mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Triangulated

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v, u_1} C
inst✝² : Preadditive C
inst✝¹ : HasBinaryBiproducts C
X₁✝ X₂✝ X₃✝ : CochainComplex C ℤ
f : X₁✝ ⟶ X₂✝
g : X₂✝ ⟶ X₃✝
inst✝ : HasZeroObject C
X₁ X₂ X₃ : CochainComplex C ℤ
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
α : mappingCone.triangle u₁₂ ⟶ mappingCone.triangle (u₁₂ ≫ u₂₃) := mappingCone.triangleMap u₁₂ (u₁₂ ≫ u₂₃) (𝟙 X₁) u₂₃ ⋯
β : mappingCone.triangle (u₁₂ ≫ u₂₃) ⟶ mappingCone.triangle u₂₃ := mappingCone.triangleMap (u₁₂ ≫ u₂₃) u₂₃ u₁₂ (𝟙 X₃) ⋯
⊢ ((quotient C (ComplexShape.up ℤ)).mapTriangle.map β).hom₂ ≫
      ((quotient C (ComplexShape.up ℤ)).mapTriangle.obj (mappingCone.triangle u₂₃)).mor₂ =
    (quotient C (ComplexShape.up ℤ)).map (mappingCone.triangle u₂₃).mor₂

Tactic:
dsimp [β]

Goal state after:
C : Type u_1
inst✝³ : Category.{v, u_1} C
inst✝² : Preadditive C
inst✝¹ : HasBinaryBiproducts C
X₁✝ X₂✝ X₃✝ : CochainComplex C ℤ
f : X₁✝ ⟶ X₂✝
g : X₂✝ ⟶ X₃✝
inst✝ : HasZeroObject C
X₁ X₂ X₃ : CochainComplex C ℤ
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
α : mappingCone.triangle u₁₂ ⟶ mappingCone.triangle (u₁₂ ≫ u₂₃) := mappingCone.triangleMap u₁₂ (u₁₂ ≫ u₂₃) (𝟙 X₁) u₂₃ ⋯
β : mappingCone.triangle (u₁₂ ≫ u₂₃) ⟶ mappingCone.triangle u₂₃ := mappingCone.triangleMap (u₁₂ ≫ u₂₃) u₂₃ u₁₂ (𝟙 X₃) ⋯
⊢ (quotient C (ComplexShape.up ℤ)).map (𝟙 X₃) ≫ (quotient C (ComplexShape.up ℤ)).map (mappingCone.inr u₂₃) =
    (quotient C (ComplexShape.up ℤ)).map (mappingCone.inr u₂₃)
Module: Mathlib.Algebra.Homology.Embedding.Boundary

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
k' : ι'
j : ι
hj : c'.Rel (e.f j) k'
hk' : ∀ (i : ι), e.f i ≠ k'
⊢ e.BoundaryLE j

Tactic:
constructor
  · simpa only [c'.next_eq' hj] using hj
  · intro k hk
    apply hk' k
    rw [← c'.next_eq' hj, c'.next_eq' hk]

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

Goal state before:
pp: case pos
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝² : Category.{v_1, u_3} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
K L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
i' j' : ι'
x✝ : c'.Rel i' j'
i : ι
hi : e.f i = i'
hj : ∃ j, e.f j = j'
⊢ extend.mapX φ (e.r i') ≫ (L.extend e).d i' j' = (K.extend e).d i' j' ≫ extend.mapX φ (e.r j')

pp: case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝² : Category.{v_1, u_3} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
K L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
i' j' : ι'
x✝ : c'.Rel i' j'
i : ι
hi : e.f i = i'
hj : ¬∃ j, e.f j = j'
⊢ extend.mapX φ (e.r i') ≫ (L.extend e).d i' j' = (K.extend e).d i' j' ≫ extend.mapX φ (e.r j')

Tactic:
· obtain ⟨j, hj⟩ := hj
        rw [K.extend_d_eq e hi hj, L.extend_d_eq e hi hj,
          extend.mapX_some φ (e.r_eq_some hi),
          extend.mapX_some φ (e.r_eq_some hj)]
        simp only [extendXIso, assoc, Iso.inv_hom_id_assoc, Hom.comm_assoc]

Goal state after:
case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝² : Category.{v_1, u_3} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
K L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
i' j' : ι'
x✝ : c'.Rel i' j'
i : ι
hi : e.f i = i'
hj : ¬∃ j, e.f j = j'
⊢ extend.mapX φ (e.r i') ≫ (L.extend e).d i' j' = (K.extend e).d i' j' ≫ extend.mapX φ (e.r j')
Module: Mathlib.Algebra.Homology.HomotopyCategory.Triangulated

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v, u_1} C
inst✝² : Preadditive C
inst✝¹ : HasBinaryBiproducts C
X₁✝ X₂✝ X₃✝ : CochainComplex C ℤ
f : X₁✝ ⟶ X₂✝
g : X₂✝ ⟶ X₃✝
inst✝ : HasZeroObject C
X₁ X₂ X₃ : CochainComplex C ℤ
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
α : mappingCone.triangle u₁₂ ⟶ mappingCone.triangle (u₁₂ ≫ u₂₃) := mappingCone.triangleMap u₁₂ (u₁₂ ≫ u₂₃) (𝟙 X₁) u₂₃ ⋯
β : mappingCone.triangle (u₁₂ ≫ u₂₃) ⟶ mappingCone.triangle u₂₃ := mappingCone.triangleMap (u₁₂ ≫ u₂₃) u₂₃ u₁₂ (𝟙 X₃) ⋯
⊢ (Triangle.mk ((quotient C (ComplexShape.up ℤ)).map α.hom₃) ((quotient C (ComplexShape.up ℤ)).map β.hom₃)
          (((quotient C (ComplexShape.up ℤ)).map (mappingCone.triangle u₂₃).mor₃ ≫
              (Functor.commShiftIso (quotient C (ComplexShape.up ℤ)) 1).hom.app (mappingCone.triangle u₂₃).obj₁) ≫
            (CategoryTheory.shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) 1).map
              ((quotient C (ComplexShape.up ℤ)).map (mappingCone.triangle u₁₂).mor₂))).mor₁ ≫
      (Iso.refl
          (Triangle.mk ((quotient C (ComplexShape.up ℤ)).map α.hom₃) ((quotient C (ComplexShape.up ℤ)).map β.hom₃)
              (((quotient C (ComplexShape.up ℤ)).map (mappingCone.triangle u₂₃).mor₃ ≫
                  (Functor.commShiftIso (quotient C (ComplexShape.up ℤ)) 1).hom.app (mappingCone.triangle u₂₃).obj₁) ≫
                (CategoryTheory.shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) 1).map
                  ((quotient C (ComplexShape.up ℤ)).map (mappingCone.triangle u₁₂).mor₂))).obj₂).hom =
    (Iso.refl
          (Triangle.mk ((quotient C (ComplexShape.up ℤ)).map α.hom₃) ((quotient C (ComplexShape.up ℤ)).map β.hom₃)
              (((quotient C (ComplexShape.up ℤ)).map (mappingCone.triangle u₂₃).mor₃ ≫
                  (Functor.commShiftIso (quotient C (ComplexShape.up ℤ)) 1).hom.app (mappingCone.triangle u₂₃).obj₁) ≫
                (CategoryTheory.shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) 1).map
                  ((quotient C (ComplexShape.up ℤ)).map (mappingCone.triangle u₁₂).mor₂))).obj₁).hom ≫
      (mappingConeCompTriangleh u₁₂ u₂₃).mor₁

Tactic:
dsimp [α, mappingConeCompTriangleh]

Goal state after:
C : Type u_1
inst✝³ : Category.{v, u_1} C
inst✝² : Preadditive C
inst✝¹ : HasBinaryBiproducts C
X₁✝ X₂✝ X₃✝ : CochainComplex C ℤ
f : X₁✝ ⟶ X₂✝
g : X₂✝ ⟶ X₃✝
inst✝ : HasZeroObject C
X₁ X₂ X₃ : CochainComplex C ℤ
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
α : mappingCone.triangle u₁₂ ⟶ mappingCone.triangle (u₁₂ ≫ u₂₃) := mappingCone.triangleMap u₁₂ (u₁₂ ≫ u₂₃) (𝟙 X₁) u₂₃ ⋯
β : mappingCone.triangle (u₁₂ ≫ u₂₃) ⟶ mappingCone.triangle u₂₃ := mappingCone.triangleMap (u₁₂ ≫ u₂₃) u₂₃ u₁₂ (𝟙 X₃) ⋯
⊢ (quotient C (ComplexShape.up ℤ)).map (mappingCone.map u₁₂ (u₁₂ ≫ u₂₃) (𝟙 X₁) u₂₃ ⋯) ≫
      𝟙 ((quotient C (ComplexShape.up ℤ)).obj (mappingCone (u₁₂ ≫ u₂₃))) =
    𝟙 ((quotient C (ComplexShape.up ℤ)).obj (mappingCone u₁₂)) ≫
      (quotient C (ComplexShape.up ℤ)).map (mappingCone.map u₁₂ (u₁₂ ≫ u₂₃) (𝟙 X₁) u₂₃ ⋯)
Module: Mathlib.Algebra.Homology.Embedding.Boundary

Goal state before:
pp: case mpr.right
p : ℤ
⊢ ∀ (i : ℕ), ¬(up ℤ).Rel ((embeddingUpIntGE p).f i) ((embeddingUpIntGE p).f 0)

Tactic:
intro i hi

Goal state after:
case mpr.right
p : ℤ
i : ℕ
hi : (up ℤ).Rel ((embeddingUpIntGE p).f i) ((embeddingUpIntGE p).f 0)
⊢ False
Module: Mathlib.Algebra.Homology.Embedding.Boundary

Goal state before:
pp: case mpr.right
p : ℤ
⊢ ∀ (k : ℕ), ¬(up ℤ).Rel ((embeddingUpIntLE p).f 0) ((embeddingUpIntLE p).f k)

Tactic:
intro i hi

Goal state after:
case mpr.right
p : ℤ
i : ℕ
hi : (up ℤ).Rel ((embeddingUpIntLE p).f 0) ((embeddingUpIntLE p).f i)
⊢ False
Module: Mathlib.Algebra.Homology.Embedding.TruncLE

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁵ : Category.{v_1, u_3} C
inst✝⁴ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝³ : e.IsTruncLE
inst✝² : ∀ (i' : ι'), K.HasHomology i'
inst✝¹ : ∀ (i' : ι'), L.HasHomology i'
inst✝ : ∀ (i' : ι'), M.HasHomology i'
i : ι
i' : ι'
hi' : e.f i = i'
hi : ¬e.BoundaryLE i
⊢ ¬e.op.BoundaryGE i

Tactic:
simpa

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁵ : Category.{v_1, u_3} C
inst✝⁴ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝³ : e.IsTruncLE
inst✝² : ∀ (i' : ι'), K.HasHomology i'
inst✝¹ : ∀ (i' : ι'), L.HasHomology i'
inst✝ : ∀ (i' : ι'), M.HasHomology i'
i : ι
i' : ι'
hi' : e.f i = i'
hi : e.BoundaryLE i
⊢ e.op.BoundaryGE i

Tactic:
simpa

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝¹ : e.IsTruncLE
inst✝ : ∀ (i' : ι'), K.HasHomology i'
i j : ι
hij : c.Rel i j
i' j' : ι'
hi' : e.f i = i'
hj' : e.f j = j'
hj : ¬e.BoundaryLE j
⊢ ¬e.op.BoundaryGE j

Tactic:
simpa

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝¹ : e.IsTruncLE
inst✝ : ∀ (i' : ι'), K.HasHomology i'
i j : ι
hij : c.Rel i j
i' j' : ι'
hi' : e.f i = i'
hj' : e.f j = j'
hj : e.BoundaryLE j
⊢ e.op.BoundaryGE j

Tactic:
simpa

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁶ : Category.{v_1, u_3} C
inst✝⁵ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝⁴ : e.IsTruncLE
inst✝³ : ∀ (i' : ι'), K.HasHomology i'
inst✝² : ∀ (i' : ι'), L.HasHomology i'
inst✝¹ : ∀ (i' : ι'), M.HasHomology i'
inst✝ : HasZeroObject C
i : ι
i' : ι'
hi' : e.f i = i'
hi : ¬e.BoundaryLE i
⊢ ¬e.op.BoundaryGE i

Tactic:
simpa

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁶ : Category.{v_1, u_3} C
inst✝⁵ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝⁴ : e.IsTruncLE
inst✝³ : ∀ (i' : ι'), K.HasHomology i'
inst✝² : ∀ (i' : ι'), L.HasHomology i'
inst✝¹ : ∀ (i' : ι'), M.HasHomology i'
inst✝ : HasZeroObject C
i : ι
i' : ι'
hi' : e.f i = i'
hi : e.BoundaryLE i
⊢ e.op.BoundaryGE i

Tactic:
simpa

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁴ : Category.{v_1, u_3} C
inst✝³ : HasZeroMorphisms C
K L : HomologicalComplex C c'
φ : K ⟶ L
e : c.Embedding c'
inst✝² : e.IsTruncLE
inst✝¹ : ∀ (i' : ι'), K.HasHomology i'
inst✝ : ∀ (i' : ι'), L.HasHomology i'
i : ι
hi : e.BoundaryLE i
i' : ι'
h : e.f i = i'
⊢ e.op.BoundaryGE i

Tactic:
simpa

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁴ : Category.{v_1, u_3} C
inst✝³ : HasZeroMorphisms C
K L : HomologicalComplex C c'
φ : K ⟶ L
e : c.Embedding c'
inst✝² : e.IsTruncLE
inst✝¹ : ∀ (i' : ι'), K.HasHomology i'
inst✝ : ∀ (i' : ι'), L.HasHomology i'
i : ι
hi : ¬e.BoundaryLE i
i' : ι'
h : e.f i = i'
⊢ ¬e.op.BoundaryGE i

Tactic:
simpa

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝¹ : e.IsTruncLE
inst✝ : ∀ (i' : ι'), K.HasHomology i'
i : ι
hi : ¬e.BoundaryLE i
⊢ ¬e.op.BoundaryGE i

Tactic:
simpa

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁵ : Category.{v_1, u_3} C
inst✝⁴ : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝³ : e.IsTruncLE
inst✝² : ∀ (i' : ι'), K.HasHomology i'
j : ι
hj : ¬e.BoundaryLE j
inst✝¹ : (K.restriction e).HasHomology j
inst✝ : (K.truncLE' e).HasHomology j
this : (K.op.restriction e.op).HasHomology j
⊢ ¬e.op.BoundaryGE j

Tactic:
simpa

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

Goal state before:
pp: ι : Type u_1
ι₁ : Type u_2
ι₂ : Type u_3
c : ComplexShape ι
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
C : Type u_4
inst✝¹ : Category.{v_1, u_4} C
inst✝ : HasZeroMorphisms C
e₁ : c₁.Embedding c
e₂ : c₂.Embedding c
ac : e₁.AreComplementary e₂
X : ι → Type u_5
x₁ : (i₁ : ι₁) → X (e₁.f i₁)
x₂ : (i₂ : ι₂) → X (e₂.f i₂)
i : ι₁ ⊕ ι₂
i₁ : ι₁
h : Sum.inl i₁ = i
⊢ e₁.f i₁ = ac.equiv i

Tactic:
by subst h; simp

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

Goal state before:
pp: ι : Type u_1
ι₁ : Type u_2
ι₂ : Type u_3
c : ComplexShape ι
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
C : Type u_4
inst✝¹ : Category.{v_1, u_4} C
inst✝ : HasZeroMorphisms C
e₁ : c₁.Embedding c
e₂ : c₂.Embedding c
ac : e₁.AreComplementary e₂
X : ι → Type u_5
x₁ : (i₁ : ι₁) → X (e₁.f i₁)
x₂ : (i₂ : ι₂) → X (e₂.f i₂)
i : ι₁ ⊕ ι₂
i₂ : ι₂
h : Sum.inr i₂ = i
⊢ e₂.f i₂ = ac.equiv i

Tactic:
by subst h; simp

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

Goal state before:
pp: case mp
ι : Type u_1
ι₁ : Type u_2
ι₂ : Type u_3
c : ComplexShape ι
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
C : Type u_4
inst✝¹ : Category.{v_1, u_4} C
inst✝ : HasZeroMorphisms C
e₁ : c₁.Embedding c
e₂ : c₂.Embedding c
ac : e₁.AreComplementary e₂
K : HomologicalComplex C c
⊢ K.IsSupportedOutside e₁ → K.IsSupported e₂

Tactic:
intro h
    exact ⟨fun i hi => by
      obtain ⟨i₁, rfl⟩ := ac.exists_i₁ i hi
      exact h.exactAt i₁⟩

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

Goal state before:
pp: case neg
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
i : ℤ
hi : ¬i ≤ n₀
⊢ (∃ i₁, (embeddingUpIntLE n₀).f i₁ = i) ∨ ∃ i₂, (embeddingUpIntGE n₁).f i₂ = i

Tactic:
obtain ⟨k, rfl⟩ := Int.exists_add_of_le (show n₁ ≤ i by lia)

Goal state after:
case neg
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
k : ℕ
hi : ¬n₁ + ↑k ≤ n₀
⊢ (∃ i₁, (embeddingUpIntLE n₀).f i₁ = n₁ + ↑k) ∨ ∃ i₂, (embeddingUpIntGE n₁).f i₂ = n₁ + ↑k
Module: Mathlib.Algebra.Homology.Embedding.CochainComplex

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
K : CochainComplex C ℤ
p q : ℤ
hpq : p ≤ q
inst✝ : K.IsStrictlyLE p
⊢ ∀ (i : ℤ), q < i → IsZero (K.X i)

Tactic:
intro i hi

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
K : CochainComplex C ℤ
p q : ℤ
hpq : p ≤ q
inst✝ : K.IsStrictlyLE p
i : ℤ
hi : q < i
⊢ IsZero (K.X i)
Module: Mathlib.Algebra.Homology.Embedding.CochainComplex

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
K : CochainComplex C ℤ
p q : ℤ
hpq : p ≤ q
inst✝ : K.IsStrictlyGE q
⊢ ∀ (i : ℤ), autoParam (i < p) isStrictlyGE_iff._auto_1 → IsZero (K.X i)

Tactic:
intro i hi

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
K : CochainComplex C ℤ
p q : ℤ
hpq : p ≤ q
inst✝ : K.IsStrictlyGE q
i : ℤ
hi : autoParam (i < p) isStrictlyGE_iff._auto_1
⊢ IsZero (K.X i)
Module: Mathlib.Algebra.Homology.Embedding.CochainComplex

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
K : CochainComplex C ℤ
p q : ℤ
hpq : p ≤ q
inst✝ : K.IsLE p
⊢ ∀ (i : ℤ), q < i → ExactAt K i

Tactic:
intro i hi

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
K : CochainComplex C ℤ
p q : ℤ
hpq : p ≤ q
inst✝ : K.IsLE p
i : ℤ
hi : q < i
⊢ ExactAt K i
Module: Mathlib.Algebra.Homology.Embedding.CochainComplex

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
K : CochainComplex C ℤ
p q : ℤ
hpq : p ≤ q
inst✝ : K.IsGE q
⊢ ∀ i < p, ExactAt K i

Tactic:
intro i hi

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
K : CochainComplex C ℤ
p q : ℤ
hpq : p ≤ q
inst✝ : K.IsGE q
i : ℤ
hi : i < p
⊢ ExactAt K i
Module: Mathlib.Algebra.Homology.Embedding.CochainComplex

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
K L : CochainComplex C ℤ
φ : K ⟶ L
e : K ≅ L
inst✝ : HasZeroObject C
A : C
n : ℤ
⊢ ∀ (i : ℤ), autoParam (i < n) isStrictlyGE_iff._auto_1 → IsZero (((single C (up ℤ) n).obj A).X i)

Tactic:
intro i hi

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
K L : CochainComplex C ℤ
φ : K ⟶ L
e : K ≅ L
inst✝ : HasZeroObject C
A : C
n i : ℤ
hi : autoParam (i < n) isStrictlyGE_iff._auto_1
⊢ IsZero (((single C (up ℤ) n).obj A).X i)
Module: Mathlib.Algebra.Homology.Embedding.CochainComplex

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
K L : CochainComplex C ℤ
φ : K ⟶ L
e : K ≅ L
inst✝ : HasZeroObject C
A : C
n : ℤ
⊢ ∀ (i : ℤ), n < i → IsZero (((single C (up ℤ) n).obj A).X i)

Tactic:
intro i hi

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
K L : CochainComplex C ℤ
φ : K ⟶ L
e : K ≅ L
inst✝ : HasZeroObject C
A : C
n i : ℤ
hi : n < i
⊢ IsZero (((single C (up ℤ) n).obj A).X i)
Module: Mathlib.Algebra.Homology.SingleHomology

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A B : C
f : A ⟶ B
⊢ (singleObjHomologySelfIso c j A).inv ≫ homologyMap ((single C c j).map f) j = f ≫ (singleObjHomologySelfIso c j B).inv

Tactic:
by
  rw [← cancel_mono (singleObjHomologySelfIso c j B).hom, assoc, assoc,
    singleObjHomologySelfIso_hom_naturality,
    Iso.inv_hom_id_assoc, Iso.inv_hom_id, comp_id]

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : CochainComplex C ℤ
n : ℤ
inst✝ : K.IsStrictlyLE n
a n' : ℤ
h : a + n' = n
⊢ ∀ (i : ℤ), n' < i → IsZero (((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K).X i)

Tactic:
intro i hi

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : CochainComplex C ℤ
n : ℤ
inst✝ : K.IsStrictlyLE n
a n' : ℤ
h : a + n' = n
i : ℤ
hi : n' < i
⊢ IsZero (((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K).X i)
Module: Mathlib.Algebra.Homology.Embedding.CochainComplex

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : CochainComplex C ℤ
n : ℤ
inst✝ : K.IsStrictlyGE n
a n' : ℤ
h : a + n' = n
⊢ ∀ (i : ℤ),
    autoParam (i < n') isStrictlyGE_iff._auto_1 →
      IsZero (((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K).X i)

Tactic:
intro i hi

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : CochainComplex C ℤ
n : ℤ
inst✝ : K.IsStrictlyGE n
a n' : ℤ
h : a + n' = n
i : ℤ
hi : autoParam (i < n') isStrictlyGE_iff._auto_1
⊢ IsZero (((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K).X i)
Module: Mathlib.Algebra.Homology.Embedding.CochainComplex

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
K : CochainComplex C ℤ
inst✝¹ : CategoryWithHomology C
n : ℤ
inst✝ : K.IsLE n
a n' : ℤ
h : a + n' = n
⊢ ∀ (i : ℤ), n' < i → ExactAt ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) i

Tactic:
intro i hi

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
K : CochainComplex C ℤ
inst✝¹ : CategoryWithHomology C
n : ℤ
inst✝ : K.IsLE n
a n' : ℤ
h : a + n' = n
i : ℤ
hi : n' < i
⊢ ExactAt ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) i
Module: Mathlib.Algebra.Homology.Embedding.CochainComplex

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
K : CochainComplex C ℤ
inst✝¹ : CategoryWithHomology C
n : ℤ
inst✝ : K.IsGE n
a n' : ℤ
h : a + n' = n
⊢ ∀ i < n', ExactAt ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) i

Tactic:
intro i hi

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
K : CochainComplex C ℤ
inst✝¹ : CategoryWithHomology C
n : ℤ
inst✝ : K.IsGE n
a n' : ℤ
h : a + n' = n
i : ℤ
hi : i < n'
⊢ ExactAt ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) i
Module: Mathlib.Algebra.Homology.Embedding.CochainComplex

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
K : CochainComplex C ℤ
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
⊢ (K.shortComplexTruncLE n₀).g ≫ K.shortComplexTruncLEX₃ToTruncGE n₀ n₁ h = K.πTruncGE n₁

Tactic:
by
  apply HomologicalComplex.g_shortComplexTruncLEX₃ToTruncGE

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

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Abelian C
K : CochainComplex C ℤ
n₀ n₁ : ℤ
inst✝² : Injective (K.X n₀)
inst✝¹ : Injective (K.X n₁)
inst✝ : K.IsStrictlyGE n₀
hK : ExactAt K n₀
h : n₀ + 1 = n₁
S : ShortComplex C :=
  { X₁ := K.X n₀, X₂ := K.X n₁, X₃ := opcycles K n₁, f := K.d n₀ n₁, g := pOpcycles K n₁, zero := ⋯ }
T : ShortComplex C := sc' K (n₀ - 1) n₀ n₁
⊢ (up ℤ).next n₀ = n₁

Tactic:
simpa

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.Fractions

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Abelian C
inst✝³ : HasDerivedCategory C
X Y : CochainComplex C ℤ
a b : ℤ
inst✝² : X.IsStrictlyLE b
inst✝¹ : Y.IsStrictlyGE a
inst✝ : Y.IsStrictlyLE b
f : Q.obj X ⟶ Q.obj Y
⊢ ∃ Y', ∃ (_ : Y'.IsStrictlyGE a) (_ : Y'.IsStrictlyLE b), ∃ g s, ∃ (x : IsIso (Q.map s)), f = Q.map g ≫ inv (Q.map s)

Tactic:
by
  obtain ⟨Y', hY', g, s, hs, fac⟩ := left_fac_of_isStrictlyGE f a
  have : IsIso (Q.map (CochainComplex.truncLEMap s b)) := by
    rw [isIso_Q_map_iff_quasiIso] at hs
    rw [isIso_Q_map_iff_quasiIso, CochainComplex.quasiIso_truncLEMap_iff]
    infer_instance
  refine ⟨Y'.truncLE b, inferInstance, inferInstance,
    inv (X.ιTruncLE b) ≫ CochainComplex.truncLEMap g b,
    inv (Y.ιTruncLE b) ≫ CochainComplex.truncLEMap s b, ?_, ?_⟩
  · rw [Q.map_comp]
    infer_instance
  · simp only [Functor.map_comp, Functor.map_inv, IsIso.inv_comp, IsIso.inv_inv, assoc, fac,
      ← cancel_mono (Q.map s), IsIso.inv_hom_id, comp_id]
    rw [← Functor.map_comp, ← CochainComplex.ιTruncLE_naturality s b,
      Functor.map_comp, IsIso.inv_hom_id_assoc,
      ← Functor.map_comp, CochainComplex.ιTruncLE_naturality g b,
      Functor.map_comp, IsIso.inv_hom_id_assoc]

Goal state after:
Module: Mathlib.CategoryTheory.Localization.SmallHom

Goal state before:
pp: C₁ : Type u₁
inst✝⁵ : Category.{v₁, u₁} C₁
W₁ : MorphismProperty C₁
C₂ : Type u₂
inst✝⁴ : Category.{v₂, u₂} C₂
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
X Y : C₁
inst✝³ : HasSmallLocalizedHom W₁ X Y
X' Y' : C₂
inst✝² : HasSmallLocalizedHom W₂ X' X'
inst✝¹ : HasSmallLocalizedHom W₂ X' Y'
inst✝ : HasSmallLocalizedHom W₂ Y' Y'
eX : Φ.functor.obj X ≅ X'
eY : Φ.functor.obj Y ≅ Y'
f : X ⟶ Y
⊢ Φ.smallHomMap' eX eY (SmallHom.mk W₁ f) = SmallHom.mk W₂ (eX.inv ≫ Φ.functor.map f ≫ eY.hom)

Tactic:
simp [smallHomMap', SmallHom.mk_comp_mk]

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

Goal state before:
pp: case pos.hR₂
C : Type u_1
ι : Type u_2
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
c : ComplexShape ι
S₁ S₂ : ShortComplex (HomologicalComplex C c)
φ : S₁ ⟶ S₂
hS₁ : S₁.ShortExact
hS₂ : S₂.ShortExact
i : ι
h₁ : Epi (homologyMap φ.τ₂ i)
h₂ : ∀ (j : ι), c.Rel i j → Epi (homologyMap φ.τ₁ j)
h₃ : ∀ (j : ι), c.Rel i j → Mono (homologyMap φ.τ₂ j)
j : ι
hij : c.Rel i j
⊢ ((δ₀Functor ⋙ δlastFunctor).obj (composableArrows₅ hS₂ i j hij)).Exact

Tactic:
exact (composableArrows₅_exact hS₂ i j hij).δ₀.δlast

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
h :
  ∀ (X Y : C) (n : ℤ),
    0 ≤ n →
      Small.{w, w'} ((singleFunctor C 0).obj X ⟶ (shiftFunctor (DerivedCategory C) n).obj ((singleFunctor C 0).obj Y))
X Y : C
a b : ℤ
hab : a ≤ b
⊢ 0 ≤ b - a

Tactic:
simpa

Goal state after: