mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.ModelTheory.Definability

Goal state before:
pp: case intro.intro.mpr.refine_1
M : Type w
A : Set M
L : Language
inst✝² : L.Structure M
α : Type u₁
β : Type u_1
s : Set (β → M)
h✝ : A.Definable L s
f : α → β
inst✝¹ : Finite α
inst✝ : Finite β
val✝¹ : Fintype α
val✝ : Fintype β
h :
  A.Definable L
    ((fun g ↦ g ∘ rangeSplitting f) ⁻¹'
      (fun g ↦ g ∘ Sum.inl) ''
        (fun g ↦ g ∘ ⇑((Equiv.refl ↑(range f)).sumCongr (Fintype.equivFin ↑(range f)ᶜ).symm)) ''
          (fun g ↦ g ∘ ⇑(Equiv.Set.sumCompl (range f))) '' s)
h' : A.Definable L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}
y : β → M
ys : y ∈ s
⊢ ((y ∘ ⇑(Equiv.Set.sumCompl (range f))) ∘ ⇑((Equiv.refl ↑(range f)).sumCongr (Fintype.equivFin ↑(range f)ᶜ).symm)) ∘
      Sum.inl =
    (y ∘ f) ∘ rangeSplitting f

pp: case intro.intro.mpr.refine_2
M : Type w
A : Set M
L : Language
inst✝² : L.Structure M
α : Type u₁
β : Type u_1
s : Set (β → M)
h✝ : A.Definable L s
f : α → β
inst✝¹ : Finite α
inst✝ : Finite β
val✝¹ : Fintype α
val✝ : Fintype β
h :
  A.Definable L
    ((fun g ↦ g ∘ rangeSplitting f) ⁻¹'
      (fun g ↦ g ∘ Sum.inl) ''
        (fun g ↦ g ∘ ⇑((Equiv.refl ↑(range f)).sumCongr (Fintype.equivFin ↑(range f)ᶜ).symm)) ''
          (fun g ↦ g ∘ ⇑(Equiv.Set.sumCompl (range f))) '' s)
h' : A.Definable L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}
y : β → M
ys : y ∈ s
a : α
⊢ (y ∘ f) a = (y ∘ f) (rangeSplitting f (rangeFactorization f a))

Tactic:
· ext
        simp [Set.apply_rangeSplitting f]

Goal state after:
case intro.intro.mpr.refine_2
M : Type w
A : Set M
L : Language
inst✝² : L.Structure M
α : Type u₁
β : Type u_1
s : Set (β → M)
h✝ : A.Definable L s
f : α → β
inst✝¹ : Finite α
inst✝ : Finite β
val✝¹ : Fintype α
val✝ : Fintype β
h :
  A.Definable L
    ((fun g ↦ g ∘ rangeSplitting f) ⁻¹'
      (fun g ↦ g ∘ Sum.inl) ''
        (fun g ↦ g ∘ ⇑((Equiv.refl ↑(range f)).sumCongr (Fintype.equivFin ↑(range f)ᶜ).symm)) ''
          (fun g ↦ g ∘ ⇑(Equiv.Set.sumCompl (range f))) '' s)
h' : A.Definable L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}
y : β → M
ys : y ∈ s
a : α
⊢ (y ∘ f) a = (y ∘ f) (rangeSplitting f (rangeFactorization f a))
Module: Mathlib.ModelTheory.Definability

Goal state before:
pp: M : Type w
A : Set M
L : Language
inst✝¹ : L.Structure M
α : Type u₁
β : Type u_1
inst✝ : Finite β
S : Set (α ⊕ β → M)
φ : L[[↑A]].Formula (α ⊕ β)
hφ : S = ofPred φ.Realize
⊢ A.Definable L {v | ∃ u, Sum.elim v u ∈ S}

Tactic:
exists φ.iExs β

Goal state after:
M : Type w
A : Set M
L : Language
inst✝¹ : L.Structure M
α : Type u₁
β : Type u_1
inst✝ : Finite β
S : Set (α ⊕ β → M)
φ : L[[↑A]].Formula (α ⊕ β)
hφ : S = ofPred φ.Realize
⊢ {v | ∃ u, Sum.elim v u ∈ S} = ofPred (Formula.iExs β φ).Realize
Module: Mathlib.ModelTheory.Satisfiability

Goal state before:
pp: L : Language
M : Type w'
inst✝ : L.Structure M
iM : Infinite M
κ : Cardinal.{w}
h1 : lift.{w, max u v} L.card ≤ lift.{max u v, w} κ
h2 : lift.{max w u v, w'} #M ≤ lift.{max w' u v, w} κ
N0 : (L.elementaryDiagram M).ModelType
hN0 : lift.{max (max u w') v w, w} κ ≤ #↑N0
⊢ lift.{w, max (max w' u) v} L[[M]].card ≤ lift.{max (max w' u) v, w} κ

Tactic:
by
        simp only [card_withConstants, lift_add, lift_lift]
        rw [add_comm, add_eq_max (aleph0_le_lift.2 (infinite_iff.1 iM)), max_le_iff]
        rw [← lift_le.{w'}, lift_lift, lift_lift] at h1
        exact ⟨h2, h1⟩

Goal state after:
Module: Mathlib.ModelTheory.Satisfiability

Goal state before:
pp: L : Language
T : L.Theory
φ : L.Sentence
M : Type u_1
inst✝² : L.Structure M
inst✝¹ : M ⊨ T
inst✝ : Nonempty M
h : ¬M ⊨ φ
⊢ (T ∪ {Formula.not φ}).IsSatisfiable

Tactic:
have : M ⊨ T ∪ {Formula.not φ} := by
    simp only [Set.union_singleton, model_iff, Set.mem_insert_iff, forall_eq_or_imp,
      Sentence.realize_not]
    rw [← model_iff]
    exact ⟨h, inferInstance⟩

Goal state after:
L : Language
T : L.Theory
φ : L.Sentence
M : Type u_1
inst✝² : L.Structure M
inst✝¹ : M ⊨ T
inst✝ : Nonempty M
h : ¬M ⊨ φ
this : M ⊨ T ∪ {Formula.not φ}
⊢ (T ∪ {Formula.not φ}).IsSatisfiable
Module: Mathlib.RingTheory.Nullstellensatz

Goal state before:
pp: k : Type u_1
K : Type u_2
inst✝² : Field k
inst✝¹ : Field K
inst✝ : Algebra k K
σ : Type u_3
I : Ideal (MvPolynomial σ k)
p : MvPolynomial σ k
hp : p ∈ I.radical
x : σ → K
⊢ x ∈ zeroLocus K I → (aeval x) p = 0

Tactic:
hx

Goal state after:
k : Type u_1
K : Type u_2
inst✝² : Field k
inst✝¹ : Field K
inst✝ : Algebra k K
σ : Type u_3
I : Ideal (MvPolynomial σ k)
p : MvPolynomial σ k
hp : p ∈ I.radical
x : σ → K
hx : x ∈ zeroLocus K I
⊢ (aeval x) p = 0
Module: Mathlib.RingTheory.Nullstellensatz

Goal state before:
pp: k : Type u_1
K : Type u_2
inst✝⁴ : Field k
inst✝³ : Field K
inst✝² : Algebra k K
σ : Type u_3
inst✝¹ : IsAlgClosed K
inst✝ : Finite σ
I : Ideal (MvPolynomial σ k)
⊢ ∀ b ∈ {J | I ≤ J ∧ J.IsMaximal}, vanishingIdeal k (zeroLocus K I) ≤ b

Tactic:
rintro J ⟨hJI, hJ⟩

Goal state after:
k : Type u_1
K : Type u_2
inst✝⁴ : Field k
inst✝³ : Field K
inst✝² : Algebra k K
σ : Type u_3
inst✝¹ : IsAlgClosed K
inst✝ : Finite σ
I J : Ideal (MvPolynomial σ k)
hJI : I ≤ J
hJ : J.IsMaximal
⊢ vanishingIdeal k (zeroLocus K I) ≤ J
Module: Mathlib.ModelTheory.Satisfiability

Goal state before:
pp: case neg
L : Language
T : L.Theory
h : ∀ (M N : T.ModelType), ↑M ≅[L] ↑N
φ : L.Sentence
M : T.ModelType
hφ : ¬↑M ⊨ φ
⊢ T ⊨ᵇ Formula.not φ

Tactic:
exact models_sentence_iff.2 fun N => (Sentence.realize_not N).2
        (mt (elementarilyEquivalent_iff.1 (h M N) φ).2 hφ)

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Divisibility

Goal state before:
pp: case refine_1
n : ℕ
⊢ ¬IsUnit ↑n ↔ ¬IsUnit n

Tactic:
simp only [isUnit_iff, Nat.isUnit_iff]

Goal state after:
case refine_1
n : ℕ
⊢ ¬↑n = 1 ↔ ¬n = 1
Module: Mathlib.FieldTheory.Cardinality

Goal state before:
pp: case inl
α : Type u
h : Fintype α
⊢ Nonempty (Field α) ↔ ℵ₀ ≤ #α ∨ ∃ n, #α = ↑n ∧ IsPrimePow n

pp: case inr
α : Type u
h : Infinite α
⊢ Nonempty (Field α) ↔ ℵ₀ ≤ #α ∨ ∃ n, #α = ↑n ∧ IsPrimePow n

Tactic:
· simpa only [Cardinal.mk_fintype, Nat.cast_inj, exists_eq_left',
      Cardinal.natCast_lt_aleph0.not_ge, false_or] using Fintype.nonempty_field_iff

Goal state after:
case inr
α : Type u
h : Infinite α
⊢ Nonempty (Field α) ↔ ℵ₀ ≤ #α ∨ ∃ n, #α = ↑n ∧ IsPrimePow n
Module: Mathlib.FieldTheory.FinTrdeg

Goal state before:
pp: case mpr
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
h : Algebra.trdeg K L < Cardinal.aleph0
s : Set L
hs : IsTranscendenceBasis K Subtype.val
fin : s.Finite
⊢ ∃ E, E.FG ∧ Algebra.IsAlgebraic (↥E) L

Tactic:
refine ⟨adjoin K s, fg_adjoin_of_finite fin, ?_⟩

Goal state after:
case mpr
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
h : Algebra.trdeg K L < Cardinal.aleph0
s : Set L
hs : IsTranscendenceBasis K Subtype.val
fin : s.Finite
⊢ Algebra.IsAlgebraic (↥(adjoin K s)) L
Module: Mathlib.FieldTheory.Finite.Polynomial

Goal state before:
pp: σ K : Type u
inst✝² : Fintype K
inst✝¹ : Field K
inst✝ : Fintype σ
f : σ →₀ ℕ
⊢ f ∈ {s | ∀ (n : σ), s n ≤ Fintype.card K - 1} ↔ Finsupp.equivFunOnFinite f ∈ {s | ∀ (n : σ), s n < Fintype.card K}

Tactic:
refine forall_congr' fun n => le_tsub_iff_right ?_

Goal state after:
σ K : Type u
inst✝² : Fintype K
inst✝¹ : Field K
inst✝ : Fintype σ
f : σ →₀ ℕ
n : σ
⊢ 1 ≤ Fintype.card K
Module: Mathlib.Topology.Separation.Connected

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : PreconnectedSpace X
inst✝ : DiscreteTopology X
⊢ Subsingleton X

Tactic:
by
  by_contra! ⟨x, y, hxy⟩
  rw [Ne, ← mem_singleton_iff, (isClopen_discrete _).eq_univ <| singleton_nonempty y] at hxy
  exact hxy (mem_univ x)

Goal state after:
Module: Mathlib.FieldTheory.Galois.NormalBasis

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : FiniteDimensional K L
inst✝ : Infinite K
e : Module.Basis (Module.Free.ChooseBasisIndex K L) K L
M : Matrix Gal(L/K) Gal(L/K) (MvPolynomial (Module.Free.ChooseBasisIndex K L) L) :=
  Matrix.of fun i j ↦ ∑ k, i.symm (j (e k)) • MvPolynomial.X k
hq : Submodule.span L (Set.range fun k j ↦ j (e k)) = ⊤
c : Module.Free.ChooseBasisIndex K L → L
hc : (MvPolynomial.eval c) M.det = 1
hM : M.det ≠ 0
⊢ ∃ b, (MvPolynomial.eval (⇑(algebraMap K L) ∘ b)) M.det ≠ 0

Tactic:
by_contra! h
    refine hM (MvPolynomial.funext_set _
      (fun _ ↦ Set.infinite_range_of_injective (algebraMap K L).injective) fun x hx ↦ ?_)
    obtain ⟨x, rfl⟩ := Set.range_piMap _ ▸ hx
    simpa using! h x

Goal state after:
Module: Mathlib.FieldTheory.Galois.NormalBasis

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : FiniteDimensional K L
inst✝ : Infinite K
e : Module.Basis (Module.Free.ChooseBasisIndex K L) K L
M : Matrix Gal(L/K) Gal(L/K) (MvPolynomial (Module.Free.ChooseBasisIndex K L) L) :=
  Matrix.of fun i j ↦ ∑ k, i.symm (j (e k)) • MvPolynomial.X k
hq : Submodule.span L (Set.range fun k j ↦ j (e k)) = ⊤
c : Module.Free.ChooseBasisIndex K L → L
hc : (MvPolynomial.eval c) M.det = 1
hM : M.det ≠ 0
b : Module.Free.ChooseBasisIndex K L → K
hb : (MvPolynomial.eval (⇑(algebraMap K L) ∘ b)) M.det ≠ 0
a : Gal(L/K) → K
ha : ∑ i, a i • i (∑ k, b k • e k) = 0
⊢ (fun x ↦ (algebraMap K L) (a x)) = fun x ↦ (algebraMap K L) 0

Tactic:
map_zero

Goal state after:
K : Type u_1
L : Type u_2
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : FiniteDimensional K L
inst✝ : Infinite K
e : Module.Basis (Module.Free.ChooseBasisIndex K L) K L
M : Matrix Gal(L/K) Gal(L/K) (MvPolynomial (Module.Free.ChooseBasisIndex K L) L) :=
  Matrix.of fun i j ↦ ∑ k, i.symm (j (e k)) • MvPolynomial.X k
hq : Submodule.span L (Set.range fun k j ↦ j (e k)) = ⊤
c : Module.Free.ChooseBasisIndex K L → L
hc : (MvPolynomial.eval c) M.det = 1
hM : M.det ≠ 0
b : Module.Free.ChooseBasisIndex K L → K
hb : (MvPolynomial.eval (⇑(algebraMap K L) ∘ b)) M.det ≠ 0
a : Gal(L/K) → K
ha : ∑ i, a i • i (∑ k, b k • e k) = 0
⊢ (fun x ↦ (algebraMap K L) (a x)) = fun x ↦ 0
Module: Mathlib.FieldTheory.Galois.Profinite

Goal state before:
pp: case h.left
k : Type u_3
K : Type u_4
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
L : IntermediateField k K
inst✝ : Normal k ↥L
N : Set Gal(↥L/k)
L' : IntermediateField k ↥L
left✝ : FiniteDimensional k ↥L'
hO : ↑L'.fixingSubgroup ⊆ N
this : Module.Finite k ↥(IntermediateField.lift L')
x : Gal(K/k)
⊢ x ∈ ↑(IntermediateField.lift L').fixingSubgroup → x ∈ ⇑(AlgEquiv.restrictNormalHom ↥L) ⁻¹' N

Tactic:
hx

Goal state after:
case h.left
k : Type u_3
K : Type u_4
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
L : IntermediateField k K
inst✝ : Normal k ↥L
N : Set Gal(↥L/k)
L' : IntermediateField k ↥L
left✝ : FiniteDimensional k ↥L'
hO : ↑L'.fixingSubgroup ⊆ N
this : Module.Finite k ↥(IntermediateField.lift L')
x : Gal(K/k)
hx : x ∈ ↑(IntermediateField.lift L').fixingSubgroup
⊢ x ∈ ⇑(AlgEquiv.restrictNormalHom ↥L) ⁻¹' N
Module: Mathlib.FieldTheory.PerfectClosure

Goal state before:
pp: K : Type u
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
n : ℕ
ih : mk K p (n, 0) = 0
this : R K p (n, 0) (n + 1, (frobenius K p) 0)
⊢ R K p (n, 0) (n + 1, 0)

Tactic:
map_zero

Goal state after:
K : Type u
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
n : ℕ
ih : mk K p (n, 0) = 0
this : R K p (n, 0) (n + 1, 0)
⊢ R K p (n, 0) (n + 1, 0)
Module: Mathlib.FieldTheory.IsPerfectClosure

Goal state before:
pp: R : Type u_1
inst✝¹ : CommSemiring R
p : ℕ
inst✝ : ExpChar R p
h : Function.Injective ⇑(frobenius R p)
x : R
x✝ : ∃ n, x ^ p ^ n = 0
n : ℕ
h✝ : x ^ p ^ n = 0
⊢ (iterateFrobenius R p n) x = (iterateFrobenius R p n) 0

Tactic:
map_zero

Goal state after:
R : Type u_1
inst✝¹ : CommSemiring R
p : ℕ
inst✝ : ExpChar R p
h : Function.Injective ⇑(frobenius R p)
x : R
x✝ : ∃ n, x ^ p ^ n = 0
n : ℕ
h✝ : x ^ p ^ n = 0
⊢ (iterateFrobenius R p n) x = 0
Module: Mathlib.FieldTheory.IsPerfectClosure

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝³ : CommSemiring K
inst✝² : CommSemiring L
i : K →+* L
p : ℕ
inst✝¹ : ExpChar L p
inst✝ : PerfectRing L p
x : K
h✝ : x ∈ pNilradical K p
n : ℕ
h : i x = (iterateFrobeniusEquiv L p n).symm 0
⊢ x ∈ ker i

Tactic:
map_zero

Goal state after:
K : Type u_1
L : Type u_2
inst✝³ : CommSemiring K
inst✝² : CommSemiring L
i : K →+* L
p : ℕ
inst✝¹ : ExpChar L p
inst✝ : PerfectRing L p
x : K
h✝ : x ∈ pNilradical K p
n : ℕ
h : i x = 0
⊢ x ∈ ker i
Module: Mathlib.FieldTheory.IsPerfectClosure

Goal state before:
pp: K : Type u_1
L : Type u_2
M : Type u_3
inst✝⁶ : CommRing K
inst✝⁵ : CommRing L
inst✝⁴ : CommRing M
i : K →+* L
j : K →+* M
p : ℕ
inst✝³ : ExpChar M p
inst✝² : ExpChar K p
inst✝¹ : PerfectRing M p
inst✝ : IsPRadical i p
x : L
n : ℕ
y : K
n' : ℕ := (Classical.choose ⋯).1
h' : i (Classical.choose ⋯).2 = x ^ p ^ n'
h✝ : (Classical.choose ⋯).2 ^ p ^ n - y ^ p ^ n' ∈ RingHom.ker i
m : ℕ
h : ((Classical.choose ⋯).2 ^ p ^ n - y ^ p ^ n') ^ p ^ m = 0
⊢ (iterateFrobeniusEquiv M p (m + n)) (j (Classical.choose ⋯).2) =
    (iterateFrobeniusEquiv M p (m + n')) ((iterateFrobeniusEquiv M p n) ((iterateFrobeniusEquiv M p n).symm (j y)))

Tactic:
RingEquiv.apply_symm_apply,

Goal state after:
K : Type u_1
L : Type u_2
M : Type u_3
inst✝⁶ : CommRing K
inst✝⁵ : CommRing L
inst✝⁴ : CommRing M
i : K →+* L
j : K →+* M
p : ℕ
inst✝³ : ExpChar M p
inst✝² : ExpChar K p
inst✝¹ : PerfectRing M p
inst✝ : IsPRadical i p
x : L
n : ℕ
y : K
n' : ℕ := (Classical.choose ⋯).1
h' : i (Classical.choose ⋯).2 = x ^ p ^ n'
h✝ : (Classical.choose ⋯).2 ^ p ^ n - y ^ p ^ n' ∈ RingHom.ker i
m : ℕ
h : ((Classical.choose ⋯).2 ^ p ^ n - y ^ p ^ n') ^ p ^ m = 0
⊢ (iterateFrobeniusEquiv M p (m + n)) (j (Classical.choose ⋯).2) = (iterateFrobeniusEquiv M p (m + n')) (j y)
Module: Mathlib.FieldTheory.IsPerfectClosure

Goal state before:
pp: K : Type u_1
L : Type u_2
M : Type u_3
inst✝⁶ : CommRing K
inst✝⁵ : CommRing L
inst✝⁴ : CommRing M
i : K →+* L
j : K →+* M
p : ℕ
inst✝³ : ExpChar M p
inst✝² : ExpChar K p
inst✝¹ : PerfectRing M p
inst✝ : IsPRadical i p
x : L
n : ℕ
y : K
n' : ℕ := (Classical.choose ⋯).1
h' : i (Classical.choose ⋯).2 = x ^ p ^ n'
h✝ : (Classical.choose ⋯).2 ^ p ^ n - y ^ p ^ n' ∈ RingHom.ker i
m : ℕ
h : ((Classical.choose ⋯).2 ^ p ^ n - y ^ p ^ n') ^ p ^ m = 0
⊢ j 0 = 0

Tactic:
map_zero

Goal state after:
K : Type u_1
L : Type u_2
M : Type u_3
inst✝⁶ : CommRing K
inst✝⁵ : CommRing L
inst✝⁴ : CommRing M
i : K →+* L
j : K →+* M
p : ℕ
inst✝³ : ExpChar M p
inst✝² : ExpChar K p
inst✝¹ : PerfectRing M p
inst✝ : IsPRadical i p
x : L
n : ℕ
y : K
n' : ℕ := (Classical.choose ⋯).1
h' : i (Classical.choose ⋯).2 = x ^ p ^ n'
h✝ : (Classical.choose ⋯).2 ^ p ^ n - y ^ p ^ n' ∈ RingHom.ker i
m : ℕ
h : ((Classical.choose ⋯).2 ^ p ^ n - y ^ p ^ n') ^ p ^ m = 0
⊢ 0 = 0
Module: Mathlib.FieldTheory.Galois.Profinite

Goal state before:
pp: k : Type u_3
K : Type u_4
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
A : Set Gal(K/k)
⊢ A ∈ 𝓝 1 ↔ ∃ L, ↑L.fixingSubgroup ⊆ A

Tactic:
rw [krullTopology_mem_nhds_one_iff_of_normal]

Goal state after:
k : Type u_3
K : Type u_4
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
A : Set Gal(K/k)
⊢ (∃ E, FiniteDimensional k ↥E ∧ Normal k ↥E ∧ ↑E.fixingSubgroup ⊆ A) ↔ ∃ L, ↑L.fixingSubgroup ⊆ A
Module: Mathlib.FieldTheory.Galois.Profinite

Goal state before:
pp: k : Type u_3
K : Type u_4
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
⊢ Continuous ⇑(mulEquivToLimit k K).symm

Tactic:
apply continuous_of_continuousAt_one _ (continuousAt_def.mpr _)
  simp only [map_one, krullTopology_mem_nhds_one_iff_of_isGalois, ← MulEquiv.coe_toEquiv_symm,
    ← MulEquiv.toEquiv_eq_coe, ← (mulEquivToLimit k K).image_eq_preimage_symm]
  intro H ⟨L, le⟩
  rw [mem_nhds_iff]
  use mulEquivToLimit k K '' L.1.fixingSubgroup
  simp only [isOpen_mulEquivToLimit_image_fixingSubgroup L]
  simpa [one_mem] using Set.image_subset_iff.mp (Set.image_mono le)

Goal state after:
Module: Mathlib.FieldTheory.IsPerfectClosure

Goal state before:
pp: K : Type u_1
L : Type u_2
M : Type u_3
N : Type u_4
inst✝⁸ : CommRing K
inst✝⁷ : CommRing L
inst✝⁶ : CommRing M
inst✝⁵ : CommRing N
i : K →+* L
j : K →+* M
k : K →+* N
f : L →+* M
g : L →+* N
p : ℕ
inst✝⁴ : ExpChar M p
inst✝³ : ExpChar K p
inst✝² : PerfectRing M p
inst✝¹ : IsPRadical i p
inst✝ : ExpChar L p
⊢ i 0 = 0

Tactic:
map_zero

Goal state after:
K : Type u_1
L : Type u_2
M : Type u_3
N : Type u_4
inst✝⁸ : CommRing K
inst✝⁷ : CommRing L
inst✝⁶ : CommRing M
inst✝⁵ : CommRing N
i : K →+* L
j : K →+* M
k : K →+* N
f : L →+* M
g : L →+* N
p : ℕ
inst✝⁴ : ExpChar M p
inst✝³ : ExpChar K p
inst✝² : PerfectRing M p
inst✝¹ : IsPRadical i p
inst✝ : ExpChar L p
⊢ 0 = 0
Module: Mathlib.FieldTheory.PerfectClosure

Goal state before:
pp: K : Type u
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
x✝¹ : PerfectClosure K p
x : ℕ × K
x✝ : IsNilpotent (mk K p x)
n m : ℕ
h : (iterateFrobenius K p m) (x.2 ^ p ^ n) = 0
⊢ (iterateFrobenius K p (n + m)) x.2 = 0

Tactic:
by simpa only [iterateFrobenius_def, pow_add, pow_mul] using h

Goal state after:
Module: Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic

Goal state before:
pp: J : Type v
inst✝ : SmallCategory J
F : J ⥤ ProfiniteGrp.{max v u}
i j : J
f : i ⟶ j
⊢ ((Functor.const J).obj (ofProfinite (Profinite.limitCone (F ⋙ forget₂ ProfiniteGrp.{max u v} Profinite)).pt)).map f ≫
      { hom' := { toFun := fun x ↦ ↑x j, map_one' := ⋯, map_mul' := ⋯, continuous_toFun := ⋯ } } =
    { hom' := { toFun := fun x ↦ ↑x i, map_one' := ⋯, map_mul' := ⋯, continuous_toFun := ⋯ } } ≫ F.map f

Tactic:
by
      simp only [Functor.const_obj_obj, Functor.comp_obj,
        Functor.const_obj_map, Category.id_comp, Functor.comp_map]
      congr
      exact funext fun x => (x.2 f).symm

Goal state after:
Module: Mathlib.FieldTheory.IsPerfectClosure

Goal state before:
pp: K : Type u_1
L : Type u_2
M : Type u_3
N : Type u_4
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
x : K
n : ℕ
h : (iterateFrobenius K p n) x = (iterateFrobenius K p n) 0
⊢ x ∈ pNilradical K p

Tactic:
map_zero

Goal state after:
K : Type u_1
L : Type u_2
M : Type u_3
N : Type u_4
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
x : K
n : ℕ
h : (iterateFrobenius K p n) x = 0
⊢ x ∈ pNilradical K p
Module: Mathlib.GroupTheory.CosetCover

Goal state before:
pp: G : Type u_1
inst✝ : Group G
ι : Type u_2
s : Finset ι
H : Subgroup G
g : ι → G
hcovers : ⋃ i ∈ s, g i • ↑H = Set.univ
hind : H.index = s.card
this✝ : Fintype (G ⧸ H)
this : Function.Bijective fun x ↦ ↑(g ↑x)
i : ι
hi : i ∈ ↑s
j : ι
hj : j ∈ ↑s
h' : i ≠ j
c : Set G
hi' : c ⊆ (fun x ↦ g x • ↑H) i
hj' : c ⊆ (fun x ↦ g x • ↑H) j
x : G
⊢ x ∈ c → x ∈ ⊥

Tactic:
hx

Goal state after:
G : Type u_1
inst✝ : Group G
ι : Type u_2
s : Finset ι
H : Subgroup G
g : ι → G
hcovers : ⋃ i ∈ s, g i • ↑H = Set.univ
hind : H.index = s.card
this✝ : Fintype (G ⧸ H)
this : Function.Bijective fun x ↦ ↑(g ↑x)
i : ι
hi : i ∈ ↑s
j : ι
hj : j ∈ ↑s
h' : i ≠ j
c : Set G
hi' : c ⊆ (fun x ↦ g x • ↑H) i
hj' : c ⊆ (fun x ↦ g x • ↑H) j
x : G
hx : x ∈ c
⊢ x ∈ ⊥
Module: Mathlib.FieldTheory.Laurent

Goal state before:
pp: R : Type u
inst✝ : CommRing R
r : R
p : R[X]
hp : p ∈ R[X]⁰
x : R[X]
⊢ x * (taylor r) p = 0 → x = 0

Tactic:
hx

Goal state after:
R : Type u
inst✝ : CommRing R
r : R
p : R[X]
hp : p ∈ R[X]⁰
x : R[X]
hx : x * (taylor r) p = 0
⊢ x = 0
Module: Mathlib.FieldTheory.KummerExtension

Goal state before:
pp: case prime_mul
p n : ℕ
hp : Nat.Prime p
IH :
  ∀ {K : Type u} [inst : Field K],
    Odd n → ∀ {a : K}, (∀ (p : ℕ), Nat.Prime p → p ∣ n → ∀ (b : K), b ^ p ≠ a) → Irreducible (X ^ n - C a)
K : Type u
inst✝² : Field K
hn : Odd (p * n)
a : K
ha : ∀ (p_1 : ℕ), Nat.Prime p_1 → p_1 ∣ p * n → ∀ (b : K), b ^ p_1 ≠ a
E : Type u
inst✝¹ : Field E
inst✝ : Algebra K E
x : E
⊢ minpoly K x = X ^ p - C a → Irreducible (X ^ n - C (AdjoinSimple.gen K x))

Tactic:
hx

Goal state after:
case prime_mul
p n : ℕ
hp : Nat.Prime p
IH :
  ∀ {K : Type u} [inst : Field K],
    Odd n → ∀ {a : K}, (∀ (p : ℕ), Nat.Prime p → p ∣ n → ∀ (b : K), b ^ p ≠ a) → Irreducible (X ^ n - C a)
K : Type u
inst✝² : Field K
hn : Odd (p * n)
a : K
ha : ∀ (p_1 : ℕ), Nat.Prime p_1 → p_1 ∣ p * n → ∀ (b : K), b ^ p_1 ≠ a
E : Type u
inst✝¹ : Field E
inst✝ : Algebra K E
x : E
hx : minpoly K x = X ^ p - C a
⊢ Irreducible (X ^ n - C (AdjoinSimple.gen K x))
Module: Mathlib.FieldTheory.KummerExtension

Goal state before:
pp: K : Type u
inst✝ : Field K
n : ℕ
hζ : (primitiveRoots n K).Nonempty
a : K
H : Irreducible (X ^ n - C a)
η : ↥(rootsOfUnity n K)
this : ↑↑η ^ n = 1
⊢ (aeval (↑↑η • root (X ^ n - C a))) (X ^ n - C a) = 0

Tactic:
rw [map_sub, map_pow, aeval_C, aeval_X, Algebra.smul_def, mul_pow, root_X_pow_sub_C_pow,
      AdjoinRoot.algebraMap_eq, ← map_pow, this, map_one, one_mul, sub_self]

Goal state after:
Module: Mathlib.FieldTheory.KummerExtension

Goal state before:
pp: case isFalse
K : Type u
inst✝¹ : Field K
n : ℕ
hζ : (primitiveRoots n K).Nonempty
a : K
H : Irreducible (X ^ n - C a)
inst✝ : NeZero n
σ : Gal(K[n√a]/K)
this✝¹ : Fact (Irreducible (X ^ n - C a)) := { out := H }
this✝ : IsDomain K[n√a] := inferInstance
this : DecidableEq K := Classical.decEq K
h✝ : ¬a = 0
⊢ (algebraMap K K[n√a]) a ≠ 0

Tactic:
rwa [Ne, map_eq_zero_iff _ (algebraMap K _).injective]

Goal state after:
Module: Mathlib.FieldTheory.KummerExtension

Goal state before:
pp: case pos
K : Type u
inst✝¹ : Field K
n : ℕ
hζ : (primitiveRoots n K).Nonempty
a : K
H : Irreducible (X ^ n - C a)
inst✝ : NeZero n
σ : Gal(K[n√a]/K)
this : Fact (Irreducible (X ^ n - C a))
h : root (X ^ n - C a) = 0
⊢ 0 = σ 0

Tactic:
map_zero

Goal state after:
case pos
K : Type u
inst✝¹ : Field K
n : ℕ
hζ : (primitiveRoots n K).Nonempty
a : K
H : Irreducible (X ^ n - C a)
inst✝ : NeZero n
σ : Gal(K[n√a]/K)
this : Fact (Irreducible (X ^ n - C a))
h : root (X ^ n - C a) = 0
⊢ 0 = 0
Module: Mathlib.FieldTheory.KummerExtension

Goal state before:
pp: case refine_1
K : Type u
inst✝⁵ : Field K
L : Type u_1
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : FiniteDimensional K L
hK : (primitiveRoots (finrank K L) K).Nonempty
inst✝¹ : IsGalois K L
inst✝ : IsCyclic Gal(L/K)
ζ : K
hζ : IsPrimitiveRoot ζ (finrank K L)
σ : Gal(L/K)
hσ : Function.Surjective fun x ↦ σ ^ x
hσ'✝ : orderOf σ = Nat.card Gal(L/K)
this : (minpoly K σ.toLinearMap).IsRoot ζ
v : L
hv : End.HasEigenvector σ.toLinearMap ζ v
hv' : ∀ (n : ℕ), (σ ^ n) v = ζ ^ n • v
n : ℕ
hσ' : σ ^ n ∈ OrderDual.ofDual ⊥
⊢ (ζ ^ finrank K L) ^ n • v ^ finrank K L = v ^ finrank K L

Tactic:
hζ.pow_eq_one,

Goal state after:
case refine_1
K : Type u
inst✝⁵ : Field K
L : Type u_1
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : FiniteDimensional K L
hK : (primitiveRoots (finrank K L) K).Nonempty
inst✝¹ : IsGalois K L
inst✝ : IsCyclic Gal(L/K)
ζ : K
hζ : IsPrimitiveRoot ζ (finrank K L)
σ : Gal(L/K)
hσ : Function.Surjective fun x ↦ σ ^ x
hσ'✝ : orderOf σ = Nat.card Gal(L/K)
this : (minpoly K σ.toLinearMap).IsRoot ζ
v : L
hv : End.HasEigenvector σ.toLinearMap ζ v
hv' : ∀ (n : ℕ), (σ ^ n) v = ζ ^ n • v
n : ℕ
hσ' : σ ^ n ∈ OrderDual.ofDual ⊥
⊢ 1 ^ n • v ^ finrank K L = v ^ finrank K L
Module: Mathlib.FieldTheory.KummerExtension

Goal state before:
pp: case refine_1
K : Type u
inst✝⁵ : Field K
L : Type u_1
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : FiniteDimensional K L
hK : (primitiveRoots (finrank K L) K).Nonempty
inst✝¹ : IsGalois K L
inst✝ : IsCyclic Gal(L/K)
ζ : K
hζ : IsPrimitiveRoot ζ (finrank K L)
σ : Gal(L/K)
hσ : Function.Surjective fun x ↦ σ ^ x
hσ' : orderOf σ = Nat.card Gal(L/K)
this : (minpoly K σ.toLinearMap).IsRoot ζ
v : L
hv : End.HasEigenvector σ.toLinearMap ζ v
hv' : ∀ (n : ℕ), (σ ^ n) v = ζ ^ n • v
⊢ v ^ finrank K L ∈ Set.range ⇑(algebraMap K L)

Tactic:
rw [← IntermediateField.mem_bot,
      ← OrderIso.map_bot IsGalois.intermediateFieldEquivSubgroup.symm]
    intro ⟨σ', hσ'⟩
    obtain ⟨n, rfl : σ ^ n = σ'⟩ := mem_powers_iff_mem_zpowers.mpr (hσ σ')
    rw [smul_pow', Submonoid.smul_def, AlgEquiv.smul_def, hv', smul_pow, ← pow_mul,
      mul_comm, pow_mul, hζ.pow_eq_one, one_pow, one_smul]

Goal state after: