mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.Eigenspace.Basic

Goal state before:
pp: R : Type v
M : Type w
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R M
f : End R M
μ₁ μ₂ : R
hμ : μ₁ ≠ μ₂
k✝ l✝ : ℕ∞
k : ℕ
property✝¹ : ↑k ≤ k✝
l : ℕ
property✝ : ↑l ≤ l✝
a✝ : Nontrivial M
p : Submodule R M := (f.genEigenspace μ₁) ↑k ⊓ (f.genEigenspace μ₂) ↑l
hp : Nontrivial ↥p
⊢ False

Tactic:
let f₁ : End R p := (f - algebraMap R (End R M) μ₁).restrict <| MapsTo.inter_inter
    (mapsTo_genEigenspace_of_comm (Algebra.mul_sub_algebraMap_commutes f μ₁) μ₁ k)
    (mapsTo_genEigenspace_of_comm (Algebra.mul_sub_algebraMap_commutes f μ₁) μ₂ l)

Goal state after:
R : Type v
M : Type w
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R M
f : End R M
μ₁ μ₂ : R
hμ : μ₁ ≠ μ₂
k✝ l✝ : ℕ∞
k : ℕ
property✝¹ : ↑k ≤ k✝
l : ℕ
property✝ : ↑l ≤ l✝
a✝ : Nontrivial M
p : Submodule R M := (f.genEigenspace μ₁) ↑k ⊓ (f.genEigenspace μ₂) ↑l
hp : Nontrivial ↥p
f₁ : End R ↥p := LinearMap.restrict (f - (algebraMap R (End R M)) μ₁) ⋯
⊢ False
Module: Mathlib.LinearAlgebra.Eigenspace.Basic

Goal state before:
pp: R : Type v
M : Type w
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R M
f : End R M
k : ℕ∞
μ₁ μ₂ : R
s : Finset R
a✝ : μ₂ ∉ s
ih : Disjoint ((f.genEigenspace μ₁) k) (s.sup fun μ ↦ (f.genEigenspace μ) k)
hμ₁₂✝ : μ₁ ∉ insert μ₂ s
hμ₁₂ : μ₁ ≠ μ₂
hμ₁ : μ₁ ∉ s
x : M
hx : x ∈ (f.genEigenspace μ₁) k
hx' : x ∈ (f.genEigenspace μ₂) k ⊔ s.sup fun μ ↦ (f.genEigenspace μ) k
this : x ∈ (f.genEigenspace μ₂) k
⊢ x ∈ (f.genEigenspace μ₁) k ⊓ (f.genEigenspace μ₂) k

Tactic:
exact ⟨hx, this⟩

Goal state after:
Module: Mathlib.Algebra.Field.ZMod

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
a : ZMod p
h : a ≠ 0
⊢ a * a⁻¹ = 1

Tactic:
obtain ⟨k, rfl⟩ := natCast_zmod_surjective a

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
k : ℕ
h : ↑k ≠ 0
⊢ ↑k * (↑k)⁻¹ = 1
Module: Mathlib.Algebra.Field.TransferInstance

Goal state before:
pp: α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
⊢ DivisionRing α

Tactic:
by
  let add_group_with_one := e.addGroupWithOne
  let inv := e.Inv
  let div := e.div
  let mul := e.mul
  let npow := e.pow ℕ
  let zpow := e.pow ℤ
  let nnratCast := e.nnratCast
  let ratCast := e.ratCast
  let nnqsmul := e.smul ℚ≥0
  let qsmul := e.smul ℚ
  apply e.injective.divisionRing _ <;> intros <;> exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Free

Goal state before:
pp: α✝ β✝ : Type u
m : Type u → Type u
inst✝⁴ : Applicative m
F : α✝ → m β✝
F✝ G✝ : Type u → Type u
inst✝³ : Applicative F✝
inst✝² : Applicative G✝
inst✝¹ : LawfulApplicative F✝
inst✝ : LawfulApplicative G✝
η : ApplicativeTransformation F✝ G✝
α β : Type u
f : α → F✝ β
x✝ x y : FreeMagma α
ih1 : (fun {α} ↦ η.app α) (traverse f x) = traverse ((fun {α} ↦ η.app α) ∘ f) x
ih2 : (fun {α} ↦ η.app α) (traverse f y) = traverse ((fun {α} ↦ η.app α) ∘ f) y
⊢ (fun {α} ↦ η.app α) (traverse f (x * y)) = traverse ((fun {α} ↦ η.app α) ∘ f) (x * y)

Tactic:
simp only [traverse_mul, functor_norm, ih1, ih2]

Goal state after:
Module: Mathlib.Algebra.Free

Goal state before:
pp: α✝ β✝ : Type u
m : Type u → Type u
inst✝⁴ : Applicative m
F : α✝ → m β✝
F✝ G✝ : Type u → Type u
inst✝³ : Applicative F✝
inst✝² : Applicative G✝
inst✝¹ : LawfulApplicative F✝
inst✝ : LawfulApplicative G✝
η : ApplicativeTransformation F✝ G✝
α β : Type u
f : α → F✝ β
x✝ : FreeSemigroup α
x : α
y : FreeSemigroup α
ih1 : (fun {α} ↦ η.app α) (traverse f (pure x)) = traverse ((fun {α} ↦ η.app α) ∘ f) (pure x)
ih2 : (fun {α} ↦ η.app α) (traverse f y) = traverse ((fun {α} ↦ η.app α) ∘ f) y
⊢ (fun {α} ↦ η.app α) (traverse f (pure x * y)) = traverse ((fun {α} ↦ η.app α) ∘ f) (pure x * y)

Tactic:
simp only [traverse_mul, functor_norm, ih1, ih2]

Goal state after:
Module: Mathlib.Algebra.GCDMonoid.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
h : IsGCDMonoid R
X : FractionRing R
x✝ : IsIntegral R X
p : R[X]
hp₁ : p.Monic
hp₂ : Polynomial.eval₂ (algebraMap R (FractionRing R)) X p = 0
⊢ ∃ y, (algebraMap R (FractionRing R)) y = X

Tactic:
by
    cases h
    obtain ⟨x, y, hg, he⟩ := IsLocalization.surj_of_gcd_domain (nonZeroDivisors R) X
    have :=
      Polynomial.dvd_pow_natDegree_of_eval₂_eq_zero (IsFractionRing.injective R <| FractionRing R)
        hp₁ y x _ hp₂ (by rw [mul_comm, he])
    have : IsUnit y := by
      rw [isUnit_iff_dvd_one, ← one_pow]
      exact
        (dvd_gcd this <| dvd_refl y).trans
          (gcd_pow_left_dvd_pow_gcd.trans <| pow_dvd_pow_of_dvd (isUnit_iff_dvd_one.1 hg) _)
    use x * (this.unit⁻¹ :)
    rw [map_mul]
    have coe_map_inv :=
      Units.coe_map_inv ((algebraMap R (FractionRing R) : R →* FractionRing R)) this.unit
    simp only [MonoidHom.coe_coe] at coe_map_inv
    rw [← coe_map_inv, eq_comm, Units.eq_mul_inv_iff_mul_eq]
    exact he

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

Goal state before:
pp: X : Type u_1
G : Type u_2
inst✝ : SMul G X
⊢ Injective toPartialEquiv

Tactic:
by
  intro ⟨_, _, _⟩ _ _
  congr

Goal state after:
Module: Mathlib.Algebra.Quaternion

Goal state before:
pp: R : Type u_3
c₁ c₂ c₃ r : R
a : ℍ[R,c₁,c₂,c₃]
inst✝ : CommRing R
⊢ a * ↑r = r • a

Tactic:
by rw [← coe_commutes, coe_mul_eq_smul]

Goal state after:
Module: Mathlib.Algebra.Quaternion

Goal state before:
pp: R : Type u_3
inst✝ : CommRing R
x : R
⊢ normSq ↑x = x ^ 2

Tactic:
rw [normSq_def, star_coe, ← coe_mul, re_coe, sq]

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

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝² : DecidableEq β
inst✝¹ : Group α
inst✝ : MulAction α β
s t : Finset β
a : α
⊢ ↑s ⊆ a • ↑t ↔ a⁻¹ • ↑s ⊆ ↑t

Tactic:
exact Set.subset_smul_set_iff

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

Goal state before:
pp: R : Type u_3
inst✝ : CommRing R
P : R[X]
x : R
j : ℕ
hP : P.natDegree < P.natDegree + j + 1
⊢ Δ_[1]^[j] (Δ_[1] fun i ↦ P.leadingCoeff • ↑P.natDegree ! i) x = 0 x

Tactic:
simp [fwdDiff_iter_eq_sum_shift]

Goal state after:
Module: Mathlib.Algebra.Group.Submonoid.Support

Goal state before:
pp: G : Type u_1
inst✝ : Group G
M : Submonoid G
h : M.mulSupport = ⊥
x : G
⊢ x ∈ M → x⁻¹ ∈ M → x = 1

Tactic:
by
    apply_fun (x ∈ ·) at h
    aesop

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
t : Set ι := {i | IsMulIndecomposable v {j | 1 < f (v j)} i}
s : Set ι := {j | 1 < f (v j) ∧ v j ∉ closure (v '' t)}
hne : s.Nonempty
i : ι
hi : MinimalFor (fun x ↦ x ∈ s) (⇑f ∘ v) i
hi₀ : 1 < f (v i)
hi₁ : v i ∉ closure (v '' t)
hi₂ : ∀ (k : ι), 1 < f (v k) → f (v k) < f (v i) → v k ∈ closure (v '' t)
hi₃ : i ∉ t
⊢ ∃ j k, ∃ (_ : 1 < f (v j)) (_ : 1 < f (v k)), v i = v j * v k

Tactic:
by
    grind [IsMulIndecomposable]

Goal state after:
Module: Mathlib.Algebra.GroupWithZero.ProdHom

Goal state before:
pp: G₀ : Type u_1
H₀ : Type u_2
inst✝² : GroupWithZero G₀
inst✝¹ : GroupWithZero H₀
inst✝ : DecidablePred fun x ↦ x = 0
⊢ (fst G₀ H₀).comp (inr G₀ H₀) = 1

Tactic:
by
  ext x
  obtain rfl | ⟨_, rfl⟩ := GroupWithZero.eq_zero_or_unit x <;>
  simp_all [WithZero.withZeroUnitsEquiv, fst, inr]

Goal state after:
Module: Mathlib.Algebra.GroupWithZero.Torsion

Goal state before:
pp: M : Type u_1
inst✝³ : CommMonoidWithZero M
inst✝² : UniqueFactorizationMonoid M
inst✝¹ : NormalizationMonoid M
inst✝ : IsMulTorsionFree Mˣ
x y : M
hy : y ≠ 0
n : ℕ
hn : n ≠ 0
u : Mˣ
hxy : u = 1
hu : x * ↑1 = y
hx : IsLeftRegular (x ^ n)
⊢ x = y

Tactic:
Units.val_one,

Goal state after:
M : Type u_1
inst✝³ : CommMonoidWithZero M
inst✝² : UniqueFactorizationMonoid M
inst✝¹ : NormalizationMonoid M
inst✝ : IsMulTorsionFree Mˣ
x y : M
hy : y ≠ 0
n : ℕ
hn : n ≠ 0
u : Mˣ
hxy : u = 1
hu : x * 1 = y
hx : IsLeftRegular (x ^ n)
⊢ x = y
Module: Mathlib.Algebra.GroupWithZero.Range

Goal state before:
pp: case refine_1
A : Type u_1
B : Type u_2
inst✝¹ : MonoidWithZero A
inst✝ : CommGroupWithZero B
f : A →*₀ B
y : Bˣ
hy : y ∈ Subgroup.closure ↑{ carrier := Units.val ⁻¹' range ⇑f, mul_mem' := ⋯, one_mem' := ⋯ }
⊢ ∃ a, f a ≠ 0 ∧ ∃ x, f a * ↑y = f x

Tactic:
induction hy using Subgroup.closure_induction with

Goal state after:
Module: Mathlib.AlgebraicTopology.MooreComplex

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X Y : SimplicialObject C
f✝ f : X ⟶ Y
n : ℕ
⊢ (objX Y n).Factors ((objX X n).arrow ≫ f.app (op ⦋n⦌))

Tactic:
cases n <;> dsimp

Goal state after:
case zero
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X Y : SimplicialObject C
f✝ f : X ⟶ Y
⊢ ⊤.Factors (⊤.arrow ≫ f.app (op ⦋0⦌))

case succ
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X Y : SimplicialObject C
f✝ f : X ⟶ Y
n✝ : ℕ
⊢ (Finset.univ.inf fun k ↦ kernelSubobject (Y.δ k.succ)).Factors
    ((Finset.univ.inf fun k ↦ kernelSubobject (X.δ k.succ)).arrow ≫ f.app (op ⦋n✝ + 1⦌))
Module: Mathlib.AlgebraicTopology.SimplexCategory.Basic

Goal state before:
pp: f : ⦋1⦌ ⟶ ⦋1⦌
e0 : (Hom.toOrderHom f) 0 = 0
e1 : (Hom.toOrderHom f) 1 = 0
⊢ (∃ a, f = ⦋1⦌.const ⦋1⦌ a) ∨ f = 𝟙 ⦋1⦌

Tactic:
refine .inl ⟨f.toOrderHom 0, ?_⟩

Goal state after:
f : ⦋1⦌ ⟶ ⦋1⦌
e0 : (Hom.toOrderHom f) 0 = 0
e1 : (Hom.toOrderHom f) 1 = 0
⊢ f = ⦋1⦌.const ⦋1⦌ ((Hom.toOrderHom f) 0)
Module: Mathlib.AlgebraicTopology.SimplexCategory.Basic

Goal state before:
pp: f : ⦋1⦌ ⟶ ⦋1⦌
e0 : (Hom.toOrderHom f) 0 = 1
e1 : (Hom.toOrderHom f) 1 = 1
⊢ (∃ a, f = ⦋1⦌.const ⦋1⦌ a) ∨ f = 𝟙 ⦋1⦌

Tactic:
refine .inl ⟨f.toOrderHom 0, ?_⟩

Goal state after:
f : ⦋1⦌ ⟶ ⦋1⦌
e0 : (Hom.toOrderHom f) 0 = 1
e1 : (Hom.toOrderHom f) 1 = 1
⊢ f = ⦋1⦌.const ⦋1⦌ ((Hom.toOrderHom f) 0)
Module: Mathlib.AlgebraicTopology.SimplexCategory.Basic

Goal state before:
pp: n : ℕ
i : Fin (n + 3)
j : Fin (n + 2)
H : i ≤ j.castSucc
⊢ i.castLT ⋯ ≤ j

Tactic:
exact H

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

Goal state before:
pp: n : ℕ
i : Fin (n + 3)
j : Fin (n + 2)
H : j.succ < i
⊢ j.succ < i

Tactic:
exact H

Goal state after:
Module: Mathlib.AlgebraicTopology.AlternatingFaceMapComplex

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
A : Type u_2
inst✝¹ : Category.{v_2, u_2} A
inst✝ : Abelian A
X : SimplicialObject A
i : ℕ
⊢ (NormalizedMooreComplex.objX X (i + 1)).arrow ≫ ((-1) ^ ↑0 • X.δ 0) +
      ∑ i_1, (NormalizedMooreComplex.objX X (i + 1)).arrow ≫ ((-1) ^ ↑i_1.succ • X.δ i_1.succ) =
    NormalizedMooreComplex.objD X i ≫ (NormalizedMooreComplex.objX X i).arrow

Tactic:
Fintype.sum_eq_zero

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
A : Type u_2
inst✝¹ : Category.{v_2, u_2} A
inst✝ : Abelian A
X : SimplicialObject A
i : ℕ
⊢ (NormalizedMooreComplex.objX X (i + 1)).arrow ≫ ((-1) ^ ↑0 • X.δ 0) + 0 =
    NormalizedMooreComplex.objD X i ≫ (NormalizedMooreComplex.objX X i).arrow

case h
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
A : Type u_2
inst✝¹ : Category.{v_2, u_2} A
inst✝ : Abelian A
X : SimplicialObject A
i : ℕ
⊢ ∀ (a : Fin (i + 1)), (NormalizedMooreComplex.objX X (i + 1)).arrow ≫ ((-1) ^ ↑a.succ • X.δ a.succ) = 0
Module: Mathlib.AlgebraicTopology.SimplexCategory.Basic

Goal state before:
pp: n : ℕ
θ : ⦋n⦌ ⟶ ⦋n + 1⦌
inst✝ : Mono θ
⊢ ¬Function.Surjective ⇑(Hom.toOrderHom θ)

Tactic:
by
    rw [← epi_iff_surjective]
    grind [→ le_of_epi]

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

Goal state before:
pp: X : SSet
s t : X.S
⊢ s ≤ t ↔ ∃ f, (ConcreteCategory.hom (X.map f.op)) t.simplex = s.simplex

Tactic:
by
  rw [le_def, Subcomplex.ofSimplex_le_iff, Subfunctor.ofSection_obj, Set.mem_ofPred_eq]
  tauto

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

Goal state before:
pp: X : SSet
n m : ℕ
f : ⦋n⦌ ⟶ ⦋m⦌
x : X _⦋m⦌
⊢ ofSimplex ((ConcreteCategory.hom (X.map f.op)) x) ≤ ofSimplex x

Tactic:
simp only [Subfunctor.ofSection_le_iff]

Goal state after:
X : SSet
n m : ℕ
f : ⦋n⦌ ⟶ ⦋m⦌
x : X _⦋m⦌
⊢ (ConcreteCategory.hom (X.map f.op)) x ∈ (ofSimplex x).obj (Opposite.op ⦋n⦌)
Module: Mathlib.AlgebraicTopology.SimplicialSet.Subcomplex

Goal state before:
pp: X : SSet
n m : ℕ
f : ⦋n⦌ ⟶ ⦋m⦌
inst✝ : Epi f
x : X _⦋m⦌
⊢ ofSimplex x ≤ ofSimplex ((ConcreteCategory.hom (X.map f.op)) x)

Tactic:
simp only [Subfunctor.ofSection_le_iff]

Goal state after:
X : SSet
n m : ℕ
f : ⦋n⦌ ⟶ ⦋m⦌
inst✝ : Epi f
x : X _⦋m⦌
⊢ x ∈ (ofSimplex ((ConcreteCategory.hom (X.map f.op)) x)).obj (Opposite.op ⦋m⦌)
Module: Mathlib.AlgebraicTopology.CechNerve

Goal state before:
pp: case id
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasTerminal C
ι : Type w
inst✝¹ : HasFiniteProducts C
inst✝ : Finite ι
X : C
i : WidePullbackShape ι
⊢ (((Functor.const (WidePullbackShape ι)).obj (∏ᶜ fun x ↦ X)).map (WidePullbackShape.Hom.id i) ≫
      Option.casesOn i (terminal.from (((Functor.const (WidePullbackShape ι)).obj (∏ᶜ fun x ↦ X)).obj none)) fun i ↦
        limit.π (Discrete.functor fun x ↦ X) { as := i }) =
    (Option.casesOn i (terminal.from (((Functor.const (WidePullbackShape ι)).obj (∏ᶜ fun x ↦ X)).obj none)) fun i ↦
        limit.π (Discrete.functor fun x ↦ X) { as := i }) ≫
      (wideCospan ι X).map (WidePullbackShape.Hom.id i)

pp: case term
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasTerminal C
ι : Type w
inst✝¹ : HasFiniteProducts C
inst✝ : Finite ι
X : C
j✝ : ι
⊢ (((Functor.const (WidePullbackShape ι)).obj (∏ᶜ fun x ↦ X)).map (WidePullbackShape.Hom.term j✝) ≫
      Option.casesOn none (terminal.from (((Functor.const (WidePullbackShape ι)).obj (∏ᶜ fun x ↦ X)).obj none)) fun i ↦
        limit.π (Discrete.functor fun x ↦ X) { as := i }) =
    (Option.casesOn (some j✝) (terminal.from (((Functor.const (WidePullbackShape ι)).obj (∏ᶜ fun x ↦ X)).obj none))
        fun i ↦ limit.π (Discrete.functor fun x ↦ X) { as := i }) ≫
      (wideCospan ι X).map (WidePullbackShape.Hom.term j✝)

Tactic:
· cases i
              all_goals simp

Goal state after:
case term
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasTerminal C
ι : Type w
inst✝¹ : HasFiniteProducts C
inst✝ : Finite ι
X : C
j✝ : ι
⊢ (((Functor.const (WidePullbackShape ι)).obj (∏ᶜ fun x ↦ X)).map (WidePullbackShape.Hom.term j✝) ≫
      Option.casesOn none (terminal.from (((Functor.const (WidePullbackShape ι)).obj (∏ᶜ fun x ↦ X)).obj none)) fun i ↦
        limit.π (Discrete.functor fun x ↦ X) { as := i }) =
    (Option.casesOn (some j✝) (terminal.from (((Functor.const (WidePullbackShape ι)).obj (∏ᶜ fun x ↦ X)).obj none))
        fun i ↦ limit.π (Discrete.functor fun x ↦ X) { as := i }) ≫
      (wideCospan ι X).map (WidePullbackShape.Hom.term j✝)
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)
⊢ Finite X.N

Tactic:
refine Finite.of_surjective (α := Σ (i : Fin d), X.nonDegenerate i)
      (f := fun ⟨i, x⟩ ↦ N.mk _ x.property) (fun x ↦ ?_)

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

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
this✝ : 0 < n
Δ : SimplexCategory
this : Nonempty (CostructuredArrow (inclusion n) Δ)
⊢ ∀ (j₁ j₂ : CostructuredArrow (inclusion n) Δ), Zigzag j₁ j₂

Tactic:
rintro ⟨⟨Δ₁, hΔ₁⟩, ⟨⟨⟩⟩, f⟩ ⟨⟨Δ₂, hΔ₂⟩, ⟨⟨⟩⟩, f'⟩

Goal state after:
n : ℕ
inst✝ : NeZero n
this✝ : 0 < n
Δ : SimplexCategory
this : Nonempty (CostructuredArrow (inclusion n) Δ)
Δ₁ : SimplexCategory
hΔ₁ : Δ₁.len ≤ n
f : (inclusion n).obj { obj := Δ₁, property := hΔ₁ } ⟶ (Functor.fromPUnit Δ).obj { as := PUnit.unit }
Δ₂ : SimplexCategory
hΔ₂ : Δ₂.len ≤ n
f' : (inclusion n).obj { obj := Δ₂, property := hΔ₂ } ⟶ (Functor.fromPUnit Δ).obj { as := PUnit.unit }
⊢ Zigzag { left := { obj := Δ₁, property := hΔ₁ }, right := { as := PUnit.unit }, hom := f }
    { left := { obj := Δ₂, property := hΔ₂ }, right := { as := PUnit.unit }, hom := f' }
Module: Mathlib.Algebra.Homology.Augment

Goal state before:
pp: 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 →
    (match i with
        | 0 => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X 0)
        | n.succ => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X (n + 1))) ≫
        C.d i j =
      ((truncate.obj C).augment (C.d 0 1) ⋯).d i j ≫
        match j with
        | 0 => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X 0)
        | n.succ => 𝟙 (((truncate.obj C).augment (C.d 0 1) ⋯).X (n + 1))

Tactic:
rcases j with (_ | _ | j) <;> cases i <;> aesop

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

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝¹⁰ : Category.{v_1, u_1} C₁
inst✝⁹ : Category.{v_2, u_2} C₂
inst✝⁸ : Category.{v_3, u_3} D
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
inst✝⁷ : HasZeroMorphisms C₁
inst✝⁶ : HasZeroMorphisms C₂
inst✝⁵ : Preadditive D
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
F : C₁ ⥤ C₂ ⥤ D
inst✝⁴ : F.PreservesZeroMorphisms
inst✝³ : ∀ (X₁ : C₁), (F.obj X₁).PreservesZeroMorphisms
c : ComplexShape J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : DecidableEq J
A : D
j : J
f : (i₁ : I₁) → (i₂ : I₂) → c₁.π c₂ c (i₁, i₂) = j → ((F.obj (K₁.X i₁)).obj (K₂.X i₂) ⟶ A)
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c (i₁, i₂) = j
⊢ K₁.ιMapBifunctor K₂ F c i₁ i₂ j h ≫ mapBifunctorDesc f = f i₁ i₂ h

Tactic:
apply HomologicalComplex₂.ι_totalDesc

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

Goal state before:
pp: case neg
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 : HomologicalComplex₂ C c₁ c₂
c₁₂ : ComplexShape I₁₂
inst✝² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹ : DecidableEq I₁₂
inst✝ : K.HasTotal c₁₂
i₁₂ i₁₂' i₁₂'' : I₁₂
h₁ : c₁₂.Rel i₁₂ i₁₂'
h₂ : c₁₂.Rel i₁₂' i₁₂''
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c₁₂ (i₁, i₂) = i₁₂
h₃ : ¬c₁.Rel i₁ (c₁.next i₁)
⊢ K.d₁ c₁₂ i₁ i₂ i₁₂' ≫ K.D₁ c₁₂ i₁₂' i₁₂'' = 0

