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

Goal state before:
pp: L : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝³ : L.Structure M
inst✝² : L.Structure N
inst✝¹ : L.Structure P
S : L.Substructure M
s : Set M
inst✝ : IsEmpty L.Constants
x : M
h : ∅ ∈ (closure L).closed
⊢ x ∉ (closure L).toFun ∅

Tactic:
← SetLike.mem_coe,

Goal state after:
L : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝³ : L.Structure M
inst✝² : L.Structure N
inst✝¹ : L.Structure P
S : L.Substructure M
s : Set M
inst✝ : IsEmpty L.Constants
x : M
h : ∅ ∈ (closure L).closed
⊢ x ∉ ↑((closure L).toFun ∅)
Module: Mathlib.ModelTheory.Semantics

Goal state before:
pp: L : Language
M : Type w
N : Type u_1
inst✝² : L.Structure M
inst✝¹ : L.Structure N
inst✝ : N ⊨ L.completeTheory M
φ : L.Sentence
h : M ⊨ Formula.not φ
⊢ N ⊨ Formula.not φ

Tactic:
exact (L.completeTheory M).realize_sentence_of_mem (mem_completeTheory.2 h)

Goal state after:
Module: Mathlib.ModelTheory.Semantics

Goal state before:
pp: case succ
L : Language
M : Type w
inst✝ : L.Structure M
α : Type u'
n✝ : ℕ
v : α → M
n : ℕ
ih : ∀ {φ : L.BoundedFormula α n}, φ.alls.Realize v ↔ ∀ (xs : Fin n → M), φ.Realize v xs
φ : L.BoundedFormula α (n + 1)
⊢ φ.alls.Realize v ↔ ∀ (xs : Fin (n + 1) → M), φ.Realize v xs

Tactic:
simp only [alls, ih, Realize]

Goal state after:
case succ
L : Language
M : Type w
inst✝ : L.Structure M
α : Type u'
n✝ : ℕ
v : α → M
n : ℕ
ih : ∀ {φ : L.BoundedFormula α n}, φ.alls.Realize v ↔ ∀ (xs : Fin n → M), φ.Realize v xs
φ : L.BoundedFormula α (n + 1)
⊢ (∀ (xs : Fin n → M) (x : M), φ.Realize v (snoc xs x)) ↔ ∀ (xs : Fin (n + 1) → M), φ.Realize v xs
Module: Mathlib.ModelTheory.Skolem

Goal state before:
pp: L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : L.Structure M
⊢ Small.{max u v, w} ↥⊥.elementarySkolem₁Reduct

Tactic:
rw [coeSort_elementarySkolem₁Reduct]

Goal state after:
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : L.Structure M
⊢ Small.{max u v, w} ↥⊥
Module: Mathlib.ModelTheory.Semantics

Goal state before:
pp: L : Language
M : Type w
inst✝ : L.Structure M
r : L.Relations 2
⊢ M ⊨ r.irreflexive ↔ Std.Irrefl fun x y ↦ RelMap r ![x, y]

Tactic:
by
  rw [irrefl_def]
  exact forall_congr' fun _ ↦ not_congr realize_rel₂

Goal state after:
Module: Mathlib.ModelTheory.Semantics

Goal state before:
pp: case refine_2
L : Language
α : Type u'
M : Type w
inst✝ : L[[α]].Structure M
s : Set α
⊢ Set.InjOn (fun i ↦ ↑(L.con i)) s →
    ∀ (φ : L[[α]].Sentence) (x x_1 : α),
      (x, x_1) ∈ s ×ˢ s ∩ (Set.diagonal α)ᶜ → ((L.con x).term.equal (L.con x_1).term).not = φ → M ⊨ φ

Tactic:
rintro h φ a b ⟨⟨as, bs⟩, ab⟩ rfl

Goal state after:
case refine_2
L : Language
α : Type u'
M : Type w
inst✝ : L[[α]].Structure M
s : Set α
h : Set.InjOn (fun i ↦ ↑(L.con i)) s
a b : α
ab : (a, b) ∈ (Set.diagonal α)ᶜ
as : (a, b).1 ∈ s
bs : (a, b).2 ∈ s
⊢ M ⊨ ((L.con a).term.equal (L.con b).term).not
Module: Mathlib.ModelTheory.Algebra.Field.IsAlgClosed

Goal state before:
pp: p : ℕ
κ : Cardinal.{u_2}
hκ : ℵ₀ < κ
M : Type u_2
struc✝¹ : Language.ring.Structure M
is_model✝¹ : M ⊨ Theory.ACF p
nonempty'✝¹ : Nonempty M
N : Type u_2
struc✝ : Language.ring.Structure N
is_model✝ : N ⊨ Theory.ACF p
nonempty'✝ : Nonempty N
hM : #↑{ Carrier := M, struc := struc✝¹, is_model := is_model✝¹, nonempty' := nonempty'✝¹ } = κ
hN : #↑{ Carrier := N, struc := struc✝, is_model := is_model✝, nonempty' := nonempty'✝ } = κ
x✝³ : Field M := fieldOfModelACF p M
this✝⁴ : M ⊨ Theory.field
x✝² : CompatibleRing M := compatibleRingOfModelField M
this✝³ : IsAlgClosed M
this✝² : CharP M p
x✝¹ : Field N := fieldOfModelACF p N
this✝¹ : N ⊨ Theory.field
x✝ : CompatibleRing N := compatibleRingOfModelField N
this✝ : IsAlgClosed N
this : CharP N p
⊢ ↑{ Carrier := M, struc := struc✝¹, is_model := is_model✝¹, nonempty' := nonempty'✝¹ } ≃+*
    ↑{ Carrier := N, struc := struc✝, is_model := is_model✝, nonempty' := nonempty'✝ }

Tactic:
apply Classical.choice

Goal state after:
p : ℕ
κ : Cardinal.{u_2}
hκ : ℵ₀ < κ
M : Type u_2
struc✝¹ : Language.ring.Structure M
is_model✝¹ : M ⊨ Theory.ACF p
nonempty'✝¹ : Nonempty M
N : Type u_2
struc✝ : Language.ring.Structure N
is_model✝ : N ⊨ Theory.ACF p
nonempty'✝ : Nonempty N
hM : #↑{ Carrier := M, struc := struc✝¹, is_model := is_model✝¹, nonempty' := nonempty'✝¹ } = κ
hN : #↑{ Carrier := N, struc := struc✝, is_model := is_model✝, nonempty' := nonempty'✝ } = κ
x✝³ : Field M := ⋯
this✝⁴ : M ⊨ Theory.field
x✝² : CompatibleRing M := ⋯
this✝³ : IsAlgClosed M
this✝² : CharP M p
x✝¹ : Field N := ⋯
this✝¹ : N ⊨ Theory.field
x✝ : CompatibleRing N := ⋯
this✝ : IsAlgClosed N
this : CharP N p
⊢ Nonempty
    (↑{ Carrier := M, struc := struc✝¹, is_model := is_model✝¹, nonempty' := nonempty'✝¹ } ≃+*
      ↑{ Carrier := N, struc := struc✝, is_model := is_model✝, nonempty' := nonempty'✝ })
Module: Mathlib.ModelTheory.Algebra.Ring.Definability

Goal state before:
pp: ι : Type u_1
K : Type u_2
inst✝¹ : Field K
inst✝ : CompatibleRing K
S : Finset (MvPolynomial ι K)
p' : ↥S → FreeCommRing ((i : ↥S) × ↥(↑i).support ⊕ ι) := genericPolyMap fun p ↦ (↑p).support
this✝ : DecidableEq ι := Classical.decEq ι
this : DecidableEq K := Classical.decEq K
i : ↥S
p : (i : ↥S) × ↥(↑i).support
⊢ coeff ↑p.snd ↑p.fst ∈ ⋃ p ∈ S, (fun m ↦ coeff m p) '' ↑p.support

Tactic:
simp only [Set.mem_iUnion]

Goal state after:
ι : Type u_1
K : Type u_2
inst✝¹ : Field K
inst✝ : CompatibleRing K
S : Finset (MvPolynomial ι K)
p' : ↥S → FreeCommRing ((i : ↥S) × ↥(↑i).support ⊕ ι) := genericPolyMap fun p ↦ (↑p).support
this✝ : DecidableEq ι := Classical.decEq ι
this : DecidableEq K := Classical.decEq K
i : ↥S
p : (i : ↥S) × ↥(↑i).support
⊢ ∃ i, ∃ (_ : i ∈ S), coeff ↑p.snd ↑p.fst ∈ (fun m ↦ coeff m i) '' ↑i.support
Module: Mathlib.ModelTheory.Algebra.Field.IsAlgClosed

Goal state before:
pp: φ : Language.ring.Sentence
T0 : Finset Language.ring.Sentence
hT0 : ↑T0 ⊆ Theory.ACF 0
h : ↑T0 ⊨ᵇ φ
ψ : Language.ring.Sentence
hψ : ψ ∈ Theory.field ∪ genericMonicPolyHasRoot '' {n | 0 < n} ∨ ∃ x ∈ {q | Nat.Prime q}, ∼(eqZero x) = ψ
⊢ { s // ∀ q ∉ s, Theory.ACF ↑q ⊨ᵇ ψ }

Tactic:
apply Classical.choice

Goal state after:
φ : Language.ring.Sentence
T0 : Finset Language.ring.Sentence
hT0 : ↑T0 ⊆ Theory.ACF 0
h : ↑T0 ⊨ᵇ φ
ψ : Language.ring.Sentence
hψ : ψ ∈ Theory.field ∪ genericMonicPolyHasRoot '' {n | 0 < n} ∨ ∃ x ∈ {q | Nat.Prime q}, ∼(eqZero x) = ψ
⊢ Nonempty { s // ∀ q ∉ s, Theory.ACF ↑q ⊨ᵇ ψ }
Module: Mathlib.ModelTheory.Satisfiability

Goal state before:
pp: L : Language
T : L.Theory
h : T.IsComplete
M : Type u_1
inst✝² : L.Structure M
inst✝¹ : M ⊨ T
inst✝ : Nonempty M
φ : L.BoundedFormula Empty 0
h_realize : M ⊨ φ
x✝ : T ⊨ᵇ φ ∨ T ⊨ᵇ Formula.not φ
⊢ T ⊨ᵇ φ

Tactic:
cases h.2 φ with

Goal state after:
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)
hI : I.IsMaximal
this✝¹ : Field (MvPolynomial σ k ⧸ I) := Quotient.field I
this✝ : Algebra.IsAlgebraic k (MvPolynomial σ k ⧸ I)
φ : MvPolynomial σ k ⧸ I →ₐ[k] K := IsAlgClosed.lift
x : σ → K := fun s ↦ φ ((Ideal.Quotient.mk I) (X s))
this : aeval x = φ.comp (Quotient.mkₐ k I)
⊢ ∃ x, I = vanishingIdeal k {x}

Tactic:
use x

Goal state after:
case h
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)
hI : I.IsMaximal
this✝¹ : Field (MvPolynomial σ k ⧸ I) := ⋯
this✝ : Algebra.IsAlgebraic k (MvPolynomial σ k ⧸ I)
φ : MvPolynomial σ k ⧸ I →ₐ[k] K := ⋯
x : σ → K := ⋯
this : aeval x = φ.comp (Quotient.mkₐ k I)
⊢ I = vanishingIdeal k {x}
Module: Mathlib.ModelTheory.Satisfiability

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

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 φ ∨ T ⊨ᵇ Formula.not φ

Tactic:
· left
      exact models_sentence_iff.2 fun N => (elementarilyEquivalent_iff.1 (h M N) φ).1 hφ

Goal state after:
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 φ ∨ T ⊨ᵇ Formula.not φ
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 J : Ideal (MvPolynomial σ k)
hJI : I ≤ J
hJ : J.IsMaximal
x : σ → K
hx : J = vanishingIdeal k {x}
y : σ → K
hy : y ∈ {x}
p : MvPolynomial σ k
hp : p ∈ I
⊢ p ∈ vanishingIdeal k {y}

Tactic:
Set.mem_singleton_iff.1 hy,

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
x : σ → K
hx : J = vanishingIdeal k {x}
y : σ → K
hy : y ∈ {x}
p : MvPolynomial σ k
hp : p ∈ I
⊢ p ∈ vanishingIdeal k {x}
Module: Mathlib.FieldTheory.AxGrothendieck

