mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplices

Goal state before:
pp: X : SSet
x : X.S
y₁ y₂ : X.N
h₁ : y₁.subcomplex = x.subcomplex
h₂ : y₂.subcomplex = x.subcomplex
⊢ y₁.subcomplex = y₂.subcomplex

Tactic:
h₁,

Goal state after:
X : SSet
x : X.S
y₁ y₂ : X.N
h₁ : y₁.subcomplex = x.subcomplex
h₂ : y₂.subcomplex = x.subcomplex
⊢ x.subcomplex = y₂.subcomplex
Module: Mathlib.Algebra.Homology.Augment

Goal state before:
pp: V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : ChainComplex V ℕ
X : V
f : C.X 0 ⟶ X
w : C.d 1 0 ≫ f = 0
i j : ℕ
h : ¬(ComplexShape.down ℕ).Rel (i + 1) (j + 1)
⊢ C.d i j = 0

Tactic:
exact C.shape i j (Nat.succ_ne_succ_iff.1 h)

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: n : ℕ
S : Finset (Fin (n + 1))
i : Fin (n + 1)
h : obj₀Equiv.symm i ∈ (face S).obj (op ⦋0⦌)
⊢ i ∈ S

Tactic:
simpa using! h

Goal state after:
Module: Mathlib.AlgebraicTopology.ExtraDegeneracy

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : Augmented C
ed : X.ExtraDegeneracy
n : ℕ
⊢ X.left.δ₀Iter ↑(Fin.last n) ⋯ ≫ X.hom.app (op ⦋0⦌) ≫ ed.s' ≫ X.left.σ₀Iter ↑(Fin.last n) ⋯ =
    X.hom.app (op ⦋n⦌) ≫ ed.section_.app (op ⦋n⦌)

Tactic:
X.δ₀Iter_hom_app_assoc _ (by grind)

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : Augmented C
ed : X.ExtraDegeneracy
n : ℕ
⊢ X.hom.app (op ⦋n⦌) ≫ ed.s' ≫ X.left.σ₀Iter ↑(Fin.last n) ⋯ = X.hom.app (op ⦋n⦌) ≫ ed.section_.app (op ⦋n⦌)
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: n : ℕ
i j : Fin (n + 2)
h : i < j
⊢ Fin n ≃ ↥{i, j}ᶜ

Tactic:
refine Equiv.ofBijective
      (fun k ↦ ⟨j.succAbove ((i.castPred (Fin.ne_last_of_lt h)).succAbove k), ?_⟩)
        ⟨fun _ _ hk ↦ ?_, fun ⟨l, hl⟩ ↦ ?_⟩

Goal state after:
case refine_1
n : ℕ
i j : Fin (n + 2)
h : i < j
k : Fin n
⊢ j.succAbove ((i.castPred ⋯).succAbove k) ∈ {i, j}ᶜ

case refine_2
n : ℕ
i j : Fin (n + 2)
h : i < j
x✝¹ x✝ : Fin n
hk :
  (fun k ↦ ⟨j.succAbove ((i.castPred ⋯).succAbove k), ⋯⟩) x✝¹ =
    (fun k ↦ ⟨j.succAbove ((i.castPred ⋯).succAbove k), ⋯⟩) x✝
⊢ x✝¹ = x✝

case refine_3
n : ℕ
i j : Fin (n + 2)
h : i < j
x✝ : ↥{i, j}ᶜ
l : Fin (n + 2)
hl : l ∈ {i, j}ᶜ
⊢ ∃ a, (fun k ↦ ⟨j.succAbove ((i.castPred ⋯).succAbove k), ⋯⟩) a = ⟨l, hl⟩
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: n m : ℕ
⊢ Function.Bijective fun x ↦
    match x with
    | ⟨x, hx⟩ => ⟨Finset.image (⇑(Hom.toOrderHom (objEquiv x))) univ, ⋯⟩

Tactic:
by
  constructor
  · rintro ⟨x₁, h₁⟩ ⟨x₂, h₂⟩ h₃
    obtain ⟨f₁, rfl⟩ := objEquiv.symm.surjective x₁
    obtain ⟨f₂, rfl⟩ := objEquiv.symm.surjective x₂
    simp only [mem_nonDegenerate_iff_mono, Equiv.apply_symm_apply,
      SimplexCategory.mono_iff_injective, SimplexCategory.len_mk] at h₁ h₂
    simp only [Set.mem_ofPred_eq, SimplexCategory.len_mk, Equiv.apply_symm_apply,
      Subtype.mk.injEq, EmbeddingLike.apply_eq_iff_eq] at h₃ ⊢
    apply SimplexCategory.Hom.ext
    rw [← OrderHom.range_eq_iff h₁ h₂]
    ext x
    simpa using congr_fun (congrArg Membership.mem h₃) x
  · intro ⟨S, hS⟩
    dsimp at hS
    let e := monoEquivOfFin S (k := m + 1) (by simpa using hS)
    refine ⟨⟨objMk ((OrderHom.Subtype.val _).comp e.toOrderEmbedding.toOrderHom), ?_⟩, ?_⟩
    · rw [mem_nonDegenerate_iff_mono, SimplexCategory.mono_iff_injective]
      intro a b h
      grind [e.injective, dsimp% h]
    · simp [e, ← Finset.image_image, Finset.image_univ_of_surjective e.surjective]

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

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝⁹ : Category.{v_1, u_1} C₁
inst✝⁸ : Category.{v_2, u_2} C₂
inst✝⁷ : Category.{v_3, u_5} C₃
inst✝⁶ : Category.{v_4, u_6} C₄
inst✝⁵ : Category.{v_5, u_3} C₁₂
inst✝⁴ : Category.{v_6, u_4} C₂₃
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
associator : bifunctorComp₁₂ F₁₂ G ≅ bifunctorComp₂₃ F G₂₃
I₁ : Type u_7
I₂ : Type u_8
I₃ : Type u_9
J : Type u_10
r : I₁ × I₂ × I₃ → J
ρ₁₂ : BifunctorComp₁₂IndexData r
ρ₂₃ : BifunctorComp₂₃IndexData r
X₁ : GradedObject I₁ C₁
X₂ : GradedObject I₂ C₂
X₃ : GradedObject I₃ C₃
inst✝³ : (((mapBifunctor F₁₂ I₁ I₂).obj X₁).obj X₂).HasMap ρ₁₂.p
inst✝² : (((mapBifunctor G ρ₁₂.I₁₂ I₃).obj (mapBifunctorMapObj F₁₂ ρ₁₂.p X₁ X₂)).obj X₃).HasMap ρ₁₂.q
inst✝¹ : (((mapBifunctor G₂₃ I₂ I₃).obj X₂).obj X₃).HasMap ρ₂₃.p
inst✝ : (((mapBifunctor F I₁ ρ₂₃.I₂₃).obj X₁).obj (mapBifunctorMapObj G₂₃ ρ₂₃.p X₂ X₃)).HasMap ρ₂₃.q
H₁₂ : HasGoodTrifunctor₁₂Obj F₁₂ G ρ₁₂ X₁ X₂ X₃
H₂₃ : HasGoodTrifunctor₂₃Obj F G₂₃ ρ₂₃ X₁ X₂ X₃
i₁ : I₁
i₂ : I₂
i₃ : I₃
j : J
h : r (i₁, i₂, i₃) = j
⊢ ιMapBifunctorBifunctor₂₃MapObj F G₂₃ ρ₂₃ X₁ X₂ X₃ i₁ i₂ i₃ j h ≫
      (mapBifunctorAssociator associator ρ₁₂ ρ₂₃ X₁ X₂ X₃).inv j =
    ((associator.inv.app (X₁ i₁)).app (X₂ i₂)).app (X₃ i₃) ≫
      ιMapBifunctor₁₂BifunctorMapObj F₁₂ G ρ₁₂ X₁ X₂ X₃ i₁ i₂ i₃ j h

Tactic:
rw [← cancel_mono ((mapBifunctorAssociator associator ρ₁₂ ρ₂₃ X₁ X₂ X₃).hom j),
    assoc, assoc, Iso.inv_hom_id_eval, comp_id, ι_mapBifunctorAssociator_hom,
    ← NatTrans.comp_app_assoc, ← NatTrans.comp_app, Iso.inv_hom_id_app,
    NatTrans.id_app, NatTrans.id_app, id_comp]

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

Goal state before:
pp: C : Type u_1
I₁ : Type u_2
I₂ : Type u_3
J : Type u_4
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c : ComplexShape J
inst✝⁴ : TotalComplexShape c₁ c₂ c
inst✝³ : TotalComplexShape c₂ c₁ c
inst✝² : TotalComplexShapeSymmetry c₁ c₂ c
inst✝¹ : K.HasTotal c
inst✝ : DecidableEq J
j j' : J
h₀ : c.Rel j j'
i₂ : I₂
i₁ : I₁
h₁ : c₂.π c₁ c (i₂, i₁) = j
h₂ : c₁.Rel i₁ (c₁.next i₁)
⊢ c.next (c₂.π c₁ c (i₂, i₁)) = j'

Tactic:
h₁,

Goal state after:
C : Type u_1
I₁ : Type u_2
I₂ : Type u_3
J : Type u_4
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c : ComplexShape J
inst✝⁴ : TotalComplexShape c₁ c₂ c
inst✝³ : TotalComplexShape c₂ c₁ c
inst✝² : TotalComplexShapeSymmetry c₁ c₂ c
inst✝¹ : K.HasTotal c
inst✝ : DecidableEq J
j j' : J
h₀ : c.Rel j j'
i₂ : I₂
i₁ : I₁
h₁ : c₂.π c₁ c (i₂, i₁) = j
h₂ : c₁.Rel i₁ (c₁.next i₁)
⊢ c.next j = j'
Module: Mathlib.Algebra.Homology.TotalComplex

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Preadditive C
I₁ : Type u_2
I₂ : Type u_3
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K L M : HomologicalComplex₂ C c₁ c₂
φ : K ⟶ L
e : K ≅ L
ψ : L ⟶ M
c₁₂ : ComplexShape I₁₂
inst✝² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹ : DecidableEq I₁₂
inst✝ : K.HasTotal c₁₂
x₁ y₁ : I₁
h₁ : x₁ = y₁
x₂ y₂ : I₂
h₂ : x₂ = y₂
i₁₂ : I₁₂
h : c₁.π c₂ c₁₂ (y₁, y₂) = i₁₂
⊢ c₁.π c₂ c₁₂ (x₁, x₂) = i₁₂

Tactic:
h₁,

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Preadditive C
I₁ : Type u_2
I₂ : Type u_3
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K L M : HomologicalComplex₂ C c₁ c₂
φ : K ⟶ L
e : K ≅ L
ψ : L ⟶ M
c₁₂ : ComplexShape I₁₂
inst✝² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹ : DecidableEq I₁₂
inst✝ : K.HasTotal c₁₂
x₁ y₁ : I₁
h₁ : x₁ = y₁
x₂ y₂ : I₂
h₂ : x₂ = y₂
i₁₂ : I₁₂
h : c₁.π c₂ c₁₂ (y₁, y₂) = i₁₂
⊢ c₁.π c₂ c₁₂ (y₁, x₂) = i₁₂
Module: Mathlib.Algebra.Homology.TotalComplex

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Preadditive C
I₁ : Type u_2
I₂ : Type u_3
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K L M : HomologicalComplex₂ C c₁ c₂
φ : K ⟶ L
e : K ≅ L
ψ : L ⟶ M
c₁₂ : ComplexShape I₁₂
inst✝² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹ : DecidableEq I₁₂
inst✝ : K.HasTotal c₁₂
x₁ y₁ : I₁
h₁ : x₁ = y₁
x₂ y₂ : I₂
h₂ : x₂ = y₂
i₁₂ : I₁₂
h : c₁.π c₂ c₁₂ (x₁, x₂) = i₁₂
⊢ c₁.π c₂ c₁₂ (y₁, y₂) = c₁.π c₂ c₁₂ (x₁, x₂)

Tactic:
h₁,

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Preadditive C
I₁ : Type u_2
I₂ : Type u_3
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K L M : HomologicalComplex₂ C c₁ c₂
φ : K ⟶ L
e : K ≅ L
ψ : L ⟶ M
c₁₂ : ComplexShape I₁₂
inst✝² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹ : DecidableEq I₁₂
inst✝ : K.HasTotal c₁₂
x₁ y₁ : I₁
h₁ : x₁ = y₁
x₂ y₂ : I₂
h₂ : x₂ = y₂
i₁₂ : I₁₂
h : c₁.π c₂ c₁₂ (x₁, x₂) = i₁₂
⊢ c₁.π c₂ c₁₂ (y₁, y₂) = c₁.π c₂ c₁₂ (y₁, x₂)
Module: Mathlib.Algebra.Homology.TotalComplexSymmetry

Goal state before:
pp: C : Type u_1
I₁ : Type u_2
I₂ : Type u_3
J : Type u_4
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c : ComplexShape J
inst✝⁴ : TotalComplexShape c₁ c₂ c
inst✝³ : TotalComplexShape c₂ c₁ c
inst✝² : TotalComplexShapeSymmetry c₁ c₂ c
inst✝¹ : K.HasTotal c
inst✝ : DecidableEq J
j j' : J
h₀ : c.Rel j j'
i₂ : I₂
i₁ : I₁
h₁ : c₂.π c₁ c (i₂, i₁) = j
h₂ : c₂.Rel i₂ (c₂.next i₂)
⊢ c.next (c₂.π c₁ c (i₂, i₁)) = j'

Tactic:
h₁,

Goal state after:
C : Type u_1
I₁ : Type u_2
I₂ : Type u_3
J : Type u_4
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c : ComplexShape J
inst✝⁴ : TotalComplexShape c₁ c₂ c
inst✝³ : TotalComplexShape c₂ c₁ c
inst✝² : TotalComplexShapeSymmetry c₁ c₂ c
inst✝¹ : K.HasTotal c
inst✝ : DecidableEq J
j j' : J
h₀ : c.Rel j j'
i₂ : I₂
i₁ : I₁
h₁ : c₂.π c₁ c (i₂, i₁) = j
h₂ : c₂.Rel i₂ (c₂.next i₂)
⊢ c.next j = j'
Module: Mathlib.Algebra.Homology.BifunctorHomotopy

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
inst✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
inst✝⁶ : Preadditive D
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K₁ L₁ : HomologicalComplex C₁ c₁
f₁ f₁' : K₁ ⟶ L₁
h₁ : Homotopy f₁ f₁'
K₂ L₂ : HomologicalComplex C₂ c₂
f₂ : K₂ ⟶ L₂
F : C₁ ⥤ C₂ ⥤ D
inst✝⁵ : F.Additive
inst✝⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
c : ComplexShape J
inst✝³ : DecidableEq J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : L₁.HasMapBifunctor L₂ F c
j : J
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c (i₁, i₂) = j
this : ∀ {X Y : D} (a b c d e f : X ⟶ Y), a = c → b = e → f = -d → a + b = c + d + (e + f)
h₃ : c₁.Rel (c₁.prev i₁) i₁
⊢ c₁.π c₂ c (c₁.prev i₁, i₂) = c.prev j

Tactic:
← ComplexShape.prev_π₁ c₂ c h₃,

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
inst✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
inst✝⁶ : Preadditive D
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K₁ L₁ : HomologicalComplex C₁ c₁
f₁ f₁' : K₁ ⟶ L₁
h₁ : Homotopy f₁ f₁'
K₂ L₂ : HomologicalComplex C₂ c₂
f₂ : K₂ ⟶ L₂
F : C₁ ⥤ C₂ ⥤ D
inst✝⁵ : F.Additive
inst✝⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
c : ComplexShape J
inst✝³ : DecidableEq J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : L₁.HasMapBifunctor L₂ F c
j : J
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c (i₁, i₂) = j
this : ∀ {X Y : D} (a b c d e f : X ⟶ Y), a = c → b = e → f = -d → a + b = c + d + (e + f)
h₃ : c₁.Rel (c₁.prev i₁) i₁
⊢ c.prev (c₁.π c₂ c (i₁, i₂)) = c.prev j
Module: Mathlib.Algebra.Homology.TotalComplexShift

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
a x a' : ℤ
h : a' = a + x
b : ℤ
⊢ (((shiftFunctor₁ C x).obj K).X a).X b = (K.X a').X b

Tactic:
subst h; rfl

Goal state after:
Module: Mathlib.CategoryTheory.Shift.CommShiftTwo

Goal state before:
pp: C₁ : Type u_1
C₁' : Type u_2
C₂ : Type u_3
C₂' : Type u_4
D : Type u_5
inst✝⁷ : Category.{v_1, u_1} C₁
inst✝⁶ : Category.{v_2, u_2} C₁'
inst✝⁵ : Category.{v_3, u_3} C₂
inst✝⁴ : Category.{v_4, u_4} C₂'
inst✝³ : Category.{v_5, u_5} D
inst✝² : Preadditive D
inst✝¹ : HasShift D ℤ
inst✝ : ∀ (n : ℤ), (shiftFunctor D n).Additive
x✝² x✝¹ x✝ : ℤ × ℤ
⊢ (-1) ^ ((x✝² + x✝¹).1 * x✝.2) * (-1) ^ (x✝².1 * x✝¹.2) = (-1) ^ (x✝².1 * (x✝¹ + x✝).2) * (-1) ^ (x✝¹.1 * x✝.2)

Tactic:
by
    dsimp
    rw [← zpow_add, ← zpow_add]
    #adaptation_note /-- After https://github.com/leanprover/lean4/pull/13593
    we need to re-enable model-based theory combination in `lia` for this to go through. -/
    lia +mbtc

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
a b y b' : ℤ
h : b' = b + y
⊢ (((shiftFunctor₂ C y).obj K).X a).X b = (K.X a).X b'

Tactic:
subst h; rfl

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
F : C ⥤ D
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroMorphisms D
inst✝ : F.PreservesZeroMorphisms
K : CochainComplex.Plus C
⊢ CochainComplex.plus D ((CochainComplex.Plus.ι C ⋙ F.mapHomologicalComplex (ComplexShape.up ℤ)).obj K)

Tactic:
obtain ⟨i, hi⟩ := K.2
    refine ⟨i, ?_⟩
    dsimp [CochainComplex.Plus.ι]
    infer_instance

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
X₁ X₂ X₃ X₄ : C
inst✝ : HasBinaryBiproduct X₂ X₃
fst : X₁ ⟶ X₂
snd : X₁ ⟶ X₃
f : X₂ ⟶ X₄
g : X₃ ⟶ X₄
sq : CommSq fst snd f g
h : IsLimit (PullbackCone.mk fst snd ⋯)
s : Fork (biprod.desc f (-g)) 0
⊢ (s.ι ≫ biprod.fst) ≫ f = (s.ι ≫ biprod.snd) ≫ g

Tactic:
rw [← sub_eq_zero, assoc, assoc, ← Preadditive.comp_sub]
          convert! s.condition <;> cat_disch

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

Goal state before:
pp: case h₁
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
X₁ X₂ X₃ X₄ : C
inst✝ : HasBinaryBiproduct X₂ X₃
fst : X₁ ⟶ X₂
snd : X₁ ⟶ X₃
f : X₂ ⟶ X₄
g : X₃ ⟶ X₄
sq : CommSq fst snd f g
h : IsLimit (PullbackCone.mk fst snd ⋯)
s : Fork (biprod.desc f (-g)) 0
⊢ (PullbackCone.IsLimit.lift h (s.ι ≫ biprod.fst) (s.ι ≫ biprod.snd) ⋯ ≫ biprod.lift fst snd) ≫ biprod.snd =
    s.ι ≫ biprod.snd

Tactic:
· simp only [assoc, biprod.lift_snd]
          apply PullbackCone.IsLimit.lift_snd h

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

Goal state before:
pp: case refine_1
C : Type u
inst✝⁶ : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type v
inst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝⁴ : ConcreteCategory C FC
inst✝³ : HasForget₂ C Ab
inst✝² : Abelian C
inst✝¹ : (forget₂ C Ab).Additive
inst✝ : (forget₂ C Ab).PreservesHomology
ι : Type u_2
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.ShortExact
i j : ι
hij : c.Rel i j
x₃ : ↑((forget₂ C Ab).obj (S.X₃.X i))
hx₃ : (ConcreteCategory.hom ((forget₂ C Ab).map (S.X₃.d i j))) x₃ = 0
x₂ : ↑((forget₂ C Ab).obj (S.X₂.X i))
hx₂ : (ConcreteCategory.hom ((forget₂ C Ab).map (S.g.f i))) x₂ = x₃
x₁ : ↑((forget₂ C Ab).obj (S.X₁.X j))
hx₁ :
  (ConcreteCategory.hom ((forget₂ C Ab).map (S.f.f j))) x₁ = (ConcreteCategory.hom ((forget₂ C Ab).map (S.X₂.d i j))) x₂
k : ι
hk : c.next j = k
⊢ (ConcreteCategory.hom ((forget₂ C Ab).map (HomologicalComplex.opcyclesMap S.g i)))
      ((ConcreteCategory.hom ((forget₂ C Ab).map (S.X₂.pOpcycles i))) x₂) =
    (ConcreteCategory.hom ((forget₂ C Ab).map (S.X₃.homologyι i)))
      ((ConcreteCategory.hom ((forget₂ C Ab).map (S.X₃.homologyπ i))) (S.X₃.cyclesMk x₃ j ⋯ hx₃))

Tactic:
rw [← ConcreteCategory.forget₂_comp_apply, ← ConcreteCategory.forget₂_comp_apply,
      HomologicalComplex.p_opcyclesMap, Functor.map_comp, ConcreteCategory.comp_apply,
      HomologicalComplex.homology_π_ι, ConcreteCategory.forget₂_comp_apply, hx₂,
      HomologicalComplex.i_cyclesMk]

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Orthogonal

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
P : ObjectProperty C
X✝ Y✝ : C
e : X✝ ≅ Y✝
h : P.rightOrthogonal X✝
X : C
f : X ⟶ Y✝
hX : P X
⊢ f = 0

Tactic:
← cancel_mono e.inv,

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
P : ObjectProperty C
X✝ Y✝ : C
e : X✝ ≅ Y✝
h : P.rightOrthogonal X✝
X : C
f : X ⟶ Y✝
hX : P X
⊢ f ≫ e.inv = 0 ≫ e.inv
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexCohomology

Goal state before:
pp: case refine_2
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
n : ℤ
f :
  (HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj K ⟶
    (HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj
      ((CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) n).obj L)
⊢ ∃ a, toHom a = f

Tactic:
· obtain ⟨f, rfl⟩ := Functor.map_surjective _ f
    exact ⟨mk (Cocycle.equivHomShift f), by simp [toHom_mk]⟩

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
K L : CochainComplex C ℤ
inst✝¹ : K.IsKInjective
inst✝ : L.IsKInjective
f : K ⟶ L
x✝ : QuasiIso f
⊢ MorphismProperty.isomorphisms (HomotopyCategory C (ComplexShape.up ℤ))
    ((HomotopyCategory.quotient C (ComplexShape.up ℤ)).map f)

Tactic:
MorphismProperty.isomorphisms.iff

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
K L : CochainComplex C ℤ
inst✝¹ : K.IsKInjective
inst✝ : L.IsKInjective
f : K ⟶ L
x✝ : QuasiIso f
⊢ IsIso ((HomotopyCategory.quotient C (ComplexShape.up ℤ)).map f)
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy

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 : HasZeroObject C
inst✝¹ : Preadditive C
K L : HomologicalComplex C c
f g : K ⟶ L
e : c.Embedding c'
inst✝ : e.IsRelIff
h : Homotopy (extendMap f e) (extendMap g e)
i : ι
⊢ f.f i =
    (((dNext i) fun i j ↦ (K.extendXIso e ⋯).inv ≫ h.hom (e.f i) (e.f j) ≫ (L.extendXIso e ⋯).hom) +
        (prevD i) fun i j ↦ (K.extendXIso e ⋯).inv ≫ h.hom (e.f i) (e.f j) ≫ (L.extendXIso e ⋯).hom) +
      g.f i

Tactic:
by
    have := h.comm (e.f i)
    simp only [extendMap_f _ _ rfl] at this
    simp only [← cancel_mono (L.extendXIso e rfl).inv,
      ← cancel_epi (K.extendXIso e rfl).hom, this, Preadditive.add_comp,
      Preadditive.comp_add, add_left_inj]
    congr 1
    · by_cases hi : c.Rel i (c.next i)
      · have hi' : c'.Rel (e.f i) (e.f (c.next i)) := by rwa [e.rel_iff]
        simp [dNext_eq _ hi, dNext_eq _ hi', K.extend_d_eq _ rfl rfl]
      · rw [dNext_eq_zero _ _ hi]
        by_cases hi' : c'.Rel (e.f i) (c'.next (e.f i))
        · simp [dNext_eq _ hi', extend_d_from_eq_zero _ _ _ _ _ rfl hi]
        · simp [dNext_eq_zero _ _ hi']
    · by_cases hi : c.Rel (c.prev i) i
      · have hi' : c'.Rel (e.f (c.prev i)) (e.f i) := by rwa [e.rel_iff]
        simp [prevD_eq _ hi, prevD_eq _ hi', L.extend_d_eq _ rfl rfl]
      · rw [prevD_eq_zero _ _ hi]
        by_cases hi' : c'.Rel (c'.prev (e.f i)) (e.f i)
        · simp [prevD_eq _ hi', extend_d_to_eq_zero _ _ _ _ _ rfl hi]
        · simp [prevD_eq_zero _ _ hi']

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
X Y : DerivedCategory C
f : X ⟶ Y
K : CochainComplex C ℤ
e₁ : X ≅ Q.obj K
h✝¹ : K.IsStrictlyLE 0
L : CochainComplex C ℤ
e₂ : Y ≅ Q.obj L
h✝ : L.IsStrictlyGE 1
⊢ f = 0

Tactic:
← cancel_epi e₁.inv,

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
X Y : DerivedCategory C
f : X ⟶ Y
K : CochainComplex C ℤ
e₁ : X ≅ Q.obj K
h✝¹ : K.IsStrictlyLE 0
L : CochainComplex C ℤ
e₂ : Y ≅ Q.obj L
h✝ : L.IsStrictlyGE 1
⊢ e₁.inv ≫ f = e₁.inv ≫ 0
Module: Mathlib.AlgebraicTopology.ModelCategory.Instances

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : CategoryWithWeakEquivalences C
inst✝ : (weakEquivalences C).HasTwoOutOfThreeProperty
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
hf : WeakEquivalence f
hfg : WeakEquivalence (f ≫ g)
⊢ WeakEquivalence g

Tactic:
rw [weakEquivalence_iff] at hf hfg ⊢
  exact of_precomp _ _ _ hf hfg

Goal state after:
Module: Mathlib.AlgebraicTopology.ModelCategory.Instances

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : CategoryWithWeakEquivalences C
inst✝² : CategoryWithCofibrations C
inst✝¹ : CategoryWithFibrations C
inst✝ : (cofibrations C).IsWeakFactorizationSystem (trivialFibrations C)
J : Type w
⊢ (cofibrations C).IsStableUnderCoproductsOfShape J

Tactic:
by
  rw [← trivialFibrations_llp]
  apply MorphismProperty.llp_isStableUnderCoproductsOfShape

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

Goal state before:
pp: case e_a
C₁ : Type u_1
C₂ : Type u_2
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝²³ : Category.{v_1, u_1} C₁
inst✝²² : Category.{v_2, u_2} C₂
inst✝²¹ : Category.{v_3, u_5} C₃
inst✝²⁰ : Category.{v_4, u_6} C₄
inst✝¹⁹ : Category.{v_6, u_4} C₂₃
inst✝¹⁸ : HasZeroMorphisms C₁
inst✝¹⁷ : HasZeroMorphisms C₂
inst✝¹⁶ : HasZeroMorphisms C₃
inst✝¹⁵ : Preadditive C₂₃
inst✝¹⁴ : Preadditive C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
inst✝¹³ : G₂₃.PreservesZeroMorphisms
inst✝¹² : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst✝¹¹ : F.PreservesZeroMorphisms
inst✝¹⁰ : ∀ (X₁ : C₁), (F.obj X₁).Additive
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₂₃ : Type u_11
ι₄ : Type u_12
inst✝⁹ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₂₃ : ComplexShape ι₂₃
c₄ : ComplexShape ι₄
inst✝⁸ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁷ : TotalComplexShape c₁₂ c₃ c₄
inst✝⁶ : TotalComplexShape c₂ c₃ c₂₃
inst✝⁵ : TotalComplexShape c₁ c₂₃ c₄
inst✝⁴ : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst✝³ : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst✝² : DecidableEq ι₂₃
inst✝¹ : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
inst✝ : HasGoodTrifunctor₂₃Obj F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄
j j' : ι₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
i₂₃ : ι₂₃ := c₂.π c₃ c₂₃ (i₂, i₃)
h₁ : ¬c₂₃.Rel i₂₃ (c₂₃.next i₂₃)
⊢ 0 = d₃ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j'

Tactic:
rw [d₃_eq_zero]
        intro h₂
        apply h₁
        simpa only [i₂₃, ComplexShape.next_π₂ c₂ c₂₃ i₂ h₂]
          using ComplexShape.rel_π₂ c₂ c₂₃ i₂ h₂

Goal state after: