mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.IntegralClosure.IntegrallyClosed

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
hRA : Function.Injective ⇑(algebraMap R A)
⊢ integralClosure R A = ⊥ ↔ IsIntegrallyClosedIn R A

Tactic:
refine eq_bot_iff.trans ?_

Goal state after:
R : Type u_1
A : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
hRA : Function.Injective ⇑(algebraMap R A)
⊢ integralClosure R A ≤ ⊥ ↔ IsIntegrallyClosedIn R A
Module: Mathlib.RingTheory.Polynomial.Eisenstein.Criterion

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
inst✝² : IsDomain R
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra R K
q f : R[X]
p : ℕ
hq_irr : Irreducible (map (algebraMap R K) q)
hq_monic : q.Monic
hf_prim : f.IsPrimitive
hfd0 : 0 < f.natDegree
hfP : (algebraMap R K) f.leadingCoeff ≠ 0
hfmodP : map (algebraMap R K) f = C ((algebraMap R K) f.leadingCoeff) * map (algebraMap R K) q ^ p
g h : R[X]
h_eq : f = g * h
P : Ideal R := ker (algebraMap R K)
hfmodP2 : map (Ideal.Quotient.mk (P ^ 2)) (f %ₘ q) ≠ 0
m : ℕ
r : R[X]
hg : g = C g.leadingCoeff * q ^ m + r
hr : map (algebraMap R K) r = 0
hm0 : m = 0 → IsUnit g
n : ℕ
s : R[X]
hh : h = C h.leadingCoeff * q ^ n + s
hs : map (algebraMap R K) s = 0
hn0 : n = 0 → IsUnit h
hm : ¬m = 0
hn : ¬n = 0
this : f %ₘ q = r * s %ₘ q
⊢ map (Ideal.Quotient.mk (P ^ 2)) (f %ₘ q) = 0

Tactic:
suffices h : map (Ideal.Quotient.mk (P ^ 2)) (r * s) = 0 by
        simp [this, h, map_modByMonic, hq_monic]
      ext n
      have h (x : ℕ × ℕ) : (Ideal.Quotient.mk (P ^ 2)) (r.coeff x.1 * s.coeff x.2) = 0 := by
        rw [eq_zero_iff_mem, pow_two]
        apply mul_mem_mul
        · rw [mem_ker, ← coeff_map, hr, coeff_zero]
        · rw [mem_ker, ← coeff_map, hs, coeff_zero]
      simp [-Polynomial.map_mul, coeff_mul, h]

Goal state after:
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
⊢ Commute (star a) a

Tactic:
commute_iff_eq,

Goal state after:
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
⊢ star a * a = a * star 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
⊢ star a * a = a * star a

Tactic:
a.star_eq_two_re_sub

Goal state after:
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
⊢ (↑(2 * a.re + c₂ * a.imI) - a) * a = a * (↑(2 * a.re + c₂ * a.imI) - a)
Module: Mathlib.RingTheory.Polynomial.Eisenstein.Criterion

Goal state before:
pp: case pos
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
f : R[X]
P : Ideal R
hP : P.IsPrime
hfl : f.leadingCoeff ∉ P
hfP : ∀ (n : ℕ), ↑n < f.degree → f.coeff n ∈ P
hfd0 : 0 < f.degree
h0 : f.coeff 0 ∉ P ^ 2
hu : f.IsPrimitive
n : ℕ
hn : n = f.natDegree
⊢ f.coeff f.natDegree - f.leadingCoeff ∈ P

Tactic:
leadingCoeff,

Goal state after:
case pos
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
f : R[X]
P : Ideal R
hP : P.IsPrime
hfl : f.leadingCoeff ∉ P
hfP : ∀ (n : ℕ), ↑n < f.degree → f.coeff n ∈ P
hfd0 : 0 < f.degree
h0 : f.coeff 0 ∉ P ^ 2
hu : f.IsPrimitive
n : ℕ
hn : n = f.natDegree
⊢ f.coeff f.natDegree - f.coeff f.natDegree ∈ P
Module: Mathlib.Algebra.Quaternion

Goal state before:
pp: R : Type u_1
inst✝² : Field R
a✝ b✝ : ℍ[R]
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
a b : ℍ[R]
⊢ 0⁻¹ = 0

Tactic:
inv_def,

Goal state after:
R : Type u_1
inst✝² : Field R
a✝ b✝ : ℍ[R]
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
a b : ℍ[R]
⊢ (normSq 0)⁻¹ • star 0 = 0
Module: Mathlib.Algebra.Quaternion

Goal state before:
pp: R : Type u_1
inst✝² : Field R
a✝¹ b✝ : ℍ[R]
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
a✝ b a : ℍ[R]
ha : a ≠ 0
⊢ a * a⁻¹ = 1

Tactic:
inv_def,

Goal state after:
R : Type u_1
inst✝² : Field R
a✝¹ b✝ : ℍ[R]
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
a✝ b a : ℍ[R]
ha : a ≠ 0
⊢ a * (normSq a)⁻¹ • star a = 1
Module: Mathlib.Algebra.Group.ForwardDiff

Goal state before:
pp: case e'_3
M : Type u_1
G : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : AddCommGroup G
h : M
f : M → G
n : ℕ
y : M
⊢ ∑ k ∈ range (n + 1), n.choose k • Δ_[h]^[k] f y =
    (∑ m ∈ range (n + 1), fwdDiffₗ M G h ^ m * 1 ^ (n - m) * ↑(n.choose m)) f y

Tactic:
· simp [Module.End.pow_apply, coe_fwdDiffₗ]

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

Goal state before:
pp: R : Type u_3
inst✝ : CommRing R
P : R[X]
x : R
⊢ (P.coeff P.natDegree • ↑P.natDegree !) x = P.leadingCoeff • ↑P.natDegree ! x

pp: R : Type u_3
inst✝ : CommRing R
P : R[X]
x : R
i : ℕ
hi : i ∈ range P.natDegree
⊢ (P.coeff i • Δ_[1]^[P.natDegree] fun i_1 ↦ i_1 ^ i) x = 0

Tactic:
leadingCoeff,

Goal state after:
R : Type u_3
inst✝ : CommRing R
P : R[X]
x : R
⊢ (P.coeff P.natDegree • ↑P.natDegree !) x = P.coeff P.natDegree • ↑P.natDegree ! x

R : Type u_3
inst✝ : CommRing R
P : R[X]
x : R
i : ℕ
hi : i ∈ range P.natDegree
⊢ (P.coeff i • Δ_[1]^[P.natDegree] fun i_1 ↦ i_1 ^ i) x = 0
Module: Mathlib.Algebra.Group.Subgroup.Finsupp

Goal state before:
pp: case inv
M : Type u_1
inst✝ : CommGroup M
ι : Type u_2
f : ι → M
x✝ x : M
hx : x ∈ closure (Set.range f)
hx' : ∃ a, x = a.prod fun x1 x2 ↦ f x1 ^ x2
⊢ ∃ a, x⁻¹ = a.prod fun x1 x2 ↦ f x1 ^ x2

Tactic:
| inv x hx hx' =>

Goal state after:
Module: Mathlib.Algebra.Group.Subgroup.Order

Goal state before:
pp: C : Type u_1
inst✝ : CommGroup C
x y z : Subgroup C
xz : x ≤ z
a : C
ha : a ∈ (x ⊔ y) ⊓ z
⊢ a ∈ x ⊔ y ⊓ z

Tactic:
by
    rw [mem_inf, mem_sup] at ha
    rcases ha with ⟨⟨b, hb, c, hc, rfl⟩, haz⟩
    rw [mem_sup]
    exact ⟨b, hb, c, mem_inf.2 ⟨hc, (mul_mem_cancel_left (xz hb)).1 haz⟩, rfl⟩

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

Goal state before:
pp: ι : Type u_1
M : Type u_2
S : Type u_4
inst✝⁴ : Monoid M
inst✝³ : LinearOrder S
inst✝² : Finite ι
inst✝¹ : CommMonoid S
inst✝ : IsOrderedCancelMonoid S
v : ι → M
f : M →* S
⊢ closure (v '' IsMulIndecomposable.baseOf v f) = closure (v '' {i | 1 < f (v i)})

Tactic:
refine le_antisymm (closure_mono (image_mono <| IsMulIndecomposable.baseOf_subset_one_lt v f))
    (closure_le.mpr ?_)
  rintro - ⟨i, hi : 1 < f (v i), rfl⟩
  by_contra hi'
  let t : Set ι := {i | IsMulIndecomposable v {j | 1 < f (v j)} i}
  let s : Set ι := {j | 1 < f (v j) ∧ v j ∉ closure (v '' t)}
  have hne : s.Nonempty := ⟨i, hi, hi'⟩
  clear! i
  obtain ⟨i, hi⟩ := s.toFinite.exists_minimalFor (f ∘ v) s hne
  have ⟨(hi₀ : 1 < f (v i)), (hi₁ : v i ∉ _)⟩ : i ∈ s := hi.prop
  have hi₂ (k : ι) (hk₀ : 1 < f (v k)) (hk₁ : f (v k) < f (v i)) : v k ∈ closure (v '' t) := by
    by_contra hk₂; exact not_le.mpr hk₁ <| hi.le_of_le ⟨hk₀, hk₂⟩ hk₁.le
  have hi₃ : i ∉ t := by contrapose hi₁; exact subset_closure <| mem_image_of_mem v hi₁
  obtain ⟨j, k, hj, hk, hjk⟩ : ∃ (j k : ι) (hj : 1 < f (v j)) (hk : 1 < f (v k)),
      v i = v j * v k := by
    grind [IsMulIndecomposable]
  have hj' : v j ∈ closure (v '' t) := hi₂ j hj <| by aesop
  have hk' : v k ∈ closure (v '' t) := hi₂ k hk <| by aesop
  replace hjk : v i ∈ closure (v '' t) := hjk ▸ mul_mem hj' hk'
  exact hi₁ hjk

Goal state after:
Module: Mathlib.Algebra.GroupWithZero.Action.Pointwise.Finset

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : DecidableEq β
inst✝¹ : Zero β
inst✝ : SMulZeroClass α β
s : Finset α
hs : s.Nonempty
⊢ 0 ⊆ s • 0

Tactic:
by simpa [mem_smul] using! hs

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

Goal state before:
pp: f : ⦋0⦌ ⟶ ⦋0⦌
⊢ f = 𝟙 ⦋0⦌

Tactic:
by
  apply Subsingleton.elim

Goal state after:
Module: Mathlib.CategoryTheory.Idempotents.FunctorCategories

Goal state before:
pp: J : Type u_1
C : Type u_2
inst✝² : Category.{v_1, u_1} J
inst✝¹ : Category.{v_2, u_2} C
P Q : Karoubi (J ⥤ C)
f : P ⟶ Q
X : J
inst✝ : IsIdempotentComplete C
F : J ⥤ C
p : F ⟶ F
hp : p ≫ p = p
hC : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p
this : ∀ (j : J), HasEqualizer (𝟙 (F.obj j)) (p.app j)
j j' : J
φ : j ⟶ j'
⊢ equalizer.ι (𝟙 (F.obj j)) (p.app j) ≫ F.map φ = (equalizer.ι (𝟙 (F.obj j)) (p.app j) ≫ p.app j) ≫ F.map φ

Tactic:
← Limits.equalizer.condition,

Goal state after:
J : Type u_1
C : Type u_2
inst✝² : Category.{v_1, u_1} J
inst✝¹ : Category.{v_2, u_2} C
P Q : Karoubi (J ⥤ C)
f : P ⟶ Q
X : J
inst✝ : IsIdempotentComplete C
F : J ⥤ C
p : F ⟶ F
hp : p ≫ p = p
hC : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p
this : ∀ (j : J), HasEqualizer (𝟙 (F.obj j)) (p.app j)
j j' : J
φ : j ⟶ j'
⊢ equalizer.ι (𝟙 (F.obj j)) (p.app j) ≫ F.map φ = (equalizer.ι (𝟙 (F.obj j)) (p.app j) ≫ 𝟙 (F.obj j)) ≫ F.map φ
Module: Mathlib.AlgebraicTopology.SimplexCategory.Basic

Goal state before:
pp: case inr.h
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
H : j.castSucc < i
k : Fin (⦋n + 1⦌.len + 1)
hik : i < k
hjk : j.castSucc < k
⊢ i ≤ (k.pred ⋯).castSucc

Tactic:
rwa [Fin.le_castSucc_pred_iff]

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✝ : HasPullbacks C
X✝ Y✝ : SimplicialObject C
x✝² : X✝ ⟶ Y✝
x✝¹ : Mono x✝²
x✝ : ℕ
⊢ Mono (((alternatingFaceMapComplex C).map x✝²).f x✝)

Tactic:
by dsimp; infer_instance

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialObject.Basic

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X✝ X : CosimplicialObject C
X₀ : C
f : X₀ ⟶ X ^⦋0⦌
w : ∀ (i : SimplexCategory) (g₁ g₂ : ⦋0⦌ ⟶ i), f ≫ X.map g₁ = f ≫ X.map g₂
⊢ ∀ ⦃X_1 Y : SimplexCategory⦄ (f_1 : X_1 ⟶ Y),
    ((const C).obj X₀).map f_1 ≫ f ≫ X.map (⦋0⦌.const Y 0) =
      (f ≫ X.map (⦋0⦌.const X_1 0)) ≫ ((𝟭 (CosimplicialObject C)).obj X).map f_1

Tactic:
intro i j g
        dsimp
        rw [Category.id_comp, Category.assoc, ← X.map_comp, w]

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialObject.Homotopy

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X Y : SimplicialObject C
f g : X ⟶ Y
H : Homotopy f g
Y' : SimplicialObject C
p : Y ⟶ Y'
n✝ : ℕ
j : Fin (n✝ + 1)
⊢ (H.h j.succ ≫ p.app (op ⦋n✝ + 1 + 1⦌)) ≫ Y'.δ j.castSucc.succ =
    (H.h j.castSucc ≫ p.app (op ⦋n✝ + 1 + 1⦌)) ≫ Y'.δ j.castSucc.succ

Tactic:
simpa using H.h_succ_comp_δ_castSucc_succ j =≫ p.app _

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

Goal state before:
pp: X : SSet
n : ℕ
x✝¹ : X _⦋n⦌
m : ℕ
f₁ : ⦋n⦌ ⟶ ⦋m⦌
inst✝ : Epi f₁
y₁ : ↑(X.nonDegenerate m)
hy₁ : x✝¹ = (ConcreteCategory.hom (X.map f₁.op)) ↑y₁
f₂ : ⦋n⦌ ⟶ ⦋m⦌
y₂ : ↑(X.nonDegenerate m)
hy₂ : x✝¹ = (ConcreteCategory.hom (X.map f₂.op)) ↑y₂
x : Fin (⦋n⦌.len + 1)
hf : SplitEpi f₁
α : Fin (m + 1) → Fin (n + 1) :=
  fun y ↦ if y = (SimplexCategory.Hom.toOrderHom f₁) x then x else (SimplexCategory.Hom.toOrderHom hf.section_) y
hα₁ : ∀ (y : Fin (m + 1)), (SimplexCategory.Hom.toOrderHom f₁) (α y) = y
hα₂ : Monotone α
x✝ : Fin (⦋m⦌.len + 1)
⊢ (SimplexCategory.Hom.toOrderHom (SimplexCategory.Hom.mk { toFun := α, monotone' := hα₂ } ≫ f₁)) x✝ =
    (SimplexCategory.Hom.toOrderHom (𝟙 ⦋m⦌)) x✝

Tactic:
apply hα₁

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

Goal state before:
pp: n : ℕ
S : Finset (Fin (n + 1))
m : ℕ
e : Fin (m + 1) ≃o ↥S
j : SimplexCategory
x✝ : (face S).toSSet.obj (op j)
x : Δ[n].obj (op j)
hx : x ∈ (face S).obj (op j)
⊢ (fun f ↦
        ⟨objMk ((OrderHom.Subtype.val fun x ↦ x ∈ S).comp (e.toOrderEmbedding.toOrderHom.comp (Hom.toOrderHom f))), ⋯⟩)
      ((fun x ↦
          match x with
          | ⟨x, hx⟩ => Hom.mk { toFun := fun i ↦ e.symm ⟨(Hom.toOrderHom (objEquiv x)) i, ⋯⟩, monotone' := ⋯ })
        ⟨x, hx⟩) =
    ⟨x, hx⟩

Tactic:
induction j using SimplexCategory.rec with | _ j
         =>  _ 

Goal state after:
case mk
n : ℕ
S : Finset (Fin (n + 1))
m : ℕ
e : Fin (m + 1) ≃o ↥S
j : ℕ
x✝ : (face S).toSSet _⦋j⦌
x : Δ[n] _⦋j⦌
hx : x ∈ (face S).obj (op ⦋j⦌)
⊢ (fun f ↦
        ⟨objMk ((OrderHom.Subtype.val fun x ↦ x ∈ S).comp (e.toOrderEmbedding.toOrderHom.comp (Hom.toOrderHom f))), ⋯⟩)
      ((fun x ↦
          match x with
          | ⟨x, hx⟩ => Hom.mk { toFun := fun i ↦ e.symm ⟨(Hom.toOrderHom (objEquiv x)) i, ⋯⟩, monotone' := ⋯ })
        ⟨x, hx⟩) =
    ⟨x, hx⟩
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

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

Tactic:
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

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
i : Fin (n + 2)
j : Fin (n + 1)
hij : j.castSucc < i
k : Fin (n + 1)
hk : j.rev = k
l : ℕ
hl : ↑i = ↑j + 1 + l
this :
  ed.s (↑k + 1) ≫ X.left.δ ⟨l + 1 + 1, ⋯⟩ =
    X.left.δ ⟨l + 1, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.homotopy._proof_24 n i j k hk l hl⟩ ≫
      ed.s ↑k
⊢ h ed j.castSucc ≫ X.left.δ i.succ = X.left.δ i ≫ h ed j

Tactic:
rw [h_eq_assoc _ _ (k + 1), h_eq _ _ k,
      X.left.σ₀Iter_δ' _ _ ⟨l + 2, by grind⟩ (by grind),
      reassoc_of% this, ← X.left.δ_δ₀Iter'_assoc _ _ i (by grind)]

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : Augmented C
ed : X.ExtraDegeneracy
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
hij : j.castSucc < i
k : Fin (n + 1)
hk : j.rev = k
l : ℕ
hl : ↑i = ↑j + 1 + l
this :
  ed.s (↑k + 1) ≫ X.left.δ ⟨l + 1 + 1, ⋯⟩ =
    X.left.δ ⟨l + 1, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.homotopy._proof_24 n i j k hk l hl⟩ ≫
      ed.s ↑k
⊢ X.left.δ i ≫ X.left.δ₀Iter ↑j.castSucc ⋯ ≫ ed.s ↑k ≫ X.left.σ₀Iter ↑j.castSucc ⋯ =
    X.left.δ i ≫ X.left.δ₀Iter ↑j ⋯ ≫ ed.s ↑k ≫ X.left.σ₀Iter ↑j ⋯
Module: Mathlib.Algebra.Homology.Bifunctor

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝⁷ : Category.{v_1, u_1} C₁
inst✝⁶ : Category.{v_2, u_2} C₂
inst✝⁵ : Category.{v_3, u_3} D
inst✝⁴ : HasZeroMorphisms C₁
inst✝³ : HasZeroMorphisms C₂
inst✝² : HasZeroMorphisms D
F : C₁ ⥤ C₂ ⥤ D
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
inst✝¹ : F.PreservesZeroMorphisms
inst✝ : ∀ (X₁ : C₁), (F.obj X₁).PreservesZeroMorphisms
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
i₁ : I₁
i₂ i₂' : I₂
h₂ : ¬c₂.Rel i₂ i₂'
⊢ (F.obj (K₁.X i₁)).map (K₂.d i₂ i₂') = 0

Tactic:
K₂.shape _ _ h₂,

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝⁷ : Category.{v_1, u_1} C₁
inst✝⁶ : Category.{v_2, u_2} C₂
inst✝⁵ : Category.{v_3, u_3} D
inst✝⁴ : HasZeroMorphisms C₁
inst✝³ : HasZeroMorphisms C₂
inst✝² : HasZeroMorphisms D
F : C₁ ⥤ C₂ ⥤ D
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
inst✝¹ : F.PreservesZeroMorphisms
inst✝ : ∀ (X₁ : C₁), (F.obj X₁).PreservesZeroMorphisms
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
i₁ : I₁
i₂ i₂' : I₂
h₂ : ¬c₂.Rel i₂ i₂'
⊢ (F.obj (K₁.X i₁)).map 0 = 0
Module: Mathlib.CategoryTheory.Shift.Twist

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
A : Type w
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
t : TwistShiftData C A
a b c : A
X : C
⊢ (shiftFunctor C c).map ((↑(t.z a b)).app X) = (↑(t.z a b)).app ((shiftFunctor C c).obj X)

Tactic:
by
  simpa using NatTrans.shift_app_comm (t.z a b).val c X

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✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₁'
inst✝⁹ : Category.{v_3, u_3} C₂
inst✝⁸ : Category.{v_4, u_4} C₂'
inst✝⁷ : Category.{v_5, u_5} D
M : Type u_6
inst✝⁶ : AddCommMonoid M
inst✝⁵ : HasShift C₁ M
inst✝⁴ : HasShift C₁' M
inst✝³ : HasShift C₂ M
inst✝² : HasShift D M
F : C₁' ⥤ C₁
inst✝¹ : F.CommShift M
G : C₁ ⥤ C₂ ⥤ D
h : CommShift₂Setup D M
inst✝ : G.CommShift₂ h
X₁' Y₁' : C₁'
f : X₁' ⟶ Y₁'
⊢ NatTrans.CommShift ((F ⋙ G).map f) M

Tactic:
by dsimp; infer_instance

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✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₁'
inst✝⁹ : Category.{v_3, u_3} C₂
inst✝⁸ : Category.{v_4, u_4} C₂'
inst✝⁷ : Category.{v_5, u_5} D
M : Type u_6
inst✝⁶ : AddCommMonoid M
inst✝⁵ : HasShift C₁ M
inst✝⁴ : HasShift C₂' M
inst✝³ : HasShift C₂ M
inst✝² : HasShift D M
F : C₂' ⥤ C₂
inst✝¹ : F.CommShift M
G : C₁ ⥤ C₂ ⥤ D
h : CommShift₂Setup D M
inst✝ : G.CommShift₂ h
X₁ Y₁ : C₁
f : X₁ ⟶ Y₁
⊢ NatTrans.CommShift ((G ⋙ (whiskeringLeft C₂' C₂ D).obj F).map f) M

Tactic:
by dsimp; infer_instance

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✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₁'
inst✝⁹ : Category.{v_3, u_3} C₂
inst✝⁸ : Category.{v_4, u_4} C₂'
inst✝⁷ : Category.{v_5, u_5} D
M : Type u_6
inst✝⁶ : AddCommMonoid M
inst✝⁵ : HasShift C₁ M
inst✝⁴ : HasShift C₂ M
inst✝³ : HasShift D M
G₁ G₂ G₃ : C₁ ⥤ C₂ ⥤ D
τ : G₁ ⟶ G₂
τ' : G₂ ⟶ G₃
h : CommShift₂Setup D M
inst✝² : G₁.CommShift₂ h
inst✝¹ : G₂.CommShift₂ h
inst✝ : G₃.CommShift₂ h
x✝ : C₁
⊢ CommShift ((𝟙 G₁).app x✝) M

Tactic:
by dsimp; infer_instance

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✝¹³ : Category.{v_1, u_1} C₁
inst✝¹² : Category.{v_2, u_2} C₁'
inst✝¹¹ : Category.{v_3, u_3} C₂
inst✝¹⁰ : Category.{v_4, u_4} C₂'
inst✝⁹ : Category.{v_5, u_5} D
M : Type u_6
inst✝⁸ : AddCommMonoid M
inst✝⁷ : HasShift C₁ M
inst✝⁶ : HasShift C₂ M
inst✝⁵ : HasShift D M
G₁ G₂ G₃ : C₁ ⥤ C₂ ⥤ D
τ : G₁ ⟶ G₂
τ' : G₂ ⟶ G₃
h : CommShift₂Setup D M
inst✝⁴ : G₁.CommShift₂ h
inst✝³ : G₂.CommShift₂ h
inst✝² : G₃.CommShift₂ h
inst✝¹ : CommShift₂ τ h
inst✝ : CommShift₂ τ' h
x✝ : C₁
⊢ CommShift ((τ ≫ τ').app x✝) M

Tactic:
by dsimp; infer_instance

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

Goal state before:
pp: case pos
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + x = n₀'
h₁ : n₁ + x = n₁'
h : n₀ + 1 = n₁
p q : ℤ
hpq : p + q = n₀
⊢ (((shiftFunctor₁ C x).obj K).d₁ (up ℤ) p q n₁ ≫
      ((shiftFunctor₁ C x).obj K).totalDesc fun p q hpq ↦ K.ιTotal (up ℤ) (p + x) q n₁' ⋯) =
    x.negOnePow • K.d₁ (up ℤ) (p + x) q n₁'

Tactic:
((shiftFunctor₁ C x).obj K).d₁_eq _ rfl _ _ (by dsimp; lia),

Goal state after:
case pos
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + x = n₀'
h₁ : n₁ + x = n₁'
h : n₀ + 1 = n₁
p q : ℤ
hpq : p + q = n₀
⊢ (((up ℤ).ε₁ (up ℤ) (up ℤ) (p, q) •
        (((shiftFunctor₁ C x).obj K).d p (p + 1)).f q ≫ ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) (p + 1) q n₁ ⋯) ≫
      ((shiftFunctor₁ C x).obj K).totalDesc fun p q hpq ↦ K.ιTotal (up ℤ) (p + x) q n₁' ⋯) =
    x.negOnePow • K.d₁ (up ℤ) (p + x) q n₁'
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 (S.X₂.pOpcycles i ≫ HomologicalComplex.opcyclesMap S.g i))) x₂ =
    (ConcreteCategory.hom ((forget₂ C Ab).map (S.X₃.homologyπ i ≫ S.X₃.homologyι i))) (S.X₃.cyclesMk x₃ j ⋯ hx₃)

Tactic:
HomologicalComplex.p_opcyclesMap,

Goal state after:
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 (S.g.f i ≫ S.X₃.pOpcycles i))) x₂ =
    (ConcreteCategory.hom ((forget₂ C Ab).map (S.X₃.homologyπ i ≫ S.X₃.homologyι i))) (S.X₃.cyclesMk x₃ j ⋯ hx₃)
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]

Goal state after:
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) ⋯ ≫ snd = s.ι ≫ biprod.snd
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy

Goal state before:
pp: case neg
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
h : Homotopy f g
e : c.Embedding c'
inst✝ : e.IsRelIff
i : ι
hi : ¬c.Rel i (c.next i)
hi' : ¬c'.Rel (e.f i) (c'.next (e.f i))
⊢ (K.extendXIso e ⋯).hom ≫ 0 ≫ (L.extendXIso e ⋯).inv = (dNext (e.f i)) (extend.hom e h.hom)

Tactic:
· simp [dNext_eq_zero _ _ hi']

Goal state after: