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

Goal state before:
pp: case mpr
R : Type u
A : Type u_4
inst✝³ : CommSemiring R
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
inst✝ : FaithfulSMul R A
M : Pic R
h : M ∈ relPic R A
e : A ⊗[R] M.AsModule ≃ₗ[A] A
f : A ⊗[R] M⁻¹.AsModule ≃ₗ[A] A
this :
  submoduleAlgebra e * submoduleAlgebra f =
    R ∙
      ↑(↑((TensorProduct.congr (submoduleAlgebraEquiv e) (submoduleAlgebraEquiv f) ≪≫ₗ ⋯.some).symm ≪≫ₗ
              tensorSubmoduleAlgebraEquivMul f (submoduleAlgebra e))
          1)
⊢ IsUnit
    (R ∙
      ↑(↑((TensorProduct.congr (submoduleAlgebraEquiv e) (submoduleAlgebraEquiv f) ≪≫ₗ ⋯.some).symm ≪≫ₗ
              tensorSubmoduleAlgebraEquivMul f (submoduleAlgebra e))
          1))

Tactic:
apply_fun (⊤ * ·) at this

Goal state after:
case mpr
R : Type u
A : Type u_4
inst✝³ : CommSemiring R
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
inst✝ : FaithfulSMul R A
M : Pic R
h : M ∈ relPic R A
e : A ⊗[R] M.AsModule ≃ₗ[A] A
f : A ⊗[R] M⁻¹.AsModule ≃ₗ[A] A
this :
  ⊤ * (submoduleAlgebra e * submoduleAlgebra f) =
    ⊤ *
      (R ∙
        ↑(↑((TensorProduct.congr (submoduleAlgebraEquiv e) (submoduleAlgebraEquiv f) ≪≫ₗ ⋯.some).symm ≪≫ₗ
                tensorSubmoduleAlgebraEquivMul f (submoduleAlgebra e))
            1))
⊢ IsUnit
    (R ∙
      ↑(↑((TensorProduct.congr (submoduleAlgebraEquiv e) (submoduleAlgebraEquiv f) ≪≫ₗ ⋯.some).symm ≪≫ₗ
              tensorSubmoduleAlgebraEquivMul f (submoduleAlgebra e))
          1))
Module: Mathlib.Combinatorics.Matroid.Basic

Goal state before:
pp: α : Type u_1
E : Set α
hE✝ : E.Finite
f : Matroid α → Set α × Set (Set α) := fun M ↦ (M.E, {B | M.IsBase B})
hf : Function.Injective f
M : Matroid α
hE : M.E ⊆ E
⊢ (f M).1 ⊆ E ∧ (f M).2 ⊆ {b | b ⊆ E}

Tactic:
exact ⟨hE, fun B hB ↦ hB.subset_ground.trans hE⟩

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

Goal state before:
pp: α : Type u_1
β : Type u_2
f : α → β
I : Set α
N : Matroid β
⊢ (N.Indep (f '' I) → ¬InjOn f I) ∧ I ⊆ f ⁻¹' N.E → ¬N.Indep (f '' I) ∧ I ⊆ f ⁻¹' N.E ∨ N.Indep (f '' I) ∧ ¬InjOn f I

Tactic:
by grind

Goal state after:
Module: Mathlib.Combinatorics.Matroid.Minor.Restrict

Goal state before:
pp: α : Type u_1
M : Matroid α
I X Y J : Set α
hI : M.IsBasis I X
hJ : M.IsBasis J Y
hJX : J ⊆ X
hIY : I ⊆ Y
⊢ M.IsBasis I Y

Tactic:
have hI' := hI.isBasis_subset (subset_inter hI.subset hIY) inter_subset_left

Goal state after:
α : Type u_1
M : Matroid α
I X Y J : Set α
hI : M.IsBasis I X
hJ : M.IsBasis J Y
hJX : J ⊆ X
hIY : I ⊆ Y
hI' : M.IsBasis I (X ∩ Y)
⊢ M.IsBasis I Y
Module: Mathlib.Combinatorics.Matroid.Rank.Finite

Goal state before:
pp: α : Type u_1
M : Matroid α
X : Set α
e : α
⊢ M.IsRkFinite (insert e X) ↔ M.IsRkFinite X

Tactic:
rw [← singleton_union, isRkFinite_singleton.isRkFinite_union_iff]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
N : Matroid β
f : α → β
h_range : N.E ⊆ range f
hf : InjOn f (f ⁻¹' N.E)
I : Set α
hI : I ⊆ f ⁻¹' N.E
⊢ (∃ I₀, (N.Indep (f '' I₀) ∧ InjOn f I₀) ∧ f '' I = f '' I₀) → N.Indep (f '' I)

Tactic:
by grind

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

Goal state before:
pp: α : Type u_1
M : Matroid α
Y : Set α
I X : Set ↑Y
⊢ M.IsBasis (↑I) (↑X ∩ M.E) ∧ ↑X ⊆ Y ↔ M.IsBasis (Subtype.val '' I) (Subtype.val '' X ∩ M.E)

pp: α : Type u_1
M : Matroid α
Y : Set α
I X : Set ↑Y
⊢ (M ↾ Y).IsBasis ↑I ↑X → I ⊆ X

Tactic:
· simp

Goal state after:
α : Type u_1
M : Matroid α
Y : Set α
I X : Set ↑Y
⊢ (M ↾ Y).IsBasis ↑I ↑X → I ⊆ X
Module: Mathlib.Combinatorics.Matroid.Loop

Goal state before:
pp: α : Type u_1
β : Type u_2
M : Matroid α
e : α
f : α → β
hf : InjOn f M.E
he : e ∈ M.E
⊢ f e ∈ f '' M.loops ↔ M.IsLoop e

Tactic:
hf.mem_image_iff M.loops_subset_ground he,

Goal state after:
α : Type u_1
β : Type u_2
M : Matroid α
e : α
f : α → β
hf : InjOn f M.E
he : e ∈ M.E
⊢ e ∈ M.loops ↔ M.IsLoop e
Module: Mathlib.Combinatorics.Matroid.Loop

Goal state before:
pp: α : Type u_1
β : Type u_2
e : α
M : Matroid β
f : α → β
⊢ M.Indep (f '' {e}) ∧ InjOn f {e} ↔ M.IsNonloop (f e)

Tactic:
image_singleton,

Goal state after:
α : Type u_1
β : Type u_2
e : α
M : Matroid β
f : α → β
⊢ M.Indep {f e} ∧ InjOn f {e} ↔ M.IsNonloop (f e)
Module: Mathlib.Combinatorics.Matroid.Loop

Goal state before:
pp: case refine_1
α : Type u_1
M : Matroid α
e : α
tfae_1_iff_2 : M.IsColoop e ↔ e ∈ M.coloops
tfae_1_iff_3 : M.IsColoop e ↔ M.IsCocircuit {e}
tfae_1_iff_4 : M.IsColoop e ↔ ∀ ⦃B : Set α⦄, M.IsBase B → e ∈ B
tfae_3_to_5 : M.IsCocircuit {e} → (∀ ⦃C : Set α⦄, M.IsCircuit C → e ∉ C) ∧ e ∈ M.E
tfae_5_to_4 : (∀ ⦃C : Set α⦄, M.IsCircuit C → e ∉ C) ∧ e ∈ M.E → ∀ ⦃B : Set α⦄, M.IsBase B → e ∈ B
h : (∀ ⦃C : Set α⦄, M.IsCircuit C → e ∉ C) ∧ e ∈ M.E
X : Set α
heX : e ∈ M.closure X
heX' : e ∉ X
⊢ False

Tactic:
obtain ⟨C, -, hC, heC⟩ := (mem_closure_iff_exists_isCircuit heX').1 heX

Goal state after:
case refine_1
α : Type u_1
M : Matroid α
e : α
tfae_1_iff_2 : M.IsColoop e ↔ e ∈ M.coloops
tfae_1_iff_3 : M.IsColoop e ↔ M.IsCocircuit {e}
tfae_1_iff_4 : M.IsColoop e ↔ ∀ ⦃B : Set α⦄, M.IsBase B → e ∈ B
tfae_3_to_5 : M.IsCocircuit {e} → (∀ ⦃C : Set α⦄, M.IsCircuit C → e ∉ C) ∧ e ∈ M.E
tfae_5_to_4 : (∀ ⦃C : Set α⦄, M.IsCircuit C → e ∉ C) ∧ e ∈ M.E → ∀ ⦃B : Set α⦄, M.IsBase B → e ∈ B
h : (∀ ⦃C : Set α⦄, M.IsCircuit C → e ∉ C) ∧ e ∈ M.E
X : Set α
heX : e ∈ M.closure X
heX' : e ∉ X
C : Set α
hC : M.IsCircuit C
heC : e ∈ C
⊢ False
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: case inl
ι : Type u_1
α : Type u_2
M✝ : Matroid α
F X Y : Set α
e f : α
M : Matroid α
hs : ∀ a ∈ ∅, M.IsFlat ↑a
⊢ M.IsFlat ↑(sInf ∅)

pp: case inr
ι : Type u_1
α : Type u_2
M✝ : Matroid α
F X Y : Set α
e f : α
M : Matroid α
s : Set ↑(Iic M.E)
hs : ∀ a ∈ s, M.IsFlat ↑a
hne : s.Nonempty
⊢ M.IsFlat ↑(sInf s)

Tactic:
· simp

Goal state after:
case inr
ι : Type u_1
α : Type u_2
M✝ : Matroid α
F X Y : Set α
e f : α
M : Matroid α
s : Set ↑(Iic M.E)
hs : ∀ a ∈ s, M.IsFlat ↑a
hne : s.Nonempty
⊢ M.IsFlat ↑(sInf s)
Module: Mathlib.Combinatorics.Matroid.Rank.ENat

Goal state before:
pp: α : Type u_1
M : Matroid α
e : α
h : M.eRk {e} = 1
⊢ M.IsNonloop e

Tactic:
rwa [← indep_singleton, indep_iff_eRk_eq_encard_of_finite (by simp), encard_singleton]

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

Goal state before:
pp: ι : Type u_1
α : Type u_2
M : Matroid α
Xs Ys : ι → Set α
A : Set ι
h : ∀ i ∈ A, M.closure (Xs i) = M.closure (Ys i)
⊢ M.closure (⋃ i ∈ A, Xs i) = M.closure (⋃ i ∈ A, Ys i)

Tactic:
← closure_biUnion_closure_eq_closure_biUnion,

Goal state after:
ι : Type u_1
α : Type u_2
M : Matroid α
Xs Ys : ι → Set α
A : Set ι
h : ∀ i ∈ A, M.closure (Xs i) = M.closure (Ys i)
⊢ M.closure (⋃ i ∈ A, M.closure (Xs i)) = M.closure (⋃ i ∈ A, Ys i)
Module: Mathlib.Combinatorics.Matroid.Loop

Goal state before:
pp: α : Type u_1
M : Matroid α
I K : Set α
hK : K ⊆ M.coloops
⊢ M.Indep (I ∪ K) ↔ M.Indep I

Tactic:
union_indep_iff_indep_of_subset_coloops hK

Goal state after:
α : Type u_1
M : Matroid α
I K : Set α
hK : K ⊆ M.coloops
⊢ M.Indep I ↔ M.Indep I
Module: Mathlib.Combinatorics.Matroid.Rank.Cardinal

Goal state before:
pp: case refine_1
α : Type u
β : Type v
f : α → β
M : Matroid α
hf : InjOn f M.E
X : Set α
hX : X ⊆ M.E
I X' : Set α
hIX : M.IsBasis I X'
hXX' : f '' X = f '' X'
hI : (M.map f hf).IsBasis (f '' I) (f '' X)
⊢ lift.{v, u} #↑I ≤ lift.{v, u} (⨆ B, #↑↑B)

Tactic:
lift_le

Goal state after:
case refine_1
α : Type u
β : Type v
f : α → β
M : Matroid α
hf : InjOn f M.E
X : Set α
hX : X ⊆ M.E
I X' : Set α
hIX : M.IsBasis I X'
hXX' : f '' X = f '' X'
hI : (M.map f hf).IsBasis (f '' I) (f '' X)
⊢ #↑I ≤ ⨆ B, #↑↑B
Module: Mathlib.Combinatorics.Matroid.Rank.Cardinal

