mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.AlgebraicIndependent.Basic

Goal state before:
pp: ι : Type u
R : Type u_2
A : Type v
x : ι → A
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Algebra R A
S : Type u_3
B : Type u_4
FRS : Type u_5
FAB : Type u_6
inst✝⁶ : CommRing S
inst✝⁵ : CommRing B
inst✝⁴ : Algebra S B
inst✝³ : FunLike FRS R S
inst✝² : RingHomClass FRS R S
inst✝¹ : FunLike FAB A B
inst✝ : RingHomClass FAB A B
f : FRS
g : FAB
H : AlgebraicIndependent S (⇑g ∘ x)
hf : Injective ⇑f
h : (algebraMap S B).comp ↑f = (↑g).comp (algebraMap R A)
⊢ AlgebraicIndependent R x

Tactic:
algebraicIndependent_iff

Goal state after:
ι : Type u
R : Type u_2
A : Type v
x : ι → A
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Algebra R A
S : Type u_3
B : Type u_4
FRS : Type u_5
FAB : Type u_6
inst✝⁶ : CommRing S
inst✝⁵ : CommRing B
inst✝⁴ : Algebra S B
inst✝³ : FunLike FRS R S
inst✝² : RingHomClass FRS R S
inst✝¹ : FunLike FAB A B
inst✝ : RingHomClass FAB A B
f : FRS
g : FAB
H : ∀ (p : MvPolynomial ι S), (aeval (⇑g ∘ x)) p = 0 → p = 0
hf : Injective ⇑f
h : (algebraMap S B).comp ↑f = (↑g).comp (algebraMap R A)
⊢ ∀ (p : MvPolynomial ι R), (aeval x) p = 0 → p = 0
Module: Mathlib.RingTheory.AlgebraicIndependent.Basic

Goal state before:
pp: ι : Type u
R : Type u_2
A : Type v
x : ι → A
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Algebra R A
S : Type u_3
B : Type u_4
FRS : Type u_5
FAB : Type u_6
inst✝⁶ : CommRing S
inst✝⁵ : CommRing B
inst✝⁴ : Algebra S B
inst✝³ : FunLike FRS R S
inst✝² : RingHomClass FRS R S
inst✝¹ : FunLike FAB A B
inst✝ : RingHomClass FAB A B
f : FRS
g : FAB
H : AlgebraicIndependent R x
hf : Surjective ⇑f
hg : Injective ⇑g
h : (algebraMap S B).comp ↑f = (↑g).comp (algebraMap R A)
⊢ AlgebraicIndependent S (⇑g ∘ x)

Tactic:
algebraicIndependent_iff

Goal state after:
ι : Type u
R : Type u_2
A : Type v
x : ι → A
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Algebra R A
S : Type u_3
B : Type u_4
FRS : Type u_5
FAB : Type u_6
inst✝⁶ : CommRing S
inst✝⁵ : CommRing B
inst✝⁴ : Algebra S B
inst✝³ : FunLike FRS R S
inst✝² : RingHomClass FRS R S
inst✝¹ : FunLike FAB A B
inst✝ : RingHomClass FAB A B
f : FRS
g : FAB
H : ∀ (p : MvPolynomial ι R), (aeval x) p = 0 → p = 0
hf : Surjective ⇑f
hg : Injective ⇑g
h : (algebraMap S B).comp ↑f = (↑g).comp (algebraMap R A)
⊢ ∀ (p : MvPolynomial ι S), (aeval (⇑g ∘ x)) p = 0 → p = 0
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
S T : Set α
hS : M.Spanning S
hST : S ⊆ T
⊢ M.closure T = M.E

Tactic:
by
  rw [← closure_inter_ground, ← spanning_iff_closure_eq]
  exact hS.superset (subset_inter hST hS.subset_ground)

Goal state after:
Module: Mathlib.GroupTheory.GroupAction.FixedPoints

Goal state before:
pp: α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
g : G
s_subset : (fixedBy α g)ᶜ ⊆ s
a : α
⊢ g • a ∈ s ↔ a ∈ s

Tactic:
by_cases a ∈ fixedBy α g

Goal state after:
case pos
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
g : G
s_subset : (fixedBy α g)ᶜ ⊆ s
a : α
h✝ : a ∈ fixedBy α g
⊢ g • a ∈ s ↔ a ∈ s

case neg
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
g : G
s_subset : (fixedBy α g)ᶜ ⊆ s
a : α
h✝ : a ∉ fixedBy α g
⊢ g • a ∈ s ↔ a ∈ s
Module: Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis

Goal state before:
pp: case pos
R : Type u_1
A : Type w
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
s : Set A
inst✝ : NoZeroDivisors A
alg : Algebra.IsAlgebraic (↥(adjoin R s)) A
h : Injective ⇑(algebraMap R A)
⊢ trdeg R A ≤ #↑s

pp: case neg
R : Type u_1
A : Type w
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
s : Set A
inst✝ : NoZeroDivisors A
alg : Algebra.IsAlgebraic (↥(adjoin R s)) A
h : ¬Injective ⇑(algebraMap R A)
⊢ trdeg R A ≤ #↑s

Tactic:
on_goal 2 => simp [trdeg_eq_zero_of_not_injective h]

Goal state after:
case pos
R : Type u_1
A : Type w
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
s : Set A
inst✝ : NoZeroDivisors A
alg : Algebra.IsAlgebraic (↥(adjoin R s)) A
h : Injective ⇑(algebraMap R A)
⊢ trdeg R A ≤ #↑s
Module: Mathlib.FieldTheory.SeparableDegree

Goal state before:
pp: F : Type u
inst✝ : Field F
f : F[X]
h : f.natDegree ≠ 0
⊢ (map (algebraMap F f.SplittingField) f).natDegree ≠ Zero.zero

Tactic:
rwa [natDegree_map]

Goal state after:
Module: Mathlib.Topology.Algebra.FilterBasis

Goal state before:
pp: case refine_1
G : Type u
inst✝ : Group G
B✝ B : GroupFilterBasis G
this : TopologicalSpace G := B.topology
basis : (𝓝 1).HasBasis (fun V ↦ V ∈ B) id
basis' : (𝓝 1 ×ˢ 𝓝 1).HasBasis (fun i ↦ i.1 ∈ B ∧ i.2 ∈ B) fun i ↦ id i.1 ×ˢ id i.2
U : Set G
U_in : U ∈ B
V : Set G
V_in : V ∈ B
hV : V * V ⊆ U
a : G
⊢ ∀ (b : G), a ∈ V → b ∈ V → a * b ∈ U

Tactic:
b

Goal state after:
case refine_1
G : Type u
inst✝ : Group G
B✝ B : GroupFilterBasis G
this : TopologicalSpace G := B.topology
basis : (𝓝 1).HasBasis (fun V ↦ V ∈ B) id
basis' : (𝓝 1 ×ˢ 𝓝 1).HasBasis (fun i ↦ i.1 ∈ B ∧ i.2 ∈ B) fun i ↦ id i.1 ×ˢ id i.2
U : Set G
U_in : U ∈ B
V : Set G
V_in : V ∈ B
hV : V * V ⊆ U
a b : G
⊢ a ∈ V → b ∈ V → a * b ∈ U
Module: Mathlib.Topology.Algebra.FilterBasis

Goal state before:
pp: case refine_1
G : Type u
inst✝ : Group G
B✝ B : GroupFilterBasis G
this : TopologicalSpace G := B.topology
basis : (𝓝 1).HasBasis (fun V ↦ V ∈ B) id
basis' : (𝓝 1 ×ˢ 𝓝 1).HasBasis (fun i ↦ i.1 ∈ B ∧ i.2 ∈ B) fun i ↦ id i.1 ×ˢ id i.2
U : Set G
U_in : U ∈ B
V : Set G
V_in : V ∈ B
hV : V * V ⊆ U
a b : G
⊢ a ∈ V → b ∈ V → a * b ∈ U

Tactic:
a_in

Goal state after:
case refine_1
G : Type u
inst✝ : Group G
B✝ B : GroupFilterBasis G
this : TopologicalSpace G := B.topology
basis : (𝓝 1).HasBasis (fun V ↦ V ∈ B) id
basis' : (𝓝 1 ×ˢ 𝓝 1).HasBasis (fun i ↦ i.1 ∈ B ∧ i.2 ∈ B) fun i ↦ id i.1 ×ˢ id i.2
U : Set G
U_in : U ∈ B
V : Set G
V_in : V ∈ B
hV : V * V ⊆ U
a b : G
a_in : a ∈ V
⊢ b ∈ V → a * b ∈ U
Module: Mathlib.Topology.Algebra.FilterBasis

Goal state before:
pp: case hmul
R✝ : Type u
inst✝¹ : Ring R✝
B✝ : RingFilterBasis R✝
R : Type u
inst✝ : Ring R
B : RingFilterBasis R
B' : AddGroupFilterBasis R := B.toAddGroupFilterBasis
this✝ : TopologicalSpace R := B'.topology
basis : (𝓝 0).HasBasis (fun V ↦ V ∈ B') id
basis' : (𝓝 0 ×ˢ 𝓝 0).HasBasis (fun i ↦ i.1 ∈ B' ∧ i.2 ∈ B') fun i ↦ id i.1 ×ˢ id i.2
this : IsTopologicalAddGroup R
U : Set R
U_in : U ∈ B'
V : Set R
V_in : V ∈ B
hV : V * V ⊆ U
a : R
⊢ ∀ (b : R), a ∈ V → b ∈ V → a * b ∈ U

Tactic:
b

Goal state after:
case hmul
R✝ : Type u
inst✝¹ : Ring R✝
B✝ : RingFilterBasis R✝
R : Type u
inst✝ : Ring R
B : RingFilterBasis R
B' : AddGroupFilterBasis R := B.toAddGroupFilterBasis
this✝ : TopologicalSpace R := B'.topology
basis : (𝓝 0).HasBasis (fun V ↦ V ∈ B') id
basis' : (𝓝 0 ×ˢ 𝓝 0).HasBasis (fun i ↦ i.1 ∈ B' ∧ i.2 ∈ B') fun i ↦ id i.1 ×ˢ id i.2
this : IsTopologicalAddGroup R
U : Set R
U_in : U ∈ B'
V : Set R
V_in : V ∈ B
hV : V * V ⊆ U
a b : R
⊢ a ∈ V → b ∈ V → a * b ∈ U
Module: Mathlib.Topology.Algebra.FilterBasis

Goal state before:
pp: case hmul
R✝ : Type u
inst✝¹ : Ring R✝
B✝ : RingFilterBasis R✝
R : Type u
inst✝ : Ring R
B : RingFilterBasis R
B' : AddGroupFilterBasis R := B.toAddGroupFilterBasis
this✝ : TopologicalSpace R := B'.topology
basis : (𝓝 0).HasBasis (fun V ↦ V ∈ B') id
basis' : (𝓝 0 ×ˢ 𝓝 0).HasBasis (fun i ↦ i.1 ∈ B' ∧ i.2 ∈ B') fun i ↦ id i.1 ×ˢ id i.2
this : IsTopologicalAddGroup R
U : Set R
U_in : U ∈ B'
V : Set R
V_in : V ∈ B
hV : V * V ⊆ U
a b : R
⊢ a ∈ V → b ∈ V → a * b ∈ U

Tactic:
a_in

Goal state after:
case hmul
R✝ : Type u
inst✝¹ : Ring R✝
B✝ : RingFilterBasis R✝
R : Type u
inst✝ : Ring R
B : RingFilterBasis R
B' : AddGroupFilterBasis R := B.toAddGroupFilterBasis
this✝ : TopologicalSpace R := B'.topology
basis : (𝓝 0).HasBasis (fun V ↦ V ∈ B') id
basis' : (𝓝 0 ×ˢ 𝓝 0).HasBasis (fun i ↦ i.1 ∈ B' ∧ i.2 ∈ B') fun i ↦ id i.1 ×ˢ id i.2
this : IsTopologicalAddGroup R
U : Set R
U_in : U ∈ B'
V : Set R
V_in : V ∈ B
hV : V * V ⊆ U
a b : R
a_in : a ∈ V
⊢ b ∈ V → a * b ∈ U
Module: Mathlib.FieldTheory.SeparableDegree

Goal state before:
pp: F : Type u
inst✝¹ : Field F
f : F[X]
q : ℕ
inst✝ : ExpChar F q
hm : f.Monic
⊢ f.natSepDegree = 1 ↔ ∃ m n y, m ≠ 0 ∧ f = (X ^ q ^ n - C y) ^ m

Tactic:
by
  refine ⟨fun h ↦ ?_, fun ⟨m, n, y, hm, h⟩ ↦ ?_⟩
  · obtain ⟨m, n, y, hm, -, h⟩ := hm.eq_X_pow_char_pow_sub_C_pow_of_natSepDegree_eq_one q h
    exact ⟨m, n, y, hm, h⟩
  simp_rw [h, natSepDegree_pow, hm, ite_false, natSepDegree_X_pow_char_pow_sub_C]

Goal state after:
Module: Mathlib.FieldTheory.SeparableClosure

Goal state before:
pp: F : Type u
E : Type v
inst✝⁶ : Field F
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
K : Type w
inst✝³ : Field K
inst✝² : Algebra F K
inst✝¹ : Algebra K E
inst✝ : IsScalarTower F K E
x : E
hx : x ∈ (separableClosureOperator F E) (IntermediateField.restrictScalars F (separableClosure K E))
⊢ x ∈ IntermediateField.restrictScalars F (separableClosure K E)

Tactic:
by
    obtain ⟨x, rfl⟩ := (separableClosure.separableClosure_eq_bot K E).le hx
    exact x.2

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

Goal state before:
pp: F : Type u_1
inst✝³ : Field F
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : FiniteDimensional F E
h : Nat.card Gal(E/F) = finrank F E
⊢ fixedField ⊤ = ⊥

Tactic:
have p : 0 < finrank (IntermediateField.fixedField (⊤ : Subgroup Gal(E/F))) E := finrank_pos

Goal state after:
F : Type u_1
inst✝³ : Field F
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : FiniteDimensional F E
h : Nat.card Gal(E/F) = finrank F E
p : 0 < finrank (↥(fixedField ⊤)) E
⊢ fixedField ⊤ = ⊥
Module: Mathlib.FieldTheory.Galois.Basic

Goal state before:
pp: F : Type u_1
inst✝⁶ : Field F
E : Type u_2
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
p : F[X]
hFE : FiniteDimensional F E
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
K : Type u_3
inst✝³ : Field K
inst✝² : Algebra F K
inst✝¹ : Algebra K E
inst✝ : IsScalarTower F K E
x : E
hx : x ∈ p.aroots E
h : IsIntegral K x
h1 : p ≠ 0
⊢ (Polynomial.aeval x) (Polynomial.map (algebraMap F K) p) = 0

Tactic:
rw [Polynomial.aeval_def, Polynomial.eval₂_map, ← Polynomial.eval_map, ←
      IsScalarTower.algebraMap_eq]

Goal state after:
F : Type u_1
inst✝⁶ : Field F
E : Type u_2
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
p : F[X]
hFE : FiniteDimensional F E
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
K : Type u_3
inst✝³ : Field K
inst✝² : Algebra F K
inst✝¹ : Algebra K E
inst✝ : IsScalarTower F K E
x : E
hx : x ∈ p.aroots E
h : IsIntegral K x
h1 : p ≠ 0
⊢ Polynomial.eval x (Polynomial.map (algebraMap F E) p) = 0
Module: Mathlib.FieldTheory.SeparableDegree

Goal state before:
pp: F : Type u
E : Type v
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
x y : E
hx : IsSeparable F x
hy : IsSeparable F y
⊢ Algebra.IsSeparable F ↥F⟮x, y⟯

Tactic:
by
  rw [← adjoin_simple_adjoin_simple]
  replace hy := IsSeparable.tower_top F⟮x⟯ hy
  rw [← isSeparable_adjoin_simple_iff_isSeparable] at hx hy
  exact Algebra.IsSeparable.trans F F⟮x⟯ F⟮x⟯⟮y⟯

Goal state after:
Module: Mathlib.LinearAlgebra.BilinearForm.DualLattice

Goal state before:
pp: R : Type u_4
S : Type u_2
M : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : Field S
inst✝⁵ : AddCommGroup M
inst✝⁴ : Algebra R S
inst✝³ : Module R M
inst✝² : Module S M
inst✝¹ : IsScalarTower R S M
B : BilinForm S M
ι : Type u_1
inst✝ : Finite ι
hB : B.Nondegenerate
b : Basis ι S M
this : FiniteDimensional S M := Basis.finiteDimensional_of_finite b
⊢ Submodule.span R (Set.range ⇑(B.flip.dualBasis ⋯ (B.dualBasis hB b))) = Submodule.span R (Set.range ⇑b)

Tactic:
dualBasis_flip_dualBasis hB

Goal state after:
R : Type u_4
S : Type u_2
M : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : Field S
inst✝⁵ : AddCommGroup M
inst✝⁴ : Algebra R S
inst✝³ : Module R M
inst✝² : Module S M
inst✝¹ : IsScalarTower R S M
B : BilinForm S M
ι : Type u_1
inst✝ : Finite ι
hB : B.Nondegenerate
b : Basis ι S M
this : FiniteDimensional S M := Basis.finiteDimensional_of_finite b
⊢ Submodule.span R (Set.range ⇑b) = Submodule.span R (Set.range ⇑b)
Module: Mathlib.LinearAlgebra.BilinearForm.DualLattice

Goal state before:
pp: R : Type u_4
S : Type u_2
M : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : Field S
inst✝⁵ : AddCommGroup M
inst✝⁴ : Algebra R S
inst✝³ : Module R M
inst✝² : Module S M
inst✝¹ : IsScalarTower R S M
B : BilinForm S M
ι : Type u_1
inst✝ : Finite ι
hB : B.Nondegenerate
hB' : B.IsSymm
b : Basis ι S M
this : FiniteDimensional S M := Basis.finiteDimensional_of_finite b
⊢ B.dualSubmodule (B.dualSubmodule (Submodule.span R (Set.range ⇑b))) = Submodule.span R (Set.range ⇑b)

Tactic:
rw [dualSubmodule_span_of_basis B hB, dualSubmodule_span_of_basis B hB,
    dualBasis_dualBasis hB hB']

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.MinpolyDiv

Goal state before:
pp: R : Type u_2
S : Type u_1
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
hx : IsIntegral R x
a✝ : Nontrivial S
this : (minpolyDiv R x * (X - C x)).leadingCoeff = (map (algebraMap R S) (minpoly R x)).leadingCoeff
⊢ (minpolyDiv R x).leadingCoeff * (X - C x).leadingCoeff ≠ 0

Tactic:
simpa using! minpolyDiv_ne_zero hx

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

Goal state before:
pp: k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
H : ClosedSubgroup Gal(K/k)
inst✝ : IsGalois k K
σ : Gal(K/k)
hσ : σ ∈ (fixedField ↑H).fixingSubgroup
h : σ ∉ ↑H
b : Set Gal(K/k)
sub : b ⊆ (fun y ↦ σ * y) ⁻¹' (↑H).carrierᶜ
gp : Subgroup Gal(K/k)
L : IntermediateField k K
eq : (fun g ↦ g.carrier) L.fixingSubgroup = b
hL : L ∈ finiteExts k K
eq' : L.fixingSubgroup = gp
this : FiniteDimensional k ↥L
L' : FiniteGaloisIntermediateField k K :=
  let __src := normalClosure k (↥L) K;
  { toIntermediateField := __src, finiteDimensional := ⋯, isGalois := ⋯ }
compl : σ • L'.fixingSubgroup.carrier ⊆ (↑H).carrierᶜ
fix : ∀ x ∈ lift (fixedField (Subgroup.map (restrictNormalHom ↥L'.toIntermediateField) ↑H)), σ x = x
⊢ (restrictNormalHom ↥L'.toIntermediateField) σ ∈ Subgroup.map (restrictNormalHom ↥L'.toIntermediateField) ↑H

Tactic:
rw [← IntermediateField.fixingSubgroup_fixedField (Subgroup.map (restrictNormalHom L') H.1)]

Goal state after:
k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
H : ClosedSubgroup Gal(K/k)
inst✝ : IsGalois k K
σ : Gal(K/k)
hσ : σ ∈ (fixedField ↑H).fixingSubgroup
h : σ ∉ ↑H
b : Set Gal(K/k)
sub : b ⊆ (fun y ↦ σ * y) ⁻¹' (↑H).carrierᶜ
gp : Subgroup Gal(K/k)
L : IntermediateField k K
eq : (fun g ↦ g.carrier) L.fixingSubgroup = b
hL : L ∈ finiteExts k K
eq' : L.fixingSubgroup = gp
this : FiniteDimensional k ↥L
L' : FiniteGaloisIntermediateField k K :=
  let __src := normalClosure k (↥L) K;
  { toIntermediateField := __src, finiteDimensional := ⋯, isGalois := ⋯ }
compl : σ • L'.fixingSubgroup.carrier ⊆ (↑H).carrierᶜ
fix : ∀ x ∈ lift (fixedField (Subgroup.map (restrictNormalHom ↥L'.toIntermediateField) ↑H)), σ x = x
⊢ (restrictNormalHom ↥L'.toIntermediateField) σ ∈
    (fixedField (Subgroup.map (restrictNormalHom ↥L'.toIntermediateField) ↑H)).fixingSubgroup
Module: Mathlib.FieldTheory.IsSepClosed

Goal state before:
pp: k : Type u
inst✝⁴ : Field k
K : Type v
inst✝³ : Field K
inst✝² : IsSepClosed k
inst✝¹ : Algebra k K
inst✝ : Algebra.IsSeparable k K
⊢ Function.Surjective ⇑(algebraMap k K)

Tactic:
refine fun x => ⟨-(minpoly k x).coeff 0, ?_⟩

Goal state after:
k : Type u
inst✝⁴ : Field k
K : Type v
inst✝³ : Field K
inst✝² : IsSepClosed k
inst✝¹ : Algebra k K
inst✝ : Algebra.IsSeparable k K
x : K
⊢ (algebraMap k K) (-(minpoly k x).coeff 0) = x
Module: Mathlib.RingTheory.Trace.Defs

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
ι : Type w
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι R S
s : S
⊢ (trace R S) s = ((leftMulMatrix b) s).trace

Tactic:
trace_apply,

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
ι : Type w
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι R S
s : S
⊢ (LinearMap.trace R S) ((lmul R S) s) = ((leftMulMatrix b) s).trace
Module: Mathlib.RingTheory.Trace.Defs

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
ι : Type w
inst✝ : Fintype ι
b : Basis ι R S
x : R
this : DecidableEq ι
⊢ (trace R S) ((algebraMap R S) x) = Fintype.card ι • x

Tactic:
trace_apply,

Goal state after:
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
ι : Type w
inst✝ : Fintype ι
b : Basis ι R S
x : R
this : DecidableEq ι
⊢ (LinearMap.trace R S) ((lmul R S) ((algebraMap R S) x)) = Fintype.card ι • x
Module: Mathlib.FieldTheory.PurelyInseparable.Basic

Goal state before:
pp: F : Type u
E : Type v
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
L : IntermediateField F E
h : IsPurelyInseparable (↥L) E
x : E
hx : x ∈ separableClosure F E
⊢ x ∈ L

Tactic:
obtain ⟨y, rfl⟩ := h.inseparable' _ <|
    IsSeparable.tower_top L (mem_separableClosure_iff.1 hx)
  exact y.2

Goal state after:
Module: Mathlib.FieldTheory.PurelyInseparable.Basic

Goal state before:
pp: case inl.add
k : Type u_1
K : Type u_2
R : Type u_3
inst✝⁶ : Field k
inst✝⁵ : Field K
inst✝⁴ : Algebra k K
inst✝³ : CommRing R
inst✝² : Algebra k R
inst✝¹ : IsPurelyInseparable k K
q : ℕ
inst✝ : ExpChar k q
a✝ : Nontrivial (R ⊗[k] K)
hq : Nat.Prime q
x y : R ⊗[k] K
this : ExpChar (R ⊗[k] K) q
h : x ∈ Subalgebra.perfectClosure R (R ⊗[k] K) q
h' : y ∈ Subalgebra.perfectClosure R (R ⊗[k] K) q
⊢ x + y ∈ Subalgebra.perfectClosure R (R ⊗[k] K) q

Tactic:
exact add_mem h h'

Goal state after:
Module: Mathlib.FieldTheory.PurelyInseparable.Basic

Goal state before:
pp: case inl.add
k : Type u_1
K : Type u_2
R : Type u_3
inst✝⁶ : Field k
inst✝⁵ : Field K
inst✝⁴ : Algebra k K
inst✝³ : CommRing R
inst✝² : Algebra k R
inst✝¹ : IsPurelyInseparable k K
q : ℕ
inst✝ : ExpChar k q
a✝ : Nontrivial (R ⊗[k] K)
hq : Nat.Prime q
x y : R ⊗[k] K
h : ∃ n, x ^ q ^ n ∈ (algebraMap R (R ⊗[k] K)).range
h' : ∃ n, y ^ q ^ n ∈ (algebraMap R (R ⊗[k] K)).range
⊢ ∃ n, (x + y) ^ q ^ n ∈ (algebraMap R (R ⊗[k] K)).range

Tactic:
| add x y h h' =>

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.IntegralClosure

Goal state before:
pp: A : Type u_1
K : Type u_2
inst✝¹⁷ : CommRing A
inst✝¹⁶ : Field K
inst✝¹⁵ : Algebra A K
inst✝¹⁴ : IsFractionRing A K
L : Type u_3
inst✝¹³ : Field L
C : Type u_4
inst✝¹² : CommRing C
inst✝¹¹ : Algebra K L
inst✝¹⁰ : Algebra A L
inst✝⁹ : IsScalarTower A K L
inst✝⁸ : Algebra C L
inst✝⁷ : IsIntegralClosure C A L
inst✝⁶ : Algebra A C
inst✝⁵ : IsScalarTower A C L
inst✝⁴ : FiniteDimensional K L
inst✝³ : IsDomain A
inst✝² : Algebra.IsSeparable K L
inst✝¹ : IsIntegrallyClosed A
inst✝ : IsNoetherianRing A
this : DecidableEq L
⊢ IsNoetherian A C

Tactic:
obtain ⟨s, b, hb_int⟩ := FiniteDimensional.exists_is_basis_integral A K L

Goal state after:
A : Type u_1
K : Type u_2
inst✝¹⁷ : CommRing A
inst✝¹⁶ : Field K
inst✝¹⁵ : Algebra A K
inst✝¹⁴ : IsFractionRing A K
L : Type u_3
inst✝¹³ : Field L
C : Type u_4
inst✝¹² : CommRing C
inst✝¹¹ : Algebra K L
inst✝¹⁰ : Algebra A L
inst✝⁹ : IsScalarTower A K L
inst✝⁸ : Algebra C L
inst✝⁷ : IsIntegralClosure C A L
inst✝⁶ : Algebra A C
inst✝⁵ : IsScalarTower A C L
inst✝⁴ : FiniteDimensional K L
inst✝³ : IsDomain A
inst✝² : Algebra.IsSeparable K L
inst✝¹ : IsIntegrallyClosed A
inst✝ : IsNoetherianRing A
this : DecidableEq L
s : Finset L
b : Basis (↥s) K L
hb_int : ∀ (x : ↥s), IsIntegral A (b x)
⊢ IsNoetherian A C
Module: Mathlib.RingTheory.IsGaloisGroup.Defs

Goal state before:
pp: G : Type u_1
A : Type u_2
A' : Type u_3
B : Type u_4
inst✝⁶ : Group G
inst✝⁵ : CommSemiring A
inst✝⁴ : Semiring B
inst✝³ : Algebra A B
inst✝² : MulSemiringAction G B
hG : IsGaloisGroup G A B
inst✝¹ : CommSemiring A'
inst✝ : Algebra A' B
e : A →+* A'
he : ∀ (a : A), (algebraMap A' B) (e a) = (algebraMap A B) a
he' : Function.Surjective ⇑e
g : G
a' : A'
⊢ ∀ (a : B), g • a' • a = a' • g • a

Tactic:
b

Goal state after:
G : Type u_1
A : Type u_2
A' : Type u_3
B : Type u_4
inst✝⁶ : Group G
inst✝⁵ : CommSemiring A
inst✝⁴ : Semiring B
inst✝³ : Algebra A B
inst✝² : MulSemiringAction G B
hG : IsGaloisGroup G A B
inst✝¹ : CommSemiring A'
inst✝ : Algebra A' B
e : A →+* A'
he : ∀ (a : A), (algebraMap A' B) (e a) = (algebraMap A B) a
he' : Function.Surjective ⇑e
g : G
a' : A'
b : B
⊢ g • a' • b = a' • g • b
Module: Mathlib.RingTheory.IsGaloisGroup.Defs

Goal state before:
pp: G : Type u_1
A : Type u_2
A' : Type u_3
B : Type u_4
inst✝⁶ : Group G
inst✝⁵ : CommSemiring A
inst✝⁴ : Semiring B
inst✝³ : Algebra A B
inst✝² : MulSemiringAction G B
hG : IsGaloisGroup G A B
inst✝¹ : CommSemiring A'
inst✝ : Algebra A' B
e : A →+* A'
he : ∀ (a : A), (algebraMap A' B) (e a) = (algebraMap A B) a
he' : Function.Surjective ⇑e
⊢ ∀ (b : B), (∀ (g : G), g • b = b) → ∃ a, (algebraMap A' B) a = b

Tactic:
intro b h
    obtain ⟨a, ha⟩ := hG.isInvariant.isInvariant b h
    exact ⟨e a, by rw [he, ha]⟩

Goal state after:
Module: Mathlib.RingTheory.Trace.Basic

Goal state before:
pp: A₁ : Type u_7
B₁ : Type u_8
A₂ : Type u_9
B₂ : Type u_10
inst✝⁵ : CommRing A₁
inst✝⁴ : CommRing B₁
inst✝³ : CommRing A₂
inst✝² : CommRing B₂
inst✝¹ : Algebra A₁ B₁
inst✝ : Algebra A₂ B₂
e₁ : A₁ ≃+* A₂
e₂ : B₁ ≃+* B₂
he : (algebraMap A₂ B₂).comp ↑e₁ = (↑e₂).comp (algebraMap A₁ B₁)
x : B₁
this : Algebra A₁ B₂ := ((↑e₂).comp (algebraMap A₁ B₁)).toAlgebra
e' : B₁ ≃ₐ[A₁] B₂ := { toEquiv := e₂.toEquiv, map_mul' := ⋯, map_add' := ⋯, commutes' := ⋯ }
⊢ (trace A₁ B₁) x = e₁.symm (e₁ ((trace A₁ B₂) (e₂ x)))

Tactic:
← Algebra.trace_eq_of_algEquiv e',

Goal state after:
A₁ : Type u_7
B₁ : Type u_8
A₂ : Type u_9
B₂ : Type u_10
inst✝⁵ : CommRing A₁
inst✝⁴ : CommRing B₁
inst✝³ : CommRing A₂
inst✝² : CommRing B₂
inst✝¹ : Algebra A₁ B₁
inst✝ : Algebra A₂ B₂
e₁ : A₁ ≃+* A₂
e₂ : B₁ ≃+* B₂
he : (algebraMap A₂ B₂).comp ↑e₁ = (↑e₂).comp (algebraMap A₁ B₁)
x : B₁
this : Algebra A₁ B₂ := ((↑e₂).comp (algebraMap A₁ B₁)).toAlgebra
e' : B₁ ≃ₐ[A₁] B₂ := { toEquiv := e₂.toEquiv, map_mul' := ⋯, map_add' := ⋯, commutes' := ⋯ }
⊢ (trace A₁ B₂) (e' x) = e₁.symm (e₁ ((trace A₁ B₂) (e₂ x)))
Module: Mathlib.FieldTheory.Finite.Basic

Goal state before:
pp: R : Type u_2
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : Fintype R
f g : R[X]
hf2 : f.degree = 2
hg2 : g.degree = 2
hR : Fintype.card R % 2 = 1
this : DecidableEq R := Classical.decEq R
hd : Disjoint (image (fun x ↦ eval x f) univ) (image (fun x ↦ eval x (-g)) univ)
⊢ f.natDegree * #(image (fun x ↦ eval x f) univ) + (-g).natDegree * #(image (fun x ↦ eval x (-g)) univ) =
    2 * #(image (fun x ↦ eval x f) univ ∪ image (fun x ↦ eval x (-g)) univ)

Tactic:
card_union_of_disjoint hd

Goal state after:
R : Type u_2
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : Fintype R
f g : R[X]
hf2 : f.degree = 2
hg2 : g.degree = 2
hR : Fintype.card R % 2 = 1
this : DecidableEq R := Classical.decEq R
hd : Disjoint (image (fun x ↦ eval x f) univ) (image (fun x ↦ eval x (-g)) univ)
⊢ f.natDegree * #(image (fun x ↦ eval x f) univ) + (-g).natDegree * #(image (fun x ↦ eval x (-g)) univ) =
    2 * (#(image (fun x ↦ eval x f) univ) + #(image (fun x ↦ eval x (-g)) univ))
Module: Mathlib.FieldTheory.Finite.Basic

Goal state before:
pp: R : Type u_2
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : Fintype R
f g : R[X]
hf2 : f.degree = 2
hg2 : g.degree = 2
hR : Fintype.card R % 2 = 1
this : DecidableEq R := Classical.decEq R
x a : R
left✝¹ : a ∈ univ
ha : eval a f = x
b : R
left✝ : b ∈ univ
hb : eval b (-g) = x
⊢ -eval b g + eval b g = 0

Tactic:
neg_add_cancel

Goal state after:
R : Type u_2
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : Fintype R
f g : R[X]
hf2 : f.degree = 2
hg2 : g.degree = 2
hR : Fintype.card R % 2 = 1
this : DecidableEq R := Classical.decEq R
x a : R
left✝¹ : a ∈ univ
ha : eval a f = x
b : R
left✝ : b ∈ univ
hb : eval b (-g) = x
⊢ 0 = 0