ι
s : Finset ι
ha : a ∉ s
ih :
  (∀ i ∈ s, f i x ≠ 0) →
    (∀ i ∈ s, DifferentiableAt 𝕜 (f i) x) → logDeriv (fun x ↦ ∏ i ∈ s, f i x) x = ∑ i ∈ s, logDeriv (f i) x
hf : f a x ≠ 0 ∧ ∀ x_1 ∈ s, f x_1 x ≠ 0
hd : DifferentiableAt 𝕜 (f a) x ∧ ∀ x_1 ∈ s, DifferentiableAt 𝕜 (f x_1) x
⊢ ∏ i ∈ s, f i x ≠ 0

Tactic:
simpa using

Goal state after:
case cons.hg
𝕜 : Type u_1
𝕜' : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
ι : Type u_3
f : ι → 𝕜 → 𝕜'
x : 𝕜
a : ι
s : Finset ι
ha : a ∉ s
ih :
  (∀ i ∈ s, f i x ≠ 0) →
    (∀ i ∈ s, DifferentiableAt 𝕜 (f i) x) → logDeriv (fun x ↦ ∏ i ∈ s, f i x) x = ∑ i ∈ s, logDeriv (f i) x
hf : f a x ≠ 0 ∧ ∀ x_1 ∈ s, f x_1 x ≠ 0
hd : DifferentiableAt 𝕜 (f a) x ∧ ∀ x_1 ∈ s, DifferentiableAt 𝕜 (f x_1) x
⊢ ∀ x_1 ∈ s, ¬f x_1 x = 0
Module: Mathlib.Analysis.Calculus.LogDeriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
⊢ logDeriv (fun x ↦ x⁻¹) x = -1 / x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
⊢ logDeriv (fun x ↦ x⁻¹) x = -1 / x
Module: Mathlib.Analysis.Calculus.LogDeriv

Goal state before:
pp: case inl
𝕜 : Type u_1
𝕜' : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NontriviallyNormedField 𝕜'
inst✝¹ : NormedAlgebra 𝕜 𝕜'
inst✝ : IsRCLikeNormedField 𝕜
f g : 𝕜 → 𝕜'
hf : DifferentiableOn 𝕜 f ∅
hg : DifferentiableOn 𝕜 g ∅
hs2 : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ∅
hsc : IsPreconnected ∅
hgn : ∀ x ∈ ∅, g x ≠ 0
hfn : ∀ x ∈ ∅, f x ≠ 0
⊢ EqOn (logDeriv f) (logDeriv g) ∅ ↔ ∃ z, z ≠ 0 ∧ EqOn f (z • g) ∅

Tactic:
simpa using

Goal state after:
case inl
𝕜 : Type u_1
𝕜' : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NontriviallyNormedField 𝕜'
inst✝¹ : NormedAlgebra 𝕜 𝕜'
inst✝ : IsRCLikeNormedField 𝕜
f g : 𝕜 → 𝕜'
hf : DifferentiableOn 𝕜 f ∅
hg : DifferentiableOn 𝕜 g ∅
hs2 : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ∅
hsc : IsPreconnected ∅
hgn : ∀ x ∈ ∅, g x ≠ 0
hfn : ∀ x ∈ ∅, f x ≠ 0
⊢ ∃ z, ¬z = 0
Module: Mathlib.Analysis.Calculus.MeanValue

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → E
C : ℝ
hf : ∀ x ∈ Icc 0 1, HasDerivWithinAt f (f' x) (Icc 0 1) x
bound : ∀ x ∈ Ico 0 1, ‖f' x‖ ≤ C
⊢ ‖f 1 - f 0‖ ≤ C

Tactic:
simpa using

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → E
C : ℝ
hf : ∀ x ∈ Icc 0 1, HasDerivWithinAt f (f' x) (Icc 0 1) x
bound : ∀ x ∈ Ico 0 1, ‖f' x‖ ≤ C
⊢ ‖f 1 - f 0‖ ≤ C
Module: Mathlib.Analysis.Calculus.MeanValue

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
C : ℝ
hf : DifferentiableOn ℝ f (Icc 0 1)
bound : ∀ x ∈ Ico 0 1, ‖derivWithin f (Icc 0 1) x‖ ≤ C
⊢ ‖f 1 - f 0‖ ≤ C

Tactic:
simpa using

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
C : ℝ
hf : DifferentiableOn ℝ f (Icc 0 1)
bound : ∀ x ∈ Ico 0 1, ‖derivWithin f (Icc 0 1) x‖ ≤ C
⊢ ‖f 1 - f 0‖ ≤ C
Module: Mathlib.Analysis.Calculus.MeanValue

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
hcont : ContinuousOn f (Icc a b)
hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x
this : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a)
⊢ ∀ x ∈ Icc a b, f x = f a

Tactic:
simpa using

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
hcont : ContinuousOn f (Icc a b)
hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x
this : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a)
⊢ ∀ x ∈ Icc a b, f x = f a
Module: Mathlib.Analysis.Calculus.MeanValue

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
hdiff : DifferentiableOn ℝ f (Icc a b)
hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0
⊢ ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0

Tactic:
simpa using

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
hdiff : DifferentiableOn ℝ f (Icc a b)
hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0
⊢ ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0
Module: Mathlib.Analysis.Calculus.MeanValue

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
hdiff : DifferentiableOn ℝ f (Icc a b)
hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0
H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0
⊢ ∀ x ∈ Icc a b, f x = f a

Tactic:
simpa using

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
hdiff : DifferentiableOn ℝ f (Icc a b)
hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0
H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0
⊢ ∀ x ∈ Icc a b, f x = f a
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
C : ℝ
s : Set E
x y : E
f' : E → E →L[𝕜] G
hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
bound : ∀ x ∈ s, ‖f' x‖ ≤ C
hs : Convex ℝ s
xs : x ∈ s
ys : y ∈ s
this✝ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this : NormedSpace ℝ G := NormedSpace.restrictScalars ℝ 𝕜 G
g : ℝ → E := ⇑(AffineMap.lineMap x y)
segm : MapsTo g (Icc 0 1) s
t : ℝ
ht : t ∈ Icc 0 1
⊢ HasDerivWithinAt (f ∘ g) ((f' (g t)) (y - x)) (Icc 0 1) t

Tactic:
simpa using

Goal state after:
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
C : ℝ
s : Set E
x y : E
f' : E → E →L[𝕜] G
hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
bound : ∀ x ∈ s, ‖f' x‖ ≤ C
hs : Convex ℝ s
xs : x ∈ s
ys : y ∈ s
this✝ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this : NormedSpace ℝ G := NormedSpace.restrictScalars ℝ 𝕜 G
g : ℝ → E := ⇑(AffineMap.lineMap x y)
segm : MapsTo g (Icc 0 1) s
t : ℝ
ht : t ∈ Icc 0 1
⊢ HasDerivWithinAt (f ∘ g) ((f' (g t)) y - (f' (g t)) x) (Icc 0 1) 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
C : ℝ
s : Set E
x y : E
f' : E → E →L[𝕜] G
hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
bound✝ : ∀ x ∈ s, ‖f' x‖ ≤ C
hs : Convex ℝ s
xs : x ∈ s
ys : y ∈ s
this✝ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this : NormedSpace ℝ G := NormedSpace.restrictScalars ℝ 𝕜 G
g : ℝ → E := ⇑(AffineMap.lineMap x y)
segm : MapsTo g (Icc 0 1) s
hD : ∀ t ∈ Icc 0 1, HasDerivWithinAt (f ∘ g) ((f' (g t)) (y - x)) (Icc 0 1) t
bound : ∀ t ∈ Ico 0 1, ‖(f' (g t)) (y - x)‖ ≤ C * ‖y - x‖
⊢ ‖f y - f x‖ ≤ C * ‖y - x‖

Tactic:
simpa using

Goal state after:
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
C : ℝ
s : Set E
x y : E
f' : E → E →L[𝕜] G
hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
bound✝ : ∀ x ∈ s, ‖f' x‖ ≤ C
hs : Convex ℝ s
xs : x ∈ s
ys : y ∈ s
this✝ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this : NormedSpace ℝ G := NormedSpace.restrictScalars ℝ 𝕜 G
g : ℝ → E := ⇑(AffineMap.lineMap x y)
segm : MapsTo g (Icc 0 1) s
hD : ∀ t ∈ Icc 0 1, HasDerivWithinAt (f ∘ g) ((f' (g t)) (y - x)) (Icc 0 1) t
bound : ∀ t ∈ Ico 0 1, ‖(f' (g t)) (y - x)‖ ≤ C * ‖y - x‖
⊢ ‖f y - f x‖ ≤ C * ‖y - x‖