Tactic:
rw [K.d₁_eq_zero c₁₂ _ _ _ h₃, zero_comp]

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

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : 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✝⁴ : TotalComplexShape c₁ c₂ c₁₂
inst✝³ : DecidableEq I₁₂
inst✝² : K.HasTotal c₁₂
inst✝¹ : L.HasTotal c₁₂
inst✝ : M.HasTotal c₁₂
⊢ map (𝟙 K) c₁₂ = 𝟙 (K.total c₁₂)

Tactic:
map_id

Goal state after:
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : 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✝⁴ : TotalComplexShape c₁ c₂ c₁₂
inst✝³ : DecidableEq I₁₂
inst✝² : K.HasTotal c₁₂
inst✝¹ : L.HasTotal c₁₂
inst✝ : M.HasTotal c₁₂
⊢ 𝟙 (K.total c₁₂) = 𝟙 (K.total c₁₂)
Module: Mathlib.Algebra.Homology.TotalComplex

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : 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✝⁴ : TotalComplexShape c₁ c₂ c₁₂
inst✝³ : DecidableEq I₁₂
inst✝² : K.HasTotal c₁₂
inst✝¹ : L.HasTotal c₁₂
inst✝ : M.HasTotal c₁₂
⊢ map (𝟙 L) c₁₂ = 𝟙 (L.total c₁₂)

Tactic:
map_id

Goal state after:
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : 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✝⁴ : TotalComplexShape c₁ c₂ c₁₂
inst✝³ : DecidableEq I₁₂
inst✝² : K.HasTotal c₁₂
inst✝¹ : L.HasTotal c₁₂
inst✝ : M.HasTotal c₁₂
⊢ 𝟙 (L.total c₁₂) = 𝟙 (L.total c₁₂)
Module: Mathlib.Algebra.Homology.TotalComplexShift

Goal state before:
pp: case neg.h₁₂
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 : ¬(up ℤ).Rel n₀ n₁
⊢ ¬(up ℤ).Rel n₀' n₁'

Tactic:
grind [up_Rel]

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

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✝²⁹ : Category.{v_1, u_1} C₁
inst✝²⁸ : Category.{v_2, u_2} C₂
inst✝²⁷ : Category.{v_3, u_5} C₃
inst✝²⁶ : Category.{v_4, u_6} C₄
inst✝²⁵ : Category.{v_5, u_3} C₁₂
inst✝²⁴ : Category.{v_6, u_4} C₂₃
inst✝²³ : HasZeroMorphisms C₁
inst✝²² : HasZeroMorphisms C₂
inst✝²¹ : HasZeroMorphisms C₃
inst✝²⁰ : Preadditive C₁₂
inst✝¹⁹ : Preadditive C₂₃
inst✝¹⁸ : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
inst✝¹⁷ : F₁₂.PreservesZeroMorphisms
inst✝¹⁶ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝¹⁵ : G.Additive
inst✝¹⁴ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
inst✝¹³ : G₂₃.PreservesZeroMorphisms
inst✝¹² : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst✝¹¹ : F.PreservesZeroMorphisms
inst✝¹⁰ : ∀ (X₁ : C₁), (F.obj X₁).Additive
associator : bifunctorComp₁₂ F₁₂ G ≅ bifunctorComp₂₃ F G₂₃
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 : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝³ : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst✝² : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst✝¹ : DecidableEq ι₁₂
inst✝ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
i₁₂ : ι₁₂
j : ι₄
h₁₂ : c₁.π c₂ c₁₂ (i₁, i₂) = i₁₂
h : c₁₂.π c₃ c₄ (i₁₂, i₃) = j
⊢ c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j

Tactic:
by rw [← h, ← h₁₂]; rfl

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

Goal state before:
pp: case neg.h₁₂
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)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + y = n₀'
h₁ : n₁ + y = n₁'
h : ¬(up ℤ).Rel n₀ n₁
⊢ ¬(up ℤ).Rel n₀' n₁'

Tactic:
grind [up_Rel]

Goal state after: