mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Generator.HomologicalComplex

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
ι : Type w
c : ComplexShape ι
inst✝² : c.HasNoLoop
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
α : Type t
X : α → C
hX : (ObjectProperty.ofObj X).IsSeparating
K L : HomologicalComplex C c
⊢ ∀ (f g : K ⟶ L),
    (∀ (G : HomologicalComplex C c), ObjectProperty.ofObj (separatingFamily c X) G → ∀ (h : G ⟶ K), h ≫ f = h ≫ g) →
      f = g

Tactic:
f

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
ι : Type w
c : ComplexShape ι
inst✝² : c.HasNoLoop
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
α : Type t
X : α → C
hX : (ObjectProperty.ofObj X).IsSeparating
K L : HomologicalComplex C c
f : K ⟶ L
⊢ ∀ (g : K ⟶ L),
    (∀ (G : HomologicalComplex C c), ObjectProperty.ofObj (separatingFamily c X) G → ∀ (h : G ⟶ K), h ≫ f = h ≫ g) →
      f = g
Module: Mathlib.CategoryTheory.Preadditive.Projective.Resolution

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
Z : C
P : ProjectiveResolution Z
n : ℕ
⊢ ExactAt P.complex (n + 1)

Tactic:
rw [← quasiIsoAt_iff_exactAt' P.π (n + 1) (exactAt_succ_single_obj _ _)]
  infer_instance

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)
i j : I
hij : c.Rel i j
⊢ (ρ_ (K.X i)).inv ≫
      K.X i ◁ (singleObjXSelf c 0 (𝟙_ C)).inv ≫ mapBifunctor.d₁ K (tensorUnit C c) (curriedTensor C) c i 0 j =
    K.d i j ≫ (ρ_ (K.X j)).inv ≫ K.X j ◁ (singleObjXSelf c 0 (𝟙_ C)).inv ≫ K.ιTensorObj (tensorUnit C c) j 0 j ⋯

Tactic:
rw [mapBifunctor.d₁_eq _ _ _ _ hij _ _ (by simp)]

Goal state after:
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)
i j : I
hij : c.Rel i j
⊢ (ρ_ (K.X i)).inv ≫
      K.X i ◁ (singleObjXSelf c 0 (𝟙_ C)).inv ≫
        (c.ε₁ c c (i, 0) •
          ((curriedTensor C).map (K.d i j)).app ((tensorUnit C c).X 0) ≫
            K.ιMapBifunctor (tensorUnit C c) (curriedTensor C) c j 0 j ⋯) =
    K.d i j ≫ (ρ_ (K.X j)).inv ≫ K.X j ◁ (singleObjXSelf c 0 (𝟙_ C)).inv ≫ K.ιTensorObj (tensorUnit C c) j 0 j ⋯
Module: Mathlib.Algebra.Homology.SpectralObject.Basic

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{u_4, u_1} C
inst✝¹ : Category.{u_3, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k : ι
f : i ⟶ j
g : j ⟶ k
fg : i ⟶ k
h : f ≫ g = fg
n₀ : ℤ
⊢ (X.H n₀).map (twoδ₂Toδ₁ f g fg h) ≫ (X.H n₀).map (twoδ₁Toδ₀ f g fg h) = 0

Tactic:
by
  subst h
  exact (X.exact₂' n₀ (mk₂ f g)).zero 0

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject.Monoidal

Goal state before:
pp: I : Type u
inst✝²⁵ : AddMonoid I
C : Type u_1
inst✝²⁴ : Category.{v_1, u_1} C
inst✝²³ : MonoidalCategory C
X₁ X₂ X₃ X₄ : GradedObject I C
inst✝²² : X₁.HasTensor X₂
inst✝²¹ : X₂.HasTensor X₃
inst✝²⁰ : X₃.HasTensor X₄
inst✝¹⁹ : (tensorObj X₁ X₂).HasTensor X₃
inst✝¹⁸ : X₁.HasTensor (tensorObj X₂ X₃)
inst✝¹⁷ : (tensorObj X₂ X₃).HasTensor X₄
inst✝¹⁶ : X₂.HasTensor (tensorObj X₃ X₄)
inst✝¹⁵ : (tensorObj (tensorObj X₁ X₂) X₃).HasTensor X₄
inst✝¹⁴ : (tensorObj X₁ (tensorObj X₂ X₃)).HasTensor X₄
inst✝¹³ : X₁.HasTensor (tensorObj (tensorObj X₂ X₃) X₄)
inst✝¹² : X₁.HasTensor (tensorObj X₂ (tensorObj X₃ X₄))
inst✝¹¹ : (tensorObj X₁ X₂).HasTensor (tensorObj X₃ X₄)
inst✝¹⁰ : X₁.HasGoodTensor₁₂Tensor X₂ X₃
inst✝⁹ : X₁.HasGoodTensorTensor₂₃ X₂ X₃
inst✝⁸ : X₁.HasGoodTensor₁₂Tensor (tensorObj X₂ X₃) X₄
inst✝⁷ : X₁.HasGoodTensorTensor₂₃ (tensorObj X₂ X₃) X₄
inst✝⁶ : X₂.HasGoodTensor₁₂Tensor X₃ X₄
inst✝⁵ : X₂.HasGoodTensorTensor₂₃ X₃ X₄
inst✝⁴ : (tensorObj X₁ X₂).HasGoodTensor₁₂Tensor X₃ X₄
inst✝³ : (tensorObj X₁ X₂).HasGoodTensorTensor₂₃ X₃ X₄
inst✝² : X₁.HasGoodTensor₁₂Tensor X₂ (tensorObj X₃ X₄)
inst✝¹ : X₁.HasGoodTensorTensor₂₃ X₂ (tensorObj X₃ X₄)
inst✝ : X₁.HasTensor₄ObjExt X₂ X₃ X₄
j i₁ i₂ i₃ i₄ : I
h : i₁ + i₂ + i₃ + i₄ = j
| (X₁ i₁ ◁ ιTensorObj₃ X₂ X₃ X₄ i₂ i₃ i₄ (i₂ + i₃ + i₄) ⋯ ≫
      ιTensorObj X₁ (tensorObj X₂ (tensorObj X₃ X₄)) i₁ (i₂ + i₃ + i₄) j ⋯) ≫
    (associator X₁ X₂ (tensorObj X₃ X₄)).inv j ≫ (associator (tensorObj X₁ X₂) X₃ X₄).inv j

Tactic:
ιTensorObj₃_eq X₂ X₃ X₄ i₂ i₃ i₄ _ rfl _ rfl,

Goal state after:
I : Type u
inst✝²⁵ : AddMonoid I
C : Type u_1
inst✝²⁴ : Category.{v_1, u_1} C
inst✝²³ : MonoidalCategory C
X₁ X₂ X₃ X₄ : GradedObject I C
inst✝²² : X₁.HasTensor X₂
inst✝²¹ : X₂.HasTensor X₃
inst✝²⁰ : X₃.HasTensor X₄
inst✝¹⁹ : (tensorObj X₁ X₂).HasTensor X₃
inst✝¹⁸ : X₁.HasTensor (tensorObj X₂ X₃)
inst✝¹⁷ : (tensorObj X₂ X₃).HasTensor X₄
inst✝¹⁶ : X₂.HasTensor (tensorObj X₃ X₄)
inst✝¹⁵ : (tensorObj (tensorObj X₁ X₂) X₃).HasTensor X₄
inst✝¹⁴ : (tensorObj X₁ (tensorObj X₂ X₃)).HasTensor X₄
inst✝¹³ : X₁.HasTensor (tensorObj (tensorObj X₂ X₃) X₄)
inst✝¹² : X₁.HasTensor (tensorObj X₂ (tensorObj X₃ X₄))
inst✝¹¹ : (tensorObj X₁ X₂).HasTensor (tensorObj X₃ X₄)
inst✝¹⁰ : X₁.HasGoodTensor₁₂Tensor X₂ X₃
inst✝⁹ : X₁.HasGoodTensorTensor₂₃ X₂ X₃
inst✝⁸ : X₁.HasGoodTensor₁₂Tensor (tensorObj X₂ X₃) X₄
inst✝⁷ : X₁.HasGoodTensorTensor₂₃ (tensorObj X₂ X₃) X₄
inst✝⁶ : X₂.HasGoodTensor₁₂Tensor X₃ X₄
inst✝⁵ : X₂.HasGoodTensorTensor₂₃ X₃ X₄
inst✝⁴ : (tensorObj X₁ X₂).HasGoodTensor₁₂Tensor X₃ X₄
inst✝³ : (tensorObj X₁ X₂).HasGoodTensorTensor₂₃ X₃ X₄
inst✝² : X₁.HasGoodTensor₁₂Tensor X₂ (tensorObj X₃ X₄)
inst✝¹ : X₁.HasGoodTensorTensor₂₃ X₂ (tensorObj X₃ X₄)
inst✝ : X₁.HasTensor₄ObjExt X₂ X₃ X₄
j i₁ i₂ i₃ i₄ : I
h : i₁ + i₂ + i₃ + i₄ = j
| (X₁ i₁ ◁
        (X₂ i₂ ◁ ιTensorObj X₃ X₄ i₃ i₄ (i₃ + i₄) ⋯ ≫ ιTensorObj X₂ (tensorObj X₃ X₄) i₂ (i₃ + i₄) (i₂ + i₃ + i₄) ⋯) ≫
      ιTensorObj X₁ (tensorObj X₂ (tensorObj X₃ X₄)) i₁ (i₂ + i₃ + i₄) j ⋯) ≫
    (associator X₁ X₂ (tensorObj X₃ X₄)).inv j ≫ (associator (tensorObj X₁ X₂) X₃ X₄).inv j
Module: Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
⊢ ∀ (r : ℤ) (pq pq' : ℤ × ℤ),
    (ComplexShape.up' (r, 1 - r)).Rel pq pq' →
      autoParam (2 ≤ r) SpectralSequenceDataCore._auto_16 → WithBotTop.coe (pq.2 - r + 2) = WithBotTop.coe (pq'.2 + 1)

Tactic:
rintro r pq hr rfl _

Goal state after:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
c : ℤ → ComplexShape κ
r₀ r : ℤ
pq : ℤ × ℤ
hr✝ : autoParam (2 ≤ r) SpectralSequenceDataCore._auto_16
⊢ WithBotTop.coe (pq.2 - r + 2) = WithBotTop.coe ((pq + (r, 1 - r)).2 + 1)
Module: Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
⊢ ∀ (r : ℤ) (pq pq' : ℤ × ℤ),
    (ComplexShape.up' (r, 1 - r)).Rel pq pq' →
      autoParam (2 ≤ r) SpectralSequenceDataCore._auto_20 → WithBotTop.coe pq.2 = WithBotTop.coe (pq'.2 + r - 1)

Tactic:
rintro r pq hr rfl _

Goal state after:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
c : ℤ → ComplexShape κ
r₀ r : ℤ
pq : ℤ × ℤ
hr✝ : autoParam (2 ≤ r) SpectralSequenceDataCore._auto_20
⊢ WithBotTop.coe pq.2 = WithBotTop.coe ((pq + (r, 1 - r)).2 + r - 1)
Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.Square

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
sq₁ sq₂ : Square C
h : sq₁.IsPullback
e : sq₁ ≅ sq₂
⊢ sq₂.IsPullback

Tactic:
by
  refine CategoryTheory.IsPullback.of_iso h
    (evaluation₁.mapIso e) (evaluation₂.mapIso e)
    (evaluation₃.mapIso e) (evaluation₄.mapIso e) ?_ ?_ ?_ ?_
  all_goals simp

Goal state after:
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
i₀ i₁ i₂ i₃ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₁₂ : i₀ ⟶ i₂
h₁₂ : f₁ ≫ f₂ = f₁₂
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ X.pOpcycles f₁₂ f₃ n₁ ≫ X.opcyclesToE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂ =
    X.toCycles f₁ f₂ f₁₂ h₁₂ n₁ ≫ X.πE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂

Tactic:
simp [opcyclesToE]

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

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq : κ
i₀ i₁ i₂ i₃ : ι
h₀ : i₀ = data.i₀ r pq ⋯
h₁ : i₁ = data.i₁ pq
h₂ : i₂ = data.i₂ pq
h₃ : i₃ = data.i₃ r pq ⋯
n₁ n₂ : ℤ
h : n₁ = data.deg pq
hn₂ : n₁ + 1 = n₂
hn₁ : n₁ - 1 + 1 = n₁
⊢ pageX X data r pq hr = X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (n₁ - 1) n₁ n₂ hn₁ hn₂

Tactic:
subst h hn₂ h₀ h₁ h₂ h₃

Goal state after:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq : κ
hn₁ : data.deg pq - 1 + 1 = data.deg pq
⊢ pageX X data r pq hr = X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (data.deg pq - 1) (data.deg pq) (data.deg pq + 1) hn₁ ⋯
Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq pq' : κ
hpq : (c r).Rel pq pq'
i₀ i₁ i₂ i₃ i₄ i₅ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₅ : i₄ ⟶ i₅
h₀ : i₀ = data.i₀ r pq' ⋯
h₁ : i₁ = data.i₁ pq'
h₂ : i₂ = data.i₀ r pq ⋯
h₃ : i₃ = data.i₁ pq
h₄ : i₄ = data.i₂ pq
h₅ : i₅ = data.i₃ r pq ⋯
n₀ n₁ n₂ n₃ : ℤ
hn₁' : n₁ = data.deg pq
hn₁ : autoParam (n₀ + 1 = n₁) pageD_eq._auto_1
hn₂ : autoParam (n₁ + 1 = n₂) pageD_eq._auto_3
hn₃ : autoParam (n₂ + 1 = n₃) pageD_eq._auto_5
⊢ data.i₀ r pq ⋯ = data.i₂ pq'

Tactic:
data.hc₀₂ r pq pq' hpq

Goal state after:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq pq' : κ
hpq : (c r).Rel pq pq'
i₀ i₁ i₂ i₃ i₄ i₅ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₅ : i₄ ⟶ i₅
h₀ : i₀ = data.i₀ r pq' ⋯
h₁ : i₁ = data.i₁ pq'
h₂ : i₂ = data.i₀ r pq ⋯
h₃ : i₃ = data.i₁ pq
h₄ : i₄ = data.i₂ pq
h₅ : i₅ = data.i₃ r pq ⋯
n₀ n₁ n₂ n₃ : ℤ
hn₁' : n₁ = data.deg pq
hn₁ : autoParam (n₀ + 1 = n₁) pageD_eq._auto_1
hn₂ : autoParam (n₁ + 1 = n₂) pageD_eq._auto_3
hn₃ : autoParam (n₂ + 1 = n₃) pageD_eq._auto_5
⊢ data.i₂ pq' = data.i₂ pq'
Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq pq' pq'' : κ
hpq : (c r).Rel pq pq'
hpq' : (c r).Rel pq' pq''
n₀ n₁ n₂ n₃ n₄ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
hn₄ : n₃ + 1 = n₄
hn₂' : n₂ = data.deg pq'
⊢ data.i₀ r pq ⋯ = data.i₂ pq'

Tactic:
data.hc₀₂ r pq pq' hpq

Goal state after:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq pq' pq'' : κ
hpq : (c r).Rel pq pq'
hpq' : (c r).Rel pq' pq''
n₀ n₁ n₂ n₃ n₄ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
hn₄ : n₃ + 1 = n₄
hn₂' : n₂ = data.deg pq'
⊢ data.i₂ pq' = data.i₂ pq'
Module: Mathlib.Order.Filter.Subsingleton

Goal state before:
pp: case refine_2
α : Type u_1
l : Filter α
⊢ (l = ⊥ ∨ ∃ a, l = pure a) → l.Subsingleton

Tactic:
rintro (rfl | ⟨a, rfl⟩)

Goal state after:
case refine_2.inl
α : Type u_1
⊢ ⊥.Subsingleton

case refine_2.inr
α : Type u_1
a : α
⊢ (pure a).Subsingleton
Module: Mathlib.RingTheory.Artinian.Module

Goal state before:
pp: R : Type u_1
M : Type u_2
P : Type u_3
N : Type u_4
inst✝⁸ : Ring R
inst✝⁷ : AddCommGroup M
inst✝⁶ : AddCommGroup P
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R M
inst✝³ : Module R P
inst✝² : Module R N
ι : Type u_5
inst✝¹ : Finite ι
α✝ : Type u_5
inst✝ : Fintype α✝
⊢ (∀ {M : α✝ → Submodule R P} [∀ (i : α✝), IsArtinian R ↥(M i)], IsArtinian R ↥(⨆ i, M i)) →
    ∀ {M : Option α✝ → Submodule R P} [∀ (i : Option α✝), IsArtinian R ↥(M i)], IsArtinian R ↥(⨆ i, M i)

Tactic:
ih

Goal state after:
R : Type u_1
M : Type u_2
P : Type u_3
N : Type u_4
inst✝⁸ : Ring R
inst✝⁷ : AddCommGroup M
inst✝⁶ : AddCommGroup P
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R M
inst✝³ : Module R P
inst✝² : Module R N
ι : Type u_5
inst✝¹ : Finite ι
α✝ : Type u_5
inst✝ : Fintype α✝
ih : ∀ {M : α✝ → Submodule R P} [∀ (i : α✝), IsArtinian R ↥(M i)], IsArtinian R ↥(⨆ i, M i)
⊢ ∀ {M : Option α✝ → Submodule R P} [∀ (i : Option α✝), IsArtinian R ↥(M i)], IsArtinian R ↥(⨆ i, M i)
Module: Mathlib.RingTheory.Artinian.Module

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : IsArtinian R M
r : R
n m : ℕ
h : n ≤ m
x : M
x✝ : x ∈ (fun n ↦ (r ^ n • LinearMap.id).range) m
y : M
hy : r ^ m • y = x
⊢ r ^ n • r ^ (m - n) • y = x

Tactic:
← smul_assoc,

Goal state after:
R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : IsArtinian R M
r : R
n m : ℕ
h : n ≤ m
x : M
x✝ : x ∈ (fun n ↦ (r ^ n • LinearMap.id).range) m
y : M
hy : r ^ m • y = x
⊢ (r ^ n • r ^ (m - n)) • y = x
Module: Mathlib.RingTheory.Artinian.Module

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsArtinianRing R
n : ℕ
this : Fintype (MaximalSpectrum R)
⊢ iInf PrimeSpectrum.asIdeal ^ n = (∏ x, x.asIdeal) ^ n

Tactic:
Ideal.prod_eq_iInf_of_pairwise_isCoprime fun I _ _ _ ↦ I.isCoprime_of_ne

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsArtinianRing R
n : ℕ
this : Fintype (MaximalSpectrum R)
⊢ iInf PrimeSpectrum.asIdeal ^ n = (⨅ i ∈ Finset.univ, i.asIdeal) ^ n
Module: Mathlib.Algebra.Lie.Subalgebra

Goal state before:
pp: case add_right
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
s : Set L
hs : ∀ x ∈ s, ∀ y ∈ s, ⁅x, y⁆ = 0
x✝ y✝ x y z : L
hx✝ : x ∈ span R s
hy✝ : y ∈ span R s
hz✝ : z ∈ span R s
hx : ⁅x, y⁆ ∈ span R s
hy : ⁅x, z⁆ ∈ span R s
⊢ ⁅x, y + z⁆ ∈ span R s

Tactic:
| add_right x y z _ _ _ hx hy =>

Goal state after:
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
J : LieIdeal R L'
⊢ map f I ≤ J ↔ I ≤ comap f J

Tactic:
rw [map_le]

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
J : LieIdeal R L'
⊢ ⇑f '' ↑I ⊆ ↑J ↔ I ≤ comap f J
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'
I₁ I₂ : LieIdeal R L
h : I₁ ≤ I₂
⊢ map f I₁ ≤ map f I₂

Tactic:
unfold 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'
I₁ I₂ : LieIdeal R L
h : I₁ ≤ I₂
⊢ LieSubmodule.lieSpan R L' ↑(Submodule.map (↑f) (toLieSubalgebra R L I₁).toSubmodule) ≤
    LieSubmodule.lieSpan R L' ↑(Submodule.map (↑f) (toLieSubalgebra R L I₂).toSubmodule)
Module: Mathlib.Algebra.Lie.Ideal

Goal state before:
pp: case h
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'
h : ∀ (x m : L') (x_1 : L), f x_1 = m → ∃ y, f y = ⁅x, m⁆
x : L'
y z : L
hz : f z = ⁅x, f y⁆
⊢ ⁅x, f y⁆ = f z

Tactic:
exact hz.symm

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
s : Set M
p : (x : M) → x ∈ lieSpan R L s → Prop
mem : ∀ (x : M) (h : x ∈ s), p x ⋯
zero : p 0 ⋯
add : ∀ (x y : M) (hx : x ∈ lieSpan R L s) (hy : y ∈ lieSpan R L s), p x hx → p y hy → p (x + y) ⋯
smul : ∀ (a : R) (x : M) (hx : x ∈ lieSpan R L s), p x hx → p (a • x) ⋯
x : M
lie : ∀ (x : L) (y : M) (hy : y ∈ lieSpan R L s), p y hy → p ⁅x, y⁆ ⋯
hx : x ∈ lieSpan R L s
⊢ p x hx

Tactic:
by
  let p : LieSubmodule R L M :=
    { carrier := { x | ∃ hx, p x hx }
      add_mem' := fun ⟨_, hpx⟩ ⟨_, hpy⟩ ↦ ⟨_, add _ _ _ _ hpx hpy⟩
      zero_mem' := ⟨_, zero⟩
      smul_mem' := fun r ↦ fun ⟨_, hpx⟩ ↦ ⟨_, smul r _ _ hpx⟩
      lie_mem := fun ⟨_, hpy⟩ ↦ ⟨_, lie _ _ _ hpy⟩ }
  exact lieSpan_le (N := p) |>.mpr (fun y hy ↦ ⟨subset_lieSpan hy, mem y hy⟩) hx |>.elim fun _ ↦ id

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
m : M
s : Set (LieSubmodule R L M)
hne : s.Nonempty
hdir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s
hsup : lieSpan R L {m} ≤ sSup s
⊢ ∃ x ∈ s, lieSpan R L {m} ≤ x

Tactic:
replace hsup : m ∈ (↑(sSup s) : Set M) := (SetLike.le_def.mp hsup) (subset_lieSpan rfl)

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
m : M
s : Set (LieSubmodule R L M)
hne : s.Nonempty
hdir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s
hsup : m ∈ ↑(sSup s)
⊢ ∃ x ∈ s, lieSpan R L {m} ≤ x
Module: Mathlib.FieldTheory.Minpoly.Basic

Goal state before:
pp: case inr
A : Type u_1
B : Type u_2
inst✝² : CommRing A
inst✝¹ : Ring B
inst✝ : Algebra A B
x : B
p : A[X]
hm : p.Monic
hp : (Polynomial.aeval x) p = 0
hl : ∀ (q : A[X]), q.degree < p.degree → q = 0 ∨ (Polynomial.aeval x) q ≠ 0
a✝ : Nontrivial A
hx : IsIntegral A x
h : (Polynomial.aeval x) (minpoly A x %ₘ p) ≠ 0
⊢ p = minpoly A x

Tactic:
exact (h <| (aeval_modByMonic_eq_self_of_root hp).trans <| aeval A x).elim

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

Goal state before:
pp: case h
A : Type u_1
B : Type u_2
inst✝² : CommRing A
inst✝¹ : Ring B
inst✝ : Algebra A B
x : B
p : A[X]
hm : p.Monic
hp : (Polynomial.aeval x) p = 0
hl : ∀ (q : A[X]), q.degree < p.degree → q = 0 ∨ (Polynomial.aeval x) q ≠ 0
a✝ : Nontrivial A
hx : IsIntegral A x
h : minpoly A x %ₘ p = 0
r : A[X]
hlead : 1 = r.leadingCoeff
hr0 : r ≠ 0
hr : (minpoly A x).natDegree = p.natDegree + r.natDegree
⊢ ?h✝ + r.natDegree ≤ ?h✝

pp: case h
A : Type u_1
B : Type u_2
inst✝² : CommRing A
inst✝¹ : Ring B
inst✝ : Algebra A B
x : B
p : A[X]
hm : p.Monic
hp : (Polynomial.aeval x) p = 0
hl : ∀ (q : A[X]), q.degree < p.degree → q = 0 ∨ (Polynomial.aeval x) q ≠ 0
a✝ : Nontrivial A
hx : IsIntegral A x
h : minpoly A x %ₘ p = 0
r : A[X]
hlead : 1 = r.leadingCoeff
hr0 : r ≠ 0
hr : (minpoly A x).natDegree = p.natDegree + r.natDegree
⊢ ℕ

Tactic:
exact hr.symm.trans_le (natDegree_le_natDegree <| min A x hm hp)

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

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝² : Field A
inst✝¹ : Ring B
inst✝ : Algebra A B
x : B
⊢ RingHom.ker (Polynomial.aeval x) = A[X] ∙ minpoly A x

Tactic:
by
  ext p
  simp_rw [RingHom.mem_ker, ← minpoly.dvd_iff, Submodule.mem_span_singleton,
    dvd_iff_exists_eq_mul_left, smul_eq_mul, eq_comm (a := p)]

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

Goal state before:
pp: case refine_1
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : CommRing L
inst✝¹ : IsDomain L
inst✝ : Algebra K L
σ : L ≃ₐ[K] L
hσ : IsOfFinOrder σ
⊢ (Polynomial.aeval σ.toLinearMap) (X ^ orderOf σ - C 1) = 0

pp: case refine_2
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : CommRing L
inst✝¹ : IsDomain L
inst✝ : Algebra K L
σ : L ≃ₐ[K] L
hσ : IsOfFinOrder σ
⊢ ∀ (q : K[X]), q.Monic → (Polynomial.aeval σ.toLinearMap) q = 0 → (X ^ orderOf σ - C 1).degree ≤ q.degree

Tactic:
· simp [← AlgEquiv.pow_toLinearMap, pow_orderOf_eq_one]

Goal state after:
case refine_2
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : CommRing L
inst✝¹ : IsDomain L
inst✝ : Algebra K L
σ : L ≃ₐ[K] L
hσ : IsOfFinOrder σ
⊢ ∀ (q : K[X]), q.Monic → (Polynomial.aeval σ.toLinearMap) q = 0 → (X ^ orderOf σ - C 1).degree ≤ q.degree
Module: Mathlib.RingTheory.Polynomial.Quotient

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
I : Ideal R
a : R[X]
ha : a ∈ map C I
n : ℕ
x✝ : n ∈ a.support
m : ℕ
⊢ (if m = 0 then (Quotient.mk I) (a.coeff n) else 0) = coeff 0 m

Tactic:
by_cases h : m = 0

Goal state after:
case pos
R : Type u_1
inst✝ : CommRing R
I : Ideal R
a : R[X]
ha : a ∈ map C I
n : ℕ
x✝ : n ∈ a.support
m : ℕ
h : m = 0
⊢ (if m = 0 then (Quotient.mk I) (a.coeff n) else 0) = coeff 0 m

case neg
R : Type u_1
inst✝ : CommRing R
I : Ideal R
a : R[X]
ha : a ∈ map C I
n : ℕ
x✝ : n ∈ a.support
m : ℕ
h : ¬m = 0
⊢ (if m = 0 then (Quotient.mk I) (a.coeff n) else 0) = coeff 0 m
Module: Mathlib.RingTheory.PowerBasis

Goal state before:
pp: S : Type u_2
inst✝³ : Ring S
A : Type u_4
inst✝² : CommRing A
inst✝¹ : Algebra A S
inst✝ : Nontrivial A
pb : PowerBasis A S
⊢ (X ^ pb.dim - ∑ i, C ((pb.basis.repr (pb.gen ^ pb.dim)) i) * X ^ ↑i).degree = ↑pb.dim

Tactic:
rw [degree_sub_eq_left_of_degree_lt]

Goal state after:
S : Type u_2
inst✝³ : Ring S
A : Type u_4
inst✝² : CommRing A
inst✝¹ : Algebra A S
inst✝ : Nontrivial A
pb : PowerBasis A S
⊢ (X ^ pb.dim).degree = ↑pb.dim

S : Type u_2
inst✝³ : Ring S
A : Type u_4
inst✝² : CommRing A
inst✝¹ : Algebra A S
inst✝ : Nontrivial A
pb : PowerBasis A S
⊢ (∑ i, C ((pb.basis.repr (pb.gen ^ pb.dim)) i) * X ^ ↑i).degree < (X ^ pb.dim).degree
Module: Mathlib.RingTheory.PowerBasis

Goal state before:
pp: S : Type u_2
inst✝² : Ring S
A : Type u_4
inst✝¹ : CommRing A
inst✝ : Algebra A S
pb : PowerBasis A S
⊢ pb.minpolyGen = minpoly A pb.gen

Tactic:
by
  nontriviality A
  refine minpoly.unique' A _ pb.minpolyGen_monic pb.aeval_minpolyGen fun q hq =>
    or_iff_not_imp_left.2 fun hn0 h0 => ?_
  exact (pb.dim_le_degree_of_root hn0 h0).not_gt (pb.degree_minpolyGen ▸ hq)

Goal state after:
Module: Mathlib.FieldTheory.Perfect

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
p n : ℕ
inst✝² : ExpChar R p
f : R[X]
inst✝¹ : PerfectRing R p
inst✝ : DecidableEq R
⊢ Finset.image (⇑(iterateFrobenius R p n)) ((expand R (p ^ n)) f).roots.toFinset = f.roots.toFinset

Tactic:
rw [Finset.image_toFinset, roots_expand_pow_map_iterateFrobenius,
    (roots f).toFinset_nsmul _ (expChar_pow_pos R p n).ne']

Goal state after:
Module: Mathlib.RingTheory.AdjoinRoot

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommSemiring S
inst✝² : Semiring T
inst✝¹ : Algebra S R
inst✝ : Algebra S T
p : R[X]
f g : AdjoinRoot p →ₐ[S] T
hAlg : f.comp (ofAlgHom S p) = g.comp (ofAlgHom S p)
hRoot : f (root p) = g (root p)
⊢ f = g

Tactic:
apply AlgHom.coe_ringHom_injective

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommSemiring S
inst✝² : Semiring T
inst✝¹ : Algebra S R
inst✝ : Algebra S T
p : R[X]
f g : AdjoinRoot p →ₐ[S] T
hAlg : f.comp (ofAlgHom S p) = g.comp (ofAlgHom S p)
hRoot : f (root p) = g (root p)
⊢ ↑f = ↑g