mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.GuitartExact.Basic

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
C₄ : Type u₄
inst✝³ : Category.{v₁, u₁} C₁
inst✝² : Category.{v₂, u₂} C₂
inst✝¹ : Category.{v₃, u₃} C₃
inst✝ : Category.{v₄, u₄} C₄
T : C₁ ⥤ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ⥤ C₄
w : TwoSquare T L R B
X₂ : C₂
X₃ : C₃
g✝ : R.obj X₂ ⟶ B.obj X₃
g : CostructuredArrow L X₃
φ : CostructuredArrow.mk g✝ ⟶ (w.costructuredArrowRightwards X₃).obj g
⊢ ∃ X₁ a b, ∃ (comm : R.map a ≫ w.app X₁ ≫ B.map b = g✝), StructuredArrow.mk φ = mk w g✝ X₁ a b comm

Tactic:
obtain ⟨X₁, b, rfl⟩ := g.mk_surjective

Goal state after:
C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
C₄ : Type u₄
inst✝³ : Category.{v₁, u₁} C₁
inst✝² : Category.{v₂, u₂} C₂
inst✝¹ : Category.{v₃, u₃} C₃
inst✝ : Category.{v₄, u₄} C₄
T : C₁ ⥤ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ⥤ C₄
w : TwoSquare T L R B
X₂ : C₂
X₃ : C₃
g : R.obj X₂ ⟶ B.obj X₃
X₁ : C₁
b : L.obj X₁ ⟶ X₃
φ : CostructuredArrow.mk g ⟶ (w.costructuredArrowRightwards X₃).obj (CostructuredArrow.mk b)
⊢ ∃ X₁_1 a b_1, ∃ (comm : R.map a ≫ w.app X₁_1 ≫ B.map b_1 = g), StructuredArrow.mk φ = mk w g X₁_1 a b_1 comm
Module: Mathlib.CategoryTheory.GuitartExact.Basic

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
C₄ : Type u₄
inst✝³ : Category.{v₁, u₁} C₁
inst✝² : Category.{v₂, u₂} C₂
inst✝¹ : Category.{v₃, u₃} C₃
inst✝ : Category.{v₄, u₄} C₄
T : C₁ ⥤ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ⥤ C₄
w : TwoSquare T L R B
hw : w.GuitartExact
X₃ : C₃
⊢ (w.costructuredArrowRightwards X₃).Final

Tactic:
by
  rw [guitartExact_iff_final] at hw
  apply hw

Goal state after:
Module: Mathlib.CategoryTheory.GuitartExact.Quotient

Goal state before:
pp: C₀ : Type u_1
C : Type u_2
H₀ : Type u_3
H : Type u_4
inst✝⁶ : Category.{v_1, u_1} C₀
inst✝⁵ : Category.{v_2, u_2} C
inst✝⁴ : Category.{v_3, u_3} H₀
inst✝³ : Category.{v_4, u_4} H
T : C₀ ⥤ H₀
L : C₀ ⥤ C
R : H₀ ⥤ H
B : C ⥤ H
inst✝² : T.EssSurj
inst✝¹ : T.Full
inst✝ : B.Full
e : T ⋙ R ≅ L ⋙ B
he :
  ∀ ⦃X₀ : C₀⦄ ⦃Y : C⦄ (f₀ f₁ : L.obj X₀ ⟶ Y),
    B.map f₀ = B.map f₁ → ∃ P, T.map P.i₀ = T.map P.i₁ ∧ Nonempty ((P.map L).LeftHomotopy f₀ f₁)
⊢ ∀ {X₂ : H₀} {X₃ : C} (g : R.obj X₂ ⟶ B.obj X₃), IsConnected (CostructuredArrowDownwards e.hom g)

Tactic:
intro Y₀

Goal state after:
C₀ : Type u_1
C : Type u_2
H₀ : Type u_3
H : Type u_4
inst✝⁶ : Category.{v_1, u_1} C₀
inst✝⁵ : Category.{v_2, u_2} C
inst✝⁴ : Category.{v_3, u_3} H₀
inst✝³ : Category.{v_4, u_4} H
T : C₀ ⥤ H₀
L : C₀ ⥤ C
R : H₀ ⥤ H
B : C ⥤ H
inst✝² : T.EssSurj
inst✝¹ : T.Full
inst✝ : B.Full
e : T ⋙ R ≅ L ⋙ B
he :
  ∀ ⦃X₀ : C₀⦄ ⦃Y : C⦄ (f₀ f₁ : L.obj X₀ ⟶ Y),
    B.map f₀ = B.map f₁ → ∃ P, T.map P.i₀ = T.map P.i₁ ∧ Nonempty ((P.map L).LeftHomotopy f₀ f₁)
Y₀ : H₀
⊢ ∀ {X₃ : C} (g : R.obj Y₀ ⟶ B.obj X₃), IsConnected (CostructuredArrowDownwards e.hom g)
Module: Mathlib.CategoryTheory.Localization.StructuredArrow

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
W : MorphismProperty C
X : C
P : StructuredArrow (W.Q.obj X) W.Q → Prop
hP₀ : P (StructuredArrow.mk (𝟙 (W.Q.obj X)))
hP₁ :
  ∀ ⦃Y₁ Y₂ : C⦄ (f : Y₁ ⟶ Y₂) (φ : W.Q.obj X ⟶ W.Q.obj Y₁),
    P (StructuredArrow.mk φ) → P (StructuredArrow.mk (φ ≫ W.Q.map f))
hP₂ :
  ∀ ⦃Y₁ Y₂ : C⦄ (w : Y₁ ⟶ Y₂) (hw : W w) (φ : W.Q.obj X ⟶ W.Q.obj Y₂),
    P (StructuredArrow.mk φ) → P (StructuredArrow.mk (φ ≫ (isoOfHom W.Q W w hw).inv))
g : StructuredArrow (W.Q.obj X) W.Q
X₀ : Paths (LocQuiver W) := { obj := X }
⊢ ∀ ⦃Y₀ : Paths (LocQuiver W)⦄ (f : X₀ ⟶ Y₀), P (StructuredArrow.mk ((Quotient.functor (relations W)).map f))

Tactic:
intro Y₀

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
W : MorphismProperty C
X : C
P : StructuredArrow (W.Q.obj X) W.Q → Prop
hP₀ : P (StructuredArrow.mk (𝟙 (W.Q.obj X)))
hP₁ :
  ∀ ⦃Y₁ Y₂ : C⦄ (f : Y₁ ⟶ Y₂) (φ : W.Q.obj X ⟶ W.Q.obj Y₁),
    P (StructuredArrow.mk φ) → P (StructuredArrow.mk (φ ≫ W.Q.map f))
hP₂ :
  ∀ ⦃Y₁ Y₂ : C⦄ (w : Y₁ ⟶ Y₂) (hw : W w) (φ : W.Q.obj X ⟶ W.Q.obj Y₂),
    P (StructuredArrow.mk φ) → P (StructuredArrow.mk (φ ≫ (isoOfHom W.Q W w hw).inv))
g : StructuredArrow (W.Q.obj X) W.Q
X₀ : Paths (LocQuiver W) := { obj := X }
Y₀ : Paths (LocQuiver W)
⊢ ∀ (f : X₀ ⟶ Y₀), P (StructuredArrow.mk ((Quotient.functor (relations W)).map f))
Module: Mathlib.CategoryTheory.Functor.Derived.PointwiseRightDerived

Goal state before:
pp: C : Type u₁
H : Type u₃
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₃, u₃} H
F : C ⥤ H
W : MorphismProperty C
inst✝ : F.HasPointwiseRightDerivedFunctor W
⊢ W.Q.HasLeftKanExtension F

Tactic:
have := F.hasPointwiseLeftKanExtension_of_hasPointwiseRightDerivedFunctor W.Q W
    infer_instance

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: case pos.ofNat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
K : ChainComplex C ℕ
L : CochainComplex C ℕ
h : ConnectData K L
m p : ℤ
hmp : m + 1 = p
n : ℕ
hnm : Int.ofNat n + 1 = m
⊢ h.d (Int.ofNat n) m ≫ h.d m p = 0

pp: case pos.negSucc.zero
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
K : ChainComplex C ℕ
L : CochainComplex C ℕ
h : ConnectData K L
m p : ℤ
hmp : m + 1 = p
hnm : Int.negSucc 0 + 1 = m
⊢ h.d (Int.negSucc 0) m ≫ h.d m p = 0

pp: case pos.negSucc.succ.zero
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
K : ChainComplex C ℕ
L : CochainComplex C ℕ
h : ConnectData K L
m p : ℤ
hmp : m + 1 = p
hnm : Int.negSucc (0 + 1) + 1 = m
⊢ h.d (Int.negSucc (0 + 1)) m ≫ h.d m p = 0

pp: case pos.negSucc.succ.succ
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
K : ChainComplex C ℕ
L : CochainComplex C ℕ
h : ConnectData K L
m p : ℤ
hmp : m + 1 = p
n : ℕ
hnm : Int.negSucc (n + 1 + 1) + 1 = m
⊢ h.d (Int.negSucc (n + 1 + 1)) m ≫ h.d m p = 0

Tactic:
· obtain rfl : m = .ofNat (n + 1) := by simp [← hnm]
    obtain rfl : p = .ofNat (n + 2) := by simp [← hmp]; lia
    simp only [Int.ofNat_eq_natCast, X_ofNat, d_ofNat, HomologicalComplex.d_comp_d]

Goal state after:
case pos.negSucc.zero
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
K : ChainComplex C ℕ
L : CochainComplex C ℕ
h : ConnectData K L
m p : ℤ
hmp : m + 1 = p
hnm : Int.negSucc 0 + 1 = m
⊢ h.d (Int.negSucc 0) m ≫ h.d m p = 0

case pos.negSucc.succ.zero
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
K : ChainComplex C ℕ
L : CochainComplex C ℕ
h : ConnectData K L
m p : ℤ
hmp : m + 1 = p
hnm : Int.negSucc (0 + 1) + 1 = m
⊢ h.d (Int.negSucc (0 + 1)) m ≫ h.d m p = 0

case pos.negSucc.succ.succ
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
K : ChainComplex C ℕ
L : CochainComplex C ℕ
h : ConnectData K L
m p : ℤ
hmp : m + 1 = p
n : ℕ
hnm : Int.negSucc (n + 1 + 1) + 1 = m
⊢ h.d (Int.negSucc (n + 1 + 1)) m ≫ h.d m p = 0
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
K : ChainComplex C ℕ
L : CochainComplex C ℕ
h : ConnectData K L
⊢ h.map h (𝟙 K) (𝟙 L) ⋯ = 𝟙 h.cochainComplex

Tactic:
by ext (m | _ | m) <;> simp; rfl

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.SpectralObject

Goal state before:
pp: C : Type u_1
ι✝ : Type u_2
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} ι✝
inst✝¹³ : HasZeroObject C
inst✝¹² : HasShift C ℤ
inst✝¹¹ : Preadditive C
inst✝¹⁰ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁹ : Pretriangulated C
D : Type u_3
inst✝⁸ : Category.{v_3, u_3} D
inst✝⁷ : HasZeroObject D
inst✝⁶ : HasShift D ℤ
inst✝⁵ : Preadditive D
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝³ : Pretriangulated D
F : C ⥤ D
inst✝² : F.CommShift ℤ
inst✝¹ : F.IsTriangulated
ι : Type u_4
inst✝ : Category.{v_4, u_4} ι
X✝ Y✝ : Triangulated.SpectralObject C ι
α : X✝ ⟶ Y✝
i✝ j✝ k✝ : ι
f : i✝ ⟶ j✝
g : j✝ ⟶ k✝
⊢ (X✝.mapTriangulatedFunctor F).δ f g ≫ (shiftFunctor D 1).map ((whiskerRight α.hom F).app (mk₁ f)) =
    (whiskerRight α.hom F).app (mk₁ g) ≫ (Y✝.mapTriangulatedFunctor F).δ f g

Tactic:
by
        have hf := (F.commShiftIso (1 : ℤ)).hom.naturality (α.hom.app (mk₁ f))
        dsimp at hf ⊢
        rw [Category.assoc, ← hf, ← F.map_comp_assoc, α.comm, F.map_comp_assoc]

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
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 n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hi : IsLimit (KernelFork.ofι (X.kernelSequenceCycles f₁ f₂ n₁ n₂ hn₂).f ⋯) := ⋯.fIsKernel
⊢ hi.lift (KernelFork.ofι (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ ⋯ ⋯).f ⋯) ≫
      Fork.ι (KernelFork.ofι (X.kernelSequenceCycles f₁ f₂ n₁ n₂ hn₂).f ⋯) =
    X.δToCycles f₁ f₂ f₃ n₀ n₁ ⋯ ≫ Fork.ι (KernelFork.ofι (X.kernelSequenceCycles f₁ f₂ n₁ n₂ hn₂).f ⋯)

Tactic:
by simpa using! hi.fac _ .zero

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
i j k l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hi : IsLimit (KernelFork.ofι (X.kernelSequenceCycles f₁ f₂ n₁ n₂ hn₂).f ⋯) := ⋯.fIsKernel
⊢ (X.leftHomologyDataShortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).f' ≫
      Fork.ι (KernelFork.ofι (X.kernelSequenceCycles f₁ f₂ n₁ n₂ hn₂).f ⋯) =
    X.δToCycles f₁ f₂ f₃ n₀ n₁ hn₁ ≫ Fork.ι (KernelFork.ofι (X.kernelSequenceCycles f₁ f₂ n₁ n₂ hn₂).f ⋯)

Tactic:
by simpa using! hi.fac _ .zero

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
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 n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hp : IsColimit (CokernelCofork.ofπ (X.cokernelSequenceOpcycles f₂ f₃ n₀ n₁ hn₁).g ⋯) := ⋯.gIsCokernel
this : hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯) = X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ hn₂
⊢ IsLimit (KernelFork.ofι (kernel.ι (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ ⋯)) ⋯)

Tactic:
refine (IsLimit.equivOfNatIsoOfIso ?_ _ _ ?_).2
        (kernelIsKernel (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂))

Goal state after:
case refine_1
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 n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hp : IsColimit (CokernelCofork.ofπ (X.cokernelSequenceOpcycles f₂ f₃ n₀ n₁ hn₁).g ⋯) := ⋯.gIsCokernel
this : hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯) = X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ hn₂
⊢ parallelPair (hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯)) 0 ≅
    parallelPair (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ ⋯) 0

case refine_2
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 n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hp : IsColimit (CokernelCofork.ofπ (X.cokernelSequenceOpcycles f₂ f₃ n₀ n₁ hn₁).g ⋯) := ⋯.gIsCokernel
this : hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯) = X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ hn₂
⊢ (Cone.postcompose (Iso.hom ?refine_1)).obj (KernelFork.ofι (kernel.ι (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ ⋯)) ⋯) ≅
    Fork.ofι (kernel.ι (X.δFromOpcycles f₁ f₂ f₃ n₁ 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
f₁₂ : i ⟶ k
h₁₂ : f₁ ≫ f₂ = f₁₂
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
A : C
x₂ : A ⟶ (X.H n₁).obj (mk₁ f₁₂)
hx₂ : x₂ ≫ X.toCycles f₁ f₂ f₁₂ h₁₂ n₁ ≫ X.πE f₁ f₂ f₃ n₀ n₁ n₂ ⋯ ⋯ = 0
A₁ : C
π₁ : A₁ ⟶ A
w✝ : Epi π₁
y₁ : A₁ ⟶ (X.cokernelSequenceCyclesE f₁ f₂ f₃ n₀ n₁ n₂ ⋯ ⋯).X₁
hy₁ : π₁ ≫ x₂ ≫ X.toCycles f₁ f₂ f₁₂ h₁₂ n₁ = y₁ ≫ (X.cokernelSequenceCyclesE f₁ f₂ f₃ n₀ n₁ n₂ ⋯ ⋯).f
⊢ ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ (X.cokernelSequenceE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂).f

Tactic:
dsimp at y₁ hy₁

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
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
A : C
x₂ : A ⟶ (X.H n₁).obj (mk₁ f₁₂)
hx₂ : x₂ ≫ X.toCycles f₁ f₂ f₁₂ h₁₂ n₁ ≫ X.πE f₁ f₂ f₃ n₀ n₁ n₂ ⋯ ⋯ = 0
A₁ : C
π₁ : A₁ ⟶ A
w✝ : Epi π₁
y₁ : A₁ ⟶ (X.H n₀).obj (mk₁ f₃)
hy₁ : π₁ ≫ x₂ ≫ X.toCycles f₁ f₂ f₁₂ h₁₂ n₁ = y₁ ≫ X.δToCycles f₁ f₂ f₃ n₀ n₁ ⋯
⊢ ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ (X.cokernelSequenceE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂).f
Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.Square

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
sq : Square C
h : sq.IsPushout
inst✝ : Epi sq.f₁₂
this : Epi sq.flip.f₁₃
⊢ Epi sq.f₃₄

Tactic:
exact h.flip.epi_f₂₄

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
A : C
x₂ : A ⟶ X.opcycles f₁₂ f₃ n₁
hx₂ : x₂ ≫ X.opcyclesToE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂ = 0
A₁ : C
π₁ : A₁ ⟶ A
w✝¹ : Epi π₁
y₂ : A₁ ⟶ (X.H n₁).obj (mk₁ f₁₂)
hy₂ : π₁ ≫ x₂ = y₂ ≫ X.pOpcycles f₁₂ f₃ n₁
A₂ : C
π₂ : A₂ ⟶ A₁
w✝ : Epi π₂
y₁ : A₂ ⟶ (X.cokernelSequenceE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂).X₁
hy₁ : π₂ ≫ y₂ = y₁ ≫ (X.cokernelSequenceE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂).f
⊢ ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ (X.cokernelSequenceOpcyclesE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂).f

Tactic:
dsimp at y₁ hy₁

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 : 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
A : C
x₂ : A ⟶ X.opcycles f₁₂ f₃ n₁
hx₂ : x₂ ≫ X.opcyclesToE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂ = 0
A₁ : C
π₁ : A₁ ⟶ A
w✝¹ : Epi π₁
y₂ : A₁ ⟶ (X.H n₁).obj (mk₁ f₁₂)
hy₂ : π₁ ≫ x₂ = y₂ ≫ X.pOpcycles f₁₂ f₃ n₁
A₂ : C
π₂ : A₂ ⟶ A₁
w✝ : Epi π₂
y₁ : A₂ ⟶ (X.H n₁).obj (mk₁ f₁) ⊞ (X.H n₀).obj (mk₁ f₃)
hy₁ : π₂ ≫ y₂ = y₁ ≫ biprod.desc ((X.H n₁).map (twoδ₂Toδ₁ f₁ f₂ f₁₂ h₁₂)) (X.δ f₁₂ f₃ n₀ n₁ ⋯)
⊢ ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ (X.cokernelSequenceOpcyclesE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂).f
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: case h₁
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
i₀' i₁' i₂' i₃' : ι
f₁' : i₀' ⟶ i₁'
f₂' : i₁' ⟶ i₂'
f₃' : i₂' ⟶ i₃'
f₁₂' : i₀' ⟶ i₂'
h₁₂' : f₁' ≫ f₂' = f₁₂'
α : mk₃ f₁ f₂ f₃ ⟶ mk₃ f₁' f₂' f₃'
β : mk₂ f₁₂ f₃ ⟶ mk₂ f₁₂' f₃'
n₀ n₁ n₂ : ℤ
h₀ : β.app 0 = α.app 0
h₁ : β.app 1 = α.app 2
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ app'
      (twoδ₁Toδ₀ f₁ f₂ f₁₂ h₁₂ ≫
        homMk₁ ((homMk₂ (α.app 0) (α.app 1) (α.app 2) ⋯ ⋯).app 1) ((homMk₂ (α.app 0) (α.app 1) (α.app 2) ⋯ ⋯).app 2) ⋯)
      1 homMk₁._proof_5 =
    app'
      (homMk₁ (β.app 0) (β.app 1) ⋯ ≫
        homMk₁ ((threeδ₁Toδ₀ f₁' f₂' f₃' f₁₂' h₁₂').app 0) ((threeδ₁Toδ₀ f₁' f₂' f₃' f₁₂' h₁₂').app 1) ⋯)
      1 homMk₁._proof_5

Tactic:
· simp [h₁]

Goal state after:
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)
⊢ nilradical R ^ n = ⨅ x ∈ ↑Finset.univ, x.asIdeal ^ n

Tactic:
PrimeSpectrum.nilradical_eq_iInf

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

Goal state before:
pp: R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
s t : Set L
h : s ⊆ t
⊢ lieSpan R L s ≤ lieSpan R L t

Tactic:
rw [lieSpan_le]
  exact Set.Subset.trans h subset_lieSpan

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'
h : f.IsIdealMorphism
y : L'
⊢ y ∈ f.idealRange ↔ ∃ x, f x = y

Tactic:
rw [f.isIdealMorphism_def] at h

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'
h : LieIdeal.toLieSubalgebra R L' f.idealRange = f.range
y : L'
⊢ y ∈ f.idealRange ↔ ∃ x, f x = y
Module: Mathlib.Algebra.Lie.Ideal

Goal state before:
pp: case a
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'
J : LieIdeal R L'
h : f.IsIdealMorphism
⊢ f.idealRange ⊓ J ≤ map f (comap f J)

Tactic:
rw [f.isIdealMorphism_def] at h

Goal state after:
case a
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'
J : LieIdeal R L'
h : toLieSubalgebra R L' f.idealRange = f.range
⊢ f.idealRange ⊓ J ≤ map f (comap f J)
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
⊢ Submodule.span R s ≤ ↑(lieSpan R L s)

Tactic:
Submodule.span_le

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
s : Set M
⊢ s ⊆ ↑↑(lieSpan R L s)
Module: Mathlib.FieldTheory.Minpoly.Basic

Goal state before:
pp: case refine_2
A : Type u_1
B : Type u_2
inst✝² : CommRing A
inst✝¹ : Ring B
inst✝ : Algebra A B
a : A
hf : Injective ⇑(algebraMap A B)
a✝ : Nontrivial A
⊢ ∀ (q : A[X]), q.degree < (X - C a).degree → q = 0 ∨ (Polynomial.aeval ((algebraMap A B) a)) q ≠ 0

Tactic:
or_iff_not_imp_left

Goal state after:
case refine_2
A : Type u_1
B : Type u_2
inst✝² : CommRing A
inst✝¹ : Ring B
inst✝ : Algebra A B
a : A
hf : Injective ⇑(algebraMap A B)
a✝ : Nontrivial A
⊢ ∀ (q : A[X]), q.degree < (X - C a).degree → ¬q = 0 → (Polynomial.aeval ((algebraMap A B) a)) q ≠ 0
Module: Mathlib.FieldTheory.Minpoly.Basic

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝⁴ : CommRing A
inst✝³ : Ring B
inst✝² : Algebra A B
x : B
inst✝¹ : IsDomain A
inst✝ : IsDomain B
hx : IsIntegral A x
f g : A[X]
hf : f.Monic
hg : g.Monic
he : f * g = minpoly A x
h : ¬IsUnit f ∧ ¬IsUnit g
heval : (Polynomial.aeval x) (f * g) = (Polynomial.aeval x) (minpoly A x)
⊢ False

Tactic:
aeval A x,

Goal state after:
A : Type u_1
B : Type u_2
inst✝⁴ : CommRing A
inst✝³ : Ring B
inst✝² : Algebra A B
x : B
inst✝¹ : IsDomain A
inst✝ : IsDomain B
hx : IsIntegral A x
f g : A[X]
hf : f.Monic
hg : g.Monic
he : f * g = minpoly A x
h : ¬IsUnit f ∧ ¬IsUnit g
heval : (Polynomial.aeval x) (f * g) = 0
⊢ False
Module: Mathlib.FieldTheory.Minpoly.Field

Goal state before:
pp: case neg
A : Type u_1
inst✝² : Field A
B : Type u_3
inst✝¹ : CommRing B
inst✝ : Algebra A B
x : B
a : A
hx : ¬IsIntegral A x
⊢ minpoly A (x + (algebraMap A B) a) = (minpoly A x).comp (X - C a)

Tactic:
minpoly.eq_zero hx,

Goal state after:
case neg
A : Type u_1
inst✝² : Field A
B : Type u_3
inst✝¹ : CommRing B
inst✝ : Algebra A B
x : B
a : A
hx : ¬IsIntegral A x
⊢ minpoly A (x + (algebraMap A B) a) = Polynomial.comp 0 (X - C a)
Module: Mathlib.FieldTheory.Minpoly.Field

Goal state before:
pp: case neg
A : Type u_1
inst✝² : Field A
B : Type u_3
inst✝¹ : Ring B
inst✝ : Algebra A B
x : B
hx : ¬IsIntegral A x
⊢ minpoly A (-x) = (-1) ^ (minpoly A x).natDegree * (minpoly A x).comp (-X)

Tactic:
minpoly.eq_zero hx,

Goal state after:
case neg
A : Type u_1
inst✝² : Field A
B : Type u_3
inst✝¹ : Ring B
inst✝ : Algebra A B
x : B
hx : ¬IsIntegral A x
⊢ minpoly A (-x) = (-1) ^ natDegree 0 * Polynomial.comp 0 (-X)
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
⊢ (Algebra.leftMulMatrix pb.basis) pb.gen =
    Matrix.of fun i j ↦ if ↑j + 1 = pb.dim then -pb.minpolyGen.coeff ↑i else if ↑i = ↑j + 1 then 1 else 0

Tactic:
by
  cases subsingleton_or_nontrivial A; · subsingleton
  rw [Algebra.leftMulMatrix_apply, ← LinearEquiv.eq_symm_apply, LinearMap.toMatrix_symm]
  refine pb.basis.ext fun k => ?_
  simp_rw [Matrix.toLin_self, Matrix.of_apply, pb.basis_eq_pow]
  apply (pow_succ' _ _).symm.trans
  split_ifs with h
  · simp_rw [h, neg_smul, Finset.sum_neg_distrib, eq_neg_iff_add_eq_zero]
    convert! pb.aeval_minpolyGen
    rw [add_comm, aeval_eq_sum_range, Finset.sum_range_succ, ← leadingCoeff,
      pb.minpolyGen_monic.leadingCoeff, one_smul, natDegree_minpolyGen, Finset.sum_range]
  · rw [Fintype.sum_eq_single (⟨(k : ℕ) + 1, lt_of_le_of_ne k.2 h⟩ : Fin pb.dim), if_pos, one_smul]
    · rfl
    intro x hx
    rw [if_neg, zero_smul]
    apply mt Fin.ext hx

Goal state after:
Module: Mathlib.RingTheory.PowerBasis

Goal state before:
pp: S : Type u_2
inst✝⁴ : Ring S
A : Type u_4
inst✝³ : CommRing A
S' : Type u_7
inst✝² : CommRing S'
inst✝¹ : Algebra A S
inst✝ : Algebra A S'
pb : PowerBasis A S
e : S ≃ₐ[A] S'
h₁ : (aeval pb.gen) (minpoly A (pb.map e).gen) = 0
h₂ : (aeval (pb.map e).gen) (minpoly A pb.gen) = 0
⊢ pb.equivOfRoot (pb.map e) h₁ h₂ = e

Tactic:
by
  ext x
  obtain ⟨f, rfl⟩ := pb.exists_eq_aeval' x
  simp [aeval_algEquiv]

Goal state after:
Module: Mathlib.RingTheory.PowerBasis

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : Ring S
inst✝ : Algebra R S
B : PowerBasis R S
⊢ Submodule.span R (Set.range ⇑B.basis) ≤ Subalgebra.toSubmodule R[B.gen]

Tactic:
Submodule.span_le

Goal state after:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : Ring S
inst✝ : Algebra R S
B : PowerBasis R S
⊢ Set.range ⇑B.basis ⊆ ↑(Subalgebra.toSubmodule R[B.gen])
Module: Mathlib.FieldTheory.KummerPolynomial

Goal state before:
pp: case inl
K : Type u
inst✝ : Field K
a : K
ha : a ≠ 0
hn : 0 < Nat.succ 0
⊢ root (X ^ Nat.succ 0 - C a) ≠ 0

pp: case inr
K : Type u
inst✝ : Field K
n : ℕ
a : K
hn✝ : 0 < n
ha : a ≠ 0
hn : Nat.succ 0 < n
⊢ root (X ^ n - C a) ≠ 0

Tactic:
· rw [pow_one]
    intro e
    refine mk_ne_zero_of_natDegree_lt (monic_X_sub_C a) (C_ne_zero.mpr ha) (by simp) ?_
    trans AdjoinRoot.mk (X - C a) (X - (X - C a))
    · rw [sub_sub_cancel]
    · rw [map_sub, mk_self, sub_zero, mk_X, e]

Goal state after:
case inr
K : Type u
inst✝ : Field K
n : ℕ
a : K
hn✝ : 0 < n
ha : a ≠ 0
hn : Nat.succ 0 < n
⊢ root (X ^ n - C a) ≠ 0
Module: Mathlib.LinearAlgebra.Determinant

Goal state before:
pp: case neg
R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : M →ₗ[R] M
hf : LinearMap.det f ≠ 1
H : ¬∃ s, Nonempty (Basis (↥s) R M)
⊢ Free R M

Tactic:
· classical simp [LinearMap.coe_det, H] at hf

Goal state after:
Module: Mathlib.LinearAlgebra.Determinant

Goal state before:
pp: M : Type u_2
inst✝³ : AddCommGroup M
A : Type u_5
inst✝² : CommRing A
inst✝¹ : Module A M
inst✝ : DecidableEq M
P : A → Prop
f : M →ₗ[A] M
hb : ∀ (s : Finset M) (b : Basis (↥s) A M), P ((toMatrix b b) f).det
h1 : P 1
s : Finset M
b : Basis (↥s) A M
⊢ P (LinearMap.det f)

Tactic:
rw [← det_toMatrix b]

Goal state after:
M : Type u_2
inst✝³ : AddCommGroup M
A : Type u_5
inst✝² : CommRing A
inst✝¹ : Module A M
inst✝ : DecidableEq M
P : A → Prop
f : M →ₗ[A] M
hb : ∀ (s : Finset M) (b : Basis (↥s) A M), P ((toMatrix b b) f).det
h1 : P 1
s : Finset M
b : Basis (↥s) A M
⊢ P ((toMatrix b b) f).det