-- Mathlib/Analysis/Analytic/IsolatedZeros.lean
theorem AnalyticAt.preimage_of_nhdsNE {x : 𝕜} {f : 𝕜 → E} {s : Set E} (hfx : AnalyticAt 𝕜 f x)
    (h₂f : ¬EventuallyConst f (𝓝 x)) (hs : s ∈ 𝓝[≠] f x) :
    f ⁻¹' s ∈ 𝓝[≠] x

Proof state:
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
h₂f : ¬EventuallyConst f (𝓝 x)
hs : s ∈ 𝓝[≠] f x
⊢ f ⁻¹' s ∈ 𝓝[≠] x

Tactic:
have : ∀ᶠ (z : 𝕜) in 𝓝 x, f z ∈ insert (f x) s := by
  filter_upwards [hfx.continuousAt.preimage_mem_nhds (insert_mem_nhds_iff.2 hs)]
  tauto

Result:
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
h₂f : ¬EventuallyConst f (𝓝 x)
hs : s ∈ 𝓝[≠] f x
this : ∀ᶠ (z : 𝕜) in 𝓝 x, f z ∈ insert (f x) s
⊢ f ⁻¹' s ∈ 𝓝[≠] x

Proof state:
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
h₂f : ¬EventuallyConst f (𝓝 x)
hs : s ∈ 𝓝[≠] f x
⊢ ∀ a ∈ f ⁻¹' insert (f x) s, f a ∈ insert (f x) s

Tactic:
tauto

Result:
no goals

Proof state:
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

Result:
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

Proof state:
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
h₂f : ¬EventuallyConst f (𝓝 x)
hs : s ∈ 𝓝[≠] f x
this : ∀ᶠ (z : 𝕜) in 𝓝 x, f z ∈ insert (f x) s
⊢ f ⁻¹' s ∈ 𝓝[≠] x

Tactic:
contrapose! h₂f with h

Result:
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
⊢ EventuallyConst f (𝓝 x)

Proof state:
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

Tactic:
intro z ⟨h₁z, h₂z⟩

Result:
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
z : 𝕜
h₁z : f z ∈ s → False
h₂z : f z ∈ insert (f x) s
⊢ f z = f x

Proof state:
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
h₂f : ¬EventuallyConst f (𝓝 x)
hs : s ∈ 𝓝[≠] f x
⊢ ∀ᶠ (z : 𝕜) in 𝓝 x, f z ∈ insert (f x) s

Tactic:
filter_upwards [hfx.continuousAt.preimage_mem_nhds (insert_mem_nhds_iff.2 hs)]

Result:
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
h₂f : ¬EventuallyConst f (𝓝 x)
hs : s ∈ 𝓝[≠] f x
⊢ ∀ a ∈ f ⁻¹' insert (f x) s, f a ∈ insert (f x) s

Proof state:
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 =ᶠ[𝓝 x] fun x_1 => f x

Tactic:
rw [EventuallyEq, ← hfx.frequently_eq_iff_eventually_eq analyticAt_const]

Result:
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

Proof state:
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
z : 𝕜
h₁z : f z ∈ s → False
h₂z : f z = f x ∨ f z ∈ s
⊢ f z = f x

Tactic:
tauto

Result:
no goals

Proof state:
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
⊢ EventuallyConst f (𝓝 x)

Tactic:
rw [eventuallyConst_iff_exists_eventuallyEq]

Result:
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
⊢ ∃ c, f =ᶠ[𝓝 x] fun x => c

Proof state:
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
⊢ ∃ c, f =ᶠ[𝓝 x] fun x => c

Tactic:
use f x

Result:
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 =ᶠ[𝓝 x] fun x_1 => f x

Proof state:
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
z : 𝕜
h₁z : f z ∈ s → False
h₂z : f z ∈ insert (f x) s
⊢ f z = f x

Tactic:
rw [Set.mem_insert_iff] at h₂z

Result:
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
z : 𝕜
h₁z : f z ∈ s → False
h₂z : f z = f x ∨ f z ∈ s
⊢ f z = f x