mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Group.Action.Equidecomp

Goal state before:
pp: X : Type u_1
G : Type u_2
inst✝¹ : Monoid G
inst✝ : MulAction G X
g f : X → X
B A : Set X
T S : Finset G
hg : IsDecompOn g B T
hf : IsDecompOn f A S
⊢ IsDecompOn (g ∘ f) (A ∩ f ⁻¹' B) (T * S)

Tactic:
intro _

Goal state after:
X : Type u_1
G : Type u_2
inst✝¹ : Monoid G
inst✝ : MulAction G X
g f : X → X
B A : Set X
T S : Finset G
hg : IsDecompOn g B T
hf : IsDecompOn f A S
a✝ : X
⊢ a✝ ∈ A ∩ f ⁻¹' B → ∃ g_1 ∈ T * S, (g ∘ f) a✝ = g_1 • a✝
Module: Mathlib.Algebra.Quaternion

Goal state before:
pp: S : Type u_1
T : Type u_2
R : Type u_3
c₁ c₂ c₃ r x y : R
a✝ b✝ : ℍ[R,c₁,c₂,c₃]
inst✝ : CommRing R
a b : ℍ[R,c₁,c₂,c₃]
⊢ star (a + b) = star a + star b

Tactic:
by ext <;> simp [add_comm]; ring

Goal state after:
Module: Mathlib.Algebra.Group.Action.Sum

Goal state before:
pp: M : Type u_1
N : Type u_2
α : Type u_3
β : Type u_4
inst✝⁵ : SMul M α
inst✝⁴ : SMul M β
inst✝³ : SMul N α
inst✝² : SMul N β
a✝ : M
b✝ : α
c : β
x✝ : α ⊕ β
inst✝¹ : SMulCommClass M N α
inst✝ : SMulCommClass M N β
a : M
b : N
x : α ⊕ β
⊢ a • b • x = b • a • x

Tactic:
cases x
    exacts [congr_arg inl (smul_comm _ _ _), congr_arg inr (smul_comm _ _ _)]

Goal state after:
Module: Mathlib.Algebra.Group.ForwardDiff

Goal state before:
pp: M : Type u_1
G : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : AddCommGroup G
h : M
R : Type u_3
inst✝¹ : Ring R
inst✝ : Module R G
f : M → R
g : G
⊢ (Δ_[h] fun y ↦ f y • g) = Δ_[h] f • fun x ↦ g

Tactic:
ext y
  simp only [fwdDiff, Pi.smul_apply', sub_smul]

Goal state after:
Module: Mathlib.Algebra.Group.Pointwise.Finset.Interval

Goal state before:
pp: α : Type u_1
inst✝⁵ : Mul α
inst✝⁴ : PartialOrder α
inst✝³ : DecidableEq α
inst✝² : MulLeftStrictMono α
inst✝¹ : MulRightStrictMono α
inst✝ : LocallyFiniteOrder α
a b c d : α
⊢ ↑(Ioc a b * Ico c d) ⊆ ↑(Ioo (a * c) (b * d))

Tactic:
simpa using Set.Ioc_mul_Ico_subset' _ _ _ _

Goal state after:
Module: Mathlib.Algebra.Group.Irreducible.Indecomposable

Goal state before:
pp: case refine_2
ι : Type u_1
G : Type u_3
S : Type u_4
inst✝⁵ : CommGroup G
inst✝⁴ : LinearOrder S
inst✝³ : Finite ι
inst✝² : InvolutiveInv ι
inst✝¹ : CommGroup S
inst✝ : IsOrderedMonoid S
v : ι → G
hv_inv : ∀ (i : ι), v i⁻¹ = (v i)⁻¹
f : G →* S
hf : ∀ (i : ι), f (v i) ≠ 1
this : univ = {i | 1 < f (v i)} ∪ {i | f (v i) < 1}
f' : G →* S := invMonoidHom.comp f
h₁ : ⇑invMonoidHom ∘ v '' IsMulIndecomposable.baseOf v f' = v '' IsMulIndecomposable.baseOf v f
h₂ : v '' {i | f (v i) < 1} = v '' {i | 1 < f' (v i)}
⊢ v '' {i | 1 < f' (v i)} ⊆ ↑(closure (Inv.inv '' v '' IsMulIndecomposable.baseOf v f'))

Tactic:
image_inv_eq_inv,

Goal state after:
case refine_2
ι : Type u_1
G : Type u_3
S : Type u_4
inst✝⁵ : CommGroup G
inst✝⁴ : LinearOrder S
inst✝³ : Finite ι
inst✝² : InvolutiveInv ι
inst✝¹ : CommGroup S
inst✝ : IsOrderedMonoid S
v : ι → G
hv_inv : ∀ (i : ι), v i⁻¹ = (v i)⁻¹
f : G →* S
hf : ∀ (i : ι), f (v i) ≠ 1
this : univ = {i | 1 < f (v i)} ∪ {i | f (v i) < 1}
f' : G →* S := invMonoidHom.comp f
h₁ : ⇑invMonoidHom ∘ v '' IsMulIndecomposable.baseOf v f' = v '' IsMulIndecomposable.baseOf v f
h₂ : v '' {i | f (v i) < 1} = v '' {i | 1 < f' (v i)}
⊢ v '' {i | 1 < f' (v i)} ⊆ ↑(closure (v '' IsMulIndecomposable.baseOf v f')⁻¹)
Module: Mathlib.Algebra.GroupWithZero.Range

Goal state before:
pp: case h₁
A : Type u_1
B : Type u_2
inst✝¹ : GroupWithZero A
inst✝ : GroupWithZero B
f : A →*₀ B
⊢ (↑f.valueMonoid)⁻¹ ⊆ ↑f.valueMonoid

Tactic:
intro _

Goal state after:
case h₁
A : Type u_1
B : Type u_2
inst✝¹ : GroupWithZero A
inst✝ : GroupWithZero B
f : A →*₀ B
a✝ : Bˣ
⊢ a✝ ∈ (↑f.valueMonoid)⁻¹ → a✝ ∈ ↑f.valueMonoid
Module: Mathlib.Order.Category.NonemptyFinLinOrd

Goal state before:
pp: A B : NonemptyFinLinOrd
f : A ⟶ B
⊢ Mono f → Function.Injective ⇑(ConcreteCategory.hom f)

Tactic:
intro _

Goal state after:
A B : NonemptyFinLinOrd
f : A ⟶ B
a✝ : Mono f
⊢ Function.Injective ⇑(ConcreteCategory.hom f)
Module: Mathlib.CategoryTheory.Idempotents.Basic

Goal state before:
pp: case mp
C : Type u_1
inst✝ : Category.{v_1, u_1} C
⊢ IsIdempotentComplete C → ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p

Tactic:
intro _

Goal state after:
case mp
C : Type u_1
inst✝ : Category.{v_1, u_1} C
a✝ : IsIdempotentComplete C
⊢ ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p
Module: Mathlib.CategoryTheory.Idempotents.Basic

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
D : Type u_2
inst✝ : Category.{v_2, u_2} D
ε : C ≌ D
h : IsIdempotentComplete C
⊢ ∀ (X : D) (p : X ⟶ X), p ≫ p = p → ∃ Y i e, i ≫ e = 𝟙 Y ∧ e ≫ i = p

Tactic:
intro X'

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
D : Type u_2
inst✝ : Category.{v_2, u_2} D
ε : C ≌ D
h : IsIdempotentComplete C
X' : D
⊢ ∀ (p : X' ⟶ X'), p ≫ p = p → ∃ Y i e, i ≫ e = 𝟙 Y ∧ e ≫ i = p
Module: Mathlib.CategoryTheory.Idempotents.FunctorCategories

Goal state before:
pp: J : Type u_1
C : Type u_2
inst✝¹ : Category.{v_1, u_1} J
inst✝ : Category.{v_2, u_2} C
P✝ Q✝ : Karoubi (J ⥤ C)
f✝ : P✝ ⟶ Q✝
X : J
P Q : Karoubi (J ⥤ C)
f : (karoubiFunctorCategoryEmbedding J C).obj P ⟶ (karoubiFunctorCategoryEmbedding J C).obj Q
j j' : J
φ : j ⟶ j'
⊢ P.X.map φ ≫ (f.app j').f = (f.app j).f ≫ (((karoubiFunctorCategoryEmbedding J C).obj Q).obj j).p ≫ Q.X.map φ

Tactic:
have h := hom_ext_iff.mp (f.naturality φ)

Goal state after:
J : Type u_1
C : Type u_2
inst✝¹ : Category.{v_1, u_1} J
inst✝ : Category.{v_2, u_2} C
P✝ Q✝ : Karoubi (J ⥤ C)
f✝ : P✝ ⟶ Q✝
X : J
P Q : Karoubi (J ⥤ C)
f : (karoubiFunctorCategoryEmbedding J C).obj P ⟶ (karoubiFunctorCategoryEmbedding J C).obj Q
j j' : J
φ : j ⟶ j'
h :
  (((karoubiFunctorCategoryEmbedding J C).obj P).map φ ≫ f.app j').f =
    (f.app j ≫ ((karoubiFunctorCategoryEmbedding J C).obj Q).map φ).f
⊢ P.X.map φ ≫ (f.app j').f = (f.app j).f ≫ (((karoubiFunctorCategoryEmbedding J C).obj Q).obj j).p ≫ Q.X.map φ
Module: Mathlib.AlgebraicTopology.SimplexCategory.Basic

Goal state before:
pp: case inl
n : ℕ
i j : Fin (n + 1)
H : i ≤ j
k : Fin (n + 1)
h : i ≤ k
hkj : k ≤ j
⊢ i ≤ j.predAbove k.castSucc

Tactic:
rwa [Fin.predAbove_of_le_castSucc _ _ (Fin.castSucc_le_castSucc_iff.mpr hkj),
          Fin.castPred_castSucc]

Goal state after:
Module: Mathlib.AlgebraicTopology.AlternatingFaceMapComplex

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
X Y : SimplicialObject.Augmented C
f : X ⟶ Y
⊢ f.left.app (op ⦋0⦌) ≫ Y.hom.app (op ⦋0⦌) = X.hom.app (op ⦋0⦌) ≫ ((ChainComplex.single₀ C).map f.right).f 0

Tactic:
ChainComplex.single₀_map_f_zero

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
X Y : SimplicialObject.Augmented C
f : X ⟶ Y
⊢ f.left.app (op ⦋0⦌) ≫ Y.hom.app (op ⦋0⦌) = X.hom.app (op ⦋0⦌) ≫ f.right
Module: Mathlib.AlgebraicTopology.SimplexCategory.Basic

Goal state before:
pp: case pos.h
n : ℕ
Δ' : SimplexCategory
θ : ⦋n + 1⦌ ⟶ Δ'
i : Fin (n + 1)
hi : (Hom.toOrderHom θ) i.castSucc = (Hom.toOrderHom θ) i.succ
x : Fin (⦋n + 1⦌.len + 1)
h' : x ≤ i.castSucc
⊢ (x.castPred ⋯).castSucc < i.succ

Tactic:
exact (Fin.castSucc_lt_succ_iff.mpr h')

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplexCategory.DeltaZeroIter

Goal state before:
pp: case pos
i n m : ℕ
h : n + (i + 1) = m
k : Fin (⦋m⦌.len + 1)
hk : ↑k ≤ i
⊢ ↑((ConcreteCategory.hom (σ₀Iter (i + 1) h)) k) =
    ↑((ConcreteCategory.hom (σ 0)) ((ConcreteCategory.hom (σ₀Iter i ⋯)) k))

pp: case neg
i n m : ℕ
h : n + (i + 1) = m
k : Fin (⦋m⦌.len + 1)
hk : i < ↑k
⊢ ↑((ConcreteCategory.hom (σ₀Iter (i + 1) h)) k) =
    ↑((ConcreteCategory.hom (σ 0)) ((ConcreteCategory.hom (σ₀Iter i ⋯)) k))

Tactic:
· rw [σ₀Iter_coe_eq_of_lt .., coe_σ]
    obtain hk | rfl := hk.lt_or_eq
    · grind [Fin.predAbove_of_le_castSucc, Fin.coe_castPred, σ₀Iter_coe_eq_of_lt]
    · grind [Fin.predAbove_of_le_castSucc, Fin.coe_castPred, σ₀Iter_coe_eq_of_ge, tsub_self]

Goal state after:
case neg
i n m : ℕ
h : n + (i + 1) = m
k : Fin (⦋m⦌.len + 1)
hk : i < ↑k
⊢ ↑((ConcreteCategory.hom (σ₀Iter (i + 1) h)) k) =
    ↑((ConcreteCategory.hom (σ 0)) ((ConcreteCategory.hom (σ₀Iter i ⋯)) k))
Module: Mathlib.AlgebraicTopology.SimplicialSet.Dimension

Goal state before:
pp: X : SSet
A : X.Subcomplex
d : ℕ
inst✝ : X.HasDimensionLT d
⊢ A = ⊤ ↔ ∀ i < d, X.nonDegenerate i ⊆ A.obj (op ⦋i⦌)

Tactic:
simp [← top_le_iff, le_iff_of_hasDimensionLT ⊤ A d]

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

Goal state before:
pp: X : SSet
A B : X.Subcomplex
⊢ A ≤ B ↔ ∀ (s : X.N), s.subcomplex ≤ A → s.subcomplex ≤ B

Tactic:
rw [Subcomplex.le_iff_contains_nonDegenerate]

Goal state after:
X : SSet
A B : X.Subcomplex
⊢ (∀ (n : ℕ) (x : ↑(X.nonDegenerate n)), ↑x ∈ A.obj (Opposite.op ⦋n⦌) → ↑x ∈ B.obj (Opposite.op ⦋n⦌)) ↔
    ∀ (s : X.N), s.subcomplex ≤ A → s.subcomplex ≤ B
Module: Mathlib.AlgebraicTopology.SimplicialSet.Finite

Goal state before:
pp: X : SSet
d : ℕ
inst✝ : X.HasDimensionLT d
h : ∀ i < d, Finite ↑(X.nonDegenerate i)
this : ∀ (i : Fin d), Finite ↑(X.nonDegenerate ↑i)
x : X.N
⊢ ∃ a,
    (fun x ↦
          match x with
          | ⟨i, x⟩ => N.mk ↑x ⋯)
        a =
      x

Tactic:
by_cases hj : x.dim < d

Goal state after:
case pos
X : SSet
d : ℕ
inst✝ : X.HasDimensionLT d
h : ∀ i < d, Finite ↑(X.nonDegenerate i)
this : ∀ (i : Fin d), Finite ↑(X.nonDegenerate ↑i)
x : X.N
hj : x.dim < d
⊢ ∃ a,
    (fun x ↦
          match x with
          | ⟨i, x⟩ => N.mk ↑x ⋯)
        a =
      x

case neg
X : SSet
d : ℕ
inst✝ : X.HasDimensionLT d
h : ∀ i < d, Finite ↑(X.nonDegenerate i)
this : ∀ (i : Fin d), Finite ↑(X.nonDegenerate ↑i)
x : X.N
hj : ¬x.dim < d
⊢ ∃ a,
    (fun x ↦
          match x with
          | ⟨i, x⟩ => N.mk ↑x ⋯)
        a =
      x
Module: Mathlib.AlgebraicTopology.SimplicialSet.Degenerate

Goal state before:
pp: case mpr
X : SSet
A : X.Subcomplex
n : ℕ
h : X.degenerate n ⊓ A.obj (op ⦋n⦌) = A.obj (op ⦋n⦌)
⊢ A.toSSet.degenerate n = ⊤

Tactic:
simp only [Set.inf_eq_inter, Set.inter_eq_right] at h

Goal state after:
case mpr
X : SSet
A : X.Subcomplex
n : ℕ
h : A.obj (op ⦋n⦌) ⊆ X.degenerate n
⊢ A.toSSet.degenerate n = ⊤
Module: Mathlib.AlgebraicTopology.SimplicialSet.Degenerate

Goal state before:
pp: X Y : SSet
n : ℕ
f : X ⟶ Y
m : ℕ
hm : m < n
g : ⦋n⦌ ⟶ ⦋m⦌
y : X _⦋m⦌
⊢ (ConcreteCategory.hom (Y.map g.op)) ((ConcreteCategory.hom (f.app (op ⦋m⦌))) y) =
    (ConcreteCategory.hom (f.app (op ⦋n⦌))) ((ConcreteCategory.hom (X.map g.op)) y)

Tactic:
rw [NatTrans.naturality_apply]

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

Goal state before:
pp: case mpr
X : Type u_1
inst✝ : PartialOrder X
n : ℕ
s : nerve X _⦋n + 1⦌
i : Fin (n + 1)
h : s.obj i.castSucc = s.obj i.succ
j : Fin ((Opposite.unop (Opposite.op ⦋n + 1⦌)).len + 1)
⊢ ((ConcreteCategory.hom (SimplicialObject.δ (nerve X) i.castSucc)) s).obj (i.predAbove j) = s.obj j

Tactic:
nerve.δ_obj

Goal state after:
case mpr
X : Type u_1
inst✝ : PartialOrder X
n : ℕ
s : nerve X _⦋n + 1⦌
i : Fin (n + 1)
h : s.obj i.castSucc = s.obj i.succ
j : Fin ((Opposite.unop (Opposite.op ⦋n + 1⦌)).len + 1)
⊢ s.obj (i.castSucc.succAbove (i.predAbove j)) = s.obj j
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: n : ℕ
⊢ face univ = ⊤

Tactic:
by
  ext
  simp only [Subfunctor.top_obj, Set.top_eq_univ, Set.mem_univ, iff_true]
  apply Finset.subset_univ

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

Goal state before:
pp: I₁ : Type u_1
I₂ : Type u_2
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₁₂ : ComplexShape I₁₂
inst✝ : TotalComplexShape c₁ c₂ c₁₂
i₁ i₁' : I₁
i₂ i₂' : I₂
h₁ : c₁.Rel i₁ i₁'
h₂ : c₂.Rel i₂ i₂'
⊢ c₁.ε₁ c₂ c₁₂ (i₁, i₂) * c₁.ε₂ c₂ c₁₂ (i₁, i₂) * 1 = -c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')

Tactic:
rw [← Int.units_mul_self (ComplexShape.ε₁ c₁ c₂ c₁₂ (i₁, i₂')), mul_assoc]
    conv_lhs =>
      arg 2
      rw [← mul_assoc, ε₂_ε₁ c₁₂ h₁ h₂]
    rw [neg_mul, neg_mul, neg_mul, mul_neg, neg_inj, ← mul_assoc, ← mul_assoc,
      Int.units_mul_self, one_mul]

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

Goal state before:
pp: case zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
i : ℕ
⊢ (ComplexShape.up ℕ).Rel i 0 →
    (match i with
        | 0 => 𝟙 (C.X 0)
        | n.succ => 𝟙 (C.X (n + 1))) ≫
        ((truncate.obj C).augment (C.d 0 1) ⋯).d i 0 =
      C.d i 0 ≫
        match 0 with
        | 0 => 𝟙 (C.X 0)
        | n.succ => 𝟙 (C.X (n + 1))

Tactic:
rcases i with - | i

Goal state after:
case zero.zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
⊢ (ComplexShape.up ℕ).Rel 0 0 →
    (match 0 with
        | 0 => 𝟙 (C.X 0)
        | n.succ => 𝟙 (C.X (n + 1))) ≫
        ((truncate.obj C).augment (C.d 0 1) ⋯).d 0 0 =
      C.d 0 0 ≫
        match 0 with
        | 0 => 𝟙 (C.X 0)
        | n.succ => 𝟙 (C.X (n + 1))

case zero.succ
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
i : ℕ
⊢ (ComplexShape.up ℕ).Rel (i + 1) 0 →
    (match i + 1 with
        | 0 => 𝟙 (C.X 0)
        | n.succ => 𝟙 (C.X (n + 1))) ≫
        ((truncate.obj C).augment (C.d 0 1) ⋯).d (i + 1) 0 =
      C.d (i + 1) 0 ≫
        match 0 with
        | 0 => 𝟙 (C.X 0)
        | n.succ => 𝟙 (C.X (n + 1))
Module: Mathlib.Algebra.Homology.Augment

Goal state before:
pp: case succ.zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
i : ℕ
⊢ (ComplexShape.up ℕ).Rel i (0 + 1) →
    (match i with
        | 0 => 𝟙 (C.X 0)
        | n.succ => 𝟙 (C.X (n + 1))) ≫
        ((truncate.obj C).augment (C.d 0 1) ⋯).d i (0 + 1) =
      C.d i (0 + 1) ≫
        match 0 + 1 with
        | 0 => 𝟙 (C.X 0)
        | n.succ => 𝟙 (C.X (n + 1))

Tactic:
rcases i with - | i

Goal state after:
case succ.zero.zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
⊢ (ComplexShape.up ℕ).Rel 0 (0 + 1) →
    (match 0 with
        | 0 => 𝟙 (C.X 0)
        | n.succ => 𝟙 (C.X (n + 1))) ≫
        ((truncate.obj C).augment (C.d 0 1) ⋯).d 0 (0 + 1) =
      C.d 0 (0 + 1) ≫
        match 0 + 1 with
        | 0 => 𝟙 (C.X 0)
        | n.succ => 𝟙 (C.X (n + 1))

case succ.zero.succ
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
i : ℕ
⊢ (ComplexShape.up ℕ).Rel (i + 1) (0 + 1) →
    (match i + 1 with
        | 0 => 𝟙 (C.X 0)
        | n.succ => 𝟙 (C.X (n + 1))) ≫
        ((truncate.obj C).augment (C.d 0 1) ⋯).d (i + 1) (0 + 1) =
      C.d (i + 1) (0 + 1) ≫
        match 0 + 1 with
        | 0 => 𝟙 (C.X 0)
        | n.succ => 𝟙 (C.X (n + 1))
Module: Mathlib.Algebra.Homology.Augment

Goal state before:
pp: case succ.succ
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
i j : ℕ
⊢ (ComplexShape.up ℕ).Rel i (j + 1 + 1) →
    (match i with
        | 0 => 𝟙 (C.X 0)
        | n.succ => 𝟙 (C.X (n + 1))) ≫
        ((truncate.obj C).augment (C.d 0 1) ⋯).d i (j + 1 + 1) =
      C.d i (j + 1 + 1) ≫
        match j + 1 + 1 with
        | 0 => 𝟙 (C.X 0)
        | n.succ => 𝟙 (C.X (n + 1))

Tactic:
rcases i with - | i

Goal state after:
case succ.succ.zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
j : ℕ
⊢ (ComplexShape.up ℕ).Rel 0 (j + 1 + 1) →
    (match 0 with
        | 0 => 𝟙 (C.X 0)
        | n.succ => 𝟙 (C.X (n + 1))) ≫
        ((truncate.obj C).augment (C.d 0 1) ⋯).d 0 (j + 1 + 1) =
      C.d 0 (j + 1 + 1) ≫
        match j + 1 + 1 with
        | 0 => 𝟙 (C.X 0)
        | n.succ => 𝟙 (C.X (n + 1))

case succ.succ.succ
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
j i : ℕ
⊢ (ComplexShape.up ℕ).Rel (i + 1) (j + 1 + 1) →
    (match i + 1 with
        | 0 => 𝟙 (C.X 0)
        | n.succ => 𝟙 (C.X (n + 1))) ≫
        ((truncate.obj C).augment (C.d 0 1) ⋯).d (i + 1) (j + 1 + 1) =
      C.d (i + 1) (j + 1 + 1) ≫
        match j + 1 + 1 with
        | 0 => 𝟙 (C.X 0)
        | n.succ => 𝟙 (C.X (n + 1))
Module: Mathlib.Algebra.Homology.TotalComplexSymmetry

Goal state before:
pp: case neg
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)
⊢ c₁.σ c₂ c i₁ i₂ • K.d₁ c i₁ i₂ j' =
    K.flip.d₂ c i₂ i₁ j' ≫ K.flip.totalDesc fun i₂ i₁ h ↦ c₁.σ c₂ c i₁ i₂ • K.ιTotal c i₁ i₂ j' ⋯

Tactic:
K.d₁_eq_zero _ _ _ _ h₂,

Goal state after:
case neg
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)
⊢ c₁.σ c₂ c i₁ i₂ • 0 = K.flip.d₂ c i₂ i₁ j' ≫ K.flip.totalDesc fun i₂ i₁ h ↦ c₁.σ c₂ c i₁ i₂ • K.ιTotal c i₁ i₂ j' ⋯
Module: Mathlib.Algebra.Homology.TotalComplexShift

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
y : ℤ
inst✝ : K.HasTotal (up ℤ)
a b n : ℤ
h : a + b = n
b' : ℤ
hb' : a + b' = n + y
⊢ K.ιTotal (up ℤ) a b' (n + y) hb' ≫
      (CochainComplex.shiftFunctorObjXIso (K.total (up ℤ)) y n (n + y) ⋯).inv ≫ (K.totalShift₂Iso y).inv.f n =
    (a * y).negOnePow • (K.shiftFunctor₂XXIso a b y b' ⋯).inv ≫ ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) a b n h

Tactic:
obtain rfl : b = b' - y := by lia

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
y : ℤ
inst✝ : K.HasTotal (up ℤ)
a n b' : ℤ
hb' : a + b' = n + y
h : a + (b' - y) = n
⊢ K.ιTotal (up ℤ) a b' (n + y) hb' ≫
      (CochainComplex.shiftFunctorObjXIso (K.total (up ℤ)) y n (n + y) ⋯).inv ≫ (K.totalShift₂Iso y).inv.f n =
    (a * y).negOnePow •
      (K.shiftFunctor₂XXIso a (b' - y) y b' ⋯).inv ≫ ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) a (b' - y) n h
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
f : X₁ ⟶ X₂
g : X₁ ⟶ X₃
inl : X₂ ⟶ X₄
inr : X₃ ⟶ X₄
h : IsPushout f g inl inr
⊢ ∀ (R : C) (g_1 : ⋯.shortComplex.X₃ ⟶ R), ⋯.shortComplex.g ≫ g_1 = 0 → g_1 = 0

Tactic:
intro _

Goal state after:
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
f : X₁ ⟶ X₂
g : X₁ ⟶ X₃
inl : X₂ ⟶ X₄
inr : X₃ ⟶ X₄
h : IsPushout f g inl inr
R✝ : C
⊢ ∀ (g_1 : ⋯.shortComplex.X₃ ⟶ R✝), ⋯.shortComplex.g ≫ g_1 = 0 → g_1 = 0
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
h : IsPullback fst snd f g
⊢ ∀ (P : C) (g_1 : P ⟶ ⋯.shortComplex'.X₁), g_1 ≫ ⋯.shortComplex'.f = 0 → g_1 = 0

Tactic:
intro _

Goal state after:
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
h : IsPullback fst snd f g
P✝ : C
⊢ ∀ (g_1 : P✝ ⟶ ⋯.shortComplex'.X₁), g_1 ≫ ⋯.shortComplex'.f = 0 → g_1 = 0
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexInduction

Goal state before:
pp: case zero
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
d : ℤ
X : ℕ → Set (Cochain K L d)
φ : (n : ℕ) → ↑(X n) → ↑(X (n + 1))
p₀ : ℤ
hφ : ∀ (n : ℕ) (x : ↑(X n)), (↑(φ n x)).EqUpTo (↑x) (p₀ + ↑n)
x₀ : ↑(X 0)
n₁ : ℕ
⊢ (↑(sequence φ x₀ n₁)).EqUpTo (↑(sequence φ x₀ (n₁ + 0))) (p₀ + ↑n₁)

Tactic:
intro _

Goal state after:
case zero
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
d : ℤ
X : ℕ → Set (Cochain K L d)
φ : (n : ℕ) → ↑(X n) → ↑(X (n + 1))
p₀ : ℤ
hφ : ∀ (n : ℕ) (x : ↑(X n)), (↑(φ n x)).EqUpTo (↑x) (p₀ + ↑n)
x₀ : ↑(X 0)
n₁ : ℕ
p✝ : ℤ
⊢ ∀ (q : ℤ) (hpq : p✝ + d = q),
    p✝ ≤ p₀ + ↑n₁ → (↑(sequence φ x₀ n₁)).v p✝ q hpq = (↑(sequence φ x₀ (n₁ + 0))).v p✝ q hpq
Module: Mathlib.CategoryTheory.Triangulated.Orthogonal

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
P : ObjectProperty C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : P.IsTriangulated
⊢ P.trW.isLocal = P.rightOrthogonal

Tactic:
ext Y

Goal state after:
C : Type u
inst✝⁶ : Category.{v, u} C
P : ObjectProperty C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : P.IsTriangulated
Y : C
⊢ P.trW.isLocal Y ↔ P.rightOrthogonal Y
Module: Mathlib.Algebra.Homology.HomotopyCategory.KInjective

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
α : Cochain K L (-1)
n : ℤ
hα : (δ (-1) 0 α).EqUpTo (Cochain.ofHom f) n
hK : HomologicalComplex.ExactAt K (n + 1)
inst✝ : Injective (L.X (n + 1))
⊢ ∃ h, (δ (-1) 0 (α + Cochain.single h (-1))).EqUpTo (Cochain.ofHom f) (n + 1)

Tactic:
let u := f.f (n + 1) - α.v (n + 1) n (by lia) ≫ L.d n (n + 1) -
    K.d (n + 1) (n + 2) ≫ α.v (n + 2) (n + 1) (by lia)

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
α : Cochain K L (-1)
n : ℤ
hα : (δ (-1) 0 α).EqUpTo (Cochain.ofHom f) n
hK : HomologicalComplex.ExactAt K (n + 1)
inst✝ : Injective (L.X (n + 1))
u : K.X (n + 1) ⟶ L.X (n + 1) :=
  f.f (n + 1) - α.v (n + 1) n ⋯ ≫ L.d n (n + 1) - K.d (n + 1) (n + 2) ≫ α.v (n + 2) (n + 1) ⋯
⊢ ∃ h, (δ (-1) 0 (α + Cochain.single h (-1))).EqUpTo (Cochain.ofHom f) (n + 1)
Module: Mathlib.CategoryTheory.Triangulated.TStructure.Basic

Goal state before:
pp: C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Preadditive C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝² : Pretriangulated C
t : TStructure C
X Y : C
f : X ⟶ Y
n₀ n₁ : ℤ
h : n₀ < n₁
inst✝¹ : t.IsLE X n₀
inst✝ : t.IsGE Y n₁
this✝¹ : t.IsLE ((shiftFunctor C n₀).obj X) 0
this✝ : t.IsGE ((shiftFunctor C n₀).obj Y) (n₁ - n₀)
this : t.IsGE ((shiftFunctor C n₀).obj Y) 1
⊢ (shiftFunctor C n₀).map f = 0

Tactic:
apply t.zero'

Goal state after:
case hX
C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Preadditive C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝² : Pretriangulated C
t : TStructure C
X Y : C
f : X ⟶ Y
n₀ n₁ : ℤ
h : n₀ < n₁
inst✝¹ : t.IsLE X n₀
inst✝ : t.IsGE Y n₁
this✝¹ : t.IsLE ((shiftFunctor C n₀).obj X) 0
this✝ : t.IsGE ((shiftFunctor C n₀).obj Y) (n₁ - n₀)
this : t.IsGE ((shiftFunctor C n₀).obj Y) 1
⊢ t.le 0 ((shiftFunctor C n₀).obj X)

case hY
C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Preadditive C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝² : Pretriangulated C
t : TStructure C
X Y : C
f : X ⟶ Y
n₀ n₁ : ℤ
h : n₀ < n₁
inst✝¹ : t.IsLE X n₀
inst✝ : t.IsGE Y n₁
this✝¹ : t.IsLE ((shiftFunctor C n₀).obj X) 0
this✝ : t.IsGE ((shiftFunctor C n₀).obj Y) (n₁ - n₀)
this : t.IsGE ((shiftFunctor C n₀).obj Y) 1
⊢ t.ge 1 ((shiftFunctor C n₀).obj Y)