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

Goal state before:
pp: R : Type u
M : Type v
N : Type u_1
P : Type u_2
inst✝⁷ : CommSemiring R
inst✝⁶ : AddCommMonoid M
inst✝⁵ : AddCommMonoid N
inst✝⁴ : AddCommMonoid P
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : Module R P
inst✝ : Module.Invertible R M
f : N →ₗ[R] P
⊢ Function.Surjective ⇑(LinearMap.rTensor M f) ↔ Function.Surjective ⇑f

Tactic:
← LinearMap.lTensor_surj_iff_rTensor_surj,

Goal state after:
R : Type u
M : Type v
N : Type u_1
P : Type u_2
inst✝⁷ : CommSemiring R
inst✝⁶ : AddCommMonoid M
inst✝⁵ : AddCommMonoid N
inst✝⁴ : AddCommMonoid P
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : Module R P
inst✝ : Module.Invertible R M
f : N →ₗ[R] P
⊢ Function.Surjective ⇑(LinearMap.lTensor M f) ↔ Function.Surjective ⇑f
Module: Mathlib.FieldTheory.Normal.Basic

Goal state before:
pp: F : Type u_1
K : Type u_2
inst✝⁴ : Field F
inst✝³ : Field K
inst✝² : Algebra F K
E E' : IntermediateField F K
inst✝¹ : Normal F ↥E
inst✝ : Normal F ↥E'
⊢ ∀ (i : Bool), Normal F ↥(Bool.rec E' E i)

Tactic:
rintro (_ | _)

Goal state after:
case false
F : Type u_1
K : Type u_2
inst✝⁴ : Field F
inst✝³ : Field K
inst✝² : Algebra F K
E E' : IntermediateField F K
inst✝¹ : Normal F ↥E
inst✝ : Normal F ↥E'
⊢ Normal F ↥(Bool.rec E' E false)

case true
F : Type u_1
K : Type u_2
inst✝⁴ : Field F
inst✝³ : Field K
inst✝² : Algebra F K
E E' : IntermediateField F K
inst✝¹ : Normal F ↥E
inst✝ : Normal F ↥E'
⊢ Normal F ↥(Bool.rec E' E true)
Module: Mathlib.Combinatorics.Matroid.Basic

Goal state before:
pp: α : Type u_1
M : Matroid α
⊢ ¬¬M.RankFinite ↔ M.RankFinite

Tactic:
not_not

Goal state after:
α : Type u_1
M : Matroid α
⊢ M.RankFinite ↔ M.RankFinite
Module: Mathlib.FieldTheory.PrimitiveElement

Goal state before:
pp: F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
α β : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
x y : F
hneq : ¬x = y
heq : F⟮α + x • β⟯ = F⟮α + y • β⟯
αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯
αyβ_in_K : α + y • β ∈ F⟮α + x • β⟯
β_in_K : β ∈ F⟮α + x • β⟯
⊢ α ∈ F⟮α + x • β⟯

Tactic:
convert! ← sub_mem αxβ_in_K (smul_mem _ β_in_K)

Goal state after:
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
α β : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
x y : F
hneq : ¬x = y
heq : F⟮α + x • β⟯ = F⟮α + y • β⟯
αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯
αyβ_in_K : α + y • β ∈ F⟮α + x • β⟯
β_in_K : β ∈ F⟮α + x • β⟯
⊢ α + x • β - ?m.327 • β = α

F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
α β : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
f : F → IntermediateField F E := ⋯
x y : F
hneq : ¬x = y
heq : F⟮α + x • β⟯ = F⟮α + y • β⟯
αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯
αyβ_in_K : α + y • β ∈ F⟮α + x • β⟯
β_in_K : β ∈ F⟮α + x • β⟯
⊢ F
Module: Mathlib.Combinatorics.Matroid.Basic

Goal state before:
pp: α : Type u_1
M : Matroid α
X : Set α
hX : X ⊆ M.E
⊢ ¬¬M.Indep X ↔ M.Indep X

Tactic:
not_not

Goal state after:
α : Type u_1
M : Matroid α
X : Set α
hX : X ⊆ M.E
⊢ M.Indep X ↔ M.Indep X
Module: Mathlib.Combinatorics.Matroid.Basic

Goal state before:
pp: α : Type u_1
M✝ M : Matroid α
E : Set α
IsBase Indep : Set α → Prop
hE : E = M.E
hB : ∀ (B : Set α), IsBase B ↔ M.IsBase B
hI : ∀ (I : Set α), Indep I ↔ M.Indep I
⊢ ExchangeProperty IsBase

Tactic:
show IsBase = M.IsBase from funext (by simp [hB])

Goal state after:
α : Type u_1
M✝ M : Matroid α
E : Set α
IsBase Indep : Set α → Prop
hE : E = M.E
hB : ∀ (B : Set α), IsBase B ↔ M.IsBase B
hI : ∀ (I : Set α), Indep I ↔ M.Indep I
⊢ ExchangeProperty M.IsBase
Module: Mathlib.FieldTheory.PrimitiveElement

Goal state before:
pp: F : Type u_3
E : Type u_4
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : FiniteDimensional F E
α : E
⊢ F⟮α⟯ = ⊤ ↔ (minpoly F α).natDegree = finrank F E

pp: F : Type u_3
E : Type u_4
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : FiniteDimensional F E
α : E
⊢ minpoly F α ≠ 0

Tactic:
primitive_element_iff_minpoly_natDegree_eq

Goal state after:
F : Type u_3
E : Type u_4
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : FiniteDimensional F E
α : E
⊢ (minpoly F α).natDegree = finrank F E ↔ (minpoly F α).natDegree = finrank F E

F : Type u_3
E : Type u_4
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : FiniteDimensional F E
α : E
⊢ minpoly F α ≠ 0
Module: Mathlib.Combinatorics.Matroid.IndepAxioms

Goal state before:
pp: α : Type u_1
P : Set α → Prop
X : Set α
n : ℕ
hP : ∀ (Y : Set α), P Y → Y.encard ≤ ↑n
I : Set α
hI : P I
hIX : I ⊆ X
⊢ (ncard '' {Y | P Y ∧ I ⊆ Y ∧ Y ⊆ X}).Finite

Tactic:
finite_iff_bddAbove,

Goal state after:
α : Type u_1
P : Set α → Prop
X : Set α
n : ℕ
hP : ∀ (Y : Set α), P Y → Y.encard ≤ ↑n
I : Set α
hI : P I
hIX : I ⊆ X
⊢ BddAbove (ncard '' {Y | P Y ∧ I ⊆ Y ∧ Y ⊆ X})
Module: Mathlib.Combinatorics.Matroid.Minor.Restrict

Goal state before:
pp: α : Type u_1
M : Matroid α
I X Y J : Set α
hIX : M.IsBasis I X
hJX : M.IsBasis J X
hXY : X ⊆ Y
hJY : M.IsBasis J Y
⊢ M.IsBasis I Y

Tactic:
rw [← isBase_restrict_iff]

Goal state after:
α : Type u_1
M : Matroid α
I X Y J : Set α
hIX : M.IsBasis I X
hJX : M.IsBasis J X
hXY : X ⊆ Y
hJY : M.IsBasis J Y
⊢ (M ↾ Y).IsBase I
Module: Mathlib.Combinatorics.Matroid.Map

Goal state before:
pp: case mp
α : Type u_1
β : Type u_2
f : α → β
M : Matroid α
hf : InjOn f M.E
D : Set β
⊢ ((∀ (x : Set α), M.Indep x → ¬D = f '' x) ∧ ∃ u ⊆ M.E, f '' u = D) → ∃ D₀, (¬M.Indep D₀ ∧ D₀ ⊆ M.E) ∧ D = f '' D₀

Tactic:
rintro ⟨h, D₀, hD₀E, rfl⟩

Goal state after:
case mp
α : Type u_1
β : Type u_2
f : α → β
M : Matroid α
hf : InjOn f M.E
D₀ : Set α
hD₀E : D₀ ⊆ M.E
h : ∀ (x : Set α), M.Indep x → ¬f '' D₀ = f '' x
⊢ ∃ D₀_1, (¬M.Indep D₀_1 ∧ D₀_1 ⊆ M.E) ∧ f '' D₀ = f '' D₀_1
Module: Mathlib.Combinatorics.Matroid.Circuit

Goal state before:
pp: α : Type u_1
M : Matroid α
C : Set α
hC : M.IsCircuit C
⊢ ∀ ⦃t : Set α⦄, t ⊂ C → ¬¬M.Indep t

Tactic:
not_not

Goal state after:
α : Type u_1
M : Matroid α
C : Set α
hC : M.IsCircuit C
⊢ ∀ ⦃t : Set α⦄, t ⊂ C → M.Indep t
Module: Mathlib.Combinatorics.Matroid.Map

Goal state before:
pp: α : Type u_1
β : Type u_2
f✝ : α → β
E I✝ : Set α
M : Matroid α
N : Matroid β
inst✝ : M.Finitary
f : α → β
hf : InjOn f M.E
I : Set β
hI : ∀ J ⊆ I, J.Finite → (M.map f hf).Indep J
e : β
he : e ∈ I
I₀ : Set α
hI₀ : M.Indep I₀
h_eq : {e} = f '' I₀
⊢ e ∈ f '' M.E

Tactic:
exact image_mono hI₀.subset_ground <| h_eq.subset rfl

Goal state after:
Module: Mathlib.Combinatorics.Matroid.Rank.ENat

Goal state before:
pp: α : Type u_1
M : Matroid α
X : Set α
⊢ M.eRk (X ∩ M.E) = M.eRk X

Tactic:
eRk_inter_ground

Goal state after:
α : Type u_1
M : Matroid α
X : Set α
⊢ M.eRk X = M.eRk X
Module: Mathlib.Combinatorics.Matroid.Rank.ENat

Goal state before:
pp: α : Type u_1
M : Matroid α
e : α
X : Set α
he : e ∉ M.E
⊢ M.eRk (X ∩ M.E) = M.eRk X

Tactic:
eRk_inter_ground

Goal state after:
α : Type u_1
M : Matroid α
e : α
X : Set α
he : e ∉ M.E
⊢ M.eRk X = M.eRk X
Module: Mathlib.Combinatorics.Matroid.Loop

Goal state before:
pp: α : Type u_1
M : Matroid α
e : α
he : e ∈ M.E
h : ¬¬M.IsLoop e
⊢ M.IsLoop e

Tactic:
not_not

Goal state after:
α : Type u_1
M : Matroid α
e : α
he : e ∈ M.E
h : M.IsLoop e
⊢ M.IsLoop e
Module: Mathlib.Combinatorics.Matroid.Loop

Goal state before:
pp: α : Type u_1
M : Matroid α
e : α
he : e ∈ M.E
⊢ ¬¬M.IsLoop e ↔ M.IsLoop e

Tactic:
not_not

Goal state after:
α : Type u_1
M : Matroid α
e : α
he : e ∈ M.E
⊢ M.IsLoop e ↔ M.IsLoop e
Module: Mathlib.Combinatorics.Matroid.Circuit

Goal state before:
pp: α : Type u_1
M : Matroid α
X : Set α
hX : M.Dep X
I : Set α
hI : M.IsBasis I X
⊢ I ≠ X

Tactic:
rintro rfl; exact hI.indep.not_dep hX

Goal state after:
Module: Mathlib.Combinatorics.Matroid.Circuit

Goal state before:
pp: α : Type u_1
M₁ M₂ : Matroid α
hE : M₁.E = M₂.E
h₁ : ∀ (C : Set α), M₁.IsCircuit C → ¬M₂.Indep C
h₂ : ∀ (C : Set α), M₂.IsCircuit C → ¬M₁.Indep C
C : Set α
hCE : C ⊆ M₁.E
hC : M₁.IsCircuit C
⊢ C ⊆ M₂.E

Tactic:
rwa [← hE]

Goal state after:
Module: Mathlib.Combinatorics.Matroid.Loop

Goal state before:
pp: α : Type u_1
M : Matroid α
e f : α
he : M.IsNonloop e
hef : e ∈ M.closure {f}
⊢ M.closure (M.closure {e}) = M.closure {e, f}

Tactic:
← insert_eq_of_mem (he.mem_closure_singleton hef),

Goal state after:
α : Type u_1
M : Matroid α
e f : α
he : M.IsNonloop e
hef : e ∈ M.closure {f}
⊢ M.closure (insert f (M.closure {e})) = M.closure {e, f}
Module: Mathlib.Combinatorics.Matroid.Rank.ENat

Goal state before:
pp: case refine_2
α : Type u_1
M : Matroid α
X : Set α
n : ℕ∞
x✝ : ∃ I ⊆ X, M.Indep I ∧ I.encard = n
I : Set α
hIX : I ⊆ X
hI : M.Indep I
hIc : I.encard = n
⊢ n ≤ M.eRk X

Tactic:
← hIc,

Goal state after:
case refine_2
α : Type u_1
M : Matroid α
X : Set α
n : ℕ∞
x✝ : ∃ I ⊆ X, M.Indep I ∧ I.encard = n
I : Set α
hIX : I ⊆ X
hI : M.Indep I
hIc : I.encard = n
⊢ I.encard ≤ M.eRk X
Module: Mathlib.Combinatorics.Matroid.Rank.ENat

Goal state before:
pp: α : Type u_1
M : Matroid α
⊢ M.eRank = ⊤ ↔ ¬M.eRank ≠ ⊤

Tactic:
not_not

Goal state after:
α : Type u_1
M : Matroid α
⊢ M.eRank = ⊤ ↔ M.eRank = ⊤
Module: Mathlib.Combinatorics.Matroid.Circuit

Goal state before:
pp: case refine_6
α : Type u_1
M : Matroid α
C₀ : Set α
hC₀ : M.IsCircuit C₀
X : Set α
S : Set (Set α)
z : α
hCS : ∀ C ∈ S, M.IsCircuit C
hXC₀ : X ⊆ C₀
hzC₀ : z ∈ C₀
hz✝ : ∀ C ∈ S, z ∉ C
C : α → Set α
hC : (∀ x ∈ X, C x ∈ S) ∧ ∀ x ∈ X, C x ∩ X = {x}
C' : Set α
hC'ss : C' ⊆ (C₀ ∪ ⋃ i, C ↑i) \ range fun x ↦ ↑x
hC' : M.IsCircuit C'
hz : z ∈ C'
⊢ ∃ C' ⊆ (C₀ ∪ ⋃₀ S) \ X, M.IsCircuit C' ∧ z ∈ C'

Tactic:
refine ⟨C', hC'ss.trans (sdiff_subset_sdiff (union_subset_union_right _ ?_) (by simp)), hC', hz⟩

Goal state after:
case refine_6
α : Type u_1
M : Matroid α
C₀ : Set α
hC₀ : M.IsCircuit C₀
X : Set α
S : Set (Set α)
z : α
hCS : ∀ C ∈ S, M.IsCircuit C
hXC₀ : X ⊆ C₀
hzC₀ : z ∈ C₀
hz✝ : ∀ C ∈ S, z ∉ C
C : α → Set α
hC : (∀ x ∈ X, C x ∈ S) ∧ ∀ x ∈ X, C x ∩ X = {x}
C' : Set α
hC'ss : C' ⊆ (C₀ ∪ ⋃ i, C ↑i) \ range fun x ↦ ↑x
hC' : M.IsCircuit C'
hz : z ∈ C'
⊢ ⋃ i, C ↑i ⊆ ⋃₀ S
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
X Y : Set α
⊢ M.closure (M.closure X ∪ M.closure Y) = M.closure (X ∪ Y)

Tactic:
closure_closure_union_closure_eq_closure_union

Goal state after:
α : Type u_2
M : Matroid α
X Y : Set α
⊢ M.closure (X ∪ Y) = M.closure (X ∪ Y)
Module: Mathlib.Combinatorics.Matroid.Rank.ENat

Goal state before:
pp: α : Type u_1
M : Matroid α
X : Set α
⊢ M✶.eRk X + M.eRank = M✶.eRk (X ∩ M✶.E) + M.eRank

Tactic:
eRk_inter_ground

Goal state after:
α : Type u_1
M : Matroid α
X : Set α
⊢ M✶.eRk X + M.eRank = M✶.eRk X + M.eRank
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
e : α
I : Set α
hI : M.Indep I
⊢ M.IsBasis I (insert e I) ↔ e ∈ M.closure I

Tactic:
hI.closure_eq_setOfPred_isBasis_insert,

Goal state after:
α : Type u_2
M : Matroid α
e : α
I : Set α
hI : M.Indep I
⊢ M.IsBasis I (insert e I) ↔ e ∈ {x | M.IsBasis I (insert x I)}
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
I : Set α
x : α
hI : M.Indep I
⊢ x ∈ M.closure I ↔ M.Dep (insert x I) ∨ x ∈ I

Tactic:
hI.closure_eq_setOfPred_isBasis_insert,

Goal state after:
α : Type u_2
M : Matroid α
I : Set α
x : α
hI : M.Indep I
⊢ x ∈ {x | M.IsBasis I (insert x I)} ↔ M.Dep (insert x I) ∨ x ∈ I
Module: Mathlib.Combinatorics.Matroid.Circuit

Goal state before:
pp: α : Type u_1
M : Matroid α
C : Set α
hC : M.IsCircuit C
⊢ M✶.IsCocircuit C

Tactic:
rwa [isCocircuit_def, dual_dual]

Goal state after:
Module: Mathlib.Combinatorics.Matroid.Circuit

Goal state before:
pp: α : Type u_1
M : Matroid α
C : Set α
e : α
K : Set α
hC : M.IsCircuit C
he : C ∩ K = {e}
heC : e ∈ C
hK : ¬M.Spanning (M.E \ K) ∧ ∀ ⦃t : Set α⦄, t ⊂ K → ¬¬M.Spanning (M.E \ t)
⊢ False

Tactic:
not_not

Goal state after:
α : Type u_1
M : Matroid α
C : Set α
e : α
K : Set α
hC : M.IsCircuit C
he : C ∩ K = {e}
heC : e ∈ C
hK : ¬M.Spanning (M.E \ K) ∧ ∀ ⦃t : Set α⦄, t ⊂ K → M.Spanning (M.E \ t)
⊢ False
Module: Mathlib.Combinatorics.Matroid.Rank.Cardinal

Goal state before:
pp: α : Type u
M : Matroid α
X Y : Set α
inst✝ : M.InvariantCardinalRank
hXY : M.closure X = M.closure Y
⊢ M.cRk (M.closure Y) = M.cRk Y

Tactic:
cRk_closure

Goal state after:
α : Type u
M : Matroid α
X Y : Set α
inst✝ : M.InvariantCardinalRank
hXY : M.closure X = M.closure Y
⊢ M.cRk Y = M.cRk Y
Module: Mathlib.RingTheory.AlgebraicIndependent.Basic

Goal state before:
pp: ι : Type u
R : Type u_2
A : Type v
x : ι → A
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
hx : ∀ (p : MvPolynomial ι R), (aeval x) p = 0 → p = 0
f : ι → MvPolynomial ι R
hf : ∀ (p : MvPolynomial ι R), (aeval f) p = 0 → p = 0
p : MvPolynomial ι R
⊢ (aeval fun i ↦ (aeval x) (f i)) p = 0 → p = 0

Tactic:
hp

Goal state after:
ι : Type u
R : Type u_2
A : Type v
x : ι → A
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
hx : ∀ (p : MvPolynomial ι R), (aeval x) p = 0 → p = 0
f : ι → MvPolynomial ι R
hf : ∀ (p : MvPolynomial ι R), (aeval f) p = 0 → p = 0
p : MvPolynomial ι R
hp : (aeval fun i ↦ (aeval x) (f i)) 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✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
f : ι → MvPolynomial ι R
H : ∀ (p : MvPolynomial ι R), (aeval fun i ↦ (aeval x) (f i)) p = 0 → p = 0
p : MvPolynomial ι R
⊢ (aeval f) p = 0 → p = 0

Tactic:
hp

Goal state after:
ι : Type u
R : Type u_2
A : Type v
x : ι → A
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
f : ι → MvPolynomial ι R
H : ∀ (p : MvPolynomial ι R), (aeval fun i ↦ (aeval x) (f i)) p = 0 → p = 0
p : MvPolynomial ι R
hp : (aeval f) 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 : ∀ (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

Tactic:
hp

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
hp : (aeval x) p = 0
⊢ p = 0
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
X : Set α
e f : α
hfE : f ∈ M.E
heE : e ∈ M.E
I : Set α
hef : e ∈ M.closure (insert f I)
he : M.Indep (insert e I) ∧ e ∉ I
hI : M.IsBasis I (X ∩ M.E)
⊢ ¬¬M.Indep (insert e (insert f I)) → f ∈ insert e I

Tactic:
not_not

Goal state after:
α : Type u_2
M : Matroid α
X : Set α
e f : α
hfE : f ∈ M.E
heE : e ∈ M.E
I : Set α
hef : e ∈ M.closure (insert f I)
he : M.Indep (insert e I) ∧ e ∉ I
hI : M.IsBasis I (X ∩ M.E)
⊢ M.Indep (insert e (insert f I)) → f ∈ insert e I
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 : ∀ (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

Tactic:
hp

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
hp : (aeval (⇑g ∘ x)) p = 0
⊢ p = 0
Module: Mathlib.RingTheory.AlgebraicIndependent.Transcendental

Goal state before:
pp: R : Type u_3
S : Type u
inst✝⁹ : CommRing R
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
inst✝⁶ : Nontrivial R
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
inst✝³ : Algebra S A
inst✝² : FaithfulSMul R S
inst✝¹ : FaithfulSMul S A
inst✝ : IsScalarTower R S A
⊢ Cardinal.lift.{u, u} (trdeg R S) + Cardinal.lift.{u, u} (trdeg S A) ≤ Cardinal.lift.{u, u} (trdeg R A)

Tactic:
exact lift_trdeg_add_le

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

Goal state before:
pp: ι : Type u
R : Type u_2
A : Type v
A' : Type v'
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : CommRing A'
inst✝¹ : Algebra R A
inst✝ : Algebra R A'
x : ι → A
f : A →ₐ[R] A'
h : Injective ⇑f
H : IsTranscendenceBasis R (⇑f ∘ x)
s : Set A
hs : AlgebraicIndepOn R _root_.id s
hs' : range x ⊆ s
this : range (⇑f ∘ x) = ⇑f '' s
⊢ range x = s

Tactic:
Set.range_comp,

Goal state after:
ι : Type u
R : Type u_2
A : Type v
A' : Type v'
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : CommRing A'
inst✝¹ : Algebra R A
inst✝ : Algebra R A'
x : ι → A
f : A →ₐ[R] A'
h : Injective ⇑f
H : IsTranscendenceBasis R (⇑f ∘ x)
s : Set A
hs : AlgebraicIndepOn R _root_.id s
hs' : range x ⊆ s
this : ⇑f '' range x = ⇑f '' s
⊢ range x = s
Module: Mathlib.FieldTheory.AlgebraicClosure

Goal state before:
pp: F : Type u_1
E : Type u_2
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : IsAlgClosed E
h : algebraicClosure F E = ⊥
p : F[X]
hmon : p.Monic
hirr : Irreducible p
⊢ ∃ x, eval x p = 0

Tactic:
obtain ⟨x, hx⟩ := IsAlgClosed.exists_aeval_eq_zero E p (degree_pos_of_irreducible hirr).ne'

Goal state after:
F : Type u_1
E : Type u_2
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : IsAlgClosed E
h : algebraicClosure F E = ⊥
p : F[X]
hmon : p.Monic
hirr : Irreducible p
x : E
hx : (aeval x) p = 0
⊢ ∃ x, eval x p = 0
Module: Mathlib.GroupTheory.GroupAction.FixedPoints

Goal state before:
pp: α : Type u_1
M : Type u_3
inst✝¹ : Monoid M
inst✝ : MulAction M α
m₁ m₂ : M
a : α
h₁ : a ∈ fixedBy α m₁
h₂ : a ∈ fixedBy α m₂
⊢ m₁ • a = a

Tactic:
h₁

Goal state after:
α : Type u_1
M : Type u_3
inst✝¹ : Monoid M
inst✝ : MulAction M α
m₁ m₂ : M
a : α
h₁ : a ∈ fixedBy α m₁
h₂ : a ∈ fixedBy α m₂
⊢ a = a
Module: Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis

Goal state before:
pp: ι : Type u
R : Type u_1
S : Type v
A : Type w
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : Algebra R S
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra S A
inst✝⁴ : IsScalarTower R S A
inst✝³ : Nontrivial R
inst✝² : NoZeroDivisors S
inst✝¹ : Algebra.IsAlgebraic S A
inst✝ : FaithfulSMul S A
x : ι → S
hx : IsTranscendenceBasis R x
f : S →ₐ[R] A := IsScalarTower.toAlgHom R S A
⊢ Algebra.IsAlgebraic (↥(adjoin R (range (⇑f ∘ x)))) A

Tactic:
Set.range_comp,

Goal state after:
ι : Type u
R : Type u_1
S : Type v
A : Type w
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : Algebra R S
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra S A
inst✝⁴ : IsScalarTower R S A
inst✝³ : Nontrivial R
inst✝² : NoZeroDivisors S
inst✝¹ : Algebra.IsAlgebraic S A
inst✝ : FaithfulSMul S A
x : ι → S
hx : IsTranscendenceBasis R x
f : S →ₐ[R] A := IsScalarTower.toAlgHom R S A
⊢ Algebra.IsAlgebraic (↥(adjoin R (⇑f '' range x))) A
Module: Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis

Goal state before:
pp: R : Type u_1
A : Type w
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : FaithfulSMul R A
inst✝ : IsDomain A
s : Set A
⊢ (∀ (x : A), IsAlgebraic (↥(adjoin R s)) x) ↔ Algebra.IsAlgebraic (↥(adjoin R s)) A

Tactic:
Algebra.isAlgebraic_def

Goal state after:
Module: Mathlib.FieldTheory.SeparableDegree

Goal state before:
pp: case refine_2.inl
F : Type u
inst✝ : Field F
f✝ f g : F[X]
h✝ : f = 0 ∨ g = 0
hf : f = 0
h : g = 0
⊢ ∃ x, C x = g

Tactic:
exact ⟨0, by rw [h, map_zero]⟩

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
H : Subgroup Gal(E/F)
inst✝ : FiniteDimensional F E
H_le : H ≤ (fixedField H).fixingSubgroup
this : Nat.card ↥H = Nat.card ↥(fixedField H).fixingSubgroup
⊢ (fixedField H).fixingSubgroup = H

Tactic:
by
    exact SetLike.coe_injective (Set.eq_of_inclusion_surjective
      ((Nat.bijective_iff_injective_and_card (Set.inclusion H_le)).mpr
        ⟨Set.inclusion_injective H_le, this⟩).2).symm

Goal state after:
Module: Mathlib.FieldTheory.SeparableDegree

Goal state before:
pp: F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : FiniteDimensional F E
heq : finSepDegree F E = finrank F E
x : E
⊢ IsSeparable F x

Tactic:
have halg := IsAlgebraic.of_finite F x

Goal state after:
F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : FiniteDimensional F E
heq : finSepDegree F E = finrank F E
x : E
halg : IsAlgebraic F x
⊢ IsSeparable F x
Module: Mathlib.FieldTheory.SeparableDegree

Goal state before:
pp: case refine_2
F : Type u
E : Type v
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
x : E
hsep : IsSeparable F x
⊢ Algebra.IsSeparable F ↥F⟮x⟯

Tactic:
· have h := IsSeparable.isIntegral hsep
    have := adjoin.finiteDimensional h
    rwa [← finSepDegree_eq_finrank_iff,
      finSepDegree_adjoin_simple_eq_finrank_iff F E x h.isAlgebraic]

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

Goal state before:
pp: case left
F : Type u_1
inst✝⁴ : Field F
E : Type u_2
inst✝³ : Field E
inst✝² : Algebra F E
K L : IntermediateField F E
inst✝¹ : IsGalois F ↥K
inst✝ : FiniteDimensional F ↥K
h : K ⊔ L = ⊤
T : F[X]
hT₁ : T.Separable
hT₂ : (Polynomial.map (algebraMap F ↥K) T).Splits ∧ adjoin F (T.rootSet ↥K) = ⊤
T' : (↥L)[X] := Polynomial.map (algebraMap F ↥L) T
⊢ (Polynomial.map (algebraMap F E) T).Splits

Tactic:
exact Polynomial.Splits.of_algHom hT₂.1 (IsScalarTower.toAlgHom _ _ _)

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussLemma

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra R K
f : R[X]
hf : f.Monic
g : K[X]
hg : g.Monic
hd : g ∣ map (algebraMap R K) f
this : ∀ (n : ℕ), (map (algebraMap K g.SplittingField) g).coeff n ∈ ↑(integralClosure R g.SplittingField).toSubring
n : ℕ
p : R[X]
hp : p.Monic
he : (algebraMap K g.SplittingField) (eval (g.coeff n) (map (algebraMap R K) p)) = 0
⊢ Function.Injective ⇑(algebraMap K g.SplittingField)

Tactic:
apply RingHom.injective

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
i : ℕ
H : (minpolyDiv R x).coeff i ∉ R[x]
this : ∀ (j : ℕ), (minpolyDiv R x).coeff (i + j) ∉ R[x]
⊢ 0 ∈ R[x]

pp: R : Type u_2
S : Type u_1
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
i : ℕ
H : (minpolyDiv R x).coeff i ∉ R[x]
this : ∀ (j : ℕ), (minpolyDiv R x).coeff (i + j) ∉ R[x]
⊢ (minpolyDiv R x).natDegree < i + ((minpolyDiv R x).natDegree + 1)

Tactic:
· exact zero_mem _

Goal state after:
R : Type u_2
S : Type u_1
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
i : ℕ
H : (minpolyDiv R x).coeff i ∉ R[x]
this : ∀ (j : ℕ), (minpolyDiv R x).coeff (i + j) ∉ R[x]
⊢ (minpolyDiv R x).natDegree < i + ((minpolyDiv R x).natDegree + 1)