awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Algebra.Homology.SpectralObject.Page
Goal:
C : Type u_1
ι : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} ι
inst_2 : Abelian.{v_1, u_1} C
X : SpectralObject.{u_1, u_2, v_1, v_2} 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
this : hi.lift (KernelFork.ofι (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ ⋯ ⋯).f ⋯) = X.δToCycles f₁ f₂ f₃ n₀ n₁ ⋯
⊢ (parallelPair (X := (KernelFork.ofι (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).f ⋯).pt)
            (hi.lift (KernelFork.ofι (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).f ⋯)) 0).map
        WalkingParallelPairHom.right ≫
      Iso.hom
        (Iso.refl
          ((parallelPair (X := (KernelFork.ofι (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).f ⋯).pt)
                (hi.lift (KernelFork.ofι (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).f ⋯)) 0).obj
            WalkingParallelPair.one)) =
    Iso.hom
        (Iso.refl
          ((parallelPair (X := (KernelFork.ofι (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).f ⋯).pt)
                (hi.lift (KernelFork.ofι (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).f ⋯)) 0).obj
            WalkingParallelPair.zero)) ≫
      (parallelPair (X.δToCycles f₁ f₂ f₃ n₀ n₁ ⋯) 0).map WalkingParallelPairHom.right

Tactic:
simp
-- Module: Mathlib.Algebra.Homology.SpectralObject.Page
Goal:
C : Type u_1
ι : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} ι
inst_2 : Abelian.{v_1, u_1} C
X : SpectralObject.{u_1, u_2, v_1, v_2} 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
this : hi.lift (KernelFork.ofι (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ ⋯ ⋯).f ⋯) = X.δToCycles f₁ f₂ f₃ n₀ n₁ ⋯
⊢ parallelPair (X := (KernelFork.ofι (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).f ⋯).pt)
      (hi.lift (KernelFork.ofι (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).f ⋯)) 0 ≅
    parallelPair (X.δToCycles f₁ f₂ f₃ n₀ n₁ ⋯) 0

Tactic:
exact parallelPair.ext (Iso.refl _) (Iso.refl _) (by simpa) (by simp)

Premises used: CategoryTheory.Iso.refl, CategoryTheory.Limits.parallelPair.ext
-- Module: Mathlib.Algebra.Homology.SpectralObject.Page
Goal:
C : Type u_1
ι : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} ι
inst_2 : Abelian.{v_1, u_1} C
X : SpectralObject.{u_1, u_2, v_1, v_2} 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
this : hi.lift (KernelFork.ofι (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ ⋯ ⋯).f ⋯) = X.δToCycles f₁ f₂ f₃ n₀ n₁ ⋯
⊢ (Cocone.precompose
          (parallelPair.ext
              (Iso.refl
                ((parallelPair (X := (KernelFork.ofι (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).f ⋯).pt)
                      (hi.lift (KernelFork.ofι (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).f ⋯)) 0).obj
                  WalkingParallelPair.zero))
              (Iso.refl
                ((parallelPair (X := (KernelFork.ofι (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).f ⋯).pt)
                      (hi.lift (KernelFork.ofι (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).f ⋯)) 0).obj
                  WalkingParallelPair.one))
              ⋯ ⋯).inv).obj
      (CokernelCofork.ofπ (cokernel.π (X.δToCycles f₁ f₂ f₃ n₀ n₁ ⋯)) ⋯) ≅
    Cofork.ofπ (P := cokernel (X.δToCycles f₁ f₂ f₃ n₀ n₁ ⋯)) (cokernel.π (X.δToCycles f₁ f₂ f₃ n₀ n₁ ⋯)) ⋯

Tactic:
exact Cofork.ext (Iso.refl _)

Premises used: CategoryTheory.Iso.refl, CategoryTheory.Limits.Cofork.ext
-- Module: Mathlib.Algebra.Homology.SpectralObject.Page
Goal:
C : Type u_1
ι : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} ι
inst_2 : Abelian.{v_1, u_1} C
X : SpectralObject.{u_1, u_2, v_1, v_2} 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
this : hi.lift (KernelFork.ofι (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ ⋯ ⋯).f ⋯) = X.δToCycles f₁ f₂ f₃ n₀ n₁ ⋯
⊢ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).LeftHomologyData

Tactic:
exact {
    K := X.cycles f₁ f₂ n₁
    H := cokernel (X.δToCycles f₁ f₂ f₃ n₀ n₁)
    i := X.iCycles f₁ f₂ n₁
    π := cokernel.π _
    wi := by simp
    hi := hi
    wπ := by rw [this]; simp
    hπ := by
      refine (IsColimit.equivOfNatIsoOfIso ?_ _ _ ?_).2
        (cokernelIsCokernel (X.δToCycles f₁ f₂ f₃ n₀ n₁))
      · exact parallelPair.ext (Iso.refl _) (Iso.refl _) (by simpa) (by simp)
      · exact Cofork.ext (Iso.refl _) }
-- Module: Mathlib.Algebra.Homology.SpectralObject.Page
Goal:
C : Type u_1
ι : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} ι
inst_2 : Abelian.{v_1, u_1} C
X : SpectralObject.{u_1, u_2, v_1, v_2} 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:
simpa using hi.fac _ .zero

Premises used: CategoryTheory.Limits.WalkingParallelPair.zero, CategoryTheory.Limits.IsLimit.fac
-- Module: Mathlib.Algebra.Homology.SpectralObject.Page
Goal:
C : Type u_1
ι : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} ι
inst_2 : Abelian.{v_1, u_1} C
X : SpectralObject.{u_1, u_2, v_1, v_2} 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' = X.δToCycles f₁ f₂ f₃ n₀ n₁ hn₁

Tactic:
exact Fork.IsLimit.hom_ext hi (by simpa using hi.fac _ .zero)

Premises used: CategoryTheory.Limits.Fork.IsLimit.hom_ext
-- Module: Mathlib.Algebra.Homology.SpectralObject.Page
Goal:
C : Type u_1
ι : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} ι
inst_2 : Abelian.{v_1, u_1} C
X : SpectralObject.{u_1, u_2, v_1, v_2} 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
⊢ (X.cyclesIso f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).inv ≫ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).iCycles = X.iCycles f₁ f₂ n₁

Tactic:
ShortComplex.LeftHomologyData.cyclesIso_inv_comp_iCycles _

Premises used: CategoryTheory.Limits.limit.π, CategoryTheory.Abelian.SpectralObject.cycles, CategoryTheory.Limits.limit, CategoryTheory.ShortComplex.LeftHomologyData.cyclesIso_inv_comp_iCycles, CategoryTheory.Abelian.SpectralObject.iCycles
-- Module: Mathlib.Algebra.Homology.SpectralObject.Page
Goal:
C : Type u_1
ι : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} ι
inst_2 : Abelian.{v_1, u_1} C
X : SpectralObject.{u_1, u_2, v_1, v_2} 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
⊢ (X.cyclesIso f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).hom ≫ X.iCycles f₁ f₂ n₁ = (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).iCycles

Tactic:
ShortComplex.LeftHomologyData.cyclesIso_hom_comp_i _

Premises used: CategoryTheory.Limits.limit.π, CategoryTheory.Abelian.SpectralObject.cycles, CategoryTheory.Limits.limit, CategoryTheory.ShortComplex.LeftHomologyData.cyclesIso_hom_comp_i, CategoryTheory.Abelian.SpectralObject.iCycles
-- Module: Mathlib.Algebra.Homology.SpectralObject.Page
Goal:
C : Type u_1
ι : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} ι
inst_2 : Abelian.{v_1, u_1} C
X : SpectralObject.{u_1, u_2, v_1, v_2} 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
⊢ X.δToCycles f₁ f₂ f₃ n₀ n₁ hn₁ ≫ (X.cyclesIso f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).inv =
    (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).toCycles

Tactic:
simp [← cancel_mono (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂).iCycles]

Premises used: CategoryTheory.cancel_mono, CategoryTheory.Abelian.SpectralObject.shortComplex, CategoryTheory.ShortComplex.iCycles
-- Module: Mathlib.Algebra.Homology.SpectralObject.Page
Goal:
C : Type u_1
ι : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} ι
inst_2 : Abelian.{v_1, u_1} C
X : SpectralObject.{u_1, u_2, v_1, v_2} 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
⊢ X.δToCycles f₁ f₂ f₃ n₀ n₁ hn₁ ≫ X.πE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂ = 0

Tactic:
simp [πE]

Premises used: CategoryTheory.Abelian.SpectralObject.πE
-- Module: Mathlib.Algebra.Homology.SpectralObject.Page
Goal:
C : Type u_1
ι : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} ι
inst_2 : Abelian.{v_1, u_1} C
X : SpectralObject.{u_1, u_2, v_1, v_2} 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
⊢ Iso.hom (Iso.refl (X.cokernelSequenceCyclesE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).X₁) ≫
      (ShortComplex.mk (X₂ := (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).cycles) (X₃ :=
          (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).homology) (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).toCycles
          (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).homologyπ ⋯).f =
    (X.cokernelSequenceCyclesE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).f ≫ Iso.hom (Iso.symm (X.cyclesIso f₁ f₂ f₃ n₀ n₁ n₂ ⋯ ⋯))

Tactic:
simp
-- Module: Mathlib.Algebra.Homology.SpectralObject.Page
Goal:
C : Type u_1
ι : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} ι
inst_2 : Abelian.{v_1, u_1} C
X : SpectralObject.{u_1, u_2, v_1, v_2} 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
⊢ Iso.hom (Iso.symm (X.cyclesIso f₁ f₂ f₃ n₀ n₁ n₂ ⋯ ⋯)) ≫
      (ShortComplex.mk (X₂ := (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).cycles) (X₃ :=
          (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).homology) (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).toCycles
          (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).homologyπ ⋯).g =
    (X.cokernelSequenceCyclesE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ≫
      Iso.hom (Iso.refl (X.cokernelSequenceCyclesE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).X₃)

Tactic:
simp [πE]

Premises used: CategoryTheory.Abelian.SpectralObject.πE
-- Module: Mathlib.Algebra.Homology.SpectralObject.Page
Goal:
C : Type u_1
ι : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} ι
inst_2 : Abelian.{v_1, u_1} C
X : SpectralObject.{u_1, u_2, v_1, v_2} 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
⊢ (X.cokernelSequenceCyclesE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).Exact

Tactic:
ShortComplex.exact_of_iso (X.cokernelSequenceCyclesEIso f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).symm
    (ShortComplex.exact_of_g_is_cokernel _ (ShortComplex.homologyIsCokernel _))

Premises used: CategoryTheory.ShortComplex.exact_of_g_is_cokernel, CategoryTheory.ShortComplex.exact_of_iso, CategoryTheory.ShortComplex.homologyIsCokernel, CategoryTheory.Iso.symm, CategoryTheory.Abelian.SpectralObject.cokernelSequenceCyclesEIso
-- Module: Mathlib.Algebra.Homology.SpectralObject.Page
Goal:
C : Type u_1
ι : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} ι
inst_2 : Abelian.{v_1, u_1} C
X : SpectralObject.{u_1, u_2, v_1, v_2} 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
dsimp : Epi (X.πE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂)
⊢ Epi (X.cokernelSequenceCyclesE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g

Tactic:
dsimp
-- Module: Mathlib.Algebra.Homology.SpectralObject.Page
Goal:
C : Type u_1
ι : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} ι
inst_2 : Abelian.{v_1, u_1} C
X : SpectralObject.{u_1, u_2, v_1, v_2} 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
⊢ Epi (X.πE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂)

Tactic:
infer_instance
-- Module: Mathlib.Algebra.Homology.SpectralObject.Page
Goal:
C : Type u_1
ι : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} ι
inst_2 : Abelian.{v_1, u_1} C
X : SpectralObject.{u_1, u_2, v_1, v_2} 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
⊢ Cofork.π (CokernelCofork.ofπ (X.cokernelSequenceOpcycles f₂ f₃ n₀ n₁ hn₁).g ⋯) ≫
      hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯) =
    Cofork.π (CokernelCofork.ofπ (X.cokernelSequenceOpcycles f₂ f₃ n₀ n₁ hn₁).g ⋯) ≫ X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ hn₂

Tactic:
simpa using hp.fac _ .one

Premises used: CategoryTheory.Limits.IsColimit.fac, CategoryTheory.Limits.WalkingParallelPair.one
-- Module: Mathlib.Algebra.Homology.SpectralObject.Page
Goal:
C : Type u_1
ι : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} ι
inst_2 : Abelian.{v_1, u_1} C
X : SpectralObject.{u_1, u_2, v_1, v_2} 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
⊢ hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯) = X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ hn₂

Tactic:
have : hp.desc (CokernelCofork.ofπ _ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).zero) =
      X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ hn₂ :=
    Cofork.IsColimit.hom_ext hp (by simpa using hp.fac _ .one)
-- Module: Mathlib.Algebra.Homology.SpectralObject.Page
Goal:
C : Type u_1
ι : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} ι
inst_2 : Abelian.{v_1, u_1} C
X : SpectralObject.{u_1, u_2, v_1, v_2} 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
⊢ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).f ≫ X.pOpcycles f₂ f₃ n₁ = 0

Tactic:
simp
-- Module: Mathlib.Algebra.Homology.SpectralObject.Page
Goal:
C : Type u_1
ι : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} ι
inst_2 : Abelian.{v_1, u_1} C
X : SpectralObject.{u_1, u_2, v_1, v_2} 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
rw :
  Eq (α :=
    kernel (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ ⋯) ⟶ (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯).pt)
    (kernel.ι (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ ⋯) ≫ X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ hn₂) 0
⊢ Eq (α :=
    kernel (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ ⋯) ⟶ (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯).pt)
    (kernel.ι (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ ⋯) ≫
      hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯))
    0

Tactic:
rw [this]
-- Module: Mathlib.Algebra.Homology.SpectralObject.Page
Goal:
C : Type u_1
ι : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} ι
inst_2 : Abelian.{v_1, u_1} C
X : SpectralObject.{u_1, u_2, v_1, v_2} 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
⊢ Eq (α :=
    kernel (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ ⋯) ⟶ (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯).pt)
    (kernel.ι (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ ⋯) ≫ X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ hn₂) 0

Tactic:
simp
-- Module: Mathlib.Algebra.Homology.SpectralObject.Page
Goal:
C : Type u_1
ι : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} ι
inst_2 : Abelian.{v_1, u_1} C
X : SpectralObject.{u_1, u_2, v_1, v_2} 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
refine_1 :
  parallelPair (Y := (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯).pt)
      (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
refine_2 :
  (Cone.postcompose
          (parallelPair.ext
              (Iso.refl
                ((parallelPair (Y := (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯).pt)
                      (hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯)) 0).obj
                  WalkingParallelPair.zero))
              (Iso.refl
                ((parallelPair (Y := (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯).pt)
                      (hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯)) 0).obj
                  WalkingParallelPair.one))
              ⋯ ⋯).hom).obj
      (KernelFork.ofι (kernel.ι (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ ⋯)) ⋯) ≅
    Fork.ofι (P := kernel (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ ⋯)) (kernel.ι (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ ⋯)) ⋯
⊢ 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))

Premises used: CategoryTheory.Limits.kernelIsKernel, Equiv.invFun, CategoryTheory.Limits.IsLimit.equivOfNatIsoOfIso, CategoryTheory.Abelian.SpectralObject.δFromOpcycles
-- Module: Mathlib.Algebra.Homology.SpectralObject.Page
Goal:
C : Type u_1
ι : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} ι
inst_2 : Abelian.{v_1, u_1} C
X : SpectralObject.{u_1, u_2, v_1, v_2} 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 (Y := (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯).pt)
            (hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯)) 0).map
        WalkingParallelPairHom.left ≫
      Iso.hom
        (Iso.refl
          ((parallelPair (Y := (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯).pt)
                (hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯)) 0).obj
            WalkingParallelPair.one)) =
    Iso.hom
        (Iso.refl
          ((parallelPair (Y := (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯).pt)
                (hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯)) 0).obj
            WalkingParallelPair.zero)) ≫
      (parallelPair (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ ⋯) 0).map WalkingParallelPairHom.left

Tactic:
simpa
-- Module: Mathlib.Algebra.Homology.SpectralObject.Page
Goal:
C : Type u_1
ι : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} ι
inst_2 : Abelian.{v_1, u_1} C
X : SpectralObject.{u_1, u_2, v_1, v_2} 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 (Y := (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯).pt)
            (hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯)) 0).map
        WalkingParallelPairHom.right ≫
      Iso.hom
        (Iso.refl
          ((parallelPair (Y := (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯).pt)
                (hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯)) 0).obj
            WalkingParallelPair.one)) =
    Iso.hom
        (Iso.refl
          ((parallelPair (Y := (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯).pt)
                (hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯)) 0).obj
            WalkingParallelPair.zero)) ≫
      (parallelPair (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ ⋯) 0).map WalkingParallelPairHom.right

Tactic:
simp
-- Module: Mathlib.Algebra.Homology.SpectralObject.Page
Goal:
C : Type u_1
ι : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} ι
inst_2 : Abelian.{v_1, u_1} C
X : SpectralObject.{u_1, u_2, v_1, v_2} 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 (Y := (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯).pt)
      (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

Tactic:
exact parallelPair.ext (Iso.refl _) (Iso.refl _) (by simpa) (by simp)

Premises used: CategoryTheory.Iso.refl, CategoryTheory.Limits.parallelPair.ext
-- Module: Mathlib.Algebra.Homology.SpectralObject.Page
Goal:
C : Type u_1
ι : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} ι
inst_2 : Abelian.{v_1, u_1} C
X : SpectralObject.{u_1, u_2, v_1, v_2} 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
          (parallelPair.ext
              (Iso.refl
                ((parallelPair (Y := (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯).pt)
                      (hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯)) 0).obj
                  WalkingParallelPair.zero))
              (Iso.refl
                ((parallelPair (Y := (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯).pt)
                      (hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯)) 0).obj
                  WalkingParallelPair.one))
              ⋯ ⋯).hom).obj
      (KernelFork.ofι (kernel.ι (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ ⋯)) ⋯) ≅
    Fork.ofι (P := kernel (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ ⋯)) (kernel.ι (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ ⋯)) ⋯

Tactic:
exact Fork.ext (Iso.refl _)

Premises used: CategoryTheory.Iso.refl, CategoryTheory.Limits.Fork.ext