Goal state before:
pp: case refine_2
α : Type u
β : Type v
f : α → β
M : Matroid α
hf : InjOn f M.E
X : Set α
hX : X ⊆ M.E
I : Set α
hI : M.IsBasis I X
⊢ lift.{u, v} #↑(f '' I) ≤ lift.{u, v} (⨆ B, #↑↑B)

Tactic:
lift_le

Goal state after:
case refine_2
α : Type u
β : Type v
f : α → β
M : Matroid α
hf : InjOn f M.E
X : Set α
hX : X ⊆ M.E
I : Set α
hI : M.IsBasis I X
⊢ #↑(f '' I) ≤ ⨆ B, #↑↑B
Module: Mathlib.Combinatorics.Matroid.Loop

Goal state before:
pp: case inl
α : Type u_1
M : Matroid α
inst✝ : M.Loopless
hI : ∅.Subsingleton
hIE : ∅ ⊆ M.E
⊢ M.Indep ∅

pp: case inr
α : Type u_1
M : Matroid α
inst✝ : M.Loopless
x : α
hI : {x}.Subsingleton
hIE : {x} ⊆ M.E
⊢ M.Indep {x}

Tactic:
· simp

Goal state after:
case inr
α : Type u_1
M : Matroid α
inst✝ : M.Loopless
x : α
hI : {x}.Subsingleton
hIE : {x} ⊆ M.E
⊢ M.Indep {x}
Module: Mathlib.Combinatorics.Matroid.Rank.Cardinal

Goal state before:
pp: case refine_1
α : Type u
β : Type v
M : Matroid β
f : α → β
X I : Set α
hI : M.IsBasis' (f '' I) (f '' X)
hfI : InjOn f I
hIX : I ⊆ X
⊢ lift.{u, v} #↑(f '' I) ≤ lift.{u, v} (⨆ B, #↑↑B)

Tactic:
lift_le

Goal state after:
case refine_1
α : Type u
β : Type v
M : Matroid β
f : α → β
X I : Set α
hI : M.IsBasis' (f '' I) (f '' X)
hfI : InjOn f I
hIX : I ⊆ X
⊢ #↑(f '' I) ≤ ⨆ B, #↑↑B
Module: Mathlib.Combinatorics.Matroid.Rank.Cardinal

Goal state before:
pp: case refine_2
α : Type u
β : Type v
M : Matroid β
f : α → β
X I₀ : Set α
hI₀X : I₀ ⊆ X
hfI₀ : InjOn f I₀
hIX : M.IsBasis' (f '' I₀) (f '' X)
⊢ lift.{v, u} #↑I₀ ≤ lift.{v, u} (⨆ B, #↑↑B)

Tactic:
lift_le

Goal state after:
case refine_2
α : Type u
β : Type v
M : Matroid β
f : α → β
X I₀ : Set α
hI₀X : I₀ ⊆ X
hfI₀ : InjOn f I₀
hIX : M.IsBasis' (f '' I₀) (f '' X)
⊢ #↑I₀ ≤ ⨆ B, #↑↑B
Module: Mathlib.RingTheory.Algebraic.MvPolynomial

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : CommRing R
i : σ
f : R[X]
⊢ Transcendental R ((Polynomial.aeval (X i)) f) ↔ Transcendental R f

Tactic:
refine ⟨?_, transcendental_polynomial_aeval_X R i⟩

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : CommRing R
i : σ
f : R[X]
⊢ Transcendental R ((Polynomial.aeval (X i)) f) → Transcendental R f
Module: Mathlib.Combinatorics.Matroid.Rank.Cardinal

Goal state before:
pp: case inl
α : Type u
M : Matroid α
h : M.RankFinite
⊢ toENat M.cRank = M.eRank

Tactic:
obtain ⟨B, hB⟩ := M.exists_isBase
    rw [← hB.cardinalMk_eq_cRank, ← hB.encard_eq_eRank, toENat_cardinalMk]

Goal state after:
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
hp : (aeval f) p = 0
⊢ (aeval x) ((aeval f) 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
⊢ (aeval x) 0 = 0
Module: Mathlib.RingTheory.Polynomial.SeparableDegree

Goal state before:
pp: F✝ : Type u_1
inst✝² : Field F✝
q✝ : ℕ
F : Type u_1
inst✝¹ : Field F
q : ℕ
hq : NeZero q
inst✝ : CharP F q
g g' : F[X]
m : ℕ
hg : g.Separable
hg' : g'.Separable
s : ℕ
h_expand : g = (expand F (q ^ s)) g'
hm : m ≤ m + s
⊢ g.natDegree = g'.natDegree

Tactic:
subst h_expand

Goal state after:
F✝ : Type u_1
inst✝² : Field F✝
q✝ : ℕ
F : Type u_1
inst✝¹ : Field F
q : ℕ
hq : NeZero q
inst✝ : CharP F q
g' : F[X]
m : ℕ
hg' : g'.Separable
s : ℕ
hm : m ≤ m + s
hg : ((expand F (q ^ s)) g').Separable
⊢ ((expand F (q ^ s)) g').natDegree = g'.natDegree
Module: Mathlib.RingTheory.AlgebraicIndependent.Basic

Goal state before:
pp: ι : Type u
R : Type u_2
A : Type v
x : ι → A
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : Nontrivial R
hx : AlgebraicIndependent R x
⊢ lift.{u, v} #↑(range x) ≤ lift.{u, v} (trdeg R A)

Tactic:
lift_le

Goal state after:
ι : Type u
R : Type u_2
A : Type v
x : ι → A
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : Nontrivial R
hx : AlgebraicIndependent R x
⊢ #↑(range x) ≤ trdeg R A
Module: Mathlib.GroupTheory.GroupAction.FixedPoints

Goal state before:
pp: α : Type u_1
G : Type u_2
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : FaithfulSMul G α
g h : G
disjoint : Disjoint (fixedBy α g)ᶜ (fixedBy α g)ᶜ
comm : Commute g h
⊢ g = 1

Tactic:
disjoint_self,

Goal state after:
α : Type u_1
G : Type u_2
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : FaithfulSMul G α
g h : G
disjoint : (fixedBy α g)ᶜ = ⊥
comm : Commute g h
⊢ g = 1
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 t : Set A
⊢ ((AlgebraicIndepOn R _root_.id s ∧ s ⊆ t) ∧ ∀ x ∉ s, ¬(AlgebraicIndepOn R _root_.id (insert x s) ∧ insert x s ⊆ t)) ∧
      t ⊆ univ ↔
    (AlgebraicIndepOn R _root_.id s ∧ s ⊆ t) ∧ ∀ a ∈ t, IsAlgebraic (↥(adjoin R s)) a

Tactic:
subset_univ,

Goal state after:
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 t : Set A
⊢ ((AlgebraicIndepOn R _root_.id s ∧ s ⊆ t) ∧ ∀ x ∉ s, ¬(AlgebraicIndepOn R _root_.id (insert x s) ∧ insert x s ⊆ t)) ∧
      True ↔
    (AlgebraicIndepOn R _root_.id s ∧ s ⊆ t) ∧ ∀ a ∈ t, IsAlgebraic (↥(adjoin R s)) 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✝ : Subsingleton A
s t : Set A
this : Subsingleton R
⊢ Maximal (fun A_1 ↦ A_1 ⊆ t) s ∧ t ⊆ univ ↔ s = t

Tactic:
subset_univ,

Goal state after:
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✝ : Subsingleton A
s t : Set A
this : Subsingleton R
⊢ Maximal (fun A_1 ↦ A_1 ⊆ t) s ∧ True ↔ s = t
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
⊢ (matroid R A).closure s = univ ∧ s ⊆ univ ↔ Algebra.IsAlgebraic (↥(adjoin R s)) A

Tactic:
subset_univ,

Goal state after:
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
⊢ (matroid R A).closure s = univ ∧ True ↔ Algebra.IsAlgebraic (↥(adjoin R s)) 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✝ : Subsingleton A
s : Set A
⊢ (∀ ⦃I X : Set A⦄, (matroid R A).IsBasis I s → (matroid R A).IsBasis I X → X ⊆ s) ∧ s ⊆ univ

Tactic:
subset_univ,

Goal state after:
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✝ : Subsingleton A
s : Set A
⊢ (∀ ⦃I X : Set A⦄, (matroid R A).IsBasis I s → (matroid R A).IsBasis I X → X ⊆ s) ∧ True
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✝ : Subsingleton A
s : Set A
⊢ s = univ ∧ s ⊆ univ ↔ s = univ

Tactic:
subset_univ,

Goal state after:
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✝ : Subsingleton A
s : Set A
⊢ s = univ ∧ True ↔ s = univ
Module: Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis

Goal state before:
pp: case inr.inr
ι : Type u
R : Type u_1
S : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : FaithfulSMul R S
inst✝ : NoZeroDivisors S
s : Set ι
i j : ι
v : ι → S
hj✝ : j ∈ insert i s
H₁ : IsTranscendenceBasis R fun x ↦ v ↑x
this✝³ : Nontrivial ↥(adjoin R (v '' (insert i s \ {j})))
this✝² : Nontrivial S
this✝¹ : IsDomain S
H₂ : v j ∈ (matroid R S).closure (v '' insert i s \ v '' {j})
this✝ : Nontrivial R
inj : InjOn v s
H' : (matroid R S).IsBase (v '' s)
hj : j ∈ s
Hj : v j ∉ (matroid R S).closure (v '' s \ {v j})
hi : i ∉ s
ne : v i ≠ v j
hi' : v i ∉ v '' s
this : InjOn v (insert i s)
⊢ (matroid R S).IsBase (v '' insert i s \ v '' {j})

Tactic:
image_singleton,

Goal state after:
case inr.inr
ι : Type u
R : Type u_1
S : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : FaithfulSMul R S
inst✝ : NoZeroDivisors S
s : Set ι
i j : ι
v : ι → S
hj✝ : j ∈ insert i s
H₁ : IsTranscendenceBasis R fun x ↦ v ↑x
this✝³ : Nontrivial ↥(adjoin R (v '' (insert i s \ {j})))
this✝² : Nontrivial S
this✝¹ : IsDomain S
H₂ : v j ∈ (matroid R S).closure (v '' insert i s \ {v j})
this✝ : Nontrivial R
inj : InjOn v s
H' : (matroid R S).IsBase (v '' s)
hj : j ∈ s
Hj : v j ∉ (matroid R S).closure (v '' s \ {v j})
hi : i ∉ s
ne : v i ≠ v j
hi' : v i ∉ v '' s
this : InjOn v (insert i s)
⊢ (matroid R S).IsBase (v '' insert i s \ {v j})
Module: Mathlib.Topology.Algebra.FilterBasis

Goal state before:
pp: G : Type u
inst✝ : Group G
B : GroupFilterBasis G
⊢ 𝓝 1 = B.filter

Tactic:
by
  rw [B.nhds_eq]
  simp only [N, one_mul]
  exact map_id

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

Goal state before:
pp: F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
⊢ ⊥.fixingSubgroup = ⊤

Tactic:
ext
  simp [mem_bot]

Goal state after:
Module: Mathlib.FieldTheory.SeparableDegree

Goal state before:
pp: case refine_1
F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Ring E
inst✝¹ : IsDomain E
inst✝ : Algebra F E
q : ℕ
hF : ExpChar F q
x : E
this✝ : ExpChar E q
this : ExpChar E[X] q
h✝ : (minpoly F x).natSepDegree = 1
n : ℕ
y : F
h : minpoly F x = X ^ q ^ n - C y
⊢ ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n

Tactic:
have hx := congr_arg (Polynomial.aeval x) h.symm

Goal state after:
case refine_1
F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Ring E
inst✝¹ : IsDomain E
inst✝ : Algebra F E
q : ℕ
hF : ExpChar F q
x : E
this✝ : ExpChar E q
this : ExpChar E[X] q
h✝ : (minpoly F x).natSepDegree = 1
n : ℕ
y : F
h : minpoly F x = X ^ q ^ n - C y
hx : (Polynomial.aeval x) (X ^ q ^ n - C y) = (Polynomial.aeval x) (minpoly F x)
⊢ ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n
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
hp : p = 0
⊢ False

Tactic:
hp,

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 ∈ Polynomial.aroots 0 E
h : IsIntegral K x
hp : p = 0
⊢ False
Module: Mathlib.Topology.Algebra.OpenSubgroup

Goal state before:
pp: G : Type u_1
inst✝³ : TopologicalSpace G
inst✝² : Group G
inst✝¹ : IsTopologicalGroup G
inst✝ : CompactSpace G
W : Set G
WClopen : IsClopen W
x : G
memW : x ∈ W
this : (x, 1) ∈ (fun p ↦ p.1 * p.2) ⁻¹' W
U V : Set G
Uopen : IsOpen[inst✝³] U
Vopen : IsOpen[inst✝³] V
xmemU : x ∈ U
onememV : 1 ∈ V
prodsub : U ×ˢ V ⊆ (fun p ↦ p.1 * p.2) ⁻¹' W
h6 : U * V ⊆ W
⊢ ∃ t ∈ 𝓝[W] x, ∃ T ∈ 𝓝 1, t * T ⊆ W

Tactic:
exact ⟨U ∩ W, ⟨U, Uopen.mem_nhds xmemU, W, fun _ a ↦ a, rfl⟩,
    V, IsOpen.mem_nhds Vopen onememV, fun _ a ↦ h6 ((mul_subset_mul_right inter_subset_left) a)⟩

Goal state after:
Module: Mathlib.FieldTheory.KrullTopology

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
U : Set Gal(L/K)
hU : U ∈ (galBasis K L).sets
⊢ U ⊆ (fun x ↦ x⁻¹) ⁻¹' U

Tactic:
rcases hU with ⟨H, _, rfl⟩
      exact fun _ => H.inv_mem'

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.dualBasis hB (B.flip.dualBasis ⋯ b))) = Submodule.span R (Set.range ⇑b)

Tactic:
dualBasis_dualBasis_flip 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.FieldTheory.Minpoly.IsIntegrallyClosed

Goal state before:
pp: case refine_1
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain R
inst✝³ : Algebra R S
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDomain S
inst✝ : IsTorsionFree R S
s : S
hs : IsIntegral R s
p : R[X]
hp : (Polynomial.aeval s) p = 0
K : Type u_1 := FractionRing R
L : Type u_2 := FractionRing S
x✝ : Algebra K L := FractionRing.liftAlgebra R L
⊢ (algebraMap S L) ((Polynomial.aeval s) p) = 0

pp: case refine_1.h
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain R
inst✝³ : Algebra R S
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDomain S
inst✝ : IsTorsionFree R S
s : S
hs : IsIntegral R s
p : R[X]
hp : (Polynomial.aeval s) p = 0
K : Type u_1 := FractionRing R
L : Type u_2 := FractionRing S
x✝ : Algebra K L := FractionRing.liftAlgebra R L
⊢ (algebraMap K L).comp (algebraMap R K) = (algebraMap S L).comp (algebraMap R S)

Tactic:
hp,

Goal state after:
case refine_1
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain R
inst✝³ : Algebra R S
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDomain S
inst✝ : IsTorsionFree R S
s : S
hs : IsIntegral R s
p : R[X]
hp : (Polynomial.aeval s) p = 0
K : Type u_1 := FractionRing R
L : Type u_2 := FractionRing S
x✝ : Algebra K L := FractionRing.liftAlgebra R L
⊢ (algebraMap S L) 0 = 0

case refine_1.h
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain R
inst✝³ : Algebra R S
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDomain S
inst✝ : IsTorsionFree R S
s : S
hs : IsIntegral R s
p : R[X]
hp : (Polynomial.aeval s) p = 0
K : Type u_1 := FractionRing R
L : Type u_2 := FractionRing S
x✝ : Algebra K L := FractionRing.liftAlgebra R L
⊢ (algebraMap K L).comp (algebraMap R K) = (algebraMap S L).comp (algebraMap R S)
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).Monic

Tactic:
leadingCoeff_mul',

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
hx : IsIntegral R x
a✝ : Nontrivial S
this : (minpolyDiv R x).leadingCoeff * (X - C x).leadingCoeff = (map (algebraMap R S) (minpoly R x)).leadingCoeff
⊢ (minpolyDiv R x).Monic

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