Goal state before:
pp: case intro
ι : Type u_1
K : Type u_2
R : Type u_3
inst✝⁴ : Field K
inst✝³ : Finite K
inst✝² : CommRing R
inst✝¹ : Finite ι
inst✝ : Algebra K R
alg : Algebra.IsAlgebraic K R
ps : ι → MvPolynomial ι R
S : Set (ι → R)
hm : Set.MapsTo (fun v i ↦ (eval v) (ps i)) S S
hinj : Set.InjOn (fun v i ↦ (eval v) (ps i)) S
is_int : ∀ (x : R), IsIntegral K x
v : ι → R
hvS : v ∈ S
val✝ : Fintype ι
s : Finset R := (univ.biUnion fun i ↦ image (fun x ↦ coeff x (ps i)) (ps i).support) ∪ image v univ
hv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s
hs₁ : ∀ (i : ι), ∀ k ∈ (ps i).support, coeff k (ps i) ∈ Algebra.adjoin K ↑s
this✝ : IsNoetherian K ↥(Algebra.adjoin K ↑s)
this : Finite ↥(Algebra.adjoin K ↑s)
S' : Set (ι → ↥(Algebra.adjoin K ↑s)) := (fun v ↦ Subtype.val ∘ v) ⁻¹' S
res : ↑S' → ↑S' := fun x ↦ ⟨fun i ↦ ⟨(eval fun j ↦ ↑(↑x j)) (ps i), ⋯⟩, ⋯⟩
hres_surj : Function.Surjective res
w : ι → ↥(Algebra.adjoin K ↑s)
hwS' : w ∈ S'
hw : res ⟨w, hwS'⟩ = ⟨fun i ↦ ⟨v i, ⋯⟩, hvS⟩
⊢ v ∈ (fun v i ↦ (eval v) (ps i)) '' S

Tactic:
refine ⟨fun i => w i, hwS', ?_⟩

Goal state after:
case intro
ι : Type u_1
K : Type u_2
R : Type u_3
inst✝⁴ : Field K
inst✝³ : Finite K
inst✝² : CommRing R
inst✝¹ : Finite ι
inst✝ : Algebra K R
alg : Algebra.IsAlgebraic K R
ps : ι → MvPolynomial ι R
S : Set (ι → R)
hm : Set.MapsTo (fun v i ↦ (eval v) (ps i)) S S
hinj : Set.InjOn (fun v i ↦ (eval v) (ps i)) S
is_int : ∀ (x : R), IsIntegral K x
v : ι → R
hvS : v ∈ S
val✝ : Fintype ι
s : Finset R := (univ.biUnion fun i ↦ image (fun x ↦ coeff x (ps i)) (ps i).support) ∪ image v univ
hv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s
hs₁ : ∀ (i : ι), ∀ k ∈ (ps i).support, coeff k (ps i) ∈ Algebra.adjoin K ↑s
this✝ : IsNoetherian K ↥(Algebra.adjoin K ↑s)
this : Finite ↥(Algebra.adjoin K ↑s)
S' : Set (ι → ↥(Algebra.adjoin K ↑s)) := (fun v ↦ Subtype.val ∘ v) ⁻¹' S
res : ↑S' → ↑S' := fun x ↦ ⟨fun i ↦ ⟨(eval fun j ↦ ↑(↑x j)) (ps i), ⋯⟩, ⋯⟩
hres_surj : Function.Surjective res
w : ι → ↥(Algebra.adjoin K ↑s)
hwS' : w ∈ S'
hw : res ⟨w, hwS'⟩ = ⟨fun i ↦ ⟨v i, ⋯⟩, hvS⟩
⊢ ((fun v i ↦ (eval v) (ps i)) fun i ↦ ↑(w i)) = v
Module: Mathlib.GroupTheory.OreLocalization.Cardinality

Goal state before:
pp: R : Type u
inst✝² : Monoid R
S : Submonoid R
inst✝¹ : OreSet S
X : Type v
inst✝ : MulAction R X
hc : ∀ (s s' : ↥S), Commute s s'
h✝ : Infinite X
key : ∀ (x : X) (s s' : ↥S), s • x = s' • x → Commute s s' → x /ₒ s = x /ₒ s'
i : X × ↥S → OreLocalization S X := fun x ↦ x.1 /ₒ x.2
hsurj : Surjective i
hi : RightInverse (surjInv hsurj) i
j : OreLocalization S X → X × X := (fun x ↦ (x.1, x.2 • x.1)) ∘ surjInv hsurj
this : Injective j
⊢ #(OreLocalization S X) ≤ lift.{u, v} #X

Tactic:
by
    have := lift_mk_le_lift_mk_of_injective this
    rwa [lift_umax.{v, u}, lift_id', mk_prod, lift_id, lift_mul, mul_eq_self (by simp)] at this

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

Goal state before:
pp: a : Cardinal.{u}
ha : ℵ₀ ≤ a
⊢ ¬Irreducible a

Tactic:
rw [irreducible_iff, not_and_or]

Goal state after:
a : Cardinal.{u}
ha : ℵ₀ ≤ a
⊢ ¬¬IsUnit a ∨ ¬∀ ⦃a_1 b : Cardinal.{u}⦄, a = a_1 * b → IsUnit a_1 ∨ IsUnit b
Module: Mathlib.FieldTheory.Finite.Extension

Goal state before:
pp: k : Type u_1
inst✝⁴ : Field k
inst✝³ : Finite k
p : ℕ
inst✝² : Fact (Nat.Prime p)
inst✝¹ : CharP k p
n : ℕ
inst✝ : NeZero n
i : ℕ
x : Extension k p n
⊢ (frob k p n ^ i) x = x ^ Nat.card k ^ i

Tactic:
by
  induction i generalizing x with
  | zero => simp
  | succ i ih =>
      rw [pow_add, pow_one, AlgEquiv.mul_apply, ih, frob_apply, ← pow_mul, ← Nat.pow_add_one']

Goal state after:
Module: Mathlib.FieldTheory.Finite.Polynomial

Goal state before:
pp: case mul_X
σ : Type u_1
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : MvPolynomial σ (ZMod p)
⊢ ∀ (p_1 : MvPolynomial σ (ZMod p)) (n : σ),
    (frobenius (MvPolynomial σ (ZMod p)) p) p_1 = (expand p) p_1 →
      (frobenius (MvPolynomial σ (ZMod p)) p) (p_1 * X n) = (expand p) (p_1 * X n)

Tactic:
simp only [expand_X, map_mul]
    intro _ _ hf; rw [hf, frobenius_def]

Goal state after:
Module: Mathlib.FieldTheory.Differential.Liouville

Goal state before:
pp: F : Type u_1
K : Type u_2
inst✝⁹ : Field F
inst✝⁸ : Field K
inst✝⁷ : Differential F
inst✝⁶ : Differential K
inst✝⁵ : Algebra F K
inst✝⁴ : DifferentialAlgebra F K
inst✝³ : CharZero F
inst✝² : FiniteDimensional F K
inst✝¹ : IsGalois F K
a : F
ι : Type
inst✝ : Fintype ι
c : ι → F
hc : ∀ (x : ι), (c x)′ = 0
u : ι → K
v : K
h : ↑a = ∑ x, ↑(c x) * logDeriv (u x) + v′
⊢ ∃ ι₀ x c₀, ∃ (_ : ∀ (x : ι₀), (c₀ x)′ = 0), ∃ u₀ v₀, a = ∑ x, c₀ x * logDeriv (u₀ x) + v₀′

Tactic:
have : CharZero K := charZero_of_injective_algebraMap
      (FaithfulSMul.algebraMap_injective F K)

Goal state after:
F : Type u_1
K : Type u_2
inst✝⁹ : Field F
inst✝⁸ : Field K
inst✝⁷ : Differential F
inst✝⁶ : Differential K
inst✝⁵ : Algebra F K
inst✝⁴ : DifferentialAlgebra F K
inst✝³ : CharZero F
inst✝² : FiniteDimensional F K
inst✝¹ : IsGalois F K
a : F
ι : Type
inst✝ : Fintype ι
c : ι → F
hc : ∀ (x : ι), (c x)′ = 0
u : ι → K
v : K
h : ↑a = ∑ x, ↑(c x) * logDeriv (u x) + v′
this : CharZero K
⊢ ∃ ι₀ x c₀, ∃ (_ : ∀ (x : ι₀), (c₀ x)′ = 0), ∃ u₀ v₀, a = ∑ x, c₀ x * logDeriv (u₀ x) + v₀′
Module: Mathlib.FieldTheory.Galois.NormalBasis

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : Finite L
this✝ : Finite K
this : Fintype K
x : L
hx :
  (toSpanSingleton K[X] (AEval' (frobeniusAlgHom K L).toLinearMap)
        ((AEval'.of (frobeniusAlgHom K L).toLinearMap) x)).ker =
    Module.annihilator K[X] (AEval' (frobeniusAlgHom K L).toLinearMap)
⊢ ∃ x, LinearIndependent K fun σ ↦ σ x

Tactic:
use x

Goal state after:
case h
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : Finite L
this✝ : Finite K
this : Fintype K
x : L
hx :
  (toSpanSingleton K[X] (AEval' (frobeniusAlgHom K L).toLinearMap)
        ((AEval'.of (frobeniusAlgHom K L).toLinearMap) x)).ker =
    Module.annihilator K[X] (AEval' (frobeniusAlgHom K L).toLinearMap)
⊢ LinearIndependent K fun σ ↦ σ x
Module: Mathlib.FieldTheory.CardinalEmb

Goal state before:
pp: case pos
F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
rank_inf : Fact (ℵ₀ ≤ Module.rank F E)
inst✝ : Algebra.IsAlgebraic F E
i : WithTop (Module.rank F E).ord.ToType
hi : IsSuccPrelimit i
f : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h : Nonempty ↑(Iio i)
x : E
hx✝ : x ∈ (filtration i).toSubalgebra
hx : x ∈ ⨆ j, filtration ↑j
⊢ ((fun f ↦ Subalgebra.iSupLift (fun x ↦ ((fun j ↦ filtration ↑j) x).toSubalgebra) ⋯ ↑f ⋯ (filtration i).toSubalgebra ⋯)
        ((fun f ↦ ⟨fun j ↦ embFunctor F E ⋯ f, ⋯⟩) f))
      ⟨x, hx✝⟩ =
    f ⟨x, hx✝⟩

Tactic:
← SetLike.mem_coe,

Goal state after:
case pos
F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
rank_inf : Fact (ℵ₀ ≤ Module.rank F E)
inst✝ : Algebra.IsAlgebraic F E
i : WithTop (Module.rank F E).ord.ToType
hi : IsSuccPrelimit i
f : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h : Nonempty ↑(Iio i)
x : E
hx✝ : x ∈ (filtration i).toSubalgebra
hx : x ∈ ↑(⨆ j, filtration ↑j)
⊢ ((fun f ↦ Subalgebra.iSupLift (fun x ↦ ((fun j ↦ filtration ↑j) x).toSubalgebra) ⋯ ↑f ⋯ (filtration i).toSubalgebra ⋯)
        ((fun f ↦ ⟨fun j ↦ embFunctor F E ⋯ f, ⋯⟩) f))
      ⟨x, hx✝⟩ =
    f ⟨x, hx✝⟩
Module: Mathlib.FieldTheory.Galois.Profinite

Goal state before:
pp: case op.op
k : Type u_1
K : Type u_2
inst✝² : Field k
inst✝¹ : Field K
inst✝ : Algebra k K
L₃ : (FiniteGaloisIntermediateField k K)ᵒᵖ
L₁ L₂ : FiniteGaloisIntermediateField k K
g : op L₂ ⟶ L₃
f : op L₁ ⟶ op L₂
⊢ finGaloisGroupMap (f ≫ g) = finGaloisGroupMap f ≫ finGaloisGroupMap g

Tactic:
| _ L₂ =>

Goal state after:
case op.op
k : Type u_1
K : Type u_2
inst✝² : Field k
inst✝¹ : Field K
inst✝ : Algebra k K
L₃ : (FiniteGaloisIntermediateField k K)ᵒᵖ
L₁ L₂ : FiniteGaloisIntermediateField k K
g : op L₂ ⟶ L₃
f : op L₁ ⟶ op L₂
⊢ finGaloisGroupMap (f ≫ g) = finGaloisGroupMap f ≫ finGaloisGroupMap g
Module: Mathlib.FieldTheory.Galois.Profinite

Goal state before:
pp: case op.op.op.mk.mk
k : Type u_1
K : Type u_2
inst✝² : Field k
inst✝¹ : Field K
inst✝ : Algebra k K
L₃ : FiniteGaloisIntermediateField k K
L₁ : IntermediateField k K
finiteDimensional✝¹ : FiniteDimensional k ↥L₁
isGalois✝¹ : IsGalois k ↥L₁
L₂ : IntermediateField k K
finiteDimensional✝ : FiniteDimensional k ↥L₂
isGalois✝ : IsGalois k ↥L₂
g : op { toIntermediateField := L₂, finiteDimensional := finiteDimensional✝, isGalois := isGalois✝ } ⟶ op L₃
f :
  op { toIntermediateField := L₁, finiteDimensional := finiteDimensional✝¹, isGalois := isGalois✝¹ } ⟶
    op { toIntermediateField := L₂, finiteDimensional := finiteDimensional✝, isGalois := isGalois✝ }
⊢ finGaloisGroupMap (f ≫ g) = finGaloisGroupMap f ≫ finGaloisGroupMap g

Tactic:
| _ L₂ =>

Goal state after:
case op.op.op.mk.mk
k : Type u_1
K : Type u_2
inst✝² : Field k
inst✝¹ : Field K
inst✝ : Algebra k K
L₃ : FiniteGaloisIntermediateField k K
L₁ : IntermediateField k K
finiteDimensional✝¹ : FiniteDimensional k ↥L₁
isGalois✝¹ : IsGalois k ↥L₁
L₂ : IntermediateField k K
finiteDimensional✝ : FiniteDimensional k ↥L₂
isGalois✝ : IsGalois k ↥L₂
g : op { toIntermediateField := L₂, finiteDimensional := finiteDimensional✝, isGalois := isGalois✝ } ⟶ op L₃
f :
  op { toIntermediateField := L₁, finiteDimensional := finiteDimensional✝¹, isGalois := isGalois✝¹ } ⟶
    op { toIntermediateField := L₂, finiteDimensional := finiteDimensional✝, isGalois := isGalois✝ }
⊢ finGaloisGroupMap (f ≫ g) = finGaloisGroupMap f ≫ finGaloisGroupMap g
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
⊢ CharP (PerfectClosure K p) p

Tactic:
constructor; intro x; rw [← CharP.cast_eq_zero_iff K]
  rw [← Nat.cast_zero, natCast_eq_iff, Nat.cast_zero]

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
⊢ (fun x ↦ ↑x j) = fun x ↦
    (F.map f).hom'.toMonoidHom
      ({ hom' := { toFun := fun x ↦ ↑x i, map_one' := ⋯, map_mul' := ⋯, continuous_toFun := ⋯ } }.hom'.toMonoidHom x)

Tactic:
exact funext fun x => (x.2 f).symm

Goal state after:
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
h : s = ∅
⊢ False

Tactic:
rw [h, ← Finset.set_biUnion_coe, Finset.coe_empty, Set.biUnion_empty] at hcovers

Goal state after:
G : Type u_1
inst✝ : Group G
ι : Type u_2
s : Finset ι
H : Subgroup G
g : ι → G
hcovers : ∅ = Set.univ
hind : H.index = s.card
h : s = ∅
⊢ False
Module: Mathlib.GroupTheory.CosetCover

Goal state before:
pp: case refine_1
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq (Subgroup G)
n : ℕ
ih :
  ∀ m < n,
    ∀ {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι},
      ⋃ i ∈ s, g i • ↑(H i) = Set.univ →
        ∀ j ∈ s,
          ⋃ i ∈ {x ∈ s | H x = H j}, g i • ↑(H i) ≠ Set.univ →
            m = (Finset.image H s).card → ∃ i ∈ s, H i ≠ H j ∧ (H i).FiniteIndex
ι : Type u_2
H : ι → Subgroup G
g : ι → G
s : Finset ι
hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ
j : ι
hj : j ∈ s
hcovers' : ⋃ i ∈ {x ∈ s | H x = H j}, g i • ↑(H i) ≠ Set.univ
hn : n = (Finset.image H s).card
x : G
hx : ∀ i ∈ s, H i = H j → ↑(g i) ≠ ↑x
y z : G
hz : z ∈ y • ↑(H j)
i : ι
hi : i ∈ s
hmem : ↑(g i) = ↑(x * (y⁻¹ * z))
hij : H i = H j
⊢ y⁻¹ * z ∈ H j

Tactic:
← SetLike.mem_coe,

Goal state after:
case refine_1
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq (Subgroup G)
n : ℕ
ih :
  ∀ m < n,
    ∀ {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι},
      ⋃ i ∈ s, g i • ↑(H i) = Set.univ →
        ∀ j ∈ s,
          ⋃ i ∈ {x ∈ s | H x = H j}, g i • ↑(H i) ≠ Set.univ →
            m = (Finset.image H s).card → ∃ i ∈ s, H i ≠ H j ∧ (H i).FiniteIndex
ι : Type u_2
H : ι → Subgroup G
g : ι → G
s : Finset ι
hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ
j : ι
hj : j ∈ s
hcovers' : ⋃ i ∈ {x ∈ s | H x = H j}, g i • ↑(H i) ≠ Set.univ
hn : n = (Finset.image H s).card
x : G
hx : ∀ i ∈ s, H i = H j → ↑(g i) ≠ ↑x
y z : G
hz : z ∈ y • ↑(H j)
i : ι
hi : i ∈ s
hmem : ↑(g i) = ↑(x * (y⁻¹ * z))
hij : H i = H j
⊢ y⁻¹ * z ∈ ↑(H j)
Module: Mathlib.FieldTheory.JacobsonNoether

Goal state before:
pp: D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Algebra.IsAlgebraic (↥k) D
H : k ≠ ⊤
p : ℕ
hp : ExpChar D p
insep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
a : D
ha : a ∉ k
ha₀ : a ≠ 0
b : D
hb1 : ((ad (↥k) D) a) b ≠ 0
m : ℕ
hm2 : ∀ (n : ℕ), p ^ m ≤ n → (⇑((ad (↥k) D) a))^[n] = 0
⊢ (⇑((ad (↥k) D) a))^[p ^ m + 1] b = 0

Tactic:
rw [hm2 (p ^ m + 1) (Nat.le_add_right _ _), Pi.zero_apply]

Goal state after:
Module: Mathlib.FieldTheory.JacobsonNoether

Goal state before:
pp: D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Algebra.IsAlgebraic (↥k) D
H : k ≠ ⊤
p : ℕ
hp : ExpChar D p
insep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
a : D
ha : a ∉ k
ha₀ : a ≠ 0
b : D
hb1 : ((ad (↥k) D) a) b ≠ 0
n : ℕ
hn : 0 < n
c : D := (⇑((ad (↥k) D) a))^[n] b
hb : c ≠ 0 ∧ (⇑((ad (↥k) D) a))^[n + 1] b = 0
hc_def : c = (⇑((ad (↥k) D) a))^[n] b
x✝ : Invertible c := { invOf := c⁻¹, invOf_mul_self := ⋯, mul_invOf_self := ⋯ }
this : ((ad (↥k) D) a) c = 0
⊢ (mulLeft (↥k) a) c - (mulRight (↥k) a) c = ((ad (↥k) D) a) c

Tactic:
simp [LieRing.of_associative_ring_bracket]

Goal state after:
Module: Mathlib.FieldTheory.MvRatFunc.Rank

Goal state before:
pp: case refine_1
σ : Type u
F : Type v
inst✝¹ : Field F
inst✝ : Nonempty σ
R : Type (max v u) := MvPolynomial σ F
K : Type (max u v) := FractionRing R
⊢ #(MvPolynomial σ F) ≤ max (max (lift.{u, v} #F) (lift.{v, u} #σ)) ℵ₀

Tactic:
MvPolynomial.cardinalMk_eq_max_lift

Goal state after:
case refine_1
σ : Type u
F : Type v
inst✝¹ : Field F
inst✝ : Nonempty σ
R : Type (max v u) := MvPolynomial σ F
K : Type (max u v) := FractionRing R
⊢ max (max (lift.{u, v} #F) (lift.{v, u} #σ)) ℵ₀ ≤ max (max (lift.{u, v} #F) (lift.{v, u} #σ)) ℵ₀
Module: Mathlib.FieldTheory.KummerExtension

Goal state before:
pp: K : Type u
inst✝ : Field K
n : ℕ
hn : Odd n
a : K
ha : ∀ (p : ℕ), Nat.Prime p → p ∣ n → ∀ (b : K), b ^ p ≠ a
⊢ Irreducible (X ^ n - C a)

Tactic:
by
  induction n using induction_on_primes generalizing K a with
  | zero => simp [← Nat.not_even_iff_odd] at hn
  | one => simpa using irreducible_X_sub_C a
  | prime_mul p n hp IH =>
    rw [mul_comm]
    apply X_pow_mul_sub_C_irreducible
      (X_pow_sub_C_irreducible_of_prime hp (ha p hp (dvd_mul_right _ _)))
    intro E _ _ x hx
    have : IsIntegral K x := not_not.mp fun h ↦ by
      simpa only [degree_zero, degree_X_pow_sub_C hp.pos,
        WithBot.natCast_ne_bot] using congr_arg degree (hx.symm.trans (dif_neg h))
    apply IH (Nat.odd_mul.mp hn).2
    intro q hq hqn b hb
    apply ha q hq (dvd_mul_of_dvd_right hqn p) (Algebra.norm _ b)
    rw [← map_pow, hb, ← adjoin.powerBasis_gen this,
      Algebra.PowerBasis.norm_gen_eq_coeff_zero_minpoly]
    simp [minpoly_gen, hx, hp.ne_zero.symm, (Nat.odd_mul.mp hn).1.neg_pow]

Goal state after: