l3lab/ntp-mathlib huggingface dataset:
/-- If `φ : S₁ ⟶ S₂` is a morphism of short complexes such that `φ.τ₁` is epi, `φ.τ₂` is an iso
and `φ.τ₃` is mono, then a left homology data for `S₂` induces a left homology data for `S₁` with
the same `K` and `H` fields. The inverse construction is `ofEpiOfIsIsoOfMono`. -/
@[simps]
noncomputable def ofEpiOfIsIsoOfMono' (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₂)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : LeftHomologyData S₁ := by
  let i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
  have wi : i ≫ S₁.g = 0 := by
    

Proof state:
C : Type u_1
inst✝⁴ : Category.{?u.177302, u_1} C
inst✝³ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h : LeftHomologyData S₂
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
⊢ i ≫ S₁.g = 0

Next tactic:
rw [assoc, ← cancel_mono φ.τ₃, zero_comp, assoc, assoc, ← φ.comm₂₃,
      IsIso.inv_hom_id_assoc, h.wi]
/-- If `φ : S₁ ⟶ S₂` is a morphism of short complexes such that `φ.τ₁` is epi, `φ.τ₂` is an iso
and `φ.τ₃` is mono, then a left homology data for `S₂` induces a left homology data for `S₁` with
the same `K` and `H` fields. The inverse construction is `ofEpiOfIsIsoOfMono`. -/
@[simps]
noncomputable def ofEpiOfIsIsoOfMono' (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₂)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : LeftHomologyData S₁ := by
  let i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
  have wi : i ≫ S₁.g = 0 := by
    rw [assoc, ← cancel_mono φ.τ₃, zero_comp, assoc, assoc, ← φ.comm₂₃,
      IsIso.inv_hom_id_assoc, h.wi]
  

Proof state:
C : Type u_1
inst✝⁴ : Category.{?u.177302, u_1} C
inst✝³ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h : LeftHomologyData S₂
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
wi : i ≫ S₁.g = 0
⊢ LeftHomologyData S₁

Next tactic:
have hi : IsLimit (KernelFork.ofι i wi) := KernelFork.IsLimit.ofι _ _
    (fun x hx => h.liftK (x ≫ φ.τ₂)
      (by rw [assoc, φ.comm₂₃, reassoc_of% hx, zero_comp]))
    (fun x hx => by simp )
    (fun x hx b hb => by rw [← cancel_mono h.i, ← cancel_mono (inv φ.τ₂), assoc, assoc,
      hb, liftK_i_assoc, assoc, IsIso.hom_inv_id, comp_id])
/-- If `φ : S₁ ⟶ S₂` is a morphism of short complexes such that `φ.τ₁` is epi, `φ.τ₂` is an iso
and `φ.τ₃` is mono, then a left homology data for `S₂` induces a left homology data for `S₁` with
the same `K` and `H` fields. The inverse construction is `ofEpiOfIsIsoOfMono`. -/
@[simps]
noncomputable def ofEpiOfIsIsoOfMono' (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₂)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : LeftHomologyData S₁ := by
  let i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
  have wi : i ≫ S₁.g = 0 := by
    rw [assoc, ← cancel_mono φ.τ₃, zero_comp, assoc, assoc, ← φ.comm₂₃,
      IsIso.inv_hom_id_assoc, h.wi]
  have hi : IsLimit (KernelFork.ofι i wi) := KernelFork.IsLimit.ofι _ _
    (fun x hx => h.liftK (x ≫ φ.τ₂)
      (by 

Proof state:
C : Type u_1
inst✝⁴ : Category.{?u.177302, u_1} C
inst✝³ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h : LeftHomologyData S₂
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
wi : i ≫ S₁.g = 0
W'✝ : C
x : W'✝ ⟶ S₁.X₂
hx : x ≫ S₁.g = 0
⊢ (x ≫ φ.τ₂) ≫ S₂.g = 0

Next tactic:
rw [assoc, φ.comm₂₃, reassoc_of% hx, zero_comp]
/-- If `φ : S₁ ⟶ S₂` is a morphism of short complexes such that `φ.τ₁` is epi, `φ.τ₂` is an iso
and `φ.τ₃` is mono, then a left homology data for `S₂` induces a left homology data for `S₁` with
the same `K` and `H` fields. The inverse construction is `ofEpiOfIsIsoOfMono`. -/
@[simps]
noncomputable def ofEpiOfIsIsoOfMono' (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₂)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : LeftHomologyData S₁ := by
  let i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
  have wi : i ≫ S₁.g = 0 := by
    rw [assoc, ← cancel_mono φ.τ₃, zero_comp, assoc, assoc, ← φ.comm₂₃,
      IsIso.inv_hom_id_assoc, h.wi]
  have hi : IsLimit (KernelFork.ofι i wi) := KernelFork.IsLimit.ofι _ _
    (fun x hx => h.liftK (x ≫ φ.τ₂)
      (by rw [assoc, φ.comm₂₃, reassoc_of% hx, zero_comp]))
    (fun x hx => by 

Proof state:
C : Type u_1
inst✝⁴ : Category.{?u.177302, u_1} C
inst✝³ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h : LeftHomologyData S₂
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
wi : i ≫ S₁.g = 0
W'✝ : C
x : W'✝ ⟶ S₁.X₂
hx : x ≫ S₁.g = 0
⊢ (fun {W'} x hx => liftK h (x ≫ φ.τ₂) (_ : (x ≫ φ.τ₂) ≫ S₂.g = 0)) x hx ≫ i = x

Next tactic:
simp
/-- If `φ : S₁ ⟶ S₂` is a morphism of short complexes such that `φ.τ₁` is epi, `φ.τ₂` is an iso
and `φ.τ₃` is mono, then a left homology data for `S₂` induces a left homology data for `S₁` with
the same `K` and `H` fields. The inverse construction is `ofEpiOfIsIsoOfMono`. -/
@[simps]
noncomputable def ofEpiOfIsIsoOfMono' (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₂)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : LeftHomologyData S₁ := by
  let i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
  have wi : i ≫ S₁.g = 0 := by
    rw [assoc, ← cancel_mono φ.τ₃, zero_comp, assoc, assoc, ← φ.comm₂₃,
      IsIso.inv_hom_id_assoc, h.wi]
  have hi : IsLimit (KernelFork.ofι i wi) := KernelFork.IsLimit.ofι _ _
    (fun x hx => h.liftK (x ≫ φ.τ₂)
      (by rw [assoc, φ.comm₂₃, reassoc_of% hx, zero_comp]))
    (fun x hx => by simp )
    (fun x hx b hb => by 

Proof state:
C : Type u_1
inst✝⁴ : Category.{?u.177302, u_1} C
inst✝³ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h : LeftHomologyData S₂
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
wi : i ≫ S₁.g = 0
W'✝ : C
x : W'✝ ⟶ S₁.X₂
hx : x ≫ S₁.g = 0
b : W'✝ ⟶ h.K
hb : b ≫ i = x
⊢ b = (fun {W'} x hx => liftK h (x ≫ φ.τ₂) (_ : (x ≫ φ.τ₂) ≫ S₂.g = 0)) x hx

Next tactic:
rw [← cancel_mono h.i, ← cancel_mono (inv φ.τ₂), assoc, assoc,
      hb, liftK_i_assoc, assoc, IsIso.hom_inv_id, comp_id]
/-- If `φ : S₁ ⟶ S₂` is a morphism of short complexes such that `φ.τ₁` is epi, `φ.τ₂` is an iso
and `φ.τ₃` is mono, then a left homology data for `S₂` induces a left homology data for `S₁` with
the same `K` and `H` fields. The inverse construction is `ofEpiOfIsIsoOfMono`. -/
@[simps]
noncomputable def ofEpiOfIsIsoOfMono' (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₂)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : LeftHomologyData S₁ := by
  let i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
  have wi : i ≫ S₁.g = 0 := by
    rw [assoc, ← cancel_mono φ.τ₃, zero_comp, assoc, assoc, ← φ.comm₂₃,
      IsIso.inv_hom_id_assoc, h.wi]
  have hi : IsLimit (KernelFork.ofι i wi) := KernelFork.IsLimit.ofι _ _
    (fun x hx => h.liftK (x ≫ φ.τ₂)
      (by rw [assoc, φ.comm₂₃, reassoc_of% hx, zero_comp]))
    (fun x hx => by simp )
    (fun x hx b hb => by rw [← cancel_mono h.i, ← cancel_mono (inv φ.τ₂), assoc, assoc,
      hb, liftK_i_assoc, assoc, IsIso.hom_inv_id, comp_id])
  

Proof state:
C : Type u_1
inst✝⁴ : Category.{?u.177302, u_1} C
inst✝³ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h : LeftHomologyData S₂
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
wi : i ≫ S₁.g = 0
hi : IsLimit (KernelFork.ofι i wi)
⊢ LeftHomologyData S₁

Next tactic:
let f' := hi.lift (KernelFork.ofι S₁.f S₁.zero)
/-- If `φ : S₁ ⟶ S₂` is a morphism of short complexes such that `φ.τ₁` is epi, `φ.τ₂` is an iso
and `φ.τ₃` is mono, then a left homology data for `S₂` induces a left homology data for `S₁` with
the same `K` and `H` fields. The inverse construction is `ofEpiOfIsIsoOfMono`. -/
@[simps]
noncomputable def ofEpiOfIsIsoOfMono' (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₂)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : LeftHomologyData S₁ := by
  let i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
  have wi : i ≫ S₁.g = 0 := by
    rw [assoc, ← cancel_mono φ.τ₃, zero_comp, assoc, assoc, ← φ.comm₂₃,
      IsIso.inv_hom_id_assoc, h.wi]
  have hi : IsLimit (KernelFork.ofι i wi) := KernelFork.IsLimit.ofι _ _
    (fun x hx => h.liftK (x ≫ φ.τ₂)
      (by rw [assoc, φ.comm₂₃, reassoc_of% hx, zero_comp]))
    (fun x hx => by simp )
    (fun x hx b hb => by rw [← cancel_mono h.i, ← cancel_mono (inv φ.τ₂), assoc, assoc,
      hb, liftK_i_assoc, assoc, IsIso.hom_inv_id, comp_id])
  let f' := hi.lift (KernelFork.ofι S₁.f S₁.zero)
  

Proof state:
C : Type u_1
inst✝⁴ : Category.{?u.177302, u_1} C
inst✝³ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h : LeftHomologyData S₂
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
wi : i ≫ S₁.g = 0
hi : IsLimit (KernelFork.ofι i wi)
f' : (KernelFork.ofι S₁.f (_ : S₁.f ≫ S₁.g = 0)).pt ⟶ (KernelFork.ofι i wi).pt :=
  IsLimit.lift hi (KernelFork.ofι S₁.f (_ : S₁.f ≫ S₁.g = 0))
⊢ LeftHomologyData S₁

Next tactic:
have hf' : f' ≫ i = S₁.f := Fork.IsLimit.lift_ι _
/-- If `φ : S₁ ⟶ S₂` is a morphism of short complexes such that `φ.τ₁` is epi, `φ.τ₂` is an iso
and `φ.τ₃` is mono, then a left homology data for `S₂` induces a left homology data for `S₁` with
the same `K` and `H` fields. The inverse construction is `ofEpiOfIsIsoOfMono`. -/
@[simps]
noncomputable def ofEpiOfIsIsoOfMono' (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₂)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : LeftHomologyData S₁ := by
  let i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
  have wi : i ≫ S₁.g = 0 := by
    rw [assoc, ← cancel_mono φ.τ₃, zero_comp, assoc, assoc, ← φ.comm₂₃,
      IsIso.inv_hom_id_assoc, h.wi]
  have hi : IsLimit (KernelFork.ofι i wi) := KernelFork.IsLimit.ofι _ _
    (fun x hx => h.liftK (x ≫ φ.τ₂)
      (by rw [assoc, φ.comm₂₃, reassoc_of% hx, zero_comp]))
    (fun x hx => by simp )
    (fun x hx b hb => by rw [← cancel_mono h.i, ← cancel_mono (inv φ.τ₂), assoc, assoc,
      hb, liftK_i_assoc, assoc, IsIso.hom_inv_id, comp_id])
  let f' := hi.lift (KernelFork.ofι S₁.f S₁.zero)
  have hf' : f' ≫ i = S₁.f := Fork.IsLimit.lift_ι _
  

Proof state:
C : Type u_1
inst✝⁴ : Category.{?u.177302, u_1} C
inst✝³ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h : LeftHomologyData S₂
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
wi : i ≫ S₁.g = 0
hi : IsLimit (KernelFork.ofι i wi)
f' : (KernelFork.ofι S₁.f (_ : S₁.f ≫ S₁.g = 0)).pt ⟶ (KernelFork.ofι i wi).pt :=
  IsLimit.lift hi (KernelFork.ofι S₁.f (_ : S₁.f ≫ S₁.g = 0))
hf' : f' ≫ i = S₁.f
⊢ LeftHomologyData S₁

Next tactic:
have hf'' : f' = φ.τ₁ ≫ h.f' := by
    rw [← cancel_mono h.i, ← cancel_mono (inv φ.τ₂), assoc, assoc, assoc, hf', f'_i_assoc,
      φ.comm₁₂_assoc, IsIso.hom_inv_id, comp_id]
/-- If `φ : S₁ ⟶ S₂` is a morphism of short complexes such that `φ.τ₁` is epi, `φ.τ₂` is an iso
and `φ.τ₃` is mono, then a left homology data for `S₂` induces a left homology data for `S₁` with
the same `K` and `H` fields. The inverse construction is `ofEpiOfIsIsoOfMono`. -/
@[simps]
noncomputable def ofEpiOfIsIsoOfMono' (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₂)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : LeftHomologyData S₁ := by
  let i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
  have wi : i ≫ S₁.g = 0 := by
    rw [assoc, ← cancel_mono φ.τ₃, zero_comp, assoc, assoc, ← φ.comm₂₃,
      IsIso.inv_hom_id_assoc, h.wi]
  have hi : IsLimit (KernelFork.ofι i wi) := KernelFork.IsLimit.ofι _ _
    (fun x hx => h.liftK (x ≫ φ.τ₂)
      (by rw [assoc, φ.comm₂₃, reassoc_of% hx, zero_comp]))
    (fun x hx => by simp )
    (fun x hx b hb => by rw [← cancel_mono h.i, ← cancel_mono (inv φ.τ₂), assoc, assoc,
      hb, liftK_i_assoc, assoc, IsIso.hom_inv_id, comp_id])
  let f' := hi.lift (KernelFork.ofι S₁.f S₁.zero)
  have hf' : f' ≫ i = S₁.f := Fork.IsLimit.lift_ι _
  have hf'' : f' = φ.τ₁ ≫ h.f' := by
    

Proof state:
C : Type u_1
inst✝⁴ : Category.{?u.177302, u_1} C
inst✝³ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h : LeftHomologyData S₂
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
wi : i ≫ S₁.g = 0
hi : IsLimit (KernelFork.ofι i wi)
f' : (KernelFork.ofι S₁.f (_ : S₁.f ≫ S₁.g = 0)).pt ⟶ (KernelFork.ofι i wi).pt :=
  IsLimit.lift hi (KernelFork.ofι S₁.f (_ : S₁.f ≫ S₁.g = 0))
hf' : f' ≫ i = S₁.f
⊢ f' = φ.τ₁ ≫ CategoryTheory.ShortComplex.LeftHomologyData.f' h

Next tactic:
rw [← cancel_mono h.i, ← cancel_mono (inv φ.τ₂), assoc, assoc, assoc, hf', f'_i_assoc,
      φ.comm₁₂_assoc, IsIso.hom_inv_id, comp_id]
/-- If `φ : S₁ ⟶ S₂` is a morphism of short complexes such that `φ.τ₁` is epi, `φ.τ₂` is an iso
and `φ.τ₃` is mono, then a left homology data for `S₂` induces a left homology data for `S₁` with
the same `K` and `H` fields. The inverse construction is `ofEpiOfIsIsoOfMono`. -/
@[simps]
noncomputable def ofEpiOfIsIsoOfMono' (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₂)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : LeftHomologyData S₁ := by
  let i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
  have wi : i ≫ S₁.g = 0 := by
    rw [assoc, ← cancel_mono φ.τ₃, zero_comp, assoc, assoc, ← φ.comm₂₃,
      IsIso.inv_hom_id_assoc, h.wi]
  have hi : IsLimit (KernelFork.ofι i wi) := KernelFork.IsLimit.ofι _ _
    (fun x hx => h.liftK (x ≫ φ.τ₂)
      (by rw [assoc, φ.comm₂₃, reassoc_of% hx, zero_comp]))
    (fun x hx => by simp )
    (fun x hx b hb => by rw [← cancel_mono h.i, ← cancel_mono (inv φ.τ₂), assoc, assoc,
      hb, liftK_i_assoc, assoc, IsIso.hom_inv_id, comp_id])
  let f' := hi.lift (KernelFork.ofι S₁.f S₁.zero)
  have hf' : f' ≫ i = S₁.f := Fork.IsLimit.lift_ι _
  have hf'' : f' = φ.τ₁ ≫ h.f' := by
    rw [← cancel_mono h.i, ← cancel_mono (inv φ.τ₂), assoc, assoc, assoc, hf', f'_i_assoc,
      φ.comm₁₂_assoc, IsIso.hom_inv_id, comp_id]
  

Proof state:
C : Type u_1
inst✝⁴ : Category.{?u.177302, u_1} C
inst✝³ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h : LeftHomologyData S₂
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
wi : i ≫ S₁.g = 0
hi : IsLimit (KernelFork.ofι i wi)
f' : (KernelFork.ofι S₁.f (_ : S₁.f ≫ S₁.g = 0)).pt ⟶ (KernelFork.ofι i wi).pt :=
  IsLimit.lift hi (KernelFork.ofι S₁.f (_ : S₁.f ≫ S₁.g = 0))
hf' : f' ≫ i = S₁.f
hf'' : f' = φ.τ₁ ≫ CategoryTheory.ShortComplex.LeftHomologyData.f' h
⊢ LeftHomologyData S₁

Next tactic:
have wπ : f' ≫ h.π = 0 := by simp only [hf'', assoc, f'_π, comp_zero]
/-- If `φ : S₁ ⟶ S₂` is a morphism of short complexes such that `φ.τ₁` is epi, `φ.τ₂` is an iso
and `φ.τ₃` is mono, then a left homology data for `S₂` induces a left homology data for `S₁` with
the same `K` and `H` fields. The inverse construction is `ofEpiOfIsIsoOfMono`. -/
@[simps]
noncomputable def ofEpiOfIsIsoOfMono' (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₂)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : LeftHomologyData S₁ := by
  let i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
  have wi : i ≫ S₁.g = 0 := by
    rw [assoc, ← cancel_mono φ.τ₃, zero_comp, assoc, assoc, ← φ.comm₂₃,
      IsIso.inv_hom_id_assoc, h.wi]
  have hi : IsLimit (KernelFork.ofι i wi) := KernelFork.IsLimit.ofι _ _
    (fun x hx => h.liftK (x ≫ φ.τ₂)
      (by rw [assoc, φ.comm₂₃, reassoc_of% hx, zero_comp]))
    (fun x hx => by simp )
    (fun x hx b hb => by rw [← cancel_mono h.i, ← cancel_mono (inv φ.τ₂), assoc, assoc,
      hb, liftK_i_assoc, assoc, IsIso.hom_inv_id, comp_id])
  let f' := hi.lift (KernelFork.ofι S₁.f S₁.zero)
  have hf' : f' ≫ i = S₁.f := Fork.IsLimit.lift_ι _
  have hf'' : f' = φ.τ₁ ≫ h.f' := by
    rw [← cancel_mono h.i, ← cancel_mono (inv φ.τ₂), assoc, assoc, assoc, hf', f'_i_assoc,
      φ.comm₁₂_assoc, IsIso.hom_inv_id, comp_id]
  have wπ : f' ≫ h.π = 0 := by 

Proof state:
C : Type u_1
inst✝⁴ : Category.{?u.177302, u_1} C
inst✝³ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h : LeftHomologyData S₂
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
wi : i ≫ S₁.g = 0
hi : IsLimit (KernelFork.ofι i wi)
f' : (KernelFork.ofι S₁.f (_ : S₁.f ≫ S₁.g = 0)).pt ⟶ (KernelFork.ofι i wi).pt :=
  IsLimit.lift hi (KernelFork.ofι S₁.f (_ : S₁.f ≫ S₁.g = 0))
hf' : f' ≫ i = S₁.f
hf'' : f' = φ.τ₁ ≫ CategoryTheory.ShortComplex.LeftHomologyData.f' h
⊢ f' ≫ h.π = 0

Next tactic:
simp only [hf'', assoc, f'_π, comp_zero]
/-- If `φ : S₁ ⟶ S₂` is a morphism of short complexes such that `φ.τ₁` is epi, `φ.τ₂` is an iso
and `φ.τ₃` is mono, then a left homology data for `S₂` induces a left homology data for `S₁` with
the same `K` and `H` fields. The inverse construction is `ofEpiOfIsIsoOfMono`. -/
@[simps]
noncomputable def ofEpiOfIsIsoOfMono' (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₂)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : LeftHomologyData S₁ := by
  let i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
  have wi : i ≫ S₁.g = 0 := by
    rw [assoc, ← cancel_mono φ.τ₃, zero_comp, assoc, assoc, ← φ.comm₂₃,
      IsIso.inv_hom_id_assoc, h.wi]
  have hi : IsLimit (KernelFork.ofι i wi) := KernelFork.IsLimit.ofι _ _
    (fun x hx => h.liftK (x ≫ φ.τ₂)
      (by rw [assoc, φ.comm₂₃, reassoc_of% hx, zero_comp]))
    (fun x hx => by simp )
    (fun x hx b hb => by rw [← cancel_mono h.i, ← cancel_mono (inv φ.τ₂), assoc, assoc,
      hb, liftK_i_assoc, assoc, IsIso.hom_inv_id, comp_id])
  let f' := hi.lift (KernelFork.ofι S₁.f S₁.zero)
  have hf' : f' ≫ i = S₁.f := Fork.IsLimit.lift_ι _
  have hf'' : f' = φ.τ₁ ≫ h.f' := by
    rw [← cancel_mono h.i, ← cancel_mono (inv φ.τ₂), assoc, assoc, assoc, hf', f'_i_assoc,
      φ.comm₁₂_assoc, IsIso.hom_inv_id, comp_id]
  have wπ : f' ≫ h.π = 0 := by simp only [hf'', assoc, f'_π, comp_zero]
  

Proof state:
C : Type u_1
inst✝⁴ : Category.{?u.177302, u_1} C
inst✝³ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h : LeftHomologyData S₂
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
wi : i ≫ S₁.g = 0
hi : IsLimit (KernelFork.ofι i wi)
f' : (KernelFork.ofι S₁.f (_ : S₁.f ≫ S₁.g = 0)).pt ⟶ (KernelFork.ofι i wi).pt :=
  IsLimit.lift hi (KernelFork.ofι S₁.f (_ : S₁.f ≫ S₁.g = 0))
hf' : f' ≫ i = S₁.f
hf'' : f' = φ.τ₁ ≫ CategoryTheory.ShortComplex.LeftHomologyData.f' h
wπ : f' ≫ h.π = 0
⊢ LeftHomologyData S₁

Next tactic:
have hπ : IsColimit (CokernelCofork.ofπ h.π wπ) := CokernelCofork.IsColimit.ofπ _ _
    (fun x hx => h.descH x (by rw [← cancel_epi φ.τ₁, ← reassoc_of% hf'', hx, comp_zero]))
    (fun x hx => π_descH _ _ _)
    (fun x hx b hx => by rw [← cancel_epi h.π, π_descH, hx])
/-- If `φ : S₁ ⟶ S₂` is a morphism of short complexes such that `φ.τ₁` is epi, `φ.τ₂` is an iso
and `φ.τ₃` is mono, then a left homology data for `S₂` induces a left homology data for `S₁` with
the same `K` and `H` fields. The inverse construction is `ofEpiOfIsIsoOfMono`. -/
@[simps]
noncomputable def ofEpiOfIsIsoOfMono' (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₂)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : LeftHomologyData S₁ := by
  let i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
  have wi : i ≫ S₁.g = 0 := by
    rw [assoc, ← cancel_mono φ.τ₃, zero_comp, assoc, assoc, ← φ.comm₂₃,
      IsIso.inv_hom_id_assoc, h.wi]
  have hi : IsLimit (KernelFork.ofι i wi) := KernelFork.IsLimit.ofι _ _
    (fun x hx => h.liftK (x ≫ φ.τ₂)
      (by rw [assoc, φ.comm₂₃, reassoc_of% hx, zero_comp]))
    (fun x hx => by simp )
    (fun x hx b hb => by rw [← cancel_mono h.i, ← cancel_mono (inv φ.τ₂), assoc, assoc,
      hb, liftK_i_assoc, assoc, IsIso.hom_inv_id, comp_id])
  let f' := hi.lift (KernelFork.ofι S₁.f S₁.zero)
  have hf' : f' ≫ i = S₁.f := Fork.IsLimit.lift_ι _
  have hf'' : f' = φ.τ₁ ≫ h.f' := by
    rw [← cancel_mono h.i, ← cancel_mono (inv φ.τ₂), assoc, assoc, assoc, hf', f'_i_assoc,
      φ.comm₁₂_assoc, IsIso.hom_inv_id, comp_id]
  have wπ : f' ≫ h.π = 0 := by simp only [hf'', assoc, f'_π, comp_zero]
  have hπ : IsColimit (CokernelCofork.ofπ h.π wπ) := CokernelCofork.IsColimit.ofπ _ _
    (fun x hx => h.descH x (by 

Proof state:
C : Type u_1
inst✝⁴ : Category.{?u.177302, u_1} C
inst✝³ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h : LeftHomologyData S₂
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
wi : i ≫ S₁.g = 0
hi : IsLimit (KernelFork.ofι i wi)
f' : (KernelFork.ofι S₁.f (_ : S₁.f ≫ S₁.g = 0)).pt ⟶ (KernelFork.ofι i wi).pt :=
  IsLimit.lift hi (KernelFork.ofι S₁.f (_ : S₁.f ≫ S₁.g = 0))
hf' : f' ≫ i = S₁.f
hf'' : f' = φ.τ₁ ≫ CategoryTheory.ShortComplex.LeftHomologyData.f' h
wπ : f' ≫ h.π = 0
Z'✝ : C
x : h.K ⟶ Z'✝
hx : f' ≫ x = 0
⊢ CategoryTheory.ShortComplex.LeftHomologyData.f' h ≫ x = 0

Next tactic:
rw [← cancel_epi φ.τ₁, ← reassoc_of% hf'', hx, comp_zero]
/-- If `φ : S₁ ⟶ S₂` is a morphism of short complexes such that `φ.τ₁` is epi, `φ.τ₂` is an iso
and `φ.τ₃` is mono, then a left homology data for `S₂` induces a left homology data for `S₁` with
the same `K` and `H` fields. The inverse construction is `ofEpiOfIsIsoOfMono`. -/
@[simps]
noncomputable def ofEpiOfIsIsoOfMono' (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₂)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : LeftHomologyData S₁ := by
  let i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
  have wi : i ≫ S₁.g = 0 := by
    rw [assoc, ← cancel_mono φ.τ₃, zero_comp, assoc, assoc, ← φ.comm₂₃,
      IsIso.inv_hom_id_assoc, h.wi]
  have hi : IsLimit (KernelFork.ofι i wi) := KernelFork.IsLimit.ofι _ _
    (fun x hx => h.liftK (x ≫ φ.τ₂)
      (by rw [assoc, φ.comm₂₃, reassoc_of% hx, zero_comp]))
    (fun x hx => by simp )
    (fun x hx b hb => by rw [← cancel_mono h.i, ← cancel_mono (inv φ.τ₂), assoc, assoc,
      hb, liftK_i_assoc, assoc, IsIso.hom_inv_id, comp_id])
  let f' := hi.lift (KernelFork.ofι S₁.f S₁.zero)
  have hf' : f' ≫ i = S₁.f := Fork.IsLimit.lift_ι _
  have hf'' : f' = φ.τ₁ ≫ h.f' := by
    rw [← cancel_mono h.i, ← cancel_mono (inv φ.τ₂), assoc, assoc, assoc, hf', f'_i_assoc,
      φ.comm₁₂_assoc, IsIso.hom_inv_id, comp_id]
  have wπ : f' ≫ h.π = 0 := by simp only [hf'', assoc, f'_π, comp_zero]
  have hπ : IsColimit (CokernelCofork.ofπ h.π wπ) := CokernelCofork.IsColimit.ofπ _ _
    (fun x hx => h.descH x (by rw [← cancel_epi φ.τ₁, ← reassoc_of% hf'', hx, comp_zero]))
    (fun x hx => π_descH _ _ _)
    (fun x hx b hx => by 

Proof state:
C : Type u_1
inst✝⁴ : Category.{?u.177302, u_1} C
inst✝³ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h : LeftHomologyData S₂
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
wi : i ≫ S₁.g = 0
hi : IsLimit (KernelFork.ofι i wi)
f' : (KernelFork.ofι S₁.f (_ : S₁.f ≫ S₁.g = 0)).pt ⟶ (KernelFork.ofι i wi).pt :=
  IsLimit.lift hi (KernelFork.ofι S₁.f (_ : S₁.f ≫ S₁.g = 0))
hf' : f' ≫ i = S₁.f
hf'' : f' = φ.τ₁ ≫ CategoryTheory.ShortComplex.LeftHomologyData.f' h
wπ : f' ≫ h.π = 0
Z'✝ : C
x : h.K ⟶ Z'✝
hx✝ : f' ≫ x = 0
b : h.H ⟶ Z'✝
hx : h.π ≫ b = x
⊢ b = (fun {Z'} x hx => descH h x (_ : CategoryTheory.ShortComplex.LeftHomologyData.f' h ≫ x = 0)) x hx✝

Next tactic:
rw [← cancel_epi h.π, π_descH, hx]