mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Idempotents.FunctorExtension

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
inst✝ : Category.{v_3, u_3} E
G : Karoubi C ⥤ Karoubi D
⊢ { app := fun P ↦ { f := (G.map P.decompId_p).f, comm := ⋯ }, naturality := ⋯ } ≫
      { app := fun P ↦ { f := (G.map P.decompId_i).f, comm := ⋯ }, naturality := ⋯ } =
    𝟙 (((whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C) ⋙ functorExtension₁ C D).obj G)

Tactic:
by
          ext P
          simpa only [hom_ext_iff, G.map_comp, G.map_id] using! G.congr_map P.decomp_p.symm

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.LeftDerived

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{v_1, u_1} D
inst✝³ : Abelian C
inst✝² : HasProjectiveResolutions C
inst✝¹ : Abelian D
X Y : C
f : X ⟶ Y
P : ProjectiveResolution X
Q : ProjectiveResolution Y
φ : P.complex ⟶ Q.complex
comm : φ.f 0 ≫ Q.π.f 0 = P.π.f 0 ≫ f
F : C ⥤ D
inst✝ : F.Additive
n : ℕ
⊢ (HomotopyCategory.homologyFunctor D (ComplexShape.down ℕ) n).map (P.isoLeftDerivedToHomotopyCategoryObj F).hom ≫
      (HomotopyCategory.homologyFunctor D (ComplexShape.down ℕ) n).map
          ((F.mapHomologicalComplex (ComplexShape.down ℕ) ⋙ HomotopyCategory.quotient D (ComplexShape.down ℕ)).map φ) ≫
        (HomotopyCategory.homologyFunctorFactors D (ComplexShape.down ℕ) n).hom.app
          ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj Q.complex) =
    (HomotopyCategory.homologyFunctor D (ComplexShape.down ℕ) n).map (P.isoLeftDerivedToHomotopyCategoryObj F).hom ≫
      (HomotopyCategory.homologyFunctorFactors D (ComplexShape.down ℕ) n).hom.app
          ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex) ≫
        HomologicalComplex.homologyMap ((F.mapHomologicalComplex (ComplexShape.down ℕ)).map φ) n

Tactic:
erw [(HomotopyCategory.homologyFunctorFactors D (ComplexShape.down ℕ) n).hom.naturality]

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{v_1, u_1} D
inst✝³ : Abelian C
inst✝² : HasProjectiveResolutions C
inst✝¹ : Abelian D
X Y : C
f : X ⟶ Y
P : ProjectiveResolution X
Q : ProjectiveResolution Y
φ : P.complex ⟶ Q.complex
comm : φ.f 0 ≫ Q.π.f 0 = P.π.f 0 ≫ f
F : C ⥤ D
inst✝ : F.Additive
n : ℕ
⊢ (HomotopyCategory.homologyFunctor D (ComplexShape.down ℕ) n).map (P.isoLeftDerivedToHomotopyCategoryObj F).hom ≫
      (HomotopyCategory.homologyFunctorFactors D (ComplexShape.down ℕ) n).hom.app
          ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex) ≫
        (HomologicalComplex.homologyFunctor D (ComplexShape.down ℕ) n).map
          ((F.mapHomologicalComplex (ComplexShape.down ℕ)).map φ) =
    (HomotopyCategory.homologyFunctor D (ComplexShape.down ℕ) n).map (P.isoLeftDerivedToHomotopyCategoryObj F).hom ≫
      (HomotopyCategory.homologyFunctorFactors D (ComplexShape.down ℕ) n).hom.app
          ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex) ≫
        HomologicalComplex.homologyMap ((F.mapHomologicalComplex (ComplexShape.down ℕ)).map φ) n
Module: Mathlib.CategoryTheory.Abelian.LeftDerived

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{v_1, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F : C ⥤ D
inst✝¹ : F.Additive
n : ℕ
X : C
inst✝ : Projective X
⊢ ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj (ProjectiveResolution.self X).complex).ExactAt (n + 1)

Tactic:
exact ShortComplex.exact_of_isZero_X₂ _ (F.map_isZero (by apply isZero_zero))

Goal state after:
Module: Mathlib.Algebra.Homology.Monoidal

Goal state before:
pp: case pos
C : Type u_1
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₁ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).obj X₁)
i₁ i₂ j : I
h₁ : c.Rel i₂ (c.next i₂)
⊢ mapBifunctor.d₂ K (tensorUnit C c) (curriedTensor C) c i₁ i₂ j = 0

Tactic:
by_cases h₂ : ComplexShape.π c c c (i₁, c.next i₂) = j
    · rw [mapBifunctor.d₂_eq _ _ _ _ _ h₁ _ h₂, single_obj_d, Functor.map_zero,
        zero_comp, smul_zero]
    · rw [mapBifunctor.d₂_eq_zero' _ _ _ _ _ h₁ _ h₂]

Goal state after:
Module: Mathlib.Algebra.Homology.Monoidal

Goal state before:
pp: C : Type u_1
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₁ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).obj X₁)
i : I
⊢ K.X i ◁ GradedObject.Monoidal.tensorUnit₀.inv ≫
      K.X i ◁ (tensorUnitIso C c).hom 0 ≫ GradedObject.Monoidal.ιTensorObj K.X (tensorUnit C c).X i 0 i ⋯ =
    K.X i ◁ (singleObjXSelf c 0 (𝟙_ C)).inv ≫ K.ιTensorObj (tensorUnit C c) i 0 i ⋯

Tactic:
← whiskerLeft_comp_assoc

Goal state after:
C : Type u_1
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₁ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).obj X₁)
i : I
⊢ K.X i ◁ (GradedObject.Monoidal.tensorUnit₀.inv ≫ (tensorUnitIso C c).hom 0) ≫
      GradedObject.Monoidal.ιTensorObj K.X (tensorUnit C c).X i 0 i ⋯ =
    K.X i ◁ (singleObjXSelf c 0 (𝟙_ C)).inv ≫ K.ιTensorObj (tensorUnit C c) i 0 i ⋯
Module: Mathlib.Algebra.Homology.SpectralObject.Differentials

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₁₂ : i ⟶ k
h₁₂ : f₁ ≫ f₂ = f₁₂
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
⊢ (X.H n₀).map (twoδ₁Toδ₀ f₁ f₂₃ (f₁ ≫ f₂ ≫ f₃) ⋯) ≫ X.toCycles f₂ f₃ f₂₃ h₂₃ n₀ ≫ X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁ = 0

Tactic:
toCycles_Ψ ..,

Goal state after:
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₁₂ : i ⟶ k
h₁₂ : f₁ ≫ f₂ = f₁₂
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
⊢ (X.H n₀).map (twoδ₁Toδ₀ f₁ f₂₃ (f₁ ≫ f₂ ≫ f₃) ⋯) ≫ X.δ f₁ f₂₃ n₀ n₁ ⋯ ≫ X.pOpcycles f₁ f₂ n₁ = 0
Module: Mathlib.Algebra.Homology.SpectralObject.Differentials

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i₀ i₁ i₂ i₃ i₄ i₅ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₅ : i₄ ⟶ i₅
n₀ n₁ n₂ n₃ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
⊢ X.toCycles f₃ f₄ (f₃ ≫ f₄) ⋯ n₁ ≫
      X.πE f₃ f₄ f₅ n₀ n₁ n₂ hn₁ hn₂ ≫ X.d f₁ f₂ f₃ f₄ f₅ n₀ n₁ n₂ n₃ hn₁ hn₂ hn₃ ≫ X.ιE f₁ f₂ f₃ n₁ n₂ n₃ hn₂ hn₃ =
    X.toCycles f₃ f₄ (f₃ ≫ f₄) ⋯ n₁ ≫ X.Ψ f₂ f₃ f₄ n₁ n₂ hn₂

Tactic:
toCycles_Ψ ..,

Goal state after:
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i₀ i₁ i₂ i₃ i₄ i₅ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₅ : i₄ ⟶ i₅
n₀ n₁ n₂ n₃ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
⊢ X.toCycles f₃ f₄ (f₃ ≫ f₄) ⋯ n₁ ≫
      X.πE f₃ f₄ f₅ n₀ n₁ n₂ hn₁ hn₂ ≫ X.d f₁ f₂ f₃ f₄ f₅ n₀ n₁ n₂ n₃ hn₁ hn₂ hn₃ ≫ X.ιE f₁ f₂ f₃ n₁ n₂ n₃ hn₂ hn₃ =
    X.δ f₂ (f₃ ≫ f₄) n₁ n₂ ⋯ ≫ X.pOpcycles f₂ f₃ n₂
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
i' j' k' l' : ι
f₁' : i' ⟶ j'
f₂' : j' ⟶ k'
f₃' : k' ⟶ l'
α : mk₃ f₁ f₂ f₃ ⟶ mk₃ f₁' f₂' f₃'
n₀ n₁ n₂ : ℤ
h₀ : IsIso ((X.H n₀).map ((functorArrows ι 2 3 3 isIso_map._proof_6 shortComplexMap._proof_3).map α))
h₁ : IsIso ((X.H n₁).map ((functorArrows ι 1 2 3 isIso_map._proof_8 isIso_map._proof_6).map α))
h₂ : IsIso ((X.H n₂).map ((functorArrows ι 0 1 3 isIso_map._proof_10 isIso_map._proof_12).map α))
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ IsIso (X.map f₁ f₂ f₃ f₁' f₂' f₃' α n₀ n₁ n₂ hn₁ hn₂)

Tactic:
have : IsIso (shortComplexMap X f₁ f₂ f₃ f₁' f₂' f₃' α n₀ n₁ n₂) := by
    apply +allowSynthFailures ShortComplex.isIso_of_isIso <;> assumption
  dsimp [map]
  infer_instance

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence

Goal state before:
pp: ι : Type u_2
κ : Type u_3
inst✝ : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq' : κ
i₀ i₁ : ι
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
⊢ i₀ ≤ i₁

Tactic:
have := data.le₀₁ r pq'
  simpa only [hi₀, hi₁] using data.le₀₁ r pq'

Goal state after:
Module: Mathlib.Algebra.Jordan.Basic

Goal state before:
pp: A : Type u_1
inst✝ : NonUnitalNonAssocCommRing A
a b c : A
⊢ ⁅L (a + b + c), L ((a + b + c) * (a + b + c))⁆ =
    ⁅L a + L b + L c, L (a * a) + L (b * b) + L (c * c) + 2 • L (a * b) + 2 • L (c * a) + 2 • L (b * c)⁆

Tactic:
rw [add_mul, add_mul]

Goal state after:
A : Type u_1
inst✝ : NonUnitalNonAssocCommRing A
a b c : A
⊢ ⁅L (a + b + c), L (a * (a + b + c) + b * (a + b + c) + c * (a + b + c))⁆ =
    ⁅L a + L b + L c, L (a * a) + L (b * b) + L (c * c) + 2 • L (a * b) + 2 • L (c * a) + 2 • L (b * c)⁆
Module: Mathlib.Algebra.Lie.OfAssociative

Goal state before:
pp: R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x y : L
z : M
n : ℕ
⊢ (φ x ^ n) ⁅y, z⁆ = ∑ ij ∈ antidiagonal n, n.choose ij.1 • ⁅((ad R L) x ^ ij.1) y, (φ x ^ ij.2) z⁆

Tactic:
induction n with
  | zero => simp
  | succ n ih =>
    rw [Finset.sum_antidiagonal_choose_succ_nsmul
      (fun i j ↦ ⁅((ad R L x) ^ i) y, ((φ x) ^ j) z⁆) n]
    simp only [pow_succ', Module.End.mul_apply, ih, map_sum, map_nsmul,
      toEnd_lie, nsmul_add, sum_add_distrib]
    rw [add_comm, add_left_cancel_iff, sum_congr rfl]
    rintro ⟨i, j⟩ hij
    rw [mem_antidiagonal] at hij
    rw [Nat.choose_symm_of_eq_add hij.symm]

Goal state after:
Module: Mathlib.Algebra.Lie.Subalgebra

Goal state before:
pp: R : Type u
L₁ : Type v
L₂ : Type w
inst✝⁴ : CommRing R
inst✝³ : LieRing L₁
inst✝² : LieRing L₂
inst✝¹ : LieAlgebra R L₁
inst✝ : LieAlgebra R L₂
f : L₁ →ₗ⁅R⁆ L₂
h : Function.Injective ⇑f
⊢ Function.Injective ⇑↑f

Tactic:
rwa [LieHom.coe_toLinearMap]

Goal state after:
Module: Mathlib.Algebra.Lie.Submodule

Goal state before:
pp: R : Type u
L : Type v
M : Type w
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : LieRingModule L M
ι : Sort u_1
p : ι → LieSubmodule R L M
x : M
⊢ x ∈ ↑(⨅ i, p i) ↔ ∀ (i : ι), x ∈ p i

Tactic:
coe_iInf,

Goal state after:
R : Type u
L : Type v
M : Type w
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : LieRingModule L M
ι : Sort u_1
p : ι → LieSubmodule R L M
x : M
⊢ x ∈ ⋂ i, ↑(p i) ↔ ∀ (i : ι), x ∈ p i
Module: Mathlib.Algebra.Lie.Ideal

Goal state before:
pp: R : Type u
L : Type v
L' : Type w₂
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieRing L'
inst✝¹ : LieAlgebra R L'
inst✝ : LieAlgebra R L
f : L →ₗ⁅R⁆ L'
I : LieIdeal R L
⊢ map f I = ⊥ ↔ I ≤ f.ker

Tactic:
rw [← le_bot_iff]

Goal state after:
R : Type u
L : Type v
L' : Type w₂
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieRing L'
inst✝¹ : LieAlgebra R L'
inst✝ : LieAlgebra R L
f : L →ₗ⁅R⁆ L'
I : LieIdeal R L
⊢ map f I ≤ ⊥ ↔ I ≤ f.ker
Module: Mathlib.Algebra.Lie.Ideal

Goal state before:
pp: R : Type u
L : Type v
L' : Type w₂
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieRing L'
inst✝¹ : LieAlgebra R L'
inst✝ : LieAlgebra R L
f : L →ₗ⁅R⁆ L'
I : LieIdeal R L
y : L'
h₁ : Function.Surjective ⇑f
h₂ : y ∈ Submodule.map ↑f ↑I
⊢ ∃ x, f ↑x = y

Tactic:
Submodule.mem_map

Goal state after:
R : Type u
L : Type v
L' : Type w₂
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieRing L'
inst✝¹ : LieAlgebra R L'
inst✝ : LieAlgebra R L
f : L →ₗ⁅R⁆ L'
I : LieIdeal R L
y : L'
h₁ : Function.Surjective ⇑f
h₂ : ∃ y_1 ∈ ↑I, ↑f y_1 = y
⊢ ∃ x, f ↑x = y
Module: Mathlib.Algebra.Lie.IdealOperations

Goal state before:
pp: R : Type u
L : Type v
M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : LieRing L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
N N' : LieSubmodule R L M
inst✝ : LieAlgebra R L
I : LieIdeal R L
⊢ lieSpan R L {x | ∃ x_1 n, ⁅↑x_1, ↑n⁆ = x} ≤ N' ↔ ∀ x ∈ I, ∀ m ∈ N, ⁅x, m⁆ ∈ N'

Tactic:
LieSubmodule.lieSpan_le

Goal state after:
R : Type u
L : Type v
M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : LieRing L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
N N' : LieSubmodule R L M
inst✝ : LieAlgebra R L
I : LieIdeal R L
⊢ {x | ∃ x_1 n, ⁅↑x_1, ↑n⁆ = x} ⊆ ↑N' ↔ ∀ x ∈ I, ∀ m ∈ N, ⁅x, m⁆ ∈ N'
Module: Mathlib.Algebra.Lie.Abelian

Goal state before:
pp: case refine_2.mem.add
R : Type u_1
L : Type u_2
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
s : Set L
h : ∀ x ∈ s, ∀ y ∈ s, ⁅x, y⁆ = 0
x✝¹ x✝ : ↥(lieSpan R L s)
x y w : L
hw : w ∈ s
u v : L
hx✝ : u ∈ lieSpan R L s
hy✝ : v ∈ lieSpan R L s
hu : ⁅⟨w, ⋯⟩, ⟨u, hx✝⟩⁆ = 0
hv : ⁅⟨w, ⋯⟩, ⟨v, hy✝⟩⁆ = 0
⊢ ⁅⟨w, ⋯⟩, ⟨u + v, ⋯⟩⁆ = 0

Tactic:
simp only [Subtype.ext_iff, coe_bracket, ZeroMemClass.coe_zero, lie_add] at hu hv ⊢
        simp [hu, hv]

Goal state after:
Module: Mathlib.Algebra.Lie.Abelian

Goal state before:
pp: R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
N : LieSubmodule R L M
I : LieIdeal R L
inst✝ : LieModule.IsTrivial L M
⊢ lieSpan R L {x | ∃ x_1 n, ⁅↑x_1, ↑n⁆ = x} ≤ ⊥

Tactic:
LieSubmodule.lieSpan_le

Goal state after:
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
N : LieSubmodule R L M
I : LieIdeal R L
inst✝ : LieModule.IsTrivial L M
⊢ {x | ∃ x_1 n, ⁅↑x_1, ↑n⁆ = x} ⊆ ↑⊥
Module: Mathlib.Algebra.Lie.IdealOperations

Goal state before:
pp: R : Type u
L : Type v
L' : Type w₂
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
inst✝¹ : LieRing L'
inst✝ : LieAlgebra R L'
f : L →ₗ⁅R⁆ L'
I₁ I₂ : LieIdeal R L
⊢ LieSubmodule.lieSpan R L {x | ∃ x_1 n, ⁅↑x_1, ↑n⁆ = x} ≤ comap f ⁅map f I₁, map f I₂⁆

Tactic:
LieSubmodule.lieSpan_le

Goal state after:
R : Type u
L : Type v
L' : Type w₂
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
inst✝¹ : LieRing L'
inst✝ : LieAlgebra R L'
f : L →ₗ⁅R⁆ L'
I₁ I₂ : LieIdeal R L
⊢ {x | ∃ x_1 n, ⁅↑x_1, ↑n⁆ = x} ⊆ ↑(comap f ⁅map f I₁, map f I₂⁆)
Module: Mathlib.Algebra.Lie.IdealOperations

Goal state before:
pp: R : Type u
L : Type v
L' : Type w₂
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
inst✝¹ : LieRing L'
inst✝ : LieAlgebra R L'
f : L →ₗ⁅R⁆ L'
I₁ I₂ : LieIdeal R L
h : Function.Surjective ⇑f
y₁ : ↥I₁
h₁ : f ↑y₁ ∈ map f I₁
y₂ : ↥I₂
h₂ : f ↑y₂ ∈ map f I₂
⊢ ↑f ⁅↑y₁, ↑y₂⁆ = ⁅↑⟨f ↑y₁, h₁⟩, ↑⟨f ↑y₂, h₂⟩⁆

Tactic:
apply f.map_lie

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

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝³ : Field A
inst✝² : Ring B
inst✝¹ : IsDomain B
inst✝ : Algebra A B
x : B
hx : IsIntegral A x
⊢ (minpoly A x).coeff 0 = 0 ↔ x = 0

Tactic:
constructor
  · intro h
    have zero_root := zero_isRoot_of_coeff_zero_eq_zero h
    rw [← root hx zero_root]
    exact map_zero _
  · rintro rfl
    simp

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

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Semiring S
inst✝ : Algebra R S
x : S
I : Ideal R
y a✝ b✝ : ↥R[x]
ha✝ : a✝ ∈ Submodule.span (↥R[x]) (⇑(algebraMap R ↥R[x]) '' ↑I)
hb✝ : b✝ ∈ Submodule.span (↥R[x]) (⇑(algebraMap R ↥R[x]) '' ↑I)
a : R[X]
ha : ∀ (i : ℕ), a.coeff i ∈ I
ha' : (aeval x) a = ↑a✝
b : R[X]
hb : ∀ (i : ℕ), b.coeff i ∈ I
hb' : (aeval x) b = ↑b✝
i : ℕ
⊢ (a + b).coeff i ∈ I

Tactic:
by simpa using add_mem (ha i) (hb i)

Goal state after:
Module: Mathlib.RingTheory.PowerBasis

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : Ring S
inst✝⁶ : Algebra R S
A : Type u_4
B : Type u_5
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
K : Type u_6
inst✝² : Field K
S' : Type u_7
inst✝¹ : CommRing S'
inst✝ : Algebra R S'
pb : PowerBasis R S
e : S ≃ₐ[R] S'
i : Fin pb.dim
⊢ ↑e (pb.gen ^ ↑i) = e pb.gen ^ ↑i

Tactic:
e.toLinearEquiv_apply,

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : Ring S
inst✝⁶ : Algebra R S
A : Type u_4
B : Type u_5
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
K : Type u_6
inst✝² : Field K
S' : Type u_7
inst✝¹ : CommRing S'
inst✝ : Algebra R S'
pb : PowerBasis R S
e : S ≃ₐ[R] S'
i : Fin pb.dim
⊢ e (pb.gen ^ ↑i) = e pb.gen ^ ↑i
Module: Mathlib.FieldTheory.Separable

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
f g : R[X]
h : (f * g).Separable
⊢ g.Separable

Tactic:
by
  rw [mul_comm] at h
  exact h.of_mul_left

Goal state after:
Module: Mathlib.LinearAlgebra.Determinant

Goal state before:
pp: A : Type u_5
inst✝⁴ : CommRing A
m : Type u_6
n : Type u_7
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : DecidableEq m
inst✝ : DecidableEq n
M : Matrix m n A
M' : Matrix n m A
N : Matrix n n A
hMM' : M * M' = 1
hM'M : M' * M = 1
⊢ (M * N * M').det = N.det

Tactic:
rw [← det_comm' hM'M hMM', ← Matrix.mul_assoc, hM'M, Matrix.one_mul]

Goal state after:
Module: Mathlib.LinearAlgebra.Determinant

Goal state before:
pp: M : Type u_2
inst✝⁴ : AddCommGroup M
ι : Type u_4
inst✝³ : DecidableEq ι
inst✝² : Fintype ι
A : Type u_5
inst✝¹ : CommRing A
inst✝ : Module A M
b : Basis ι A M
f : M →ₗ[A] M
⊢ (detAux (Trunc.mk b)) f = ((toMatrix b b) f).det

Tactic:
#adaptation_note /-- Proof repaired after leanprover/lean4#13492.
  The first line below was previously just `rw [detAux]`.
  The replacement proof is a short-term fix, and we request that the authors/maintainers of
  this file review the proof, and either approve it by removing this note, revise
  the proof or the prerequisites appropriately, or minimize a problem in lean4 that still
  needs addressing. -/

Goal state after:
M : Type u_2
inst✝⁴ : AddCommGroup M
ι : Type u_4
inst✝³ : DecidableEq ι
inst✝² : Fintype ι
A : Type u_5
inst✝¹ : CommRing A
inst✝ : Module A M
b : Basis ι A M
f : M →ₗ[A] M
⊢ (detAux (Trunc.mk b)) f = ((toMatrix b b) f).det
Module: Mathlib.FieldTheory.Separable

Goal state before:
pp: case h.inr.succ
F : Type u
inst✝ : Field F
p : ℕ
HF : CharP F p
hp : Nat.Prime p
f : F[X]
hf : Irreducible f
h1 : ¬f.Separable
g : F[X]
hg : Irreducible g
hgf : (expand F p) g = f
N : ℕ
ih : ∀ m < N + 1, ∀ {f : F[X]}, Irreducible f → f.natDegree = m → ∃ n g, g.Separable ∧ (expand F (p ^ n)) g = f
hn : f.natDegree = N + 1
⊢ ∃ n g, g.Separable ∧ (expand F (p ^ n)) g = f

Tactic:
have hg1 : g.natDegree * p = N.succ := by rwa [← natDegree_expand, hgf]

Goal state after:
case h.inr.succ
F : Type u
inst✝ : Field F
p : ℕ
HF : CharP F p
hp : Nat.Prime p
f : F[X]
hf : Irreducible f
h1 : ¬f.Separable
g : F[X]
hg : Irreducible g
hgf : (expand F p) g = f
N : ℕ
ih : ∀ m < N + 1, ∀ {f : F[X]}, Irreducible f → f.natDegree = m → ∃ n g, g.Separable ∧ (expand F (p ^ n)) g = f
hn : f.natDegree = N + 1
hg1 : g.natDegree * p = N.succ
⊢ ∃ n g, g.Separable ∧ (expand F (p ^ n)) g = f
Module: Mathlib.LinearAlgebra.Determinant

Goal state before:
pp: M : Type u_2
inst✝⁴ : AddCommGroup M
A : Type u_5
inst✝³ : CommRing A
inst✝² : Module A M
N : Type u_7
inst✝¹ : AddCommGroup N
inst✝ : Module A N
f : M →ₗ[A] M
e : M ≃ₗ[A] N
⊢ LinearMap.det (↑e ∘ₗ f ∘ₗ ↑e.symm) = LinearMap.det f

Tactic:
by_cases H : ∃ s : Finset M, Nonempty (Basis s A M)
    · rcases H with ⟨s, ⟨b⟩⟩
      rw [← det_toMatrix b f, ← det_toMatrix (b.map e), toMatrix_comp (b.map e) b (b.map e),
        toMatrix_comp (b.map e) b b, ← Matrix.mul_assoc, Matrix.det_conj_of_mul_eq_one]
      · rw [← toMatrix_comp, LinearEquiv.comp_coe, e.symm_trans_self, LinearEquiv.refl_toLinearMap,
          toMatrix_id]
      · rw [← toMatrix_comp, LinearEquiv.comp_coe, e.self_trans_symm, LinearEquiv.refl_toLinearMap,
          toMatrix_id]
    · have H' : ¬∃ t : Finset N, Nonempty (Basis t A N) := by
        contrapose H
        rcases H with ⟨s, ⟨b⟩⟩
        exact ⟨_, ⟨(b.map e.symm).reindexFinsetRange⟩⟩
      simp only [coe_det, H, H', MonoidHom.one_apply, dif_neg, not_false_eq_true]

Goal state after:
Module: Mathlib.FieldTheory.Separable

Goal state before:
pp: case inr
F : Type u
inst✝ : Field F
p : ℕ
HF : CharP F p
f : F[X]
hf : Irreducible f
hp : 0 < p
n₁ n₂ : ℕ
this :
  ∀ {F : Type u} [inst : Field F] (p : ℕ) [HF : CharP F p] {f : F[X]},
    Irreducible f →
      0 < p →
        ∀ (n₁ n₂ : ℕ),
          n₁ ≤ n₂ →
            ∀ (g₁ : F[X]),
              g₁.Separable →
                (expand F (p ^ n₁)) g₁ = f →
                  ∀ (g₂ : F[X]), g₂.Separable → (expand F (p ^ n₂)) g₂ = f → n₁ = n₂ ∧ g₁ = g₂
hn : ¬n₁ ≤ n₂
⊢ ∀ (g₁ : F[X]),
    g₁.Separable →
      (expand F (p ^ n₁)) g₁ = f → ∀ (g₂ : F[X]), g₂.Separable → (expand F (p ^ n₂)) g₂ = f → n₁ = n₂ ∧ g₁ = g₂

pp: F✝ : Type u
inst✝¹ : Field F✝
p✝ : ℕ
F : Type u
inst✝ : Field F
p : ℕ
HF : CharP F p
f : F[X]
hf : Irreducible f
hp : 0 < p
n₁ n₂ : ℕ
hn : n₁ ≤ n₂
⊢ ∀ (g₁ : F[X]),
    g₁.Separable →
      (expand F (p ^ n₁)) g₁ = f → ∀ (g₂ : F[X]), g₂.Separable → (expand F (p ^ n₂)) g₂ = f → n₁ = n₂ ∧ g₁ = g₂

Tactic:
· intro g₁ hg₁ Hg₁ g₂ hg₂ Hg₂
    simpa only [eq_comm] using this p hf hp n₂ n₁ (le_of_not_ge hn) g₂ hg₂ Hg₂ g₁ hg₁ Hg₁

Goal state after:
F✝ : Type u
inst✝¹ : Field F✝
p✝ : ℕ
F : Type u
inst✝ : Field F
p : ℕ
HF : CharP F p
f : F[X]
hf : Irreducible f
hp : 0 < p
n₁ n₂ : ℕ
hn : n₁ ≤ n₂
⊢ ∀ (g₁ : F[X]),
    g₁.Separable →
      (expand F (p ^ n₁)) g₁ = f → ∀ (g₂ : F[X]), g₂.Separable → (expand F (p ^ n₂)) g₂ = f → n₁ = n₂ ∧ g₁ = g₂
Module: Mathlib.FieldTheory.Separable

Goal state before:
pp: F : Type u_1
L : Type u_2
inst✝² : CommRing F
inst✝¹ : Ring L
inst✝ : Algebra F L
S : Subalgebra F L
⊢ Algebra.IsSeparable F ↥S ↔ ∀ x ∈ S, IsSeparable F x

Tactic:
Algebra.isSeparable_def,

Goal state after:
F : Type u_1
L : Type u_2
inst✝² : CommRing F
inst✝¹ : Ring L
inst✝ : Algebra F L
S : Subalgebra F L
⊢ (∀ (x : ↥S), IsSeparable F x) ↔ ∀ x ∈ S, IsSeparable F x
Module: Mathlib.RingTheory.AdjoinRoot

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
f : R[X]
inst✝ : IsDomain R
p : R
hp : f ∣ C p
h : ¬f = 0
h_contra : p ≠ 0
⊢ C p ≠ 0

Tactic:
by rwa [Ne, C_eq_zero]

Goal state after:
Module: Mathlib.RingTheory.AdjoinRoot

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
K : Type u_5
inst✝ : CommRing R
g : R[X]
hg : g.Monic
x : Fin g.natDegree → R
i : Fin g.natDegree
a✝ : Nontrivial R
⊢ ((∑ i, (monomial ↑i) (x i)) %ₘ g).coeff ↑i = x i

Tactic:
(modByMonic_eq_self_iff hg).mpr,

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
K : Type u_5
inst✝ : CommRing R
g : R[X]
hg : g.Monic
x : Fin g.natDegree → R
i : Fin g.natDegree
a✝ : Nontrivial R
⊢ (∑ i, (monomial ↑i) (x i)).coeff ↑i = x i

R : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
K : Type u_5
inst✝ : CommRing R
g : R[X]
hg : g.Monic
x : Fin g.natDegree → R
i : Fin g.natDegree
a✝ : Nontrivial R
⊢ (∑ i, (monomial ↑i) (x i)).degree < g.degree